Numerical modelling of an enhanced perpendicular transport regime in the scrape-off layer of ASDEX Upgrade

The considered experimental scenarios were taken from the AUG discharge #33341 (figure 1), which is the main experiment discussed in [13]. It is a pure deuterium L-mode density ramp, i.e. in which the fueling was gradually increased. In this way a wide range of values in divertor collisionality was covered. It featured a toroidal field of −2.52 T and a plasma current of 0.80 MA.

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We tried to reproduce two different time intervals during the ramp, at 'low' and 'high' density respectively. The corresponding measurements were taken as input data for the simulations.

For low density/transport, from here on denoted as 'Case A', we considered the time t = 1.77 s. For high density/transport, from here on denoted as 'Case B', we considered the time t = 3.56 s. These are the same times at which the measurements discussed in [13] were taken. The second case featured a highly collisional divertor and a strong increase in intensity of radial filament propagation at the outer SOL midplane and, at the same time, the formation of a density shoulder and the achievement of a partially detached divertor plasma.

The employed computational grid, prepared for the considered scenarios, is shown in figure 2.

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The grid has a resolution of of 48 poloidal cells and 18 radial cells. The grid cells are properly aligned with the equilibrium magnetic flux surfaces of the AUG discharge #33341 at the considered time slices. The reconstructed equilibrium was not exactly the same for the two times, but the small differences in the parameters relevant to exhaust physics such as the connection length, being less than 10%, justified the use of one single grid for both cases. As 'outer midplane' we set the row of cells at a poloidal location slightly higher than the equatorial midplane (blue cells in figure 2). Here most of the plasma edge diagnostics, including the ones considered in this work, actually measure the radial plasma profiles [13].

The source for the deuterium molecules is located in the divertor region, in front of the dome baffle (green arrow in figure 2). This location resembles the position of the valves employed in the experiment for the gas puff. The source of neutrals at the puffing location is mono-energetic, with injected neutrals having a fixed energy equal to the thermal energy at room temperature. Active pumping takes place at the actual pumping surfaces in the divertor region, i.e. at the turbo- and cryopump (green lines in figure 2). We employed a pumping model for the neutrals which was experimentally validated at AUG. Neutral absorption on the turbo- and cryopump surfaces was modelled by albedo coefficients of 0.993 and 0.7 respectively. These values roughly reproduce the experimentally observed pumping speeds for deuterium neutrals at both pumping surfaces. The finite conductances for neutral transport existing between the various chambers in the divertor region were emulated by imposing fictitious ducts.

The standard set of atomic and molecular processes in EIRENE for pure deuterium SOLPS runs was used (see table 1). This set includes electron-impact ionization (EI), recombination processes (RC), molecular dissociation processes (DS), charge-exchange collisions (CX) and elastic scattering (EL).

Table 1. Atomic and molecular volumetric processes modelled by EIRENE in the performed simulations. All these are modelled using the rate coefficients present in the databases AMJUEL and HYDHEL.

Reaction	Type
$D + e \rightarrow D^+ + 2e$	EI
$D_2 + e \rightarrow D_2^+ + 2e$	EI
$D_2 + e \rightarrow D + D^+ + 2e$	DS-EI
$D_2^+ + e \rightarrow D^+ + D^+ + 2e$	DS-EI
$D^+ + e \rightarrow D$	RC
$D_2^+ + e \rightarrow D + D$	DS-RC
$D_2 + e \rightarrow D + D + e$	DS
$D_2^+ + e \rightarrow D^+ + D + e$	DS
$D + D^+ \rightarrow D^+ + D$	CX
$D_2 + D^+ \rightarrow D_2^+ + D$	CX
$D_2 + D^+ \rightarrow D_2 + D^+$	EL

Some relevant boundary conditions which were used are constant values of electron/ion heat fluxes crossing the core plasma boundary (see next subsection) and sheath conditions at the plasma/target interfaces in the divertor (i.e. parallel flow velocity larger than or equal to the local sound speed). Additionally, the default choice for the parallel heat flux limiters was used [18].

The main physical inputs to be chosen as input parameters for SOLPS simulations are input power, density level and an assumption for the perpendicular transport coefficients.

Setting the input power in the computational domain was straightforward. We imposed given values of electron/ion heat fluxes on the core plasma boundary in order to model the power leaving the core plasma region. The sum of these corresponds to the total power entering the edge plasma region. We assumed that such power is carried half by electrons and half by ions. The total heating power for the cases A and B is, respectively, 620 and 820 kW (see figure 1). We considered the same estimate for the fraction of radiated power in the core stated in [13], i.e. 0.15 for the case A and 0.25 for the case B. So, the total input power in the computational domain for the two cases was assumed as 520 and 610 kW respectively.

More challenging were the specification of the density and a choice for the perpendicular transport coefficients. A number of uncertainties had to be faced in this regard.

First, we chose to impose the density through a feedback scheme in the simulations. Namely, arbitrarily chosen values for the separatrix electron density n_e,sep at the outer midplane had to be defined: these could be achieved through a feedback mechanism regulating the gas puff source during the run. However, the existing methods of magnetic reconstruction at AUG allow the knowledge of the position of the separatrix only to within an error of several mm. This implies a strong uncertainty in the estimation of the separatrix density n_e,sep at the outer midplane in the considered experimental scenarios. To handle this issue, we considered the position of the separatrix R_sep as unknown a priori, and took the experimental radial plasma profiles at the outer midplane as function of the major radius R as only experimental basis. We chose the signals provided by the integrated data analysis (IDA) for the electron density profiles, which are derived from the combination of different heterogeneous diagnostics [23, 24].

Second, SOLPS-ITER assumes an anomalous perpendicular transport. So, empirically-derived radial profiles for the perpendicular transport coefficients need to be defined by the user (namely, for all radial locations of the computational domain). The most crucial issue was achieving a correct fit for the experimental radial density profiles for all modelled cases. Therefore, since first-principle-based theories for predicting the transport coefficients required for modelling filamentary transport are missing, a fine tuning of the employed perpendicular particle transport was needed.

In detail, SOLPS-ITER can make use of a particle diffusivity D_n and an anomalous pinch velocity v_n [20]. The first one defines a diffusive transport component which is proportional to the radial density gradient. The second one, instead, defines a real advective transport component proportional to the density itself. We chose to model the increase of transport solely through the imposition of an adequate radial profile of the particle diffusivity D_n as the use of an anomalous pinch velocity would be numerically problematic. In the simulations such a coefficient intends then to average the effects of the intermittent filament-driven advective transport on a sufficiently long time scale, assuming the meaning of an 'effective' particle diffusivity. In other words, it was used to simulate any processes related to radial particle transport, regardless of their actual nature (which might be diffusive of advective).

A diffusive component of the perpendicular energy transport was also assumed, by means of thermal diffusivities χ_e , χ_i . The assumed χ_e , χ_i profiles were based upon the setup of similar SOLPS simulations of L-mode discharges at AUG [25], without carrying out a further fine tuning. Indeed, it is generally accepted that the radial propagation of filaments does not affect the temperature-gradient-driven thermal transport [13].

In order to deal with these issues we developed a fitting procedure for such input parameters, which is described in detail in the appendix A. We satisfactorily resolved the uncertainty on the position of the separatrix and the arbitrariness of the choice of the perpendicular particle transport coefficients. The employed strategy was coupling the experimental knowledge of the radial plasma profiles at the outer midplane with the results of the simulations in a self-consistent way. This allowed us to choose the most appropriate estimate for the position of the separatrix (and hence for n_e,sep ) and reproduce the experimental radial density profiles at the outer midplane. The latter was done through an algorithmic optimization of the radial profiles of D_n .

The results of the algorithm included the most reliable density levels to be imposed in the final simulations, and the relative uncertainties (i.e. the values of n_e,sep and the relative lower and upper bounds). This allowed us to present the results for all modelled cases in confidence bands resulting from the simulations being run for the 'central' n_e,sep values as well as for the relative lower/upper bounds. These are depicted as shadowed areas in the plots presented later in the text.

Figure 3 shows the good agreement achieved between modelled and experimental radial density and temperature profiles at the outer SOL midplane for both cases. For the density profiles, this was precisely the goal of the employed fitting algorithm: the deviation of the modelled values from the experimental values is not larger than 10% at all radial locations. The diffusivity profiles required for this, resulting from the employed fitting algorithm, are also reported in the top plots. We can note that:

• For case A (low density, no density shoulder) the best assumption for the position of the separatrix corresponded to a separatrix density of 0.78(±0.06) × 10¹⁹ m⁻³. The nearly exponentially decaying radial density profile in the SOL could be reproduced by means of a quite flat profile for the particle diffusivity, with values of about 0.5–1 m² s⁻¹, i.e. of the order of the Bohm scaling.
• For case B (high density, fully formed density shoulder) the best assumption for the position of the separatrix corresponded to a separatrix density of 1.52(±0.18) × 10¹⁹ m⁻³. The region with flattened density could be reproduced by means of a strong localized increase of the particle diffusivity, up to several tens of m² s⁻¹.

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The unusually high values of the perpendicular transport coefficient used for the case B are supported by a recently published transport model by Manz et al [26]. This model interprets the advective radial propagation of microscopic structures with some typical radial velocity (such as filaments) through a diffusive approximation, including the effects of time intermittency by means of some auto-correlation time. It suggests that a perpendicular transport dominated by the fast radial ballistic propagation of filaments could be described through a time-averaged steady-state effective particle diffusivity of the order of tens of m² s⁻¹. Such order-of-magnitude estimate is, indeed, in reasonable agreement with what achieved with the present SOLPS results.

The first matter of investigation was the possible impact of the increase of radial particle transport on the radial energy transport at the outer SOL midplane. An increase of radial energy transport was indeed observed experimentally in the considered high density/transport scenario [13]. This was attributed to the fact that, in their radially-directed motion, filaments might directly carry some energy with them towards the far SOL.

SOLPS-ITER is suitable for investigating the possible impact of filament-driven transport on the energy fluxes. The modelled radial energy flux density is [20]

$$\begin{equation} q_{\perp} = - n_e \left(\chi_e \frac{\partial T_e}{\partial r} + \chi_i \frac{\partial T_i}{\partial r} \right) + \frac{5}{2}\Gamma_{\perp}(T_e+T_i). \end{equation} \tag{ 1 }$$

The first component, i.e. the conductive component, which is not expected to be directly affected by the filament propagation, is not related to the particle transport intensity, but only described by the thermal diffusivities χ_e , χ_i . The second one, i.e. the convective component, which is the one expected to increase, is indeed sensitive to an increase of the radial particle flux. Therefore, an increase of the last component following an increase of the particle diffusivity is plausible.

We investigated the possible impact of the increase of radial particle transport on radial energy transport at the LFS (low field side) of the SOL by performing an energy balance on a domain radially extended in the SOL and poloidally extended about the outer midplane (yellow volume in figure 4).

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At any radial location r − r_sep , the decrease of the amount of energy $Q_{\perp}$ which is still present at the midplane is balanced by the cumulative parallel losses $Q_{\parallel}$ and the cumulative volumetric energy losses, i.e. occurring between r − r_sep  = 0 and that given radial location.

The energy still present at the midplane at each radial location is computed as

$$\begin{equation} Q_{\perp}(r-r_{sep}) = \sum_{poloidal} q_{\perp} A_{\perp} \end{equation} \tag{ 2 }$$

i.e. as the product between modelled radial energy flux density $q_{\perp}$ entering each cell from its left boundary and surface area $A_{\perp}$ of such boundary. The sum is performed over all the cells present in the poloidal extension of the considered domain.

The calculation yielding the cumulative parallel losses is

$$\begin{equation} \begin{split} Q_{\parallel}(r-r_{sep})& = \left[ 2\pi \int_{0}^{r-r_{sep}} \left( \frac{B_{\theta}}{B}q_{\parallel} \right)R(r^{^{\prime}})\,dr^{^{\prime}}\right]_{\textrm{div,in}} \\ & \quad +\left[ 2\pi \int_{0}^{r-r_{sep}} \left( \frac{B_{\theta}}{B}q_{\parallel} \right)R(r^{^{\prime}})\,dr^{^{\prime}}\right]_{\textrm{div,out}}. \end{split} \end{equation} \tag{ 3 }$$

The poloidal projection of the modelled parallel energy flux density $q_{\parallel}$ is integrated in the toroidal and radial directions. The net parallel energy loss is obtained summing the energy flows calculated in this way crossing top and bottom boundaries of the considered domain, i.e. leaving the domain and directed to inner and outer divertor respectively.

The competition between radial energy transport and parallel energy losses resulting from such balance is visible in figure 5 for both cases. All values are normalized to the input power $P_{\textrm{SOL}}$ in the balance domain, i.e. the energy ejected from the core plasma through the separatrix and crossing the left boundary of the domain. $P_{\textrm{SOL}}$ corresponds to 508 and 538 kW for cases A, B respectively, out of a total input power in the computational domain of 520 and 610 kW respectively.

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At the separatrix, $Q_{\perp}/P_{\textrm{SOL}}$ is equal to one by definition. Then, going further the fraction of energy which is still present at the midplane SOL decreases as the cumulative parallel losses towards the divertors and the volumetric losses increase. At high density/transport a large fraction of energy ejected from the core evidently remains at the midplane SOL also several cm far from the separatrix. As expected, this is because of a strong increase of the fraction of energy radially convected in the first cm of the SOL, i.e. precisely where the radial particle transport was increased. Because of this, the fraction of energy $Q_{\textrm{wall}}/P_{\textrm{SOL}}$ which reaches the outer boundary of the domain increases from less than 10% at low density/transport to about 30% at high density/transport. This result agrees very well with the experimental power balance of the same scenarios presented in [13] and more in detail in [27].

One aspect possibly related to the shoulder formation is the local ionization at the SOL midplane. We focused then on evaluating a possible increase of the local particle source at the midplane, between cases A, B, by ionization of the neutrals.

The total neutral density n_n and the total particle source by ionization S_i at the midplane cells of the computational domain were extracted for both cases (figure 6). The ionization source is plotted as dashed lines, normalized to the density at the separatrix to account for the increase of the ionizations merely related to the increase of density.

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At low density/transport S_i increases continuously towards the separatrix, following a decrease of the ionization mean free path (caused in turn by the increase of plasma density and temperature). At high density/transport, instead, an evident 'ionization peak' appears. Interestingly, the radial location of such peak corresponds exactly to where the flattening starts degrading (top right plot in figure 3). This is consistent with the neutral population (solid lines) decaying much faster, when approaching the separatrix, at high density/transport rather than at low density/transport.

The second figure of merit for the numerical investigation was the impact of the enhanced transport on the divertor conditions. Such impact was experimentally revealed [13]: while raising the density in the AUG discharge #33341, the one under investigation, the divertor plasma was seen to transit from a high-recycling regime to a partially detached regime, at least for the outer target. This was manifested after observing the rollover of the ion flux onto the target.

We analyzed then the outer divertor target conditions in order to see if this was reproduced by the performed simulations. Figure 7 shows the total particle (i.e. ion) flux density and heat flux density striking the outer target surface for all cases, including the 'intermediate' case characterized by a high density without the transport enhancement. A quantitative reproduction of the divertor conditions was out of the scope of this work, especially since the simulations were performed without considering the impact of drifts. Nevertheless, the modelled ion flux density profiles are in good qualitative agreement with the Langmuir probe data from the modelled discharges [13]. The modelled heat flux density profiles, instead, slightly underestimated the experimental ones [13].

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At high density, high transport, both ion and heat flux profiles onto the target are visibly flattened. Their peak value is decreased and radially shifted outwards. This is sufficient to say that a pronounced detachment state has been achieved by the simulations in this scenario [28]. In the high density scenario without enhanced transport, instead, the heat flux profile is already strongly flattened, while the ion flux profile is still peaked close to the separatrix strikepoint. This is a strong indication that the enhancement of perpendicular transport has played a crucial role in dropping the ion flux onto the target, over the mere density increase: only after the increase of transport, n_e,sep being the same, the peak ion flux is decreased roughly by half.

As final task of our exploration of the numerical plasma solutions, we tried to quantify the dissipation of momentum and power along the SOL and characterize the relative loss mechanisms. The goal was understanding whether the main role in achieving the experimentally revealed highly dissipative regime in the high transport scenario [13] was played by the mere increase of density or specifically by the increase of perpendicular transport.

First, we quantified the power losses occurring in the SOL plasma before reaching the targets. This was done by extracting, from the plasma solutions, the energy flows crossing the main boundary surfaces of the computational domain. We took the total power $P_{\textrm{SOL}}$ crossing the separatrix as reference for the input power into the SOL. Part of this power, $P_{\textrm{wall}}$, is radially transported towards the main chamber wall. Another part, $P_{\textrm{targets}}$, is parallel transported towards the divertor targets. The remaining part, $P_{\textrm{vol}} = P_{\textrm{SOL}}-P_{\textrm{wall}}-P_{\textrm{targets}}$ is the total volumetric power loss.

In [29] a power loss factor was defined as the total fraction of the SOL input power being dissipated before reaching the targets, i.e. as

$$\begin{equation} f_{\textrm{power}} \equiv 1 - \frac{P_{\textrm{targets}}}{P_{\textrm{SOL}}}. \end{equation} \tag{ 4 }$$

We calculated this factor for all cases, obtaining $f_{\textrm{power}} = 0.36$ for the low density, low transport case, $f_{\textrm{power}} = {0.83}$ for the high density, low transport case, and $f_{\textrm{power}} = 0.91$ for the high density, high transport case. Power losses are, as expected, quite relevant at high density, reaching a dissipation of over the 80% of the total input power into the SOL. This is mainly by means of an increase of radiation losses by plasma-neutral interactions. However, the value which was calculated for the 'real' high density, high transport case is not much larger than the one which would be achieved in the same scenario but without transport enhancement (0.91 vs. 0.83). Therefore, power losses are achieved mainly with the mere increase of density rather than with the enhancement of transport.

Finally, we investigated the intensity of momentum losses occurring in the plasma flow from an upstream location (outer midplane) to a downstream location (outer target) and their correlation with the increase of transport. To do this, we looked directly at the fluid equations solved by SOLPS-ITER.

We employed a two-point-model (2PM)-like approach, relating the plasma profiles between the upstream and downstream locations. Integral two-point relations were derived considering the same treatment developed by Kotov and Reiter in [30]. Namely, we integrated the parallel momentum balance equation solved by B2.5 between such two poloidal locations, assuming a steady-state situation (i.e. assuming the numerical convergence being reached). The difference in total pressure

$$\begin{equation} p_{\textrm{tot}} = n_e(T_e+T_i) + m_in_ev_{\parallel}^2 \end{equation} \tag{ 5 }$$

i.e. the sum of static and dynamic pressure, between downstream and upstream, for each flux tube, results in a total momentum source term (i.e. integrated between upstream and downstream). A brief derivation of such momentum balance analysis is given in the appendix B. The results for all cases are shown in figure 8.

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While at low density momentum losses are negligible, moving on to high density a strong increase of the total loss was found, mainly concentrated in the near SOL. After the transport enhancement, profile and relative intensity of momentum losses are similar. However, the peak of the pressure profile on the target is quite radially shifted to the far SOL. As a result, pressure is completely removed from the flux tubes close to the separatrix when approaching the target, differently from the high density, low transport case, where some amount of pressure was still present on the near-strikepoint portion of the target. In section 5 the physical mechanisms leading to such momentum losses will be investigated in more detail.

No agreement has been reached in the literature yet about the mechanisms relating perpendicular transport and shoulder formation. It is accepted that changes in the divertor collisionality $\Lambda_{\textrm{div}}$ lead to changes in filament characteristics and, hence, to a transition in the global perpendicular SOL transport intensity [11]. However, it is still debated if this is sufficient for forming the shoulder.

In the already mentioned studies performed with pure deuterium L-mode plasmas at AUG by Carralero et al [12], an increase of $\Lambda_{\textrm{div}}$ over some threshold value appears as a necessary condition for the shoulder formation. The transition in perpendicular transport occurring at high collisionality was seen to directly contribute, in first instance, to bring more particles towards the far SOL. Following Lipschultz et al [31], it was assumed that the consequent rise of density in the far SOL would make it more 'opaque' to the neutrals. This would contribute to ionize much more neutrals close to the wall rather than close to the separatrix. Consequently, the local volumetric particle source would be enhanced, increasing even further the far SOL density and making the plasma here even more opaque. This 'ionization feedback loop' would sustain the elevated plasma density in the far SOL.

Similar studies performed with L-mode plasmas at JET by Wynn et al [14] point out that some additional mechanism would be needed. The increase of $\Lambda_{\textrm{div}}$ (and hence of perpendicular filament-driven transport) was found to be not sufficient for sustaining the shoulder formation. Increasing $\Lambda_{\textrm{div}}$ through a ramped fueling, the flattening of the profiles was seen to scale with the collisionality, in agreement with the AUG results. Increasing, instead, $\Lambda_{\textrm{div}}$ by cooling down the divertor through an increased nitrogen seeding, no shoulder was observed. The only quantity which was seen to correlate well with the presence of a shoulder in a wide range of scenarios is the outer divertor Balmer $D_{\alpha}$ line emission magnitude. This is a proxy for the amount of ionization processes in the divertor, i.e. for the intensity of neutral recycling at the targets. This led to assume that any mechanisms involved in the shoulder formation would need to be related with divertor recycling.

The results about far SOL midplane ionization (figure 6) confirm the presence of an ionization peak in the far SOL, as assumed by Carralero et al. This would effectively prevent neutrals from reaching the separatrix. The localized increase of ionization processes would provide the required additional plasma particles. This finding supports the assumption of an ionization feedback loop playing a role in forming the shoulder.

Wynn et al pointed out that ionization in the far SOL might be enhanced as a consequence of an increased density in the far SOL, rather than being its cause. If this was true, another physical mechanism would be needed for the 'existence' of more particles in the far SOL. Let's consider a radial balance equation for particles in the simple form [29]

$$\begin{equation} \frac{\partial n_e}{\partial t} = D_n \frac{\partial^2 n_e}{\partial r^2} - \frac{n_e}{\tau_{\parallel}}+S_i \end{equation} \tag{ 6 }$$

with parallel loss term $\frac{n_e}{\tau_{\parallel}}$, being $\tau_{\parallel}$ a characteristic parallel loss time. If the local particle source by ionization S_i really does not play a causal role, then such a balance could be altered only through a direct increase of the radial particle flux (first term on the R.H.S. of the equation (6)) over the parallel losses (second term on the R.H.S. of the equation (6)). We found out that, while the radial particle flux at the outer midplane increased when increasing the particle diffusivity, the ratio of radial-to-parallel particle flux is not substantially changed. This means that the increase of radial particle transport at the midplane is compensated by a concurrent increase of the parallel losses towards the divertor. What 'feeds' the shoulder, at least in the performed simulations, cannot be anything other than an increase of S_i .

This assumption is further supported by the strong decrease of the ion temperature in the far SOL when the shoulder is present (bottom right plot in figure 3). This is consistent with the picture of hot ions ejected from the separatrix joining a population of cold ions being originated in the far SOL by ionization of neutrals, already discussed in [12].

Additional aspects of the numerical plasma solutions support the assumption of the ionization loop in the far SOL in conditions of enhanced perpendicular transport. The increased radial energy transport (figure 5) might be invoked as a further, if not the main, feature triggering the formation of the ionization peak. Namely, only after the transport enhancement a larger amount of energy can be transported towards the far SOL (where a larger population of neutrals is present). Therefore, there is more possibility of supplying the energy cost needed for each ionization event, implying an increase of the number of ionizations per electron in the far SOL.

This assumption is further supported after decomposing the volumetric energy losses (from the electron channel) in the high-transport solution: in the far SOL they are predominantly due to electron-impact ionizations.

Additionally, this might even explain why Wynn et al encountered highly collisional scenarios without a density shoulder in nitrogen-seeded plasmas. Namely, reaching high values of $\Lambda_{\textrm{div}}$ might have been sufficient to provide a strong filament-driven radial particle transport. However, this could not translate into a larger amount of energy transported towards the far SOL because of increased impurity radiation due to nitrogen. As there was not enough energy available in the far SOL to account for the cost due to more ionization, the activation of the ionization feedback loop might have been not possible.

After suggesting the first necessary 'ingredient' for the ionization loop, i.e. the energy, we investigated the possible origin of the second one, i.e. the neutrals.

It is usually assumed that, in conditions of enhanced perpendicular transport, an important source of neutrals in the far SOL is due to the onset of a recycling condition at the main chamber wall. Namely, the larger fraction of ions striking the wall would result in a larger amount of recycled neutrals emitted towards the plasma. Such situation, first observed in Alcator C-Mod by LaBombard et al [32], is referred to as main wall recycling.

SOLPS allows to characterize the volumetric sources driven by atomic/molecular processes, computed by EIRENE, in terms of the physical origin of the involved neutrals. So we were able to decompose the ionization source profiles of plasma particles at the outer midplane cells of the domain (dashed lines in figure 6) in terms of the origin of the neutrals which were ionized (figure 9).

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In both cases the ionization source originates only to a small part from neutrals recycled at the wall. The gas puff source is also negligible in contributing to the total ionization source (about 1%). Most of the ionized neutrals come from divertor recycling. An increase of main wall recycling, in absolute quantity, is actually present at high density/transport, following the increase of the ion flux striking the wall, in agreement with many experimental observations. However, it is still not dominant in providing the neutrals which are ionized at the midplane (less than 10% of the total). Therefore, the existence of a localized increase of the ionization source in the case where the shoulder is present seems to not be strongly affected by the presence of main wall recycling.

This result, however, is in contrast with already published EMC3-EIRENE modelling works, such as the one performed by Lunt et al [33]. There it was argued that the presence (i.e. the activation) of the main wall recycling mechanism is a necessary condition for forming a density shoulder. A possible underestimate of this contribution in SOLPS-ITER could be explained by the fact that, in the modelled case, the computational grid cannot be actually extended up to the physical walls, while in EMC3-EIRENE this is possible. Thus, we think that this aspect would need a more careful evaluation.

The major relevance of the (outer) target recycling source, for the neutrals ionized at the midplane, is consistent with the assumption of Wynn et al of any major mechanism responsible for shoulder formation to be related with divertor recycling. In order to investigate this in more detail we compared, for the case where the shoulder is present, the ionization front of the neutrals recycled at the outer target and the poloidal dependence of the total pressure drop (the last one evaluated in the section 3.5). Figure 10 shows the contour levels describing the fraction of neutrals recycled at the outer target which have ionized after being emitted towards the SOL plasma. The poloidal dependence of the total pressure drop is shown plotting a 'local' momentum loss factor, defined in a similar way as in [29], as

$$\begin{equation} f_{\textrm{mom,local}}(x) \equiv 1 - \frac{p_{\textrm{tot}}(x)}{p_{\textrm{tot,up}}}. \end{equation} \tag{ 7 }$$

This quantifies the decrease of total pressure (equation (5)) while travelling from upstream (i.e. outer midplane) towards the target along a poloidal coordinate x for each flux tube, with respect to the total pressure upstream $p_{\textrm{tot,up}}$ at that flux tube.

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The strong momentum losses which occur in this case along the SOL, described in the section 3.5, are concentrated close to the targets, where plasma-neutral interactions are most relevant (see next section). Therefore, at the poloidal location at which most of the neutrals recycled from the target have been ionized, the pressure is more or less the same as at the midplane.

The fact that pressure is roughly conserved between midplane and ionization front of the target-recycled neutrals suggests a correlation between density shoulder and divertor neutral recycling, as first assumed by Wynn et al [14]: it is the recycling profile at the outer target which might determine the midplane profile.

Pressure conservation between midplane and divertor ionization front implies, indeed, the conservation of the shape of the radial density profile [29]. At the location of the ionization front of the target-recycled neutrals, this is already flattened. This suggests that the shape of the midplane profile is merely controlled by the plasma conditions at the ionization front. And what takes place at the ionization front of the target-recycled neutrals must be, in turn, correlated to the target recycling profile (through divertor geometry and neutrals opacity).

While the correlation of plasma profiles between midplane and ionization front is easily explained by pressure conservation, the exact correlation between ionization front and divertor target cannot be established with accuracy, because of the strong momentum losses occurring between the two locations. However, in view of the presented results, an at least qualitative connection between midplane plasma profiles and target recycling profiles appears plausible. This is, indeed, consistent with most of the neutrals contributing to the far SOL ionization source upstream being originated from target recycling (see figure 9). Additionally, this is also consistent with the experimental observation, made not only at JET [14] but also at AUG [34] of the $D_{\alpha}$ emission in the outer divertor region moving further from the separatrix following the onset of the shoulder formation.

Summarizing, the results presented in this section are consistent with different findings reported in the literature. In agreement with the numerical EMC3-EIRENE results presented by Carralero et al [12], SOLPS-ITER as well indicates the existence of an enhanced ionization in the far SOL when a density shoulder is present. Moreover, consistently with the experimental observations made at JET by Wynn et al [14], SOLPS-ITER indicates the presence of an interplay between shoulder formation and divertor neutral recycling processes.

The performed analysis of various aspects of the numerical plasma solutions suggests the following mechanism for the shoulder formation at the outer SOL midplane, pictured in figure 11:

• (1)  
  The enhancement of perpendicular transport in the SOL causes, in first instance, a reshaping of the ion flux profile onto the target towards flux tubes further away from the separatrix (see left plot in figure 7).
• (2)  
  The modified recycling profile may imply the presence of a larger neutral population in the far SOL flux tubes. The larger amount of energy transported towards the far SOL (see right plot in figure 5) is able to supply the energy cost needed for ionizing more neutrals, triggering an enhanced far SOL ionization (see figures 6 and 9).
• (3)  
  This implies an elevated plasma density in the far SOL upstream. At the same time a large ion flux striking the target far from the separatrix is sustained, finally closing a 'loop' which had been first initiated by the enhanced perpendicular transport.

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In the literature it is widely accepted that the mutually complementary effects of the increase of radiation power losses and volumetric recombination are the processes causing a drop in the parallel ion flow before it reaches the targets in detached conditions [35]. According to a simple 2PM analysis of the SOL plasma the access of a detached regime is also intimately related to an increase of the momentum losses in the plasma from upstream to target [29]. The ion flux striking the target can be shown, indeed, to be reduced by a factor which scales with the square of the total pressure drop between upstream and target. Such a correlation between increase of momentum losses and detachment onset has been confirmed both experimentally and numerically by Paradela Pérez et al [36].

However, it is still debated if a causality relationship exists between momentum losses and reduction of the ion flux. According to Krasheninnikov and Kukushkin [15], momentum losses would be a mere complementary feature arising from processes (e.g. CX collisions) which are effective at low temperatures, i.e. the same temperatures which facilitate recombination processes. A more advanced analytical model recently developed by Stangeby [16, 17] suggests that the increase of momentum losses along the SOL is, instead, critical in directly causing the ion flux rollover.

On the other hand, the correlation between collisionality regimes, intensity of perpendicular transport and detachment onset has been already investigated numerically by Wiesen et al [37]. The ion flux rollover during a fueling ramp was seen to be facilitated by the application of a model in which the intensity of perpendicular transport scales with the collisionality. This already showed a direct impact which an increased transport has in causing the detachment. However, no thorough investigation was performed about the physical mechanisms relating these two aspects.

Analyzing the numerical plasma solutions we tried to identify and quantify the physical mechanisms which lead to the detachment in conditions of enhanced transport, and, at the same time, to establish role and relevance of momentum losses in achieving it.

According to figure 7, increasing the density was sufficient to achieve the 'power detachment', i.e. the complete flattening of the heat flux density profile at the outer target. This is consistent with a power loss factor for the SOL (equation (4)) having reached already a large and almost saturated value in the modelled intermediate high density, low transport case. On the other hand, the achievement of the 'plasma detachment', i.e. a relevant flattening of the ion flux profile, was possible only after the increase of transport. In the intermediate case, the ion flux onto the target was still large and peaked close to the separatrix.

For the ion flux to drop in a relevant way from low transport to high transport cases, in spite of these having the same density regime (i.e. same n_e,sep ), the intensity of recombination must have increased. Plotting the 2D distributions of the recombination rate in the divertor (figure 12) reveals, indeed, that recombination is much more relevant after the increase of transport. This is especially true for the flux tubes close to the separatrix, i.e. precisely where the ion flux onto the target drops.

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We investigated then which is the physical mechanism, mainly effective only after the increase of transport, which might make the recombination rise. Looking at figure 8 reminds that the relative intensity of momentum losses is similar between high density, low transport case and high density, high transport case. However, the last one features a radial shift of the total pressure peak at the outer target, as a consequence of pressure being completely removed from the near-separatrix flux tubes. This is consistent with the ion flux being flattened precisely on the near-strikepoint portion of the target (left plot in figure 7).

We checked what makes the pressure completely disappear from the near-separatrix flux tubes only after the transport enhancement. For doing this we decomposed the total poloidally-integrated momentum source term considered in the section 3.5, i.e. the yellow lines in figure 8, in all its individual components as computed by SOLPS-ITER. The results for all cases are shown in figure 13. See again appendix B for more details.

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As one could expect, the bulk increase of momentum losses when moving from low to high density is driven by a stronger presence of plasma-neutral interactions (e.g. CX collisions). This mainly originates from the cooling down of the divertor after the density increase, which lowers the temperature sufficiently to promote these processes, while simultaneously sustaining high recycling levels.

However, the complete drop off of pressure onto the near-strikepoint portion of the target, occurring only after the transport enhancement, can be attributed to another loss component, whose intensity is represented by the yellow lines in figure 13. This is caused by the mechanism of radial momentum transport. This consists in momentum being gained in or removed by each flux tube directly due to perpendicular particle transport. Increased flows of particles radially directed towards the far SOL, indeed, can remove from the near SOL the momentum which is carried by such particles.

This term is always negligible in the two low transport cases with respect to the total loss. In the high transport case, instead, its relative contribution in the total loss is not negligible: it removes up to 30% of the total momentum from the flux tubes close to the separatrix, resulting in an even stronger momentum removal than the one solely caused by plasma-neutral interactions. Albeit remaining secondary in relative magnitude with respect to the loss due to plasma-neutral interactions, this mechanism is effective enough to allow for some redistribution of momentum from the near SOL to the far SOL. This explains the radial shift to the far SOL of the peak of the pressure profile onto the target.

After the power balance performed in the section 3.4, we found that the transport enhancement alone does not contribute much in removing more power from the plasma. This analysis is consistent with the heat flux profile at the outer target being already flattened after the mere increase of density, i.e. even without increasing the transport (right plot in figure 7). The increase of power losses, arising mainly after the density increase, is not sufficient alone in driving the processes (such as recombination) which make the ion flux drop before striking the targets. Indeed, they are not able to detach the plasma from the target in the high density, low transport scenario.

According to the results presented in this section, the transport enhancement does contribute instead in a relevant way in removing more pressure at least in the near-separatrix flux tubes, redistributing momentum towards the far SOL. We believe that this feature likely plays a key role in the achievement of a detached regime, which was experimentally observed and numerically reproduced. This is consistent with a drop in the ion flux onto the target occurring only in the high density, high transport scenario (left plot in figure 7).

We assume that what physically occurs is the following. The increase of power dissipation by radiation, following the increase of density, cools down the plasma near the targets. The complete removal of pressure in the near-separatrix flux tubes, occuring after the increase of transport, contributes to slow down the plasma flow in these flux tubes. Such situation finally increases the residence time of electrons and ions in a cold divertor. This ultimately promotes a stronger presence of recombination processes (see figure 12), which makes a large quantity of plasma particles 'disappear' before reaching the targets, at least close to the strikepoint.

The combination of both strong momentum and power losses is then necessary to detach the divertor plasma, at least in the performed simulations. Without the further pressure drop occurring after transport enhancement in the near-separatrix flux tubes, driven by momentum redistribution, recombination would not have been strong enough to drop the ion flux onto the near-strikepoint portion of the target.

Overall, these results confirm the presence of a direct link between perpendicular particle transport in the SOL and the access to a detached regime for the divertor, as already suggested by Wiesen et al [37]. Additionally, they support the idea, motivated by the analytical calculations performed by Stangeby [16, 17], of momentum losses being a necessary ingredient for the detachment to take place, in addition to the power losses.