# Reduction in benefits of total flux expansion on divertor detachment due to
parallel flows

M.Carpita1444 Author to whom any correspondence should be addressed., O.
Février1, H. Reimerdes1, C. Theiler1,
B. P. Duval1, C. Colandrea1, G. Durr-Legoupil-Nicoud1,
D. Galassi1, S. Gorno1, E. Huett1, J. Loizu1, L. Martinelli1,
A. Perek1, L. Simons1, G. Sun1, E. Tonello1, C. Wüthrich1
and the TCV team111 See author list of Reimerdes et al. 2022, Nucl. Fusion 62
042018 1 EPFL-SPC, CH-1015 Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract

The Super-X divertor (SXD) is an alternative divertor configuration leveraging
total flux expansion at the outer strike point (OSP). According to the
extended 2-point model (2PM), the key attractive features of the SXD are
facilitated detachment access and control, but this is not always retrieved
experimentally. However, parallel flows are not consistently included in the
2PM. In this work, the 2PM is refined to overcome this limitation: the role of
total flux expansion on the pressure balance is made explicit, by including
the effect of parallel flows. Consequentially, the effect of total flux
expansion on detachment access and control is weakened, compared to
predictions of the 2PM. This new model partially explains discrepancies
between the 2PM and experiments performed on TCV, in ohmic L-mode scenarios,
which are particularly apparent when scanning the OSP major radius Rt. In core
density ramps in lower single-null (SN) configuration, the impact of Rt on the
CIII emission front movement in the divertor outer leg - used as a proxy for
the plasma temperature – is substantially weaker than 2PM predictions.
Furthermore, in OSP radial sweeps in lower and upper SN configurations, in
ohmic L-mode scenarios with a constant core density, the peak parallel
particle flux density at the OSP is almost independent of Rt, while the 2PM
predicts a linear dependence. Finally, analytical and numerical modelling of
parallel flows in the divertor is presented. It is shown that an increase in
total flux expansion can favour supersonic flows at the OSP. Parallel flows
are also shown to be relevant by analysing SOLPS-ITER simulations of TCV.

  * June 2023

Keywords: power exhaust, divertor, detachment, total flux expansion, Mach
number, parallel flows

##  1 Introduction

Power exhaust is a key challenge for the realisation of a magnetic confinement
fusion reactor, such as tokamaks, as identified by the European roadmap for
fusion energy [1]. In a future power plant, large power losses from the
confined plasma must be exhausted in a very narrow scrape-off-layer (SOL)
region. The peak power density at the target, if unmitigated, are predicted to
greatly exceed material limits [2]. Moreover, avoiding excessive long-term
erosion on the reactor vessel components requires sufficiently low plasma
target temperature [3].

Diverted plasma configurations are employed for power exhaust, with the
ability to support a large plasma temperature gradient between the confined
plasma and the divertor targets. At sufficiently low electron temperature
$T_{e}$, radiation by hydrogen and low-Z impurities becomes more efficient
($T_{e}\lesssim 10~{}eV$), and the cross-sections for charge exchange
($T_{i}\lesssim 5~{}eV$) and volumetric recombination ($T_{e}\lesssim 1~{}eV$)
increase, redistributing the exhausted power more isotropically, and
transferring some of the plasma momentum and energy to neutrals [4, 5]. This
allows the divertor to enter the detached regime, greatly reducing the power
and particle fluxes to the targets.

The lower-single-null (LSN) is currently the reference configuration for most
operating tokamaks and is the chosen configuration for ITER [6]. Nonetheless,
the extrapolation of this configuration to future reactors, with higher power
and particle fluxes, cannot be safely assumed, in particular with respect to
the integration of a detached divertor with a reactor-relevant core plasma.
Alternative divertor configurations (ADCs) are, therefore, studied as
potential solutions to this problem. ADCs’ foreseen benefits include easier
access to detachment, higher total radiated power capabilities, and better
control over the location of the radiation front [7]. However, ADCs must be
judged in the light of increased machine complexity, hence their assessment,
through experiments and modeling, is crucial [8].

Among ADCs, one considered concept for future reactors is the Super-X divertor
(SXD) [9]. Its main feature is an increase of the outer strike point (OSP)
major radius $R_{t}$, which comes with an increased total flux expansion. The
increase of $R_{t}$ increases the cross-sectional area of a flux tube
$A_{\perp,t}$ (as the total magnetic field $B_{tot}$ is proportional to the
inverse of the major radius $R^{-1}$) and, as a result, decreases the parallel
power densities at the target, $q_{\parallel,t}$. For a constant grazing angle
at the outer target, an increase in $R_{t}$ corresponds exactly to a linear
increase of the target wetted area. The power density at the OSP then scales
as $R_{t}^{-1}$, which has been demonstrated experimentally [7]. According to
the extended 2-point model (2PM) [10, 11], the key attractive features of the
SXD are facilitated detachment access and control. However, in some cases,
these predictions were neither retrieved experimentally [7, 12, 13] nor
numerically [14]. In specific cases, it was argued that this disagreement with
analytical predictions was explained by several possible effects, e.g. target
geometry [12], neutral compression [15, 16] and / or the divertor being in a
sheath-limited regime [14]. However, a general understanding of the
discrepancy has still not been obtained.

In this paper, the role of $R_{t}$ is discussed, both in terms of target
conditions and for detachment access and control. Section 2 presents the 2PM,
its predictions with respect to total flux expansion effects on detachment
access and control and its modification to make the effect of parallel flows
on the total pressure balance explicit, leading to predictions of weaker total
flux expansion effects compared to the original ones. Section 3 presents SXD
experiments on TCV tokamak [13] to investigate the role of $R_{t}$. Finally,
in section 4, the analytical and numerical modelling of parallel flows in the
divertor is presented, showing that an increase in total flux expansion can
favour supersonic flows at the OSP and that parallel flows are relevant by
analysing SOLPS-ITER [17] simulations of TCV. A summary and conclusions are
presented in section 5.

##  2 2PM extension accounting for parallel flows

The 2PM [10, 18] is a reduced model which relates target quantities (electron
temperature $T_{e,t}$, electron density $n_{e,t}$, parallel particle flux
density $\Gamma_{t}$) with upstream control parameters for the SOL plasma,
e.g. the total plasma pressure $p_{tot,u}$ and the parallel power density
$q_{\parallel,u}$ at the upstream location. These quantities pertain to one
individual flux tube in the SOL and are linked together by momentum and power
balances. The upstream location, labeled by $u$, is somewhat arbitrary, and
can refer to the X-point location, the outer mid-plane (OMP), etc. It is
usually taken as a stagnant location, i.e. where $v_{\parallel}=0$. In the
following, when needed, this location will be specified.

In the interest of completeness, in the 2PM, the parallel power density
$q_{\parallel}$ is defined as the total parallel plasma power density, i.e. in
the simplest form (taking $n_{e}=n_{i}=n$ for densities and $T_{e}=T_{i}=T$
for temperatures)

$q_{\parallel}=(5nT+\frac{1}{2}m_{i}nv_{\parallel}^{2})v_{\parallel}+q_{\parallel}^{heat,cond}$
(1)

with $q_{\parallel}^{heat,cond}$ the total conducted heat flux density, $T$
the plasma temperature, $n$ the plasma density and $v_{\parallel}$ the
parallel plasma velocity.

###  2.1 2PM predictions for target quantities and their dependence on $R_{t}$

The most general 2PM expressions for target quantities are reported by
Stangeby in (15)-(17) of [10]. These are equivalent to expressions obtained by
Kotov and Reiter in [19] that were derived from the steady-state version of
the equations solved by the 2D multi-species plasma fluid code B2.

These expressions are reported here, assuming the following simplifying
hypotheses: (S-I) only hydrogenic ion species (i.e. $n=n_{e}=n_{i}$) and no
net current (i.e. $v_{\parallel}=v_{e,\parallel}=v_{i,\parallel})$; (S-II)
thermal equilibration is achieved along the flux tube (i.e. $T=T_{e}=T_{i}$);
(S-III) the plasma flow at the target is sonic (i.e. $M_{t}=1$, where
$M=v_{\parallel}/c_{s}$ is the Mach number and
$c_{s}=\sqrt{(T_{e}+T_{i})/m_{i}}=\sqrt{2T/m_{i}}$ the sound speed, and the
subscript $t$ representing the target in what follows). Hypothesis (S-III) and
its link to the total flux expansion effects are discussed in section 4.1.
These assumptions, introduced for simplicity, can be easily relaxed and do not
limit the following discussion. However, an additional assumption required in
the derivation of the following 2PM expressions is: (A-I) target quantities
are evaluated at the sheath entrance (i.e.
$q_{\parallel,t}=q_{\parallel,se}=\gamma n_{t}T_{t}c_{s,t}$). Further details
are provided in appendix A. The expressions are

$\displaystyle T_{t}^{2PM}=$
$\displaystyle\frac{8m_{i}}{\gamma^{2}}\cdot\frac{q_{\parallel,u}^{2}}{p_{tot,u}^{2}}\cdot\frac{(1-f_{cooling})^{2}}{(1-f_{mom-
loss})^{2}}$ $\displaystyle\cdot\left(\frac{R_{u}}{R_{t}}\right)^{2}$
$\displaystyle n_{t}^{2PM}=$
$\displaystyle\frac{\gamma^{2}}{32m_{i}}\cdot\frac{p_{tot,u}^{3}}{q_{\parallel,u}^{2}}\cdot\frac{(1-f_{mom-
loss})^{3}}{(1-f_{cooling})^{2}}$
$\displaystyle\cdot\left(\frac{R_{u}}{R_{t}}\right)^{-2}$
$\displaystyle\Gamma_{t}^{2PM}=$
$\displaystyle\frac{\gamma}{8m_{i}}\cdot\frac{p_{tot,u}^{2}}{q_{\parallel,u}}\cdot\frac{(1-f_{mom-
loss})^{2}}{(1-f_{cooling})}$
$\displaystyle\cdot\left(\frac{R_{u}}{R_{t}}\right)^{-1}$

where $m_{i}$ is the ion mass, $\gamma\approx 8.5$ the sheath heat
transmission coefficient [18] and $R_{u/t}$ are the upstream and target major
radii respectively. The power and momentum loss factors, $f_{cooling}$ and
$f_{mom-loss}$, are

$\displaystyle\frac{q_{\parallel,t}}{q_{\parallel,u}}\cdot\frac{R_{t}}{R_{u}}\equiv
1-f_{cooling}$ (5) $\displaystyle\frac{p_{tot,t}}{p_{tot,u}}\equiv 1-f_{mom-
loss}$ (6)

and the total plasma pressure is

$p_{tot}=2nT+m_{i}nv_{\parallel}^{2}=2(1+M^{2})nT$ (7)

The ratio $(R_{u}/R_{t})$ in (2.1)-(2.1) explicitly relates target quantities
to total flux expansion. Both experiments and simulations [7, 12, 14] were
done to test such specific explicit dependencies of target quantities on
$R_{t}$, showing several discrepancies.

###  2.2 Explicit dependence of $f_{mom-loss}$ on $R_{t}$ and the effective
Mach number $M_{eff}$

The loss factors $f_{cooling}$ and $f_{mom-loss}$ are lumped parameters
accounting for a variety of complex physical processes [12, 14, 15, 16]. These
processes can be separated into two main groups: (1) volumetric sources and
cross-field transport effects; (2) geometrical effects, related to flux tube
cross-sections. This work focuses mainly on the latter, as they can be
explicitly linked to total flux expansion effects, as shown in the following.

While $f_{cooling}$ relates only to processes pertaining to group (1),
$f_{mom-loss}$ accounts also for geometrical effects. To show this, the total
power and parallel momentum steady-state balances in a flux tube element are
taken

$\displaystyle\frac{1}{A_{\perp}}\partial_{s}(A_{\perp}q_{\parallel})=S_{pwr}$
(8)
$\displaystyle\frac{1}{A_{\perp}}\partial_{s}(A_{\perp}mnv_{\parallel}^{2})=-\partial_{s}(2nT)+S_{mom}$
(9)

where $s$ is a length coordinate along the flux tube and $S_{pwr/mom}$ are
effective sources (or sinks) within the flux tube, respectively for power and
momentum, related to processes pertaining to group (1). As in a flux tube
$A_{\perp}\propto B_{tot}^{-1}\propto R$, rearranging (8)-(9) gives

$\displaystyle\frac{1}{q_{\parallel}}\partial_{s}(q_{\parallel})=\frac{S_{pwr}}{q_{\parallel}}-\frac{1}{R}\partial_{s}(R)$
(10)
$\displaystyle\frac{1}{p_{tot}}\partial_{s}(p_{tot})=\frac{S_{mom}}{p_{tot}}-\frac{\kappa}{R}\partial_{s}(R)$
(11)

where $\kappa=mnv_{\parallel}^{2}/p_{tot}=M^{2}/(1+M^{2})$ is the local ratio
of dynamic and total pressure in the flux tube. Integrating (10)-(11) from
upstream to target, rearranging and using (5)-(6) gives

$\displaystyle\frac{q_{\parallel,t}}{q_{\parallel,u}}\cdot\frac{R_{t}}{R_{u}}$
$\displaystyle=exp\left(\int_{u}^{t}\frac{S_{pwr}}{q_{\parallel}}ds\right)\equiv$
(12) $\displaystyle\equiv 1-f_{cooling}$
$\displaystyle\frac{p_{tot,t}}{p_{tot,u}}$
$\displaystyle=exp\left(\int_{u}^{t}\left[\frac{S_{mom}}{p_{tot}}-\frac{\kappa}{R}\partial_{s}(R)\right]ds\right)\equiv$
(13) $\displaystyle\equiv 1-f_{mom-loss}$ (14)

It thus becomes apparent that $f_{mom-loss}$ includes geometrical effects,
whereas $f_{cooling}$ does not. In literature, the influence of geometrical
effects on $f_{mom-loss}$ was recognized, but was not investigated in detail,
as it was considered negligible or avoided for simplicity [10, 14].

To explicitly highlight the effect of total flux expansion on the total
pressure variation, it is useful to rewrite (13) in a form similar to (12). A
constant $\kappa_{eff}$ is introduced, which satisfies

$\int_{u}^{t}\frac{\kappa}{R}\partial_{s}(R)ds=\kappa_{eff}\int_{u}^{t}\frac{1}{R}\partial_{s}(R)ds$
(15)

$\kappa_{eff}$ is then the average of the ratio of dynamic to total pressure,
weighted by the local relative variation of the flux tube area, between
upstream and target. (13) then becomes

$\displaystyle 1-f_{mom-loss}$
$\displaystyle\equiv\frac{p_{tot,t}}{p_{tot,u}}=$
$\displaystyle=\left(\frac{R_{u}}{R_{t}}\right)^{\kappa_{eff}}exp\left(\int_{u}^{t}\frac{S_{mom}}{p_{tot}}ds\right)$

This equation now explicitly shows the effect of total flux expansion on the
total pressure variation. It also shows the explicit dependence of $f_{mom-
loss}$ on total flux expansion.

An additional quantity can be defined to substitute $\kappa_{eff}$ in (2.2),
termed the effective Mach number $M_{eff}$

$M_{eff}=\sqrt{\frac{\kappa_{eff}}{1-\kappa_{eff}}}~{}\leftrightarrow~{}\kappa_{eff}=\frac{M_{eff}^{2}}{1+M_{eff}^{2}}$
(17)

From here, $M_{eff}$ will be used. Further insights on $\kappa_{eff}$ and
$M_{eff}$, and their physical interpretation, are provided in appendix B.

###  2.3 Consequence on target quantities scaling

The result obtained in (2.2) is now considered together with (2.1)-(2.1). For
the sake of clarity, the following notation is introduced

$\displaystyle 1-f_{cooling}\equiv(1-f_{cooling}^{S})$ (18) $\displaystyle
1-f_{mom-loss}\equiv(1-f_{mom-loss}^{S})$ (19)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot\left(\frac{R_{u}}{R_{t}}\right)^{\frac{M_{eff}^{2}}{1+M_{eff}^{2}}}$

so the newly defined factors $f_{cooling}^{S}$ and $f_{mom-loss}^{S}$ are
accounting for the same physics, i.e. volumetric sources and cross-field
effects only. With this new definition of loss factors, (2.1)-(2.1) then
become

$\displaystyle T_{t}^{mod}=$
$\displaystyle\frac{8m_{i}}{\gamma^{2}}\cdot\frac{q_{\parallel,u}^{2}}{p_{tot,u}^{2}}\cdot\frac{(1-f_{cooling}^{S})^{2}}{(1-f_{mom-
loss}^{S})^{2}}$
$\displaystyle\cdot\left(\frac{R_{u}}{R_{t}}\right)^{2-\frac{2M_{eff}^{2}}{1+M_{eff}^{2}}}$
$\displaystyle n_{t}^{mod}=$
$\displaystyle\frac{\gamma^{2}}{32m_{i}}\cdot\frac{p_{tot,u}^{3}}{q_{\parallel,u}^{2}}\cdot\frac{(1-f_{mom-
loss}^{S})^{3}}{(1-f_{cooling}^{S})^{2}}$ (21) $\displaystyle $
$\displaystyle\cdot\left(\frac{R_{u}}{R_{t}}\right)^{-2+\frac{3M_{eff}^{2}}{1+M_{eff}^{2}}}$
$\displaystyle\Gamma_{t}^{mod}=$
$\displaystyle\frac{\gamma}{8m_{i}}\cdot\frac{p_{tot,u}^{2}}{q_{\parallel,u}}\cdot\frac{(1-f_{mom-
loss}^{S})^{2}}{(1-f_{cooling}^{S})}$
$\displaystyle\cdot\left(\frac{R_{u}}{R_{t}}\right)^{-1+\frac{2M_{eff}^{2}}{1+M_{eff}^{2}}}$

The dependence of target quantities on $R_{u}/R_{t}$ now varies with
$M_{eff}$, figure 1, and is weakened with increasing $M_{eff}$. The
qualitative dependence of $\Gamma_{t}^{mod}$ and $n_{t}^{mod}$ on $1/R_{t}$
even reverses for $M_{eff}\geq 1$ and $\geq\sqrt{2}$, respectively. When
$M_{eff}=0$, the dependence of target quantities on $R_{u}/R_{t}$ recovers the
original ones in (2.1)-(2.1).

Figure 1: Exponents of $R_{u}/R_{t}$ for target temperature $T_{t}^{mod}$
(red), target density $n_{t}^{mod}$ (blue) and parallel particle flux density
$\Gamma_{t}^{mod}$ (green), plotted against the effective Mach number
$M_{eff}$.

###  2.4 Consequence on detachment window

It has been predicted that the Super-X configuration allows for an increased
detachment front stability and control [8, 11, 20], e.g. a larger control
parameters interval over which a detached regime can be achieved in the
divertor while maintaining a tolerable impact on the core performance. This is
a consequence of the negative parallel power density gradient that total flux
expansion establishes towards the target, providing opposing response with
respect to an upstream detachment front movement, i.e. opposing radiation
cooling. In terms of the operational window for detachment, Lipschultz et al.
(see (30) of [11]) provided an analytical estimate for the dependence of the
detachment window on $B_{tot}\propto R_{t}^{-1}$

$\frac{\zeta_{x}}{\zeta_{t}}=\left[\frac{R_{t}}{R_{x}}\right]^{\beta}$ (23)

where $\zeta_{x,t}$ are the values of a control parameter
$\zeta=[p_{u},f_{I},P_{SOL}]$, that corresponds to the detachment front***The
detachment front is intended here as the location where, ideally, all of the
power loss occurs in a flux tube [11, 20], separating a hot, attached upstream
portion of the flux tube and a cold, detached downstream portion in front of
the target. being at the X-point or at the target, respectively. The three
control parameters considered in this work are the upstream static pressure
$p_{u}=2n_{u}T_{u}$ (instead of $n_{u}$, as used in [11] \- see appendix C),
the impurity fraction $f_{I}$ and the power entering the SOL in the flux tube
of interest $P_{SOL}$. $R_{x,t}$ are the X-point and the target major radii,
respectively. $\beta=[1,2,-1]$ is a specific exponent related to the
considered control parameter.

The derivation of (23) uses a momentum balance equivalent to the one in the
2PM and does not account explicity for any $p_{u}$ variation from upstream to
target, i.e. flux expansion effects and/or total pressure redistribution
between dynamic and static contributions. When taken into account, the
dependence of the detachment window on $B_{tot}\propto R_{t}^{-1}$ becomes

$\frac{\zeta_{x}}{\zeta_{t}}=\left[\left(\frac{R_{t}}{R_{x}}\right)^{1-\frac{M_{eff}^{2}}{1+M_{eff}^{2}}}\cdot\frac{1+M_{x}^{2}}{1+M_{t}^{2}}\right]^{\beta}$
(24)

where the first factor in (24) accounts for the total flux expansion and the
second factor accounts for the total pressure redistribution. Further details
on the derivation of (24) are provided in appendix C. The inclusion of total
flux expansion and redistribution effects on total pressure reveals that the
static pressure $p$ can include a gradient towards the target. In particular,
$p$ is proportional to the radiated power in the detachment front, as shown in
(48). Consequentially, a negative gradient of the static pressure, as opposed
to parallel power density, provides a positive feedback for the upstream
movement of the detachment front and, hence, weakens the total flux expansion
dependence of the detachment window.

###  2.5 Summary of the effects of parallel flows on total flux expansion

The impact of accounting for total flux expansion effects on momentum balance
was shown and the following important points were highlighted:

  * •

The total pressure variation along a flux tube, see (2.2), can be linked
explicitly to total flux expansion via $M_{eff}$, a lumped parameter
characterising flows in the flux tube of interest.

  * •

For negligible $M_{eff}$, this variation and its related effects are
negligible.

  * •

Increasing $M_{eff}$ generally weakens the dependence on $R_{t}$ of target
quantities, see (2.3)-(2.3), and detachment window, see (24), compared to
predictions by the 2PM. In the case of “effective supersonic” flows
($M_{eff}\geq 1$), some dependencies also qualitatively reverse, starting with
the particle flux.

  * •

$M_{eff}$ depends on both the flow patterns in the flux tube and the
geometrical design of the leg, in particular on the change of relative flux
expansion along field lines, i.e. $R^{-1}\partial_{s}(R)$, see (15) and (17).
Two different divertor geometries, characterized by the same flow patterns and
total flux expansion, can exhibit different behaviour with respect to their
sensitivity to $R_{t}$. In appendix B this point is discussed in detail.

##  3 SXD experiments in TCV and comparison with 2PM predictions

Experiments to investigate the SXD configuration are carried out in the
Tokamak à Configuration Variable (TCV) at EPFL [21, 13], testing the 2PM
predictions presented in section 2.1, regarding total flux expansion effects
on detachment access and control. TCV is a medium-sized tokamak ($R_{0}\sim
0.88~{}\mathrm{m},~{}B_{0}<1.45~{}\mathrm{T},~{}a\sim 0.25~{}\mathrm{m}$) with
a highly elongated open vessel structure and a set of 16 independently-powered
poloidal field coils, allowing for unique shaping capabilities that can
generate many divertor configurations. The almost complete coverage of the
vessel surfaces with graphite tiles allows for flexible placement of particle
and power loads.

###  3.1 Key diagnostics and experimental approach

Different plasma geometries, characterized by varying OSP major radius
$R_{t}$, are employed in this study, figure 2. A set of polycrystalline
graphite tiles, characterized by a longer structure on the low-field side
compared to the high-field side (SILO baffles), is also employed in some
experiments. They are designed to increase divertor closure, whilst
maintaining good compatibility with alternative divertor configurations [22,
23].

Figure 2: Examples of baffled geometries used in the experimental work (large
and small $R_{t}$). (a) The red dots indicate the position of wall-embedded
Langmuir probes, while the cyan line indicates the FIR chord used for the
feedback control of fuelling. (b) The black rectangles indicate the poloidal
location of fuelling valves, the orange lines indicate the lines of sight of
the DSS and the green dots indicate Thomson scattering measurement locations
(intercepts between the laser and spectrometer lines of sight).

$\mathrm{D}_{2}$ fuelling can employ alternative valves either on the floor,
the inner wall or the ceiling of the vessel, figure 2b, allowing, among other
things, to test the possible impact of fuelling locations on the results. The
flow rates are feedback controlled and can be adjusted according to the line-
averaged density $\langle n_{e}\rangle$ measurements by a far-infrared (FIR)
interferometer, along a vertical chord, figure 2a. Density and temperature
measurements in the core and across the separatrix are also measured by
Thomson scattering [24], figure 2b. Thomson scattering measurements also allow
to compute $\langle n_{e}\rangle$ in the core. Wall-embedded Langmuir probes
(LP) [25] cover a large part of the poloidal perimeter of the vessel, figure
2a. These were operated with a triangular voltage sweep (from $-120$ to
$80~{}\mathrm{V}$ at $\sim 330~{}\mathrm{Hz}$ and $\sim 990~{}\mathrm{Hz}$
frequencies), in order to obtain temperature measurements as well as particle
flux. Details on their analyses are provided in [26]. The orange lines in the
right panel show the lines of sight of a divertor spectroscopy system (DSS)
[27]. Line radiation and their distributions are also obtained from a system
of filtered optical cameras, MANTIS [28], that provide 2D poloidal map
inversions of the emissivity for selected radiating spectral lines. This work
focuses, in particular, on the CIII ($465.8~{}\mathrm{nm}$) line emission to
obtain emissivity profiles. In previous TCV studies, the CIII radiation front
along a divertor leg (determined as the location where the emissivity profile
along the outer leg drops by $50\%$ with respect to the peak) was shown to
provide a convenient estimation of the detachment status of the divertor. Due
to a strong dependency on the local electron temperature, the CIII radiation
front is a reliable proxy to identify the low temperature region along the
outer leg [23, 29]. A system of 64 gold foil bolometers, then substituted with
a new system of 120 channels [30], is used to obtain radiation emissivity maps
across a TCV poloidal section, by tomographically inverting their line
integrated chord intensities. Finally, LIUQE [31] is used to reconstruct the
magnetic equilibrium across the discharges.

Two different scenarios are explored in this work, both with a plasma current
$I_{p}\sim 250~{}\mathrm{kA}$ and the ion $\nabla B$ drift directed away from
the X-point into the core, to avoid accessing H-mode [7]. The first employs
ohmically-heated L-mode core density ramps $\langle
n_{e}\rangle\simeq[4.0\rightarrow 10.0]\cdot 10^{19}~{}\mathrm{m}^{-3}$
(corresponding to $f_{g}\simeq[0.20\rightarrow 0.55]$, $f_{g}$ being the
Greenwald fraction). The density ramp is performed separately for two LSN
configurations with small and large $R_{t}$, respectively. SILO baffles are
employed to increase divertor closure, that is expected to improve the match
between the 2PM predictions and experimental results, according to SOLPS-ITER
simulations of TCV [15]. Fuelling is performed from either the floor, inner
wall (IW) or ceiling valves. The second scenario employs ohmically-heated
L-mode OSP target radius $R_{t}$ scans at constant density $\langle
n_{e}\rangle\simeq 5.5\cdot 10^{19}~{}\mathrm{m}^{-3}$ ($f_{g}\simeq 0.31$).
This scenario is repeated in both Lower-Single-Null (LSN) or Upper-Single-Null
(USN) configurations, with either SILO baffles or without, and floor-only
fuelling.

###  3.2 Density ramps at constant $R_{t}$

Shot | $R_{t}$ | Fuel. | $(1/B_{tot})^{OSP}~{}(T^{-1})$ | $L_{\parallel}^{OSP}~{}(m)$ | $f_{x,pol}^{OSP}$
---|---|---|---|---|---
70202 | Small | IW | 0.50 | 14.2 | 2.79
70201 | Large | IW | 0.80 | 14.7 | 2.36
63935 | Small | Floor | 0.50 | 13.6 | 2.82
63917 | Large | Floor | 0.82 | 14.3 | 2.38
63925 | Small | Ceiling | 0.50 | 13.8 | 2.83
63934 | Large | Ceiling | 0.85 | 12.4 | 2.57
Table 1: Density ramps at constant $R_{t}$ \- SOL geometry quantities at the
OSP: inverse of the total magnetic field $(1/B_{tot}^{OSP})\propto
R_{t}^{OSP}$, parallel connection length $L_{\parallel}^{OSP}$ (measured from
the OMP, 5 mm from the separatrix) and poloidal flux expansion
$f_{x,pol}^{OSP}$ (measured at 5 mm from the separatrix).

Two values of $R_{t}$ are investigated during core density ramps: $R_{t}\simeq
0.62~{}\mathrm{m}$ (small $R_{t}$) and $R_{t}\simeq 1.03~{}\mathrm{m}$ (large
$R_{t}$). When ramping the core density, the temperature in the divertor
gradually reduces. Using the CIII front as proxy for low temperature region in
the divertor, the 2PM prediction on temperature is tested in these
experiments.

Figure 3: Density ramps at constant $R_{t}$ \- (a) line-averaged density
$\langle n_{e}\rangle$ variation in time; (b) ohmic power $P_{OHM}$ variation
against $\langle n_{e}\rangle$; (c) power to the SOL $P_{SOL}$ variation
against $\langle n_{e}\rangle$.

The discharges have a similar time evolution for $\langle n_{e}\rangle$ and
input ohmic power $P_{OHM}$ dependence on $\langle n_{e}\rangle$, figures 3a
and 3b. The power to the SOL, $P_{SOL}$, is defined as the difference between
$P_{OHM}$ and the power radiated from the core, computed from bolometry,
excluding a $5~{}\mathrm{cm}$ circular region centered around the X-point,
figure 4. $P_{SOL}$ dependence on $\langle n_{e}\rangle$ shows significant
differences in cases with inner wall fuelling (up to $25\%$), figure 3c.
Tomographic reconstruction of the emissivities for this fuelling location,
figure 4, suggests that this difference can be ascribed to increased radiation
inside the confined plasma region at higher $\langle n_{e}\rangle$. Thomson
scattering measurements (not shown) also show that the density and temperature
in the core and near the separatrix remain comparable in all cases. Relevant
SOL geometry quantities are reported in table 1.

Figure 4: Density ramps at constant $R_{t}$, inner wall (IW) fuelling cases -
Emissivity maps $(\mathrm{W}/\mathrm{m}^{3})$ at $\langle
n_{e}\rangle=6.75\cdot 10^{19}~{}\mathrm{m}^{-3}$. The colormap is saturated
at $2.1\cdot 10^{6}~{}\mathrm{W}/\mathrm{m}^{3}$, to better highlight features
of the emissivity maps away from the X-point. The red circle defines the
$5~{}\mathrm{cm}$ radial area centered around the X-point, excluded from core
radiation computation. Figure 5: Density ramps at constant $R_{t}$ \- CIII
front position analyses from MANTIS along the outer leg - CIII front position
is defined in terms of relative (%) poloidal distance from the X-point, where
100% is the target position. The expected shifts of large $R_{t}$ cases curves
with respect to small $R_{t}$ cases are also plotted, computed according to
(29).

The CIII front movement at the outer leg against $\langle n_{e}\rangle$, taken
from inversions of MANTIS measurements, is analysed to compare small and large
$R_{t}$ configurations, figure 5a. Similar results are obtained by the DSS
(not shown). However, variations in $P_{SOL}$ and $L_{\parallel}^{OSP}$ can
influence the CIII front behaviour between compared cases, as the front
position is strictly related to the temperature in the divertor leg. According
to the 2PM, the OSP target temperature $T_{t}$ (see (2.1), when changing the
upstream control parameter from total pressure $p_{tot,u}$ to density $n_{u}$
[10, 18]) is proportional to

$\displaystyle T_{t}^{2PM}$
$\displaystyle\propto\frac{1}{R_{t}^{2}}\cdot\frac{q^{10/7}_{\parallel,u}}{n_{u}^{2}L_{\parallel}^{4/7}}$
(25)

and taking

$\displaystyle n_{u}\propto\langle n_{e}\rangle$ (26) $\displaystyle
q_{\parallel,u}\propto\frac{P_{SOL}}{\lambda_{sol,u}2\pi
R_{u}B_{pol,u}/B_{tot,u}}$

one can write

$\displaystyle
T_{t}^{2PM}\propto\frac{1}{R_{t}^{2}}\cdot\frac{P_{SOL}^{10/7}}{\langle
n_{e}\rangle^{2}(L_{\parallel}^{OSP})^{4/7}}$ (27)

Note that this reasoning does not account for differences in other quantities
between compared cases, such as: I) the geometrical location and features of
the upstream location (e.g. the scrape-off layer width $\lambda_{sol,u}$); II)
in-out power sharing; III) the conducted-to-total power density ratio
$f_{cond}$, IV) the ratio $n_{u}/\langle n_{e}\rangle$.

From (27), the parameter

$C\equiv\frac{\langle
n_{e}\rangle(L_{\parallel}^{OSP}/L_{\parallel}^{ref})^{2/7}}{(P_{SOL}/P_{SOL}^{ref})^{5/7}}$
(28)

can be defined as a corrected density. Plotting the CIII front movement
against $C$ allows to consistently account for $P_{SOL}$ and
$L_{\parallel}^{OSP}$ variations between compared cases, according to the 2PM.
Here, $L_{\parallel}^{ref}=10~{}\mathrm{m}$ and $P_{SOL}^{ref}=2.5\cdot
10^{5}~{}\mathrm{W}$ are considered. This is done in figure 5b. From (27), the
large $R_{t}$ configuration should see lower target temperatures for the same
value of $C$. The CIII front movement from the target should thus happen at
lower $C$ values for the higher $R_{t}$ cases. Given a specific front position
obtained at values $C_{(small~{}R_{t})}$ in the small $R_{t}$ cases, the
expected reduced values for $C_{(large~{}R_{t})}^{~{}expected}$ in the
corresponding large $R_{t}$ cases can be computed as

$C_{(large~{}R_{t})}^{~{}expected}=C_{(small~{}R_{t})}\cdot\frac{R_{t}^{(small)}}{R_{t}^{(large)}}$
(29)

This is, however, not retrieved in the results shown in figure 5b. For all the
different fuelling cases, the variation in CIII front position with different
$R_{t}$ is much weaker than predicted by the 2PM.

Figure 6: $R_{t}$ sweeps at approximately constant line-averaged density
$\langle n_{e}\rangle$ \- $\langle n_{e}\rangle$ (a), ohmic power $P_{OHM}$
(b), power to the SOL $P_{SOL}$ (c) and OSP major radius $R_{t}$ (d)
variations in time. In between $\sim 1.05$ and $1.35~{}s$, the OSP is
localised on a vessel segment for which a complete LPs coverage can not be
achieved and, therefore, not of interest for the analyses and not reported
here. For the USN case (black curve), only the outward sweep is available due
to an early disruption.

Figure 7: $R_{t}$ sweeps at approximately constant line-averaged density
$\langle n_{e}\rangle$ \- (a) separatrix geometries for the unbaffled cases,
showing the minimum and maximum $R_{t}$ achieved; (b) parallel connection
length $L_{\parallel}^{OSP}$ (taken at the outboard midplane, $5~{}mm$ from
the separatrix) variation against $R_{t}$.

###  3.3 $R_{t}$ scans at constant density

Here the opposite scenario is investigated by sweeping poloidally the OSP at
constant core density, scanning a range of $R_{t}$ values
($R_{t}\simeq[0.7\leftrightarrow 1.05]~{}\mathrm{m}$), with both outward and
inward sweeps. When $R_{t}$ is modified, the target particle flux is also
expected to vary according to the 2PM (see (2.1)). This prediction is tested
in these experiments, using target LPs measurements.

During the strike-point sweeps, $\langle n_{e}\rangle$, $P_{OHM}$ and
$P_{SOL}$ are kept approximately constant, figure 6, with observed variation
of up to $10-20\%$. For all cases, $\langle n_{e}\rangle$ is always below
$\sim 6.0\cdot 10^{19}~{}\mathrm{m}^{-3}$. At this density, for these
experimental conditions, the CIII front in the outer leg remains close to the
target and an attached state is maintained, as shown by the density ramps in
3.2, figure 5a. Thomson scattering measurements (not shown) show that the
density and the temperature in the core and near the separatrix remain
comparable across the strike-point sweeps. Figure 7 plots the plasma geometry
and $L_{\parallel}^{OSP}$ against $R_{t}$.

Figure 8a shows the variation of the peak parallel particle flux density at
the OSP $\Gamma_{t}$ against $R_{t}$. $\Gamma_{t}$ is taken from LPs
measurements. However, variations in $\langle n_{e}\rangle$, $P_{SOL}$ and
$L_{\parallel}^{OSP}$ can influence $\Gamma_{t}$ variations. According to the
2PM the OSP peak parallel particle flux density $\Gamma_{t}$ (see (2.1), when
changing the upstream control parameter from total pressure $p_{tot,u}$ to
density $n_{u}$ [10, 18]) is proportional to

$\Gamma_{t}^{2PM}\propto
R_{t}\cdot\frac{n_{u}^{2}L_{\parallel}^{4/7}}{q_{\parallel,u}^{3/7}}\propto
R_{t}\cdot\frac{\langle
n_{e}\rangle^{2}(L_{\parallel}^{OSP})^{4/7}}{P_{SOL}^{3/7}}$ (30)

Here, the same approximations (see (26)) employed in section 3.2 are used.
From (30), the variable

$F\equiv\frac{\Gamma_{t}~{}(P_{SOL}/P_{SOL}^{ref})^{3/7}}{(\langle
n_{e}\rangle/\langle
n_{e}\rangle^{ref})^{2}(L_{\parallel}^{OSP}/L_{\parallel}^{ref})^{4/7}}\propto
R_{t}$ (31)

can be defined as a corrected parallel particle flux density. Plotting $F$
against $R_{t}$ consistently accounts for $\langle n_{e}\rangle$, $P_{SOL}$
and $L_{\parallel}^{OSP}$ variations between compared cases, according to the
2PM. Here, $\langle n_{e}\rangle^{ref}=5.5\cdot 10^{19}~{}\mathrm{m}^{-3}$,
$L_{\parallel}^{ref}=10~{}\mathrm{m}$ and $P_{SOL}^{ref}=2.5\cdot
10^{5}~{}\mathrm{W}$ are considered. From (31), $F$ is expected to increase
linearly with $R_{t}$ which is, however, not observed in the experiments,
figure 8b. For all the different cases, the variation of $F$ with $R_{t}$ is
much weaker than predicted by the 2PM. Significant discrepancies from the 2PM
predictions, consistent with this result, are also observed for the integrated
particle flux (not shown).

Figure 8: $R_{t}$ sweeps at approximately constant line-averaged density
$\langle n_{e}\rangle$ \- peak parallel particle flux density $\Gamma_{t}$ (a)
and variable $F$ (b) against $R_{t}$. For shots in the LSN configuration (cyan
and brown lines), two lines are reported representing the two sweeps performed
(outward and inward).

##  4 Modelling of parallel flows in the divertor

SXD experiments in TCV, section 3.2 \- 3.3, showed much weaker total flux
expansion effects than predicted by the 2PM. Parallel flows can potentially
explain part of this discrepancy, section 2.3 \- 2.4. As a direct, reliable
measurement of parallel flows was not available in the experiments, analytical
and numerical modeling are presented in this section to assess if this effect
can be significant in the experimental conditions.

###  4.1  Mach number evolution and possibility for supersonic flows

The impact of parallel flows on total flux expansion effects increases with
higher values of the Mach number $M$ in the divertor, by definition of
$M_{eff}$, (15) - (17). The evolution equation for $M$ along a SOL flux tube
is presented here, obtained by combining particle and momentum balances. The
simple case of a single hydrogenic ion species ($n_{e}=n_{i}=n$) is considered

$\displaystyle(1-M^{2})\partial_{s}(M)=$
$\displaystyle\frac{1+M^{2}}{nc_{s}}S_{par}$
$\displaystyle+\frac{M(1+M^{2})}{c_{s}}\partial_{s}(c_{s})$
$\displaystyle+A_{\perp}M\partial_{s}(\frac{1}{A_{\perp}})-\frac{M}{m_{i}nc_{s}^{2}}S_{mom}$

where $s$ is a length coordinate along the flux tube, increasing from upstream
to target ($s=s_{t}$). $S_{par,mom}$ are effective sources/sinks in the flux
tube, respectively for particles and momentum, related to volumetric sources
and cross-field effects, see (8)-(9). $c_{s}=\sqrt{T_{e}+T_{i}/m_{i}}$ is the
local sound speed. The derivation of (4.1) is shown in appendix D. It is
important to note that (4.1) must satisfy the Bohm condition [32], i.e. $M\geq
1$, at the target, as the target corresponds to the sheath entrance in this
fluid model. Qualitatively, (4.1) shows that:

  * •

four main drivers are responsible for $M$ variation along a field line:
particle and momentum effective sources/sinks (both volumetric sources and
cross-field effects), sound speed $c_{s}$ variation and total flux expansion.

  * •

the effect of these drivers is reversed when $M$ is lower or higher than $1$,
i.e. whether the plasma flow is subsonic or supersonic.

  * •

a necessary (but not sufficient) condition for a supersonic transition is a
change of sign of the right-hand-side in (4.1).

Moreover, the constraint provided by the Bohm condition at the target allows
to extract a sufficient (but not necessary) condition for the development of
supersonic flows at the target. Taking a region $[s_{t}-\Delta s,s_{t}]$
before the target: if, in this region, the right-hand-side of (4.1) is
negative, then the flow must be supersonic.

This case is interesting for the SXD configuration. Considering the ideal case
where $S_{par,mom}$ and $\partial_{s}(c_{s})$ are negligible across the region
$[s_{t}-\Delta s,s_{t}]$, the right-hand-side of (4.1) is then negative in the
same region in the outer leg, due to total flux expansion, and supersonic
flows would arise. The idea that the convergent-divergent magnetic structure
of a flux tube, such as in the outer leg of a SXD configuration, can favour
supersonic flows at target has already been addressed before [33, 34, 35]. The
possibility for supersonic flows to arise at the OSP for the SXD configuration
was already demonstrated numerically before [36, 37]. In consequence,
$M_{eff}$ and parallel flow effects on total flux expansion are suggested to
be potentially significant for the SXD configuration.

Moreover, when the other drivers are considered, for low target temperature
(i.e. $T_{t}\lesssim 5~{}eV$) as required in detached conditions, in front of
the target:

  * •

$S_{par}$ is negative: at low temperatures the ionisation front moves away
from the target and the the only effective particle sources/sinks will be
radial transport†††Here and in the following radial particle and momentum
transport are considered negative contributions to $S_{par,mom}$. This is
generally true for the hottest channels in the common flux region of the SOL.
and recombination that both make $S_{par}$ negative.

  * •

$\partial_{s}(c_{s})$ is negative.

  * •

$S_{mom}$ is negative due to charge exchange, recombination and radial
transport (thus $-S_{mom}$ will be positive).

In the outer leg of the SXD configuration, 3 out of 4 terms on the right-hand-
side of (4.1) are therefore negative in the outer leg of a detached SXD
configuration. This type of analysis can be also applied to other divertor
configurations, even with negligible total flux expansion, and supersonic
flows can arise for similar target conditions [38, 39, 40, 41].

###  4.2 SOLPS-ITER modelling of SXD experiments in TCV

A SOLPS-ITER simulation of TCV is used to study the patterns of parallel flows
and $M_{eff}$ in the divertor region. SOLPS-ITER (B2.5-Eirene) is a transport
code that couples the B2.5 multi-fluid solver with the kinetic Monte Carlo
model neutral code Eirene [17, 42]. SOLPS-ITER is one of the most established
SOL plasma simulators and it has been used for the design of the ITER divertor
[43, 6]. The simulation discussed in this work was already presented in [44],
where details of the simulation setup are reported. The simulation features a
baffled LSN geometry, figure 9, with parameters typical of an ohmically-heated
L-mode plasma in TCV, such as the experiments presented in section 3. Drift
effects are not included in this work, so radial transport is purely anomalous
and incorporated by an artificial cross-field diffusion. The fuelling rate is
varied to allow the analyses of different divertor conditions.

Figure 9: B2.5 and Eirene meshes for the SOLPS simulation. The pink arrow
indicates the location for the fuelling. The green shaded area indicates the
domain considered for the analyses of the outer leg.

At the targets, a Dirichlet boundary condition satisfying the marginal Bohm
criterion [32] is applied, i.e. the parallel ion velocity at the sheath
entrance is forced to match the plasma sound velocity (accounting for carbon
impurities resulting from wall sputtering). This means that a Mach number
$M=1$ at the target is imposed, excluding, a priori, supersonic flows at the
target (see section 4.1). This implies that the following evaluation of
$M_{eff}$ is conservative: $M_{eff}$ could potentially have higher values in
reality.

To compute $M_{eff}$, the common flux region of the outer leg is considered in
the simulation, taking as the upstream location the divertor entrance, figure
9. This is also a conservative choice: the value of $M_{eff}$ usually has a
minimum for a choice of upstream location which is close to the X-point (see
appendix B).

For each flux tube in the analysed domain, $M_{eff}$ is evaluated according to
(15) - (17). Its value varies both with the radial position of the flux tube,
figure 10b, and with divertor conditions, figure 10a, as higher values are
achieved for lower target temperatures. For the intermediate and higher
fuelling rates, where divertor conditions are similar to the experiments
presented in section 3, $M_{eff}\geq 0.5$ for all the flux tubes. In
consequence, this SOLPS-ITER simulation suggests that $M_{eff}$ and parallel
flow effects on total flux expansion are significant in these conditions, even
with the conservative choices in the present analyses.

Figure 10: (a) Mach number $M=v_{\parallel}/c_{s}$ map in the divertor region,
for the intermediate fuelling rate case, where $v_{\parallel}$ is the parallel
velocity of the main plasma species $D^{+}$ and $c_{s}$ is the plasma sound
speed, accounting for C impurities resulting from wall sputtering. (b)
effective Mach number $M_{eff}$ for different flux tubes, in the case of
intermediate fuelling rate, mapped against their radial distance from the
separatrix at the OMP $dr_{sep}^{OMP}$. The flux tube with higher target
temperature is indicated by the red vertical line. (c) effective Mach number
$M_{eff}$ for the flux tube with higher target temperature
($\simeq~{}\mbox{Min }M_{eff}$), against $\mbox{Max }T_{e}^{OT}$.

##  5 Conclusions

In this paper, the role of total flux expansion on the total pressure balance,
neglected in the 2PM, is made explicit, by including the effect of parallel
flows. This effect is quantified by a newly defined lumped parameter, the
effective Mach number $M_{eff}$, characterising each flux tube. Its
introduction allows to decouple geometrical effects from cross-field and
sources/sinks effects in the momentum loss factor $f_{mom-loss}$. In
consequence, 2PM target quantity expressions can be rewritten and their
dependence on total flux expansion, through the ratio $R_{u}/R_{t}$, is now
modified and varying with $M_{eff}$. For increasing $M_{eff}$, total flux
expansion effects on target quantities is reduced and eventually qualitatively
reversed, starting with the particle flux. The same modifications are applied
to the detachment model by Lipschultz et al., showing how the dependence of
the detachment window on total flux expansion weakens for increasing
$M_{eff}$. Physically, this is ascribed to the fact that a negative static
pressure gradient is established towards the target due to total flux
expansion.

Experiments on the SXD configuration are carried out in the TCV tokamak,
testing 2PM predictions. These are all ohmically-heated L-mode discharges, in
SN configuration, with $I_{p}\sim 250kA$ and the ion $\nabla B$ drift directed
away from the X-point. In core density ramps, featuring a baffled geometry and
different fuelling locations, the CIII front movement in the outer leg, used
as a proxy for the plasma temperature, shows variations with the outer strike
point radius $R_{t}$ much weaker than 2PM predictions, especially when
variations in $P_{SOL}$ and $L_{\parallel}$ are taken into account. In OSP
sweeps, at approximately constant core density, the peak particle flux density
at the OSP remains rather independent of $R_{t}$ variations, while a linear
increase was predicted by the 2PM.

To understand if parallel flow effects can be significant in the experiments
presented in this work, in the absence of experimental parallel flow
measurements, both analytical and numerical modeling are employed. It is shown
that supersonic flows, and therefore larger values of $M_{eff}$, are favoured
in a SXD configuration due to the convergent-divergent magnetic structure of
flux tubes in the outer leg. Moreover, the analyses of a SOLPS-ITER simulation
of a baffled LSN geometry in TCV, with parameters typical of an ohmically-
heated L-mode plasma, show that $M_{eff}\geq 0.5$ in the outer leg, for
divertor conditions similar to the ones present in the experiments, even with
conservative choices in its evaluation. The modeling then suggests that
parallel flows are, at least partially, causing the discrepancy between the
2PM predictions and the experiments.

## Acknowledgements

This work has been carried out within the framework of the EUROfusion
Consortium, via the Euratom Research and Training Programme (Grant Agreement
No 101052200 — EUROfusion) and funded by the Swiss State Secretariat for
Education, Research and Innovation (SERI). Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union, the European Commission, or SERI. Neither the European
Union nor the European Commission nor SERI can be held responsible for them.

##  Appendix A Derivation of 2PM expressions for target quantities

In this appendix the expressions (2.1)-(2.1) are derived. To simplify the
final expressions with respect to those reported in [10], it is assumed:

  * •

(S-I) only hydrogenic ion species (i.e. $n=n_{e}=n_{i}$) and no net current
(i.e. $v_{\parallel}=v_{e,\parallel}=v_{i,\parallel})$.

  * •

(S-II) thermal equilibration is achieved in the flux tube (i.e.
$T=T_{e}=T_{i}$).

An additional general assumption is needed:

  * •

(A-I) the target corresponds to the sheath entrance (i.e.
$q_{\parallel,t}=\gamma n_{t}T_{t}M_{t}\sqrt{2T_{t}/m_{i}}$, where
$M_{t}=v_{\parallel,t}/c_{s,t}=v_{\parallel,t}/\sqrt{2T_{t}/m_{i}}$ is the
Mach number at the target and $\gamma$ is the sheath heat transmission
coefficient). Note that, by Bohm condition at the sheath entrance, $M_{t}\geq
1$ must hold.

Introducing the standard definitions of power and momentum loss factors
(5)-(6) and using the above assumptions

$\displaystyle(1-f_{cooling})q_{\parallel,u}R_{u}=$ $\displaystyle~{}\gamma
n_{t}T_{t}M_{t}\sqrt{\frac{2T_{t}}{m_{i}}}R_{t}$ (33) $\displaystyle(1-f_{mom-
loss})p_{tot,u}=$ $\displaystyle~{}2(1+M_{t}^{2})n_{t}T_{t}$ (34)

where
$p_{tot,t}=p_{tot,t}^{e}+p_{tot,t}^{i}=2n_{t}T_{t}+m_{i}n_{t}v_{\parallel,t}^{2}=2n_{t}T_{t}(1+M_{t}^{2})$.

The factor $n_{t}T_{t}$ is isolated in (34) and substituted into (33), before
isolating $T_{t}$ to obtain

$\displaystyle T_{t}=$
$\displaystyle\frac{2m_{i}(1+M_{t}^{2})^{2}}{\gamma^{2}M_{t}^{2}}\cdot\frac{q_{\parallel,u}^{2}}{p_{tot,u}^{2}}$
$\displaystyle\cdot\frac{(1-f_{cooling})^{2}}{(1-f_{mom-
loss})^{2}}\cdot\left(\frac{R_{u}}{R_{t}}\right)^{2}$

$n_{t}$ is then obtained from (34) and (A)

$\displaystyle n_{t}=$
$\displaystyle\frac{\gamma^{2}M_{t}^{2}}{4m_{i}(1+M_{t}^{2})^{3}}\cdot\frac{p_{tot,u}^{3}}{q_{\parallel,u}^{2}}$
$\displaystyle\cdot\frac{(1-f_{mom-
loss})^{3}}{(1-f_{cooling})^{2}}\cdot\left(\frac{R_{t}}{R_{u}}\right)^{2}$

Finally, $\Gamma_{t}$ is obtained as
$n_{t}v_{\parallel,t}=M_{t}n_{t}\sqrt{2T_{t}/m_{i}}$

$\displaystyle\Gamma_{t}=$ $\displaystyle\frac{\gamma
M_{t}^{2}}{2m_{i}(1+M_{t}^{2})^{2}}\cdot\frac{p_{tot,u}^{2}}{q_{\parallel,u}}$
$\displaystyle\cdot\frac{(1-f_{mom-
loss})^{2}}{(1-f_{cooling})}\cdot\left(\frac{R_{t}}{R_{u}}\right)$

Note that $\gamma=\gamma(M_{t})\simeq 7.5+M_{t}^{2}$ [18, 19], therefore
$\gamma_{0}=\gamma(M_{t}=1)\simeq 8.5$. The target quantities can then be
rewritten, by grouping the terms directly depending on $M_{t}$ as factors of
unitary value when $M_{t}=1$

$\displaystyle T_{t}=$
$\displaystyle\left(\frac{8.5(1+M_{t}^{2})}{2(7.5+M_{t}^{2})M_{t}}\right)^{2}\cdot\frac{8m_{i}}{\gamma_{0}^{2}}\cdot\frac{q_{\parallel,u}^{2}}{p_{tot,u}^{2}}$
$\displaystyle\cdot\frac{(1-f_{cooling})^{2}}{(1-f_{mom-
loss})^{2}}\cdot\left(\frac{R_{u}}{R_{t}}\right)^{2}$ $\displaystyle n_{t}=$
$\displaystyle\left(\frac{8(7.5+M_{t}^{2})^{2}M_{t}^{2}}{8.5^{2}(1+M_{t}^{2})^{3}}\right)\cdot\frac{\gamma_{0}^{2}}{32m_{i}}\cdot\frac{p_{tot,u}^{3}}{q_{\parallel,u}^{2}}$
$\displaystyle\cdot\frac{(1-f_{mom-
loss})^{3}}{(1-f_{cooling})^{2}}\cdot\left(\frac{R_{t}}{R_{u}}\right)^{2}$
$\displaystyle\Gamma_{t}=$
$\displaystyle\left(\frac{4(7.5+M_{t}^{2})M_{t}^{2}}{8.5(1+M_{t}^{2})^{2}}\right)\cdot\frac{\gamma_{0}}{8m_{i}}\cdot\frac{p_{tot,u}^{2}}{q_{\parallel,u}}$
$\displaystyle\cdot\frac{(1-f_{mom-
loss})^{2}}{(1-f_{cooling})}\cdot\left(\frac{R_{t}}{R_{u}}\right)$

These expressions recover (2.1)-(2.1)-(2.1) when $M_{t}=1$, that is hypothesis
(S-III) in section 2.1.

##  Appendix B Further comments and insights on total flux expansion effects
on momentum balance and on the effective Mach number $M_{eff}$

The synergy between parallel flows and total flux expansion on total pressure
balance

A short insight on the physical intuition behind the synergy between parallel
flows and total flux expansion is provided here, highlighting the difference
with the power balance.

Starting from (11) it is possible to notice that, contrary to the power
balance expression (10), the total flux expansion effect
$-R^{-1}\partial_{s}(R)$ on the local total pressure variation is weighted by
$\kappa=mnv_{\parallel}^{2}/p_{tot}=M^{2}/(1+M^{2})$. In other words, the
local flux expansion effect is re-scaled according to the local parallel flow
conditions, in terms of $M$. In particular, for $M\ll 1$, total flux expansion
effects can be neglected.

The physical intuition is that the only component of the total pressure which
is subject to the effect of locally varying cross-section in the flux tube is
the dynamic pressure $mnv_{\parallel}^{2}$. This is because, in this work, the
static pressure ($p=nT$) is considered isotropic‡‡‡This assumption may be
questionable in some conditions, and the anisotropy of pressure, especially
for ions, in parallel and radial directions might play a direct role on total
flux expansion effects [37]., while the dynamic pressure is anisotropic, with
a preferential direction along the flux tube. Mathematically, this is
reflected in (9) by the fact that dynamic pressure enters the balance via the
divergence operator while the static pressure via the gradient operator.

Mathematical definition of $M_{eff}$, counter-intuitive values and its
dependence on the upstream location

From (15) - (17), $M_{eff}$ (or $\kappa_{eff}$) can be defined as: the value
of $M$ (or $\kappa$) which, when constant from upstream to target, would
provide the same total pressure variation $p_{tot,t}/p_{tot,u}$ due to total
flux expansion. Despite $\kappa=M^{2}/(1+M^{2})\in[0,1)$, from (15) it is
clear that $\kappa_{eff}$ can in principle take on any real value, due to the
averaging process against $R^{-1}\partial_{s}(R)$. This reflects in
$M_{eff}\rightarrow+\infty$ for $\kappa_{eff}\rightarrow 1^{-}$ or $M_{eff}$
assuming imaginary values for $\kappa_{eff}\notin[0,1]$ (see (17)). Despite
being counter-intuitive, this does not pose a direct problem to the
mathematical formulation: $M_{eff}$ always enters the expressions presented in
this work as $M_{eff}^{2}/(1+M_{eff}^{2})=\kappa_{eff}\in\mathbb{R}$. The
remaining problem is when $\kappa_{eff}\rightarrow\pm\infty$, that can happen
for $R_{u}\rightarrow R_{t}$. In this case, in the expressions presented in
this work, the indeterminate form $(R_{u}/R_{t})^{\kappa_{eff}}$ would appear.
This is a consequence of forcing the geometrical term in the total pressure
variation $(1-f_{mom-loss})$ to take the form of a power of $(R_{u}/R_{t})$
(see (13) - (2.2)). However, this was necessary to maintain a simple form
compatible with the algebraic expressions of the 2PM.

Here, a pathological example is provided to discuss the meaning of infinite or
imaginary values for $M_{eff}$, which could be difficult to understand in
terms of the $M_{eff}$ definition provided above. This also shows how the
$M_{eff}$ value depends on the upstream location. Consider a LSN geometry and
focus on computing $M_{eff}$ for a flux tube in the outer divertor leg,
varying the upstream location from the OSP to the OMP. A parallel length
coordinate $s$ is defined, increasing from $s=s_{OMP}$ at the OMP to $s=s_{t}$
at the OSP. Assume that:

  * •

the Mach number is unitary between the X-point and the OSP and null elsewhere,
that is $M=1\cdot\chi[s_{x},s_{t}]$

  * •

$R_{x}<R_{t}<R_{OMP}$, where $R$ is the major radius.

Figure 11: Graphical illustration of the example taken into consideration and
variation of major radius $R$, Mach number $M$ and $\kappa=M^{2}/(1+M^{2})$
against upstream poloidal location. The vertical dotted red line indicates the
location at which $R=R_{t}$, between OMP and X-point. Figure 12: Variation of
$\kappa_{eff}$, effective Mach number $M_{eff}$ and geometrical factor in the
total pressure variation $(p_{tot,u}/p_{tot,t})_{geom}$ against upstream
poloidal location in the example taken into consideration. The vertical dotted
red line indicates the location at which $R=R_{t}$, between OMP and X-point.
The vertical green line indicates the location at which $k_{eff}=1$. Figure
13: Variation of major radius $R$, Mach number $M$ and effective Mach number
$M_{eff}$ against upstream poloidal location in the simulation presented in
4.2. The intermediate fuelling case and the flux tube with the highest target
temperature are considered. The vertical dotted red line indicates the
location at which $R=R_{t}$, between OMP and X-point.

Figure 11 shows a graphical visualisation of this example. Despite the $R$
variation, the total pressure $p_{tot}$ does not vary due to total flux
expansion where $M=0$, i.e. between the OMP and the X-point. $p_{tot}$ then
gradually decreases, due to total flux expansion, between the X-point and the
OSP, as $M=1$ and $R$ increases (see (11)). In this simple case,
$\kappa_{eff}$ can be computed analytically for varying upstream location
$s_{u}\in[s_{OMP},s_{t}]$ by using (15)

$\displaystyle
k_{eff}=0.5\cdot\frac{\ln\left(R_{t}/R_{x}\right)}{\ln\left(R_{t}/R_{u}\right)}~{}~{}$
$\displaystyle\mbox{ for }s_{u}\in[s_{OMP},s_{x})$ $\displaystyle k_{eff}=0.5$
$\displaystyle\mbox{ for }s_{u}\in[s_{x},s_{t}]$

$M_{eff}$ can be then computed by (17), together with the geometrical factor
in the total pressure variation $p_{tot,t}/p_{tot,u}$ (see (2.2))

$\displaystyle\left(\frac{p_{tot,t}}{p_{tot,u}}\right)_{geom}$
$\displaystyle=\left(\frac{R_{u}}{R_{t}}\right)^{0.5\cdot\ln(R_{t}/R_{x})/\ln(R_{t}/R_{u})}=$
$\displaystyle=e^{-0.5\cdot\ln(R_{t}/R_{x})}=$
$\displaystyle=\left(\frac{R_{x}}{R_{t}}\right)^{0.5}~{}~{}\mbox{ for
}s_{u}\in[s_{OMP},s_{x})$
$\displaystyle\left(\frac{p_{tot,t}}{p_{tot,u}}\right)_{geom}$
$\displaystyle=\left(\frac{R_{u}}{R_{t}}\right)^{0.5}~{}~{}\mbox{ for
}s_{u}\in[s_{x},s_{t}]$

The results are represented in figure 12. For a choice of upstream location
between the OSP and the X-point, where $M=1$, $M_{eff}$ and $k_{eff}$ are
constants. The total pressure variation, due to total flux expansion, is
reflected in the variation in $(p_{tot,u}/p_{tot,t})_{geom}$. When the
upstream location is shifted beyond the X-point and towards the OMP, as the
total pressure no longer varies due to total flux expansion,
$(p_{tot,u}/p_{tot,t})_{geom}$ is constant. However, as $(R_{u}/R_{t})$ keeps
varying in this region (in this example, increasing towards the OMP),
$M_{eff}$ also varies to accommodate this change. When $(R_{u}/R_{t})$
increases above a given threshold (where $\kappa_{eff}=1$), a positive
$M_{eff}$ can no longer accommodate this variation and imaginary values are
obtained. This is understandable in terms of the definition provided above:
taking for example the OMP as the upstream location, for which
$R_{u}/R_{t}>1$, there exists no constant value of $M\in\mathbb{R}$ which
would result in a total pressure decrease towards the target, as in this
example.

Similar results can be obtained in more realistic cases, such as for example
the SOLPS-ITER simulation analysed in section 4.2. Also in this case, flux
tubes feature a convergent-divergent magnetic structure, between the OMP and
the OSP, and a monotonically increasing $M$ towards the OSP, figures 10a and
13. These conditions tend to push the minimum for $M_{eff}$ close to the
poloidal location where $R$ is minimum, that is often the X-point location for
the standard geometry of outer legs in diverted configurations, figure 13.
This justifies why the choice of the divertor entrance, as the upstream
location to evaluate $M_{eff}$ in section 4.2, was termed as conservative.

Dependence of $M_{eff}$ on the divertor leg geometry

$M_{eff}$ is derived, through $\kappa_{eff}$, from a weighted average of
$\kappa=M^{2}/(1+M^{2})$ along the flux tube, where the weighting factor is
the local relative variation of the flux tube area $R^{-1}\partial_{s}(R)$
(see (15) - (17)). This implies that for a given $M$ distribution, from
upstream to target, the local flux expansion distribution along the leg
influences the value of $M_{eff}$ and, therefore, the magnitude by which total
flux expansion effects are reflected on total pressure variation, target
quantities and detachment window. In other words, the divertor leg geometry
influences the sensitivity to total flux expansion effects.

Here, a couple of pathological examples are provided to better highlight this
point. Two cases, with the same total flux expansion $R_{t}/R_{u}$, are
considered in which the local flux expansion is constant and focused only:
(case A) in the region where $M=0$; (case B) in the region where $M=1$.
Consider a SOL flux tube and a field aligned length coordinate $s=[0,L]$,
where $s=0$ corresponds to the upstream position (with major radius $R_{u}$)
and $s=L$ corresponds to the target position (with major radius $R_{t}$).
Assume the following profiles for the Mach number along the flux tube

$\displaystyle M=1\cdot\chi[L-\Delta,L]$

and for local relative flux expansion

(case A)
$\displaystyle~{}~{}\frac{1}{R}\partial_{s}(R)=\ln{\frac{R_{t}}{R_{u}}}\cdot\frac{\chi[0,L-\Delta]}{(L-\Delta)}$
(case B)
$\displaystyle~{}~{}\frac{1}{R}\partial_{s}(R)=\ln{\frac{R_{t}}{R_{u}}}\cdot\frac{\chi[L-\Delta,L]}{\Delta}$

where $\chi[s_{1},s_{2}]$ is a function which equals 1 in between $s_{1}$ and
$s_{2}$ and 0 elsewhere, and $\Delta\in(0,L)$. In practice, it is imposed that
$M$ will increase instantaneously from $0$ to $1$ in the portion
$[L-\Delta,L]$ of the flux tube in front of the target. Notice that in both
cases the total flux expansion $R_{t}/R_{u}$ is the same. Figure 14 shows a
graphical visualisation of these examples, for the outer leg of a single-null
configurations (taking the X-point as upstream).

Figure 14: Graphical illustration of the two examples taken into consideration
(case A on the left, case B on the right). Upstream location is here the
X-point and the flux tube of interest is in the outer leg.

Computing now $\kappa_{eff}$ by (15) and $M_{eff}$ by (17), it is found

$\displaystyle\mbox{(case A)}~{}~{}~{}\kappa_{eff}=0~{}$
$\displaystyle\rightarrow~{}~{}~{}M_{eff}=0$ $\displaystyle\mbox{(case
B)}~{}~{}~{}\kappa_{eff}=0.5~{}$ $\displaystyle\rightarrow~{}~{}~{}M_{eff}=1$

It is then clear the drastic change in $M_{eff}$ depending on the geometry of
the flux tube, considering the same total flux expansion and flows pattern.

##  Appendix C Derivation of detachment window expression

The derivation of (24) is presented. This is similar to the original
derivation reported in [11]. In addition, the same hypothesis of thermal
equilibration in the flux tube (i.e. $T=T_{e}=T_{i}$) is adopted, as in
appendix A. Therefore, the plasma static pressure is $p=2nT$.

Consider the total steady-state energy balance in a flux tube. Assume (a)
cross-field transport effects are negligible. Assume also that (b) the ratio
$f_{cond}$ of conducted to total parallel power density is constant, and (c)
Spitzer’s formulation for parallel heat conductivity can be used:
$\kappa_{\parallel}=\kappa_{0}T^{5/2}$. The power balance is then

$H=-\frac{1}{f_{cond}}B\partial_{s}\left(\frac{\kappa_{\parallel}}{B}\partial_{s}(T)\right)$
(41)

where $s$ is the length coordinate along a field line from target to upstream,
here considered as corresponding to the X-point ($s:~{}[0,s_{x}]$).

It is assumed that (d) $H=-n^{2}f_{I}Q(T)$, which means the local effective
power sources/sinks can be approximated with their only radiation-related
component. Here $n$ is the plasma density, $f_{I}$ is the impurity fraction
($f_{I}=n_{I}/n$) and $Q(T)$ is a radiation efficiency function. The radiation
efficiency $Q(T)$ is assumed (e) to be a function which peaks sharply just in
a range $[T_{c},T_{h}]$ (with $T_{c}<T_{h}$) and it’s null outside of it.

The following change of variable is introduced

$dz=\frac{B_{x}}{B}ds$ (42)

Practically,
$z=\int_{0}^{z}dz^{\prime}=\int_{0}^{s(z)}\frac{B_{x}}{B}ds^{\prime}$ will be
the volume ($ds/B\propto dV$) of the flux tube contained from the target
($s,z=0$) up to the point of interest, normalized by a reference perpendicular
area ($\propto 1/B_{x}$), where the upstream/X-point is taken as this
reference.

Defining

$\kappa=\kappa_{\parallel}\frac{B_{x}^{2}}{B^{2}}$ (43)

(41) becomes

$\partial_{z}q=H$ (44)

with

$q=-\frac{1}{f_{cond}}\kappa\partial_{z}T$ (45)

$q=(1/f_{cond})q_{\parallel,cond}B_{x}/B$ is then the total parallel power
$Q_{\parallel}\propto(1/f_{cond})q_{\parallel,cond}/B$ normalized by the same
reference perpendicular area $\propto 1/B_{x}$.

Taking (44) and multiplying both sides by $q$, then integrating from
$z(T_{c})$ to $z(T_{h})$ (note that $z(T_{h})>z(T_{c})$, in the chosen
coordinate system)

$[q^{2}]_{z(T_{c})}^{z(T_{h})}=-\int_{T_{c}}^{T_{h}}\frac{2}{f_{cond}}\kappa(T^{\prime})H(T^{\prime})dT^{\prime}$
(46)

Using assumptions (b)-(e), the square root of the integral on the right hand
side of this equation becomes

$\Delta
q_{rad}\equiv\sqrt{\frac{2\kappa_{0}}{f_{cond}}\int_{T_{c}}^{T_{h}}\frac{B_{x}^{2}}{B^{2}}T^{5/2}n^{2}f_{I}Q(T)dT}$
(47)

Assume (f) the radiation region (i.e. the region in between $z(T_{c})$ and
$z(T_{h})$) is so narrow that $B$ and $f_{I}$ variations are negligible in it.
Assuming also that (g) volumetric processes and cross-field transport effects
on momentum balance and total pressure redistribution are negligible in this
region, this implies $p^{2}=4n^{2}T^{2}=p_{tot}^{2}/(1+M^{2})^{2}$ can be
taken out of the integral as its variation will be then linked just to total
flux expansion effects (hence, $B$ variation), negligible by assumption (f).
Therefore

$\Delta
q_{rad}=\frac{B_{x}}{B_{z(T_{h})}}p_{z(T_{h})}\sqrt{\frac{\kappa_{0}}{2f_{cond}}f_{I}\mathcal{F}}$
(48)

with $\mathcal{F}=\int_{T_{c}}^{T_{h}}\sqrt{T}Q(T)dT$.

The pressure at the detachment front entrance $p_{z(T_{h})}$ is linked with
pressure upstream/at the X-point $p_{u}$ using (2.2), substituting $B\propto
R^{-1}$. It is assumed that (h) volumetric processes and cross-field transport
effects on momentum balance are negligible in the region between the X-point
and the detachment front entrance. It then holds

$\displaystyle\frac{p_{tot,z(T_{h})}}{p_{tot,x}}$
$\displaystyle=\frac{1+M^{2}_{z(T_{h})}}{1+M_{x}^{2}}\frac{p_{z(T_{h})}}{p_{u}}=$
$\displaystyle=\left(\frac{B_{z(T_{h})}}{B_{x}}\right)^{\kappa_{eff}^{x\rightarrow
z(T_{h})}}$

(48) becomes then

$\displaystyle\Delta q_{rad}$ $\displaystyle=$
$\displaystyle\frac{B_{x}}{B_{z(T_{h})}}\frac{1+M_{x}^{2}}{1+M^{2}_{z(T_{h})}}$
(50) $\displaystyle $ $\displaystyle~{} $
$\displaystyle\cdot\left(\frac{B_{z(T_{h})}}{B_{x}}\right)^{\kappa_{eff}^{x\rightarrow
z(T_{h})}}p_{u}\sqrt{\frac{\kappa_{0}}{2f_{cond}}f_{I}\mathcal{F}}=$
$\displaystyle=$
$\displaystyle\frac{1+M_{x}^{2}}{1+M^{2}_{z(T_{h})}}\left(\frac{B_{x}}{B_{z(T_{h})}}\right)^{1-\kappa_{eff}^{x\rightarrow
z(T_{h})}}$ $\displaystyle $ $\displaystyle~{} $ $\displaystyle\cdot
p_{u}\sqrt{\frac{\kappa_{0}}{2f_{cond}}f_{I}\mathcal{F}}$

Finally, to obtain a model for the operational window for different control
parameters, it is assumed that (i) the power leaving the cold detachment front
is negligible. This will imply $q_{z(T_{h})}=-\Delta q_{rad}$ by (46). The
power entering the hot detachment front must then match the power entering
upstream/at the X-point, thanks to assumption (d), and the latter can be
expressed as $q_{i}=-P_{SOL}$, by definition of $q$.

Now one can equate $q_{z(T_{h})}$ and $q_{i}$ and solve in terms of the
control parameters $\zeta=[p_{u},f_{I},P_{SOL}]$. The front position
$z(T_{h})$ is then set to be at the X-point first and then at the target to
find the corresponding values $\zeta_{x,t}$ (leaving the others parameters
constant). Dividing these two values, the detachment window is obtained

$\frac{\zeta_{x}}{\zeta_{t}}=\left(\left(\frac{B_{tot,x}}{B_{tot,t}}\right)^{1-k_{eff}}\frac{1+M_{x}^{2}}{1+M_{t}^{2}}~{}\right)^{\beta}$
(51)

with $\beta=[1,2,-1]$.

##  Appendix D Derivation of Mach number evolution equation

Consider the steady-state ion particle balance and plasma momentum balance
along a flux tube

$\displaystyle B\partial_{s}\left(\frac{nv_{s}}{B}\right)$
$\displaystyle=S_{par}$ (52) $\displaystyle
B\partial_{s}\left(\frac{m_{i}nv^{2}_{s}}{B}\right)$
$\displaystyle=-\partial_{s}(nT^{*})+S_{mom}$ (53)

where $s$ is a length reference coordinate along the flux tube and
$S_{par,mom}$ includes contributions from volumetric sources and cross-field
transport effects. A single hydrogenic ion species and quasi-neutrality
($n_{e}=n_{i}=n$) are considered. For the sake of simplicity in the notation,
$T^{*}=T_{e}+T_{i}$ is introduced.

Start rewriting the pressure term in (53)

$\displaystyle B\partial_{s}\left(\frac{m_{i}nv^{2}_{s}}{B}\right)=$
$\displaystyle-B\partial_{s}\left(\frac{nT^{*}}{B}\right)$ (54) $\displaystyle
$ $\displaystyle-\frac{nT^{*}}{B}\partial_{s}(B)+S_{mom}$

In both (52) and (54), isolate $\partial_{s}(n)$

$\displaystyle\partial_{s}(n)=\frac{S_{par}}{v_{s}}-\frac{nB}{v_{s}}\partial_{s}\left(\frac{v_{s}}{B}\right)$
(55) $\displaystyle
m_{i}v_{s}^{2}\partial_{s}(n)+nB\partial_{s}\left(\frac{mv_{s}^{2}}{B}\right)=-T^{*}\partial_{s}(n)$
(56) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}
-nB\partial_{s}\left(\frac{T}{B}\right)-\frac{nT^{*}}{B}\partial_{s}(B)+S_{mom}$

Reordering and inserting (55) into (56)

$\displaystyle-\frac{nB}{v_{s}}(m_{i}v_{s}^{2}+T^{*})\partial_{s}\left(\frac{v_{s}}{B}\right)+(m_{i}v_{s}^{2}+T^{*})\frac{S_{par}}{v_{s}}=$
$\displaystyle=-nB\partial_{s}\left(\frac{m_{i}v_{s}^{2}+T^{*}}{B}\right)$
$\displaystyle~{}~{}~{}-\frac{nT^{*}}{B}\partial_{s}(B)+S_{mom}$ (57)

Introducing $c_{s}=\sqrt{T^{*}/m_{i}}$ and reordering

$\displaystyle-B(v_{s}^{2}+c_{s}^{2})\partial_{s}\left(\frac{v_{s}}{B}\right)+v_{s}B\partial_{s}\left(\frac{v_{s}^{2}+c_{s}^{2}}{B}\right)=$
$\displaystyle=-(v_{s}^{2}+c_{s}^{2})\frac{S_{par}}{n}-v_{s}\frac{c_{s}^{2}}{B}\partial_{s}(B)+\frac{v_{s}S_{mom}}{m_{i}n}$
(58)

The left-hand-side of this equation is equivalent to

$-(c_{s}^{2}-v_{s}^{2})\partial_{s}(v_{s})+2v_{s}c_{s}\partial_{s}(c_{s})$
(59)

Exploiting this and introducing $M=v_{s}/c_{s}$, one obtains

$\displaystyle\frac{1-M^{2}}{c_{s}}\partial_{s}(v_{s})=2\frac{M}{c_{s}}\partial_{s}(c_{s})$
(60) $\displaystyle~{}~{}~{}~{}
+(1+M^{2})\frac{S_{par}}{nc_{s}}+\frac{M}{B}\partial_{s}(B)-\frac{MS_{mom}}{m_{i}nc_{s}^{2}}$

which can be rewritten as

$\partial_{s}(v_{s})=\partial_{s}(Mc_{s})=M\partial_{s}(c_{s})+c_{s}\partial(M)$
(61)

Exploiting this in (60) and using $B\propto(A_{\perp})^{-1}$, it is finally
possible to retrieve (4.1)

$\displaystyle(1-M^{2})\partial_{s}(M)=$
$\displaystyle\frac{1+M^{2}}{nc_{s}}S_{par}$ (62)
$\displaystyle+\frac{M(1+M^{2})}{c_{s}}\partial_{s}(c_{s})$
$\displaystyle+A_{\perp}M\partial_{s}(\frac{1}{A_{\perp}})$
$\displaystyle-\frac{M}{m_{i}nc_{s}^{2}}S_{mom}$

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