Emergence of two inertial sub-ranges in solar wind turbulence: dependence on heliospheric distance and solar activity
Shiladittya Mondal Supratik Banerjee Luca Sorriso-Valvo



































































0009-0001-4841-1103]Shiladittya Mondal
Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

0000-0002-3746-0989]Supratik Banerjee
Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India


0000-0002-5981-7758]Luca Sorriso-Valvo
Institute for Plasma Science and Technology (ISTP), CNR, Bari, Italy
Space and Plasma Physics, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden
















§ ABSTRACT 

The solar wind is highly turbulent, and intermittency effects are observed for fluctuations within the inertial range. By analyzing magnetic field spectra and fourth-order moments, we perform a comparative study of intermittency in different types of solar wind measured during periods of solar minima and a maximum. 
Using eight fast solar wind intervals measured during solar minima between 0.3 au and 3.16 au, we found a clear signature of two inertial sub-ranges with f^-3/2 and f^-5/3 power laws in the magnetic power spectra. 
The intermittency, measured through the scaling law of the kurtosis of magnetic field fluctuations, further confirms the existence of two different power laws separated by a clear break. 
A systematic study on the evolution of the said sub-ranges as a function of heliospheric distance shows correlation of the break scale with both the turbulence outer scale and the typical ion scales. 
During solar maximum, we analyzed five intervals for each of Alfvénic fast, Alfvénic slow and non-Alfvénic slow solar wind. Unlike the case during the solar minima, the two sub-ranges are no longer prominent and 
the Alfvénic slow wind is found to be in an intermediate state of turbulence compared to that of the fast wind and the usual non-Alfvénic slow wind.




























§ INTRODUCTION


The solar wind is the most accessible natural laboratory for studying turbulence in space plasmas <cit.>. 
The dynamic solar activity and the diversity of the originating regions produce solar wind with a variety of characteristics, the most evident being the plasma speed. 
While the fast solar wind (FSW, >550  km  s^-1) mainly emanates from the polar coronal holes, the slow solar wind (SSW, <400  km  s^-1) is believed to be originated from equatorial streamers <cit.>. 
During high solar activity, however, both FSW and SSW are distributed at all latitudes instead of being confined exclusively to polar and equatorial regions, respectively. 
Another interesting feature of FSW is the high Alfvénicity i.e. high correlation (or anti-correlation) between velocity fluctuations () and  (=𝐁/√(μ_0ρ), where B the magnetic field fluctuation and ρ the mass density) in contrast with the SSW comprising of weak v-b correlations. 
This one-to-one correspondence, however, does not strictly hold during high solar activity, as a third type of wind is also observed. This wind, termed as Alfvénic slow solar wind (ASSW), has speed similar to that of the slow wind but is surprisingly permeated with high Alfvénicity <cit.>. 
The degree of Alfvénicity influences the nature of turbulence in different types of solar winds. 
A high degree of Alfvénicity represents an imbalance between the Elsässer variables, 𝐳^± = ±, thus leading to less developed turbulence whereas low Alfvénicity corresponds to comparatively more developed turbulence owing to the balance between them. 
At scales greater than the ion-inertial length (d_i), a longer k^-5/3 energy power spectrum is therefore observed for the slow wind whereas a comparatively shorter k^-5/3 spectrum is observed in the Alfvénic fast wind <cit.>.

These observed spectra are universal in solar wind turbulence and are consistent with self-similar energy cascade (Kolmogorov phenomenology) within the inertial range. In physical space, universal energy cascade is obtained in terms of the linear scaling law for the third-order moments of velocity and magnetic field fluctuations <cit.>. In order to assure a self-similar cascade, the kurtosis K (the normalised fourth-order moment of the fluctuations) should be scale invariant. For a turbulent flow, one such possibility is the case of quasi-Gaussian PDFs where the third-order moment (skewness) is non-zero but the K is roughly equal to that of a Gaussian distribution.  

However, careful studies in turbulent fluids and plasmas consistently show a departure from self-similarity as one moves towards the smaller length scales within the inertial range. 
This departure, known as inertial-range intermittency, is characterised by the large tails of the PDFs at those scales. 
In particular, intermittency effects are quantified by the deviation from the self-similar scaling laws of the higher-order moments <cit.>. 
Instead of using arbitrary higher-order moments, the kurtosis is often used as a practical measure of intermittency and a higher probability of extreme events leads to its increase with decreasing length scale ℓ. 
From a physical point of view, this implies that the small-scale coherent structures, such as the vortices, current sheets, etc., generated due to nonlinear interactions, do not fill the available space in a self-similar way nor are randomly distributed, but rather tend to form inhomogeneously distributed clusters of bursts <cit.>.

While the solar wind expands and accelerates through the heliosphere, the turbulence becomes more developed, with the fluctuations being majorly energized by the nonlinear interactions between the oppositely propagating Alfvén waves <cit.>, switchbacks <cit.>, large-scale structures, and instabilities <cit.>. 
Studies based on spacecraft observations have shown a variation in spectral indices of the magnetic and velocity power spectra <cit.>, a decrease in - correlations and a broadening of the inertial range <cit.>. 
Recently, using high resolution in-situ data of the Parker Solar Probe it has been suggested that the magnetic spectral index evolves from -3/2 near the Sun (as close as 0.17 au) to a more developed -5/3 at 1 au <cit.>. 
These observations are consistent with the idea of radial evolution of solar wind turbulence into more developed states and the non-adiabatic heating of the medium with increasing heliospheric distance <cit.>. 

In addition, a power law behaviour for the kurtosis <cit.> and an increase in intermittency in solar wind turbulence have been observed at greater heliospheric distances <cit.>. 
Using the magnetic data of Helios 2, <cit.> observed a break in the scaling of K of the magnetic field fluctuations in FSW, during solar minimum. They provided a plausible explanation suggesting this observed break to be associated with the f^-1 break in the magnetic power spectrum of FSW. However, clear disparity is observed between the scales corresponding to the breaks in kurtosis scaling and the power spectra. 

A break in both the spectral density and the higher-order structure functions has also been observed <cit.>.
However, the nature of such break and its implications on the dynamics of the solar wind turbulence have not been investigated in detail yet.

In this paper, we revisit the aforementioned problem and carry out a systematic study to provide an explanation for the break observed in the kurtosis scaling. 
Using the in-situ data of Helios and Ulysses during solar minima, we show the kurtosis break is primarily associated with an observed break between two inertial sub-regimes of magnetic power spectra, having -3/2 and -5/3 spectral indices, respectively. 
In addition, we also study the radial evolution of the break scale to characterise the solar wind turbulence as a function of the heliospheric distance. 
During a solar maximum, however, breaks are not prominent in the scaling of K. Nevertheless, a comparative study of FSW, ASSW and SSW shows a clear distinction of these three types of wind according to the degree of turbulence and the degree of intermittency, along with some insights on the origin of ASSW. 
In Sections <ref> and <ref>, we briefly describe the data and methodologies used for the analysis. Section <ref> provides the results obtained in our study during solar minima (<ref>) and maxima (<ref>), respectively. Finally, in Section <ref>, we summarize our findings and conclude.




§ DATA SELECTION













For our analysis, we have used in-situ data from the Helios and Ulysses spacecraft data repository publicly available at NASA CDAWeb (https://cdaweb.gsfc.nasa.gov/https://cdaweb.gsfc.nasa.gov) and AMDA science analysis system (https://amda.irap.omp.eu/https://amda.irap.omp.eu).
The plasma data for Helios and Ulysses have been obtained from the E1 Plasma Experiment instrument and the Solar Wind Observations Over the Poles of the Sun (SWOOPS) instrument, respectively. For magnetic power spectrum and kurtosis scaling, we use 6 s resolution magnetic-field data from the E3 Flux-gate Magnetometer (FGM) onboard Helios and 1 s resolution magnetic field data from the Vector Helium Magnetometer (VHM) onboard Ulysses spacecraft. 
During a declining phase of solar activity near a solar minimum between 1975 and 1976, Helios 1 and 2 recorded several streams of FSW from a coronal hole (or the same source), which sustained through nearly two solar rotations <cit.>. 
Several intervals of the fast wind expelled from this coronal hole were also identified by <cit.>. 
In particular, for our current analysis, we use the streams - A3, A6, A7, and A8, ranging from 0.3 au to 1 au, mentioned therein.
Each chosen interval (i) contains negligibly small amount of data gaps, (ii) is free of any considerable mean trend, and (iii) turns up to be reasonably stationary. 
The stationarity is assured by the approximate constant average of sub-intervals of different lengths. Extending our analysis beyond 1 au, we use four intervals of FSW at varying heliospheric distances (F1 - F4 as listed in Table - <ref>), recorded by Ulysses during the years 1995-1996. Typical features of certain FSW intervals in the inner and outer heliosphere used in our analysis with high v-b correlations are shown in Fig. <ref>.

In order to interpret our findings, we also need to compute the co-spectra of cross-helicty σ_c (see Section <ref>), for which we have used the 40.5 s resolution magnetic field and proton velocity data from the E3 FGM and the E1 Plasma Experiment instrument onboard Helios. We use degraded resolution for the magnetic field data in order to keep coherence with the available plasma data from the data repository. 
A similar analysis cannot be done using the plasma data of Ulysses where the data resolution is 240 s, and hence cannot be used to capture the required length scales of our interest. 

During solar maximum, five Ulysses intervals each for the three types of solar wind were selected following similar methods prescribed in <cit.> based on their speed, proton density, and Alfvénic correlations (see Table <ref>). 
A particular case study represents several properties of the different types of wind within a 20-day interval (see Fig. <ref>). While ASSW looks very similar to SSW with respect to the flow speed (< 400 km/sec), it is characterised by low proton density (∼ 1 particle/cm^3) and high Alfvénicity (∼ 0.6) similar to FSW. 
These findings are in agreement with previous studies <cit.>.




§ ANALYSIS METHOD



Our analysis is mainly based on the computation of (i) the kurtosis (K) or the normalized fourth-order moments of magnetic field fluctuations, (ii) the magnetic power spectral density (PSD), and (iii) the cross-helicity co-spectra (σ̂_̂ĉ). All the data sets were made evenly sampled by interpolating the data gaps before using for any of the computations. 

Since all the intervals used in our study are permeated by superalfvénic solar wind, one can practically use Taylor's hypothesis, which means if the turbulent fluctuations are much smaller than the bulk speed, they can be considered as frozen (or slowly evolving) as the flow sweeps the probe <cit.>. 
When using single-point measurements in the form of a time series, the only accessible direction for the increments is along the bulk flow.
This provides an equivalence between the longitudinal (along the flow) length scale ℓ and the corresponding time scale τ as ℓ = V_swτ, where V_sw is the mean solar wind speed.
Therefore, we define the increments of the i^th component (with i=r,t,n) of the magnetic field as Δ b_i(t,τ)=b_i(t+τ)-b_i(t).
In order to capture both magnitudinal and directional fluctuations of , we define the n^th order structure function as:


    S_n(τ) = ⟨[ ∑_i (Δ b_i )^2 ]^n/2⟩,

where ⟨·⟩ represents the ensemble average <cit.>.
The corresponding kurtosis (K) is then calculated using the standard expression:


    K(τ) = S_4(τ)/[S_2(τ)]^2.


Note that, when each Δ b_i follows a Gaussian distribution with zero mean, K (τ) is equal to 5/3 (see appendix Section <ref>). For a self-similar, non intermittent flow, in the inertial range of scales (namely much smaller than the energy-injection scales and larger than the dissipative scales) the n^th order structure function is expected to scale as S_n(τ) ∝τ^np, where p is a phenomenological constant <cit.>. 
It is therefore straightforward to see that K becomes independent of τ. 
However, in the presence of intermittency, this linear scaling does not hold any longer and the simplest intermittency model can be given as S_n(τ) ∝τ^np + q(n), where q(n) is a nonlinear correction accounting for the intermittent structures. 
For the kurtosis, this leads to a power-law scaling K(τ) ∼τ^-κ, with κ=q(4)/2q(2). 
Such a scaling, universally observed in fluid turbulence, has recently been described in the case of solar wind turbulence as well <cit.>. 
In this work, we study the scaling properties of K of the magnetic field fluctuations at different heliospheric distances.

Finally, the magnetic energy spectra and cross-helicity co-spectra are defined by, PSD = b̂_̂î^†b̂_̂î and σ̂_̂ĉ = (b̂_̂î^†v̂_̂î + v̂_̂î^†b̂_̂î)/2 respectively, where b̂_̂î and v̂_̂î are the Fast Fourier transforms (FFT) of the magnetic and velocity field components b_i and v_i, respectively, with summation being intended over the repeated indices (where i=r,t,n).







§ RESULTS AND DISCUSSIONS





 §.§ Observations during Solar Minimum








During a period of solar minimum in 1976, using data from Helios spacecraft, we study FSW streams in the inner heliosphere (at 0.3, 0.41, 0.65, and 0.98 au) from a sustained coronal hole near the ecliptic plane. 
Beyond 1 au, FSW streams are studied using Ulysses data collected during the 1995-1996 solar minimum, at varying heliospheric distances (at 1.44, 2.1, 2.75, and 3.16 au), which were also measured at different latitudes.



In Fig. <ref>, we have drawn the magnetic power spectral traces, estimated using nearly equispaced frequencies in logarithmic scales. 
Top panels refer to Helios intervals, while bottom panels to Ulysses. 
As typically observed in the Alfvénic solar wind, at low frequencies we can identify a large-scale, energy-containing range (white background in the figure), where the power decays as ∼ f^-1. Fitted power laws and the corresponding scaling exponents are shown as green lines. 
A break identifies a clear change in the power-law scaling exponent, as indicated by vertical dashed lines. 
Such break can be associated with the correlation scale of the turbulence. 
The low-frequency range is clearly visible in Helios data, while it is only indicatively present in the Ulysses intervals. This is consistent with the well-known shift of the correlation scale towards lower frequency with increasing R in the solar wind <cit.>.
The f^-1 range is followed by the usual inertial range of turbulence, where the spectrum roughly follows an f^-5/3 power law dependence <cit.>.
However, a more accurate inspection shows that a further break emerges within such range, indicated by the vertical dot-dashed lines separating the light and deeper blue shaded areas in Fig. <ref>. 
Although the dynamical range of frequencies is relatively short, for intervals other than that at 3.16 au, it is possible to identify two different sub-ranges with different power laws as demonstrated by the red and blue lines, with the associated scaling exponents indicated nearby. 
In the lower-frequency range (light blue background), the spectral index approaches -3/2, whereas at larger frequencies (deep blue background), the spectra show a transition to a -5/3 spectral index usually observed in non-Alfvénic solar wind <cit.>.

In isotropic turbulence, whereas an f^-5/3 scaling often represents an energy cascade by eddy fragmentation in strong turbulence, f^-3/2 scaling can possibly be explained by an energy cascade through the sporadic interaction of Alfvénic wave packets in MHD turbulence <cit.>.
However, -5/3 and -3/2 power laws can also be obtained under various circumstances if anisotropy is taken into account <cit.>.
Irrespective of the true nature of energy cascade, a single power law is often assumed for the magnetic power spectra in the frequency range 10^-4 - 10^-1 Hz <cit.>, 
although a few studies have found variation in the power law exponents in the inertial range of magnetic power spectra <cit.> as well as the scaling of higher order structure functions <cit.>.
In our study, the co-existence of the two sub-regimes (with -3/2 and -5/3 spectral indices) within the turbulence spectra of FSW has been consistently observed at various heliospheric distances both in the inner as well as the outer heliosphere.
The break scale between those two sub-ranges, f_b, appears to shift towards lower frequencies (approaching the correlation scale) with increasing heliospheric distance. 
This is consistent with the fact that a -3/2 scaling has been observed for solar wind close to the sun, whereas a steeper -5/3 power law is obtained at and beyond 1 au <cit.>.
Finally, in the Ulysses intervals, the ion-scale breaks are visible, separating the MHD range from the sub-ion range, where Hall effects and other kinetic effects start to affect the cascade (white background) <cit.>. 
Such break is usually observed at frequencies ∼ 10^-1 Hz, which is the upper cut-off for the MHD range.
However, similar breaks do not turn up in the Helios intervals, due to the low cadence of the data used here.




To further investigate on the sub-inertial range spectral break, f_b, we study the kurtosis K(τ) for all of the eight FSW intervals.
The scaling of K(τ) defined in Section <ref> for Helios and Ulysses data are depicted in Fig. <ref> top and bottom panels, respectively, for each R.

To inspect on the general radial trend of intermittency we have drawn a consolidated plot for the Helios and Ulysses intervals (see Fig. <ref>). From this figure one can conclude that the value of K at all scales increases with increasing R, thus implying higher intermittency with increasing heliospheric distance, in agreement with previous studies <cit.>.
At each given distance R, K is systematically found to decrease as one moves towards the larger scales. This is consistent with the notion that deviation from Gaussian statistics increases at smaller scales 
<cit.>.
Upon reaching the typical correlation scales of the flow (τ≃ 10^4 s), corresponding to the f^-1 power law in energy spectrum (see Fig. <ref>), the kurtosis saturates to a constant value K≃ 1.67, representing a quasi-Gaussian distribution (with a non-zero skewness) of the fluctuations of the magnetic field components (see appendix Section <ref>).
Within the inertial range, from the nature of K(τ) in Fig. <ref>, a clear signature of broken power law is observed. 
While two breaks are visible for Ulysses data (with 1 s resolution), the small-scale break at around τ∼ 10 s is missing for the intervals using Helios magnetic field data with 6 s resolution. 
This break, corresponding to a frequency of ∼ 10^-1 Hz, is associated with the transition from the ordinary MHD range to the sub-ion kinetic or Hall MHD regime <cit.>. 
The other break which occurs at a larger τ (solid vertical lines) is clearly visible for both Helios and Ulysses data. 
In particular this break scale (τ_K) shifts towards larger τ as R increases.
Within the distance range of 0.3 - 2.75 au, τ_K is found to increase from ∼ 100 s to ∼ 1500 s. 
It is to be emphasized here that except for certain cases, the appearance of the break τ_K is persistent in the component-wise K scaling as well (see Figs. <ref> and <ref> in appendix Section <ref>).
A detailed list of the break scale τ_K as a function of R is given in Table <ref>. 
As it is evident from Fig. <ref> and <ref>, τ_K separates the steeper power law (K∼τ^-κ with κ≃ 0.37 averaged over the eight intervals) at smaller scales (dashed lines) from the less steeper one (κ≃ 0.11 on average) at large scales (dotted lines), but with an exception.
Note that for the Ulysses interval at R=3.16 au, K(τ) reaches the Gaussian regime without going through the large τ break, suggesting that the turbulence has fully developed that transforms the shallower scaling range at large scale into the steeper power law at smaller scales. 
We will elucidate this point in the following.









As mentioned in the introduction, similar broken power law behaviour for K(τ) in FSW has already been observed by <cit.>. 
However, those authors suggested that τ_K might correspond to the break between low-frequency f^-1 regime to Kolmogorov f^-5/3 regime in the magnetic power spectra. 
This was inspired by the fact that f^-1 regime is exclusively found in FSW intervals and the f^-1 break also shows nearly similar behaviour to 1/τ_K as R changes <cit.>. 
Instead, for all the intervals where the break is observed, it is systematically found in our study that 1/τ_K occurs at a higher frequency (roughly by a factor ∼ 10) than the f^-1 break scale (see Fig. <ref>). 
The inverse of τ_K is typically corresponding to f_b, although with some consistent small discrepancy that could be due to the different frequency response of Fourier transform and scale-dependent increments (see Fig. <ref> where both τ_K, solid lines, and 1/f_b, dashed lines, are drawn). 

The two scaling ranges in the kurtosis therefore approximately correspond to the two inertial sub-ranges observed in the spectrum. 
Since PSD and kurtosis are related quantities, the observation of a double power law in both supports the robustness of the break, and therefore indicates the emergence of a new characteristic scale in the inertial range that marks the transition from f^-3/2 to f^-5/3 regime.











Summarizing, from the existence of the two turbulent inertial sub-regimes it is clear that as we move from the larger towards the smaller scales the nature of turbulence also varies.
This variation becomes more apparent when we examine the cross-helicity spectrum for the FSW intervals within the inner heliosphere (Fig. <ref> top). 
The same could not be computed for the FSW beyond 1 au due to the limitation in terms of low plasma data resolution, as mentioned in Section <ref>. 
Nevertheless, for all the FSW intervals in the inner heliosphere we see that the σ̂_̂ĉ power decreases as we move from larger to smaller scales (see Fig. <ref>). Thus, with the forward progression of the turbulent cascade, the imbalance between the inward and outward Alfvén modes propagating along the mean magnetic field decreases to a more balanced state. 
While recent studies have shown the transition from a weak to a strong turbulence regime on moving towards smaller scales <cit.>, a transition from imbalanced (|z^+2|≫ |z^-2|, or vice-versa) to a balanced (|z^+2|∼|z^-2|) turbulent state could as well be associated with the steepening of the spectra from the low frequency f^-3/2 regime to the higher frequency f^-5/3 regime.
A similar gradual change from an imbalanced towards a relatively balanced state is also evident with increasing heliospheric distance R. Even though σ_c shows sufficiently higher values being associated with FSW, it declines slowly as understood from the straight line fit having a slope α=-0.05 (see Fig. <ref> bottom). 
This is again consistent with the absence of the f^-3/2 regime at R=3.16 au and recent observations of change in the inertial range spectral index from -3/2 to -5/3 with increasing R <cit.>.












We further determine the evolutionary nature of the break scale, τ_K, with R and have investigated its relationship with the typical ion and correlation scales. 

In Fig. <ref> (top), we show the radial evolution of τ_K, appearing in the scaling of K, converted from time scale to length scale (l_K) via Taylor's hypothesis as mentioned in Section <ref>. 
Clearly, l_K shift towards larger scales with R, as evident from Fig. <ref> and Table <ref> as well.

We see that a strong power-law relation exists between R and l_K, with l_K evolving as l_K∝ R^ 1.18 for R<1 au and l_K∝ R^ 1.87 for R>1 au.

The central panel in Fig. <ref> shows how the break scale behaves with R when normalized to the ion-inertial length scale, d_i = c/ω_pi (where ω_pi=√(ne^2/ϵ_0 m) is the plasma frequency). 
The ion-inertial scale has been found to vary between ∼45 to ∼500 km for R ranging from R ≃ 0.3—3.2 au. 
After normalization, we find that the evolutionary nature is nearly lost for FSW intervals in the inner heliosphere near the ecliptic plane, with a residual weak R^ 0.13 dependence, and l_K is ∼10^3 times d_i. 
A similar pattern was observed (but not shown) after normalization with the ion gyro-radius ρ_i = v_th^⊥ / Ω_i, in the inner heliosphere (the ρ_i in the outer heliosphere could not be computed again due to data limitations).  
Note that the typical ion scales have an approximately linear radial increase up to 5 au <cit.> which might explain the constant radial trend of the normalized break scale.
However, beyond 1 au, it is to be noted that even after normalization, the evolutionary nature of l_K still persists, so that only the radial trend of the break decouples from that of the ion scales. 
The residual power law could be associated with the variation in heliospheric latitude (and to the associated variation of the angle between the large-scale magnetic field B and the bulk speed V_sw) at which the FSW streams were sampled, indicated in the labels in Fig. <ref>. 
Understanding this variation of l_K with latitude and V_sw-B angle would be interesting for a future study but is currently beyond the scope of this paper. 

In order to compare the break scale l_K and the correlation scale L_c, we have drawn l_K normalized to L_c as a function of R (see Fig. <ref> bottom). 
It is evident from the plot that, for R<1, a small power-law exponent is observed, l_K/L_c ∼ R^ -0.14, so that the normalization to the correlation scale removes the radial dependence, similar to what we observe when normalized to the ion scale.
Moreover, in this case, l_K is ∼ 0.4 times L_c and certainly does not correspond to scales within the f^-1 power law regime in the spectrum, contrary to what has been suggested previously <cit.>.
For R>1, l_K approaches L_c, thereby explaining the absence of the f^-3/2 regime in the R=3.16 au interval and supporting recent observations of spectral steepening of the inertial range with increasing R <cit.>.
Note that, in the inner heliosphere, break scales normalized to both the characteristic ion scale and the correlation scale follow weak radial dependence of R^ 0.13 and R^ -0.14, respectively.









 §.§ Observations during Solar Maximum


We now perform a similar spectral and intermittency analysis using the set of intervals recorded during the solar maximum (see Table <ref>).
While the previous section was confined to only analyzing FSW, in this section we take into consideration the three main solar wind types, namely FSW, SSW and the ASSW.
Previous studies on spectra and intermittency mostly focused on FSW and SSW <cit.>. 
More recently, the spectral properties of ASSW, which exclusively permeates the heliosphere during periods of high solar activity, were also examined <cit.>. However, such studies did not include intermittency. Moreover, a comparative analysis between FSW, SSW and ASSW at solar maxima has not yet been conducted. 
Thus, in this section, we examine the intermittency properties of ASSW <cit.> in comparison with the other two types of wind using Ulysses data, during the ascending phase of solar cycle 23 (year 2001), at R≃1.5 au. 

In Fig. <ref>, we show examples of the magnetic field power spectral density for three intervals at solar maximum, one for each type of solar wind. 
Unlike the previous case during solar minima, in this case, there is no clear emergence of two sub-ranges, and the inertial range is consistently showing the typical Kolmogorov scaling, close to -5/3. In the fast stream only, it is possible to see a shallower range at low frequency, compatible with the usual f^-1 range. This is also not visible in the two slow wind types. Similar spectra were observed for all the other intervals (not shown). 

The variation of K (defined in Section <ref>) as a function of τ is shown in Fig. <ref> for all the intervals of FSW, ASSW and SSW listed in Table <ref>. 
Similar to what has been observed during solar minima, K is found to be scale dependent, decreasing with the time scale τ and approaching the Gaussian value K≃1.67 at τ>∼10^4. 
This is again a clear indication of the non-universal nature of the distribution function of the magnetic field increments.
However, just like for the spectra, in this case no clear break within the inertial range is evident for any of the three types of solar wind. 


The consolidated plot shown in Fig. <ref> allows to perform a comparative study of intermittency among those three types of solar wind.
As evident from the plots, turbulence in ASSW is moderately intermittent, characterized by a value of K which is intermediate between that of the SSW with the strongest intermittency and that of the FSW having the weakest intermittency.
Our observations are in agreement with the fact that, in the outer heliosphere, the SSW is in a state of more developed turbulence than ASSW and FSW. 
This can also be inferred from the broad inertial range in the magnetic power spectra exhibited by SSW extending to much lower frequencies compared to FSW (see Fig. <ref>), for which we observe a f^-1 break <cit.>. 
Several studies have observed similar spectral characteristics for these two types of wind <cit.>. 
The power spectrum of ASSW shows similar nature as SSW, with a broad inertial range. However, it is to be noted that a recent study conducted by <cit.> did observe a f^-1 break in the spectra of ASSW at 1 au, which hints how the turbulence develops in ASSW understood by the broadening of the inertial range as it evolves with R. 
While studies by <cit.> explain the high  Alfvénicity of ASSW as due to its generation from coronal hole boundaries based on its composition and micro-physics, our findings on ASSW with an intermediate state of turbulence hints that the low speed of ASSW may be due to the intermixing of FSW and SSW inside the heliosphere.





§ SUMMARY AND CONCLUSION



In this paper, we report the existence of two distinct sub-regimes for the inertial range in the magnetic power spectrum of solar wind turbulence within and beyond 1 au.
Although a single inertial range spectral power law has been traditionally observed <cit.>, a few studies have also identified variations in the spectral indices of the magnetic power spectrum <cit.> and in the scaling exponents of higher-order structure functions <cit.>. 
Additionally, <cit.> observed a break (τ_K) in the scaling of kurtosis (K) within FSW intervals, suggesting a possible connection between this break and the f^-1 break due to their similar behavior with R as discussed by <cit.>.
However, our findings show that τ_K in the kurtosis scaling closely coincides with the break (f_b) observed in magnetic spectra separating the two sub-regimes characterized by f^-3/2 and f^-5/3 spectral power laws in both the inner as well as the outer heliosphere (see Fig. <ref>).
The appearance of a double power-law in both the magnetic power spectrum and kurtosis (or normalized fourth-order moments) supports the robustness of this break, indicating the existence of a previously unidentified characteristic scale within the inertial range. 


Whereas the most probable explanation for the f^-5/3 regime can be obtained by the isotropic Kolmogorov phenomenology or anisotropic mhd turbulence with a weak v-b alignment in non-Alfvénic solar wind, the f^-3/2 regime could be reasonably associated with the anisotropic spectra along the strong v-b alignment <cit.>. Note that, we consciously eliminate the possibility of a -3/2 spectra by Iroshnikov-Kraichnan phenomenology which is valid only for balanced MHD and cannot explain the emergence of -3/2 spectra when there is a strong v-b correlation.

A recent study by <cit.> provided evidence of a transition from a weak to a strong turbulence regime as one moves from larger to smaller scales.
In our study, an inspection of the cross-helicity co-spectra revealed that the turbulence in FSW shifts from a highly imbalanced state (|z^+2|≫|z^-2|, or vice-versa) at larger scales to a relatively balanced one (|z^+2|∼|z^-2|) on moving towards the smaller scales (see Fig. <ref>).
These observations may explain the broken power-law behavior of the spectrum and the kurtosis indicating a transition in the nature of turbulence as the cascade progresses towards the smaller scales.


We have further investigated the dependence of the sub-inertial regime break (τ_K) on the heliospheric distance (R) in comparison with the ion and correlation scales. Our findings indicate a power-law behavior for l_K <cit.> with R, which upon normalization with the typical ion scales (e.g. the ion-inertial scale d_i and the ion gyro-radius ρ_i) and the correlation scale (L_c) practically disappears in the inner heliosphere (see Fig. <ref>). Therefore, both the correlation scale and the characteristic ion scale appear to control the location of the break.
Interestingly, though, l_K appears to approach the correlation scale shifting towards larger scales as R increases, resulting in the absence of the f^-3/2 regime at 3.16 au. 
This observation could explain the transition of the inertial range magnetic spectral slope from -3/2 near the Sun to -5/3 farther away <cit.>.
Note that a residual power-law radial dependence of the break scale still persists in the outer heliosphere, possibly due to variations in the latitude at which the FSW streams were sampled.
This residual behaviour of the normalized l_K must be studied in depth in a future study as functions of the latitude and also the large-scale magnetic field angle, which determines the degree of anisotropy in the measured turbulence.

Our analysis, extended to a period of year 2001 for different types of solar wind intervals, shows that the two inertial sub-ranges are no longer surviving during the period of high solar activity.
Nevertheless, the study enables us to characterize the state of turbulence in the Alfvénic slow solar wind <cit.>, as compared to traditional fast and slow winds. 
We showed that during this period, when ASSW is found in abundance near the ecliptic plane, it is in an intermediate state of turbulence between those typical of fast and slow streams. 
This also gives us insights on the position of the break separating the integral (f^-1) and inertial ranges <cit.> in the case of ASSW.
While <cit.> found the f^-1 break to be occurring at the same frequency for FSW as well as ASSW at 1 au, in our study it occurs at a much lower frequency for the ASSW compared to the FSW at distances greater than 1.5 au (see Fig. <ref>), thus suggesting a plausible explanation for the `slowness' of the ASSW due to strong intermixing between the FSW and the SSW during the high solar activity.





§ ACKNOWLEDGMENTS



S.M. was supported by Students-Undergraduate Research Graduate Excellence (SURGE) summer internship program at Indian Institute of Technology Kanpur. 
S.B. acknowledges the financial support from the grant by Space Technology Cell-ISRO (STC/PHY/2023664O). 
L.S.-V. received support by the Swedish Research Council (VR) Research Grant N. 2022-03352 and by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #23-591 (Evolution of Turbulence in the Expanding Solar Wind). 




§ DATA AVAILABILITY


For our study, we have used publicly available data from NASA CDAWeb (https://cdaweb.gsfc.nasa.gov/https://cdaweb.gsfc.nasa.gov) and AMDA science analysis system (https://amda.irap.omp.eu/https://amda.irap.omp.eu).





§ ESTIMATION OF KURTOSIS OF MAGNETIC FIELD FLUCTUATIONS FOLLOWING A GAUSSIAN DISTRIBUTION


Following the definition of the n^th order structure function (S_n) given by eqn. (<ref>), the expression of S_4 and S_2 takes the form:

    S_4 = ((Δ b_r)^2 + (Δ b_t)^2 + (Δ b_n)^2)^2,

and

    S_2 = ((Δ b_r)^2 + (Δ b_t)^2 + (Δ b_n)^2),

respectively. Now considering that the fluctuations follow a zero mean gaussian distribution f(Δ b_i) having a standard deviation σ such that

    f(Δ b_i) = 1/√(2πσ^2)exp[(Δ b_i)^2/2σ^2],

we have

    S_4 = ∫∫∫ ((Δ b_r)^2 + (Δ b_t)^2 + (Δ b_n)^2)^2 f(Δ b_r) f(Δ b_t) f(Δ b_n)  d(Δ b_r)  d(Δ b_t)  d(Δ b_n),

and

    S_2 = ∫∫∫ ((Δ b_r)^2 + (Δ b_t)^2 + (Δ b_n)^2) f(Δ b_r) f(Δ b_t) f(Δ b_n)  d(Δ b_r)  d(Δ b_t)  d(Δ b_n),

which seem a bit rigorous but can be solved easily to obtain S_4 = 15σ^4 and S_2=3σ^2. Thus, the kurtosis defined by eqn. (<ref>) takes the value K=5/3 ≃ 1.67.





§ COMPONENT-WISE KURTOSIS OF THE MAGNETIC FIELD FLUCTUATIONS IN FSW INTERVALS DURING SOLAR MINIMUM


In this appendix we show the kurtosis K for each individual RTN magnetic field component, using four example intervals from both Helios and Ulysses database, at eighth different distances from the Sun. Different colors refer to the different intervals. 
Whenever present, a power law is shown as colored dashed line, and the corresponding scaling exponents are indicated in each panel. 
Two power laws can be identified in all of the Helios and most of the Ulysses intervals, with the exception of the radial component at 2.75 au and of all components at 3.16 au. 
The timescale τ_K of the break between the two power laws is indicated by a solid vertical grey line, while the dashed grey vertical lines indicate the location of the spectral break, 1/f_b.





*

aasjournal










