Paper Title: First Access to ELM-free Negative Triangularity at Low Aspect Ratio.







[Corresponding author: ]a.o.nelson@columbia.edu
Columbia University, New York City, New York, USA


UKAEA-CCFE, Culham Science Centre, Abingdon, Oxon, UK


General Atomics, San Diego, California, USA


Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA


Princeton Plasma Physics Laboratory, Princeton, New Jersey, USA


Columbia University, New York City, New York, USA


York Plasma Institute, Heslington, York, UK


General Atomics, San Diego, California, USA





UKAEA-CCFE, Culham Science Centre, Abingdon, Oxon, UK


UKAEA-CCFE, Culham Science Centre, Abingdon, Oxon, UK


UKAEA-CCFE, Culham Science Centre, Abingdon, Oxon, UK














Columbia University, New York City, New York, USA


Columbia University, New York City, New York, USA


Columbia University, New York City, New York, USA


Columbia University, New York City, New York, USA



§ ABSTRACT 
A plasma scenario with negative triangularity (NT) shaping is achieved on MAST-U for the first time. While edge localized modes (ELMs) are eventually suppressed as the triangularity is decreased below δ≲-0.06, an extended period of H-mode operation with Type-III ELMs is sustained at less negative δ even through access to the second stability region for ideal ballooning modes is closed. This documents a qualitative difference from the ELM-free access conditions documented in NT scenarios on conventional aspect ratio machines. The electron temperature at the pedestal top drops across the transition to ELM-free operation, but a steady rise in core temperature as δ is decreased allows for similar normalized β in the ELM-free NT and H-mode positive triangularity shapes.



First Access to ELM-free Negative Triangularity at Low Aspect Ratio
    the MAST-U Team
    August 5, 2024
===================================================================









§ INTRODUCTION


Tokamak scenarios exploring negative triangularly (NT) shaping, where the plasma x-points are situated at a larger radius than the magnetic axis, have recently gathered significant interest from the international community due to their favorable confinement and power-handling properties <cit.>. In machines with conventional aspect ratios A=R_0/a_minor∼2.5-4, where R_0 and a_minor are respectively the major and minor radii, NT experiments are often characterized by reduced electron heat flux throughout their volumes compared to similar discharges with positive triangularity (PT), which comes from a reduction in core turbulent transport <cit.>. This leads to an enhancement of the core pressure gradient in NT plasmas that improves confinement over matched PT discharges <cit.>. Notably, these same plasmas do not induce edge localized modes (ELMs), even at high heating powers <cit.>. This ELM-free behavior allows NT fusion power plant designs to avoid issues associated with plasma-wall interactions during ELM events <cit.>. The larger major radius of the x-point in NT scenarios also provides additional room for divertor heat flux spreading, highlighting the potential power-handling benefits of NT plasmas <cit.>.

While NT scenarios show tangible promise as an power plant candidate in conventional aspect ratio designs, they have not yet been fully explored in spherical tokamaks (STs) with lower aspect ratio (A≲2). Initial gyrokinetic simulations of NT scenarios at low A, both with  <cit.> and  <cit.>, predict that the confinement enhancement over PT discharges observed at conventional aspect ratio might be lost in NT ST scenarios, particularly in regimes dominated by trapped electron modes. In addition, NT plasmas are predicted to be strongly unstable to long-wavelength kinetic ballooning modes (KBMs) across the plasma in high-β, reactor-relevant STs <cit.>. This results from a closure of access to the so called “second stability” region for ballooning modes, potentially limiting the maximum achievable β for NT scenarios in STs <cit.>. 

Notably, the loss of second stability access has also been strongly correlated with the prevention of ELMs in conventional aspect ratio NT discharges <cit.>. Access to second stability has long been associated with large H-mode pedestals <cit.>, so it is initially unsurprising that closing the stability window for ideal ballooning modes would prevent H-mode access. On DIII-D, second stability access can be closed by directly scanning the triangularity δ below some critical value (δ_crit, DIII-D∼-0.15) while holding other plasma and machine parameters (such as the injected power, plasma current, toroidal magnetic field and density) constant <cit.>. This shape change modifies the local shear near the plasma edge, closing access to the second stability regime and preventing the formation of an ELMy H-mode regime by severely limiting the allowable edge pressure gradient <cit.>. Negative triangularity plasmas on TCV follow a similar pattern, where closure of the second stability region is encountered in ELM-free NT plasmas <cit.>.

As noted in the literature, however, closure of the second stability region is not itself always sufficient for H-mode prevention in PT plasmas, particularly at lower aspect ratios <cit.>. Further, experimentally measured gradients from DIII-D ELM-free plasmas with δ∼-0.5 are not always indicative of being directly limited by the ideal ballooning mode <cit.>. These two facts suggest (1) that closure of the second ballooning stability region alone may not be sufficient to prevent ELMs, especially in regimes STs where first-stable H-modes are more common, and (2) that additional transport mechanisms that further limit the pressure gradient below the ballooning limit may be present even in conventional aspect ratio machines, especially when δ≪0. 

In this Letter, we present initial observations and modeling from the first ELM-free NT discharge achieved on MAST-U. Importantly, this work demonstrates a distinct decorrelation between the closure of the second stability region and the prevention of ELMy H-mode operation that has not yet been observed in conventional aspect ratio machines. Section <ref> details the access conditions for NT operation on MAST-U, leading to documentation of the stability modeling in section <ref>. In section <ref>, the potential consequences of this result are briefly discussed, particularly with respect to the construction of a predictive model for the NT pedestal height, which may ultimately set the performance of NT reactor scenarios. 








§ NT OPERATION ON THE MAST-U SPHERICAL TOKAMAK





The magnetic equilibrium for MAST-U discharge #49463, which features diverted negative triangularity operation
, is reconstructed by the  code <cit.> in figure <ref>. This discharge was achieved during a snowflake divertor development experiment in which the  code <cit.> was employed to reduce the plasma squareness at high elongation <cit.>. During this particular discharge, the lower x-point (subject to vertical position control) left the x-point control region as the plasma elongation was reduced, at which point the plasma control system continued to push the plasma in feedback. Luckily, this resulted in a smooth triangularity scan at constant power and near constant density, providing an excellent first NT discharge for the initial physics investigations presented in this work. Further details on this control scheme, including techniques that can be employed to create stable NT discharges on MAST-U, will be the subject of a subsequent publication.



Figure <ref> shows major time traces for this discharge, highlighting the stable nature of the plasma as the shape is changed. The discharge has constant plasma current (I_p=600kA) and beam heating power (P_inj = 3MW). A shape scan starts around t=200ms, as can be seen in the δ scan presented in figure <ref>. We note that the strong dip in δ near t=420ms is a result of the inner strike-line moving on to the outboard portion of the vacuum vessel.  The line-integrated average density (figure <ref>) grows until t=400ms, at which point an ELMy H-mode appears. The density then remains roughly constant for the remainder of the discharge, although some slight rise in the density can be observed after the transition to the ELM-free state, presumably linked to reduced pumping in the absence of ELMs (as also occurs before H-mode is achieved.) 


Evidence of the ELMs can be seen in the D_α traces presented in figure <ref>. Notably, the plasma remains in H-mode with rapid Type-III ELMs as the triangularity drops below δ=0 until a transition to an ELM-free state is made at around t_crit∼710ms when δ_crit≲-0.06.  
The observation of ELM-free behavior below a critical triangularity δ_crit is qualitatively consistent with expectations concerning the behavior of NT scenarios from the DIII-D and TCV tokamaks <cit.>. However, it remains to be conclusively demonstrated that this phenomena is a robust property of MAST-U NT discharges. Interestingly, infinite-n stability modeling suggests that access to this ELM-free scenario may be governed by different physics than in DIII-D or TCV, as is discussed in section <ref>. 

Over the course of discharge #49463, the normalized confinement H_98y2, as calculated with the  code <cit.>, decreases steadily with decreasing κ (figure <ref>) from H_98y2∼0.9 at 300ms to H_98y2∼0.66 during the ELM-free phase. No significant changes in H_98y2 are observed near δ=0 or δ=δ_crit. The normalized pressure β_N (figure <ref>) drops significantly from β_N∼2.4 to β_N∼2 when the inner strikeline moves across the divertor opening but remains constant at β_N∼2 for the rest of the discharge. Notably, little change in the impurity content is observed across this discharge, as indicated by the carbon emission measurements captured with the SPRED diagnostic shown in figure <ref> <cit.>. Slight (∼10-20%) increases in the line-averaged density and the C and He SPRED emissions are observed during the ELM-free phase, potentially suggestive of reduced impurity transport under ELM-free conditions. 









Representative traces of the electron temperature (T_e), density (n_e) and pressure (p_e) at the plasma core (ψ_N∼0.25) and near the pedestal top (ψ_N∼0.92) are shown in figure <ref>. Near the pedestal top, T_e,ped drops significantly both when the inner strikeline transitions to the outboard side of the vessel around t∼420ms and then again when the ELM-free state is achieved at t∼710ms. The edge density n_e,ped stays roughly constant as the strikeline shifts, but drops upon the transition to the ELM-free state before growing again towards the end of the discharge. Notably, the core traces display markedly different trends: while n_e,core remains roughly constant after H-mode is achieved, T_e,core actually rises steadily as δ is decreased, reaching its highest values in the ELM-free portion of the discharge when δ is the most negative. This behavior is reproduced qualitatively in initial simulations with the  code <cit.>, which recovers an increase in the core T_e as δ is decreased at fixed density and injected power that results from a reduced  SAT-2 <cit.> electron heat flux at more negative triangularity. The rise in core temperature at constant density translates to a steady rise in the core pressure as δ is decreased, as seen in figure <ref>. Similar qualitative behavior demonstrating a decrease in edge pressure but a retention of high core pressures has been observed in NT plasmas on DIII-D <cit.> and TCV <cit.> and is often associated with a reduction in the electron thermal conductivity in NT scenarios.

Further characterization of the plasma performance (including transport mechanisms, β_N limits and confinement scalings) as a function of δ<0 requires additional investigations over a dedicated set of NT discharges on MAST-U and are thus not considered here. Exploration of the heat-flux width, ELM behavior and shaping limits will also be included in future studies.



§ DECORRELATION OF SECOND STABILITY CLOSURE AND ELM-FREE NT OPERATION



On the DIII-D and TCV tokamaks, which have conventional aspect ratios, access to ELM-free conditions via NT shaping occurs suddenly at a critical triangularity (δ_crit). This phenomena has been strongly correlated with the closure of the second stability region for infinite-n ballooning modes, which introduces an additional transport mechanism at the plasma edge, preventing the growth of steep pressure gradients in the NT edge <cit.>. However, as mentioned in <cit.>, it is in general not necessarily sufficient in tokamak plasmas that second stability access be closed to ensure L-mode operation. Broadly speaking, if the collisionality is high enough to suppress the bootstrap current, local shear can remain high enough to keep the edge in the first stable region even when weak peeling-ballooning modes are destabilized. This effect is particularly evident in devices with low aspect ratio, and it is not uncommon for MAST-U H-mode discharges with positive triangularity to form pedestals that reside in the first-stable region <cit.>. On NSTX, the prevalence of first-stable H-modes can be observed in the bifurcation of pedestal widths into `wide' and `narrow' branches, which are in turn set by variations in the stability of kinetic ballooning modes <cit.>. 



The ideal ballooning stability of tokamak discharges can be easily assessed using the  code <cit.>, which requires as input well-converged kinetic equilibrium reconstructions. For this work, kinetic equilibria were generated every 10ms using the kineticEFITtime workflow provided in the OMFIT framework <cit.>. This workflow allows for iteration between profile fits to the electron and ion profiles and a free-boundary EFIT reconstruction <cit.>. The stability results from this study are presented in figure <ref>, which shows the normalized pressure gradient (α) corresponding to the stability boundaries (dashed lines) and experimental conditions (solid lines) for five key times throughout discharge #49463. 

In figure <ref>, the first four times (yellow – 300ms, orange – 400ms, magenta – 480ms and red – 590ms) are all from periods of the discharge when the plasma is firmly in a Type-III ELMy regime. No stability boundary is found by the  code during the earliest times (t≲400ms), allowing the pressure gradient at the edge to grow to high values. However, the stability boundary rapidly closes around t∼480ms when the average triangularity measured at the last closed flux surface (LCFS) nears zero. Throughout the evolution of this discharge, the peak pressure gradient in the edge region decreases as δ is decreased. ELMy H-mode behavior is robustly sustained until a triangularity of δ∼-0.06 is reached around t=710ms (black lines in figure <ref>). An ELM-free time slice (t=800ms) is shown in blue for comparison. Notably, max(α) remains similar for the entire ELM-free phase (t=720-830ms) and does not continue to drop as δ is further decreased during this time.

Strikingly, this analysis reveals a decorrelation between the closure of ballooning stability window and the prevention of ELMs by NT shaping. While previously suggested as a possible complication to the no-ELM NT scenario <cit.>, this has not yet been observed on either the DIII-D or TCV tokamaks, where closure of second stability typically coincides with the loss of H-mode in NT <cit.>. 

To compensate for any uncertainties in the equilibrium reconstructions that could impact this result, over 500 variations of the kinetic equilibrium representing the plasma at t=500ms were generated by solving the fixed-boundary using the  code <cit.> with arbitrary profile modifications, following the workflow established in <cit.>. Scaling the edge pressure gradient, collisionality and current separately by factors of five was not sufficient to open access to the second stability region in this shape, suggesting that the conclusion that this MAST-U discharge achieves a first-stable H-mode regime at low values of δ<0 is not an artifact of any workflow. 

As seen in figure <ref>, an ELM-free state is still achieved in this low aspect ratio discharge, suggesting that additional physics may be needed to fully explain the transition to ELM-free states via NT shaping over a wider variety of plasma conditions. As the plasma conditions (n_e, P_inj and impurity content) were held constant over this scan, it is likely that the transition to an ELM-free state in discharge #49463 is still largely a result of the shape changes. Interestingly, and as a result of the unique shapes achieved in this discharge, while the triangularity measured at the last closed flux surface drops below zero around t=590ms, the triangularity just inside of the LCFS (up to ψ_N∼0.9) remains positive until the ELM-free transition at t=710ms. This may suggest that the effects of NT must extend over a significant portion of the edge region to inhibit the transition to H-mode, which is likely true in conventional aspect ratio scenarios as well. 



To begin to assess possible mechanisms for pedestal height reduction responsible for the ELM-free behavior observed at strong NT on MAST-U, the  code <cit.> was utilized to determine the Gyrokinetic Critical Pedestal (GCP) in discharge #49463. This model combines linear gyrokinetics with self-consistent pedestal equilibrium variation to explicitly determine stability limits stemming from the Kinetic Ballooning Mode (KBM) in the plasma edge and has recently been utilized to reproduce experimental measurements of the pedestal width-height scalings in the NSTX tokamak, which also has low-aspect-ratio <cit.>. This GCP is calculated for an ELMing case with δ>0 (600ms) in figure <ref>, showing excellent agreement with the experimentally measured pedestal height (β_θ,ped) and width (Δ_ped), which are calculated from a modified-tanh fit to the edge pressure profile. Notably, the Ballooning Critical Pedestal (BCP), which is calculated directly from the infinite-n ballooning stability boundary, lies well above this curve, in agreement with the  modeling presented in figure <ref>. 

This result carries several important implications. First, it offers insight into the Type-III ELMy nature of these discharges. Initial investigations with the  code <cit.> suggest that these plasmas sit well within the nominal stability region for Type-I ELMs – a reduction in the diamagnetic stabilization factor by a factor of ∼1/10 from typical MAST-U values is needed to localize the experimental measurements along the predicted stability boundary. If the plasma is indeed limited by a KBM constraint that sits below the peeling-ballooning boundary, as is suggested by , it would offer an justification for the reduced pedestal height and Type-III ELMy operation. Second, the prevalence of this KBM limit lends some credibility to the theoretical arguments made in <cit.>, which finds that the KBM will be a primary source of transport in ST NT scenarios with reactor-relevant parameters. Further study of the role of KBMs in setting the β limits for NT scenarios on STs should be the subject of future work.

Finally, the reduction of the modeled GCP limit from the BCP provides a potential explanation for the decorrelation of second stability closure and ELM-free NT operation that distinguishes this regime from counterparts on conventional aspect ratio tokamaks. While the mechanisms for ELM suppression are still unclear, reduction of the pedestal height below the first stability limit for infinite-n ballooning modes occurs already in the ELMy phase of this discharge as a result of the KBM limit. While potentially more applicable in low-aspect ratio devices <cit.>, such a reduction due to kinetic effects could also be present in conventional aspect ratio devices and offer a potential avenue for gradient reduction below the ideal ballooning limit, as observed in <cit.>. Future work, including similar KBM analysis on conventional aspect ratio machines and further study of the ELM-free NT regime on MAST-U across a broader parameter range, is needed to fully explore the validity of this model.




§ DISCUSSION AND CONCLUSION



This Letter reports on the first NT plasma achieved in a spherical tokamak. While further experiments, including on other STs like NSTX-U and SMART, are needed to fully characterize the performance and stability of this regime, several interesting observations can already be made due to the fortuitously stable plasma conditions achieved in this discharge. First, the qualitative behavior expected from NT operation on other machines concerning a sharp transition to an ELM-free regime without significant degradation of plasma performance is achieved. While H_98y2 decreases across the shape scan with decreasing κ, no obvious decrease in β_N is associated with the transition to a shape with δ<0 or when achieving H-mode inhibition. This optimistically suggests that NT operation of low aspect ratio devices follow the intuition developed on larger aspect ratio devices, though gyrokinetic predictions do predict degradation of plasma performance at very low A <cit.>. Appropriate characterization of the confinement properties of the plasmas demands additional experimental investigation.

Perhaps more interestingly, at less negative triangularities an extended period of Type-III ELMing H-mode persists even though second stability for ideal ballooning modes is closed. The decoupling of second stability closure and access to the NT ELM-free state has not yet been observed on either DIII-D or TCV, which typically transition directly from a Type-I ELMy H-mode into the NT ELM-free state when the ballooning stability bifurcates. Access to first-stable ELMy H-mode at small absolute values of δ<0 could have significant consequences on the feasibility of this approach as a pilot plant scenario for low aspect ratio machines. This supports arguments presented in reference <cit.> regarding consideration of the scalings presented therein only as bounding values on the required shaping parameters for access to the ELM-free state. If first-stable H-modes with Type-III ELMy can be achieved in a given configuration, more negative values of δ may be needed to achieve robust H-mode inhibition. 

We note further that this key result – that additional physics mechanisms beyond the ideal ballooning stability likely impact the edge gradient limits in NT plasmas – is not necessarily inconsistent with results obtained on other machines at larger aspect ratio <cit.>. On DIII-D, while access to ELM-free conditions is strongly correlated with the closure of ideal ballooning stability <cit.>, a majority of ELM-free NT discharges sustain gradients well below the maximum gradient allowed by first stability limit for ballooning modes. This can be observed clearly in figure 3 of reference <cit.>, which shows numerous ELM-free DIII-D equilibria sitting well under the first stability limit. Taken together, these results are indicative of additional transport mechanisms present in the NT edge that can both limit the edge gradient and robustly prevent ELMs. Future work on this subject should aim to identify this physics to enable reliable predictions of the NT edge under the conditions expected in future fusion power plants.



§ ACKNOWLEDGMENTS


Some calculations performed in this study were completed through the OMFIT framework <cit.>, and the generated data are available upon request. This material is based upon work supported by the U.S. Department of Energy, Office of Science, under Awards DE-SC0022272, DE-SC0018991, DE-AC05-00OR22725 and DE-FC02-04ER54698. This work has been funded in part by the EPSRC Energy Programme [grant number EP/W006839/1]. To obtain further information on the data and models underlying this paper please contact PublicationsManager@ukaea.uk.


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 Paper Title: Kinetic development of low-temperature propane oxidation in a repetitively-pulsed nanosecond discharge.


	






























1]Zhenyang Li
2]Bo Yin
1]Qifu Lin

2]Yifei Zhu*
yifei.zhu.plasma@gmail.com
[*]Corresponding author.

2,3]Yun Wu
					
[1]organization=Institute of Energy, Hefei Comprehensive National Science Center,
						city=Hefei,
						postcode=230026,
						country=China
[2]organization=National Key Lab of Aerospace Power System and Plasma Technology, Xi'an Jiaotong University,
						city=Xi'an,
						postcode=710049,
						country=China
[3]organization=National Key Lab of Aerospace Power System and Plasma Technology, Airforce Engineering University,
						city=Xi'an,
						postcode=710038,
						country=China


























§ ABSTRACT 
	
	The kinetics of plasma assisted low temperature oxidation of C_3H_8/O_2/Ar mixtures have been studied in a wide specific deposition energy with the help of nanosecond repetitively pulsed discharge. Two types of nanosecond pulsed plasma sources, the nanosecond capillary discharge (nCD) and dielectric barrier discharge (DBD) combined with the synchrotron photoionization mass spectrometry are investigated. The electron impact reaction rate of propane dissociation and some combustion chemical reactions rate constants are updated according to the nCD and DBD experiment results, and uncertainty of the reactions are analyzed in detail. Compared to the existing model, the updated model's prediction accuracy has great improvement in species H_2O, CO, CO_2, CH_4, CH_2O, CH_3OH, C_2H_2, C_2H_4, C_2H_6, C_2H_5OH, C_2H_5OOH, C_3H_4-A, C_3H_4-P, C_2H_5CHO, i-C_3H_7OH and C_3H_7OOH. The propane oxidation processes assisted by DBD and nCD were compared under different single pulse deposition energy (SPDE) conditions while maintaining the same total deposition energy. The reduced electric field in nCD is concentrated at 150-200 Td and 450-500 Td, whereas in DBD it ranges from 0-25 Td and 50-250 Td. Notably, SPDE shows minimal influence on the propane oxidation process, which is primarily controlled by total deposition energy and demonstrates little dependence on the discharge type (DBD or nCD).
		













	Plasma-assisted combustion Low-temperature oxidation Propane Kinetic modeling













§ NOVELTY AND SIGNIFICANCE STATEMENT

The kinetic of plasma assisted C_3H_8/O_2/Ar oxidation is developed and studied in a wide specific deposition energy of nanosecond repetitively pulsed discharge. Two types of nanosecond pulsed plasma sources (nCD and DBD) are used to validate the kinetic mechanism. The electron impact reaction rate of propane dissociation and some combustion chemical reactions rate constants are updated according to the nCD and DBD experiment results. The prediction accuracy has great improvement, and uncertainty of the reactions are analyzed in detail. The propane oxidation processes assisted by DBD and nCD were compared, and the propane oxidation processes is primarily controlled by total deposition energy and demonstrates little dependence on the discharge type (DBD or nCD).
	


§ AUTHORS CONTRIBUTIONS

Zhenyang Li: Writing-original draft, Formal analysis, Data curation. Bo Yin: Investigation, Experiment development. Qifu Lin: Investigation, Supervision. Yifei Zhu: Writing-review & editing, Conceptualization. Yun Wu: Funding acquisition, Project administration.




§ INTRODUCTION


Non-equilibrium plasma-assisted combustion (PAC) offers an effective approach for controlling ignition in internal combustion engines, industrial burners, and aviation engines, thereby enhancing flame stability, reducing pollutant emissions, and expanding ignition limits <cit.>. Understanding the detailed mechanism of PAC requires studying the kinetic pathways of PAC and quantifying their reaction rates. The kinetics of non-equilibrium plasma-assisted small molecules like methane have been extensively studied <cit.>.

Propane, as the smallest n-alkane with low-temperature reactivity, serves as a typical molecule for studying the kinetics of large alkanes <cit.>, attracting significant attention from researchers. Adamovich et al. <cit.> proposed a plasma-assisted propane-air combustion kinetic model, which showed good agreement with experimental temperature and time-resolved OH density measurements. They further studied the plasma-assisted propane oxidation kinetics in C_3H_8/O_2/Ar mixtures <cit.>, validated the kinetic model with experiments on alkane small molecule density at different temperatures, and found large discrepancies for C_2H_2 and CH_3CHO. Chen et al. <cit.> employed molecular beam mass spectrometry with tunable synchrotron vacuum ultraviolet photoionization (SVUV-PIMS) to measure hydrocarbons and oxygenated intermediates during plasma-assisted C_3H_8/O_2/Ar processes, reporting numerous oxygenated intermediates and establishing a kinetic model. However, significant discrepancies existed between the model predictions and experimental measurements for certain hydrocarbons and intermediates. Ban et al. <cit.> developed a propane-air plasma combustion mechanism and validated it against OH density measurements from experiments <cit.> that lacked sufficient validation. Based on the above analysis, current kinetic models still exhibit significant discrepancies with experimental measurements in predicting certain hydrocarbons and oxygenated intermediates. Furthermore, current kinetic models are validated under specific single pulse deposition energies, necessitating further validation under larger deposition energy scales.

To validate the kinetics of the plasma-assisted oxidation process in C_3H_8/O_2/Ar mixtures under large deposition energy scales, stable discharge characteristics and spatiotemporal uniformity are required in experiments. Fast ionization wave (FIW), a nanosecond capillary discharge at moderate pressures, is characterized by spatiotemporal uniformity of discharge, high reduced electric field, high electron energy, and fast propagation speed <cit.>. FIW serves as an ideal platform for investigating fundamental plasma processes under various discharge parameters and energy deposition conditions <cit.>. Due to its uniformity and high energy deposition, FIW is suitable for investigating plasma-assisted fuel cracking and oxidation. FIW has two discharge types depending on the presence or absence of a metal screen. In the absence of a metal screen (referred to as nanosecond capillary discharge, nCD), the discharge propagation pattern shifts from single-direction to dual-direction propagation, improving discharge channel uniformity. Consequently, nCD is particularly suitable for validating the accuracy of C_3H_8/O_2/Ar plasma combustion mechanisms under varying energy deposition conditions.

In this study, we investigate the mechanism of plasma-assisted low-temperature oxidation of propane using numerical simulations and capillary discharge experiments. Section <ref> outlines the numerical model, the kinetic schemes, and the experimental setups. The validation of the kinetic schemes against capillary discharge experiments and DBD experiments <cit.> is detailed in Section <ref>. An uncertain analysis of the kinetic schemes is performed using path analysis in Section <ref>. Section <ref> compares the differences between nCD and DBD in the process of propane oxidation. Finally, we provide conclusions in Section <ref>.



§ EXPERIMENT AND MODELLING METHOD




 §.§ The experiment settings



The experimental setup of the nCD-assisted propane oxidation is depicted in Figure <ref>. This setup primarily comprises a gas supply and control system, a plasma discharge system, and a gas collection and analysis system. The discharge system includes a high-voltage electrode, a ground electrode, a quartz tube, and insulating rubber gaskets. The cross-sectional diameter and length of the discharge region are 4.5 mm and 50 mm, and the distance between the two electrodes is 44 mm.



The gas mixture in the cylinder consists of Ar, O_2, and C_3H_8 in a ratio of 92.46:6.33:1.21. The gas flow rate is controlled using a flowmeter (ASERT AST10-ALC 100 sccm) and is set to 30 sccm. Gas flow is controlled using a flowmeter (ASERT AST10-ALC 100 sccm) set at 30 sccm, and pressure is maintained at 3000 Pa by a rotary vane pump, monitored by a vacuum gauge (AILEIKE TT&C Equipment Co., Ltd., ALKC400HJ, 0-1 atm absolute pressure gauge, 0.05%FS). As shown in Fig. <ref>,  valve 3 is adjusted to stabilize the gas pressure before collecting the gas. During collection, valves 3 and 2 are closed, and a compression tank with variable volume stabilizes and stores the low-pressure gas, which is later pressurized to atmospheric pressure and analyzed using a gas chromatograph (GC) (Agilent 7890B) for species concentrations.

The discharge is powered by a nanosecond pulse discharge power supply (Xi'an Lingfengyuan Electronic Technology Co., Ltd., HVP-20P), providing continuous pulse outputs for steady-state gas collection. Each pulse has triangular wave shape with 50 ns rise and fall times, operating at a frequency of 1 kHz and a voltage range of 4.2 kV to 5 kV. Discharge voltage is monitored with a high-voltage probe (Tektronix, P6015A), and discharge current is measured with a current coil (Pearson 6585). Both voltage and current data are recorded using an oscilloscope (Siglent SDS2354X 350 MHz).



 §.§ Modelling methods
 

A new coupled plasma and combustion chemistry solver, CPCC, has been developed by integrating the ZDPlasKin code <cit.> with a custom-built combustion code in F90 format. CPCC supports sensitivity and path flux analysis and interfaces with popular post-processing software such as QtPlaskin <cit.> and PumpKin <cit.>. Detailed descriptions of CPCC can be found in <cit.>.The electron impact process is solved using the Boltzmann equation solver BOLSIG+ <cit.>, which is integrated within ZDPlasKin. The plasma kinetic mechanism and ground state kinetic mechanism are alternately solved using the splitting method. For efficient solution of large combustion kinetic mechanisms, CPCC utilizes the semi-implicit ODE solver VODPK <cit.> and the fifth-order implicit Runge-Kutta ODE solver, RADAU5 <cit.>.

The key parameter of the reduced electric field (E/N) is derived from experimental current data using the following equation:

    I = n _  e μ ( E/N ) E e S

where n_e is the electron density (initially set to 1.0×10^9 cm^-3), I is the electron mobility dependent on the reduced electric field, N is the natural particle density of last time, E is the electric field strength, e is the elementary charge, and S is the cross-sectional area of the plasma. E is iteratively obtained using equation <ref> through the Newton iteration method. The intial electron density is given a empirical value, because E is insensitive to the intial electron density, which will iterate consistently to a reasonable value in the first few time steps without affecting the result of the calculation. Then E/N is obtained by dividing E by N. 

In present model, as the Poisson equation is not solved, the deposition energy is assumed to be an adjustable parameter and controls the discharge duration <cit.>. For DBD experiment, the present model's deposition energy stands at 1.85 × 10^-5 cm^-3, significantly lower than the experimentally measured value. Furthermore, the deposition energy control is not implemented. For nCD experiment, To ensure good agreement with the experimental results, the model utilized 40% of the experimental deposited energy to regulate the discharge time.

To account for primary energy loss due to conduction to the quartz channel walls, similar to the approach in Ref. <cit.>, the model includes surface heat transfer between the plasma and the quartz wall. In the energy equation for the nCD experiment, an additional energy loss term is introduced:
  
    ρ C _  v  d  T _  gas / d  t  = - ∑ _  i = 1  ^  i _ max H _  i ω _  i  -  T _  gas  - T _  wall /(  1 / A _  1  h _  1  + ln( r _  2  / r _  1 ) / 2 πλ L ) V _  p

where ρ is the gas density, C_v is the specific heat capacity at constant volume of the gas, H_i and ω _i represent the enthalpy of formation and production/loss rate of component i, respectively. T_wall is the outer wall temperature of the quartz tube (set to 300K), A_1 is the contact area between the plasma and the inner wall of the quartz tube, h_1 is the heat transfer coefficient between the plasma and the inner wall of the quartz tube (which has been adjusted to 10 W/(m^2 K) based on experimental results due to the cylindrical shape and small diameter of this structure, making empirical formulas unsuitable for estimating the heat transfer coefficient), r_2 and r_1 are the outer and inner radius of the quartz tube, λ is the thermal conductivity of quartz (which is 1.09 W/(m K)), L is the length of the discharge region, and V_p is the volume of the plasma.

For a DBD configuration where the discharge region is rectangular, the energy loss term in the gas energy equation is adjusted as follows:

    ρ C _  v  d  T _  gas / d  t  = - ∑ _  i = 1  ^  i _ max H _  i ω _  i  -  T _  gas  - T _  wall /(  1 / A _  2  h _  2  + δ/ A _  2 λ) V _  p


    h _  2  = λ/ L N_u


    N_u = 0.664  R_e ^  1 / 2  P_r  ^  1 / 3


    P_r  =  C _  p μ/λ


    R_e  = ρ v L /μ

where A_2 is the contact area between the plasma and the upper and lower walls, h_2 is the surface heat transfer coefficient of the walls (calculated using equation <ref>), and the Nusselt number N_u is calculated using equation <ref> from the book <cit.>. P_r is the Prandtl number (calculated by equation <ref>), R_e is the Reynolds number (calculated by equation <ref>), μ is the dynamic viscosity, v is the airflow velocity, and C_p is the specific heat capacity at constant pressure of the gas.

This kinetic model includes both plasma chemistry and classical combustion chemistry. The combustion chemistry model incorporates a total of 2521 reactions, which have been updated based on the scheme proposed in Ref. <cit.>. Specific modifications derived from experimental data are outlined in Table <ref>. In addition, several combustion reactions have been added and replaced, drawing from various established mechanisms such as HP-Mech v3.3 <cit.>, AramoMech3.0 <cit.>, NUIGMech1.1 <cit.> (see supplementary material of Table S1). The plasma chemistry model consists of 2202 reactions, involving vibrationally excited species such as C_3H_8(v1)-C_3H_8(v27), CH_4(v2,4), CH_4(v1,3), O_2(v1)-O_2(v4); electronically excited species like O_2 (a^1 Δ _g ), O_2 (b^1 Σ_g^+ ), O_2(4.5eV)(summation of O_2 ( c ^1 Σ _ u ^ -), O_2 ( C ^3 Δ  _ u), O_2 ( A ^3 Σ _ u ^ +)), O(^1D), O(^1S), Ar(1s_2), Ar(1s_4), Ar(11.55eV), Ar(2p_1); ions like O^+, O_2^+, O^-, O_2^-, O_3^-, CH_4^+, CH_3^+, CH_2^+, CH_5O^+, C_2H_3O^+, CHO_3^-, CO_3^-, CHO_2^-, OH^-, C_2H_4^+, C_2H_5^+, C_2H^+, C_2H_3^+, H_5C_2O^+, C_3H_7^+, C_4H_9^+, C_3H_8^+, and electron. The plasma chemistry model includes chemical reactions involving vibrationally excited states, vibrational relaxation reactions, charge transfer reactions, and reaction channels involving O(^1D), O(^1S), O_2 (a^1 Δ _g ) , O_2 (b^1 Σ_g^+ ), O_2 (4.5eV), Ar(1s_2), Ar(1s_4), Ar(11.55eV), Ar(2p_1) with fuel fragments. The electron collision dissociation cross-section for C_3H_8 is sourced from Ref. <cit.>, while reactions involving electron collision dissociation for CH_4, including dissociation and ionization, are adopted from Ref. <cit.>.  The plasma chemistry of Ar and O_2 are taken from Ref. <cit.>.



The original model in Section <ref> and Section <ref> is defined as the plasma chemistry and classical combustion chemistry in which the reaction rate coefficients are from literature without any modifications. Compared to the model of Ref. <cit.>, the original model reconstruct the plasma chemistry and has the same combustion chemistry. As some reactions are not validated with experiments or quantum chemical calculations, the original model predictions for some species have large uncertainties. So the electron impact cross sections of propane, branch ratios of self-/cross-reactions for i-/n-C_3H_7O_2 and H-abstraction of i-/n-C_3H_7O_2 are fitted within uncertainties according to experimental data and path flux analysis (named as updated model in Section <ref> and Section <ref>). The updated reactions are listed in Table <ref>. Detailed descriptions and uncertainty analysis can be found in Section <ref>. The species overall predictions of the updated model are highly improved compared to the original model in DBD experiment <cit.> and nCD experiment conditions. Even so, the quantum chemical calculations or fundamental kinetic experiment are still requred to improve the model's prediction capacity.



§ VALIDATION OF THE MECHANISM



 §.§ Validation I:DBD assisted propane oxidation




Model validation is performed by comparing the model predicted and experimental molar fractions of speices in a repetitively-pulsed nanosecond DBD <cit.>, as shown in Fig. <ref>. The experimental conditions involve a deposition energy of 1.64× 10^-4 J/cm^3, at 30 Torr and a discharge region temperature of 340 K, with a C_3H_8/O_2/Ar mixture (mole fraction 4:20:76). The updated model's predictions for several species, including H_2O, CO, CH_2O, CH_3OH, C_2H_2, C_2H_4, C_2H_5OH, C_2H_5OOH, C_3H_4-A, C_3H_4-P, C_3H_6, CH_3COCH_3, C_2H_5CHO, i-C_3H_7OH, and C_3H_7OOH are found to be in good agreement with experimental data, falling within the range of experimental measurement errors. The updated model's predictions for species CO_2, CH_4, C_2H_6, CH_2CO, CH_3CHO, and n-C_3H_7OH are within 5 times of the experimental measurement. However, the updated model's predictions for species CH_3, CH_3OOH, cC_3H_6, C_2H_3CHO, C_3H_6O_1-3, and C_3H_6O_1-2 are over an order of magnitude off. The updated model demonstrates noticeable improvements in prediction accuracy for many species compared to the original model and a previously published model <cit.>.

The reasons for the lower predictions of C_3H_6O_1-2 and C_3H_6O_1-3 in the updated model are as follows: In Ref. <cit.>, C_3H_6 is consumed by an O atom to form C_3H_6O_1-2 and C_3H_6O_1-3 intermediates through the addition of an O atom to the C=C bonds in the center and end carbon atoms of C_3H_6. Then, C_3H_6O_1-2 and C_3H_6O_1-3 intermediates decompose into CH_3, C_2H_5, CH_2O, and H, among others. The reactions provided in Ref. <cit.> do not consider these intermediate products (C_3H_6O_1-2 and C_3H_6O_1-3) but instead directly provide the reactions leading to the final products. This ultimately results in the underprediction of C_3H_6O_1-2 and C_3H_6O_1-3.





 §.§ Validation II: nCD assisted propane oxidation


The DBD experiment was conducted under conditions with a deposition energy of 1.64× 10^-4 J/cm^3. To further validate the accuracy of the mechanism, nCD experiments were conducted with deposition energies ranging from 1.7× 10^-4 J/cm^3  to 1.53× 10^-3 J/cm^3. The uniformity of the nCD was evaluated using images captured by a high-speed ICCD camera operated at 5.0 kV, which were processed using an Abel transformator, as depicted in Fig. <ref>. Due to the enhanced electric field near the electrode tip, localized discharge enhancement was observed on the ground electrode. Conversely, a phenomenon of discharge weakening was noted at distances ranging from 9 to 20 mm away from the high-voltage electrode. These localized effects were considered negligible in comparison to the overall discharge characteristics. Overall, the nCD was found to exhibit relatively uniform behavior across the tested conditions.



Figure <ref> presents a comparison between experimental measurements and predictions from different kinetic schemes in the nCD conditions. The original model exhibits poor predictions for species such as CO, H_2, CO_2, CH_4, C_2H_6, C_2H_2, C_3H_6. In contrast, the updated model shows significant improvements across these species. However, despite these enhancements, notable discrepancies remain in the predicted mole fractions of CO_2, C_2H_4, and CO, which are under-predicted by 55%, 45%, and 45%, respectively.



The comparison between measurement and predictions from different kinetics scheme in nCD is shown in Fig. <ref>. The original model has poor predictions in species CO, H_2, CO_2, CH_4, C_2H_6, C_2H_2, C_3H_6. The updated model has great improvement in above mentioned species. However, compared to experimental measurements, the updated model still has significant differences in the mole fractions of CO_2, C_2H_4, and CO. The mole fractions of updated model are under-predicted by 55%, 45%, and 45%, respectively. The predictions of remaining species are generally well. 

The updated model's predictions for C_3H_8, H_2, CH_4, and C_2H_6 generally fall within the experimental error range. For C_3H_6, while it is overpredicted by 30% at low deposition energy, the agreement improves at higher deposition energies. The trend of C_2H_2 predicted by the updated model aligns with experimental data, albeit with molar fractions approximately 30% lower than measured values. This suggests that while the added reaction pathways in the updated model are correct, further research is needed to refine the cross-sectional data used.







§ RESULTS AND DISCUSSION




 §.§ Uncertain analysis in the plasma and combustion kinetic mechanism
 

The updated model was derived from the original model through adjustments to several reactions based on experimental findings and pathway flux analysis. Specifically, the modifications listed in Table <ref> are discussed in detail as follows.


In the original model, significant over-predictions of i-/n-C_3H_7OOH were observed during the DBD-assisted propane oxidation process, as depicted in Fig. <ref>. The generation rate of i-/n-C_3H_7OOH was found to be two orders of magnitude larger than its consumption rate. Pathway flux analysis based on the original model identified that the primary generation pathways for i-/n-C_3H_7OOH involved reactions i-/n-C_3H_7O_2 with HO_2 (reactions R7 and R8, respectively), as illustrated in Fig. <ref>. However, the rate coefficients for R7 and R8 used in the original model were adapted from analogous reactions such as those between CH_3O_2 and HO_2 reported in Ref. <cit.>. These rate coefficients are valid within a temperature range below 338 K and are subject to an error of up to 5. Given the temperature range of 340-500 K in the current study and the inherent uncertainties associated with the analogy method, significant uncertainties were anticipated for R7 and R8. Therefore, adjustments were made to the rate coefficients of R7 and R8, reducing them by a factor of 0.034 based on insights from the DBD experiment <cit.>. The updated model now indicates that 87% of n-C_3H_7OOH is generated from the reaction of n-C_3H_7O with OH, while 84% of i-C_3H_7OOH is produced via reaction R8.



Both the model by Chen et al <cit.> and the original model exhibit under-prediction of CH_2O and over-prediction of CH_3OH. As depicted in Fig. <ref>, pathway flux analysis of the original model indicates that 97% of CH_3OH is generated via the reaction of CH_3O with HO_2 (R6). However, it was noted that the reaction rate coefficients of R6 were sourced from Ref. <cit.>, which states that the reaction between CH_3OH and O_2 produces CH_2OH and HO_2, not CH_3O and HO_2. Consequently, in the updated model, reaction R6 has been removed. The updated model's prediction for CH_3OH now aligns well with experimental measurements, with 63% of CH_3OH being produced by the self-reaction of CH_3O (Reaction <ref>), and 34% being generated through Reaction <ref>.


    CH_3O+CH_3O↔ CH_3OH+CH_2O


    CH_3O_2+OH↔ CH_3OH+O_2


To address the lack of certain CH_3O consumption pathways, the CH_3O sub-mechanism was supplemented by referencing the HP-Mech v3.3 mechanism <cit.> (as shown in Table <ref>). The removal of R6 and the supplement of CH_3O consumption reactions led to significant changes in the CH_3O consumption pathway, as illustrated in Fig. <ref>. In the original model, CH_3O was almost exclusively consumed by the reverse reaction of R6. In contrast, in the updated model, half of CH_3O is consumed to form CH_3, while the remaining CH_3O is consumed to form CH_2O and CH_2O. Consequently, the prediction of CH_2O in the updated model has improved and now aligns well with experimental measurements due to the removal of R6 and the supplement of CH_3O consumption reactions.


    n-C_3H_7O→ C_2H_5+CH_2O


    C  _  3  H  _  5 - T  +  O  _  2 ↔ CH  _  3  CO  +  CH  _  2  O


    CH  _  3  +  O ↔ CH  _  2  O  +  H


    CH_3O+O_2/H/OH/O ↔ CH_2O+HO_2/H_2/H_2O/OH




As depicted in Fig. <ref>, the original model exhibited over-predictions of n-C_3H_7OH and under-predictions of i-C_3H_7OH. It was observed that nearly all i-C_3H_7OH and almost half of n-C_3H_7OH were produced through self- and cross-reactions of n-/i-C_3H_7O_2 (Reactions R10-R13), as illustrated in Fig. <ref>. The inaccuracies in the branching ratios of these reactions likely contributed to the over-prediction of n-C_3H_7OH and the under-prediction of i-C_3H_7OH. The rate coefficients for Reactions R10-R13 were originally derived from Ref. <cit.>. However, the temperature-dependent rate constants used in Ref. <cit.>, sourced from Ref. <cit.>, lacked detailed pressure and temperature dependencies. Moreover, the products of Reactions R10-R13 were not experimentally validated in Ref. <cit.>, contributing to significant uncertainty in the branch ratios. To improve prediction accuracy, this model adjusted the branching ratios of Reactions R10-R13 based on DBD experimental results. This adjustment notably enhanced the prediction accuracy for i-C_3H_7OH, although a slight increase in n-C_3H_7OH mole fraction compared to the original model was observed. This increase can be attributed to the enhanced reaction rate of n-C_3H_7O with HO_2, reflecting the increased mole fraction of HO_2 resulting from the deletion of Reaction R6.

Additionally, a formation pathway for C_2H_5OH was introduced based on the NUIGMech1.1 mechanism <cit.>, using rate coefficients derived from self- and cross-reactions similar to those of n-/i-C_3H_7O_2. This inclusion brought the model's prediction for C_2H_5OH within the experimental measurement error range, as depicted in Fig. <ref>.

Furthermore, the original model showed under-prediction of C_2H_5OOH compared to experimental measurements, with almost all C_2H_5OOH formed via the reaction between C_2H_5O_2 and HO_2. The increased mole fraction of HO_2 in the updated model led to an enhanced reaction rate between C_2H_5O_2 and HO_2, aligning the predicted mole fraction of C_2H_5OOH more closely with experimental measurements. This further supports the decision to delete Reaction R6.



The discrepancies of CH_4, C_2H_2, C_2H_4, C_2H_6 and C_3H_6 between experimental measurements and the original model are observed in the DBD experiment and the nCD experiment. Given the large energy deposition scales of the nCD experiment, it was chosen as the primary reference for adjusting the kinetic model. A discharge peak voltage of 5.0 kV was selected for pathway flux analysis, as shown in Fig. <ref>. In the original model, CH_4, C_2H_4, C_2H_6, and C_3H_6 are produced from the electron impact dissociation of C_3H_8. The dissociation cross sections for C_3H_8 were obtained from Ref. <cit.>, which provided estimates based on the total experimental dissociation cross sections for both ionized and neutral fragments (e + C_3H_8 → A + B + e and e + C_3H_8 → A^+ + B + 2e). However, the cross-section branch ratios were estimated using empirical formulas, leading to significant uncertainties. Consequently, the reaction rate coefficients for C_3H_8 dissociation were adjusted according to experimental results.

First, in the original model, Reaction (R1) is the primary production pathway for C_3H_6, and its prediction was overestimated across all discharge voltage scales in the nCD experiment. To improve accuracy, the cross section for Reaction (R1) was reduced by a factor of 0.8, which enhanced the prediction accuracy for C_3H_6, as shown in Fig. <ref>.

Second, reaction (R2) is responsible for the formation of C_2H_6. The original model significantly under-predicted C_2H_6 in the nCD experiment. An adjustment factor of 12 was applied to the cross section for Reaction (R2), aligning the C_2H_6 predictions with the experimental results.

Furthermore, reaction (R3) is the primary pathway for the formation of both CH_4 and C_2H_4. The plasma sub-mechanism for CH_4 was adopted from Ref. <cit.> with experimental validation, indicating that the CH_4 consumption rates in the original model were accurate. Despite this, both CH_4 and C_2H_4 were over-predicted in the DBD experiment, and CH_4 was also over-predicted in the nCD experiment. This suggested that the cross section for Reaction (R3) was overestimated. An adjustment to the cross section for Reaction (R3) was made to achieve better agreement for CH_4 in the nCD experiment and C_2H_4 in the DBD experiment.

Besides, A discrepancy in the prediction of C_3H_4 between the original model and the DBD experiment was noted. Reaction (R4) is responsible for 93% of the formation of C_3H_4-A and C_3H_4-P. Thus, the rate of Reaction (R4) was adjusted to better match the DBD experimental results.

Finally, the original model under-predicted C_2H_2 in both the nCD and DBD experiments, as shown in Fig. <ref> and Fig. <ref>. In the original model, C_2H_2 is generated through reactions between excited argon and C_2H_4 and between O and C_3H_4-A. However, with a mole fraction of C_2H_2 around 600 ppm in the nCD experiment compared to a C_2H_4 mole fraction of less than 400 ppm, C_2H_4 alone cannot account for the C_2H_2 formation. Similarly, C_3H_4-A is present at too low a concentration to significantly contribute to C_2H_2 production. Therefore, electron-impact dissociation of C_3H_8 (Reaction R5) was considered as a primary source for C_2H_2. Adding Reaction (R5) to the updated model significantly improved the accuracy of C_2H_2 predictions, with the updated model's C_2H_2 predictions closely matching experimental results from both the nCD and DBD experiments.

As stated above, The model undergoes significant updates through pathway flux analysis informed by experiments conducted under nCD and DBD conditions. By adjusting key reaction mechanisms according to experimental data, including C_3H_6, C_2H_6, CH_4, C_2H_4, C_2H_2, and C_3H_4, the updated model now demonstrates markedly improved predictive accuracy. This enhancement is particularly evident in its ability to better replicate experimental observations in both nCD and DBD experiments.



 §.§ The comparisons of C_3H_8 oxidation process assisted by DBD and nCD


To explore the impact of the single pulse deposition energy (SPDE) on the oxidation of C_3H_8, we have designed three distinct cases, each tailored to mimic specific energy deposition conditions. As depicted in Table <ref>, these cases are denoted as Case1, Case2, and Case3, with corresponding SPDE values in the model set at 1.85×10^-5 J/cm^3, 9.08×10^-5 J/cm^3, and 3.9×10^-4 J/cm^3, respectively. Notably, the total deposition energy in the model is standardized at 7.36×10^-3 J/cm^3 by adjusting the discharge frequency and pulse number, ensuring consistency across the cases. Furthermore, the gas temperature is maintained at a constant 400 K, while the discharge current values for the DBD and nCD in the models are sourced from Ref. <cit.> and the nCD experiment, respectively. These cases are designed to provide insights into the role of SPDE on C_3H_8 oxidation, offering valuable comparisons under varying energy deposition conditions.



In Fig. <ref>(a), the average reduced electric field evolution for DBD and nCD is depicted, where the reduced electric field is averaged over all discharge pulses. Notably, the average reduced electric field for nCD experiences a rapid increase to 450-500 Td before 3 ns, followed by a dramatic decrease to 150-200 Td at 8 ns, maintaining this level until the end of the discharge. In contrast, the average reduced electric field for DBD fluctuates repeatedly from 0 Td to 400 Td. As illustrated in Fig. <ref>(b), the reduced electric field distribution for nCD is concentrated at 150-200 Td and 450-500 Td. In comparison, the reduced electric field distribution for DBD predominantly ranges from 0-25 Td and 50-250 Td, accounting for approximately 55% and 35%, respectively, indicating a lower reduced electric field distribution compared to nCD.



Fig. <ref> presents a comparison of species density in the C_3H_8 oxidation process assisted by DBD and nCD. Interestingly, the density of species such as H_2O, CO, CO_2, C_2H_2, and C_2H_4 in the C_3H_8 oxidation process remains similar between DBD and nCD, despite varying SPDE values. This suggests that SPDE has minimal influence on the C_3H_8 oxidation process when the total deposition energy remains constant. Furthermore, Fig. <ref> highlights the time evolution of C_3H_6 and CH_4 in the C_3H_8 oxidation process assisted by DBD and nCD. Both C_3H_6 and CH_4 are produced during nanosecond pulse discharge and subsequently consumed in the discharge gaps. Notably, the increase in C_3H_6 and CH_4 during a given period correlates with the rise in SPDE. These observations collectively suggest that the C_3H_8 oxidation process is primarily controlled by total deposition energy and displays limited correlation with the specific discharge type, be it DBD or nCD.




As depicted in Fig. <ref>, the consumption pathways of C_3H_8 in DBD and nCD environments were analyzed. In DBD, 57% of C_3H_8 is consumed through vibrationally excited reactions, attributed to the concentrated reduced electric field distribution of 0-25 Td, which enhances efficient vibrational excitation. Conversely, in nCD, a lower proportion of C_3H_8 is consumed via vibrationally excited reactions, with a higher contribution from species such as O(^1D), Ar^+, e, and OH. C_3H_8(v) is quenched back to C_3H_8 through relaxation reactions, contributing thermally to the process. Although a significant portion of C_3H_8 in DBD is excited to C_3H_8(v), the energy deposited into C_3H_8(v) remains relatively low due to the limited vibrational excitation energy. In DBD, the majority of energy deposition occurs within the 100-250 Td range, similar to processes occurring at 150-200 Td and 500 Td in nCD. Consequently, the species density in the C_3H_8 oxidation process, whether assisted by DBD or nCD, exhibits comparable values. Additionally, the time evolutions of C_3H_6 and CH_4 densities in both nCD and DBD environments display similar tendencies.







§ CONCLUSIONS


This study investigates the kinetics of plasma-assisted propane oxidation in nanosecond pulse discharge. A zero-dimensional plasma-combustion coupled solver equipped with sensitivity and pathway flux analysis modules was developed, based on the ZDPlaskin package and a custom-built combustion chemistry code. A kinetic model was proposed and validated against the nCD experiment using gas chromatography and the nanosecond DBD experiment using synchrotron photoionization mass spectrometry. The updated model demonstrates predictions for 15 species falling within the range of DBD experimental measurement errors, with significantly improved prediction accuracy for 17 species. Notably, the updated model exhibits good prediction accuracy with the nCD experiment for C_3H_8, H_2, CH_4, and C_2H_6 under large deposition energy scales. However, the model-predicted fractions of CO_2, C_2H_4, and CO are 55%, 45%, and 45% lower than those measured in the nCD experiment, respectively.

Several modifications to the reactions were made based on experiments and pathway flux analysis. This included fitting the reaction rate coefficients of i -/n-C_3H_7O_2 with HO_2 (R7, R8) due to the large uncertainty of the analogy method. Additionally, the deletion of the erroneous addition reaction CH_3OH+O_2↔ CH_3O+HO_2 resulted in improved model predictions for CH_3OH. The supplementation of the CH_3O sub-mechanism from the HP-Mech v3.3 mechanism enhanced the updated model's prediction accuracy for CH_2O. Furthermore, adjustments to the C_3H_8 dissociation reaction rate coefficients, owing to large uncertainties, greatly improved the prediction accuracy for H_2, CH_4, C_2H_6, C_2H_2, and C_3H_6.

Comparisons of C_3H_8 oxidation processes assisted by DBD and nCD under varying SPDE but with the same total deposition energy revealed that the process is primarily controlled by total deposition energy, demonstrating minimal influence from SPDE and displaying limited correlation with the specific discharge types, DBD and nCD.

Overall, these findings illustrated the intricate interplay between plasma and combustion chemistry, offering valuable insights into the mechanisms of plasma-assisted propane oxidation.












§ ACKNOWLEDGMENTS

The work is supported by the National Natural Science Foundation of China (No.52277168, No.52025064, No.52207151), the National Key Research and Development Program of China (No.2023YFB4005700, No.2023YFB4005705, and No.2023YFB4005702-03), the University Synergy Innovation Program of Anhui Province (No.GXXT-2022025), and the independent project of the Energy Research Institute of Hefei Comprehensive National Science Center (Anhui Energy Laboratory) (No.22KZZ525, No.23KZS402, No.22KZS301, and No.22KZS304). The authors are thankful to the young research group in Atelier des Plasmas for fruitful discussions.

elsarticle-num-names















 Paper Title: Asymptotic windings, surface helicity and their applications in plasma physics.









	Wadim Gerner[E-mail address: mailto:wadim.gerner@inria.frwadim.gerner@inria.fr]




Sorbonne Université, Inria, CNRS, Laboratoire Jacques-Louis Lions (LJLL), Paris, France


Abstract: 
In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper we investigate properties of surface helicity and in particular answer two open questions posed in the aforementioned work: (i) We give a precise mathematically rigorous physical interpretation of surface helicity in terms of linking of distinct field lines. (ii) We prove that surface helicity is non-trivial if and only if the underlying surface has non-trivial topology (i.e. at least one hole).

We then focus on toroidal surfaces which are of relevance in plasma physics and express surface helicity in terms of average poloidal and toroidal windings of the individual field lines of the underlying vector field which enables us to provide a connection between surface helicity and rotational transform. Further, we show how some of our results may be utilised in the context of coil designs for plasma fusion confinement devices in order to obtain coil configurations of particular "simple" shape.

Lastly, we consider the problem of optimising surface helicity among toroidal surfaces of fixed area and show that toroidal surfaces admitting a symmetry constitute global minimisers.
Keywords: Submanifold helicity, Surface helicity, Biot-Savart operator, Plasma physics, Coil winding surface, Rotational transform
2020 MSC: 14J81, 37E35, 49Q10, 78A30, 78A55


§ INTRODUCTION

	In the realm of 3-dimensional (magneto-)hydrodynamics there is a well-known preserved physical quantity called helicity. To set the stage, we let Ω⊂ℝ^3 be a bounded smooth domain and we let u be a smooth vector field which is divergence-free on Ω and tangent to its boundary. The 3-d Biot-Savart helicity of u may then be defined as follows
	
    ℋ(u):=1/4π∫_Ω×Ωu(x)·(u(y)×x-y/|x-y|^3)d^3yd^3x.

	An alternative way to express helicity consists of defining the Biot-Savart potential of u by 
    BS_Ω(u)(x):=1/4π∫_Ωu(y)×x-y/|x-y|^3d^3y

	so that ℋ(u) is just the L^2-inner product of u and BS_Ω(u), c.f. <cit.>. From a physical perspective helicity may be regarded as a measure of the average linking of distinct field lines of the underlying vector field u, see <cit.>,<cit.>,<cit.>. The relevance of helicity is that it is preserved under the action of volume preserving diffeomorphisms, c.f. <cit.>, <cit.>,<cit.> and in fact is in some sense the only such invariant <cit.>. In the context of the incompressible Euler equations the velocity field v satisfies the equations
	
    ∂_tv+∇_vv=∇ p, div(v)=0 in Ω, v∥∂Ω,

	where p denotes the pressure function. Upon applying the curl operator to the first equation one verifies that the vorticity ω of the fluid, i.e. the curl of the velocity field v, is a solution to
	
    ∂_tω=-[v,ω],
 
	where [·,·] denotes the Lie-bracket of vector fields. This implies that ω(t,x)=((ψ_t)_*ω_0)(x), where ψ_t denotes the flow of v, (ψ_t)_* denotes the pushforward with respect to ψ_t and ω_0(x):=ω(0,x)=curl(v(0,x)) is the initial vorticity. Since v∥∂Ω and div(v)=0 we conclude that ψ_t is well-defined and volume-preserving. We hence conclude from our preceding discussion that the helicity of the vorticity vector field is preserved in time. In the context of ideal magnetohydrodynamics, one can show in a similar spirit that the helicity of the underlying magnetic field is preserved in time. Even more, helicity also provides a lower bound on the L^2-norm of the underlying vector field, which in the context of ideal magnetohydrodynamics corresponds to the magnetic energy of the underlying magnetic field, c.f. <cit.>, <cit.>
	
    |ℋ(u)|≤ c(Ω)∫_Ω|u(x)|^2d^3x

where c(Ω)>0 is a constant independent of u.
It is further interesting to understand which other diffeomorphisms, beyond the volume-preserving diffeomorphisms, preserve helicity. This question has been investigated in <cit.> and it turns out that in general there exist non-volume preserving diffeomorphisms which preserve helicity.

In <cit.> Cantarella and Parsley introduced shortly the notion of submanifold helicity, a special case of which is the surface helicity, which corresponds to the (0,2,3)-helicity in the terminology of <cit.>. As we have seen, the notion of 3-d helicity is an important quantity in the context of fluid dynamics. It is therefore of interest to understand its 2-d analogue since vector fields tangent to surfaces appear naturally in many applications, some of which we will discuss in the next sections.
The main goal of the present manuscript is to illuminate our understanding of surface helicity and establish connections to other known dynamical quantities.
Outline of the paper: In the next section we present the main results which we divided in subsections to increase readability. We first discuss answers to some open questions raised in <cit.> regarding the surface helicity and provide in particular a general interpretation of surface helicity in terms of linkage of distinct field lines of the underlying vector field. In <Ref> we provide an alternative interpretation on toroidal surfaces which connects surface helicity to other important dynamical quantities on the torus. Based on this alternative interpretation we connect in <Ref> the notion of surface helicity to the notion of rotational transform in the context of plasma fusion. We then deal in <Ref> with the question of optimising surface helicity among toroidal surfaces of prescribed area and prove that toroidal surfaces admitting a symmetry are global minimisers while the existence of global maximisers is stated as an open problem. Finally, in <Ref> we discuss how some of the results obtained in the present manuscript can be used to construct "simple" surface currents on a coil winding surface which approximate desired target magnetic fields in a given plasma domain, which is of relevance in stellarator plasma fusion confinement devices. The proofs of the theorems are contained in <Ref>.



§ MAIN RESULTS

Before we state the main results we establish some notation.
Notation: For a given, C^1,1-regular, closed, connected surface Σ⊂ℝ^3 we let L^p(Σ), 1≤ p≤∞, denote the standard space of L^p-integrable functions on Σ and we define the vectorial counterpart by

    L^p𝒱(Σ):={v∈(L^p(Σ))^3| v(x)·𝒩(x)=0 a.e. x∈Σ},

where 𝒩 denotes the outward pointing unit normal on Σ, i.e. the unit normal pointing in the direction of the unbounded component of ℝ^3∖Σ. In addition we let C^0,1𝒱(Σ) denote the Lipschitz-continuous vector fields on Σ which are tangent to Σ at every point. We say that v∈ L^p𝒱(Σ) is divergence-free if ∫_Σ v·∇_Σα dσ=0 for all α∈ C^1(Σ), where ∇_Σ denotes the surface gradient of α which can be computed as ∇_Σα=∇α̃-(𝒩·∇α̃)𝒩, where α̃ is an arbitrary C^1-extension of α to Ω and ∇ denotes the standard Euclidean gradient, and where dσ denotes the standard surface measure. We denote by L^p𝒱_0(Σ) the subspaces of L^p𝒱(Σ) which consist of divergence-free vector fields. Lastly, we say that v∈ L^p𝒱(Σ) is curl-free if ∫_Σv· (∇_Σα×𝒩)dσ=0 for all α∈ C^1(Σ).


 §.§ Properties of Surface helicity

We start by defining the 2-dimensional analogue of the Biot-Savart operator.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Given 1<p<∞, the following operator is a well-defined, continuous operator called the surface Biot-Savart operator
	
    BS_Σ:L^p𝒱(Σ)→ L^q𝒱(Σ), v↦1/4π(∫_Σv(y)×x-y/|x-y|^3dσ(y))^∥

	where q:=2p/2-p if 1<p<2, where 1≤ q<∞ can be taken arbitrary if p=2 and q=∞ if 2<p≤∞.

In order to define the "tangent part" ·^∥ we cover Σ by any collection of open domains U_k which support local orthonormal frames {E^k_1,E^k_2} of class C^0,1. On each such U_k we define (we drop the superscript k)

    (∫_Σ v(y)×x-y/|x-y|^3dσ(y))^∥:=∑_i=1^2(∫_Σv(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x).

The well-definedness of <Ref> is shown in <Ref>. In particular we show that the defined vector fields agree on the overlaps of the U_k almost surely and that any two covers U_k and Ũ_k induce the same vector field.
We come now to the definition of (cross-)helicity.

	
	Let Σ⊂ℝ^3 be a connected, closed, C^1,1-surface. Then we define the cross-helicity as follows
	
    ℋ_c:L^4/3𝒱(Σ)× L^4/3𝒱(Σ)→ℝ, (v,w)↦∫_Σv(x)·BS_Σ(w)(x)dσ(x)
    		
    
    		=1/4π∫_Σ×Σv(x)·(w(y)×x-y/|x-y|^3)dσ(y)dσ(x).

Further, we define the surface helicity of a vector field by

    ℋ:L^4/3𝒱(Σ)→ℝ, v↦ℋ_c(v,v).



	
	Due to the continuity of BS_Σ:L^4/3𝒱(Σ)→ L^4𝒱(Σ) we see that cross helicity (and therefore helicity) is continuous and that there is some constant c(Σ)>0 such that for all v,w∈ L^4/3𝒱(Σ) we have |ℋ_c(v,w)|≤ c_0v_L^4/3(Σ)w_L^4/3(Σ).

In <cit.> Cantarella and Parsley introduced the notion of helicity of closed forms on (sub-)manifolds of Euclidean spaces. Of particular interest are the (1,3,3)-helicity and the (0,2,3)-helicity which correspond to helicities of 2-forms on domains in ℝ^3 and of 1-forms on embedded surfaces in ℝ^3 respectively. It is shown <cit.> that the (1,3,3)-helicity coincides with the classical helicity of divergence-free fields, tangent to the boundary of a given domain Ω upon an identification of a given closed 2-form α with a vector field v by contraction of the corresponding volume form.

Regarding the (0,2,3)-helicity the authors in <cit.> mention that it should be possible to regard it as a measure of the linkage of distinct field lines of a vector field associated with a given closed 1-form α and they raise the question of whether or not the (0,2,3)-helicity may attain non-trivial values (i.e. whether there exists v∈ C^∞𝒱_0(Σ) with ℋ(v)≠ 0). They conjecture that this should be the case whenever Σ has a non-trivial homology.

To summarise, the following two central problems regarding the (0,2,3) helicity were open <cit.>


	
  * Show that if Σ has non-trivial homology, then the (0,2,3)-helicity is not identically zero.
	
  * Provide a mathematically rigorous physical interpretation of (0,2,3)-helicity.

In the upcoming result we denote by ℋ_(0,2,3) the (0,2,3)-helicity on (closed) 1-forms and by ℋ the (vector field) helicity as introduced in <Ref>. Given some v∈ C^∞𝒱(Σ) we denote by ι_vω_Σ the contraction of the area form ω_Σ via the vector field v. In particular, if v is divergence-free, the associated 1-form is closed. Since the authors in <cit.> deal with smooth domains and surfaces we formulate the following result in the smooth setting.

	
	Let Σ⊂ℝ^3 be a smooth, connected, closed surface and let v∈ C^∞𝒱_0(Σ). Then
	
    ℋ_(0,2,3)(ι_vω_Σ)=ℋ(v).


We exploit this relationship to provide an answer to both of the open problems raised in <cit.> regarding (0,2,3)-helicity.

The following provides an answer to the first open problem regarding surface helicity.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then there exists some v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0 if and only if g(Σ)≥ 1, where g(Σ) denotes the genus of Σ. Further, if v∈ L^2𝒱_0(Σ) is co-exact, i.e. it can be written as v=grad_Σ(f)×𝒩, for some function f∈ H^1(Σ) and where 𝒩 denotes the outward unit normal, then ℋ_c(v,w)=0 for all w∈ L^2𝒱_0(Σ).

Regarding the second question posed above we intend to follow the ideas of <cit.> and <cit.> regarding the physical interpretation of the standard 3-d helicity (which corresponds to (1,3,3)-helicity in the notation of <cit.>).

Note that in general Σ might contain a large "flat region" (a copy of a large 2-d disc) and then if the field lines of a given vector field v∈ C^0,1𝒱_0(Σ) are not necessarily periodic, we might attempt, in the same spirit as in <cit.>, to close the field lines by geodesics. However, at least as long as the endpoints of the considered field lines are contained in the flat region, the geodesics will be lines passing through the corresponding endpoints of the distinct field line segments. But then these artificially added line segments will intersect with "high probability" (since the directions of the lines passing through the pairs of endpoints will be "generically" not parallel). Thus, one should expect that there is a set of positive measure in Σ×Σ for which the "artificially" closed field lines will intersect if we close field line segments by means of geodesics. In addition, the artificially added parts may intersect the original field line segments, see <Ref>.


Therefore it is necessary to find a new way to appropriately close the field lines of the underlying vector field to be able to obtain an interpretation of helicity in terms of average linking numbers.

Given some v∈ C^0,1𝒱_0(Σ) and x∈Σ we let γ_x(t) denote the (unique) field line of v starting at x. Now if we are given some x∈Σ and T>0 it can happen that γ_x(T)=x=γ_x(0) in which case the field line γ_x is closed (possibly a point). In general however we will find γ_x(T)≠ x=γ_x(0) and therefore it is necessary to close the field line in a suitable way to give an interpretation of helicity in terms of linking of suitable closed curves. It is well-known that, if Σ∈ C^2, there exists some τ_0(Σ)>0 such that the map Ψ_τ(x):=x+τ𝒩(x), x∈Σ, is a diffeomorphism onto its image for every |τ|<τ_0 where 𝒩 denotes the outward pointing unit normal, <cit.>. We define Σ_τ:=Ψ_τ(Σ) so that in particular Σ_0=Σ. Now it is the case that for almost every (x,y)∈Σ×Σ the images of the corresponding field lines γ_x(ℝ) and γ_y(ℝ) are disjoint. Given T,S>0 we then consider the curves γ_x:[0,T]→Σ and γ_y:[0,S]→Σ. In order to close them (in case they are not already closed) we proceed as follows: We connect γ_x(T) with Ψ_τ(γ_x(T)) for a suitably chosen τ (which we will be able to choose independent of x,y,T and S) by following the path Ψ_t(γ_x(T)) from t=0 until t=τ. Similarly, we can connect Ψ_τ(x) and x=γ_x(0) by inverting the path Ψ_t(x) from t=0 until t=τ. We note that Ψ_τ(γ_x(T)),Ψ_τ(x)∈Σ_τ and that we are left with connecting Ψ_τ(γ_x(T)) and Ψ_τ(x). We connect these two points by any curve contained in Σ_τ which is of unit speed and whose length can be uniformly bounded independently of x and T. For instance, we can connect these two points by any (length minimising) geodesic on Σ_τ. We close the curve γ_y:[0,S]→Σ in a similar fashion, but in contrast we connect y and γ_y(S) with their corresponding points on Σ_-τ and connect the corresponding two points on Σ_-τ by a geodesic on Σ_-τ. Let us denote these two closed curves obtained in that way by σ^τ_x,T and σ^-τ_y,S, see <Ref> for an illustration. By construction these two closed curves intersect if and only if γ_x([0,T]) and γ_y([0,S]) intersect, which we will show does not happen for almost every (x,y)∈Σ×Σ.
Note also that we explicitly leave the surface Σ to close our field lines.
By convention we set the linking number of a closed curve and any point not lying on this curve to zero.




With these preliminary observations we obtain the following result where we assume the surface Σ to be of class C^2 to avoid technicalities.

	
	Let Σ⊂ℝ^3 be a closed, connected C^2-surface. Then there exists some τ(Σ)>0 such that for every v∈ C^0,1𝒱_0(Σ) and every sequences (T_n)_n,(S_n)_n⊂ (0,∞) which are strictly increasing and diverging to ∞, the functions 
	
    Σ×Σ→ℝ, (x,y)↦lk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_n

	are well-defined elements of L^1(Σ×Σ) for every n and converge to a limit function in L^1(Σ×Σ). Further, we have the identities
	
    ℋ(v)=lim_n→∞∫_Σ×Σlk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_ndσ(x)dσ(y)=∫_Σ×Σlim^L^1_n→∞lk(σ^τ_x,T_n,σ^-τ_y,S_n)/T_nS_ndσ(x)dσ(y).


Here lim^L^1 indicates that we consider the L^1-limit (and not the pointwise limit which exists only upon passing to appropriate subsequences). Therefore surface helicity may be regarded as either the "asymptotic average linking number of distinct field lines of v" or, as is done more commonly in the 3-d case, the "average asymptotic linking number of distinct field lines of v".

	
	If all field lines of v are periodic and there exists some τ>0 such that all field lines of v have a period of at most τ, then we can instead close the field line segments of v by continuing to follow the flow of v until we return to the respective starting points. In this periodic situation no additional technical issues arise by assuming that Σ∈ C^1,1.



 §.§ Surface helicity on toroidal surfaces


We have seen in <Ref> that on an arbitrary (closed) surface Σ, we can interpret helicity as a measure of entanglement of distinct field lines of the underlying vector field. In the 3-dimensional situation entanglement of distinct field lines does not allow to infer information about the dynamical properties of individual field lines. However, in the 2-dimensional situation closed field lines are co-dimension 1 submanifolds and therefore the dynamics of individual field lines may very well influence the ability of the remaining field lines to link. To illustrate this point further, we imagine we are given a toroidal surface Σ≅ T^2 in ℝ^3 and a (regular) vector field v tangent to Σ. We suppose that v has a closed field line γ which wraps once in poloidal direction around Σ. Since distinct field lines do not cross we infer that the remaining field lines of v must be contained within Σ∖γ. But we can continuously "unfold" Σ∖γ and we see that in fact the remaining field lines may be viewed as being contained in a cylinder so that no linking may take place, c.f. <Ref>. Consequently, the helicity of v must be zero by means of the linking interpretation of helicity, <Ref>. We therefore see that the dynamics of individual field lines may have a strong influence on surface helicity and one may thus expect that an alternative interpretation of surface helicity in terms of averages of dynamical properties of individual field lines should be possible.




We focus here on C^1,1-surfaces Σ⊂ℝ^3 which bound solid tori, i.e. Σ=∂Ω for a domain Ω⊂ℝ^3 which is diffeomorphic to the solid torus D^2× S^1, where D^2 is the unit disc in ℝ^2 and S^1 is its boundary. We note that these type of surfaces are the most relevant in plasma physics, as they appear as the boundaries of plasma domains or model the surfaces along which the coils wind which generate the plasma confining magnetic field <cit.>.

Before we state the main result of this subsection let us introduce some notions.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3, i.e. Ω≅ D^2× S^1 and ∂Ω=Σ∈ C^1,1. We say that a simple closed C^1-curve σ_p on Σ is purely poloidal if σ_p is contractible when viewed as curve within Ω, but represents a non-trivial element of the first fundamental group of Σ. We call a simple closed C^1-curve σ_t in Σ toroidal if it generates the first fundamental group of Ω once viewed as a curve in Ω.

We note first that, up to orientation and a standard identification, any two purely poloidal curves are path-homotopic within Σ. Similarly, up to orientation, all toroidal curves are homotopic within Ω. However, two toroidal curves may not be (even up to orientation) homotopic within Σ. In addition, any choice of σ_p and σ_t forms a pair of generators of the first fundamental group of Σ.

To motivate the next definition we consider an arbitrary simple closed C^1-curve σ contained in Σ and for a fixed pair σ_p and σ_t of a purely poloidal and a toroidal curve we can express σ=Pσ_p⊕ Qσ_t where ⊕ denotes the concatenation of curves and P,Q∈ℤ denote the amount of times the curves σ_p and σ_t are traversed. We observe first that there is a unique pair γ_p,γ_t∈ℋ(Σ):={v∈ W^1,2𝒱(Σ)|curl_Σ(v)=0=div_Σ(v)} which forms a basis of ℋ(Σ) and satisfies

    ∫_σ_pγ_t=0=∫_σ_tγ_p, ∫_σ_pγ_p=1=∫_σ_tγ_t.

We observe further, which for example follows from an identical reasoning as in <cit.>, that in fact ℋ(Σ)⊂⋂_1<p≤∞W^1,p𝒱(Σ)⊂⋂_0<α<1C^0,α𝒱(Σ) and therefore the line integrals in (<ref>) make sense in a classical sense. Now since σ is homotopic to Pσ_p⊕ Qσ_t, we find

    ∫_σγ_t=P∫_σ_pγ_t+Q∫_σ_tγ_t=Q

and further

    lim_T→∞1/T∫_σ[0,T]γ_t=Q/τ,

where τ>0 is the period of σ and σ[0,T] is the path [0,T]→Σ, t↦σ(t) (which is not simple for T≥τ). Similarly we find P=∫_σγ_p and lim_T→∞1/T∫_σ[0,T]γ_p=P/τ. This motivates the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a toroidal curve σ_t and a purely poloidal curve σ_p we define the weighted asymptotic toroidal windings resp. weighted asymptotic poloidal windings of a (not necessarily periodic) C^1-curve σ:[0,∞)→Σ by
	
    q̂(σ):=lim_T→∞1/T∫_σ[0,T]γ_t, p̂(σ):=lim_T→∞1/T∫_σ[0,T]γ_p.



	
	We note that γ_t in (<ref>) is independent of the chosen purely poloidal curve σ_p and it only depends on the toroidal curve σ_t via its orientation, i.e. if two toroidal curves are oriented such that they are homotopic within Ω then the corresponding γ_t will be the same and if they are oriented in opposite direction, then the corresponding γ_t will differ by a minus sign. Consequently, q̂(σ) also depends only on the chosen orientation of the toroidal curve. However, γ_p will in general depend on the homotopy type of σ_t when viewed as a curve in Σ so that even if two toroidal curves are homotopic within Ω, the corresponding γ_p may be distinct (but will be the same if the toroidal curves are homotopic within Σ).
		
The following is the natural generalisation to vector fields.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a toroidal curve σ_t, a purely poloidal curve σ_p and a vector field v∈ C^0,1𝒱_0(Σ), we define the weighted asymptotic toroidal windings and the weighted asymptotic poloidal windings of v at a point x∈Σ as follows
	
    q̂(x):=q̂(γ_x), p̂(x):=p̂(γ_x),

	where γ_x denotes the field line of v starting at x. We further define the average weighted asymptotic toroidal windings and the average weighted asymptotic poloidal windings of v by
	
    Q(v):=∫_Σq̂(x)dσ(x)/|Σ|, P(v):=∫_Σp̂(x)dσ(x)/|Σ|,

	where |Σ| denotes the area of Σ.

We note that in order to compute the average one should have q̂,p̂∈ L^1(Σ). This however follows from standard ergodic theoretical arguments. We state it as a separate lemma since it is of importance when establishing a connection between surface helicity and toroidal and poloidal averages.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve respectively. If v∈ C^0,1𝒱_0(Σ), then q̂,p̂∈ L^1(Σ) and we have
	
    Q(v)=∫_Σv(x)·γ_t(x)dσ(x)/|Σ|, P(v)=∫_Σv(x)·γ_p(x)dσ(x)/|Σ|.



	One natural way to define purely toroidal simple closed curves is to demand that a given toroidal loop is trivial within ℝ^3∖Ω, see also <cit.>. Another useful way to choose a toroidal curve σ_t can consist in making sure that the resulting induced element γ_p is L^2(Σ)-orthogonal to γ_t so that the quantities Q(v) and P(v) essentially correspond to the L^2(Σ)-orthogonal projection of v onto the space ℋ(Σ) in the sense that π(v)=|Σ|(Q(v)γ_t/γ_t^2_L^2(Σ)+P(v)γ_p/γ_p^2_L^2(Σ)) where π denotes the projection onto ℋ(Σ) and v∈ C^0,1𝒱_0(Σ). Both choices can have advantages in distinct situations and therefore we allow general toroidal curves in our considerations.

To establish a connection between the quantities Q and P, which make statements about the average behaviour of individual field lines, and helicity which is a measure of the average linking of distinct field lines, we need to introduce one final notion.
We define the space ℋ_N(Ω) of harmonic Neumann fields on a bounded C^1,1-domain Ω⊂ℝ^3 as follows

    ℋ_N(Ω):={Γ∈ H^1(Ω,ℝ^3)|curl(Γ)=0=div(Γ), Γ∥∂Ω}.

We note first, c.f. <cit.>, that ℋ_N(Ω)⊂⋂_1≤ p<∞W^1,p𝒱(Ω) and thus, due to Sobolev embeddings ℋ_N(Ω)⊂⋂_0<α<1C^0,α𝒱(Ω). Since curl(Γ)=0 it follows that the restriction Γ|_Σ satisfies curl_Σ(Γ|_Σ)=0 in the weak sense and according to the Hodge-decomposition theorem <cit.> we can therefore write

    Γ|_Σ=∇_Σα+γ

for suitable α∈ H^1(Σ) and γ∈ℋ(Σ). Here γ is the L^2-orthogonal projection of Γ|_Σ onto ℋ(Σ). We note that since ℋ_N(Ω) is L^2(Ω)-orthogonal to the gradient fields and because Γ=curl(A) for a suitable vector potential A (for instance A may be taken to be the Biot-Savart potential of Γ) we must have γ≠ 0 whenever Γ≠ 0.

We are now finally ready to give an alternative interpretation of surface helicity in terms of the "dynamics of individual field lines".

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p be a purely poloidal curve and σ_t be a toroidal curve. Further, fix any Γ∈ℋ_N(Ω) such that the L^2(Σ)-orthogonal projection γ of Γ|_Σ onto ℋ(Σ) is L^2(Σ)-normalised. In addition let γ̃:=γ×𝒩. Then, given any v∈ C^0,1𝒱_0(Σ), we have
	
    ℋ(v)/|Σ|^2=Q(v)P(v)∫_σ_tγ∫_σ_pγ̃+[ℋ(γ)(∫_σ_tγ)^2+∫_σ_tγ∫_σ_tγ̃]Q^2(v).

	If, moreover, σ_t bounds a bounded C^1,1-surface 𝒜 outside of Ω, i.e. 𝒜⊂ℝ^3∖Ω and ∂𝒜=σ_t, then
	
    ℋ(v)/|Σ|^2=Q(v)P(v)∫_σ_tγ∫_σ_pγ̃.



	
	We observe that (ℋ_N(Ω))=1 whenever Ω is a solid torus, <cit.>. Hence the condition of L^2(Σ)-normalisation of γ determines Γ uniquely up to a minus sign. If a chosen Γ is replaced by -Γ we see that γ as well as γ̃ both are also changed by a minus sign and hence (since ℋ(-γ)=ℋ(γ)) the above formula is independent of that choice, as one would expect. Similarly, if σ_p and σ̃_p are purely poloidal curves on Σ then either they are homotopic and thus the integrals involving σ_p are unchanged or otherwise σ_p and σ̃_p are oppositely oriented, in which case ∫_σ̃_pγ̃=-∫_σ_pγ̃, but P(v) also changes sign (since it is defined via γ_p which in turn is defined via σ_p). In addition, γ_t is independent of the chosen σ_p and thus Q is also independent of that choice. Therefore the right hand side of (<ref>) is independent of the chosen purely poloidal curve. Finally, we observe that Γ|_Σ and γ differ only by a gradient field such that ∫_σ_tγ=∫_σ_tΓ from which it follows that ∫_σ_tγ only depends on the orientation of σ_t (considered as a curve in Ω) and it is clear once again that changing the orientation of σ_t leaves the right hand side of (<ref>) invariant. Since changing orientation of σ_t leaves the right hand side invariant, we are left with understanding how the quantities ∫_σ_tγ̃, P(v) and Q(v) change if we replace σ_t by a toroidal curve σ̃_t which is homotopic to σ_t in Ω. It follows easily from (<ref>) that γ_t and hence Q(v) remains unchanged. On the other hand, since σ_t and σ_p form a basis of the fundamental group of Σ, we may express σ̃_t=σ_t⊕ Pσ_p, where we used that we may assume that σ̃_t is homotopic to σ_t. Then ∫_σ̃_tγ̃=∫_σ_tγ̃+P∫_σ_pγ̃ and one can verify that γ̃_p-γ_p=-Pγ_t so that it follows from <Ref> that P(v)-P(v)=-PQ(v) (where γ̃_p, P(v) are computed with respect to σ_p and σ̃_t). We hence see that the additional terms appearing through the change of P(v) and ∫_σ_tγ̃ when σ_t is replaced by σ̃_t cancel each other. We conclude that the right hand side of (<ref>) is independent of the chosen curves σ_p and σ_t, as it should be because the left hand side of (<ref>) is of course independent of any such choices.



 §.§ Surface helicity and rotational transform


Assume once more that T^2≅Σ⊂ℝ^3 bounds a C^1,1-solid torus Ω⊂ℝ^3 and suppose that we fixed some purely poloidal curve σ_p and a toroidal curve σ_t on Σ. Now, if σ:ℝ→Σ is any other closed C^1-curve, then we can write it as σ=Pσ_p⊕ Qσ_t, where Q may be viewed as the "toroidal" turns of σ and P may be viewed as the "poloidal" turns of σ. We recall that according to (<ref>) we can write

    Q/τ=lim_T→∞1/T∫_σ[0,T]γ_t, P/τ=lim_T→∞∫_σ[0,T]γ_p,

where γ_p,γ_t are determined by the equations in (<ref>). Consequently, if we are interested in the ratio of "poloidal" twists by "toroidal" twists P/Q, we obtain the identity (assuming Q≠ 0)

    ι(σ):=P/Q=lim_T→∞1/T∫_σ[0,T]γ_p/lim_T→∞1/T∫_σ[0,T]γ_t=p̂(σ)/q̂(σ),

where p̂(σ),q̂(σ) denote the "asymptotic windings" of σ. These observations motivate the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve respectively and let v∈ C^0,1𝒱(Σ). If q̂(x)≠ 0 for a given x∈Σ, c.f. <Ref>, we define the rotational transform of v at x as
	
    ι_v(x):=p̂(x)/q̂(x),

provided the limits in the definition of p̂(x) and q̂(x) exist.


	
	
		
  * According to <Ref>, if v∈ C^0,1𝒱_0(Σ), then p̂(x) and q̂(x) exist for a.e. x∈Σ w.r.t. the standard surface measure.
		
  * We observe that the definition of the rotational transform depends on the choice of σ_p and σ_t in the following way: We recall first the defining equations (<ref>) of γ_p and γ_t
		
    ∫_σ_pγ_t=0=∫_σ_tγ_p, ∫_σ_pγ_p=1=∫_σ_tγ_t.

		If σ̃_p is any other closed, purely poloidal curve, then either σ̃_p is homotopic to σ_p or otherwise it is homotopic to -σ_p (the curve obtained from σ_p by inverting the orientation). In both cases γ_t remains unchanged, while γ_p changes sign and so q̂ remains unchanged, while p̂ changes sign, i.e. at most the sign of ι is changed, depending whether σ̃_p is oriented in the same way as σ_p or oppositely oriented. On the other hand, if we change σ_t to some σ̃_t, then by definition either σ̃_t will be homotopic to σ_t or -σ_t as a curve in Ω (but not necessarily within Σ). Since σ_p and σ_t generate the first fundamental group of Σ, we can then however express σ̃_t=±σ_t⊕ Pσ_p for a suitable P∈ℤ, where the sign in ± depends on whether σ̃_t is homotopic to σ_t or -σ_t within Ω. Consequently, we see that γ_t at most changes sign, while γ̃_p=γ_p∓ Pγ_t. Thus, q̂ may change its sign, depending whether σ_t and σ̃_t are oriented in the same, or opposite, way within Ω and p̂̃̂(x)=p̂(x)∓ Pq̂(x), where p̂̃̂ indicates that this quantity is computed with respect to the curve σ̃_t. In conclusion ι̃_v(x)=±ι_v(x)∓ P, where once more the tilde indicates that ι_v is computed with respect to the toroidal curve σ̃_t. In particular, if σ_p,σ̃_p are two purely poloidal curves of the same orientation and σ_t,σ̃_t are two poloidal curves of the same orientation (within Ω), then ι̃_v(x)=ι_v(x)+β for some constant β∈ℤ, which is independent of x and the vector field v under consideration. Hence, if we wish to study a problem which aims to maximise ι (possibly under some additional constraints), then the maximisers will be independent of the chosen curves (as long as they are oriented in the same way). Finally, also notice that the property q̂(x)≠ 0 at a given x∈Σ is independent of the chosen curves.
		

We will now first state the main mathematical result of this section, before we shortly discuss the relevance of rotational transform in the context of plasma fusion confinement devices.

In order to do so we state here the (standard) definition of rectifiability of a vector field on a 2-torus T^2=ℝ^2ℤ^2.

	
	Let T^2≅Σ be a C^1,1-surface, which is diffeomorphic to the 2-torus. A vector field v∈ C^0,1𝒱(Σ) is said to be rectifiable if there exists a diffeomorphisms ψ:Σ→ T^2 such that ψ_*v=a∂_1+b∂_2 for suitable constants a,b∈ℝ and the standard coordinate fields ∂_1,∂_2 on T^2, where ψ_*v denotes the corresponding pushforward vector field.

With this we can formulate the main result of this subsection.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p and σ_t be a given purely poloidal and a toroidal curve on Σ respectively. Further, let γ_p,γ_t be a basis of ℋ(Σ) determined by the condition (<ref>). Assume that v∈ C^0,1𝒱(Σ) is rectifiable and that there exists some f∈ C^0,1(Σ,(0,∞)) such that w:=fv∈ C^0,1𝒱_0(Σ), i.e. div_Σ(w)=0. If Q(w)≠ 0, then the rotational transforms of v and w are well-defined at a.e. x∈Σ and there holds
	
    ι_v(x)=ι_w(x)=∫_Σw·γ_pdσ/∫_Σw·γ_t dσ=P(w)/Q(w),

	where the subscript indicates whose rotational transform is considered. In particular, ι_v(x) is independent of x.


	
	In general, for a given vector field v∈ C^0,1𝒱(Σ), even if the rotational transform ι_v(x) is an element of L^1(Σ), it is not true that 1/|Σ|∫_Σι(x)dσ(x)=P(v)/Q(v) because the average of a quotient does not need to coincide with the quotient of the respective averages.

As an immediate consequence of <Ref> and <Ref>, we obtain the following relation between surface helicity and rotational transform.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Let σ_p be a purely poloidal curve on Σ and let σ_t be a toroidal curve on Σ which bounds a bounded C^1,1-surface 𝒜 outside of Ω. Assume further that v∈ C^0,1𝒱(Σ) is rectifiable and that there exists some f∈ C^0,1(Σ,(0,∞)) such that w:=fv∈ C^0,1𝒱_0(Σ), i.e. div_Σ(w)=0. If Q(w)≠ 0, then
	
    ℋ(w)/|Σ|^2=ι_wQ^2(w)(∫_σ_tγ)(∫_σ_pγ̃),

	where ι_w is the constant rotational transform of w and γ,γ̃ are defined as in <Ref> and in particular are independent of w.


	
	
		
  * We point out shortly that a corresponding formula for the vector field v does not need to hold necessarily, since we require the divergence-free property in order to invoke <Ref>, which is the reason why the conclusion is solely formulated for the vector field w.
		
  * It is also customary to call a vector field v∈ C^0,1𝒱(Σ) semi-rectifiable, provided there exists a (strictly) positive C^0,1-function f such that fv is rectifiable, c.f. <cit.>. Hence <Ref> can be reformulated by assuming that a given w∈ C^0,1𝒱_0(Σ) is semi-rectifiable.
	


	
	In the realm of plasma physics one is interested in steady solutions of the equations of ideal magnetohydrodynamics with a resting plasma <cit.>. Such plasma equilibria are determined by the PDE
	
    B×curl(B)=∇ p and div(B)=0 in 𝒫, B∥∂𝒫

	where 𝒫 denotes the region containing the plasma ("plasma region"), B is the corresponding magnetic field and p is the pressure. According to Arnold's structure theorem, <cit.>, <cit.>, if B is analytic and B and curl(B) are not everywhere collinear, the domain 𝒫 will decompose, after removing a 2-dimensional analytic subset from it, into a finite number of chambers, each of which is invariant under the flow of B and is one of the following two types:
	
		
  * Either the chamber foliates further into toroidal surfaces, each of which is again invariant under the flow of B and such that on each such surface B is rectifiable and hence in particular either all field lines of B are closed (with the same period) or all of the field lines of B are dense within the surface.
		
  * Or the chamber foliates into cylindrical surfaces diffeomorphic to S^1×ℝ, each of which is invariant under the flow of B and such that all field lines of B are closed on any such surface.
	
	Of particular interest are the invariant toroidal surfaces. In particular, in order to counter the plasma particle drift in a fusion plasma it is desirable to have a high rotational transform <cit.>, see also <Ref> for a precise relationship between the different notions of rotational transform. We therefore wish to understand the quantity ι_B on regular toroidal level sets of the pressure p. We note that if Σ is a regular toroidal level set of p, then |∇ p|≠ 0 on all of Σ and in fact, letting f:=1/|∇ p| one can verify that B:=fB is a divergence-free field on Σ (divergence-free with respect to the surface measure on Σ), c.f. <cit.>. It follows therefore from <Ref> that we can compute the rotational transform of B on a given regular toroidal level set of Σ as
	
    ι_B(x)=∫_ΣB·γ_pdσ/∫_ΣB·γ_tdσ

	for a fixed purely poloidal curve σ_p and a toroidal curve σ_t bounding a surface outside of the domain enclosed by the level set and their corresponding induced harmonic fields γ_p,γ_t∈ℋ(Σ). Note once more, that the rotational transform of B (on any fixed regular toroidal surface) is constant.



 §.§ Surface helicity optimisation


In <cit.> Cantarella, DeTurck, Gluck and Teytel considered the following optimisation in 3-dimensional space: First, given a smooth, bounded domain Ω⊂ℝ^3, they show that there exists some smooth vector field v on Ω with div(v)=0, v∥∂Ω such that

    ℋ_3D(v)/v^2_L^2(Ω)=sup_w≠ 0
    div(w)=0
     w∥∂Ωℋ_3D(w)/w^2_L^2(Ω)=:λ(Ω),

where ℋ_3D denotes the standard 3-dimensional Biot-Savart helicity, see <cit.> for a proof of the above fact. Further, <cit.>, implies that |λ(Ω)|≤ c_0√(|Ω|) for an absolute constant c_0 independent of Ω. In view of this it is natural to ask whether there exists a bounded, smooth domain Ω which maximises λ(Ω) among all other domains of the same volume. Physically this amounts to finding a domain Ω which supports a magnetic field (tangent to the boundary of Ω) whose field lines are (on average) as entangled as possible among all other magnetic fields and all other domains of the same volume.

Some necessary conditions, assuming the existence of optimal domains, have been derived in <cit.>. Up to this day the question of existence of optimal domains for this problem has remained open, even though some recent advances have been made within certain classes of domains, c.f. <cit.>,<cit.>.
Here we consider a corresponding problem for the surface helicity. The first result is the following.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then there exists some v_±∈ L^2𝒱_0(Σ) with v_±_L^2(Σ)=1 and such that
	
    ±ℋ(v_±)=max_w≠ 0
    
    		w∈ L^2𝒱_0(Σ)±ℋ(w)/w^2_L^2(Σ)=:Λ_±(Σ).

	Further,
	
		
  * If Σ≅ S^2, then every non-zero element of L^2𝒱_0(Σ) realises the above Rayleigh quotients.
		
  * If Σ≇S^2, then v_±∈ℋ(Σ) (i.e. v_± are additionally curl-free).
	

To state the next result we denote by π the L^2-orthogonal projection from L^2𝒱(Σ) onto L^2𝒱_0(Σ).

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then the operator π∘BS_Σ:L^2𝒱_0(Σ)→ L^2𝒱_0(Σ) is compact and self-adjoint and admits a largest positive and smallest negative eigenvalue. Further, v∈ L^2𝒱_0(Σ) maximises helicity among all other fields w∈ L^2𝒱_0(Σ) of the same L^2-norm if and only if v is an eigenfield of π∘BS_Σ corresponding to the largest positive eigenvalue. A corresponding result holds for the helicity minimisation problem.


	In general, it is highly non-trivial to compute the corresponding eigenfields of π∘BS_Σ which realise the Rayleigh quotients in (<ref>) even for rotationally symmetric surfaces. However, we provide in <Ref> an, in comparison, easy way to compute these eigenfields as well as the corresponding values Λ_+(Σ), Λ_-(Σ) if the surface Σ is toroidal. This will allow us to obtain exact solutions in the case of rotationally symmetric surfaces.

We now deal with the question of optimising the following quantity, recall (<ref>),

    Λ(Σ):=max{Λ_+(Σ),Λ_-(Σ)}=max_w≠ 0
    
    			w∈ L^2𝒱_0(Σ)|ℋ(w)|/w^2_L^2(Σ)

among C^1,1-closed surfaces. Note that according to <Ref> we have Λ(Σ)≥ 0 with equality if and only if Σ≅ S^2. Hence, Σ is a global minimiser of Λ(Σ) if and only if Σ≅ S^2. Here, due to their relevance in plasma physics, we want to focus on surfaces Σ≅ T^2.

	
	Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω be the domain of finite volume bounded by Σ. Then
	
    Λ(Σ)≥1/2.

	Equality holds if and only if ℋ(γ)=0, where γ is the L^2-orthogonal projection of Γ|_Σ onto ℋ(Σ) and Γ is any fixed non-zero harmonic Neumann field of Ω, i.e. any H^1-regular, curl-free, div-free vector field on Ω which is tangent to ∂Ω=Σ.


	With some more effort it should be possible to extend the same lower bound to surfaces of arbitrary (strictly positive) genus using similar methods as in the proof of <Ref> by working with a suitable basis of ℋ_N(Ω) in order to obtain an appropriate matrix representation of the surface Biot-Savart operator. However, we do not pursue such a possible extension further and instead list this as an open problem below.

Using the linking interpretation of helicity, <Ref> and <Ref>, we will be able to conclude the following.

	
	Let Σ⊂ℝ^3 be a closed, connected, C^1,1-regular, axisymmetric torus. Then Λ(Σ)=1/2. Consequently
	
    Λ(Σ)=min_T^2≅Σ∈ C^1,1Λ(Σ).



	
	We note that in <Ref> and <Ref> we did not specify the area of Σ. As we shall see, the scaling properties of the functional Λ imply that minimising or maximising Λ among surfaces with prescribed area is equivalent to optimising Λ among surfaces without an area constraint.

We recall that the corresponding 3-d problem motivated to try to find a global maximiser of Λ(Σ) rather than a global minimiser. We pose this as a remaining open problem.
Open Problems:

	
  * Does there exist a constant 0<c<∞ such that Λ(Σ)≤ c for all T^2≅Σ which are C^1,1-regular?
	
  * If the answer to (i) is positive, does there exist some T^2≅Σ such that Λ(Σ)=max_T^2≅Σ∈ C^1,1Λ(Σ)?
	
  * Do the answers to (i) and (ii) change if we drop the assumption T^2≅Σ and instead allow Σ to have arbitrary genus?
	
  * Does <Ref> extend to surfaces of arbitrary genus, i.e. is it true that every closed, connected C^1,1-surface Σ⊂ℝ^3 of genus g≥ 1 satisfies Λ(Σ)≥1/2?



 §.§ Simple Surface currents


When building plasma fusion confinement devices one wishes to generate magnetic fields in order to confine the fusion plasma. In particular, in so called stellarator devices, one uses a complex coil structure to generate the fields <cit.>. One way to model these complex coil structures is the so called coil winding surface (CWS) model, <cit.>, where one assumes that the coils are infinitely thin and wind around a given closed surface. The induced current is then modelled as a vector field tangent to the CWS and is assumed to be divergence-free in view of Maxwell's equations. Given a C^1,1-regular CWS Σ⊂ℝ^3 and a divergence-free current j on the CWS, the corresponding magnetic field induced by j within the domain Ω enclosed by Σ is given by the Biot-Savart law

    BS(j)(x)=1/4π∫_Σj(y)×x-y/|x-y|^3dσ(y), x∈Ω.

Now, for a fixed current j, there exist in general further currents j^' which induce the same magnetic field within Ω. In fact, since BS is a linear operator, we must have that j-j^'∈Ker(BS). To be more precise, we may view BS as a bounded operator from L^2𝒱_0(Σ) into L^2𝒱(Ω) (the square integrable vector fields on Ω), c.f. <cit.>. It has been further shown, see <cit.> that (Ker(BS))=g(Σ), where g(Σ) denotes the genus of Σ. In our applications, the CWSs are toroidal, i.e. T^2≅Σ, and thus (Ker(BS))=1. This gives us some flexibility and hence we may look for currents, and consequently coil configurations, which have a "simple" form. To this end we make the following definition.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Given a purely poloidal curve σ_p and a toroidal curve σ_t we further let γ_t∈ℋ(Σ) denote the induced harmonic field. Given a current j∈ L^2𝒱_0(Σ) we define Q(j):=∫_Σj·γ_t dσ/|Σ| and say that j simple if Q(j)=0. 


	
	
		
  * In view of <Ref>, <Ref> coincides with <Ref> whenever j is of class C^0,1.
		
  * We have previously explained in <Ref> that γ_t is independent of the choice of σ_p and that changing σ_t may at most change γ_t to -γ_t and therefore the definition of a simple current is independent of the chosen curves σ_p and σ_t.
		
  * If j∈ C^0,1𝒱_0(Σ), then according to <Ref> we see that j is simple if and only if ∫_Σq̂(x)dσ(x)=0, or in other words if and only if, on average, the weighted asymptotic toroidal windings of the field lines of j are zero. Therefore, in an oversimplified manner of speaking, the field lines of a simple current should not form any "knotted" structures and be expected to tend to be either more poloidal or contractible in nature.
		
  * A word of caution: It might turn out that the field lines of j are all toroidal curves and that simply on average the corresponding field lines point as often in positive toroidal direction as they point in negative toroidal direction. As an example consider Σ⊂ℝ^3 to be a standard rotationally symmetric torus and let ϕ denote the toroidal coordinate and θ the poloidal coordinate. We may then define j:=sin(θ)e_ϕ, where e_ϕ denotes the standard normalised toroidal field. Then j∈ C^0,1𝒱_0(Σ) and Q(j)=0, as follows easily from <Ref>, and hence j is simple. However, j has only a toroidal component, see <Ref>.
	





Here is our main result regarding the existence of simple current configurations.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Then for every j∈ L^2𝒱_0(Σ) there exists a unique j^'∈ L^2𝒱_0(Σ) such that BS(j^')=BS(j) in Ω and j^' is simple. Further, if j∈ L^p𝒱_0(Σ) for some 2≤ p≤∞ or j∈ C^0,α𝒱_0(Σ) for some 0≤α<1, then so is j^', where C^0,0≡ C^0 denotes the continuous currents.

In the context of plasma fusion confinement devices one of the situations one faces is the following: We suppose we are given a CWS Σ and a plasma region P which is assumed to be precompact within the finite domain Ω bounded by the CWS and is thought of to contain the fusion plasma. Further, we assume we are given a square integrable, harmonic, target magnetic field B_T∈ L^2ℋ(P):={X∈ L^2(P,ℝ^3)|curl(X)=0=div(X)} which we wish to generate by a surface current in order to confine the fusion plasma. It is known, see <cit.>, that under reasonable assumptions, the image of BS, with domain L^2𝒱_0(Σ), is contained in and L^2(P)-dense in the space L^2ℋ(P). We therefore obtain the following important corollary from <Ref>, where the slightly technical assumptions in the upcoming results are depicted in <Ref>.





	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Assume further, that we are given another C^1,1-solid torus P which is contained in Ω and of positive distance to Σ and that there exists a C^2-disc D⊂Ω such that ∂ D⊂Σ is a purely poloidal C^1-curve in Σ and such that D∩ P is also a disc bounded by a purely poloidal C^1-curve contained in ∂ P. Then for every B_T∈ L^2ℋ(P) and every ϵ>0 there exists a current j∈ L^2𝒱_0(Σ) with Q(j)=0 and BS(j)-B_T_L^2(P)≤ϵ.


Our final result provides a minimisation procedure which allows one to obtain the desired approximating simple currents as minimisers of a specific "energy functional". The procedure is a modification  of a minimisation procedure which has already been used successfully in the past. We first recall the already established procedure, c.f. <cit.>. Given a positive parameter λ>0 one can consider the following minimisation problem

    C(λ;B_T):=inf_j∈ L^2𝒱_0(Σ)(BS(j)-B_T^2_L^2(P)+λj^2_L^2(Σ)).

It follows from standard variational techniques and the convexity of the functional involved that the infimum is in fact a minimum and that there exists a unique current j_λ∈ L^2𝒱_0(Σ) which realises C(λ;B_T). Using the density of the image of the Biot-Savart operator within L^2ℋ(P) it has been shown that BS(j_λ)-B_T_L^2(P)→ 0 as λ↘ 0, see <cit.>. In general, there is no reason to expect that the obtained currents j_λ satisfy Q(j_λ)=0. However, one can enforce this condition by observing that 

    L^2𝒱^Q=0_0(Σ):={j∈ L^2𝒱_0(Σ)|Q(j)=0}

is an L^2-closed subspace, and hence a Hilbert space in its own right, and by setting up a corresponding minimisation procedure

    C_0(λ;B_T):=inf_j∈ L^2𝒱^Q=0_0(Σ)(BS(j)-B_T^2_L^2(P)+λj^2_L^2(Σ)).

Just like in the case of C(λ;B_T) it follows from standard arguments that the infimum in C_0(λ;B_T) is actually achieved and that the global minimiser is unique. Our final result is the following

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Assume further that we are given another C^1,1-solid torus P which is contained in Ω and of positive distance to Σ and that there exists a C^2-disc D⊂Ω such that ∂ D⊂Σ is a purely poloidal C^1-curve in Σ and such that D∩ P is also a disc bounded by a purely poloidal C^1-curve contained in ∂ P. Given any B_T∈ L^2ℋ(P) we denote by j^0_λ the (unique) global minimiser realising C_0(λ;B_T) in (<ref>) for fixed λ>0. Then BS(j^0_λ)-B_T_L^2(P)→ 0 as λ↘ 0.



§ PROOFS OF MAIN RESULTS




 §.§ Proof of <Ref>

Before we come to the proof let us recall how the (0,2,3)-helicity is defined. Given a closed 1-form α on a surface Σ we can consider the inclusion ι: (Σ×Σ)∖{x=y}→Σ×Σ and the projections π_x:Σ×Σ→Σ,(x,y)↦ x and π_y:Σ×Σ→Σ,(x,y)↦ y. We define the Gauss map g:(Σ×Σ)∖{x=y}→ S^2, (x,y)↦x-y/|x-y| and we let ω^2_S^2 denote the standard normalised(!) area form on S^2. We can then define α_x:=(π_x)^#α, α_y:=(π_y)^#α (here f^# denotes the pullback via a map f). Now, strictly speaking, the space C_2(Σ):=(Σ×Σ)∖{x=y} is first compactified by means of the Fulton-MacPherson compactification, c.f. <cit.>, and it is shown that the forms α_x, α_y as well as the Gauss map extend smoothly to the compactification <cit.>. Further, the compactification has a manifold structure <cit.>. Denoting by C_2[Σ] the compactification and with a slight abuse of notation we can define the helicity as

    ℋ_(0,2,3)(α):=∫_C_2[Σ]α_x∧α_y∧ g^#ω^2_S^2.

We note that the space C_2[Σ] is 4-dimensional and so it makes sense to integrate a 4-form. The same reasoning as in <cit.> tells us that it is enough to prove that for all (x,y)∈ (Σ×Σ)∖{x≠ y} we have the identity

    α_x∧α_y∧ g^#ω^2_S^2=1/4πv(x)·(v(y)×x-y/|x-y|^3)ω_Σ(x)∧ω_Σ(y)

where ω_Σ is the area-form on Σ induced by the Euclidean metric and where v is the unique vector field on Σ satisfying ι_vω_Σ=α.

	We prove more generally that if α and β are two 1-forms on Σ, then
	
    β_x∧α_y∧ g^#ω^2_S^2=1/4πw(x)·(v(y)×x-y/|x-y|^3)ω_Σ(x)∧ω_Σ(y)

	where w is the vector field associated with β and v is the vector field associated with α by means of contracting the area form. We fix any x,y∈Σ with x≠ y and let μ,η be charts of Σ around x and y respectively. Additionally, we choose μ×η as a chart around (x,y)∈ (Σ×Σ)∖{x=y}. Further, we let p,q be any two points contained in the domains of μ and η respectively. We see that, if we denote by dμ^i,dη^j the induced covector bases at p,q∈Σ respectively and with a slight abuse of notation by dμ^i(p,q),dη^j(p,q) the induced basis at (p,q)∈ (Σ×Σ)∖{x=y}, then, if we express β(p)=β_i(p)dμ^i(p), we find 
	
    β_x(p,q)=((π_x)^#β)(p)=∑_i=1^2β_i(p)dμ^i(p,q).

	Similarly, α_y(p,q)=∑_j=1^2α_j(q)dη^j(p,q). As for the remaining term g^#ω^2_S^2 we recall that we have the identity
	
    ω^2_S^2=1/4πι^#(x dy∧ dz-ydx∧ dz+zdx∧ dy)

	where ι denotes the inclusion S^2↪ℝ^3. Hence, we may view the Gauss map g as a map into ℝ^3 rather than S^2 and are left with computing the pullback
	
    G^#(x dy∧ dz-ydx∧ dz+zdx∧ dy)(p,q)

	where G:(Σ×Σ)∖{x=y}→ℝ^3,(x,y)↦x-y/|x-y|. In order to find an appropriate expression for the coefficients of this 2-form we observe that
	
    G_*(p,q)∂_μ^i(p,q)=∂_μ^i(p)+(p-q)·∂_μ^i(p)/|p-q|^2(q-p)/|p-q|
    
    		G_*(p,q)∂_η^j(p,q)=-∂_η^j(q)+(p-q)·∂_η^j(q)/|p-q|^2(q-p)/|p-q|

	where the vectors tangent to Σ are viewed as vectors in ℝ^3. Defining ω(x,y,z):=xdy∧ dz-ydx∧ dz+z dx∧ dy, we observe that by definition of the pullback we have
	
    (G^#ω)(p,q)(∂_μ^1(p,q),∂_μ^2(p,q))=ω(p-q/|p-q|)(G_*(p,q)∂_μ^1(p,q),G_*(p,q)∂_μ^2(p,q)).

	Letting r̂:=p-q/|p-q|, ν:=∂_μ_1(p) and ξ:=∂_μ_2(p) we note that we need therefore to compute
	
    ω(r̂)(ν-(r̂·ν)r̂,ξ-(r̂·ξ)r̂)

	and an explicit computation yields
	
    ω(r̂)(ν-(r̂·ν)r̂,ξ-(r̂·ξ)r̂)=(ν×ξ)·r̂.

	Using this formula we can compute all the coefficient functions and eventually arrive at
	
    (G^#ω)(p,q)
    		
    
    		=p-q/4π|p-q|^3·(∂_μ^1(p)×∂_μ^2(p)dμ^1(p,q)∧ dμ^2(p,q)+∂_η^1(q)×∂_η^2(q)dη^1(p,q)∧ dη^2(p,q).
    		
    . -∑_i,j=1^2∂_μ^i(p)×∂_η^j(q)dμ^i(p,q)∧ dη^j(p,q))

	where we again view the tangent vectors at Σ as vectors in ℝ^3. We observe that the 2-form β_x∧α_y only contains basis vectors of the form dμ^i(p,q)∧ dη^j(p,q) so that all the dμ^1∧ dμ^2 and dη^1∧ dη^2 terms from G^#ω will vanish in the product β_x∧α_y∧ G^#ω because at least one of the covectors dμ^i or dη^j in these expressions must repeat, so that only the mixed terms are of relevance. We obtain
	
    β_x(p,q)∧α_y(p,q)∧ g^#ω^2_S^2(p,q)
    		
    
    		=q-p/4π|p-q|^3·∑_i,j,k,m=1^2β_k(p)α_m(p)(∂_μ^i(p)×∂_η^j(q))dμ^k(p,q)∧ dη^m(p,q)∧ dμ^i(p,q)∧ dη^j(p,q)
    		
    
    		=q-p/4π|p-q|^3·((β_2(p)∂_μ_1(p)-β_1(p)∂_μ_2(p))× (α_1(q)∂_η^2(q)-α_2(q)∂_η^1(q)))dμ^1∧ dμ^2∧ dη^1∧ dη^2.

	We recall that we started with two points x,y∈Σ with x≠ y and μ and η were any arbitrary charts centred around these two points. In particular, we may pick our charts such that g_ij(x)=δ_ij=g̃_ij(y) at these two points where g_ij and g̃_ij denote the metric tensor coefficients in the respective coordinate charts. Further, with the right choice of orientation on Σ we may assume that ∂_μ^1(x)×∂_μ^2(x)=𝒩(x) where 𝒩 denotes the outward pointing unit normal. Using the musical isomorphism we can then identify the forms α and β with vector fields X and Z respectively (note that this identification differs from identifying a 1-form by contracting the area-form). We can then expand (x-y) in the basis ∂_μ^1(x),∂_μ^2(x),𝒩(x) and using the properties of the chart we find
	
    (x-y)× (β_2(x)∂_μ^1(x)-β_1(x)∂_μ^2(x))=[(y-x)· Z(x)]𝒩(x)+[(x-y)·𝒩(x)]Z(x).

	Applying the cyclic properties of the inner product we find
	
    (x-y)· [(β_2(x)∂_μ^1(x)-β_1(x)∂_μ^2(x))× (α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))]
    		
    
    		=(α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))·([(y-x)· Z(x)]𝒩(x)+[(x-y)·𝒩(x)]Z(x)).

	Now we note that if a∈ℝ^3 is an arbitrary vector, then upon expanding it in the (orthonormal) basis {∂_η^1(y),∂_η^2(y),𝒩(y)}, we obtain
	
    𝒩(y)· [X(y)× a]=α_1(y)a^2-α_2(y)a^1=(α_1(y)∂_η^2(y)-α_2(y)∂_η^1(y))· a.

	We hence arrive at
	
    β_x(x,y)∧α_y(x,y)∧ g^#ω^2_S^2(x,y)
    		
    
    		=𝒩(y)/4π|x-y|^3·[[((y-x)· Z(x))𝒩(x)+((x-y)·𝒩(x))Z(x)]× X(y)]ω_Σ(x)∧ω_Σ(y)
    		
    
    		=X(y)×𝒩(y)/4π|x-y|^3· [((y-x)· Z(x))𝒩(x)+((x-y)·𝒩(x))Z(x)]ω_Σ(x)∧ω_Σ(y)
    		
    
    		=X(y)×𝒩(y)/4π|x-y|^3· [(Z(x)×𝒩(x))× (y-x)]ω_Σ(x)∧ω_Σ(y)

	where we used the vector-triple product rule in the last step. Lastly, we observe that the vector field associated with a 1-form α by contracting the area-form can be obtained from the vector field associated with the 1-form α via the musical isomorphism by taking the cross product with the outward pointing unit normal. So if we let v,w be the vector fields associated with the 1-forms α and β by contracting the area-form respectively, we arrive at
	
    β_x(x,y)∧α_y(x,y)∧ (g^#ω^2_S^2)(x,y)=v(y)·(w(x)×y-x/4π|x-y|^3)ω_Σ(x)∧ω_Σ(y).

	Consequently
	
    ∫_C_2[Σ]β_x∧α_y∧ g^#ω^2_S^2=∫_Σ v(y)·BS_Σ(w)(y)dσ(y)

	and hence ℋ_(0,2,3)(α)=ℋ(v).



 §.§ Proof of <Ref>

We recall that we want to prove that for a given closed, connected surface Σ⊂ℝ^3 there exists some v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0 if and only if g(Σ)≥ 1, where g(Σ) denotes the genus of Σ. In addition, we also need to show that ℋ_c(v,w)=0 for all w∈ L^2𝒱_0(Σ) whenever v is a co-exact vector field, where ℋ_c denotes the cross-helicity and ℋ denotes the helicity of vector fields.

	
	Step 1: We first prove that if g(Σ)≥ 1, then there exists some element v∈ L^2𝒱_0(Σ) with ℋ(v)≠ 0.
	We first observe that each closed surface Σ bounds a bounded domain Ω⊂ℝ^3 with ∂Ω=Σ. We then have the well-known relation (ℋ_N(Ω))=g(Σ), where ℋ_N(Ω) denotes the space of curl-free, div-free, H^1-vector fields on Ω which are tangent to the boundary ∂Ω and g(Σ) denotes the genus of Σ, <cit.>. In our situation g(Σ)≥ 1 and so we can fix some non-zero Γ∈ℋ_N(Ω). Since Γ is tangent to Σ we may view it as a vector field on Σ. The fact that Γ is curl-free on Ω implies that so is its restriction to Σ (in the weak sense). We recall that ℋ_N(Ω)⊂⋂_0<α<1C^0,α𝒱(Ω) and hence we can write according to the Hodge decomposition theorem Γ|_Σ=grad_Σ(f)+γ for suitable f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ) where ℋ(Σ) denotes the space of curl-free and div-free fields on Σ of class ⋂_1≤ p<∞W^1,p. Since Γ is non-zero so must be γ (since if the restriction of a harmonic Neumann field to the boundary is a gradient field, then the harmonic Neumann field is already identically zero). We can now let 𝒩 denote the outward pointing unit normal and define γ̃:=γ×𝒩 (so that in turn γ=𝒩×γ̃). This new field γ̃ is still divergence-free and curl-free (in the language of differential forms the 1-form associated with γ̃ by the musical isomorphism can be obtained from that of γ by applying the Hodge-star operator). If either ℋ(γ)≠ 0 or ℋ(γ̃)≠ 0, then the theorem is proven (since both vector fields are div-free). If on the other hand ℋ(γ)=0=ℋ(γ̃), then we can consider the vector field v:=γ̃+γ and notice that, due to the symmetry property of the Biot-Savart operator, we have
	
    ℋ(v)=ℋ(γ)+ℋ(γ̃)+2ℋ_c(γ̃,γ)=2ℋ_c(γ̃,γ)

	where we recall that ℋ_c(γ̃,γ)=(γ̃,BS_Σ(γ))_L^2(Σ) is the "cross-helicity". We note that since  γ̃=γ×𝒩 we get by means of the cyclic properties of the Euclidean product
	
    4πℋ_c(γ̃,γ)=∫_Σ∫_Σ(γ̃(x)×γ(y))·x-y/|x-y|^3dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ((γ(x)×𝒩(x))×γ(y))·x-y/|x-y|^3dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ [(γ(x)·γ(y))𝒩(x)-(γ(y)·𝒩(x))γ(x)]·x-y/|x-y|^3dσ(y)dσ(x)
    		
    
    		=∫_Σ∫_Σγ(y)·[(𝒩(x)·x-y/|x-y|^3)γ(x)-(γ(x)·x-y/|x-y|^3)𝒩(x)]dσ(y)dσ(x).

	We first claim that
	
    ℝ^3→ℝ, y↦∫_Σ(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩̃(y))dσ(x)

	is a continuous function, where 𝒩̃ is any C^0,1-regular (compactly supported) extension of 𝒩 to ℝ^3. But this follows easily from the generalised dominated convergence theorem because we have the estimate
	
    |(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩̃(y))|≤c/|x-y|

	for a suitable constant c>0 which is independent of y and from the fact that the function ℝ^3→ℝ, y↦∫_Σ1/|x-y|dσ(x) is continuous.
	
	Therefore, if for some fixed y∈Σ we let (y_n)_n⊂Ω be any sequence converging to y, we find
	
    γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)𝒩(x)dσ(x)
    		
    
    		=γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)(𝒩(x)-𝒩(y))dσ(x)
    		
    
    		=lim_n→∞γ(y)·∫_Σ(γ(x)·x-y_n/|x-y_n|^3)(𝒩(x)-𝒩̃(y_n))dσ(x)

	where we used that γ(y)·𝒩(y)=0 because γ is a tangent field of Σ. We observe that since y_n∉Σ and since 𝒩̃(y_n) is independent of x, the term involving 𝒩̃(y_n) vanishes because
	
    ∫_Σγ(x)·x-y_n/|x-y_n|^3dσ(x)=-∫_Σγ(x)·∇^Σ_x(1/|x-y_n|)dσ(x)=0 for all n

	where we used that γ is divergence-free and that x↦1/|x-y_n| defines a C^1-function on Σ for all fixed n because each y_n has a positive distance to Σ. We conclude
	
    γ(y)·∫_Σ(γ(x)·x-y/|x-y|^3)𝒩(x)dσ(x)=lim_n→∞γ(y)·∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x).

	On the other hand we have the well-known jump formula, <cit.> which tells us that we can find a suitable sequence (y_n)_n⊂Ω converging to y such that
	
    ∫_Σ(𝒩(x)·x-y/|x-y|^3)γ(x)dσ(x)=-4π/2γ(y)+lim_n→∞∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x).

	Combining our considerations so far, we see that for all y∈Σ there exists some sequence (y_n)_n⊂Ω converging to y such that
	
    4πℋ_c(γ,γ̃)=-4π/2γ^2_L^2(Σ)
    
    		+∫_Σlim_n→∞γ(y)·(∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x)-∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x))dσ(y).

	We observe now that γ admits a curl- and div-free extension to Ω, i.e. there exists some ṽ∈⋂_1≤ p<∞W^1,p𝒱(Ω) with curl(ṽ)=0, div(ṽ)=0 in Ω and ṽ^∥=γ, where ṽ^∥:=ṽ-(𝒩·ṽ)𝒩. Indeed, we recall that Γ|_Σ=grad_Σ(f)+γ for a suitable function f∈⋂_0<α<1C^1,α(Σ), and thus if we let f̃ denote the harmonic extension of f, c.f. <cit.>, then Γ-grad(f̃)∈⋂_1≤ p<∞W^1,p𝒱(Ω) is the desired curl- and div-free extension. One can then use Gauss' theorem and the fact that y_n∈Ω for fixed n, to obtain
	
    ∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)γ(x)dσ(x)-∫_Σ(γ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x)
    		
    
    		=∫_Σ(𝒩(x)·x-y_n/|x-y_n|^3)ṽ(x)dσ(x)-∫_Σ(ṽ(x)·x-y_n/|x-y_n|^3)𝒩(x)dσ(x)
    		
    
    		=4πṽ(y_n)+∫_Ωdiv(ṽ)(x)x-y_n/|x-y_n|^3d^3x+∫_Ωy_n-x/|y_n-x|^3×curl(ṽ)(x)d^3x=4πṽ(y_n),

	where the first identity follows from the fact that the terms involving the normal part of ṽ cancel each other. Taking the limit n→∞ we conclude
	
    4πℋ_c(γ,γ̃)=4π/2γ^2_L^2(Σ).

	We recall that ℋ(γ+γ̃)=2ℋ_c(γ,γ̃) and therefore we arrive at
	
    ℋ(γ+γ̃)=2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)>0

	because γ≢0. Hence, in any case, there must exist some γ̂∈ℋ(Σ) satisfying ℋ(γ̂)≠ 0.
	Step 2: In this step we prove that if w∈ L^2𝒱_0(Σ) and f∈ H^1(Σ), then ℋ_c(∇^⊥ f,w)=0, where ∇^⊥ f:=∇_Σf×𝒩. In particular, as ℋ(∇^⊥ f)=ℋ_c(∇^⊥ f,∇^⊥ f) we will get ℋ(∇^⊥ f)=0. Since on a genus zero surface all divergence-free fields are co-exact this will conclude the proof.
	First we observe that by a density argument we may without loss of generality assume that w,∇_Σf∈⋂_1≤ p<∞W^1,p𝒱(Σ). We follow the arguments of the first step. We simply replace γ(y) by w(y) and γ(x) by ∇_Σ f(x) in the arguments. The only step where the arguments need to be modified is in (<ref>). More specifically, the argument that γ(y)·𝒩̃(y_n) ∫_Σ(γ(x)·x-y_n/|x-y_n|^3)dσ(x) vanishes no longer applies because in our new situation γ(x) has to be replaced by ∇_Σ f(x) which is no longer divergence-free. However, we see that
	
    ∫_Σ(∇_Σf(x)·x-y_n/|x-y_n|^3)dσ(x)=-∫_Σ∇_Σ f(x)·∇^Σ_x1/|x-y_n|dσ(x)=∫_ΣΔ_Σf(x)/|x-y_n|dσ(x)

	where Δ_Σ:=div_Σ∘∇_Σ denotes the Laplace operator on Σ. We note that the function ℝ^3→ℝ, y↦∫_Σh(x)/|x-y|dσ(x) is continuous for any function h∈ L^p(Σ), for some p>2, and therefore
	
    lim_n→∞w(y)·𝒩̃(y_n)∫_Σ(∇_Σ f(x)·x-y_n/|x-y_n|^3)dσ(x)=w(y)·𝒩(y)∫_ΣΔ_Σ f(x)/|x-y|dσ(x)=0

	because 𝒩̃(y)=𝒩(y) (since 𝒩̃ is an extension of 𝒩) and because w(y) is tangent to Σ. The remaining arguments apply verbatim which leads us to
	
    2ℋ_c(∇^⊥ f,w)=( ∇_Σ f,w)_L^2(Σ)=0

	because w is divergence-free.



 §.§ Proof of <Ref>


	We recall that v∈ C^0,1𝒱_0(Σ) is a div-free Lipschitz vector field on Σ. Further, we denote by γ_x the (unique) integral curve of v starting at x and for given T>0 we denote by γ_x[0,T] the path along the field line γ_x from time t=0 until t=T.
	Step 1: In the first step we prove that for every T,S∈ (0,∞) the two path γ_x[0,T] and γ_y[0,S] are disjoint almost surely (and hence by construction the artificially closed path σ^τ_x,T and σ^-τ_y,S will be also almost surely disjoint and have a well-defined linking number).
	
	For given T,S>0 we consider the set B_T,S:={(x,y)∈Σ×Σ|γ_x[0,T]∩γ_y[0,S]≠∅}. Now we note that (x,y)∈ B_T,S⇔∃ z∈γ_x[0,T]∩γ_y[0,S].	
	
	In conclusion we must have γ_x(τ)=z=γ_y(λ)⇔ y=γ_x(τ-λ) for suitable 0≤τ≤ T, 0≤λ≤ S, where we used the properties of the flow. Hence, for any T,S>0 we have
	
    B_T,S⊂⋃_n∈ℕ{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}.

	We observe that the right hand side is independent of T and S and that by sigma-subadditivity it is enough to prove that {(x,y)∈Σ×Σ|y∈γ_x(-n,n)} is a null-set in order to show that for almost every (x,y)∈Σ×Σ the sets γ_x[0,T] and γ_y[0,S] do not intersect (in fact this null set is independent of the choice of T and S).
	
	Denoting by μ the Riemannian measure on Σ we find, using Fubini's theorem,
	
    (μ×μ)({(x,y)∈Σ×Σ|y∈γ_x(-n,n)})=∫_Σ∫_Σχ_{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}dσ(y)dσ(x).

	We observe that
	
    χ_{(x,y)∈Σ×Σ|y∈γ_x(-n,n)}(x,y)=χ_B_x(y)

	where B_x:={y∈Σ|y∈γ_x(-n,n)} so that
	
    ∫_Σχ_B_x(y)dσ(y)=μ(B_x)=μ(γ_x(-n,n))=0

	where we used in the last step that γ_x(-n,n) is either a point or otherwise a local embedding of a 1-manifold and hence has Hausdorff-dimension at most 1.
	Step 2: In this step we prove the remaining claims of the theorem.
	In order to specify our choice of τ(Σ)>0, we observe first that there exists some c=c(Σ)>0 such that, c.f. <cit.>,
		
    |𝒩(x)· (x-y)|≤ c|x-y|^2 for all x,y∈Σ.

		Further, we note that the function
		
    κ:Σ×Σ→ℝ, (x,y)↦𝒩(x)·𝒩(y)

		is uniformly continuous and thus
		
    𝒩(x)·𝒩(y)≥1/2 for all x,y∈Σ, |x-y|<δ(Σ)

		for a suitable δ(Σ)>0. We can then choose τ(Σ)>0 such that τ(Σ)≤min{δ/3,1/6c} and such that Ψ_t(x):=x+t𝒩(x) is a diffeomorphism from Σ onto its image for every |t|≤τ.
		
		In order to obtain the desired linking interpretation we observe first that, c.f. <cit.>:
		
		For every f∈ L^1(Σ×Σ), the sequence of functions ∫_0^S_n∫_0^T_nf(γ_x(t),γ_y(s))dtds/T_nS_n admits a well-defined L^1-limit denoted by f̂∈ L^1(Σ×Σ) and 
    ∫_Σ×Σf(x,y)dσ(x)dσ(y)=∫_Σ×Σf̂(x,y)dσ(x)dσ(y).

		We define f(x,y):=1/4π(v(x)× v(y))·x-y/|x-y|^3∈ L^1(Σ×Σ), which can be seen by writing v(x)× v(y)=v(x)× (v(y)-v(x)), using the Lipschitz property of v and the fact that 1/|x-y| is integrable over Σ×Σ. We therefore arrive at
		
    ℋ(v)=1/4π∫_Σ×Σlim^L^1_n→∞∫_0^T_n∫_0^S_n(γ̇_x(t)×γ̇_y(s))·γ_x(t)-γ_y(s)/|γ_x(t)-γ_y(s)|^3dtds/T_nS_ndσ(x)dσ(y).

		We recall the linking integral for two piecewise differentiable closed curves σ_1,σ_2 with σ_i(0)=σ_i(T_i)
		
    lk(σ_1,σ_2)=1/4π∫_0^T_1∫_0^T_2(σ̇_1(t)×σ̇_2(s))·σ_1(t)-σ_2(s)/|σ_1(t)-σ_2(s)|^3dtds.

		Now, as discussed before the statement of <Ref>, we close the curves γ_x([0,T_n]) and γ_y([0,S_n]) and obtain closed curves σ^τ_x,T_n and σ^-τ_y,S_n. For notational simplicity we drop the index n in T_n and S_n. According to (<ref>) and (<ref>) we have to show that the additional terms involved in the expression of lk(σ^τ_x,T,σ^-τ_y,S) converge to zero in the L^1(Σ×Σ)-sense.
		
		To this end we first let a∈Σ∖γ_x(ℝ), 0≤ s≤τ be any fixed elements and we observe that
		
    |a-s𝒩(a)-γ_x(t)|^2=|a-γ_x(t)|^2+s^2-2s(a-γ_x(t))·𝒩(a).

		Now, since a,γ_x(t)∈Σ we can use the estimate |(a-γ_x(t))·𝒩(a)|≤ c|a-γ_x(t)|^2 so that
		
    |a-s𝒩(a)-γ_x(t)|^2≥ |a-γ_x(t)|^2+s^2-2cs|a-γ_x(t)|^2
    			
    ≥ |a-γ_x(t)|^2+s^2/2-2c^2|a-γ_x(t)|^4

		where we used the elementary inequality 2db≤d^2/ϵ+ϵ b^2 for all d,b∈ℝ, ϵ>0 with d=s, b=c|a-γ_x(t)|^2 and ϵ=2. Now we distinguish two cases: Either 2c^2|a-γ_x(t)|^2≤1/2⇔ |a-γ_x(t)|^2≤1/4c^2, in which case we obtain
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2+s^2/2≥|a-γ_x(t)|^2+s^2/4,

		otherwise we find |a-γ_x(t)|^2≥1/4c^2. But in this case we can use the estimate |2s(a-γ_x(t))·𝒩(a)|≤ 2s^2+|a-γ_x(t)|^2/2 and therefore in this case
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2/2-s^2≥|a-γ_x(t)|^2/4+1/16c^2-s^2.

		To estimate the last term we recall that we are interested only in the situation 0≤ s≤τ and thus we can use the fact that τ≤1/6c, i.e. 1/36c^2≥τ^2 and therefore 1/16c^2≥ 2τ^2≥ 2s^2 which yields
		
    |a-s𝒩(a)-γ_x(t)|^2≥|a-γ_x(t)|^2+s^2/4

		in this case. We conclude that in any case we can estimate a part of the linking integral as follows:
		
    |∫_0^T∫_0^τ(v(γ_x(t))×𝒩(a))·a-s𝒩(a)-γ_x(t)/|a-s𝒩(a)-γ_x(t)|^3dsdt|≤ 4c_v∫_0^T∫_0^τ1/|a-γ_x(t)|^2+s^2dsdt

		for a suitable constant c_v>0 which is given by the (finite) C^0(Σ) norm of v. We can further explicitly compute and estimate
		
    ∫_0^τ1/|a-γ_x(t)|^2+s^2ds=arctan(τ/|a-γ_x(t)|)/|a-γ_x(t)|≤π/2|a-γ_x(t)|.

		We recall that in our application we want to connect either the point y or γ_y(S) with the corresponding points on Σ_-τ via a straight line following the normal direction of the starting point. We note that γ_y(S)=ψ_S(y) where ψ denotes the flow of v and because v is divergence-free, the corresponding flow is area-preserving. Hence we want to consider the situation where a=ψ(y) for a suitable area preserving diffeomorphism of Σ. But then we can estimate
		
    ∫_Σ×Σ|∫_0^T∫_0^τ(v(γ_x(t))×𝒩(ψ(y)))·ψ(y)-s𝒩(ψ(y))-γ_x(t)/|ψ(y)-s𝒩(ψ(y))-γ_x(t)|^3dsdt|dσ(x)dσ(y)
    			
    ≤ 2π c_v∫_Σ×Σ∫_0^T1/|ψ(y)-γ_x(t)|dtdσ(x)dσ(y)=2π c_vT∫_Σ×Σ1/|x-y|dσ(x)dσ(y)

		where we used Fubini's theorem in the last step and the fact that ψ as well as γ_x(t)=ψ_t(x) are area preserving diffeomorphisms for every t. Therefore, if we divide the above expression by T and S and take the limit T,S→∞ we see that the corresponding part of the linking integral converges to zero in the L^1-sense.
		
		Now we connect Ψ_-τ(y) and Ψ_-τ(γ_y(S)) on the surface Σ_-τ by a curve γ which is of unit speed and whose length is uniformly bounded independently of y and S, for instance we can connect them by a length minimising geodesic on Σ_-τ in which case its length is always bounded by the intrinsic diameter of Σ_-τ. In that case we can use the fact that Σ_-τ has a positive distance to Σ and therefore |γ_x(t)-γ(s)|≥ d(-τ)>0 for a suitable d which is independent of x,t,y,S and s. Consequently one can use the rough, pointwise, upper bound
		
    |v(γ_x(t))×γ̇(s)·γ_x(t)-γ(s)/|γ_x(t)-γ(s)|^3|≤c_v/|γ_x(t)-γ(s)|^2≤c_v/d^2

		in order to see that the corresponding part of the linking integral converges to zero in the L^1-sense.
		The only remaining part of the linking integral which is not covered by the considerations made so far is of the following type
		
    ∫_0^τ∫_0^τ𝒩(b)×𝒩(a)·b+t𝒩(b)-a+s𝒩(a)/|b+t𝒩(b)-a+s𝒩(a)|^3dsdt

		where in our application a∈{y,γ_y(S)} and b∈{x,γ_x(T)}. But here we can argue similar in spirit as in the first case, namely we can consider
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2=|(b-a)+t𝒩(b)+s𝒩(a)|^2
    			
    
    			=|b-a|^2+|s𝒩(a)+t𝒩(b)|^2+2s𝒩(a)· (b-a)+2t𝒩(b)· (b-a).

		We note that, since a,b∈Σ and 0≤ t≤τ, |2t𝒩(b)· (b-a)|≤ 2cτ |b-a|^2. We recall that τ≤1/6c and thus |2t𝒩(b)· (b-a)|≤|b-a|^2/3 and so we arrive at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2/3+|s𝒩(a)+t𝒩(b)|^2=|b-a|^2/3+t^2+s^2+2st𝒩(a)·𝒩(b).

		We recall that if |a-b|≤δ(Σ) then 𝒩(a)·𝒩(b)≥1/2 and hence, since s,t≥ 0 we arrive in this case at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|a-b|^2/3+t^2+s^2≥|a-b|^2+t^2+s^2/6≥|a-b|^2+t^2/6.

		On the other hand, if |a-b|≥δ, we can estimate 2st≤t^2/2+2s^2 and thus
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2/3+t^2/2-s^2.

		Further, |b-a|^2/6≥δ^2/6≥τ^2≥ s^2 for all 0≤ s≤τ because by choice of τ we have τ≤δ/3. Consequently we arrive at
		
    |a-s𝒩(a)-(b+t𝒩(b))|^2≥|b-a|^2+t^2/6 for all a,b∈Σ.

		From here on out we can argue identically as in the first part of the proof that the corresponding part of the linking integral converges to zero in the L^1-sense. Hence, combining our findings with (<ref>) and the definition of the linking integral (<ref>) we conclude the validity of the theorem.



 §.§ Proof of <Ref>

We start by proving <Ref>.

	We will prove the statement only for Q(v), since an identical reasoning applies to P(v). We recall first that for given v∈ C^0,1𝒱_0(Σ) we denote its integral curves starting at a point x∈Σ by γ_x. Further we have by definition q̂(x)=lim_T→∞1/T∫_γ_x[0,T]γ_t where γ_t∈ℋ(Σ) is a harmonic field on Σ defined by the relations ∫_σ_pγ_t=0 and ∫_σ_tγ_t=1, where σ_p and σ_t are some fixed purely poloidal and a toroidal closed curves respectively. Finally, if q̂∈ L^1(Σ), we had set Q(v):=∫_Σq̂(x)dσ(x)/|Σ|.
	To see that q̂∈ L^1(Σ) and Q(v)=∫_Σv(x)·γ_t(x)dσ(x)/|Σ| we spell out the definition of the line integral, which leads us to
	
    ∫_γ_x[0,T]γ_t=∫_0^T γ̇_x(τ)·γ_t(γ_x(τ))dτ=∫_0^Tv(γ_x(τ))·γ_t(γ_x(τ))dτ.

	Now we define the function f:Σ→ℝ, f(x):=v(x)·γ_t(x)∈ L^1(Σ) (recall that γ_t∈ L^2𝒱(Σ)). We note that γ_x(τ)=ψ_τ(x) where ψ_τ denotes the area-preserving flow of v. Hence, standard ergodic theoretical results, c.f. <cit.>, imply that q̂∈ L^1(Σ) and that Q(v)=∫_Σf(x)dσ(x)/|Σ|=∫_Σv(x)·γ_t(x)dσ(x)/|Σ| as was to be shown.

Before we come to the proof of <Ref> we will need one additional lemma. To this end we recall that ℋ_N(Ω) denotes the space of curl-free, div-free, vector fields tangent to the boundary of a given domain Ω.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω⊂ℝ^3 be the bounded domain enclosed by Σ, ∂Ω=Σ. Given any Γ∈ℋ_N(Ω) we let γ∈ℋ(Σ) denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto the space of harmonic fields ℋ(Σ). Then ℋ(γ×𝒩)=0, where 𝒩 denotes the outward pointing unit normal.


	We follow the proof of <Ref>. Let us set γ̃:=γ×𝒩. Then
	
    4πℋ(γ̃)=∫_Σ∫_Σγ̃(x)·(γ̃(y)×x-y/|x-y|^3)dσ(x)dσ(y)
    		
    
    		=∫_Σ∫_Σ(γ̃(x)×γ̃(y))·x-y/|x-y|^3dσ(x)dσ(y)=∫_Σ∫_Σ((γ(x)×𝒩(x))×γ̃(y))·x-y/|x-y|^3dσ(y)dσ(x).

	From here on all the arguments in the computations of 4πℋ_c(γ,γ̃) apply verbatim, with the only caveat that γ(y) has to be replaced by γ̃(y) in the appropriate places during the computations. This leads us to the identity 2ℋ(γ̃)=∫_Σγ̃(y)·γ(y)dσ(y)=0 because γ̃ and γ are pointwise everywhere orthogonal to each other.

For potential future reference we also state the following immediate consequence.

	
	Let Σ⊂ℝ^3 be a closed, connected C^1,1-surface and let Ω⊂ℝ^3 be the bounded domain enclosed by Σ, ∂Ω=Σ. Given any Γ∈ℋ_N(Ω) we have Γ×𝒩∈ L^2𝒱_0(Σ) and ℋ(Γ×𝒩)=0.


	As mentioned in the proof of <Ref> we can decompose Γ|_Σ as Γ|_Σ=∇_Σf+γ for a suitable function f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ). Since taking the cross product with 𝒩 corresponds, in the language of differential forms, to applying the Hodge star operator, it is immediate that Γ×𝒩∈ L^2𝒱_0(Σ). In addition, Γ×𝒩 and γ×𝒩 differ only by a co-exact vector field, so that the corollary follows from <Ref> and <Ref>.


	Since by assumption T^2≅Σ we know from standard Hodge-theory that (ℋ(Σ))=2. Further, if we fix any Γ∈ℋ_N(Ω)∖{0} where Ω⊂ℝ^3 is the solid torus enclosed by Σ, we see that its L^2-orthogonal projection γ onto ℋ(Σ) on Σ defines a non-zero element of ℋ(Σ). Setting γ̃:=γ×𝒩 we note that γ̃∈ℋ(Σ) because taking the cross product with the outer unit-normal corresponds to applying the Hodge star operator to the corresponding 1-forms (obtained from the musical isomorphism). Now, if v∈ C^0,1𝒱_0(Σ) is an arbitrary vector field, we can decompose v by means of the Hodge decomposition theorem as v=∇_Σf×𝒩+αγ+βγ̃ for a suitable C^1-function f∈ C^1(Σ) and suitable constants α,β∈ℝ. We observe that γ and γ̃ have the same L^2-norms, so that γ̃ is L^2-normalised if γ is. Assuming that γ is L^2-normalised and keeping in mind that the Hodge decomposition is L^2-orthogonal we find
	
    α=∫_Σγ· vdσ(x), β=∫_Σγ̃(x)· v(x)dσ(x).

	Utilising <Ref> we conclude
	
    ℋ(v)=ℋ(αγ+βγ̃)=α^2ℋ(γ)+2αβℋ_c(γ,γ̃)+β^2ℋ(γ̃).

	According to <Ref> we have ℋ(γ̃)=0. In addition, we have shown in the proof of <Ref> that 2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)=1 due to the normalisation of γ. We arrive at
	
    ℋ(v)=α^2ℋ(γ)+αβ.

	Now we recall that we are given a purely poloidal curve σ_p and a toroidal curve σ_t which uniquely determine elements γ_p,γ_t∈ℋ(Σ) according to the relations ∫_σ_pγ_t=0=∫_σ_tγ_p and ∫_σ_pγ_p=1=∫_σ_tγ_t. We claim that γ_t=γ/∫_σ_tγ. To see this we use the fact that Γ|_Σ-γ is the gradient of a C^1-function so that ∫_σ_pγ=∫_σ_pΓ. Further, we may assume that the C^1-curve σ_p, being purely poloidal, bounds a C^1 disc D⊂Ω, ∂ D=σ_p. The idea now is to apply Stokes' theorem to deduce ∫_σ_pΓ=∫_Dcurl(Γ)· n dσ=0 where n denotes the corresponding normal to D and where we used that Γ is curl-free. However, due to the boundary regularity, we only know that Γ∈⋂_1≤ p<∞W^1,p𝒱(Ω) so that the classical Stokes' theorem is not immediately applicable. To bypass this problem one can exploit the fact that Γ is in fact analytic within Ω, since it is a weak solution of ΔΓ=0 in Ω. We can then fix a (compactly supported) C^∞-vector field X defined on ℝ^3 which is everywhere inward pointing along Σ. If we let Ξ_τ denote the flow of X we obtain a C^1-curve σ_p(τ):=Ξ_τ∘σ_p⊂Ω which still bounds a C^1-disc within Ω since X is inward pointing. It then follows from an application of Stokes' theorem and interior regularity of Γ that ∫_σ_p(τ)Γ=0 for all τ>0. On the other hand, Sobolev embeddings tell us that Γ is continuous up to the boundary, so that it is not hard to see that ∫_σ_pΓ=lim_τ↘ 0∫_σ_p(τ)Γ=0 as previously suggested so that ∫_σ_pγ=0. Finally, if ∫_σ_tγ=0 would be true, it would follow, because σ_p and σ_t generate the first fundamental group, that ∫_σγ=0 for every closed loop σ so that a standard construction would imply that γ is a gradient field implying that γ=0 since γ is div-free and hence L^2-orthogonal to the gradient fields. Since γ≠ 0 we conclude that γ/∫_σ_tγ satisfies the defining equations of γ_t and so by uniqueness γ_t=γ/∫_σ_tγ. Using this, we obtain from (<ref>) and <Ref>
	
    α=Q(v)|Σ|∫_σ_tγ.

	Now, since γ and γ̃ form a basis of ℋ(Σ) we can also express γ_p=μγ+λγ̃ for suitable μ,λ∈ℝ. We find 1=∫_σ_pγ_p=λ∫_σ_pγ̃ so that λ=1/∫_σ_pγ̃ where ∫_σ_pγ̃≠ 0 because otherwise γ and γ̃ would be linearly dependent. Lastly, 0=∫_σ_tγ_p=μ∫_σ_tγ+∫_σ_tγ̃/∫_σ_pγ̃. We can solve this for μ and conclude from (<ref>) and <Ref>
	
    β=∫_Σγ_p/λ· v-μ/λγ· vdσ(x)=|Σ|(∫_σ_pγ̃)(P(v)+∫_σ_tγ̃/∫_σ_pγ̃Q(v))

	where we used that γ_t=γ/∫_σ_tγ. Inserting the expressions (<ref>) and (<ref>) into (<ref>) yields the result.
	
	Finally, if σ_t bounds a surface 𝒜 outside of Ω, the result follows from <Ref>.



 §.§ Proof of <Ref>


	We will divide the proof in several steps. We at first consider the divergence-free vector field w∈ C^0,1𝒱_0(Σ). We recall that by assumption there exists a positive function f∈ C^0,1(Σ,(0,∞)) such that v=w/f is rectifiable. Then in the last step we show that the rotational transform of v coincides with that of w which will conclude the proof.
	Step 1: We observe that v being rectifiable implies that either all of its field lines are closed or none of them are. Assume in the first step that the field lines of v are all closed. Then by the rectifiability property it follows further that all field lines are of the same type, i.e. if σ_x is a field line of v starting at x∈Σ and we write σ_x=P_xσ_p⊕ Q_xσ_t for suitable P_x,Q_x∈ℤ (and where σ_p,σ_t denote the fixed purely poloidal and toroidal curve respectively), then in fact P_x and Q_x are independent of the chosen point x. Since w and v have the same field lines, which are merely traced out with a different speed, we conclude that also all field lines of w are of the type Pσ_p⊕ Qσ_t with the same P,Q∈ℤ (which are independent of x and we thus drop the subscript x). We recall (<ref>) and see that if for a given x∈Σ we let τ_x>0 denote the period of the field line of w starting at x, then q̂(x)=Q/τ_x and with the same reasoning p̂(x)=P/τ_x where q̂,p̂ are the quantities associated with w. Further, by assumption, Q(w)≠ 0 which by definition implies 0≠∫_Σq̂(x)dσ(x)=Q∫_Σ1/τ_xdσ(x) and consequently Q≠ 0 so that w has a well-defined rotational transform at every x∈Σ and by definition ι_w(x)=p̂(x)/q̂(x)=P/Q since q̂(x)=Q/τ_x, p̂(x)=P/τ_x. Finally, since p̂(x)=P/Qq̂ and P/Q is a constant, we conclude P(w)=P/QQ(w) so that ι_w(x)=P(w)/Q(w) is independent of x and the statement follows from <Ref>.
	Step 2: Now we assume that the field lines of w are not closed. We recall that if we let μ denote the normalised surface measure on Σ, then an area preserving flow ψ_t is ergodic if for any (Borel-)measurable A⊂Σ with ψ_-t(A)⊂ A for all t≥ 0 we have μ(A)=0 or μ(A)=1.
	
	We claim first that the flow ψ_t of w is ergodic. Since w is div-free the flow is clearly area preserving. On the other hand, since v=w/f where f is a strictly positive function, it is easy to see that if Ψ_t denotes the flow of v, the condition ψ_-t(A)⊂ A for all t≥ 0 implies Ψ_-t(A)⊂ A for all t≥ 0. We have to prove that this implies that either A or Σ∖ A is a null-set. However, since diffeomorphisms preserve null-sets it is enough, due to the rectifiability of v, to see that the linear flow with irrational slope is ergodic on the flat torus which is a well-known fact, see <cit.>. Then the ergodicity and the ψ_t invariance of q̂ and p̂ respectively imply that q̂ and p̂ are both constant a.e.. Then by definition we find P(w)=p̂(x), Q(w)=q̂(x) for a.e. x and since by assumption Q(w)≠ 0 we see that w has a well-defined rotational transform given by ι_w(x)=p̂(x)/q̂(x)=P(w)/Q(w) so that the statement now follows once more from <Ref>.
	Step 3: Here we prove that ι_v(x)=ι_w(x) for a.e. x∈Σ, which will complete the proof.
	
	We recall the definition p̂_v(x)=lim_T→∞1/T∫_σ_x^v[0,T]γ_p, where the subscript indicates that p̂ refers to the poloidal twists of v and where σ_x^v denotes the field line of v starting at x. Further recall that w=fv for a strictly positive function f∈ C^0,1(Σ,(0,∞)). We can then consider 
	
    f̂(x):=lim_S→∞1/S∫_0^Sf(σ_x^w(t))dt,

where σ_x^w denotes the field line of w starting at x. By standard ergodic theory, c.f. <cit.>, this defines an element of L^1(Σ) since the flow of w is area-preserving. We observe that f is bounded above and away from zero so that 0≤ c_1≤f̂(x)≤ c_2<∞ for suitable constants 0<c_1≤ c_2 and all x∈Σ. Further, since v and w have the same field lines we may parametrise σ^v_x(t) via w so that we can write σ_x^v[0,T]=σ_x^w[0,S] for a suitable S>0 (in case that σ_x^v is periodic we pick S such that σ_x^w[0,S] passed x the same amount of times as σ_x[0,T] did). We therefore obtain
	
    p̂_v(x)=lim_T→∞1/T∫_σ_x^v[0,T]γ_p=lim_T→∞1/T∫_σ_x^w[0,S]γ_p

	where we note that S depends on T. To establish a relation between T and S we observe that T=∫_σ_x^v[0,T]v/|v|^2 and therefore we can write by choice of S
	
    T=∫_σ_x^w[0,S]v/|v|^2=∫_0^Sf(σ_x^w(t))dt

	where we used that w=fv. Since f is bounded away from zero and bounded above we see that T→∞⇔ S→∞. Inserting this relation into (<ref>) we find
	
    p̂_v(x)=lim_T→∞S/T·1/S∫_σ_x^w[0,S]γ_p=p̂_w(x)/f̂(x).

	In the same way one proves that q̂_v(x)=q̂_w(x)/f̂(x) so that in particular v has a well-defined rotational transform if and only if w has a well-defined rotational transform and we obtain ι_v(x)=p̂_v(x)/q̂_v(x)=p̂_w(x)/q̂_w(x)=ι_w(x) as claimed.



 §.§ Proof of <Ref>


	Case (i): If Σ≅ S^2 it follows from <Ref> that ℋ(w)=0 for all w∈ L^2𝒱_0(Σ) which implies the claim.
	Case (ii): Given any w∈ L^2𝒱_0(Σ) we can express w by means of the Hodge decomposition theorem as w=grad_Σ(f)×𝒩+γ for a suitable f∈ H^1(Σ) and γ∈ℋ(Σ). It then follows from <Ref> that if ℋ(w)≠ 0, we must have γ≠ 0. In addition, if ℋ(w)>0, the L^2(Σ)-orthogonality of the Hodge-decomposition implies that if grad_Σ(f)≠ 0 we must have ℋ(w)/w^2_L^2(Σ)<ℋ(w)/γ^2_L^2(Σ)=ℋ(γ)/γ^2_L^2(Σ) and that therefore w could not have been a maximiser. We will now at first show that there exists some w∈ L^2𝒱_0(Σ) with ℋ(w)>0 which by the previous argument will allow us to look for a maximiser within the space ℋ(Σ). To see this we let Ω denote the bounded domain bounded by Σ and fix any element Γ∈ℋ_N(Ω)∖{0} (the space of curl- and div-free H^1(Ω) vector fields which are tangent to Σ). Further, we let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ) and we define γ̃:=γ×𝒩. We recall that we have shown in the proof of <Ref> that 2ℋ_c(γ,γ̃)=γ^2_L^2(Σ) (ℋ_c being the cross-helicity) and that <Ref> states that ℋ(γ̃)=0. We can now let β≠ 0 be any constant and set α:=1/|β|. We define γ̂:=αγ+βγ̃ and observe that the previous considerations yield
	
    ℋ(γ̂)=α^2ℋ(γ)+2αβℋ_c(γ,γ̃)+β^2ℋ(γ̃)=α^2ℋ(γ)+αβγ^2_L^2(Σ)=1/β^2ℋ(γ)+sign(β)γ^2_L^2(Σ)

	where sign(β) equals +1 if β>0 and equals -1 if β<0. Letting β≫1 or β≪ -1 we see that there is some γ̂_± with ℋ(γ̂_±)· (±1)>0. Then we are left with observing that the functional {γ̂∈ℋ(Σ)|γ̂_L^2(Σ)=1}→ℝ,γ̂↦ℋ(γ̂) admits a (necessarily positive) global maximum since the unit sphere in finite dimensional vector spaces is compact. The claim follows then from the scaling properties of the helicity, i.e. ℋ(λ v)=λ^2ℋ(v) for all λ∈ℝ and v∈ L^2𝒱_0(Σ).



 §.§ Proof of <Ref>


	We recall that we let π:L^2𝒱(Σ)→ L^2𝒱_0(Σ) denote the L^2(Σ)-orthogonal projection from the space of square integrable vector fields on Σ into the space of square-integrable div-free fields on Σ. We prove first that π∘BS_Σ:L^2𝒱_0(Σ)→ L^2𝒱_0(Σ) is self-adjoint with respect to L^2(Σ). This is easy to see because for every v,w∈ L^2𝒱_0(Σ) we have ⟨ v,(π∘BS_Σ)(w)⟩_L^2(Σ)=⟨ v,BS_Σ(w)⟩_L^2(Σ) and one can then write out the definitions, use the cyclic property of the Euclidean inner product and Fubini's theorem to conclude the self-adjointness.
	
	We will now show that the image of π∘BS_Σ is finite dimensional and therefore this operator must be compact. Indeed, it follows from <Ref> that for any f∈ H^1(Σ) and w∈ L^2𝒱_0(Σ), we have ⟨grad_Σ(f)×𝒩,(π∘BS_Σ(w))⟩_L^2(Σ)=⟨grad_Σ(f)×𝒩,BS_Σ(w)⟩_L^2(Σ)=ℋ_c(grad_Σ(f)×𝒩,w)=0 and so the Hodge decomposition theorem implies that (π∘BS_Σ)(w)∈ℋ(Σ) for every w∈ L^2𝒱_0(Σ) which is a finite dimensional space. It now follows from the spectral theorem for compact, self-adjoint operators that π∘BS_Σ admits a discrete spectrum which accumulates at most at zero and a corresponding eigenbasis which together with Ker(π∘BS_Σ) spans the space L^2𝒱_0(Σ). According to the proof of <Ref> we see that there exist elements v,w∈ L^2𝒱_0(Σ) with ℋ(v)>0 and ℋ(w)<0. Since ℋ(v)=⟨ v,(π∘BS_Σ)(v)⟩_L^2(Σ) and similarly for w, it must be the case that π∘BS_Σ admits a positive and negative eigenvalue and since the spectrum accumulates at most at zero there must exist a largest positive and smallest negative eigenvalue. It then follows from the eigenspace decomposition that v∈ L^2𝒱_0(Σ) with v_L^2(Σ)=1 maximises helicity among all other such fields if and only if v is an eigenfield of π∘BS_Σ corresponding to the largest positive eigenvalue. Due to the scaling properties of helicity the same remains true for the maximisation of the Rayleigh quotient in (<ref>).
	
	

 §.§ Proof of <Ref>

	Before we come to the proof of <Ref> we prove the claim in <Ref>.
	
		
		Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-surface. Then
		
    inf_T^2≅Σ∈ C^1,1Λ(Σ)=inf_T^2≅Σ∈ C^1,1,|Σ|=1Λ(Σ) and sup_T^2≅Σ∈ C^1,1Λ(Σ)=sup_T^2≅Σ∈ C^1,1,|Σ|=1Λ(Σ).

	
	
	We first prove that for given Σ and λ>0 we have Λ(Σ)=Λ(Σ_λ) where Σ_λ:={λ x| x∈Σ}. To see this we can start with an arbitrary vector field w∈ L^2𝒱_0(Σ) and define w_λ(x):=w(x/λ) for x∈Σ_λ. One easily verifies that w_λ∈ L^2𝒱_0(Σ_λ) and that accordingly the map w↦ w_λ gives rise to an isomorphism between L^2𝒱_0(Σ) and L^2𝒱_0(Σ_λ) (note that Σ=(Σ_λ)_1/λ). Then, using the change of variable formula and noting that ψ_λ:Σ→Σ_λ, x↦λ x provides an (orientation-preserving) diffeomorphism between Σ and Σ_λ, we find w_λ^2_L^2(Σ_λ)=λ^2w^2_L^2(Σ) for all w∈ L^2𝒱_0(Σ). As for helicity, we can write, keeping in mind that we perform a double integration,
	
    ℋ(w_λ)=1/4π∫_Σ_λ∫_Σ_λw_λ(x)·(w_λ(y)×x-y/|x-y|^3)dσ(y)dσ(x)
    		
    
    		=λ^4/4π∫_Σ∫_Σw(q)·(w(p)×λ q-λ p/|λ q-λ p|^3)dσ(p)dσ(q)=λ^2ℋ(w).

	Since w↦ w_λ defines an isomorphism we obtain
	
    Λ_±(Σ_λ)=max_w̃≠ 0
    w̃∈ L^2𝒱_0(Σ_λ)±ℋ(w̃)/w̃^2_L^2(Σ_λ)=max_w≠ 0
     w∈ L^2𝒱_0(Σ)±ℋ(w_λ)/w_λ^2_L^2(Σ_λ)=max_w≠ 0
     w∈ L^2𝒱_0(Σ)±ℋ(w)/w^2_L^2(Σ)=Λ_±(Σ)

	and therefore Λ(Σ_λ)=Λ(Σ) for all λ>0. We can now take sequences (Σ_n)_n approaching either the infimum or the supremum on the left hand side of the statement of the lemma respectively and rescale the Σ_n by appropriate λ_n>0 to normalise their area which in combination with Λ(Σ_λ)=Λ(Σ) immediately implies the claim.
	
	
		We recall that according to <Ref> and <Ref> the L^2(Σ)-normalised vector fields γ_+,γ_- realising the Rayleigh quotient (<ref>) are elements of ℋ(Σ) and eigenfields of π∘BS_Σ where π:L^2𝒱(Σ)→ L^2𝒱_0(Σ) denotes the L^2(Σ)-orthogonal projection onto L^2𝒱_0(Σ). In addition, while proving the compactness of π∘BS_Σ during the course of the proof of <Ref>, we also showed that π∘BS_Σ maps L^2𝒱_0(Σ) onto ℋ(Σ). Therefore, the operator π∘BS_Σ:ℋ(Σ)→ℋ(Σ) is well-defined and the Rayleigh quotient (<ref>) is realised by some eigenfields of this restricted operator. However, the Hodge isomorphism <cit.>, implies (ℋ(Σ))=(H^1_dR(Σ))=2, where H^1_dR(Σ) denotes the first de Rham cohomology group and where we used that Σ≅ T^2. It follows further from <Ref> that π∘BS_Σ admits at least one positive and one negative eigenvalue. Consequently, π∘BS_Σ|_ℋ(Σ) is a self-adjoint operator on the 2-dimensional Hilbert space (ℋ(Σ),⟨·,·⟩_L^2(Σ)). It therefore admits precisely two eigenvalues and thus one eigenvalue is strictly positive λ_+(Σ)>0 and one eigenvalue is negative λ_-(Σ)<0 with corresponding (L^2(Σ)-orthonormal) eigenfields γ_+(Σ) and γ_-(Σ) respectively. It follows immediately from the eigenspace decomposition that Λ_+(Σ)=ℋ(γ_+)=λ_+(Σ) and Λ_-(Σ)=-ℋ(γ_-)=-λ_-(Σ)=|λ_-(Σ)|. In order to obtain a relation between λ_+(Σ) and λ_-(Σ) we will now find an explicit matrix representation of (π∘BS_Σ)|_ℋ(Σ). To this end we let Ω⊂ℝ^3 denote the bounded domain which is bounded by Σ, ∂Ω=Σ. Then we can fix any Γ∈ℋ_N(Ω)∖{0} and as usual let γ∈ℋ(Σ) denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ). Upon rescaling we may assume that γ is L^2(Σ)-normalised. We then define γ̃:=γ×𝒩 where 𝒩 is the outward unit normal and note that γ̃∈ℋ(Σ) is also L^2(Σ)-normalised. We then observe that
		
    (π∘BS_Σ)(γ)=⟨ (π∘BS_Σ)(γ),γ⟩_L^2(Σ)γ+⟨ (π∘BS_Σ)(γ),γ̃⟩_L^2(Σ)γ̃=ℋ(γ)γ+ℋ_c(γ,γ̃)γ̃.

		Similarly we find (π∘BS_Σ)(γ̃)=ℋ_c(γ̃,γ)γ+ℋ(γ̃)γ̃ and therefore π∘BS_Σ has the following matrix representation with respect to the basis ℬ:={γ,γ̃}
		
    ℳ:=(π∘BS_Σ)_ℬ^ℬ=[      ℋ(γ) ℋ_c(γ,γ̃); ℋ_c(γ̃,γ)     ℋ(γ̃) ].

		It now follows from <Ref> that ℋ(γ̃)=0 and it follows from the first step in the proof of <Ref> and the symmetry of the Biot-Savart operator that 2ℋ_c(γ̃,γ)=2ℋ_c(γ,γ̃)=γ^2_L^2(Σ)=1 due to the normalisation of γ. We obtain ℳ=[ ℋ(γ)  1/2;  1/2    0 ] and conclude 
		
    λ_+(Σ)λ_-(Σ)=(ℳ)=-1/4⇔Λ_+(Σ)Λ_-(Σ)=1/4.

		Now if both Λ_+(Σ) and Λ_-(Σ) would be strictly smaller than 1/2 it would contradict (<ref>) and therefore we must have Λ(Σ)≥1/2.
		
		Lastly we observe that Λ_+(Σ)-Λ_-(Σ)=λ_+(Σ)+λ_-(Σ)=Trace(ℳ)=ℋ(γ). So if ℋ(γ)=0 we see that Λ_+(Σ)=Λ_-(Σ) and then (<ref>) becomes Λ^2(Σ)=1/4⇒Λ(Σ)=1/2. Conversely, if Λ(Σ)=1/2 we note that (<ref>) can be equivalently expressed as Λ(Σ)min{Λ_+(Σ),Λ_-(Σ)}=1/4 and that therefore min{Λ_+(Σ),Λ_-(Σ)}=1/2=Λ(Σ) and so Λ_+(Σ)=Λ_-(Σ) so that 0=Trace(ℳ)=ℋ(γ).
	
	

 §.§ Proof of <Ref>

	
		According to <Ref> we only need to prove that ℋ(γ)=0 whenever Σ is a C^1,1-rotationally symmetric torus where we recall that γ is the L^2(Σ)-orthogonal projection of any element Γ∈ℋ_N(Ω)∖{0} onto ℋ(Σ) and where Ω denotes the bounded domain with boundary Σ. Upon applying isometries to Σ we may assume that the axis of symmetry of Σ is the z-axis. One can then consider the vector field Y(x,y,z):=(-y,x,0)^tr defined on all of ℝ^3 and which induces isometries. Since Σ is a rotationally symmetric regular torus, it does not intersect the axis of symmetry so that in particular Ω does not intersect it either. One can therefore consider the vector field Γ:=Y/|Y|^2 on Ω which can be easily verified to satisfy curl(Γ)=0 and div(Γ)=0. In addition, since Y generates rotations around the z-axis, we find Y∥Σ and hence Γ∥Σ so that Γ∈ℋ_N(Ω)∖{0}. Since Γ is curl-free and tangent to Σ it is standard that Γ|_Σ is also curl-free as a vector field on Σ. Finally we note that Y induces isometries so that it is a Killing field. But the restriction of Killing fields to invariant surfaces remain Killing so that one finds that Y|_Σ is div-free. In addition, a direct calculation yields Y·grad(|Y|^2)=0 on all of ℝ^3 so that the product rule gives us div_Σ(Γ|_Σ)=0. Consequently Γ|_Σ∈ℋ(Σ) and so γ=Γ|_Σ. Lastly we note that Γ|_Σ and Y|_Σ have the same field lines which are all mutually unlinked and periodic with a uniformly bounded period. It then follows from the linking interpretation of helicity, <Ref> and <Ref>, that ℋ(γ)=0 so that the claim follows from <Ref>.
	
	

 §.§ Proof of <Ref>

	
		We recall that we call an element j∈ L^2𝒱_0(Σ) simple, if it satisfies ∫_Σj·γ_tdσ=0 or equivalently, c.f. <Ref>, Q(j)=0 where γ_t is the unique element of ℋ(Σ) satisfying ∫_σ_pγ_t=0 and ∫_σ_tγ_t=1 (recall also that the notion of being simple is independent of the chosen curves σ_p,σ_t). We have to show that for any given j∈ L^2𝒱_0(Σ) there exists some j^'∈ L^2𝒱_0(Σ) which is simple and satisfies BS(j)(x)=BS(j^')(x) for all x∈Ω, where BS(j)(x):=1/4π∫_Σj(y)×x-y/|x-y|^3dσ(y) corresponds physically to the magnetic field inside the bounded domain Ω bounded by Σ which is induced by the current density j∈ L^2𝒱_0(Σ). So let j∈ L^2𝒱_0(Σ) be given, it then follows from <cit.> that there exists some j_0∈Ker(BS)∖{0}⊂ L^2𝒱_0(Σ)∩⋂_0<α<1C^0,α𝒱(Σ) so that BS(j+α j_0)=BS(j) for every α∈ℝ. Further, it was shown in the first step of the proof of <cit.> that if we fix any Γ∈ℋ_N(Ω)∖{0}, then j_0 can be chosen such that ∫_Σj_0(y)·Γ(y)dσ(y)=Γ^2_L^2(Ω)≠ 0. Now we observe first that curl(Γ)=0 in Ω implies that Γ|_Σ is curl-free on Σ and hence we can use the Hodge decomposition theorem to express Γ=grad_Σ(f)+γ for suitable f∈⋂_0<α<1C^1,α(Σ) and γ∈ℋ(Σ). Since j_0 is div-free it is L^2(Σ)-orthogonal to all gradient fields, which yields ∫_Σj_0·γ dσ=∫_Σj_0·Γ dσ≠ 0. Finally we recall that we have shown in the proof of <Ref> that γ_t=γ/∫_σ_tγ and consequently ∫_Σj_0·γ_tdσ≠ 0. Hence, by linearity, we may always find a suitable α_j∈ℝ such that ∫_Σ(j+α_j j_0)·γ_tdσ=0 so that we may set j^':=j+α_jj_0 and the regularity claims follow immediately from the regularity of j_0. Finally, we need to show that such a j^' is unique. To this end, suppose j^',ĵ∈ L^2𝒱_0(Σ) are simple and satisfy BS(j^')=BS(ĵ). Then j̅:=j^'-ĵ is also simple and contained in the kernel of BS. Therefore j̅∈Ker(BS) and consequently there exists some β∈ℝ with j̅=β j_0 where j_0 is chosen like above. In particular, ∫_Σj_0·γ_tdσ≠ 0 so that the simplicity of j̅ implies β=0 and consequently j^'=ĵ.
	
	

 §.§ Proof of <Ref>

	
		It follows from <cit.> that for every B_T∈ L^2ℋ(P) and every ϵ>0 there exists some j∈ L^2𝒱_0(Σ) with BS(j)-B_T_L^2(P)≤ϵ. Then <Ref> implies that there exists j^'∈ L^2𝒱_0(Σ) which is simple and with BS(j)=BS(j^') which concludes the proof.
	
	

 §.§ Proof of <Ref>

	
		The proof is in essence identical to the proof of <cit.>. For convenience of the reader we present it here nonetheless. Given B_T∈ L^2ℋ(P) we can fix any ϵ>0. According to <Ref> we can find some j_ϵ∈ L^2𝒱_0(Σ) which is simple and such that BS(j_ϵ)-B_T_L^2(P)^2≤ϵ. Then, by definition, j_λ^0 is the unique global minimiser of C_0(λ;B_T) (recall (<ref>)) so that
		
    C_0(λ;B_T)≤BS(j_ϵ)-B_T^2_L^2(P)+λj_ϵ^2_L^2(Σ)≤ϵ+λj_ϵ^2_L^2(Σ)

		by choice of j_ϵ. Further, we note that
		
    BS(j_λ^0)-B_T^2_L^2(P)≤BS(j_λ^0)-B_T^2_L^2(P)+λj^0_λ^2_L^2(Σ)=C_0(λ;B_T)

		and hence
		
    BS(j^0_λ)-B_T^2_L^2(P)≤ϵ+λj_ϵ^2_L^2(Σ).

		We recall that j_ϵ depends only on B_T and ϵ but is independent of λ. Therefore we may take the limit and obtain
		
    0≤lim inf_λ↘ 0BS(j^0_λ)-B_T^2_L^2(P)≤lim sup_λ↘ 0BS(j^0_λ)-B_T^2_L^2(P)≤ϵ.

		Since ϵ>0 was arbitrary we conclude lim_λ↘ 0BS(j^0_λ)-B_T_L^2(P)=0.
	


§ ACKNOWLEDGEMENTS

I would like to thank Ugo Boscain and Mario Sigalotti for fruitful discussions about the interpretation of surface helicity, in particular for suggesting to try to artificially close the field lines as depicted in <Ref>. This work has been supported by the Inria AEX StellaCage.


§ CONTINUITY OF SURFACE BIOT-SAVART OPERATOR



	
	<Ref> is well-defined, i.e. the Biot-Savart operator indeed defines a q-integrable vector field for any choice of orthonormal frame and it is independent of the chosen frame.


	First let us suppose that we have shown that
	
    ∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x)

	is finite for a.e. x in the domain of definition of any fixed orthonormal frame E_i. Then, if E_i and Ẽ_j are any two orthonormal frames whose domains of definition overlap, then on this overlap the integrals will exist almost surely for any x in the overlap. For any such fixed x we then have
	
    ∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))E_i(x)
    		
    
    		=∑_j=1^2∑_i=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)· E_i(x))(E_i(x)·Ẽ_̃j̃(x))Ẽ_j(x)
    		
    
    		=∑_j=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)·∑_i=1^2[(E_i(x)·Ẽ_̃j̃(x))E_i(x)])Ẽ_j(x)
    		
    
    		=∑_j=1^2(∫_Σ v(y)×x-y/|x-y|^3dσ(y)·Ẽ_j(x))Ẽ_j(x)

	and so the value of the Biot-Savart operator is independent of the chosen local orthonormal frames.
	
	In order to establish the q-integrability we define u(y):=𝒩(y)× v(y) and fix any w∈ T_xΣ. We observe that because v(y) is tangent to Σ we have v(y)=u(y)×𝒩(y). Then, using the vector triple product rule, we find
	
    ∫_Σ v(y)×x-y/|x-y|^3dσ(y)· w=∫_Σ(u(y)·x-y/|x-y|^3)𝒩(y)-(𝒩(y)·x-y/|x-y|^3)u(y)dσ(y)· w.

	It is well-known that there is some c>0 satisfying |x-y/|x-y|^3·𝒩(y)|≤c/|x-y| for all y,x∈Σ with y≠ x, c.f. <cit.>. On the other hand, we have, since w is orthogonal to 𝒩(x),
	
    |(u(y)·x-y/|x-y|^3)(𝒩(y)· w)|=|(u(y)·x-y/|x-y|^3)(𝒩(y)-𝒩(x))· w|≤ L|v(y)||w|/|x-y|

	where L is the Lipschitz character of 𝒩 and we used that |u(y)|=|v(y)|. Therefore, if |w|=1, then we see that there is some constant c_0>0 which is independent of v such that
	
    |∫_Σ v(y)×x-y/|x-y|^3dσ(y)· w|≤ c_0∫_Σ|v(y)|/|x-y|d σ(y).

	It follows easily from the dominated convergence theorem that if 1≤β <2 then the map Σ→ℝ, x↦∫_Σ1/|x-y|^βdσ(y) is continuous so that it is globally bounded on Σ. Hence, the above bound ensures that BS_Σ(v)∈ L^∞𝒱(Σ) for any v∈ L^p𝒱(Σ) with p>2 by an application of Hölder's inequality.
	
	If v∈ L^2𝒱(Σ), then in particular v∈ L^p𝒱(Σ) for any 1<p<2 and so the claim will follow from the case p<2 after observing that 2p/2-p converges to infinity as p↗ 2.
	
	Suppose now that 1<p<2. In that case one can use a partition of unity to localise the problem and hence reduce the situation to an application of the standard Hardy-Littlewood-Sobolev inequality, c.f. <cit.>.



§ PHYSICAL ROTATIONAL TRANSFORM


To simplify the discussion, we consider here only "unknotted" smooth solid tori Ω⊂ℝ^3 in the sense that we demand that there exists an orientation preserving C^∞-diffeomorphism Ψ:ℝ^3→ℝ^3 such that Ω=Φ(Ω_S) where Ω_S:={(x,y,z)^tr∈ℝ^3| (√(x^2+y^2)-2)^2+z^2≤1} is the standard rotationally symmetric solid torus bounded by a torus with minor radius r=1 and major radius R=2. We have the following parametrisation of Σ_S:=∂Ω_S, (ϕ,θ)↦μ(ϕ,θ)=((2+cos(θ))cos(ϕ),(2+cos(θ))sin(ϕ),sin(θ))^tr, 0≤ϕ,θ≤ 2π where ϕ is called the "toroidal" angle and θ is called the "poloidal angle". Further, we consider the following curves σ^S_p(τ):=(2+cos(τ),0,sin(τ))^tr, σ^S_t(τ):=(cos(τ),sin(τ),0)^tr, 0≤τ≤ 2π and call them the standard purely poloidal and standard purely toroidal loops. We then refer to σ_p:=Φ∘σ^S_p and σ_t:=Φ∘σ^S_t as a purely poloidal and a purely toroidal curve on Σ=∂Ω.

In addition, we observe that for any fixed ϕ∈ [0,2π), the curve σ_ϕ(τ):=μ(ϕ,τ), 0≤τ≤ 2π, is a closed circle (which bounds a disc within Ω_S). We call the family of images, Σ^ϕ_S:=σ_ϕ([0,2π]), (Σ^ϕ_S)_0≤ϕ<2π the standard foliation of Σ_S into poloidal sections. Accordingly we call (Σ_ϕ)_0≤ϕ<2π with Σ_ϕ:=Φ(Σ^S_ϕ) a foliation of Σ into poloidal sections.

Now one often assumes in plasma physics that the magnetic field lines of plasma equilibria B are transversal to the poloidal sections, i.e. never tangent to any such section, recall also <Ref> regarding the flow structure of plasma equilibria. Then, given a field line of B starting at some initial poloidal section Σ_ϕ_0, it will, due to the transversality assumption, continue to move along these poloidal sections Σ_ϕ until it returns to the initial poloidal section Σ_ϕ_0. During this toroidal turn, the field line will also have made a certain amount of poloidal turns (Δθ)_1. More precisely, we may write σ(τ)=Φ(σ_S(τ)) where σ_S(τ) is a curve on Σ_S parametrised by σ_S(τ)=μ(ϕ(τ),θ(τ)) for suitable smooth functions ϕ(τ) and θ(τ). The transversality condition demands that ϕ(τ) is a strictly increasing function in τ. Therefore there will be a unique 0<τ_1<∞ with ϕ(τ_1)=2π which corresponds to a full toroidal turn upon which we returned to Σ_ϕ_0. Then we set (Δθ)_1:=θ(τ_1). Similarly we can continue to follow our field line until we arrive again at Σ_ϕ_0 which uniquely determines a smallest time τ_1<τ_2<∞ with ϕ(τ_2)=4π. Accordingly we define (Δθ)_2:=θ(τ_2)-θ(τ_1) to be the amount of poloidal turns made during the second toroidal turn. Similarly we define (Δθ)_k to be the amount of poloidal turns during the k-th toroidal turn.

One way to define "physical rotational transform" of a magnetic field line σ starting at Σ_ϕ_0 is then as follows, c.f. <cit.>,

    ι_P(σ):=lim_n→∞∑_k=1^n(Δθ)_k/2π n

whenever the limit exists.

Suppose for now that σ is periodic, then it will be homotopic to Pσ_p⊕ Qσ_t for suitable P,Q∈ℤ. We observe that this means that σ performs precisely Q toroidal turns and P poloidal turns within one period. We are assuming (without loss of generality) that σ moves in the same toroidal direction as σ_t which corresponds to Q≥ 0 and the transversality assumption ensures Q≥ 1. We know that after every Q toroidal turns, we performed P poloidal turns so that ∑_i=1^kQ(Δθ)_i=2π kP. Thus, if kQ≤ n<(k+1)Q, we find ∑_i=1^n(Δθ)_i=2π kP+∑_i=kQ+1^n(Δθ)_i and we observe that -2π P≤∑_i=kQ+1^n(Δθ)_i≤ 2π P because we made less than Q additional toroidal turns and hence at most P poloidal turns. We note further that Q≤n/k≤ (1+1/k)Q so that n/k→ Q as n→∞. Hence, we overall arrive at

    ι_P(σ)=lim_n→∞∑_k=1^n(Δθ)_k/2π n=lim_n→∞(k/nP+R(n)/n)=P/Q

where R(n) is the remainder term which we have shown to be bounded in absolute value by 2π P. We recall that with our "mathematical" rotational transform ι we have, c.f. <Ref>, ι(σ)=∫_ΣB·γ_pdσ/∫_ΣB·γ_t dσ where B=fB and f>0 is a suitable positive function related to the pressure function of the plasma equilibrium and B is the underlying magnetic field. Most importantly, a field line σ of B starting at some point in Σ is of type Pσ_p⊕ Qσ_t if and only if the same is true for B. But then <Ref> and (<ref>) imply that ι(σ)=P/Q=ι_P(σ) so that the rotational transform agrees in this case, recall also <Ref>.

If the field line σ is not closed, we can close it artificially. The idea is that at time τ_n>0 when we return for the n-th time to the initial poloidal section Σ_ϕ_0 we close the curve by following the path σ_ϕ_0 from σ(τ_n) along the poloidal section until we reach the first point on Σ_ϕ_0 which was intersected by σ previously (recall that we will have visited Σ_ϕ_0 by that time precisely (n-1)-times resulting (due to the non-periodicity) in (n-1) intersection points and since Σ_ϕ_0 is a circle, we must necessarily arrive at such a point, provided n≥ 2). The important observation is that by means of this closing procedure we add at most an angle |α_n|≤ 2π to the total toroidal angle. If we denote by σ̃_n=σ[0,τ_n]⊕σ_ϕ_0 the corresponding concatenated path, we see that it will have made precisely n toroidal turns and P_n∈ℤ poloidal turns. Hence, if we let ι^n_P(σ):=∑_k=1^n(Δθ)_k/2π n then we obtain

    ι^n_P(σ)=P_n/n-α_n/2π n, |α_n|≤ 2π, ι_P(σ)=lim_n→∞ι^n_P(σ).

On the other hand, with the "mathematical" definition of rotational transform we find

    ι(σ)=lim_T→∞∫_σ[0,T]γ_p/∫_σ[0,T]γ_t

and know that this limit exists for a.e. x∈Σ at which the field line σ starts and that the limit is independent of the chosen sequence T→∞, c.f. <Ref>. We can therefore consider the sequence T=τ_n which diverges to +∞ as n→∞. But we observe that ∫_σ[0,τ_n]γ_p=∫_σ̃_nγ_p-∫_σ_ϕ_0[t^1_n,t^2_n]γ_p=P_n-r_n where we define the remainder r_n:=∫_σ_ϕ_0[t^1_n,t^2_n]γ_p and where we used that σ̃_n is closed and makes P_n poloidal turns. We observe further that |r_n| can be uniformly bounded because γ_p is globally bounded on Σ, σ̇_ϕ is globally bounded along any poloidal section and because by construction t^2_n-t^1_n=|α_n|≤ 2π is the additional toroidal angle which we need to sweep out to close our curve. Identically we can argue that ∫_σ[0,τ_n]γ_t=n+R_n where R_n is a uniformly (in n) bounded error term and we made use of the fact that σ̃_n makes n toroidal turns. We therefore arrive at

    ∫_σ[0,τ_n]γ_p/∫_σ[0,τ_n]γ_t=P_n-r_n/n+R_n=P_n/n-r_n/n/1+R_n/n=ι^n_P(σ)+α_n/2π n-r_n/2π/1+R_n/n.

Since the limit on the left hand side exists we conclude that also the physical rotational transform is well-defined and that ι(σ)=lim_n→∞∫_σ[0,τ_n]γ_p/∫_σ[0,τ_n]γ_t=lim_n→∞ι^n_P(σ)=ι_P(σ) and therefore the physical rotational transform and mathematical rotational transform always coincide and hence <Ref> provides a way to compute the rotational transform of a plasma equilibrium B, see also <Ref>.


§ EIGENFIELDS OF THE SURFACE BIOT-SAVART OPERATOR



	
	Let T^2≅Σ⊂ℝ^3 be a closed, connected C^1,1-torus and let Ω⊂ℝ^3 be the unique bounded domain whose boundary is Σ. Let further Γ∈ℋ_N(Ω)∖{0} and let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ) and set γ̃:=γ×𝒩 where 𝒩 denotes the outward unit normal. Further we scale Γ such that γ is L^2(Σ)-normalised. Then the set of vector fields v_±∈ℋ(Σ) realising the Rayleigh quotient Λ_±(Σ), c.f. (<ref>), are 1-dimensional subspaces E_±(Σ) of ℋ(Σ) respectively and spanned by the following vector fields
	
    E_+(Σ)=span{(√(ℋ^2(γ)+1)+ℋ(γ))γ+γ̃}, E_-(Σ)=span{γ-(√(ℋ^2(γ)+1)+ℋ(γ))γ̃}

	and we have
	
    Λ_+(Σ)=√(ℋ^2(γ)+1)+ℋ(γ)/2, Λ_-(Σ)=√(ℋ^2(γ)+1)-ℋ(γ)/2.



	We recall that we have shown in the proof of <Ref> that we may restrict attention to the operator (π∘BS_Σ)|_ℋ(Σ) which maps into ℋ(Σ) and has the following matrix representation with respect to the basis ℬ:={γ,γ̃}
	
    ℳ:=(π∘BS_Σ)^ℬ_ℬ=[ ℋ(γ)  1/2;  1/2    0 ].

	Computing the eigenvalues and eigenvectors of this matrix is a standard exercise and yields the result upon recalling that Λ_+(Σ)=λ_+(Σ) and Λ_-(Σ)=-λ_-(Σ) where λ_+(Σ),λ_-(Σ) denote the positive and negative eigenvalue of ℳ respectively and that the eigenvectors of ℳ precisely coincide with the vector fields v_± of interest.


	We note that we have already argued in the proof of <Ref> that ℋ(γ)=0 whenever Σ is a rotationally symmetric C^1,1-torus. In addition, upon applying isometries and therefore assuming that the z-axis is the symmetry axis, we had found that γ=Γ|_Σ where Γ=Y/|Y|^2 and Y(x,y,z)=(-y,x,0)^tr it the vector field inducing rotations around the z-axis. Hence <Ref> tells us that in this case Λ_±(Σ)=1/2 and that E_±(Σ)={γ±γ̃} where γ is explicitly known and γ̃=γ×𝒩 is determined by the outward unit normal field of Σ which can for instance be obtained from a globally defining function of Σ.

The downside of <Ref> is that we still need to compute ℋ(γ) which is again a non-trivial task. The following proposition reduces the problem to compute certain (simple) line and surface integrals.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3 and suppose that σ_t is a toroidal curve which bounds a bounded C^1,1-surface 𝒜 outside of Ω, i.e. 𝒜⊂ℝ^3∖Ω and ∂𝒜=σ_t. Let Γ∈ℋ_N(Ω)∖{0} be fixed and let γ denote the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ). Further let γ̃:=γ×𝒩. Then we have
	
    ℋ(γ)=-∫_σ_tγ̃/∫_σ_tγγ^2_L^2(Σ).



	We follow the ideas of the proof of <Ref>. We observe that we want to compute 4πℋ(γ)=4πℋ_c(γ,γ). We recall that γ̃=γ×𝒩 and consequently γ=𝒩×γ̃. We can then follow the reasoning of Step 1 in the proof of <Ref> where we computed ℋ_c(γ̃,γ). We observe that since γ=-(γ̃×𝒩) we will get an additional minus sign in the final expression. Apart from that only γ(x) has to be replaced by γ̃(x) throughout the computations which eventually leads us to the following identity
	
    4πℋ(γ)=-∫_Σγ(y)·[4πγ̃(y)/2+∫_Ωdiv(ṽ)(x)x-y/|x-y|^3d^3x+∫_Ωcurl(ṽ)(x)×x-y/|x-y|^3d^3x]dσ(y)
    		
    
    		=-∫_Σγ(y)·[∫_Ωdiv(ṽ)(x)x-y/|x-y|^3d^3x+∫_Ωcurl(ṽ)(x)×x-y/|x-y|^3d^3x]dσ(y)

	where we used that γ and γ̃ are perpendicular and where ṽ is any H^1-vector field on Ω such that ṽ^∥=γ̃. We note that if 𝒩×ṽ=γ, then due to the relation of γ and γ̃ this implies ṽ^∥=γ̃. It then follows from <cit.> and its proof and the regularity of γ that there exists some ṽ∈ W^1,2𝒱(Ω) satisfying 𝒩×ṽ=γ, div(ṽ)=0 in Ω and curl(ṽ)=Γ where Γ∈ℋ_N(Ω) satisfies Γ^2_L^2(Ω)=∫_Σγ·Γdσ. We note that in particular ∫_Σγ·Γdσ≥ 0 and since ∫_Σγ·Γ dσ=γ^2_L^2(Σ)>0 we must have, since ℋ_N(Ω) is spanned by Γ, Γ=Γ_L^2(Ω)/Γ_L^2(Ω)Γ from which we conclude Γ_L^2(Ω)=∫_Σγ·Γdσ/Γ_L^2(Ω)=∫_Σγ·Γ dσ/Γ_L^2(Ω). We hence arrive at curl(ṽ)=∫_Σγ·Γ dσ/Γ^2_L^2(Ω)Γ=γ^2_L^2(Σ)/Γ^2_L^2(Ω)Γ where we used that γ is the L^2(Σ)-orthogonal projection of Γ|_Σ onto ℋ(Σ).
	
	Combining these considerations, we arrive at
	
	
	ℋ(γ)=γ^2_L^2(Σ)/Γ^2_L^2(Ω)∫_Σγ(y)·BS_Ω(Γ)(y)dσ(y)

	where BS_Ω(Γ)(y)=1/4π∫_ΩΓ(x)×y-x/|y-x|^3d^3x denotes the volume Biot-Savart operator.
	
	Since Γ is div-free and tangent to the boundary of Ω it is standard, <cit.>, that curl(BS_Ω(Γ))=Γ in Ω and curl(BS_Ω(Γ))=0 outside of Ω. We further observe that since curl(BS_Ω(Γ))=Γ∥Σ, the tangent part of the restriction of BS_Ω(Γ) to Σ is curl-free. In particular, according to the Hodge decomposition theorem it follows that the L^2(Σ)-orthogonal projection of (BS_Ω(Γ))^∥ onto ℋ(Σ) differs from the original tangential field (BS_Ω(Γ))^∥ only by a gradient field. Consequently, if we let σ_t denote the toroidal curve bounding a surface 𝒜 outside of Ω, we obtain, by virtue of Stokes' theorem,
	
		
		∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=∫_σ_tBS_Ω(Γ)=∫_𝒜curl(BS_Ω(Γ))·n dσ=0,
	
	where π_ℋ(Σ) denote the projection onto ℋ(Σ), where we used that σ_t is tangent to Σ, where n denotes the corresponding unit normal and where we used that curl(BS_Ω(Γ))=0 on ℝ^3∖Ω.
	
	Due to the scaling properties of helicity we may now without loss of generality assume that γ (and therefore γ̃) are L^2(Σ)-normalised and hence form an L^2(Σ)-orthonormal basis of ℋ(Σ). We can then express
	
		
		π_ℋ(Σ)((BS_Ω(Γ))^∥)=∫_Σγ·BS_Ω(Γ)dσγ+∫_Σγ̃·BS_Ω(γ)dσγ̃
	
	where we used that γ and γ̃ are tangent to Σ. Combining (<ref>),(<ref>) and (<ref>) (keeping in mind that we assume γ to be normalised) we arrive at
	
		
		0=∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Γ^2_L^2(Ω)ℋ(γ)∫_σ_tγ+∫_Σγ̃·BS_Ω(Γ)dσ∫_σ_tγ̃.
	
	Finally we recall that (BS_Ω(Γ))^∥ is curl-free in the weak sense on Σ and that so is Γ|_Σ because curl(BS_Ω(Γ))=Γ and curl(Γ)=0 in Ω are both tangent to Σ. Accordingly Γ×𝒩 and γ×𝒩=γ̃ differ only by a co-exact vector field and keeping in mind that (BS_Ω(Γ))^∥ is L^2(Σ)-orthogonal to the co-exact fields (as it is weakly curl-free on Σ) we may calculate
	
		∫_Σγ̃·BS_Ω(Γ)dσ=∫_Σγ̃·(BS_Ω(Γ))^∥dσ=∫_Σ(Γ×𝒩)·(BS_Ω(Γ))^∥dσ=∫_Σ(Γ×𝒩)·BS_Ω(Γ)dσ
		
		
		=∫_Σ(BS_Ω(Γ)×Γ)·𝒩dσ=∫_Ωdiv(BS_Ω(Γ)×Γ)d^3x=Γ^2_L^2(Ω),
	
	where we used the cyclic properties of the Euclidean product, that γ̃ it tangent to Σ and the standard calculus formula div(X× Y)=curl(X)· Y-X·curl(Y) for arbitrary C^1-vector fields X,Y on Ω. We can insert this identity in (<ref>) and finally arrive at
	
    0=Γ^2_L^2(Ω)(ℋ(γ)∫_σ_tγ+∫_σ_tγ̃)

	which immediately implies the claimed identity.

The following final result allows us to express the helicity of a vector field through Γ alone.

	
	Let T^2≅Σ⊂ℝ^3 bound a C^1,1-solid torus Ω⊂ℝ^3. Suppose that σ_p is a purely poloidal curve which bounds a C^1,1-disc D⊂Ω and suppose that σ_t is a toroidal curve bounding a C^1,1-surface 𝒜⊂Ω^c. Let further Γ∈ℋ_N(Ω)∖{0} and let γ denote the projection of Γ|_Σ onto ℋ(Σ) and set γ̃:=γ×𝒩. Then
	
		
		∫_σ_tγ=∫_σ_tΓ, ∫_σ_pγ̃=Flux(Γ)/Γ^2_L^2(Ω)γ^2_L^2(Σ)
	
	where Flux(Γ):=∫_DΓ· n dσ and n is the unit normal on the disc D compatible with a chosen orientation of σ_p with regards to Stokes' theorem. In particular, if v∈ C^0,1𝒱_0(Σ), then
	
		
		ℋ(v)/|Σ|^2=Q(v)P(v)(∫_σ_tΓ)Flux(Γ)/Γ^2_L^2(Ω)=∫_Σv·Γdσ∫_Σv·BS_Ω(Γ)dσ/Γ^2_L^2(Ω)|Σ|^2.
	


	The first identity in (<ref>) follows from the fact that Γ|_Σ is curl-free in the weak sense and therefore Γ|_Σ and γ differ only by a gradient field, so that the fact that σ_t is a closed curve yields the result. We assume now without loss of generality that γ is L^2(Σ)-normalised. Then for the second identity in (<ref>) we may utilise (<ref>),(<ref>) and the fact that ∫_σ_pγ=0 to obtain
	
    Flux(Γ)=∫_DΓ· ndσ=∫_Dcurl(BS_Ω(Γ))· ndσ=∫_σ_pBS_Ω(Γ)=∫_σ_pπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Γ^2_L^2(Ω)∫_σ_pγ̃

	where we used that D⊂Ω, ∂ D=σ_p, that σ_p is tangent to Σ and that (BS_Ω(Γ))^∥ is curl-free in the weak sense.
	
	The first equality in (<ref>) is a direct consequence of (<ref>) and <Ref>. As for the second identity we recall that we have argued on several occasions that (∫_σ_tγ) γ_t=γ and we have seen that ∫_σ_pπ_ℋ(Σ)((BS_Ω(Γ))^∥)=Flux(Γ) and according to (<ref>) ∫_σ_tπ_ℋ(Σ)((BS_Ω(Γ))^∥)=0 so that by the defining properties of γ_p we find Flux(Γ)γ_p=π_ℋ(Σ)((BS_Ω(Γ))^∥). We can then utilise <Ref> and use the fact that v is tangent to Σ and div-free and that Γ and γ as well as (BS_Ω(Γ))^∥ and π_ℋ(Σ)((BS_Ω(Γ))^∥) only differ by a gradient field respectively which yields the last equality.

plain



 Paper Title: Turbulent features near the X point of a DTT-like tokamak plasma.



Turbulent features near the X point..]Turbulent features near the X point of a DTT-like tokamak plasma



^1 ENEA, Nuclear Department, C. R. Frascati, Via E. Fermi 45, 00044 Frascati (Roma), Italy
^2 Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 Roma, Italy
francesco.cianfrani@enea.it, giovanni.montani@enea.it




  July 2024


§ ABSTRACT 
The background magnetic geometry at the edge of a tokamak plasma has to be designed in order to mitigate the particle and energy looses essentially due to turbulent transport. The Divertor-Tokamak-Test (DTT) facility under construction at ENEA Frascati will test several magnetic configurations and mitigation strategies, that are usually based on the realization of nontrivial topologies in which one or more X points are present. In order to get a clear understanding of turbulent transport near one of such X points, we perform 3D electro-static fluid simulations of tokamak edge plasma for a DTT-like scenario. We will outline: i) the resulting turbulent spectral features and their dependence on some model parameters (the background pressure gradients and diffusivity) and on the magnetic geometry through a comparative analysis with the results of the companion paper <cit.>, ii) the connection between small scale poloidal structures and toroidal asymmetries, iii) the formation of quiescent regions, iv)the crucial role of radial Dirichlet boundary conditions for the excitation of zonal flows that can screen the radial component of the magnetic geometry.    



















§ INTRODUCTION

The geometry of the magnetic field is crucial for energy and particle confinement in a tokamak plasma. The main component is directed toroidally and generated by external (toroidal field) coils, while a secondary poloidal contribution is inductively produced through a discharge in the main plasma column. These two contributions sums up to form an helical magnetic field winding around the main tokamak axis and confining particles in the collisionless limit. However, collisional turbulent transport provides heat and particle looses that must be kept under control to prevent damages of the plasma-facing components (see for instance <cit.>). The technical solution is to drive those particles exiting the tokamak core towards some targets capable of sustaining larger heat and particle fluxes than the rest of the chamber walls. This is achieved by realizing nontrivial topological magnetic configurations in the so-called divertor region <cit.> through some external (poloidal field) coils generating one or more X points where the poloidal magnetic field vanishes.  

The Italian Divertor-Tokamak-Test facility (DTT) <cit.> is under construction at ENEA laboratories in Frascati with the aim to test different divertor configurations for ITER and DEMO. The reactor design and the forthcoming experimental campaigns must be supported by an effort to simulate realistic plasma conditions under all the feasible magnetic geometries in order to understand and predict the impact of turbulent transport and the resulting heat and particle fluxes across the targets. The integration of core and scrape-off-layer physics is an hard multi-scale problem requiring extremely costly (gyrokinetic) numerical simulations <cit.>. Instead due to the relatively low temperature and density (the magnetic connection length is much larger than the particle mean free path) the investigation of the edge plasma region alone can be done using Braginskii fluid models <cit.> with drift ordering <cit.> that are the basis of the high-performance turbulence codes for tokamak reactors (as for instance BOUT++<cit.>, GBS <cit.>, TOKAMAK3X <cit.> and GRILLIX <cit.>, see Ref.<cit.> for a recent review). These codes have been validated with experimental data <cit.> showing good agreement with the profiles near the midplane and poorer agreement in the divertor region that could be ascribed to the adopted values of dissipative parameters, sheat boundary conditions or to the missing contribution of neutrals, on which the subsequent investigations focused (see for instance Refs.<cit.>). 

In Ref.<cit.> we presented a complementary approach to investigate the physics of tokamak edge plasma turbulence: we performed spatially local 3D electro-static simulations for a DTT like scenario varying some model parameters (background pressure gradients and diffusivity) and the magnetic geometry. The resulting spectral features outlined strong dependence on the parameter values, with the limiting case of large (neoclassical) diffusivity and weak background pressure gradients providing a decaying turbulence scenario and the emergence of a sheared radial electric field. In all the examined cases even though axisymmetric n=0 modes dominate, 3D transitions from n≠ 0 to n=0 played a crucial role in establishing the saturated profiles, as anticipated by the pioneering work in Ref.<cit.>, and the resulting fluxes have the same magnitude as 2D transitions (n=0→ n=0) and the linear contributions (those due to background pressure gradient and the dissipative ones). The inclusion of a poloidal magnetic component breaks the formal isotropy in the poloidal plane of the drift reponse term and the resulting spectra show strong anisotropy with an excess of energy into m=0 modes (zonal flow). A similar effect but less intense is also produced by varying the boundary conditions (BCs) along the radial direction from periodic to vanishing Dirichlet ones. Finally, a configuration with a radially sheared poloidal component has been investigated, showing intermediate spectral properties with anisotropic features due to the poloidal drift contribution and average profiles that roughly follow those in SLAB.              

We now extend the analysis in <cit.> to a X-point magnetic configuration. The squeeze of the magnetic flux tube near the X-point is expected to break the correlation with the rest of SOL fluctuations, where elongated structures are generated, providing instead ribbon-like structures <cit.> and quiescent regions among them. The presence of such regions in the low field side has been confirmed experimentally <cit.> and derived also in TOKMAK3X <cit.> and GRILLIX <cit.> numerical simulations. In <cit.> continuous measurements of Langumir probe saturation current from low to high field side SOL at ASDEX upgrade were presented and the spectral analysis revealed the presence of two quiescent regions: one at LFS SOL close to the separatrix and one in the PFR near the separatrix HFS leg. While the LFS quiescent region is characterized by low absolute and relative fluctuations, the PFR one has large relative fluctuations and the physical mechanism behind that is quite obscure.   
		
In this work, we perform the same analysis of Ref.<cit.> for the magnetic geometry near the X point and we make a comparative analysis to emphasize the impact of the model parameters and of the magnetic field morphology on the development of toroidaly and poloidally symmetric configurations and on the resulting saturated energy spectra. We will outline: i) the dual role of toroidal transitions, with a direct cascade at long wave-lengths proceeding in connection with an inverse cascade at short wave-lengths; ii) the energy flow due to the drift response that acts as the main source contribution for the electrostatic potential. These two aspects of the flux analysis will be used to explain the observed correlation between short scale poloidal structures and toroidal asymmetries. We will also discuss the different scenarios for periodic and radially vanishing Dirichlet BCs. In the former case, the spectra are quite isotropic and their steepness is strongly dependent on the considered model parameters and different from the results for the magnetic configurations in Ref.<cit.>. Moreover, the frequency analysis with spectrograms will reveal the presence of some model dependent quiescent regions below the X point. For neoclassical diffusivity the frequency range is very wide (up to 500KHz) and the quiescent region is in PF region which is both underdense and a local maximum of the electro-static potential. Instead for classical diffusivity quiescence is restricted to two narrow regions to the left of the separatrix legs in correspondence to local fluctuations minima. For vanishing radial Dirichlet BCs, a quite different turbulent scenario will arise with the enhancement of poloidally symmetric m=0 modes that provides energy spectra very similar to those obtained in Ref.<cit.> for a radially sheared poloidal field, i.e. ignoring the radial magnetic component of the X point configuration.  

Therefore, some unique features of the physics of turbulence near the X point magnetic geometry are here addressed: 


  * the relevance of the toroidal cascades and of the drift response for establishing nonlinear saturation, 

  * the dependence of the spectral features on the model parameters, that could be relevant in connection with turbulence measurements to test some of the assumptions in the modelization of a realistic tokamak edge plasma, 

  * the formation of quiescent regions due to a local reduction of fluctuations or to the background field dynamics,   

  * the development of zonal flows in conjunction with the mathematical assumption of radial vanishing Dirichlet BCs and their role in hiding the radial magnetic field component effectively spoiling the full X-point morphology.  


The organization of the work is as follows. In section <ref> the model equations and the numerical scheme are presented, outlining the physical assumptions, the morphology of the magnetic geometry and the adopted technical tools. The results of the simulations are shown in section <ref>, with the degree of toroidal and poloidal symmetry, the energy spectra and the analysis of fluxes discussed  for periodic and vanishing Dirichlet BCs for the radial coordinate in subsections <ref> and <ref>, respectively. Brief concluding remarks follow in section <ref>.      



§ BASIC EQUATIONS AND METHODOLOGY

We consider the fluid description of edge tokamak plasma in the low frequency limit (ω<<Ω_i, Ω_i being ion Larmor frequency) with drift-ordering approximation assuming equal and constant electron and ion temperatures T_e=T_i=T. Neglecting parallel velocity and magnetic fluctuations (electro-static limit), drift turbulence can be described in terms of two variables, the electrostatic potential ϕ and the pressure p=p_0+p̃, which we split into a background component p_0 accounting for the background triggering for linear instability and the fluctuation p̃. The model is derived from the conditions of charge neutrality and energy balance <cit.> neglecting magnetic drifts and the compressibility of ion diamagnetic velocity, resulting in the following system of equations 

    {[         ∂_t Δ_ϕ + {ϕ,Δ_ϕ} = BF/η ∇_∥^2 (p̃-ϕ)  + μ Δ_^2ϕ; ∂_t p̃ + {ϕ,p̃} = -k̂ δ_yϕ + DF/η ∇_∥^2 (p̃-ϕ) + χ_ Δ_p̃ ].

The two equations above are for vorticity Δ_ϕ and perturbed pressure p̃. They have a similar structure with the nonlinear contributions due to the E × B advection expressed in terms of some Poisson brackets (see below) at the left-hand side, while the right-hand side contains the drift response and the dissipative contributions, written in terms of the perpendicular Laplacian Δ_ and the squared parallel gradients ∇_∥ with respect to the magnetic field. For a generic magnetic configuration, these operators can be written in terms of the magnetic unit vector b̂=B⃗/|B⃗| as 

    ∇_∥=b̂·∂⃗    Δ_=Δ-(∇_∥)^2 .

The only formal difference between the two equations in (<ref>) is due to the first term on the right-hand side of the second one: it is due to the Poisson brackets {ϕ,p_0} and its coefficient k̂ is proportional to the inverse background pressure gradient length 1/ℓ_0=∂_xp_0/p_0. A reduced model with one single equation has been considered in Refs.<cit.> (see also <cit.>) neglecting this contribution and identifying the dissipative parameters. Instead we will discuss in this work the relevance of ℓ_0 for the resulting drift turbulent scenario close to the X point.

The Poisson brackets are given in terms of some generalized derivative operators δ_x and δ_y living in the perpendicular plane to the magnetic field

    {f,g}=δ_x f δ_y g - δ_x g δ_y f

where δ_x and δ_y read 

    δ_x=1/√(b_y^2+b_z^2)[(1-b_x^2)∂_x-b_x(b_y∂_y+b_z∂_z] 
    δ_y=1/√(b_y^2+b_z^2)(b_z∂_y - b_y∂_z)

 
In what follows, we take some coordinates x and y in the poloidal plane that are close to the X point (they extend over a 2cm×2cm square centered at the X point position) and correspond to radial and poloidal directions, respectively, while the third coordinate z covers the whole toroidal axis. We neglect toroidicity, so the resulting system of coordinates is effectively a Cartesian one and all the effects we will discuss are entirely due to the X point magnetic geometry, which we choose as follows 

    B_x/B_0=α y     B_y/B_0=α x     B_z=B_0 ,

with α=1.5× 10^-3 and the field lines in the poloidal plane are shown in Fig.<ref>.


Viscosity μ, resistivity η and the additional parameters B, D and F are fixed according with Braginskii values and the chosen normalization. In particular, poloidal coordinates x and y and the toroidal one z are rescaled so that the corresponding dimensionless values range in [-5,5], while electro-static potential and pressure are normalized with e/K_BT, K_B being the Botzmann constant, and n_0K_BT, respectively, for a typical DTT-like scenario <cit.> with

    n_0=5· 10^13cm^-3   T= 100eV,   B_0=3T   R=2.14m   a=68cm.


The perpendicular diffusivity χ_ and the background pressure gradient length ℓ_0 are the two parameters that are varied to test the sensitivity of the model. In particular, the classical Braginskii value χ_class=110 cm^2/s and a value ten times larger χ_neocl=10χ_class, that is representative of the neoclassical value, are considered for χ_, while two cases are discussed for ℓ_0: strong ℓ_0=2cm and weak ℓ_0=10cm gradients.  

The numerical integration scheme is discussed with details in Ref.<cit.>. We adopt the explicit 4^th order Runge-Kutta method in Fourier space. At each step, vorticity and perturbed pressure are advanced, while the electro-static potential is calculated through the numerically pre-comupted inverse Laplacian operator. Extensive use is made of Fast Fourier Transforms (FFTs) to implement the pseudospectral method for all those (nonlinear) terms involving the product of two or three fields, such as E× B advection and drift response contributions. Half of the modes are zero-padding to avoid aliasing error and we use  
 56× 56× 24 modes covering physical length scales down to 3 times the Larmor radius (for l,m=± 14, l and m being radial and poloidal mode numbers). The simulations are run in python and the basic adopted packages are scipy and numpy. 

In what follows, we implement the model equations (<ref>) in two ways: 


  * the standard formulation in FS resulting in fully periodic boundary conditions along all the coordinates,

  * a nontrivial formulation for x simulation domain with sine transform and a background configuration that is symmetric under the reflection over the plane passing through the midpoint, so obtaining skew-symmetric fields under the same reflection and realizing vanishing Dirichlet boundary conditions on half of the domain (see Ref.<cit.> for a more detailed explanation).   


The initial conditions are shown in Fig.<ref>: they are 3D Gaussian wave packets centered inside the simulation domain with ϕ=p̃ minimizing the initial drift response. The height and width are chosen such that turbulence is self-sustained <cit.> and no linear phase occurs.


 



§ RESULTS


The simulation results are presented in this section for the four considered cases (a)χ_=χ_neocl and ℓ_0=2cm in red, b)χ_=χ_class and ℓ_0=2cm in green, c)χ_=χ_neocl and ℓ_0=10cm in blue, d)χ_=χ_class and ℓ_0=10cm in yellow) with fully periodic and vanishing radial Dirichlet boundary conditions. It is performed a comparative analysis of the energy spectra defined as  

    E_ϕ(k⃗)=1/2 |∇_ϕ|^2(k⃗)     E_p̃(k⃗)=1/2|p̃(k⃗)|^2

 and of the corresponding fluxes, i.e.

    {[          ∂_t E_ϕ = T_ϕ + Γ_drift,ϕ  - Γ_visc; ∂_t E_p̃ = T_p̃ + Γ_k̂ + Γ_drift,p̃ - Γ_diff ].

whose definition is given in Ref.<cit.>.  




 §.§ Fully periodic boundary conditions



The multi-panel figure Tab.<ref> shows the profiles of p̃ and ϕ at a given time t=0.335ms (in normalized units τ=Ω_i t= 50000 with the simulation time step dτ=0.1). The two upper panels describe the cases with strong background pressure gradient (ℓ_0=2cm) and the lower ones are for weak gradients (ℓ_0=10cm) with neoclassical/classical diffusivity coefficients χ_ to the left/right. In each panel the figures above correspond to the poloidal plane (x,y) (where magnetic field lines are also plotted) and the figures below are for the radial-toroidal plane (x,z).



It is worth noting that by decreasing diffusivity (left to right) and/or increasing ℓ_0 (top to bottom) one sees a lower degree of toroidal symmetry and smaller vortex structures in the poloidal plane, in qualitative agreement with the results of Ref.<cit.> obtained with different magnetic configurations (SLAB, constant and radially sheared poloidal magnetic field).  

The fractions of n=0 (solid lines) and n=m=0 (dash-dotted lines) modes vs time are plotted in figure <ref>. Saturated values for p̃ (left) solid lines are reached quite fast and range from practically 1 (full toroidal symmetry) for the case (blue) with χ_=χ_neocl and ℓ_0=10cm to ∼ 60% for χ_=χ_class and ℓ_0=2cm (green). The same curves for ϕ show a more complex behavior for weak gradients with longer time to saturation (blue) and a sort of fast decay (yellow) that could be part of an oscillation. In all cases, the degree of toroidal symmetry stays quite high, with more that 50% of modes being axisymmetric.  

 


The dash-dotted lines reach lower saturated values signaling that zonal flows are present but they are not leading the dynamics. This is confirmed by the scatter plot of n=0 energy spectra in Fig.<ref> in which m=0 modes are outlined with black edges and they do not reach highest values than the other points with the same k_. For a comparison one can look at the same plots for constant and radially sheared poloidal magnetic fields in Ref.<cit.> (see Figs. 9 and 20), where m=0 modes stand out from other spectral points. The reason for that was found in the drift term breaking the formal symmetry between the coordinates in the poloidal plane, resulting in an effective energy pumping to m=0 modes. On the contrary, the drift response due to the X point configuration does not break the formal symmetry between x and y and it does not produce any enhancement of m=0 modes.   



Although the fluctuations at fixed k_ are not completely negligible, one can approximate the spectra as being isotropic and use the reference curves in figure. This way, ones sees how the profiles become steeper by increasing diffusivity (from green to red) and/or decreasing background pressure gradients (from green to yellow), while in the blue case they are not even stationary and become more and more stiff with a decaying turbulence scenario. Despite these qualitative similarities with the profiles in Ref.<cit.>, in particular with those in SLAB (see Fig.4), the logarithmic spectral steepness parameters are very different: while p̃ spectra go like 1/k_^4 and 1/k_^2 in the two SLAB cases with ℓ_0=2cm (χ_=χ_neocl (red) and χ_=χ_class (green)) and 1/k_^3 for χ_=χ_class and ℓ_0=10cm (yellow), the same curves from Fig.<ref> are 1/k_^3 (red), 1/k_ (green) and 1/k_^4 (yellow), respectively. Similarly for ϕ the spectra in SLAB are ∼ 1/k_^3 (red), ∼ 1/k_ (green) and ∼ 1/k_^2 (yellow) and the same here go as 1/k_^1.5 (red), const. (green) and 1/k_^2 (yellow). We see how just ϕ spectral profiles for case d) (yellow) is on average the same as in SLAB, while the X point magnetic geometry generically provides more gentle profiles.

This findings outline how the local field spectral profiles are strongly correlated with both the model parameters and the morphology of the X point magnetic geometry.

In order to distinguish toroidal from poloidal cascades the nonlinear transfer functions T_p̃ and T_ϕ in Eq.(<ref>) can be split into toroidal transitions from n ≠ 0 to n=0 and 2D ones within n=0 modes:

    T_.. = T^(n≠0→ n=0)_.. +  T^(2D)_.. .
 

T^(n≠0→ n=0)_p̃ (left) and T^(n≠0→ n=0)_ϕ (right) are shown in Fig.<ref> by normalizing with the corresponding dissipative terms, i.e. the diffusive Γ_diff and viscous Γ_visc fluxes, respectively, and using a logarithmic sampling for (k_ρ)^2. 



One can see how in all cases but the blue one the direction of the toroidal transitions for p̃ (left) depends on k_, with a direct cascade at large wave-lengths (T^(n≠0→ n=0)_p̃<0), partially compensating the source due to background pressure gradients (dashed lines), and an inverse cascade at large wave-lengths (T^(n≠0→ n=0)_p̃>0). For ϕ (right), a direct cascade occurs in most of the spectral domain modulo some fluctuations in the two cases with classical diffusivity (yellow and green). A more complex pattern is shown by 2D transitions in Fig.<ref>, with essentially a direct poloidal cascade from very large wave-lengths for p̃, whose magnitude is of the same order as toroidal transitions, and strong oscillations around zero for ϕ.  
  


The drift contributions are shown in Fig.<ref>: they provide an energy flow from p̃ (left), for which they act as sink terms (Γ_drift, p̃<0), to ϕ (right) where they mostly play the role of an energy source (Γ_drift, ϕ>0).  




The inverse toroidal cascade for p̃ at small poloidal wave-lengths means that non-axisymmetric modes combine to generate small scale axisymmetric modes. Hence, it can explain the observed correlation between non-axisymmetric modes and small vortex structures in the poloidal plane noted in the multi-panel figure Tab.<ref> for p̃. Such a correlation extends to ϕ as well thanks to the drift term providing an energy flow from p̃ to ϕ. 

The differences between the considered cases are outlined by the spectral analysis in time. This is done by computing the spectrogram, obtained by splitting the considered time window into overlapping intervals along which FFTs in time are computed and the mode square amplitude is shown. This procedure provides a visualization of the signal frequency content vs time. In what follows, we will fix a radial position below the X point (x=1.64cm) and plot the time average of the spectrogram at saturation while changing y from SOL across PFR to SOL again but on the other side (we do not have here a distinction between LFS and HFS SOL since toroidicity is ignored, B_φ=B_0=const.). 

In case a) (χ_=χ_neocl and ℓ_0=2cm) the spectrogram for p̃ (left) and ϕ (right) together with the corresponding time average (white curves) and fluctuations (violet curves) is shown in Fig.<ref>. 

It is worth noting how the frequency spectrum is very wide, from zero up to the limit given by the time resolution of the output file (∼500KHz). It is clearly visible a reduction of the spectral intensity in PFR, in correspondence to an under-dense region (the time average of p̃<0) and a peak of the electrostatic potential, while the fluctuations are nearly constant across all the considered poloidal region. 

The same quantities are shown in Fig.<ref> for the case b) with the same ℓ_0 and classical χ_. 

The spectrogram has now a rapid decrease of the intensity above 100KHz. Moreover, one can notice two narrow quiescent regions for p̃ (left) to the left of the separatrix (black), one in SOL and the other in PFR, in correspondence to the minima of the fluctuations. A similar but less intense spectrogram is obtained in the other case (d) with classical diffusivity and it is not shown here, while the analysis in case c) is not so relevant since turbulence is quenched, as discussed before. 

Therefore, the spectral analysis in time outlines an unexpected feature of the turbulent spectra while changing diffusivity: the involved frequencies are much higher while increasing χ_. Moreover, for neoclassical diffusivity a quiescent region appears in PFR that seems to be due to the background dynamics, while in the classical case two narrow quiescent region are placed just to the left of the separatrix and they are characterized by a local suppression of the fluctuations. 



 §.§ Vanishing radial Dirichlet BCs


We present here the results of the simulations with vanishing radial Dirichlet BCs. Each panel of Tab.<ref> corresponds to one of the four considered cases with strong/weak background pressure gradients for the top/bottom panels and neoclassical/classical diffusivity to the left/right. In each panel the fields profiles (p̃ to the left and ϕ to the right) are shown in the poloidal plane (x,y) (top figures) and in the radial-toroidal plane (x,z) (bottom figures). The full simulation domain is shown with the physical domain restricted to just the first half of the x axis, while the second half contains specular copies of the field profiles. This means that only those modes corresponding to the sine transform are present and the vanishing Dirichlet boundary conditions has been correctly implemented.  



By comparing the panels in Tab.<ref> with those in Tab.<ref> it is clear how in the former case with fully periodic BCs the X magnetic geometry has a strong impact on the field profiles, while in the latter one sees that formation of elongated structures along y and generically the obtained scenarios resemble those with a radially sheared poloidal field in Ref.<cit.>. This means that the imposition of the radial boundary conditions affects significantly the resulting drift turbulence scenario in X-point magnetic geometry. 
  
A similar conclusion is obtained looking at the plots in Fig.<ref>, in which the ratio of n=0 (solid lines) and n=m=0 (dash-dotted lines) mode amplitudes to the total amplitude is shown in time, and comparing with those for radially sheared poloidal magnetic filed in Ref.<cit.> (Fig.25) that exhibit the same qualitative and to a large extent also quantitative behavior.



In particular, the electro-static potential (right) reaches saturated mode structures that are essentially toroidally and poloidally symmetric in all cases but b) (green), for which 60% of mode amplitude is toroidally symmetric and less than 10% of them are also poloidally symmetric. The scenario for p̃ (left) is very different: the fraction of n=0 modes is ∼ 60% for cases a), b) and d) and more than 1/3 of them have also m=0. Case c) (blue) is peculiar as toroidally symmetric modes are also poloidally symmetric and their amplitude decays with time. 

A clear distinction with the fully periodic BCs discussed above (see Fig.<ref>) is found in the 2D anisotropic spectra in Fig.<ref> characterized by the enhancement of poloidally symmetric m=0 modes (black edges). 
It is also quite impressive the similarity with the spectra for a radially sheared poloidal magnetic field in Ref.<cit.> (see Fig.26). 



Fluxes have the same qualitative spectral features as in the case with fully periodic boundary conditions:


  * the toroidal transitons for p̃ provide a direct cascade at large wave-lengths and an inverse cascade at small wave-lengths (see Fig.<ref>),

  * the fluxes due to 2D transitions (see Fig.<ref>) are of the same order as those for toroidal ones,

  * the drift terms (see Fig.<ref>) tend to drive energy from p̃ to ϕ.  





 



The spectral analysis in time does not give any significant insight into the turbulent scenarios, with the spectrograms being suppressed above 100kHZ similarly to that in Fig.<ref> but without any quiescent region. 

Therefore, the impact of the X-point magnetic configuration is greatly reduced for the simulations with vanishing radial Dirichlet BCs and the resulting spectral features resemble those of the radially sheared magnetic field discussed in Ref.<cit.>. The reason for that could be ascribed to the leading role of m=0 modes, such that the field profiles are insensitive to those background structure that are not constant in y, such as the radial component of the magnetic field B_x∝ y (see eq.(<ref>)), and they effectively see just the radially sheared poloidal magnetic field component.    



§ CONCLUSIONS

We presented electro-static simulations of edge tokamak plasma for a DTT-like scenario near the X point in the low-frequency limit, taking the electro-static potential ϕ and the perturbed pressure p̃ as field variables. In order to interpret the results, we made a comparison with the outcomes of the companion paper Ref.<cit.> where a similar analysis is performed with simpler magnetic configurations, namely SLAB, a constant and a radially sheared poloidal field in addition to the toroidal component.   

We discussed the relevance of two model parameters for the emerging turbulent scenario. These are the background pressure-gradient length scale ℓ_0 fixing the free-energy source of the system and perpendicular diffusivity χ_, that determines the amount of dissipation which is complementary to the drift response. We found the same qualitative results as in Ref.<cit.>, i.e. a correlation between increasing 1/ℓ_0 and/or decreasing χ_ with the emergence of toroidal asymmetries and small scale structures in the poloidal plane. The analysis of fluxes outlined the crucial role of toroidal transitions, thus the necessity of 3D simulations, for the resulting saturated spectra even though the leading modes are axisymmetric ones (n=0). In fact, looking at p̃ energy flow a direct toroidal cascade at large wave-lengths co-exists with an inverse toroidal one at small wave-lengths. This finding explains the connection between toroidal asymmetries and small scale poloidal structures for p̃. The same connection extends also to ϕ due to the drift term, that acts as a sink for p̃ and as a source for ϕ, thus transferring energy between the two fields preserving the wave-length. 

The combination of all these effects with other contributions (such as the mostly direct toroidal cascade for ϕ and 2D transitions) provide the saturated 2D spectral profiles (see Fig.<ref>) that are quite isotropic in the case with fully periodic boundary conditions. This achievement must be compared with the strong anisotropy obtained in those cases with a constant or sheared poloidal magnetic field component discussed in Ref.<cit.>, in which the enhancement of poloidally symmetric m=0 modes (zonal flow) was predicted. The different outcomes can be ascribed to the formal symmetry of the drift term near the X point under exchange of the coordinates x and y in the poloidal plane. Therefore, the X point morphology does not induce the excitation of zonal flows, while the more conventional magnetic geometry with just a poloidal field component in addition to the toroidal one is expected to generate them, posing the fundamental question of how the two mechanisms can co-exist in a global framework and how they affect the propagation of macroscopic structures (filaments, blobs).   

A spectral analysis in time (spectrograms) has also been performed along a line below the X-point and it showed the different characteristics of the turbulent modes when changing χ_: in the case with neoclassical diffusivity a wide frequency range is covered, up to 500KHzs, and we noticed a quiescent PFR, while for classical diffusivity those modes with frequencies above 100KHz are suppressed and two narrow quiescent regions are predicted near the separatrix legs, one in PFR and the other in SOL. The nature of these quiescent regions appear also very different, with the latter due to a local reduction of the fluctuations level, while the former is correlated with the formation of an underdense region and of a local maximum of the potential.   

The imposition of vanishing Dirichlet BCs along the radial direction x provides significant differences compared to those simulations with fully periodic boundary conditions. In particular, the spectra are not anymore anisotropic with poloidally symmetric m=0 modes being enhanced than the other modes with the same k_ (see the energy spectra in Fig.<ref>). Moreover, the results are very close to those obtained with a radially sheared poloidal magnetic field in Ref.<cit.>. This is an indication of the prominent role of the radial BCs in suppressing m≠ 0 modes to the extent that the background y-dependent magnetic structure, i.e. the radial component B_x, is not relevant at all and the fields “see” an effective magnetic geometry with just a radially sheared poloidal field B_y. Therefore, the choice of the BCs is crucial for the spectral features of the emerging turbulent scenario near the X point magnetic geometry and the investigation of more general conditions (sheat BCs) will be the subject of further developments of this approach.    

This paper and the companion one <cit.> provide a good understanding of the main physical picture behind drift turbulence in the edge of a tokamak plasma near the X point in terms of dependence on the model parameters, energy fluxes, relevance of the toroidal transitions, excitation of zonal flows, role of BCs. The inclusion of some additional terms accounting for the curvature contribution, electro-magnetic fluctuations, diamagnetic velocity or gyrofluid corrections will be performed straightforwardly by updating the simulation code. Similarly, future investigations will focus to other magnetic configurations (snowflake, Super-X, Double Super-X) that can be realized in forthcoming Italian DTT machine <cit.>. 

However, the main drawback of this approach is its locality in space, while global effects are expected to be relevant for the description of turbulent transport in a reactor. Therefore, in parallel some technical and modelization efforts will be devoted to the extension of the presented analysis to poloidally global simulations that could give us a realistic picture on the development of macroscopic structures (zonal flows, blobs and filaments) in view of the comparison with the achievement of those advanced codes <cit.> focused on reproducing the experimental conditions.     
 





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 Paper Title: Full Helicon thruster modeling.



On the use of calibration data in error-aware compilation techniques for NISQ  devices
    Handy Kurniawan


Universidad Complutense de Madrid

Madrid, Spain 

handykur@ucm.es
Laura Rodríguez-Soriano


Universitat Politècnica de València

Valencia, Spain 

lrodsor1@disca.upv.es
Daniele Cuomo


Universitat Politècnica de València

Valencia, Spain 

cuomo.daniele@outlook.com
Carmen G. Almudever


Universitat Politècnica de València
Valencia, Spain 

cargara2@disca.upv.es
Francisco García Herrero


Universidad Complutense de Madrid
Madrid, Spain 

francg18@ucm.es

    August 5, 2024
================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 

This research work is part of a PhD project aiming to design a new helicon plasma thruster for cargo space missions, within a collaboration between the School of Aerospace Engineering of La Sapienza University and the ENEA Frascati research center. 

A 0-D global design model of the thruster is first established to identify the main operational conditions of the thruster, starting from a targeted performance in terms of specific impulse and thrust. These operational conditions are tested with dedicated simulations on the electromagnetic plasma-wave interaction within the ionization chamber, and on the electrostatic plasma acceleration within the magnetic nozzle. The former are carried out with a finite element electromagnetic code and confirm the technical feasibility of the identified operational RF power with a predefined chamber geometry. The latter are obtained with an electrostatic particle-in-cell code, which allows to estimate more precisely the achieved thruster performance. A sub-optimal thruster with an absorbed plasma power of around 1 kW, a thrust of around 10 mN and a specific impulse higher than 1000 s is successfully simulated. The established simulation framework will serve as the basis for the next optimization of the thruster geometry and operational conditions, with the ultimate goal of building and testing a new helicon thruster prototype.









§ NOMENCLATURE





ṁ_i, ṁ      = ion and total mass flow rates (kg/s)  

m_i, m_e        = ion and electron elementary masses (kg) 

e             = elementary electronic charge (C) 

c, c_s             = speeds of light and of plasma sound waves (m/s) 

C_r, C_z     = plasma density ratios along the radial and axial directions at the sheath edge 

μ_0             = vacuum magnetic permeability (H/m) 

ϵ_0             = vacuum dielectric constant (C^2/(Nm^2)) 

F             = thrust force (N) 

f_z,mag             = axial magnetic force density (N/m^3) 

I_sp             = specific impulse in time units (s) 

g_0             = standard gravity acceleration (m/s^2) 

η_m          = propellant utilization efficiency (-) 

η_RF          = RF power transmission efficiency to the plasma (-) 

η_t, η_div      = total thruster efficiency and plume divergence efficiency (-) 

u_i, u_z,i           = ion fluid velocity and its axial component (m/s) 

P_RF,       = input antenna power (W) 

P_wall, P_ion, P_exc, P_Ω
    = power to wall, ionization and excitation powers, power absorbed from wave (W) 

r, z      = radial and axial coordinates (m) 

T_e, T_e0     = electron temperature (in energy units) and its peak value (J, eV) 

n_e, n_i, n_e0       = electron and ion densities and their peak value (m^-3) 

n̅_e, n̅_e,l, n_e,f       = plasma densities averaged over respectively the full chamber, the lateral chamber surface and the chamber exit surface (m^-3)

A_f,A_l    = frontal and lateral surfaces of the ionization chamber (m^2)  

L, R      = ionization chamber length and radius (m) 

Δ V      = ionization chamber volume (m^3) 

n̅_n      = average neutral density inside the chamber (m^-3) 

R_ion,R_exc      = ionization and excitation rates (m^3/s)  

E_ion,E_exc      = ionization and excitation threshold energies (J, eV)  

γ_tot, γ_tot,l, γ_tot,f     = energy transfer coefficient and its average over the lateral and front surfaces (-) 

ψ      = angle between the magnetic field line and the surface normal (rad, deg) 

k, k_∥, k_⊥      = total, parallel and perpendicular wave numbers (m^-1) 

λ      = parallel wave length (m) 

ω      = wave angular frequency (rad/s) 

ω_p,e,ω_c,e      = electron plasma and cyclotron frequencies (rad/s) 

a, L_a      = antenna radius and length (m) 

B_0      = magnetostatic induction field (T, G) 

E, B      = Electric and magnetic induction field vectors (V/m) 

j      = Electric current density vector (A/m^2)

E^*      = complex conjugate of the wave electric field vector (V/m) 

ϕ, ϕ_∞, ϕ_t, ϕ_exit      = electrostatic potential, its value at infinity, at nozzle throat, and at electrons domain exit point (V) 

E_tot      = total electron mechanical energy (J, eV) 

v_e      = electron velocity (m/s) 

I_e ∞, I_i ∞     = electron and ion currents to infinity (A)

C_∞     = equivalent electric capacity (F) 

v_th,e     = thermal electron velocity (m/s) 

σ_n     = neutral cross section (m^2) 

Z_i     = ion charge number (-)











§ INTRODUCTION

The helicon thruster is a relatively new electric thruster in which the thrust is produced by accelerating, through
a magnetic nozzle, a propulsive plasma. This is obtained by coupling RF power with helicon waves inside a ionization
chamber (a dielectric tube where the propellant is injected) using a special type of antenna, known as helicon
antenna <cit.>, that is wrapped around the tube. The helicon thruster is an electrodeless thruster as it ionizes the
propellant without electrodes and requires no external cathode to neutralize the emitted plume, which is already
globally ambipolar. This simplifies the design and allows for a lighter thruster, not exposed to sputtering, and
therefore characterized by a longer lifetime. The ionization mechanism has already been tested experimentally
and helicon thrusters are designed to be mounted on small satellites with feeding powers from a few hundred W up to few kWs and thrust levels from few mN up to hundreds of mN.

The main goal of this research study is to model the entire thruster, simulating the plasma within both the helicon thruster ionization chamber and the magnetic nozzle, in order to estimate the overall propulsive performance of the device. 

First of all, a 0D global performance model is introduced in order to determine a first guess of sub-optimal operating conditions (electron temperature, mass flow, average plasma and neutral density), as a function of the target thruster performance (mainly thrust, specific impulse and total operating RF power).

Once the average plasma density and electron temperature in the source is known, the electromagnetic antenna-plasma interaction is modeled by a full-wave approach, with the goal of designing in realistic geometry the required  antenna to couple the correct power to the plasma.
In particular, the coupling of the helicon antenna with the plasma is solved using a 3D finite element method (FEM) electromagnetic code, accounting for a realistic geometry of both the antenna structure and the ionization chamber. 
The plasma is modeled as an homogeneous, anisotropic, equivalent dielectric with permittivity given by the Stix dielectric tensor <cit.> with collisional corrections. Electron-ion and electron-neutral collisions are considered in the plasma dielectric tensor using correction terms proportional to the collisional frequency <cit.>. This is equivalent to consider the plasma as a conductive, anisotropic, locally
homogeneous, collisional fluid, magnetized by an external magnetostatic field. The cold plasma approximation is considered adequate in describing the propagation of fast waves (helicons) in the cold plasma of the helicon thruster <cit.>. 
Although it has been shown that a non-uniform magnetic induction field might be relevant for an accurate estimation of the wave absorption <cit.>, in these full-wave simulations a constant and uniform axial field has been considered, given its negligible spatial variations within the ionization chamber (with a small aspect ratio R/L as in the present case). 

Then, the plasma properties at the thruster exit section are considered by a particle-in-cell (PIC) code, called PICCOLO (PIC COde of LOw temperature plasma discharges), which simulates the entire magnetic expansion of the nozzle, thus estimating the produced thrust force. PICCOLO is based on the electrostatic PIC technique, in which ions and electrons are simulated as macro-particles subject to Newton’s equation and their charge is weighted at the nodes of a dedicated mesh (hence the name particle-in-cell to obtain the spatial distribution of electric charge density <cit.>). This is used by a Poisson’s equation solver to obtain the self-consistent electric field used to advance the charged particles into the next time step.

The simulation results obtained in this study (from both the full-wave and PIC codes) will be used in the future to better refine the parameters of the global performance model.



Therefore, this work will serve as the basis for the optimized design, manufacture and testing of a helicon thruster prototype. Future comparison of the experimental data with the numerical predictions will finally allow to further tune and validate the developed numerical models.

In the following, the 0-D performance model is presented in Sec. <ref>, the plasma-wave interaction model in Sec. <ref> and the PIC model for the magnetic nozzle in Sec. <ref>. The simulation results are discussed in Sec. <ref>, and finally conclusions are drawn in Sec. <ref>.












§ 0-D GLOBAL DESIGN MODEL


In order to have a first-guess sizing of the Helicon plasma thruster, a 0-D global equilibrium model is used in order to compute operating parameters such as the  electron density n_e0, the electron temperature T_e0 inside the source, and the total mass flow rate ṁ. This 0-D model assumes:



  * A given thrust force F, as a target design parameter;

  * a given thruster specific impulse (I_sp), as a target design parameter;

  * a given geometry of the ionization chamber, defined in terms of both its radius R and length L. Such geometrical parameters may be changed in a iterative design for the global optimization of the thruster. However, this design procedure is not pursued here and is demanded to a successive work.


The total mass flow rate of the thruster can be directly estimated from the target parameters as:



    ṁ = F/g_0 I_sp = F/η_m u_z,i,

where I_sp is the specific impulse in time units, g_0 is the standard gravity acceleration, u_z,i is the axial velocity of the ions (at infinity) and η_m = ṁ_̇i̇/ṁ is the propellant utilization efficiency. In this equation, we have assumed a negligible neutral exit velocity (compared to ions that are typically 10-20 times faster). 



From the energy conservation across the magnetic nozzle expansion, we can compute the final ion exit velocity from the predicted electric potential drop Δϕ_MN. In fact, in a magnetic nozzle, the thermal electron energy is transformed into directed ion kinetic energy: 



    1/2 m_i u_z,i^2 =     e Δϕ_MN,
        
    Δϕ_MN =    T_e0/e( -ln√(2π m_e/m_i) + 0.5 ) = C_shd T_e0/e,


where T_e0 is the electron temperature (in energy units), C_shd≈ 5 is a non-dimensional constant depending on the electron to ion mass ratio, and the factor 0.5 models the ion acceleration in the pre-sheath inside the ionization chamber (where ions are still subsonic). Eq. <ref> represents the electric potential drop over a classical Debye sheath, which is actually a conservative approach, as shown in recent studies suggesting higher potential drops <cit.>. Combining Eqs. <ref> and <ref>, and knowing that g_0 I_sp≈η_m  u_z,i, it is possible to obtain a conservative estimate of the required electron temperature to achieve the target specific impulse:


    T_e0 = 1/2m_i/η_m^2 C_shd (g_0 I_sp)^2.




Referring to Fig. <ref>, and assuming that the plasma is singularly ionized, quasi-neutral everywhere (except at the thin Debye sheaths close to the walls), and that ions exit from the ionization chamber with the Bohm's velocity (or sonic velocity), we can estimate the average plasma density at the exit section n̅_ef as:


    ṁ_̇i̇ = m_in̅_ef c_s A_f = m_in̅_ef A_f√(T_e0/m_i),
 

where c_s = √(T_e0/m_i) is the ion sonic velocity and A_f = π R^2 is the exit surface area. This 0-D model assumes symmetric and quadratic decays along both z and r of the plasma density n_e from the chamber center. This means that:


    n_e (r,z)     = n_e0[ (1/(1-C_r)-r^2/R^2)( 1- C_r)] [4/L^2( C_z -1 )z^2 +1 ]  = 
        = n_e0[ 4/L^2(C_z - 1 ) (C_r -1 )/R^2r^2z^2 + ( C_r - 1 ) /R^2r^2 +  4/L^2(C_z - 1)z^2 + 1],


where n_e0 is the plasma density at the ionization chamber center (r=0,z=0), and C_r,C_z are constants that can be obtained from 1D PIC simulations of the plasma-wall interaction. In particular, for Ar ions and a collisionless plasma with no electron emission from the walls, the  ratio between the sheath edge density and the peak density at the symmetry point (cylinder axis) are assumed to be C_z=0.5 (which corresponds to an unmagnetized scenario case) and C_r=0.35. 

The following averaged plasma densities are then introduced: the chamber averaged plasma density n̅_e, the averaged lateral density n̅_e,l at r=R, and the exit section averaged density n̅_e,f at z=L/2, computed as:


    n̅_e =    1/V∬_r,z n_e(r,z)2π r drdz = 2n_e0[ (  C_z - 1 ) ( C_r - 1 )/12 + (  C_r - 1 )/4  +(  C_z - 1 )/6  + 1/2],
    
    n̅_e,l =    1/A_l∫_z=-L/2^L/2 n_e(r=R,z)2π R dz = C_r[ (  C_z -1 )1/3  + 1] n_e0,
    
    n̅_e,f =    1/A_f∫_r=0^R n_e(r, z=L/2) 2 π r dr =  C_z[ (  C_r -1 )1/2  + 1] n_e0,

where Δ V is the volume of the ionization chamber, and A_l = 2 π R L is its total lateral surface. Therefore, combining Eqs. <ref> and <ref>, we can obtain the peak plasma density n_e0 as a function of the required mass flow rate:


    n_e0 = 1/ C_z[ (  C_r -1 )1/2  + 1]√(m_i/T_e0)ṁ_i/m_i A_f.











An average neutral density 

can be obtained by equating the ions production rate in the volume with the rate of ion loss through the lateral and frontal boundary surfaces of the ionization chamber. Such an equilibrium requires that:


    n̅_en̅_n R_ion(T_e0)Δ V = c_s( n̅_e,l A_l + 2 n̅_e,f A_f)














where R_ion=<σ_ionv_e> is the ionization rate, depending on the electron temperature, that can be found by integrating the ionization cross section over a Maxwellian electron distribution. In particular, the ionization cross sections are obtained from Ref. haya03.



















 







The electron temperature inside the chamber is estimated by solving a power balance equation in which the source power is the RF power deposited in the plasma from helicon waves, and the dissipated power is the power lost from the plasma through the three main dissipation mechanisms: the particle loss on the walls, the ionization of the gas and the excitation of neutrals. The first term is dominant and can be estimated through a fitting law obtained from 1D PIC simulations of the plasma wall interaction in a partially magnetized scenario like this one (where electrons are magnetized but ions are not). In particular, knowing the mean plasma density at the sheath edge n̅_e,l, n̅_e,f (depending on the considered surface), it is possible to estimate this power as:


    P_wall = 2 A_f n_e,f c_sγ_tot,f T_e0 + A_l n_e,l c_sγ_tot,l T_e0

where γ_tot represents a total energy transfer coefficient (summing the contribution of both electrons and ions) that depends on the considered surface and more precisely on the angle ψ of the magnetic induction field relative to the surface normal direction. In particular, ψ=0 for the two frontal surfaces, and ≈ 89^o for the lateral surfaces (nearly parallel incidence). In general, for cold ions (T_i0=0), the following fitting can be used:


    γ_tot = 5/2 - ln(√(2π m_e/m_i) C_γ(ψ,r_Le/λ_De, m_e/m_i)cosψ)


where C_γ is a parameter that, in general, depends on the magnetic field angle, on the ratio between the electron Larmor radius and the Debye length λ_De, and on the electron to ion Larmor radius. In a plasma with Ar ions and an electron Larmor radius comparable with the Debye length (λ_De/r_Le=O(1)), we obtain C_γ,f=1 for the frontal surfaces (ψ=0), and C_γ,l=0.4 for the lateral surface (ψ≈ 89^o). Again, the above fittings and coefficients have been obtained from 1D PIC simulations of a non-emitting wall surface interacting with a collisionless plasma, so that the effects of secondary electrons and charged particles collisions with neutrals are neglected.

Concerning the other two loss mechanisms, the expressions of the ionization and excitation power are the following:


    P_exc = n̅_nn̅_e R_exc(T_e0)E_excΔ V,



    P_ion = n̅_nn̅_e R_ion(T_e0)E_ionΔ V,

where the rates are obtained from integrating over a Maxwellian electron distribution the cross sections available in Refs.haya03,lxcat for respectively ionization and excitation.

The ionization and the excitation energy thresholds are respectively 15.76 and 11.5 eV for argon. 


The equilibrium temperature T_e0 of the helicon thruster is then computed by assuming a given absorbed power and equating it to the total power losses. This means that:


    P_Ω = P_wall (T_e0)+ P_ion (T_e0) + P_exc (T_e0),

where the terms on the right hand side depend on the electron temperature.

The targeted absorbed power is then the requested input for the finite-element full-wave model (described in the following section), which has the goal of validating the design of a helicon antenna capable of delivering that power to the plasma. 

The helicon antenna is designed according to the helicon wave theory <cit.>. The helicon antenna is a Nagoya type-III antenna which accelerates free electrons in a neutral gas immersed in a magnetostatic field, ionizing it through energy deposition. 
It is assumed, as stated in the previous equation, that the deposited energy in the plasma by helicon waves is a fraction of the RF input power and heat the plasma through Joule heating. The size of the helicon antenna is determined by combining the quadrature relation for the total wave number k of the helicon wave and the plasma dispersion relation for helicons in a magnetized plasma, as follows:


    k^2 = k_⊥^2 + k_∥^2, 
    
       k_∥ = ωkω_p,e^2ω_c,ec^2,


where ω_c,e is the electron cyclotron frequency, c is the speed of light and ω_p,e is the electron plasma frequency. From this linear system, the wave numbers of the helicon wave are computed, such as the total wave number k, the parallel wave number k_∥  and the transversal wave number k_⊥. 
Then, the size of the antenna is determined by assuming that the antenna is resonant with an helicon mode propagating in the magnetized plasma. The antenna is assumed of cylindrical shape and therefore it is determined from the antenna radius a and the antenna length L_a. The antenna radius a, is computed from the ionization chamber radius R, assuming a thickness of the ionization chamber equal to 5 mm. The design conditions for the Nagoya type-III antenna are 


    k_⊥ = 3.83R, 
    
      L_a = λ2.



Therefore, once the antenna shape (a,L_a) is chosen and the chamber averaged plasma density n̅_e (Eq. <ref>) is known, the required magnetostatic field can be computed from the plasma dispersion relation according to the following equation:


    B_0 = ωk_∥μ_0 e n̅_ek


and therefore the main output parameters of the 0D model 

are fully determined.






§ FULL-WAVE SIMULATION MODEL FOR THE IONIZATION CHAMBER



The plasma in the ionization chamber is simulated with a in-house 3D FEM code which solves Maxwell's equations for a realistic 3D geometry in a cartesian reference frame. The obtained full-wave solutions allow to compute the electromagnetic field coupled by the helicon antenna in the plasma and to estimate the power transferred from the helicon antenna to the plasma. The simulation model assumes that a feeding power (as requested from the 0D model) is provided to the helicon antenna and a fraction of this power is absorbed by the plasma through collisional damping. 

The plasma parameters (plasma density, the neutrals density, the electron temperature) and the geometrical parameters (the helicon antenna dimensions and the size of the ionization chamber) are selected in accordance with the 0D model of the thruster described above. The magnetostatic field, necessary for the helicon wave propagation, is derived from the dispersion relation of helicon waves <cit.> (see Eq. <ref>).

The plasma, already ionized, is modeled as an equivalent homogeneous anisotropic dielectric with permettivity given by the Stix dielectric tensor <cit.> with collisional corrections. Electron-ion and electron-neutral collisions are computed according to the following formulas:


    ν_ei =     3.9 · 10^-6n_iZ_i^2lnΛ/T_e0^3/2, 
    ν_en =    σ_n n_n v_th,e,


where Z_i=1 is the ion charge number, lnΛ is the Coulomb logarithm, σ_n is a constant neutral atom cross section, and v_th,e=√(T_e0/m_e) is the electron thermal velocity in the ionization chamber. For the considered case, ν_ei≈ 3.9 MHz, and ν_en≈ 4.2 MHz. Such frequencies are considered in the plasma dielectric tensor using correction terms proportional to the collisional frequency <cit.>. The helicon electric field and the induced current density in the plasma are obtained from the full-wave simulation results and they allow the computation of the deposited power in the plasma:


    P_Ω (T_e) = 1/2∫_VRe(j·E^*) dV


where j and E^* are complex vectors representing the current density vector and the complex conjugate of the helicon electric field vector and V is the volume of the ionization chamber. The deposited power P_Ω is the source power in the power balance equation used in the 0-D model for the electron temperature evaluation (Eq. 16).

The main results obtained from the full-wave simulation are the verification of the helicon antenna design for the target absorbed power (as required by the 0D model) and the computation of the exact RF input power level required for the desired operative point of the antenna. Last, but not least, the verification of the RF coupling with a propagating helicon mode in the target magnetized plasma.



§ PARTICLE-IN-CELL MODEL OF THE MAGNETIC NOZZLE EXPANSION


The collisionless expansion within the magnetic nozzle is simulated with an electrostatic particle-in-cell code <cit.> named PICCOLO <cit.>. This choice is justified by the fact that most of the electromagnetic wave absorption takes place inside the ionization chamber since the helicon wave becomes evanescent in the nozzle, not far away from the exit surface of the chamber.

Ion and electron macro-particles (each one representing a large number of elementary particles, here referred to as macro-particle weight) are moved according to Newton's equation and the local electric and magnetic fields E and B:


    m dv/dt = q ( E + v×B),

where m,q are the elementary particle mass and charge. A classical leap-frog algorithm <cit.> is used to advance particles in time. 

While the magnetic induction field is given as an input map (from the 0D global model and the antenna design constraints) and is fixed throughout the simulation, the electrostatic field E= - ∇ϕ is obtained by differentiation of the self-consistent electric potential ϕ, which is the solution of Poisson's equation in the considered cylindrical coordinates r and z:

    1/r∂ϕ/∂ r + ∂^2 ϕ/∂ r^2 + ∂^2 ϕ/∂ z^2 = - e/ϵ_0( n_i-n_e),

where n_i represents the singly charged Ar ion density, and ϵ_0 is the vacuum permittivity. 

Neither neutral particles/background nor charged particle collisions are included in the simulations shown here (collisionless plasma).



The boundary conditions for ion and electron macro-particles are then shown in Fig. <ref>. 
In particular, the domain starts at the exit section of the ionization chamber (with a radius R), and an ambipolar flux of ions and electrons is injected from a region extending from r=0 to r=R with a Maxwellian-flux velocity distribution<cit.>, and a spatial distribution given by the parabolic radial profile of Eq. <ref>. At z=L/2 and for r>R, a dielectric material surface is assumed, meaning that ions and electrons are lost there and accounted for in the local surface charge density σ_c. While quasi-neutrality is automatically satisfied at the injection surface by refluxing all particles that cross the injection surface (z=L/2, r<R) towards the source, at the open free-loss boundaries (dashed blue lines) all ions are lost and electrons are either reflected or not, according to their mechanical energy E_tot (relative to infinity),

    E_tot = 1/2 m_e v_e^2 - e ( ϕ_exit - ϕ_∞),

where v_e is the electron velocity, ϕ_exit is the local electric potential at the outflow position, and ϕ_∞ is the potential at infinity.
All electrons with E_tot<0 are specularly reflected backwards toward the simulation domain, while all electrons with E_tot>0 are lost and charge up a virtual capacitor at infinity, according to:

    dϕ_∞/dt =  I_i∞ + I_e∞/C_∞,

where I_i∞,I_e∞ are the ion and electron currents that leave the domain to infinity, and C_∞ is an equivalent electric capacity.




§ SIMULATION RESULTS






 §.§ 0-D model sizing results


The global design model assumes a target propulsive performance of the helicon plasma thruster, with an achieved thrust (12 mN) and specific impulse (1200 s), and with an absorbed plasma power of around 1 kW. The main parameters of the 0D global sizing model are reported in Tab.  <ref>. The reported data are split in inputs and outputs of the model. 

The geometry of the antenna and ionization chamber are chosen to obtain the above mentioned target absorbed power. 
The results of the 0-D model are the required inputs for both the full-wave and PIC simulations. In fact, the thermodynamic state in terms of pressure, density and temperature of both plasma and neutrals, and moreover the size of the helicon plasma thruster and the magnetostatic field are provided as input to the electromagnetic simulation software to define the geometry of the thruster and to compute the helicon mode solution and the power transfer between the helicon antenna and the plasma. In particular, a magnetostatic field of around 90 G is required and in this case it is generated by two coils of radius 12 cm located axially at the two ends of the ionization chamber (z=± L/2).

The efficiency of the designed thruster is quite low (3%), although it is here emphasized that the 0-D model provides a conservative estimate of this figure, and higher achievable values may be obtained.

The thruster efficiency is computed as: 


    η_t = F g_0 I_sp/2P_RF


where P_RF is the total RF power provided in input to the helicon antenna, which relates to the absorbed plasma power according to P_Ω = η_RF P_RF where η_RF is the RF power transmission efficiency (from RF amplifier to plasma). Here a conservative estimate equal to 70% is assumed. Note that η_t is the total thruster efficiency, considering the input power at the RF amplifier while, in some cases, the thrust efficiency is only referred to the thruster chamber (with the absorbed plasma power in the denominator).







 §.§ Full-wave simulation of the plasma-wave interaction


The full-wave simulations are made with an in-house electromagnetic 3D FEM code that, for a given geometry, is capable of simulating the electromagnetic wave field excited in the plasma by the helicon antenna. The geometry considered here is the Nagoya type-III antenna that is fed by a coaxial cable connecting the RF generator with the antenna feeding port, a spatial gap in one of the two antenna legs. When fed by a RF source, the Nagoya type-III antenna can excite an electromagnetic plasma mode corresponding to a propagating helicon wave with poloidal number m = 1 <cit.>. While the helicon wave is propagating, the wave deposits its power in the plasma through collisional damping and the plasma temperature is determined by Joule heating. As stated previously, the heating power is computed from the volume integral of the scalar product between the current density vector and the complex conjugate of the electric field vector. 

The operational point of the thruster is obtained fine tuning the RF input power at about 2 kW at the target electron temperature and it corresponds to the intersection between the deposited RF power and the total loss power when they are computed as a function of the electron temperature, as shown in Figure <ref>. Indeed, the intersection of the two curves is the steady state operational point of the helicon plasma thruster at the target electron temperature (as computed from the 0D model), i.e. where the deposited power in the plasma 

is equal to the total power lost.














 §.§ Full particle-in-cell simulation of the magnetic nozzle


The PIC simulation of the magnetic nozzle considers the parameters summarized in Tab. <ref>. 



The time step and the cell size are chosen to respect the PIC constraints on Debye length, plasma frequency and cyclotron frequency <cit.>. Ions are injected with sonic conditions with a fully axial fluid velocity, and a temperature of 0.5 eV, resembling a realistic velocity dispersion due to the fact that they are created at different positions inside the ionization chamber. An artificially increased permittivity by a factor of 9 is used to reduce the simulation cost, which corresponds to simulating a Debye length 3 times as large as the real one. This, however, should not impact significantly the obtained results, since the plasma is quasi-neutral nearly everywhere and the total magnetic nozzle potential drop is not affected by the value of ϵ_0, as shown in Eq. <ref>.

The magnetic field intensity and streamlines is shown in Fig. <ref>. 



At the exit surface, the intensity is slightly lower than 90 G, and decreases to just a few G downstream. The intensity peaks at the exit surface at r=12 cm, as that is the radius of one of the two coils that generate this magneto-static field (the other coil is located at z=-L/2, i.e. at the other end of the ionization chamber).

The infinity potential evolution with time is shown in Fig. <ref>. 



In particular, this represents the infinity potential (Eq. <ref>) relative to the nozzle throat (r=0, z = L/2) and its oscillations are mainly due to the PIC noise at nozzle throat (the centerline in axisymmetric simulations is always affected by a large PIC noise, see Ref. domi18). Nevertheless, the absolute drop from nozzle throat to infinity should tend asymptotically to the one predicted by Eq. <ref>, which is around 40 V in this case. In this case, the predicted drop is slightly larger than expected (approx. 50 V), since Eq. <ref> underestimates the total potential drop across the magnetic nozzle, as also confirmed in other studies <cit.>.

The 2D maps of several variables of interest are finally reported in Fig. <ref>. 



The electrostatic potential and field of subplots (a,b) clearly show the effect of the magnetic nozzle, which is that of reducing the divergence of the emitted plume by focusing the ions axially. A plume boundary effect can be observed in both subplots and tends to propagate downstream as well. This effect is produced by both the injection strategy (purely axial ion injection velocity) and by the abrupt density drop from C_zC_rn_e0 to 0 at r=R, z= L/2.

Fig. <ref> (c,d) show the electron density and temperature in the meridional plane. Clearly, since electrons are magnetized and the plasma is collisionless, they cannot traverse radially the magnetic field, so that their density drops to 0 beyond the most external magnetic field line (from the inner ionization chamber), shown by a black solid line. The electron temperature, on the other hand, presents a minimum close to the plume boundary (similarly to what happens in a classical Debye sheath) and slowly decreases within the magnetic nozzle expansion, consistently with the expectations that electron thermal energy is converted into ion directed kinetic energy.

Fig. <ref> (e) then shows the relative charge density, highlighting the non-neutral regions of the expansion. A positively charged region can be observed close to the plume boundary due to the fact that ions are essentially unmagnetized and can easily traverse magnetic field lines, while electrons cannot. In the near totality of the core plasma, however, quasi-neutrality holds well, including at the injection boundary z=L/2, thus validating the simulated injection approach.

Finally, Fig. <ref> (f) shows the axial magnetic thrust force density (j×B)_z, i.e. the magnetic thrust per unit volume. Since the magnetic induction field has no azimuthal component, this is given by f_z,mag = - j_θ B_r, and is mainly generated at the peripheral plume regions. Peak values close to 30 N/m^3 can be observed. Since the plasma plume is essentially quasi-neutral everywhere (see Fig. <ref>(e)), this volumetric force is the only direct contribution to thrust of the magnetic nozzle.

The total thrust force is obtained by integrating the total axial momentum flow through the simulation boundaries (with the exception of the injection boundary r<R, z=L/2) as:


    F =    ∬_S_down[ ( n_iu_i·1_⊥) m_i u_z,i 
        + ( n_eu_e·1_⊥) m_e u_z,e + (p_i+p_e) 1_⊥·1_z] dS
        + ∬_S_up[ ( n_iu_i·1_⊥) m_i u_z,i 
        + ( n_eu_e·1_⊥) m_e u_z,e + (p_i+p_e) 1_⊥·1_z] dS,

where 1_⊥ is the unit vector normal to the boundaries and directed outward (away from the simulation domain), 1_z is the unit vector along the axial direction z, S_down is the downstream open boundary surface, and S_up is the upstream back-flow surface (r>R, z=L/2), as shown in Fig. <ref>. Note that for the second integral, both the average axial momentum and pressure contributions are negative, since 1_⊥ = - 1_z. After a simulation time of 41.5 μs, the total thrust force amounts to 10.9 mN and is almost completely stationary. This can be split between 7.6 mN of injected momentum flow (from the ionization chamber), 3.4 mN due to the integrated axial magnetic force density over the simulated volume (∭_vol -j_θ B_rd V), and a small negative contribution of -0.1 mN due to the integrated electric force (∭_volρ_c E_zd V).
This thrust value is close to the one predicted by the 0-D global model (12 mN), which did not assume plume divergence losses. These can be estimated as the ratio between the z-directed and the total (in any direction) energy flows through the surfaces S_down and S_up. Neglecting pressure and heat flux effects at the downstream section (where the plasma density has decreased significantly), this plume divergence efficiency can be roughly estimated from the ion fluid properties as:


    η_div≈∬_S_down[ 
        n_iu_i·1_⊥ m_i u_z,i^2 ] dS   - ∬_S_up[ n_iu_i·1_⊥ m_i u_z,i^2 ] dS ∬_S_down+S_up[ 
        n_iu_i·1_⊥ m_i u_i^2 ] dS ,

i.e. as the ratio between the z-directed ion kinetic energy flow (with sign) and the total ion kinetic energy flow through the loss surfaces. At the latest available time step, this divergence efficiency is almost stationary and equal to approx. 87%. Clearly, such a high value can be ascribed to both the magnetic topology and the non-modeled ion/electron collisions with the neutral background, which should act to reduce it slightly.



§ CONCLUSIONS



In this work, a 0D global model has been developed to size the main components of a helicon thruster and the main operating plasma and magnetic circuit parameters. The sizing model is based on fundamental propulsive and physical models that allow to dimension the helicon thruster for a target performance in terms of thrust and specific impulse. This 0-D model has been coupled with advenced simulations that predict both the helicon mode coupling by the antenna and plasma fields generated inside the thruster ionization chamber and the plasma expansion of the magnetic nozzle, in order to estimate the effective performance of the thruster.

In particular, the electromagnetic simulations of the plasma-wave interaction allow to verify that a propagating helicon mode is coupled by the antenna to the plasma and to compute the RF input power and the realistic dimensions of the helicon antenna required to meet the desired operative point of the thruster. The electrostatic PIC simulations of the magnetic nozzle expansion, on the other hand, cast light on the acceleration process and permit to estimate the propulsive performance of the thruster. Simulations are in good agreement with the prediction of the 0D global model, both for the electromagnetic and electrostatic parts. Therefore, it can be stated that the proposed 0D model can provide a reasonable preliminary sizing of the helicon thruster.

Based on these results, an optimization study of the helicon thruster geometry and operational conditions will be pursued in a successive work. Additional magnetic nozzle PIC simulations (featuring also particle collisions, neglected here) will permit to optimize the magnetic circuit in terms of the achieved thrust (i.e. the one maximizing the divergence efficiency of the plume). 
At a second level, when a real helicon prototype will be available for vacuum testing, by combining the numerical predictions with real experimental measurements, the 0D global model parameters will be further tuned and, for given constraints, a helicon thruster with an optimal thrust efficiency will be designed.

aiaa




 Paper Title: Highly-efficient electron ponderomotive acceleration in underdense plasmas.



APS/123-QED





lorenzo.martelli295@gmail.com
LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France
THALES AVS - MIS, 2 Rue Marcel Dassault, Vélizy-Villacoublay, 78140, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France
Institut Curie, Inserm U 1021-CNRS UMR 3347, University Paris-Saclay, PSL Research University, Centre Universitaire, Orsay, 91471, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France






FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, Vilnius, LT-2300, Lithuania


THALES AVS - MIS, 2 Rue Marcel Dassault, Vélizy-Villacoublay, 78140, France

cedric.thaury@ensta-paris.fr
LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

§ ABSTRACT 
Laser-plasma accelerators represent a promising technology for future compact accelerating systems, enabling the acceleration of tens of pC to above 1GeV over just a few centimeters. Nonetheless, these devices currently lack the stability, beam quality and average current of conventional systems. While many efforts have focused on improving acceleration stability and quality, little progress has been made in increasing the beam's average current, which is essential for future laser-plasma-based applications. In this paper, we investigate a laser-plasma acceleration regime aimed at increasing the beam average current with energies up to few-MeVs, efficiently enhancing the beam charge. We present experimental results on configurations that allow reaching charges of 5-30nC and a maximum conversion efficiency of around 14%. Through comprehensive Particle-In-Cell simulations, we interpret the experimental results and present a detailed study on electron dynamics. From our analysis, we show that most electrons are not trapped in a plasma wave; rather, they experience ponderomotive acceleration. Thus, we prove the laser pulse as the main driver of the particles' energy gain process.


Highly-efficient electron ponderomotive acceleration in underdense plasmas
    C. Thaury
    August 5, 2024
==========================================================================





§ INTRODUCTION

Since their proposal <cit.>, laser-plasma accelerators (LPA) have interested the scientific community for their ability to produce accelerating gradients 10^3 - 10^4 times those of conventional systems (i.e., ∼100MV/m). The extremely intense electric field would allow for a scaling-down of future accelerators, representing a cost-effective alternative to state-of-the-art linear accelerators (LINACs) and radio frequency cavities. The production of stable, low-emittance, and highly energetic monochromatic LPA electron beams <cit.>, as well as recent optimization studies <cit.>, have dominated the literature. While these results demonstrate the potential of laser-plasma accelerators, their beam properties are still far from those of conventional systems. For instance, current medical electron LINACs can produce up to 10-10^2 μA, while LPAs deliver only tens-nA <cit.>. Naturally, one direct method of increasing the average current of LPAs comes from increasing the beam's charge. Using high-Z gases such as nitrogen and argon is proven as an effective way of enhancing the charge up to the nC-level <cit.>. Recently, using a 27TW, Ti:Sapphire laser and a pure nitrogen plasma with a density n_e≈3.6×10^19cm^-3, it was possible to accelerate around 15nC to few-MeVs, with divergences exceeding 100s mrad <cit.>. This potentially paves the way for μA-level LPAs, marking an important milestone for laser-plasma-based applications. The acceleration of such beams is believed to be associated with the ionization injection of electrons in multiple plasma periods. Once trapped, the particles are subsequently accelerated by the plasma electric field, producing large energy spectra up to a few-tens of MeV.

In this work, we explore the interaction between a superintense laser pulse and a pure nitrogen plasma with densities in excess of n_e=0.01 n_c, where n_c(cm^-3) = 1.1×10^27/λ_0^2(nm) and here λ_0=800nm. Specifically, through a parametric experimental campaign performed at “Laboratoire d'Optique Appliquée" (LOA), the charge-per-Joule metric serves as a straightforward method in gauging the efficiency of the different configurations of interest. By varying the laser energy, plasma density and gas nozzles, we study the conditions to produce highly divergent (i.e., >100s mrad), few-MeV electron beams with charges of 5-30nC and a maximum laser-to-electron energy conversion efficiency around 14%. Through further investigation of the experimental configurations using Fourier-Bessel Particle-In-Cell (FBPIC) <cit.> simulations, we identify three acceleration mechanisms: Ponderomotive Acceleration (PA), Wakefield Acceleration (WA) and Direct Laser Acceleration (DLA). Specifically, the numerical analysis underlines that, upon interaction with nitrogen, the driver pulse radially expels most electrons through its ponderomotive force. Consequently, our study challenges the wakefield's role in accelerating electrons within the configurations of interest, as commonly attributed.










§ HIGHLY-EFFICIENT ACCELERATION OF CHARGED ELECTRON BEAMS

The experiment was performed using Salle Jaune's 60TW Ti:Sapphire laser system able to produce linearly polarized pulses of λ_0 = 813nm central wavelength and a Full Width at Half Maximum (FWHM) duration of 30fs, the driver beam in Figs. <ref>(a) and (b). The laser pulse is focused using an f/4 Off-Axis Parabola (OAP) on the gas target, as shown in Fig. <ref>(b), leading to a FWHM focal spot of 5± 0.3 μm. We estimate that the maximum laser energy on target is 1.7±0.2J of which ∼ 57% is contained within the central spot, corresponding to the first minima ring. This corresponds to a peak laser intensity I_0 ≈ 1.5×10^20W cm^-2 and a maximum normalized vector potential a_0 ≈ 8.5. We use pure nitrogen and two different cylindrically symmetric supersonic gas nozzles of 2mm and 0.4mm exit diameter. The latter is a fused silica nozzle obtained via hybrid 3D laser machining technique <cit.> produced at the Center for Physical Sciences and Technology in Lithuania. As depicted in Fig. <ref>(b), using a Phasics wavefront sensor <cit.> and a probe laser pulse we are able to perform plasma density measurements.

This figure also illustrates the Beam Profile Monitor (BPM), allowing to perform shot-to-shot charge and beam divergence measurements. Specifically, the beam charge is determined using an absolutely calibrated tritium capsule <cit.>. For each configuration, we calculate the average beam charge over ten consecutive shots, with the statistical error defined as the Root Mean Square Deviation (RMSD). The BPM consists of a motorized Lanex Regular Carestream screen with a diameter of 75mm, positioned on the laser axis, and a 16-bit CCD camera to collect the electron beam within a given solid angle. To measure the energy spectra, instead, we remove the BPM and position a motorized electron spectrometer on the laser axis, as shown in Fig. <ref>(b). Each spectrum is determined as the average over ten shots, with the corresponding RMSD. The spectrometer is composed of a magnetic dipole (B_max = 0.44T) with a 2mm diameter pinhole at the entrance. The selected electrons are subsequently deviated on a Lanex screen calibrated from 1.2MeV to 14.7MeV.


In Fig. <ref>(a) the continuous curves represent the charge-per-Joule (Q_/J,Ω) achieved with the 0.4mm nozzle within a ∼0.25 sr solid angle of the BPM, for three different plasma densities. Here, E_L represents the estimated laser central spot energy. The PIC simulations (dashed lines) reproduce the charges obtained under these experimental conditions. Further details about this set of simulations can be found in the Supplementary Materials (Figs. S1 and S2). In the experimental measurements, at n_e = 0.01 n_c we notice that the charge-per-Joule from 7.5nC/J/sr at E_L=0.24J increases to around 10nC/J/sr for E_L=0.48J. At higher plasma densities, instead, we observe a stronger dependency on the laser energy. Specifically, for n_e=0.02 n_c the charge-per-Joule increases with the laser energy, until reaching E_L>0.24J, where it reaches a maximum of ∼69nC/J/sr and subsequently starts to decrease. Similarly, the curve at n_e=0.05 n_c tends to decrease with the laser energy. The simulations show that this behavior is due to the limited BPM collecting angle: the dimension of the electron beams produced at these laser energies exceeds that of the BPM scintillating screen. Hence, some electrons are not co    llected by the diagnostics, causing a decrease in the charge-per-Joule. 

Furthermore, in Fig. <ref>(a) we also notice that the curves at n_e = 0.02 n_c and n_e = 0.05 n_c superimpose, highlighting the presence of a charge-per-Joule saturation effect. However, the numerical study seems to confirm that the superposition of these curves is somehow magnified by the limited collecting angle. Indeed, even an increase in the plasma density yields more divergent electrons that fall out of the BPM measuring cone. With this nozzle, we estimate a maximum of 82nC/J/sr at E_L = 0.12J (i.e., corresponding to a total charge of 2.5nC), with a conversion efficiency η = 6 %. Here, we define the conversion efficiency as the ratio of the total electron energy to the total laser energy on target. 

In Fig. <ref>(b), instead, we show the results obtained with the 2mm nozzle at n_e = 0.29 n_c and n_e = 0.59 n_c. In this figure, we move the BPM closer to the nozzle, allowing the collecting cone to increase to around 0.5sr. In Fig. <ref>(b) we notice very similar tendencies and values to what we have previously discussed. With this nozzle, we measure a maximum charge-per-Joule of 89nC/J/sr with n_e = 0.59 n_c at E_L = 0.12J (i.e., 5.7nC), corresponding to an energy conversion efficiency η=14.4%. At full laser energy and at n_e = 0.22 n_c, we reach 55nC/J/sr, allowing to measure a maximum charge of ∼ 28 nC and an efficiency η = 9.2%. The same efficiency is also estimated for the same laser energy at n_e=0.59 n_c.

Fig. <ref>(c) and (d) illustrate examples of electron energy spectra measured around the beam center. The curves of Fig. <ref>(c) were obtained with the 2mm nozzle at n_e=0.29 n_c for different laser energies. Thus, we observe that the shape of the energy spectrum and the beam's average energy remain constant across different laser energies ranging from 0.12J to 1J, with an average energy of approximately 5.25±0.12MeV, calculated within the range 1.2MeV and 14.7MeV. We also measured the electron beam energy at different horizontal angles as shown in Fig. <ref>(d). From this figure, it is possible to conclude that the most divergent electrons are also the least energetic. Indeed, the average energy drops from ∼ 6MeV at the central position (i.e., Θ_x = 0) to ∼4.5MeV at the extremes (i.e., Θ_x = ±245mrad), within the range 3-10MeV.

In Fig. <ref>(a-d) we present some single-shot images of the BPM scintillating screen. Specifically, these images refer to the 2 mm nozzle at n_e = 0.29 n_c and n_e = 0.59 n_c for two different laser energies. The white lines represent the transverse and longitudinal beam profiles passing through the maxima of each image. Coherent with our prior discussion, Figs. <ref>(a-d) prove that increasing either the plasma density or the laser energy leads to larger beam sizes. Indeed, from these figures, we estimate FWHM divergences between 200mrad and 440mrad, highlighting the influence of varying plasma density and laser energy on the beam divergence.








§  NUMERICAL ANALYSIS



 §.§ Simulation setup

In this section, we present a numerical study that allows us to interpret the experimental results. We first investigate the charge-per-Joule saturation effect and then discuss the different acceleration mechanisms.

The numerical study was performed employing the 3D code FBPIC, which uses a cylindrical grid with azimuthal decomposition. Concerning the simulation setup, we define a 30 μm density ramp, allowing the laser pulse to focus with limited energy losses at the beginning of a 1500 μm plateau (Fig. S3 in Supplementary Materials). We assume the nitrogen gas is preionized up to N^+3, corresponding to the first three L-shell electrons, for numerical ease. The investigation considers four different plateau plasma densities between n_e=0.02 n_c and n_e=0.18 n_c, corresponding to the full L-shell ionization (i.e., N^+5). The laser considered is a 30 fs-Gaussian beam propagating along z and polarized in the x direction, with energies (E_L) ranging from 0.05J to 1J and a waist w_0=3 μm. Regarding the numerical parameters, we employ a (r,z) mesh with Δ z = λ_0 / 24 and Δ r = 5 Δ z, where λ_0 = 800 nm is the laser wavelength. Finally, three azimuthal modes (m=0-2) are considered, and the macroparticles per cell along r, z and θ are set to 1, 1 and 4 respectively.



 §.§ Efficiency saturation effect


We now intend to explain the charge-per-Joule saturation effect observed experimentally and discussed in the Section <ref>. Thus, in Fig. <ref> we present the charge-per-Joule (Q_/J) within a 4πsr solid angle as a function of the laser energy and plasma density, derived from linearly interpolated numerical results. We only consider electrons with a minimum energy E=2MeV, which can be relevant for a number of low-energy applications such as industrial X-ray tomography. The white dashed lines in Fig. <ref>, instead, refer to charge-per-Joule isolines. From this figure, it is possible to notice that for E_L≳0.12 J and n_e≳0.03 n_c the charge-per-Joule slowly increases from 40nC/J to around 50nC/J with the laser energy. This region is outlined by black dotted lines for visual reference. In accordance with the experimental findings discussed in Section <ref>, we observe that within this region, increasing the plasma density at a fixed laser energy does not result in higher charges-per-Joule.

The numerical analysis underlines that this effect is due to the saturation of the conversion efficiency with respect to the plasma density. More precisely, the laser interacts with the plasma over a characteristic length approximately equal to half the pump depletion length, L_pd∝ n^-1_e <cit.>. Simultaneously, increasing the plasma density results in more electrons being accelerated per unit length. Therefore, despite the reduced interaction length at higher densities, the faster ionization ultimately leads to the same amount of charge being accelerated. This analysis is corroborated by the energy spectra in Fig. <ref>(b), where we consider electrons with energies in the range 2 MeV<E<15 MeV, comprising over 80% of the charge above 2MeV. Indeed, we notice that the curves at E_L=1J for n_e=0.03 n_c and n_e=0.06 n_c overlap, presenting an average energy of 4.7MeV. For these configurations, we estimate the efficiency to be η∼21%. A similar behavior is also observed at E_L=0.12J, where we estimate the average energy to be 5.7MeV and the conversion efficiency is η∼18% for both densities. In other words, regardless of variations in plasma density, the same amount of laser energy ionizes and accelerates the same number of electrons to the same average energy.



 §.§ Electron dynamics and acceleration mechanisms

Before discussing the acceleration mechanisms in detail, we intend to briefly describe the plasma structures that develop during the interaction. Thus, Fig. <ref>(a1) depicts the plasma density (top half) and corresponding radial plasma field (bottom half) obtained with n_e=0.03 n_c and E_L=1J. In this figure, we notice the presence of a long and rapidly-changing channel-like structure, with a characteristic length L≫λ_p, where λ_p=4.7 μm is the plasma wavelength. This massive sheath is formed by nitrogen L-shell electrons and it is filled with K-shell electrons, continuously accumulating and flowing through the channel.

In the bottom half of Fig. <ref>(a1), we highlight that the radial focusing field generated by the L-shell electron sheath confines K-shell electrons close to the laser axis. Consequently, this dense concentration of particles on axis shields the longitudinal wakefield and hinders the formation of ion cavities. Nonetheless, inside of this structure, we can still find rapidly changing density modulations, which can generate a longitudinal wakefield. As we will discuss in the following, these modulations can contribute to the electron acceleration process. Close to the laser intensity peak (i.e., z-ct≈90 μm), instead, the ponderomotive force allows for effective charge separation and the formation of a cavity in front of the channel structure.

In order to understand how electrons gain energy, we calculated the work performed by the laser and plasma electric fields. This task was carried out employing the numerical tool FBPIC-Electric Work Profiler (FBPIC-EWP) <cit.>. This code allows to estimate the work exerted on FBPIC-tracked electrons by both the laser and plasma electric fields, exploiting FBPIC modal decomposition. Hence, if we consider a single electron, in the time interval [0, t] an electric field 𝐄_W,L performs the work

    W_W,L(t) = -e∫^t_0𝐄_W,L·𝐯 dt',

where the subscripts W and L denote the wakefield and laser field contribution respectively, 𝐯 is the electron velocity and e is its charge.

Fig. <ref>(a1) illustrates two examples of K-shell electron trajectories (black curves) on the (z-ct, x) plane experiencing ponderomotive acceleration. L-shell electrons exhibit a similar dynamic as will be discussed in the following and in the Supplementary Materials (Fig. S6). Considering the case denoted by the continuous trajectory in Fig. <ref>(a1), we estimate that the particle crosses N_oc = 12 laser optical cycles before being expelled at the instant t^*=50fs, denoted by a circle in Fig. <ref>(a1). Here, t=0 corresponds to the electron ionization time. Similar to what is observed in other acceleration regimes <cit.>, this sort of dynamic is typical of ponderomotive electrons: they slip through several optical cycles experiencing low energy gains, as shown in Fig. <ref>(a2). Here, we plot the electron kinetic energy and, for simplicity, the work done by the plasma and the laser along the radial direction, defined as W^r_W,L=W^x_W,L + W^y_W,L. We estimate that the laser provides 6.8MeV in radial push. Simultaneously, the electron loses 3.2MeV crossing regions of space where the focusing radial wakefield opposes the ponderomotive push (i.e., at z-ct≈85 μm). Concerning the longitudinal dynamics, instead, the numerical analysis underlines that at t=22fs (i.e., z-ct≈93 μm) the electron crosses the wakefield decelerating region in the front cavity, causing a 1MeV loss, while also the laser performs a negative work around 2.1MeV. Finally, the electron leaves the laser with an energy E=0.5MeV.

The particle denoted with the dashed trajectory in Fig. <ref>(a1) experiences a somewhat similar acceleration process. It slips through the laser field oscillating over N_oc = 21 optical cycles, before leaving the laser field at t^*=83fs, as denoted by the circle on the electron trajectory. From Fig. <ref>(a3) we deduce that in this case, the laser also provides most of the energy, with a radial push of 2.7MeV, while the wakefield exerts a pull of 1.1MeV. Longitudinally, the particle receives 0.7MeV from the plasma field, since it experiences its accelerating phase at t=50fs (i.e., z-ct≈86 μm), while the laser performs a negative work of 0.8MeV. The electron leaves the laser pulse with an energy of E=1.5MeV. Moreover, for t>83fs in Fig. <ref>(a1) we note that this particle remains closer to the laser axis and is radially trapped in the channel-like structure discussed above. Here, it performs radial oscillations until leaving the channel without a significant difference in energy.


Having discussed the ponderomotive acceleration through two examples, we now present a statistically relevant study distinguishing between the number of electrons accelerated by the laser or the wakefield. Moreover, we provide an overview of other acceleration mechanisms we identified, and more details can be found in the Supplementary Materials (Figs. S4 and S5). Therefore, for each tracked electron at the instant t, we now define the laser-to-plasma work ratio as

    χ(t) = |W_L(t)/W_W(t)| - 1,

allowing to compare the laser and plasma field contributions. A positive χ value clearly indicates the laser's central role in driving the electron acceleration process. For coherence with Section <ref>, in Fig. <ref>(b1) we plot the beam angular distribution for both K- and L-shell electrons with E>2MeV at the last iteration. Each electron in the top half of this figure is colored based on its χ value. Thus, we define two criteria for χ and we identify the corresponding electron populations. In the range -1≤χ<0 we find yellow electrons that gain most of their energy from wakefield acceleration.

The red particles, instead, have χ≥0 and are primarily accelerated by the laser. In this population, we find ponderomotive electrons, displaying dynamics similar to the two examples in Fig. <ref>(a1). As discussed below, some of these electrons can also experience direct laser acceleration <cit.>. We estimate that around 70% of the particles exhibit positive values of χ, proving the laser as the primary driver in particle acceleration. In the bottom half of Fig. <ref>(b1), instead, the color represents the energy of each particle. The average energy for E>2MeV is around 9MeV for both laser- and wakefield-accelerated electrons, and we notice a concentration of higher energies closer to the laser axis, which is consistent with the experimental observation. For instance, if we consider a solid angle of 0.25sr (white circle in Fig. <ref>(b1)), the average energy increases to 14MeV. 

Fig. <ref>(b2) depicts the (z-ct, E) phase space of laser- and wakefield-accelerated K-shell electrons at the final iteration of the simulation. This figure highlights the different behaviors of laser-accelerated electrons. As previously mentioned, in this population we find particles undergoing ponderomotive acceleration. We estimate that around 85% of laser-accelerated K-shell electrons gain up to m_ec^2a_p≈8MeV through the ponderomotive push <cit.>. Here, a_p≈16 is the maximum laser normalized vector potential in plasma. Instead, the remaining 15% experiences direct laser acceleration. Fig. <ref> presents an example of electron trajectory undergoing DLA (continuous curve). These particles are trapped in the ion cavity in front of the channel structure via ionization injection. Once trapped, they overlap with the driver pulse and perform oscillations along the laser polarization direction. Subsequently, these oscillations can lead to a gain in longitudinal momentum via the e𝐯×𝐁 term of the laser. The longitudinal wakefield reduces the dephasing between the electrons and the laser, ultimately allowing these particles to reach 10s-100s MeV <cit.>. In the example of Fig. <ref>, the electron reaches a maximum energy of 85MeV, with 79MeV attributed to the laser.

In Fig. <ref>(b2) we can also observe wakefield-accelerated K-shell electrons (yellow dots). Specifically, the numerical analysis highlights the presence of “traditional" wakefield acceleration, where the particles are trapped in phase with the wakefield within the front ion cavity long enough to reach 10s-100sMeV. Additionally, we recognize wakefield accelerated particles displaying a behavior somewhat similar to the dashed trajectory of Fig. <ref>(a1). Once ionized, they initially receive most of their energy from the laser ponderomotive push and they subsequently slip into the channel, where they radially oscillate until escaping the structure with few-MeVs in energy. However, once inside the channel, they undergo acceleration in the longitudinal wakefield phase, induced by the plasma density modulations mentioned earlier. The numerical analysis shows that this plasma field contribution exceeds the laser initial push and, in this sense, they undergo wakefield acceleration. Nonetheless, unlike traditional wakefield acceleration, where particles are trapped in phase with the wakefield within the front ion cavity, these electrons continuously slip through the channel. In Fig. <ref> we provide an example of this sort of non-traditional wakefield acceleration (dashed curve). At the end of the simulation, the particle has gained 4.3MeV in energy, of which ∼3MeV are provided by the wakefield.

Analogously, Fig. <ref>(b3) displays the (z-ct, E) phase space of laser- and plasma-accelerated L-shell electrons at the final iteration of the simulation. We estimate that around 83% of L-shell electrons with E>2MeV undergo PA, while the remaining fraction is mainly accelerated via WA while momentarily crossing the longitudinal wakefield accelerating phase.



§  CONCLUSION AND OUTLOOK

In this paper, we studied a regime allowing the increase of the average current of laser-plasma accelerators with energies at few-MeVs, through the enhancement of the beam charge. With an extensive experimental campaign, we were able to produce charges of 5-30nC, with average energies around 5MeV. Notably, employing a 0.12J laser pulse, we achieve a charge of 5.3nC with a conversion efficiency of 14.4%, one of the highest recorded to date. Improving the laser focal spot quality could lead to efficiencies exceeding 20%, as shown by the numerical study. These results are promising for future LPAs, paving the way for unprecedented average currents. Novel J-class lasers with 100W in average power are emerging <cit.> and they would allow to exceed the μA-level. Considering, for instance, the configuration discussed above (i.e., 5.7nC at 0.12J), we can easily estimate that the maximum achievable average current is ∼5μA, assuming 100W in laser average power. With such capabilities, this electron source emerges as a promising candidate for various applications needing few-MeV electrons, including X-ray tomography <cit.> and irradiation studies.

In configurations similar to those studied here, electron trapping via ionization injection followed by wakefield acceleration is typically considered the dominant acceleration process. However, in this paper, we were able to show that most particles are not injected in plasma cavities, as also discussed in other works present in the literature <cit.>. More specifically, we proved that the electrons primarily gain energy through ponderomotive acceleration, establishing the laser pulse as the main driver in the energy gain mechanism.


This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement n°101020100.


apsrev4-2




 Paper Title: Matched Guiding and Controlled Injection in Dark-Current-Free, 10-GeV-Class, Channel-Guided Laser Plasma Accelerators.



APS/123-QED



apicksley@lbl.gov
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
University of California, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
University of California, Berkeley, California 94720, USA

Institute of Research in Electronics and Applied Physics and Dept. of Physics, University of Maryland, College Park, MD 20742, USA

Institute of Research in Electronics and Applied Physics and Dept. of Physics, University of Maryland, College Park, MD 20742, USA

Institute of Research in Electronics and Applied Physics and Dept. of Physics, University of Maryland, College Park, MD 20742, USA

Institute of Research in Electronics and Applied Physics and Dept. of Physics, University of Maryland, College Park, MD 20742, USA
Dept. of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
University of California, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA



§ ABSTRACT 
We measure the high intensity laser propagation throughout meter-scale, channel-guided LPAs by adjusting the length of the plasma channel on a shot-by-shot basis, showing high quality guiding of 500TW laser pulses over 30 cm in a hydrogen plasma of density n_0 ≈1e17cm^-3.
We observed transverse energy transport of higher-order modes in the first ≈12cm of the plasma channel, followed by quasi-matched propagation, and the gradual, dark-current-free depletion of laser energy to the wakefield. We quantify the laser-to-wake transfer efficiency limitations of currently available PW-class laser systems, and demonstrate via simulation how control over the laser mode can significantly improve accelerated beam parameters. Using just 21.3J of laser energy, and triggering localized electron injection into the accelerator, we observed electron bunches with single, quasimonoenergetic peaks, relative energy spreads as low as 3% and energy up to 9.2GeV with charge extending beyond 10GeV.



Matched Guiding and Controlled Injection in Dark-Current-Free, 10-GeV-Class, Channel-Guided Laser Plasma Accelerators 


    A. J. Gonsalves
    August 5, 2024
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Acceleration of particles in plasma waves driven by intense laser pulses <cit.> has attracted significant attention due to the ultra-high accelerating gradients that can be achieved (up to 100s GV/m). These compact accelerators are attractive for applications such as free-electron lasers <cit.>, Thomson sources <cit.>, and high-energy physics colliders <cit.>. Significant progress has been made towards producing high-energy (> 1GeV) <cit.>, high-quality <cit.> electron bunches in laser-plasma accelerators (LPAs). 

Maximizing electron beam energy for a given laser energy is critical for these applications. This requires maintaining laser intensity over several tens of centimeters, much longer than the typical Rayleigh length z_R∼1cm of focused PW-class laser systems. Hence, laser pulses must be guided via relativistic self-focusing <cit.> or using a preformed plasma channel <cit.>. The former requires operating in the bubble regime of the LPA <cit.>, at a plasma density n_0 high enough to remain above the critical power for self-focusing. High-energy electrons can be produced but high laser pulse energy is required <cit.>; for example Aniculaesei et al. <cit.> demonstrated an energy gain of ∼10GeV using 118J of laser energy. Preformed plasma channels have transverse electron density profiles with a minimum on-axis such that the refractive index is peaked, much like gradient-index optical fibers. They allow operation of LPAs at lower n_0 where the product of acceleration gradient and accelerator length is higher.


Matching the drive laser to the fundamental transverse mode of the plasma channel maximises the efficiency of laser-to-wake energy transfer without degrading the accelerated bunch parameters, and minimizes the accelerator dark current. For a parabolic plasma channel where n_e(r)-n_0 ∝ r^2, a low-intensity laser pulse with a Gaussian transverse profile is perfectly matched to the fundamental channel mode and propagates at constant spot-size if w_0 = w_m, where w_m is the matched spot-size, a measure of the steepness of the parabolic profile <cit.>. Future applications of 10-GeV-class LPAs require matched propagation of PW-class lasers with w_m≲50 m at n_0 ≈1.0e17cm^-3 <cit.>. 

Capillary-discharge waveguides <cit.> have previously been employed in multi-GeV LPAs <cit.>. Whilst they are approximately parabolic, and can provide per-mille level stability of w_m and n_0 <cit.>, w_m is too large for sufficient confinement at n_0 ≈e17cm^-3. Even with the addition of an auxillary laser to reduce w_m <cit.>, effective guiding was only achieved above optimal n_0, limiting energy gain to 7.8GeV <cit.>.

Building upon the previous demonstrations of hydrodynamically formed channels that employed collisional heating <cit.> and were limited to n_0 ∼e19cm^-3, laser heating with ≲100fs pulses has been employed <cit.>. The heating mechanism, optical field ionization, is effective at lower density <cit.>. A cylindrical column of plasma is formed in the line focus of an ultrashort laser pulse, and expands radially. When an intense drive laser is focused into the plasma channel, the leading edge ionizes the neutral gas surrounding the expanding shock front to create a deep, thick-walled plasma channel <cit.>. Steep channels formed by hydrodynamic expansion of optical field-ionized (HOFI) plasmas (20 m≲ w_m≲50 m) with n_0 ≲e17cm^-3 can be generated <cit.>. Guiding of high intensity pulses <cit.>, and electron acceleration <cit.> have been previously demonstrated, with ∼5GeV the highest energy to date <cit.>. In such experiments, guiding is assessed at the end of the accelerator, and propagation of the laser pulse throughout the plasma is inferred from simulations. Laser mode beating and evolution theory <cit.> was recently analyzed by Shrock et al. <cit.> via PIC simulations showing mismatched laser guiding linked to measured electron spectra.

In this Letter, we measure high-intensity laser propagation throughout meter-scale LPAs by adjusting the length of the accelerator on a shot-by-shot basis for the first time, showing high quality guiding of 500TW laser pulses throughout a 30-cm-long hydrogen plasma of density n_0 ≈1e17cm^-3. Using extensive optical diagnostics, we show laser pulse coupling into the higher-order channel modes and energy loss through mode-filtering, followed by quasi-matched propagation of the fundamental mode, and the gradual, dark-current-free depletion of laser energy to the plasma wave. Then, by triggering electron injection via the localized addition of nitrogen to the plasma, electron bunches with single, quasi-monoenergetic peaks up to 9.2GeV and charge extending to > 10GeV is achieved using just 21.3(3)J of laser energy. We quantify the laser-to-wake transfer efficiency limitations of currently available PW-class laser systems, and demonstrate via simulation how control over the laser mode can result in ≳13GeV bunches for the same plasma channel. This work provides in-depth understanding of non-linear laser evolution and electron acceleration in 10-GeV-class LPA stages suitable for future compact accelerators. 


The Ti:Sapphire-based BELLA petawatt laser <cit.> produces pulses of full-width-half-maximum (FWHM) duration ∼40fs at a central wavelength λ_0 = 815nm. Recent upgrades allow the amplified laser to be split into two separately compressed beamlines <cit.> with control over the relative timing, wavefront, and focusing geometry. Figure <ref>(a) shows a schematic of the experimental setup. In the channel-forming beamline, 1.3(3)J was focused by an axicon lens and reflected by a mirror with a hole drilled in the center into the gas target <cit.>. The peak intensity I_ax as a function of distance from the entrance of the gas jet, z, is shown (black) in figure <ref>(b). The drive pulse was focused to a spot-size w_0 = 53(1) m at the entrance of the gas target. The energy was varied up to a maximum of ℰ_0 = 21.3(3)J, corresponding to a peak normalized vector potential a_0 ≈ 0.85 λ [ m] √(I_0 [e18W.cm^-2]) of up to a_0 ≈ 2.2. Here, I_0 is the peak intensity of the drive pulse. 





The drive laser and electron diagnostics have been described previously <cit.>. The input and guided mode of the drive laser could be imaged over a range of ∼60cm; the propagated drive was also imaged at the plane of the third wedge ≈10m downstream of the channel exit. The optical spectrum was measured using fiber-based spectrometers covering the range 400nm≲λ≲2200nm. The energy transmission T(z) = ℰ(z)/ℰ_0 was retrieved by integrating the counts on the detector at z ≈10m, and then using the measured optical spectrum to correct for the spectral response of the detector. 

For this experiment, a 30-cm-long gas target was developed <cit.>. The jet comprised an elongated, converging-diverging nozzle operated with hydrogen, or hydrogen with a ≤5% nitrogen dopant. The length could be varied by blocking the flow of gas above the nozzle. Figure <ref>(b) shows the molecular density as a function of distance along the gas jet measured using method outlined in <cit.>. The delay between the channel-forming beam and the drive beam was set to Δτ = 6ns, and the jet was operated 12mm below the laser axis to avoid blocking the channel-forming beam (which had a radius of ≈12mm at z=0). For these conditions, two-color interferometry measurements <cit.> shown in figure <ref>(c) indicated an axial plasma density n_0 ≈1e17cm^-3 and matched spot-size w_m≈37 m. We note that due to the large uncertainties associated with two-color interferometry, even though the channel expanded further with increasing Δτ, the calculated w_m was unchanged for delays 57ns used in this work.



The evolution of key drive laser parameters for two different laser energies ℰ_0 = 6.0(1)J (a_0 ≈ 1.3) and 19.6(4)J (a_0 ≈ 2.2) is shown in figure <ref>. Only shots for which the transverse position of the laser focus with respect to the channel entrance Δ R < 25 m were included, inferred from a non-destructive centroid diagnostic <cit.>. This condition was satisfied for 71% of shots. 
Figure  <ref>(a) shows representative transverse fluence profiles of the drive laser for several different channel lengths ≈10m downstream of the waveguide exit. For L_ch≈7cm the drive laser mode had transformed from the top-hat-like input mode (that is typical of currently available PW-class systems based on bulk crystal) to near-Gaussian; the super-Gaussian <cit.> fit order reduced from ≈ 6 to ≈ 2. Through z ≈12cm, the propagated mode exhibited rings outside the central fluence peak. As the channel was lengthened, a single, approximately Gaussian transverse fluence profile was always observed when the drive laser was well-aligned to the channel. Figure <ref>(b) shows a waterfall plot of lineouts from the re-imaged exit mode, with the channel length varied in steps of 0.2cm up to L_ch≈5cm. The dashed line indicates the measured peak of n(r). Figure <ref>(c) shows re-imaged exit mode images for channel lengths up to L_cm≈30cm. For z ≳12cm, a well-confined, near-Gaussian drive mode was observed for all channel lengths and for both laser intensities. 


The dominant mechanisms behind laser pulse propagation <cit.> can be understood from guiding measurements presented in figure <ref>. The channel supports several quasi-bound transverse modes which we denote as (p,m) referring to the radial and azimuthal mode number respectively. Their structure are determined by n_e(r). HOFI plasma channels are finite in extent and not radially parabolic; only a finite number of low-order modes can propagate, of which the fundamental (0,0) mode is close to Gaussian. Since the input laser mode [shown in figure <ref>(a)] was not the fundamental channel mode, energy was coupled into higher-order modes, observed directly from the ring structure in figure <ref>(a) for L_ch≈7cm. Figure <ref>(b) shows rapid leakage of higher-order modes out of the bound region at oblique angles over a few cm <cit.>. These modes do not contribute to wakefield generation. After this initial period of mode-filtering, which occurred over z ≲12cm, higher-order mode content in the guided mode was severely reduced, evidenced by figure <ref>(c,i). Remaining higher-order modes slip behind the fundamental due to group velocity dispersion, and eventually become separated longitudinally such that they also do not contribute to wakefield generation <cit.>. The distance over which the (1,0) and (2,0) modes separate by 40fs was calculated to be 21.6cm, and 9.4cm respectively. For z ≳12cm, the measured exit mode remained approximately Gaussian with measured spot-size oscillation ≲ 6 %, demonstrating approximately matched propagation of the drive in the fundamental mode.


Coupling and propagation losses were evaluated quantitatively for the laser and channel used here. The mode coupling efficiency η of the input drive laser into the measured fundamental mode was calculated as the overlap integral between the two modes. For the measured input spot (w_0 ≈53 m), η≈60%. The focus of a perfect flat-top-like laser with the same w_0 yields η≈72%, and for a size of w_0≈47 m, η can be as high as ≈85%. Measured propagation loss [see figure <ref>(d)] occurred when light exited the channel outside the acceptance angle of our diagnostics (≳0.75deg) or when higher-order modes that remained at the end of the channel diffracted. For the latter, it was calculated that > 99%, 85% and 28% of light remaining in the (0,0), (1,0) and (2,0) modes respectively was captured inside the diagnostic acceptance angle. For ℰ_0 = 6.0(1)J, measured T(z) indicates 31(4)% losses in the first 16cm, consistent with our predicted coupling losses. Further losses for 16cm≲ z ≲30cm were ≲10% indicating mild coupling of remaining energy to the wake. The calculated attenuation through leakage of light remaining in the (1,0) and (0,0) modes was small over 16cm≲ z ≲30cm. Since wakefield generation is mainly driven by energy in the (0,0) mode, and higher-order modes are filtered, control over the channel length directly illustrates how laser-to-wake transfer efficiency is limited in LPAs due to currently available PW-class systems, and can be maximized by careful matching of the laser size to the channel.




Driving a wakefield suitable of generating multi-GeV beams on the length scales of our gas jet required increased laser intensity <cit.>. Similar behavior of the drive laser evolution for laser energy 19.6(4)J is observed in figure <ref>(b, ii), with the exception of decreased transmission associated with wakefield generation. This was confirmed by gradual laser redshifting shown in figure <ref>(a). Simulations of this case were performed using the code INF&RNO <cit.>. The measured parameters for the energy, temporal profile, and transverse profile of the drive laser (which was recovered using a Gerchberg-Saxton algorithm) were input, and n(r) was set to the measured channel for Δτ [see figure <ref>(c)]. Figure <ref>(b) shows the calculated optical spectrum during propagation. The wavelength at which the spectrum reduces to 5% of the peak, λ_R is shown in black and matched closely to experiment for all L_ch, explicitly demonstrating continual depletion of energy from the laser to the plasma wave. The calculated average field was E_z ≈30GV.m^-1. Unlike for ℰ_0 = 6.0(1)J, at high-intensity T(z) continuously reduced as energy was coupled to the wake <cit.>. 

Figure <ref>(c, blue) shows the calculated evolution of the normalized peak laser intensity â(z) for parameters of figure <ref>(b). Oscillations in â(z) are due to mode beating and cause periodic changes in the longitudinal and transverse structure of the wakefield <cit.>. Changes in the longitudinal structure result from relativistic effects (i.e., the dependence of the plasma wavelength on the laser peak strength), while changes in the transverse structure result from laser mode evolution (i.e., the shape of the laser mode varies because of mode beating, and this affects the transverse component of the ponderomotive force). The latter can result in a wakefield that is unsuitable for the transport of electron beams if, for instance, the laser mode acquires a sufficiently deep minimum on axis. Mode filtering and subsequent mode dispersion reduce the visibility of oscillations for z ≳12cm, consistent with the diminishing ring structures in figure <ref>(a). Low visibility oscillations caused by beating between the (0,0) and (1,0) modes are present as the laser self-steepens and redshifts. 









No electron beams were generated for the experiments presented above [see figure <ref>(a)], demonstrating that laser pulses can be well-guided for densities below the self-trapping threshold at laser intensities sufficient to generate high amplitude plasma waves. Electron beams were generated by introducing a nitrogen dopant to the gas jet <cit.>. A 1% dopant extending throughout the jet triggered injection at several points, and resulted in electron bunch spectra with a broad distribution. Electrons injected after short propagation distance experience the plasma wave over a longer distance and reach higher energies, whilst electrons injected later in the plasma channel experience less energy gain. An example bunch with conditions similar to figure <ref> (but with Δτ = 7ns) is shown in figure <ref>(b). A peak in the tail of the distribution was observed at ∼9.4GeV with charge extending ≳10GeV. 

To study the acceleration of single, quasimonoenergetic bunches, we restricted the dopant region 0 ≤ z ≲ L_dop within the gas jet <cit.>. For L_dop≈6cm, high energy electrons were not observed. Figure <ref>(c) shows generated beams for L_dop≈12cm, Δτ = 5ns, and ℰ_0 = 21.3(3)J. Singly-peaked electron bunches were observed, indicating injection in the region 6cm≲ z ≲12cm. The mean energy, and FWHM spread for the examples in figure <ref>(c) were 8.67(0.48)GeV, 7.70(0.88)GeV, 7.96(0.44)GeV and 9.15(1.80)GeV. Shot-to-shot stability was dominated by transverse offset of the laser focus at the plasma channel entrance, and by variations of ≳20% in the pulse duration. Due to pointing variations and limited acceptance of the spectrometer, not all of the charge recorded by the phosphor screen was captured by the magnetic spectrometer. For each example, the measured charge within the quasimonoenergetic bunch and percentage of charge captured is shown. The bottom panel in figure <ref>(c) shows results from INF&RNO simulations with the same conditions. The simulation confirmed ionization of nitrogen occurred throughout the dopant region, z ≤12cm, however changes in the wake structure noted in figure <ref>(a) prevented the trapping of electron bunches with a significant charge for z ≲8.6cm <cit.>. A portion of the electrons ionized within 8.6cm≲ z ≲12cm were accelerated to 9.3GeV (FWHM energy spread 1.3GeV, bunch charge 6pC). 


Maximal wake-to-bunch energy transfer also requires beam loading of the bunch current <cit.>. For bunch currents with strong beamloading, the overall acceleration gradient is reduced. Increasing the dopant concentration to 5% evidenced this [see figure <ref>(d)]. The estimated total charge increased by a factor of ≳ 3, but resulted in a maximum bunch energy of 7.44GeV (0.25GeV FWHM energy spread). 

We note that future, high repetition-rate laser systems (e.g., those based on fiber lasers <cit.>) allow for precise control over the transverse laser modes as opposed to bulk-crystal flat-top-like beams currently available. For the same laser energy and channel density, although with the pulse length optimized for electron beam trapping (≈70fs) <cit.>, the orange curve in figure <ref>(c) shows the laser intensity evolution for a linearly-matched input mode. Mode beating was greatly reduced, but not eliminated because at these intensities the plasma refractive index itself is modified, slice-by-slice along the pulse, by relativistic self-focusing and ponderomotive self-channeling <cit.>. This simulation resulted in the production of a 13.0GeV, 65pC electron bunch, trapped in the region z ≲4cm, after a 50cm propagation in plasma. This was then compared to the green curve of figure <ref>(c) where the transverse laser mode on each longitudinal slice of the pulse was varied so that it remained matched to the ponderomotively perturbed channel (super-matching) <cit.>. Mode beating, and oscillations in â(z) were completely eliminated, permitting the wake structure to remain constant throughout. The simulated electron bunch was  13.1GeV, 102pC after a z ≈50cm. This demonstrates clearly that control over the laser mode maximizes coupling efficiency to the (0,0) channel mode, and minimizes the distance over which over which the wakefield is unsuitable for electron beam transport. 
The marginal difference in energy gain between a laser pulses initiated with a super-matched or a linearly-matched input mode is critically encouraging, since it could mitigate the necessity for super-matching. 
We also note that tailoring of the plasma channel at the entrance of the plasma channel <cit.> to filter p,m > 0 modes more rapidly could also reduce the mode-beating distance in current systems. 


In conclusion, unprecedented insight into the mechanisms of laser propagation in meter-scale LPAs has been gained through varying the accelerator length on a shot-by-shot basis for the first time. Using extensive optical diagnostics, we observed laser coupling into high-order channel modes and their energy loss through mode-filtering, followed by quasi-matched propagation of the fundamental mode, and non-linear depletion of laser energy to the wakefield. We quantified the reduction in laser-to-wake efficiency and electron energy gain caused by the laser mode of currently available PW-class laser systems, and showed how precise control over the mode can result in a significant increase of the bunch energy and charge for the same channel. Matched guiding at n_0 ≈e17cm^-3 suppressed self-trapping of electrons throughout the structure. With the introduction of nitrogen dopant, electron beams were with single, quasimonoenergetic peaks to 9.2GeV were generated with 21.3J of laser energy. We note that previous demonstrations of acceleration to ∼7.8GeV using the same laser required energy of 31J and produced electron spectra containing several lower energy peaks <cit.>. This work opens the door for advanced injection techniques to trap ultra-low emittance bunches <cit.> using plasma structures well-suited to repetition rates exceeding 1kHz <cit.>, meeting vital requirements for future compact accelerators. 







This work was supported by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, the Defense Advanced Research Projects Agency, and used the computational facilities at the National Energy Research Scientific Computing Center (NERSC). E. Rockafellow is supported by NSF GRFP (DGE 1840340). We greatly acknowledge technical support from Zac Eisentraut, Mark Kirkpatrick, Federico Mazzini, Nathan Ybarrolaza, Derrick McGrew, Teo Maldonado Mancuso, Art Magana and Joe Riley. The authors would like to thank Nathan Cook,  Jens Osterhoff, Davide Terzani, Remi Lehe, Liona Fan-Chiang, Lieselotte Obst-Huebl, Marlene Turner, and Aodhan McIlvenny for useful discussions. We thank Samantha Trieu (LBNL Creative Services) and Chetanya Jain for design support in Figure <ref>.











 Paper Title: Effects of plasma nonuniformity on zero frequency zonal structure generation by drift Alfven wave instabilities in toroidal plasmas.










^1 Key laboratory of frontier physics in controlled nuclear fusion and  Institute of Plasma Physics, Chinese Academy of Sciences, Hefei 230031, P.R.C

^2 Center for Nonlinear Plasma Science and ENEA,   C. R. Frascati,  Italy

^3 Institute for Fusion Theory and Simulation, School of Physics, Zhejiang University, Hangzhou 310027, P.R.C

^4 Department of   Physics and Astronomy,  University of California, Irvine CA 92697-4575, U.S.A.

^5 Southwest Institute of Physics, P.O. Box 432, Chengdu 610041, P.R.C 



§ ABSTRACT 
Effects of plasma nonuniformity on zero frequency zonal structure (ZFZS)   excitation  by drift Alfvén wave (DAW) instabilities in toroidal plasmas are investigated using nonlinear gyrokinetic theory. The governing  equations describing    nonlinear interactions among ZFZS and DAWs are derived, with the contribution of DAWs self-beating and radial modulation accounted for on the same footing.   The obtained equations  are then used to derive the nonlinear dispersion relation, which is then  applied to   investigate ZFZS generation in several scenarios. In particular, it is found that, the condition for zonal flow excitation by kinetic ballooning mode (KBM) could be sensitive to plasma parameters, and more detailed investigation is needed to understand   KBM nonlinear saturation, crucial for bulk plasma transport in  future reactors.



52.30.Gz, 52.35.Bj,52.35.Fp, 52.35.Mw

Effects of plasma nonuniformity on zero frequency zonal  structure generation by drift  Alfvén wave instabilities in toroidal plasmas
    Zhiyong Qiu^1,2, Guangyu Wei^3,  Liu Chen^2,3,4,    and Ruirui Ma^2,5
    August 5, 2024
=====================================================================================================================================








§ INTRODUCTION



Shear Alfvén waves (SAWs) are fundamental electromagnetic oscillations in magnetized plasmas <cit.>. In magnetically confined fusion devices such as tokamaks, SAW instabilities driven unstable by energetic particles (EPs) including fusion alpha-particles could cause  significant redistribution and transport loss of EPs, which can lead to degradation of the confinement of both EPs and thermal components in future reactors <cit.>. Thus, in-depth understanding of SAW related physics, including linear excitation, nonlinear evolution,  and  saturation, is crucial for understanding of fusion reactor performance. Among various channels for SAW instability nonlinear saturation, nonlinear excitation of   zonal field structures (ZFS) is an important route <cit.>, and has drawn research interest by both analytical theory <cit.> and large-scale simulations <cit.> in the past decade.  

Zonal field structures (ZFS) correspond to radial corrugations of plasma equilibrium, and  are characterized by toroidally symmetric and usually poloidally symmetric structures in toroidal plasmas. ZFS are linearly stable to expansion free energy, and can be nonlinearly excited by microscopic drift wave (DW) turbulences including drift Alfvén waves (DAWs), and in this process, scatter DW/DAW  into linearly stable short radial wavelength regime. Spontaneous excitation of zero-frequency zonal structure (ZFZS) by toroidal Alfvén eigenmode (TAE) <cit.> was initially investigated in Ref. <cit.> using modulational instability methodology, where the contribution of zonal current (ZC) and zonal flow (ZF) were accounted for on the same footing. It was found that, for typical plasma parameters, ZF generation is possible as the pure Alfvénic state is broken, i.e., the nonlinear Reynolds and Maxwell stresses don't exactly cancel each other,  by toroidicity; and ZC generation is preferred with a much lower threshold condition, and can be estimated by |δ B_r/B_0|∼ O(10^-4), comparable to typical electromagnetic fluctuation level in present day tokamak experiments <cit.>.   Here, δ B_r is the perturbed magnetic field associated with the primary TAE, and B_0 is the equilibrium on-axis toroidal magnetic field. The  ZFZS growth rate is approximately  proportional to the amplitude of the pump TAE as the nonlinear drive overcomes the threshold due to frequency mismatch.

It was further found that, in the presence of resonant EP drive to TAE, ZF can be  excited by TAE even when TAE amplitude is small, with the ZF growth rate being approximately  twice of instaneous TAE growth rate <cit.>. This thresholdless   “forced driven" (also called “passive excitation" in some literatures)   process was shown to be dominated  by the contribution of EPs to the curvature coupling term, and was  observed by  simulations of Alfvén instability nonlinear dynamics using both hybrid code <cit.> and particle-in-cell codes <cit.>. A unified theory containing both the “forced driven" process and the spontaneous excitation was presented in Ref. , which shows that these two processes occur as the corresponding nonlinear contribution of EPs  or thermal plasmas take over the nonlinear coupling. It is also found that, the forced-driven process can also occur as plasma nonuniformity associated with diamagnetic drift is accounted for, with the generated ZF amplitude being proportional to the local diamagnetic frequency <cit.>.  In the following discussion,  we will term the “forced driven" as “beat driven" based on recent advances of understanding <cit.>. 

The above analysis and the obtained understanding, using TAE as a paradigm, can be applied to other SAW instabilities, with proper understanding of their respective properties.  For example, it is shown in Ref. <cit.> that, ZFZS can be excited by beta-induced Alfvén eigenmode (BAE) <cit.>, with ZF dominating due to vanishing k_∥ B. Here, k_∥ B is the parallel wavenumber of BAE. The theory of Ref. <cit.> may represent a series of low frequency Alfvén modes (LFAMs) <cit.> that can be excited by both EPs and thermal plasmas in different spatiotemporal scales, with plasma nonuniformity expected to play important roles with their frequencies comparable to diamagnetic frequency ω_* due to plasma nonuniformity.    In this work, we will extend the theory of ZFZS generation to nonuniform plasma, motivated by the recent discovery that, system nonuniformity can both qualitatively and quantitatively affect the parametric decay of kinetic Alfvén waves (KAW) due to the diamagnetic effects <cit.>.   Here, for nonuniformity, we mean the plasma profile associated with ω_*, as the previous analysis   labelled “uniform", is intrinsically nonuniform due to, e.g., toroidicity of magnetic geometry.  The following analysis,  using general WKB representation, can be applied to DAWs of a  broad frequency range, from TAE down to  kinetic ballooning mode (KBM)  frequency range   <cit.>, and is expected to be crucial for   KBM with 
 ω∼ω_* and higher toroidal mode numbers with k_⊥ρ_i≲ O(1).  Here, ω is the mode frequency,  ω_* is the characteristic ion diamagnetic frequency due to profile nonuniformity, k_⊥ is the perpendicular wavenumber, and ρ_i is the ion Larmor radius. The present work, is thus, of particular importance to future reactor scale tokamaks with thermal to magnetic pressure ratio significantly higher than present day machines, and KBM  is expected to be crucial for thermal plasma transport, while there is still ongoing debate on effects of ZFS in saturating KBM <cit.>.  The analysis, however,  is  general, and can be applied to other DAW instabilities, e.g., TAE, though the effects of system nonuniformity  are expected to be less important for TAE, in that |ω_*/ω| ≪ 1 for typical scenarios.

The rest of the manuscript is organized as follows. In Sec. <ref>, the theoretical model and governing equations are introduced, which are then used in Sec. <ref> to derive the modulational dispersion relation for ZFZS generation by DAWs. The obtained nonlinear dispersion relation, is applied to study the effects of plasma nonuniformity on ZFZS generation in Sec. <ref>. Finally, summary and discussion are presented in Sec. <ref>. 






§ THEORETICAL MODEL



For the nonlinear excitation of ZFZS by DAW instabilities in nonuniform plasmas, the analysis follows closely that of Ref. <cit.>, using the methodology of modulational instability. The ballooning representation  is adopted for the pump DAW and its lower/upper sidebands due to ZFZS modulation:

    δϕ_0    =    A_0 e^i(n_0ϕ-m̂_0θ-ω_0 t)∑_j e^-ijθΦ_0(x-j)+c.c.,
    δϕ_±   =    A_± e^± i(n_0ϕ-m̂_0θ-ω_0 t) e^i(k_Z r-ω_Zt)
          ×∑_j e^∓ ijθΦ_0(x-j)+c.c.,

while ZFZS   potential can be taken as

    δϕ_Z=A_Z e^i(k_Z r-ω_Z t).

In the above expressions, A is the mode envelope amplitude, n is the toroidal mode number, m̂ is the reference poloidal mode number, m=m̂+j is the poloidal mode number, Φ_0(x-j) is the parallel mode structure, subscripts “0", “±" and “Z" represent pump DAW, its upper/lower sidebands and ZFZS,  and frequency/wave number matching conditions for nonlinear mode coupling are applied.

The analysis follows closely the standard modulational instability approach, where the nonlinear ZFZS and DAW sidebands equations are derived, which then couple and yield the nonlinear modulational instability dispersion relation for ZFZS excitation. While “modulational instability” is used, the process can go beyond spontaneous excitation by modulational instability, and beat-driven process can also be included in the general equations <cit.>. The perturbed distribution function, δ f_s with s=e,i, obeys the nonlinear gyrokinetic equation <cit.>:

    δ f_s=-(e/T)_sδϕ F_0s +exp(-ρ_s·∇)δ H_s,

with the nonadiabatic particle response δ H_s derived from nonlinear gyrokinetic equation <cit.>

    (∂_t +  v_∥∂_l  +𝐯_d·∇)δ H_s   =   -i (e/T)_s (ω-ω^t_*) F_0s  J_kδ L_k
          - Λ^k'_k”J_k'δ L_k'δ H_k”.

Here,  l is the arc length along the equilibrium magnetic field line,  𝐯_d≡𝐛̂×[(v^2_⊥/2)∇ln B_0 + v^2_∥𝐛̂·∇𝐛̂] is the magnetic drift velocity, 𝐛̂≡𝐁_0/B_0 is  the unit vector along the equilibrium magnetic field line, J_k≡ J_0(k_⊥ρ_i) is Bessel function of zero index accounting for finite Larmor radius (FLR) effects, δ L_k≡δϕ-(k_∥v_∥/ω)δψ with δψ≡ωδ A_∥/(ck_∥) and  δψ=δϕ corresponding to vanishing parallel electric field δ E_∥ in the ideal MHD condition, ω^t_*=ω_*[1+η (v^2/v^2_th-3/2)] is the diamagnetic drift frequency associated with plasma nonuniformity with ω_*=-ck_θT/(q B L_n), η=L_n/L_T,  L_n and L_T are the scale lengths of density and temperature nonuniformity, respectively, and   Λ^k'_k”≡ (c/B_0)𝐛̂·𝐤”×𝐤'  denotes the perpendicular nonlinearity with   the matching condition 𝐤=𝐤'+𝐤”.

The governing field equations, in the β≪1 limit with magnetic compression being negligible,  can be derived from  the quasi-neutrality condition

    n_0e^2/T_i(1+T_i/T_e)=∑_s=e,i⟨ e_s J_kδ H_k⟩,

and nonlinear gyrokinetic vorticity equation <cit.>

    c^2/4πω^2_kB∂/∂ lk^2_⊥/B∂/∂ lδψ_k +n_0e^2/T_i(1-ω_*i/ω)(1-Γ_0)δϕ_k
          -e^2/T_iω_*i/ωη_i⟨ F_0 J^2_0 (v^2/v^2_th-3/2)⟩δϕ_k
          -∑_s=e,i⟨q/ω_kJ_kω_dδ H_k ⟩_s
       =   -i/ω_kΛ^k'_k” [ c^2/4πk”^2_⊥∂_lδψ_k'∂_lδψ_k”/ω_k'ω_k”.
          .+ ⟨ e(J_kJ_k'-J_k”)δ L_k'δ H_k”⟩].

The terms on the left hand side of equation (<ref>) are the field line bending, inertial, and curvature coupling terms, with the SAW dispersion relation being straightforwardly obtained from the balance of the former two, while the curvature coupling term plays crucial role in the low frequency SAW spectrum including KBM  <cit.>. The terms on the right hand side correspond to nonlinear Maxwell stress (MX)  and gyrokinetic Reynolds stress (RS), respectively.
In the following analysis, electron force balance equation, i.e., nonlinear Ohm's law, is sometimes used alternatively to simplify the analysis.




§ GENERAL EQUATION FOR ZFZS GENERATION



The zonal current equation can be derived from the parallel component of the electron force balance equation,

    δ E_∥+𝐛̂·δ 𝐮_⊥×δ 𝐁_⊥=0,

describing ZC generation due to dynamo effects.  Here, noting δ 𝐮_⊥≡∇δϕ×𝐛̂/B_0 and δ 𝐁_⊥=∇×δ A_∥𝐛̂, and that for ZFZS with n/m=0/0 and thus δ E_∥ Z=-∂_tδ A_∥ Z, one has, for ZC generation due to pump DAW and its upper/lower sidebands coupling,

    iω_Zδ A_∥ Z   =   c/B_0k_Zk_θ 0k_∥ 0[(σ_* -/ω_0-ω_Z-σ_* 0/ω_0)δϕ_0δϕ_- .
          .-(σ_* +/ω_0+ω_Z-σ_* 0/ω_0)δϕ_0^*δϕ_+ ],

with the expression of σ_*k=δψ_k/δϕ_k given by

    σ_*k≡1+τ-τΓ_k +τ⟨ (F_0/n_0)J^2_0(ω^t_*i/ω)⟩_k/(1-ω_*e/ω)_k,

and  again, σ_*k=1 corresponding to ideal MHD condition.

Noting σ_*±=σ_*0(ω_0±ω_Z,k_Z), one has

    σ_*±/ω_0±ω_Z-σ_*0/ω_0=±∂/∂ω_0(σ_*0/ω_0)ω_Z + 1/2∂^2/∂ k^2_r(σ_*0/ω_0) k^2_Z.

Defining δ A_∥ Z=ck_∥ 0δψ_Z/ω_0 using the frequency and parallel wavenumber of the pump DAW, one has

    δψ_Z   =   i/Bk_Zk_θ 0ω_0 [∂/∂ω_0(σ_*0/ω_0) (δϕ_-δϕ_0 + δϕ_+δϕ_0^*).
          . - 1/2ω_Z∂^2/∂ k^2_r(σ_*0/ω_0) k^2_Z (δϕ_-δϕ_0 - δϕ_+δϕ_0^*)  ].

The first term on the right hand side  of equation (<ref>) comes from the frequency difference between the DAW sideband to that of the pump, and corresponds to the “beat-driven" of ZC that has been extensively studied <cit.>. The second term, on the other hand, comes from plasma nonuniformity and kinetic effects, will contribute to “spontaneous excitation" of ZC, and is not accounted for in previous publications, focusing on TAE frequency where   ω_* effects can be neglected due to the |ω_*/ω|≪1 ordering typical of TAE <cit.>. 



The ZF equation can be derived from the nonlinear vorticity equation, by substituting the particle responses to DAWs  into RS, and one obtains:

    χ_iZδϕ_Z = - ick_Zk_θ0/B_0 ω_Z(α_-δϕ_0δϕ_–α_+δϕ_0^*δϕ_+),

with

    χ_iZ≡ 1-Γ_Z  -T_i/n_0e^2δϕ_Z∑_s=e,i⟨q/ωJ_Zω_dδ H_Z⟩

being the neoclassical inertial enhancement dominated by trapped particle contribution  <cit.>, and

    α_±=      ⟨ (J_ZJ_±J_0-J^2_±)F_0/n_0(1-ω^t_*i/ω)_±⟩
       -   ⟨ (J_ZJ_±J_0-J^2_0)F_0/n_0(1-ω^t_*i/ω)_0⟩
       -   c^2/4π (k^2_⊥±-k^2_⊥0)k_∥±k_∥0σ_*0σ_*±/ω_0ω_±.

 In deriving equation (<ref>),  the linear particle response to DAW   has been used <cit.>, i.e., 

    δ H^L_Ai   =   e/T_i(1-ω^t_*i/ω)_A J_A F_Miδϕ_A,
    δ H^L_Ae   =   -e/T_e(1-ω^t_*e/ω)_A F_Meδψ_A.

Here, the superscript “L" denotes linear response, and the subscript “A" denotes DAW, which can be used for both the pump DAW and its lower/upper sidebands. 

In the long wavelength |k_⊥ρ_i|≪1 and uniform plasma |ω^t_*i/ω|≪1 limit, equation (<ref>) recovers the results of Ref. , with α_± denoting  MX and RS  competition, and χ_iZ≃ 1.6 b_Z q^2/√(ϵ) with b_Z≡ k^2_Zρ^2_i.

 


The governing equations describing the DAW upper sideband generation due to pump DAW and ZFZS coupling, can be derived from the quasi-neutrality condition and the nonlinear vorticity equation. The governing equations for the DAW  lower sideband generation, can be derived similarly. 
The nonlinear electron response to δϕ_+, can be derived from the nonlinear gyrokinetic equation. Noting the  k_∥+v_e≫ω_+ ordering, one has,

    δ H^NL_e +≃  i  ck_Zk_θ0/B_0ω_0e/T_e F_0δψ_0[δϕ_Z -(1- ω^t_*e/ω)_0   δψ_Z  ],

with the superscript “NL" denoting nonlinear particle response. 
The nonlinear ion response to δϕ_+, on the other hand,  can be derived noting the ω∼ω_*≫ k_∥v_i, ω_d ordering, and one obtains

    δ H^NL_i +≃ -ick_Zk_θ 0/B_0ω_+  J_0J_Ze/T F_0 (ω^t_*i/ω)_0 δϕ_0δϕ_Z.

Substituting the derived particle responses  into the quasi-neutrality condition, one obtains

    δψ_+    =   σ_*+δϕ_+
       +    i(c/B_0)k_Zk_θ 0/ω_0(1-ω_*e/ω)_+δϕ_0 [ -σ_*0( 1- ω_*e/ω)_0 δψ_Z.
       +   .(σ_*0+τ⟨F_0/n_0 J_0J_ZJ_+(ω^t_*i/ω)_0⟩)δϕ_Z  ].





The other equation can be derived from vorticity equation. Substituting the linear ion  response  to ZFZS and pump DAW  into RS,  we have

    b_+[-k^2_∥ +V^2_A/ω^2_+δψ_+   + 1-Γ_+/b_+(1-ω_*i/ω)_+ δϕ_+. 
       -   .η_iω_*i/ω_+ b_+⟨F_0/n_0 J^2_+(v^2/v^2_th-3/2)⟩δϕ_+ 
    +T_i/e b_+⟨F_0/n_0 J_+ω_d/ω_+δ H^L_+i⟩] 
       ≃   -ic/B_0ω_+k_Zk_θ0{-(b_Z-b_0)k^2_∥0V^2_A/ω^2_0δϕ_0δψ_Z.
       +   .[ Γ_0-Γ_Z+⟨ (J_+J_ZJ_0 - J^2_0)(ω^t_*i/ω)_0F_0/n_0⟩]δϕ_Zδϕ_0}.


Substituting equation (<ref>) into (<ref>), one obtains, the equation describing δϕ_+ evolution due to δϕ_0 and δϕ_Z coupling

    b_+ϵ_A+δϕ_+  ≃ -ic k_Zk_θ0/B_0ω_+( β_+  δϕ_Z  + γ_+ δψ_Z   )δϕ_0,

with

    ϵ_A+≡  -k^2_∥ +V^2_A/ω^2_+σ_*+  +T_i/e b_+δϕ_+⟨F_0/n_0 J_+ω_d/ω_+δ H^L_+i⟩
           +  1-Γ_+/b_+(1-ω_*i/ω)_+ + η_iω_*i/ω_+ b_+⟨F_0/n_0 J^2_+(v^2/v^2_th-3/2)⟩

being the DAW  upper sideband dispersion relation in the WKB limit,

    β_+   =   Γ_0-Γ_Z+⟨ (J_+J_ZJ_0 - J^2_0)(ω^t_*i/ω)_0F_0/n_0⟩
       -   k^2_∥ +V^2_A b_+/ω_+(ω -ω_*e)_+( σ_*0+τ⟨ J_+J_0J_Z(ω^t_*i/ω)_0F_0/n_0⟩),

and

    γ_+≡   -[k^2_∥0V^2_A/ω^2_0(b_Z-b_0)-k^2_∥0V^2_A/ω^2_+b_+]σ_*0.

Here, the terms proportional to b_+  are from the parallel electric field through equation (<ref>), while the rest are  from the generalized gyrokinetic RS and MX.


The lower sideband equation can be derived similarly as

    b_- ϵ_A-δϕ_-   =   -ick_Zk_θ0/B_0|ω_-|(β_- δϕ_Z+γ_-δψ_Z)δϕ_0^*,

with β_- and γ_- defined similarly to β_+ and γ_+. 




§ MODULATIONAL DISPERSION RELATION FOR ZFZS GENERATION BY DAW



The nonlinear  dispersion relation for   ZFZS excitation by DAW, can be derived from equations (<ref>), (<ref>), (<ref>) and (<ref>).  Noting the major  difference between β_+ and β_-, and γ_+ and γ_-  are multiplied by a k^2_∥0 factor, which vanishes for low frequency range where effects associated with plasma nonuniformities could be important, we can safely take γ̂≡γ_+=γ_- and β̂≡β_+=β_- from now on.    Taking (<ref>)×β̂/χ_iZ + (<ref>))×γ̂, and substituting δϕ_± from equations (<ref>) and (<ref>), one obtains

    (c/B_0k_Zk_θ0)^2|δϕ_0|^2/b_+[γ̂∂/∂ω_0(σ_*0/ω_0)(1/ϵ_A-+1/ϵ_A+).
        +   . γ̂k^2_Z /2ω_Z∂^2/∂ k^2_r(σ_*0/ω_0) (1/ϵ_A+-1/ϵ_A-).
        +   . β̂/ω_Zω_0χ_iZ( α_+/ϵ_A+-α_-/ϵ_A-) ]=1.

Equation (<ref>) is the desired modulational dispersion relation for ZFZS generation by DAW, with the effects of system nonuniformity and kinetic effects such as finite orbit width and FLR   effects  systematically accounted for. It also includes the contribution of beat-driven and spontaneous excitation on the same footing, which is reflected in, e.g., equation (<ref>) for zonal current and equation (<ref>) for zonal flow  noting the dependence of “α_±" on ω_Z and k_Z.    Equation (<ref>) is general and needs numerical solution for  proper understanding of the ZFZS generation by DAW. It can, however, be analyzed in various limits, by using of the knowledge from previous works.



 §.§ Uniform plasma  limit with k^2_∥ 0V^2_A/ω^2_0≃ 1


We   start from the TAE frequency mode in the  uniform plasma (ω_*=0)    ideal MHD (σ_*0=1) limit.  Noting |ω_Z|≪ |ω_0| and k^2_∥0V^2_A≃ω^2_0 to the leading order,  one has α_+=α_-≃ b_Z (1-k^2_∥0V^2_A/ω^2_0), γ̂≃ 2b_0, and β̂≃ -2 b_0.  We further note ϵ_A±≡ (∂ϵ_A0/∂ω_0) (±ω_Z+Δ_T), with  Δ_T≡ (∂^2ϵ_A0/∂ k^2_r) k^2_Z/(2∂ϵ_A0/∂ω_0) being frequency mismatch and ω_Z=iγ_Z with γ_Z being the nonlinear growth rate of ZFZS, we then  obtain

    γ^2_Z=-Δ^2_T   +   (c/B_0k_Zk_θ0)^22 b_0 |A_0|^2/b_+ω_0(∂ϵ_A0/∂ω_0)
       ×   [Δ_T/ω_0 + 4(1-k^2_∥0V^2_A/ω^2_0)/χ̂_iZ].

Equation (<ref>) recovers the result of Ref. <cit.>,  describing ZFZS generation as the nonlinear drive overcomes the threshold due to frequency mismatch, with the two terms in the square brackets representing the contribution from ZC and ZF, respectively.  It can be seen that, the contribution of ZF is limited by two factors, i.e., neoclassical shielding (χ̂_iZ≡χ_iZ/b_Z≃ 1.6 q^2/√(ϵ)≫1) as well as RS & MX cancellation with 1-k^2_∥0V^2_A/ω^2_0∼ O(ϵ), rendering  ZC generation   preferred with much lower threshold for modes in the TAE frequency range with ω^2_0≃ k^2_∥0V^2_A if  Δ_T/ω_0>0.  On the other hand, if Δ_T/ω_0<0, ZFZS generation is still possible, given the pump TAE is excited in the upper half of the toroidicity induced SAW continuum gap, and the frequency mismatch is small enough. It, however, requires a relatively higher pump TAE amplitude to overcome the threshold due to frequency mismatch. The  physics picture is extensively discussed in Ref. <cit.> on ZFZS excitation by TAE, and interested readers may refer to the original publication for more details. 



 §.§ Nonuniform plasma with predominant  ZC generation


If we consider the general case with χ̂_iZ≫1  and ZC generation (proportional to γ̂ in equation (<ref>))    is dominant,  the modulational dispersion relation can be written as

    γ^2_Z   =    -Δ^2_T  +  2 (c/B_0k_Zk_θ0)^2 γ̂|A_0|^2/b_+(∂ϵ_A0/∂ω_0)
       ×   [∂/∂ω_0(σ_*0/ω_0)Δ_T -  1/2∂^2/∂ k^2_r(σ_*0/ω_0) k^2_Z  ].

Noting the expression of ϵ_A, one has,

    ∂^2/∂ k^2_r(σ_*0/ω_0)    ≃    -ω_0/k^2_∥0V^2_A∂^2ϵ_A0/∂ k^2_r, 
    ∂/∂ω_0(σ_*0/ω_0)    ≃   σ_*0/ω^2_0 -ω_0/k^2_∥0V^2_A∂ϵ_A0/∂ω_0.

On the other hand, γ̂≃ 2b_0σ_*0,   and we have

    γ^2_Z=-Δ^2_T    +    4 (c/B_0k_Zk_θ0)^2 b_0/b_+Δ_T σ^2_*0|A_0|^2/ω^2_0(∂ϵ_A0/∂ω_0),

i.e., compared to the uniform plasma case (neglecting contribution of ω_*) with predominant ZC generation, the nonlinear drive is quantitatively enhanced by σ^2_*0, while the qualitative picture is not altered.



 §.§ ZF generation by DAW with k^2_∥0V^2_A≪ω^2_0


An important parameter regime for plasma nonuniformity to play crucial role to SAW instability is the low frequency range with mode frequency comparable to  the diamagnetic  and/or ion transit frequency range <cit.>, where the modes are characterized by ω^2_0≫ k^2_∥0V^2_A. It is shown for the BAE case that, ZF generation dominates due to vanishing k^2_∥V^2_A/ω^2 such that the pure SAW state is broken due to  RS dominance over MX <cit.>.  To analyze this parameter regime, we take k_∥0=0 limit in equation (<ref>) and focus on the effects associated with plasma nonuniformity.  In this parameter regime, we have  γ̂=0, 

    β_±   =   Γ_0-Γ_Z+⟨F_0/n_0(J_±J_0J_Z-J^2_0)(ω^t_*i/ω)_0⟩,
    α_±   =   ⟨(J_ZJ_0J_±-J^2_±)F_0/n_0(1-ω^t_*i/ω)_±⟩
          -⟨(J_ZJ_0J_±-J^2_0)F_0/n_0(1-ω^t_*i/ω)_0⟩,

the modulational dispersion relation, equation (<ref>), then becomes:

    (c/B_0k_Zk_θ0)|A_0|^2β̂/b_+ω_Zω_0χ_iZ[α_+/ϵ_A+-α_-/ϵ_A-]=1.

Noting again, ϵ_±=(∂ϵ_A0/∂ω_0)(±ω_Z+Δ), and 

    α_++α_-   ≃   2ω_Z/ω_0⟨(J_ZJ_0J_Z-J^2_+)F_0/n_0ω^t_*i0/ω_0⟩,
    α_+-α_-   ≃   -2⟨(J^2_+-J^2_0)F_0/n_0(1-ω^t_*i/ω)_0⟩,

we then have

    γ^2_Z   =   -Δ^2 +(c/B_0k_Zk_θ)^2|A_0|^2β̂/b_+ω_0χ_iZ2/∂ϵ_A0/∂ω_0
       ×   [-iΔ_T/γ_Z⟨ (J_+J_ZJ_0-J^2_+)(ω^t_*i/ω)_0F_0/n_0⟩.
          . +⟨ (J^2_+-J^2_0) (1-ω^t_*i/ω)_0F_0/n_0⟩].







Equation (<ref>) is derived in the k_∥0=0 limit, with the two terms in the square bracket  corresponding to ZF generation by DAW self-beating and radial envelope modulation, and can be used for studying the condition for ZF generation by KBM. [It is noteworthy that, the  KBM dispersion relation and polarization  can be sensitive to plasma parameters including ω_* and η=L_n/L_T. KBMs may have finite k_∥  in certain parameter regimes, as carefully investigated in Ref. . In this sense, the investigation here can be more straightforwardly applied to BAEs.  However,  in this work  we will still take the |k_∥V_A/ω|≪1 limit for KBM, while leaving the more detailed analysis for a separate publication.]   It is worth noting that, in the uniform plasma limit, it will recover the results of ZF excitation by BAE <cit.>.  Equation (<ref>) can be re-written as

    γ^3_Z + Aγ_Z +iB=0,

with the coefficients given by

    A    ≡   Δ^2 - (c/B_0k_Zk_θ)^2|A_0|^2β̂/b_+ω_0χ_iZ2/∂ϵ_A0/∂ω_0
          ×⟨ (J^2_+-J^2_0) (1-ω^t_*i/ω)_0F_0/n_0⟩,
    
    B   ≡   (c/B_0k_Zk_θ)^2|A_0|^2β̂/b_+ω_0χ_iZ2/∂ϵ_A0/∂ω_0
          ×⟨ (J_+J_ZJ_0-J^2_+)(ω^t_*i/ω)_0F_0/n_0⟩.

The conditions for ZF excitation by KBM, can thus be derived, by noting the expression of the pump KBM dispersion relation ϵ_A0.   For the special case with  B=0, it is straightforward to see that ZF can   be driven unstable, as the nonlinear drive overcomes the threshold due to frequency mismatch. 
From the general  case with B≠0, the condition for equation (<ref>) to have a root with positive real part, can be determined from the    properties of cubic equations with one variable.  We, however, will only illustrate briefly the results from numerical solution of equation (<ref>) in Fig. <ref>. In Fig. 1(a), the dependence of the real part of γ_Z with the  biggest real part on parameters A and B is given, and the regions for ZFZS exponentially growing is shown by the white region of Fig. 1(b), while the black region corresponds to ZFZS marginally stable with (γ_Z)=0. It is worth noting that, the boundary separating  the white and black regions, is given by B^2/4=A^3/27, from the properties of cubic equations. 
The  detailed analysis, however, will need more careful investigation with realistic plasma parameters and global KBM dispersion relation, and is, beyond the scope of the present work to formulate the general dispersion relation for ZFZS generation by DAW. The detailed analysis of ZFZS generation by KBM, of particular interest for turbulence transport in reactor scale tokamaks with plasma to magnetic pressure ratio significantly higher than present day machines, will be reported in a future publication.  









§ SUMMARY AND DISCUSSION



In this work, general equations  for zero frequency zonal structure (ZFZS) nonlinear excitation by drift Alfvén waves (DAWs) are derived, with contribution  of plasma nonuniformity and kinetic effects accounted for on the same footing.  It is found that, the   finite coupling between DAWs to effectively generate ZFZS, may from the radial modulation as the DAWs having different radial wavenumber, and self-beating where no difference of radial wavenumber  is required, corresponding to spontaneous excitation  and beat-driven  <cit.>, respectively.  This can be clearly seen from equation (<ref>) for zonal current (ZC) generation, and equation (<ref>) for zonal flow generation.  

The obtained nonlinear dispersion relation can, thus, be applied to study ZFZS generation by DAWs covering a broad frequency range.   In the first application, it is shown that the general dispersion relation can recover that of ZFZS excitation by TAE as effects associated with  ω_* is neglected <cit.>. For the modes in the TAE frequency range and ZC generation dominant, it is shown that, plasma nonuniformity will quantitatively modify the ZFZS generation process, while the qualitative picture is not changed.  For DAWs in the KBM frequency range with frequency comparable to diamagnetic frequency, where effect  of plasma nonuniformity is expected to be crucial, it is found that, the contribution of self-beating and radial envelope modulation renders the final nonlinear dispersion relation into a cubic equation of ZF growth rate, which will yield parameter regions for ZF excitation and marginally stable.  The detailed analysis with realistic KBM dispersion relation, however, is beyond the scope of the present work to formulate the general dispersion relation for ZFZS generation by DAW instabilities, and will be reported in a future publication. 










§ ACKNOWLEDGEMENT

This work was  supported by the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDB0790000,  the National Science Foundation of China under Grant Nos. 12275236 and 12261131622, and  Italian Ministry for Foreign Affairs and International Cooperation Project under Grant  No. CN23GR02. The authors acknowledge   Dr. Fulvio Zonca (CNPS-ENEA and ZJU)  for fruitful discussions. 
 








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