Energy and particle balance during plasma detachment in a long-leg divertor configuration

Comprehensive studies of energy and particle balances in the transition to plasma detachment in an alternative divertor configuration with long outer legs are shown. Numerical simulations are performed with the 2D code suite SOLPS 4.3, using a disconnected double null grid with narrow, tightly baffled long poloidal leg divertors at the outer lower target and outer upper target. A particle count scan is performed using the ‘closed gas box’ model, where the tunable parameter in the simulations is the total number of deuterium particles in the simulation space and all other parameters are held fixed, including a constant input power and trace neon impurity radiation, to assess the physics of the transition to detachment in the system as the particle count increases. Three main aspects of the physics of divertor detachment are addressed: the criteria for the local onset of divertor detachment in each of the divertors, the distribution of heat flux and other plasma parameters between the four divertors as each divertor transitions to detachment, and the role of perpendicular transport in the transition to the detached regime. A synergistic mechanism by which the cross-field transport is reduced by factors associated with the onset of plasma recombination effects is identified. These results are compared to the existing understanding of the physics of the transition to plasma detachment in standard divertors.


Introduction
Managing heat and particle exhaust in tokamak devices remains an open challenge in the development of nextgeneration fusion machines. Present understanding indicates that the operational regime best suited for handling the extreme conditions of the plasma exhaust by allowing for both * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. heat removal and suppression of incoming particle and heat fluxes is the so-called 'detached' divertor regime. Virtually all present-day tokamaks and future reactors will operate in this detached divertor regime, where a collection of atomic and radiative processes nearly extinguishes plasma heat and particle flux before it reaches the divertor target.
While many aspects of the complex and multifaceted edge plasma remain poorly understood, the general physics of the mechanisms of the 'detachment' of plasma from the material surfaces in the 'standard' divertor regime have been explained [1,2]. However, a new class of magnetic divertor configurations have been proposed, the design of which has been guided by the underlying physics principles of plasma detachment in standard divertors. Alternative divertor configurations, the so-called 'advanced' divertors, are designed to minimize heat flux to the targets and improve detachment stability by introducing geometric variations in the magnetic field topology with characteristics that are thought to be favorable to stable detachment conditions. To do this, one common feature to many of these alternative divertor configurations is to manipulate the magnetic geometry of the plasma to poloidally extend the outer leg of the separatrix, such that the plasma-material interface at the outer target is far removed from the main plasma [3]. This long leg allows for a longer connection length, increased divertor volume, and, in some cases, larger wetted target area relative to standard divertor configurations.
Long legs are features of several proposed alternative divertor designs, shown in simulations of Super-X, XPTD, and others, and demonstrated in experiments, including the TCV tokamak, the KSTAR tokamak, and recent experiments in MAST-U [4][5][6][7]. These results have shown promise in the long leg as a novel divertor improvement, and simulations suggest the existence of stable, fully detached solutions for a multitude long-leg configurations in reactor-relevant conditions, making the long divertor leg configuration an appealing candidate for exhaust systems in future fusion pilot plants [8][9][10]. However, although these results are encouraging, no validation of the existing analytical models describing the transition to detachment has been performed for these long leg regimes. It is not yet fully known if and how the physics of detachment in standard divertors translates to these long legs, nor is there a definitive understanding of the impact of the long leg on plasma and impurity transport, meaning that stability and long-term operational outcomes of these long leg regimes remains unclear.
In this work, the SOLPS4.3 code is used to assess the transition to the detached divertor regime in a long leg divertor configuration. We hold the total number of particles in the simulation space constant in the so-called 'closed gas box' approximation of the tokamak divertor to assess the physics of the transition to detachment over a scan of increasing particle count. These simulations feature a deuterium plasma with a trace neon impurity, impinging on a tungsten target. Over the course of this scan, we find that the criteria for detachment in a local flux tube is similar to theory-based predictions and existing numerical simulations of 'standard' divertor targets. We observe asymmetry between the up-down and in-out divertors, with favorable characteristics in the long outer legs that are largely driven by cross-field transport. These simulations show that the transition to detachment is largely similar to plasma detachment in standard divertors.

Fundamental physics of divertor detachment
Fundamental physics of divertor detachment have been extensively analyzed and detailed in other works [1,2]. However, it is useful to briefly elucidate a few aspects of the existing understanding of the fundamental physics of detachment to emphasize how this understanding translates to the new magnetic geometries shown in this work.
The divertor is the exhaust and fueling system for the tokamak; field lines divert the exhausted heat and particles away from the fragile core plasma and into targets designed to handle the extreme conditions induced by the plasma exhaust. Initially, the plasma ions will hit the target plates and neutralize back into neutrals, which will form a protective cushion of neutral gas that will mitigate some of the incoming heat flux to the divertor target by expending energy on ionizing these newly-formed neutrals. This process will continue, eventually becoming so effective that the plasma-neutral-plasma ionization 'loop' will become the primary mechanism of dissipating heat in the divertor volume, and the divertor enters what can be characterized as the 'high-recycling regime'. An important characteristic of the high-recycling regime is that the plasma flux generated from the ionization of the newly-formed neutrals in the divertor volume greatly exceeds the plasma flux from the scrape-off layer (SOL) or the influx of particles from neutral puffing by several orders of magnitude. Under these conditions, the scrape-off layer plasma influx can be ignored, and the SOL and divertor region can be considered as a 'selforganized dissipative closed plasma box' [1]. This means that effectively, the SOL/divertor region contains a fixed number of deuterium or hydrogen particles and impurity that is sustained by a fixed heat flux (Q SOL ) coming across the separatrix and into the divertor region. The plasma-neutral-plasma ionization cycle in the divertor volume sustains the upstream plasma since there is virtually no upstream plasma source under these conditions.
The detached divertor regime is a natural extension of the high-recycling regime, where the dissipative processes in the divertor volume become so effective that the plasma exhaust can be virtually extinguished before it hits the target. Three synergistic atomic processes work together to reduce and eliminate plasma flux to the target: impurity radiation (which works as an energy sink, but can contaminate or reduce performance of the core plasma), ionization of neutrals (which works as an energy sink, but a source of plasma), and volumetric plasma recombination (which works as an extremely effective particle sink, but only 'activates' at single eV electron temperatures for molecular-activated recombination (MAR) processes sub-eV electron temperatures for electron-ion three body recombination). Because these processes are largely dependent on temperature and each process is optimized at different temperatures, the divertor region can be imagined as a series of spatially stratified regions where heat flux is dissipated by different processes and temperature progressively reduced along the divertor leg before reaching the target, as in figure 1. While ion-neutral friction is important for the momentum balance and cooling the divertor to sub-eV temperatures, it cannot alter the flux to the target. Instead, flux to the wall is characterized by the relation [11,12] where Γ t is the flux to the target and Γ rec is the plasma sink from recombination. The plasma source from ionization is characterized by the first term, where Q SOL is the incoming Figure 1. A cartoon imagining a simplified divertor. Each 'region' is designated by a different color and characterized by a different temperature-dependent dissipative process: impurity radiation, neutral ionization, and plasma recombination (both molecule activated recombination, MAR, and electron-ion recombination, EIR).
heat flux from the scrape-off layer, Q imp is the heat loss related to impurity radiation, E ion is the ionization energy needed per ionization event, which characterizes the plasma source as the total energy available for ionization with its corresponding energy 'cost' [13]. This relation explains the behavior of the familiar 'rollover curve', observed extensively in experiments and simulations, where detachment can be characterized as a 'rollover' of the ion flux, or saturation current, to the material surfaces as a parameter related to the dissipation (e.g. density or particle count) is varied. Ion flux increases (as expected in the high recycling regime) then decreases, as the dissipative processes become effective enough to cool the divertor low enough to trigger volumetric plasma recombination processes, and the protective cloud of neutrals that forms at the divertor target will 'detach' the plasma from its contact with the surface.
Fundamentally, equation (1) describes the flux to the divertor targets as a function of the plasma source due to ionization and plasma sink due to recombination. However, this formulation neglects the effects of perpendicular transport, which might be important for long leg divertor configurations, as the increase in connection length and divertor volume in the long leg could significantly impact turbulence in the divertor leg. Simulations of the ADX conceptual tokamak (with a longleg divertor design) using the UEDGE plasma edge modeling code showed that perpendicular transport to the side walls was significant for a steady-state detached plasma [8,10], but the full scope of the impact of this substantial influence of perpendicular transport and side wall interaction on the edge plasma in the transition to detachment is unclear, and warrants further investigation. As such, equation (1) should be modified for long leg divertor configurations to consider the effects of perpendicular transport, or where Q ⊥ refers to the heat lost due to perpendicular transport onto the main chamber and divertor walls in the main chamber and divertor. We distinguish this perpendicular heat flux leaving the plasma along the separatrix with Q ⊥ , while the heat load to the target will be denoted later with Q target .

Detachment onset criterion
Rollover of the ion saturation current can be indicative of detachment, but does not necessarily tell the full story; it is possible that other mechanisms, such as particle losses to side walls [14][15][16] or variations in Langmuir probe measurements, might create an artificial impression of the detachment of a plasma when the actual physical mechanisms of true plasma detachment are not actually engaged. Similarly, the 'Degree of Detachment' parameter [17], which relies on measurements of the saturation current, could face similar issues with interpretation of the characterization of plasma detachment. As such, it is useful to articulate a physics-based scaling relation for the local onset of detachment to appropriately characterize the detachment state of a particular plasma and to make meaningful comparisons between different machines and operating conditions. Such a parameter was identified [18] and confirmed in a DIII-D-like plasma with simulations [19] as the ratio of the upstream pressure P up to the specific heat flux entering the recycling region q recycle in a single flux tube, or P up /q recycle , which indicates detachment when it meets and exceeds some critical value (approximately 20 N MW −1 for deuterium plasmas). This is because maintaining that upstream plasma pressure requires some energy flux above the critical level to sustain the recycling, and therefore provide the sufficient inventory of neutrals to maintain the stability of radiation, ionization, and recombination processes in the divertor plasma.

SOLPS4.3
To assess the physics of detachment in a long leg divertor configuration, a scan of simulations was performed with the SOLPS4.3 code [20,21]. The SOLPS4.3 code is a suite of plasma transport codes, composed of the 2D multifluid B2 code to solve the plasma state along and across magnetic flux surfaces coupled with the 3D multi-species Monte Carlo EIRENE code to solve the state of the neutral particles. Here, the behavior of electrons and ions (deuterium and all charge states of neon) are calculated with the B2 fluid code and used as the background plasma for EIRENE calculations of the trajectory of the neutrals (deuterium and neon). The B2 code solves transport equations for electrons and ions, including continuity, momentum, and energy balance equations in the magnetic configuration of a tokamak. Equations solved in B2 are derived from a modified set of the Braginskii equations, where transport along field lines is assumed to be classical, but includes some ad hoc processes (such as anomalous or turbulence-driven transport coefficients) that allow for The primary divertor has a short inner leg (outlined in dashed red) and long outer leg (solid red). The primary separatrix is drawn as a dark black line, and the secondary separatrix is drawn as a dashed black line. (c) The secondary divertor also has a short inner leg (dashed blue) and a long outer leg (solid blue). The vessel wall is drawn in orange on each figure; the overlap of the vessel wall and the simulation grid on the inner primary divertor is a consequence of a limitation of the parameterization used by the plotting software to create a closed shape with no overlapping lines, and does not reflect the real geometry of the vessel at that point (which actually sits just outside the plasma simulation grid).
cross-field transport. The EIRENE code calculates individual trajectories of neutral particles, and includes atomic processes, such as ionization, charge exchange, elastic collisions, and recombination. MAR processes (MAR) are not included since the MAR process is not expected to contribute significantly to effective recombination of the divertor plasma for high power and high density cases [22], but electron-ion recombination processes are included. Drifts were not included in these simulations, as they are not included in the SOLPS4.3 code package.
It is important to note that drifts have been shown to have a profound impact on plasma detachment in simulations in multiple plasma edge codes [23,24], where the dynamics of the plasma (and resultant neutral dynamics in response to changes in the plasma behavior) are shown to be strongly influenced by the presence of drifts. However, it was observed that the presence of drifts had little impact on the steady-state solutions of SOLPS-ITER simulations with disconnected double-null geometries with and without drifts, while similar simulations in connected double-null and single-null geometries showed significant differences between comparative simulations with and without drifts [25]. All simulations presented in this work feature a computational grid with a disconnected double-null configuration, and up-down asymmetries in plasma parameters that are sometimes ascribed to the presence of drifts are evident in the simulation results. While the lack of a model for drifts in the SOLPS4.3 code package prohibits the study of the impact of drifts on these simulations in the first place, it is reasonable that the more physically realistic geometry of the disconnected double-null (and corresponding increased complexity of the computational domain) might mitigate the impact of drifts on the qualitative nature of the steady-state of the simulations presented in this work.

Setup
The computational grid features a disconnected double null configuration with a long, tightly baffled outer divertor leg with a tilted target at the outer divertors (both upper and lower) and short, tightly baffled inner divertor legs, shown in figure 2. Unlike a symmetric double null configuration, where the two X-points lie on the same flux surface and are connected by one separatrix, the disconnected double null reflects a more realistic consideration where there is no exact magnetic connection between the upper and lower divertor X-points [26]. In this situation, there is a primary X-point on the last-closed flux surface, with a secondary X-point on another magnetic flux surface just outside the last closed flux surface [27]. The divertor with both primary and secondary separatrices will receive incoming fluxes from both the separatrices, while the other divertor will only receive fluxes from the secondary separatrix, which results in an imbalance in the overall magnitude of the integrated fluxes directed towards each divertor. In these simulations, the primary X-point in the disconnected double null configuration is in the lower divertor. As such, to avoid a loss of generality in this work, the lower divertor will be denoted as the 'primary' divertor and the upper divertor will be denoted as the 'secondary' divertor, referring to the intersection of the principal separatrix with each target plate. This nomenclature used for identification of each divertor is shown in figure 2.
These simulations are performed with a total of 30 MW input power, carried evenly between ions and electrons, and fixed across all simulations. Cross-field transport coefficients are constant and set at χ ⊥ = 1.0 m 2 s −1 and D ⊥ = 0.3 m 2 s −1 , which were chosen to emulate a plasma regime with strong turbulence and no edge transport barrier in a compact (R 0 = 1.85 m), high-field (12.2 T) tokamak with high input power [28], resulting in a heat flux width λ q = 0.2 mm. Unlike the SOLPS-ITER simulations presented in that reference, the SOLPS4.3 code package does not have the capability to incorporate spatially varying transport coefficients, so these transport coefficients are consistent with the ones used outside of the imposed transport barrier that is present in the simulations shown in that work (the transport coefficients listed in that article only describe the near-SOL transport barrier values). They are also equivalent to a 0-D estimate that is consistent with the projected separatrix temperature listed in that reference.
The SOL width (from the separatrix to the edge of the plasma domain) is 5 mm, and the distance between the primary and secondary separatrices is 2 mm. At the midplane, there is a 7 mm gap between the primary separatrix and the vessel wall; the SOLPS code only simulates neutrals, not plasma, all the way to the vessel wall, but the rather wide SOL width included in the plasma simulation domain covers most of this gap, and the divertor side walls in the vessel geometry are close to the plasma simulation domain in the divertor legs. As such, all fluxes out of the north and south boundaries (with boundary conditions of a 3 cm radial decay length for temperature and density) are assumed as losses to the walls.
Boundary conditions were selected to achieve a 'closed box' setup to emulate the high-recycling regime, where the total number of deuterium particles is held constant in a simulation as the simulation is run to steady state. The next simulation in the scan is generated by replicating the steady-state simulation, holding all input parameters constant, incrementally modifying the total number of particles in the edge of the existing steady-state simulation (denoted in this work as N edge D ), and running the new simulation to steady-state. This process is repeated to generate a bank of steady-state simulations, each with a slightly different and gradually increased inventory of deuterium particles, that can be analyzed as individual 'snapshots' that collectively emulate the transition to detachment. This approach, in contrast to the more common methodology of varying upstream density at the core-edge interface to generate a parameter scan, allows for a more natural evolution of plasma parameters and their response to varying input conditions that is more likely to capture behavior (like hysteresis or bifurcations) that might be obscured or otherwise not captured by modulating the upstream settings [2]. We note that using the total number of particles as the control parameter of the simulation is not repeatable in experiments; we emphasize that these studies are intended as a framework for rigorous theoretical analysis and a thorough examination of the physics of plasma detachment and are not intended to be replicated on any real tokamak machine.
To achieve the 'closed box' configuration, recycling is set to 100% at the targets and walls, where all ions impacting the material surfaces and boundaries are reflected as neutral particles. There is no active puff or pump included in these simulations, but particle count of both species is maintained using a feedback system implemented in the SOLPS4.3 code package that controls an effective particle puff that is sourced at all material surfaces on the outboard side and the core to make up for inherent and unavoidable particle losses which result from numerical coupling between the implicit fluid and explicit kinetic codes. The neon impurity is fully contained in this simulation, meaning an initial inventory of particles was included in the simulation with no sink or source of neon particles. The Bohm sheath condition is applied at the target, and a zero-flux boundary condition is applied at the core.

Up-down asymmetries
In these simulations, there are pronounced differences in the primary and secondary divertor heat fluxes, which are consistent with observations of up-down asymmetries in 'standard' divertor configurations. Asymmetries are characterized as imbalances of the heat and particle loads between the different divertor regions (upper and lower divertors, in the case of a double null, or inner and outer, for each divertor). These asymmetries are frequently observed in experiments and simulations, and can arise from the ballooning nature of anomalous transport, drift effects, impurity radiation, geometric effects, or as a result of other instabilities [1,29]. Such asymmetries are observed in these simulations: the primary and secondary divertors display vastly different characteristics, and the inner and outer divertors in each of the primary and secondary divertors also display significant asymmetries (which will be identified and discussed in later sections). Figure 3 shows the distribution of the heat flux entering into each of the four divertor legs, which reflect the asymmetry in fluxes directed towards each of the divertors.
The primary cause of these up-down asymmetries are the transport mechanisms associated with the disconnected double null magnetic topology, since other up-down symmetrybreaking effects, such as the influence of drifts or impurity radiation, are not applicable here, as these simulations do not include drift effects and include only trace impurity radiation. This effect, which is well-understood and observed in 'standard' disconnected double null divertors, also manifests in the long leg geometry; the disconnected double null configuration of these simulation grids feature the primary X-point (and therefore the primary divertor) at the lower part of the device. In this magnetic configuration, there are two separatrices in the lower divertor, and rapid longitudinal transport along or very close to these separatrices amounts to a total flux that is much greater than the flux to the secondary divertor [26].

Plasma detachment
The strong disbalance in heat fluxes induced by the asymmetric magnetic topology causes plasma detachment to proceed with different dynamics in each of the upper and lower divertors. The results of a simulation set with incrementally increasing particle content N edge D shows a rollover of plasma flux to the divertor targets, shown in figure 4. Across the simulation set, both the inner and outer secondary divertors detach first and second, respectively, as particle count is increased. The outer primary divertor detaches third, and the inner primary divertor never detaches.  Despite very different plasma conditions in each of the individual divertor legs (which are detailed in subsequent sections) the rollover of the plasma flux to the target starts at (or never surpasses) the same P up /q recycle value of 20 N MW −1 , shown in figure 5, which is consistent with existing theory and understanding of divertor physics. To calculate the P up /q recycle ratio, the procedure detailed in [19] is followed. A simulation with a strongly attached plasma in all four divertors is identified from figure 4, and the most loaded flux tube on each target plate in this simulation is determined. The four different flux tubes that are identified are unique to each of the four divertors. Each of these four flux tubes is then analyzed across the entire simulation set to see how the plasma parameters evolve in the same flux tube to obtain the different profiles for each of the divertors. The upstream pressure, P up , is the plasma pressure in each flux tube at the midplane. The q recycle value is determined by taking the specific heat flux entering the recycling region, where the recycling region is defined as the region where 85% of the total ionizations in that flux tube are taking place. This is calculated by taking the integral of the total number of ionizations in the flux tube from the target up to the heat flux stagnation point, and then identifying the cell where the integral of the ionization events becomes ⩾85% of the total ionizations. The parallel heat flux entering the cell where this region begins, which can be interpreted as the input energy to facilitate the recycling process that maintains the upstream plasma, is q recycle . Figure 5 shows the dependence of the saturation current to the target on the P up /q recycle ratio for all simulations in the particle count scan preceding the ion flux rollover observed in figure 4 in each divertor. All simulation results are included for the primary inner divertor, which never displays a rollover in target ion flux; notably, the P up /q recycle ratio approaches, but never exceeds the 'critical' value of 20 N MW −1 predicted for plasma detachment. The outer secondary divertor rolls over with P up /q recycle value of roughly 28 N MW −1 , which is higher than the other divertors and visible as the peak on the solid blue line in figure 5. This is not abnormal, since the physics basis for this ratio assumes a set of simplifying assumptions and minor variations in adherence to this parameter, similar to this one, have been observed in other simulation sets [19,30]. In this case, the P up /q recycle value is likely higher than the prediction since most of the fluxes are directed towards the primary divertor, and these strong up-down asymmetries mean the effects of the primary divertor dominate, so heat flux to the outer secondary divertor is lower and the stagnation of the upstream pressure on the outboard side does not occur until the primary outer target detaches (shown in figure 6).
We also notice a modest decrease in the average deuterium ionization cost, consistent with observations in other works [31,32]. This effect is shown in figure 7, where the averaged electron energy dissipation per ionization of each deuterium atom (accounting for both the potential energy required to ionize the neutral atom and the radiative energy losses from excitation collisions before the ionization, divided by the total number of ionizations) is shown. The average energy per  ionization is dependent on density and temperature, and significant changes in this value might have an impact on the overall energy balance as the plasma transitions to the detached regime. The small variation in the ionization energy cost does not play a significant role in the energy balance in the transition to plasma detachment shown in these simulation results.

In-out asymmetry
In each of the primary and secondary divertors, there is significant poloidal asymmetry between the inner and outer legs. The secondary divertor, which is positioned at the top of the device and receives lower heat and particle fluxes relative to the primary divertor at the bottom, shows lower heat and particle fluxes to the secondary inner divertor and lower temperatures than the secondary outer divertor across the whole simulation set. The plasma at the secondary inner divertor shows the 'roll over' of the saturation current to the target and detaches first, at lower number of particles N edge D , than the secondary outer divertor. The blue lines in figures 8(a), (b) and (c) show the peak heat flux, the target electron temperature just outside the separatrix, and the total heat load, respectively, for the secondary inner (dashed blue) and secondary outer (solid blue) divertors. The peak heat flux and total heat load to the secondary outer target are consistently higher than the secondary inner target, while the electron temperature at the secondary inner target decreases to volumetric recombination temperatures before the secondary outer target does. Volumetric recombination increases first at the secondary inner target, shown in figure 9, which is consistent with the rollover of ion flux observed in figure 4. This distribution of in-out symmetry, where the inner divertor detaches before the outer one, is common and expected in 'standard' divertor configurations, as higher fluxes are expected at the outboard side due to ballooning effects [1].
However, the opposite asymmetric effect occurs for the primary divertor; while there is still significant asymmetry in the heat and particle fluxes between both the primary divertor legs, the long legged primary outer divertor 'rolls over' and detaches first, while the ion flux to the short primary inner divertor reaches a level of saturation and the plasma stays attached. The red lines across the quantities shown in figure 8 show a cooler outer divertor and lower heat fluxes (red solid line) at high densities relative to the inner divertor (red dashed line).
This 'opposite' asymmetric effect appears to be a unique feature of the long leg in the primary divertor, which has a larger power load than the secondary divertor. Experiments and modeling of lower single null L-mode deuterium plasmas with a short inner leg and long outer leg on the KSTAR tokamak also achieve plasma detachment in the long outer leg before the short inner leg [33]. Analogous results from similar experimental setups were also observed on the TCV tokamak, but these discharges used reverse B T and featured a much longer connection length in the outer leg relative to the KSTAR results or the simulations presented here [34,35]. In the KSTAR experiments and simulations, the magnetic geometry of the KSTAR tokamak features a lower single null divertor with an open, vertical inner target with a short poloidal connection length and a much longer outer leg onto an open, inclined target plate. The TCV geometry is similar to the KSTAR geometry, but with a highly-flared leg on a flat outer target. The simulations presented in this work feature a similar short inner leg and longer outer leg in the lower divertor, but include tight baffling along each of the divertor leg channels, unlike the open target plates used in the KSTAR or TCV tokamaks. The KSTAR study attributes this asymmetry to target geometry, while the TCV study concludes that the asymmetry is a consequence of flux flaring effects at the target.
While the specifics of target geometry might play a role in enhancing the detachment of the long leg, it is not clear that this is the dominant mechanism driving this significant asymmetry, since the same effect is observed in only one of the divertors in this simulation set for a geometric configuration with matching angled, closed divertors with long outer legs in the upper and lower divertor regions. In these simulations, the secondary divertor targets are also angled with closed geometries and a long outer leg, but detachment proceeds with the inner before the outer divertor in the particle count scan, consistent with detachment in standard divertor configurations, but opposite to that of other long leg discharges. This is likely due to both the tight baffling of the narrow divertor leg channel, in contrast to the open divertors at the inner target in the KSTAR and TCV tokamaks, and the lower power entering the secondary inner leg. The tight channel in the inner divertor facilitates neutral trapping and promotes detachment at the inner target. A similar effect (although less pronounced than shown in these simulations, and only after the detachment of the outer target) was observed on TCV, where modular baffles were installed at the inner target to create a more closed inner divertor region and enabled detachment at the inner target [36].
However, while the primary divertor also features tight baffling at the short inner leg that would enhance neutral confinement, the plasma in the primary inner divertor never detaches from the target in this simulation set. This seems to be related to the magnitude of the heat fluxes directed towards each divertor; due to up-down asymmetry effects, more heat flux is directed to the primary divertor target, and any improvement in neutral confinement due to this tight slot geometry is mitigated by the shorter connection length and smaller overall divertor volume that do not appear to have the capacity to dissipate the incoming power. The decomposition of the heat flux into major sources/sinks and identification of spatial characteristics of each divertor leg in the simulation with the maximum particle flux to the outer divertor leg is summarized in table 1. The relevant quantities relating to the energy available for ionization in equation (2) are shown, where Q in is the component of Q SOL that is directed towards each divertor leg (from figure 3), Q ⊥ is the perpendicular losses to the side walls (from figure 11), and Q ion is the total energy available for ionization (from figures 7 and 9(b)), such that the energy hitting the target Q target (from figure 8(c)) is: The volume and surface area for each divertor leg are also included.
As in the KSTAR and TCV tokamaks, the outer target in the primary divertor does detach, while the inner one remains attached. The magnitude of the heat flux received by the outer divertors is consistently larger relative to the inner ones, as shown in figure 3, but the outer divertor volume itself is also much larger than the inner one, shown in table 1. This equates to a larger space for a higher particle inventory for more dissipative processes to occur, but it also increases the surface area of the divertor walls. The cumulative perpendicular heat loss to the divertor side wall, Q ⊥ , is a significant energy sink in the power balance in the outer leg. While the baffled, angled target of the various long leg divertor configurations likely enhances the detachment of the plasma, as mentioned previously, it cannot explain the mismatch in the observations of the detachment in the primary and secondary divertors observed in these simulations. Instead, the energy balance indicates that it is likely that the dominant mechanism behind the strong inout asymmetry in detachment is the influence of perpendicular transport.

Perpendicular transport
4.4.1. Divertor heat flux. The effects of perpendicular transport appear to be extremely important in the transition to detachment in the long leg in several ways. To help clarify the following discussion, a schematic of the outer divertor leg and associated heat fluxes and sinks is shown in figure 10. As described in the previous section, a significant portion of the divertor heat flux is directed out of the plasma and onto the   material surfaces on the tightly baffled divertor leg, acting as an additional energy sink identified in equation (2) and represented as the blue surfaces in figure 10. These losses are displayed in figure 11 and identified in table 1. The perpendicular transport enhances heat flux spreading and delocalization in the outer divertor leg. The total cross-field heat transport directed across the separatrix, illustrated as the small purple arrows in figure 10, is shown in figure 12. The heat flux entering the private flux region from the SOL is shown in figure 12(a), while the heat flux passing through the near SOL and entering the outer SOL is shown in figure 12(b). The majority of the heat flux directed into each primary divertor region is carried by the primary separatrix, and passes through the near SOL region (the orange-shaded channel in figure 10), flanked by the primary and secondary separatrix legs. The heat flux passing into the near SOL in each primary divertor leg is shown in figure 13, while the total heat flux entering the divertor legs is shown in figure 3; roughly  three quarters of the total power to each leg travels into this region, so this region will be the focus of this analysis. Across the particle count scan, roughly 1 MW of the incoming heat flux to this near SOL region is consistently transported across the separatrix into the private flux region (the yellow-shaded channel in figure 10), providing a strong reduction in the heat flux directed towards the near SOL divertor target. In comparison, the cross-field transport across the separatrix to the private flux region of the inner leg is much weaker relative to the outer leg; the near SOL in the inner divertor consistently receives a portion of incoming heat flux equivalent to approximately 75% of the power to the near SOL in the outer leg, but only loses an equivalent of approximately 50% of the power lost to the private flux region from cross-field transport in the outer leg.
At the secondary separatrix in the primary divertor, there is significant asymmetry in the cross-field transport crossing from the near SOL to the outer SOL. At very low densities, there is essentially no transport across the separatrix (there is even a small flux in the reverse direction, back towards the near SOL, at the lowest densities in the outer divertor). However, as the particle count increases and the secondary divertors detach, the cross-field transport towards the outer SOL in the outer divertor increases significantly, and continues to rise as the outer primary divertor detaches. The inner primary divertor cross-field transport to the outer SOL remains very low and never increases.
The spread of the overall heat load in the near SOL helps to reduce power density to a tolerable level where either the cross-field transport can continue to carry the heat flux to the side walls of the leg, or the dissipative processes can deplete the remainder of the heat flux before the plasma starts to recombine. In contrast, the heat flux to the inner target remains fairly localized at the primary separatrix, which seems to prevent detachment altogether.
To assess whether perpendicular transport plays a significant role in the energy balance in the divertor heat flux spreading in the detached plasma, additional simulations were generated with transport coefficients reduced by one third to χ ⊥ = 0.33 m 2 s −1 and D ⊥ = 0.1 m 2 s −1 . A comparison between the electron temperatures in the primary divertor for the simulation at the highest N edge D value with the 'original' (a) and 'reduced' (b) transport coefficients is shown in figure 14, dramatic increase in temperature all along the separatrix and to the target that is incompatible with the detachment processes described earlier in the paper. In this simulation, which displays the 'most' detached outer leg with the original transport coefficients, the perpendicular heat flux interacting with the main chamber side walls reduces from around 16 MW to 7.6 MW, greatly increasing the heat flux received by each of the divertor legs, and the plasma fully reattaches at the primary outer leg with modified transport coefficients. As shown in the figure 14(b), the simulation with the modified perpendicular transport coefficients has considerably higher electron temperatures all the way down the leg to the target in both the near SOL and outer SOL than shown in the original simulation. In the primary outer leg, the heat flux entering into the divertor leg increases from 6.7 MW to 10.4 MW, the perpendicular heat flux to the divertor side wall Q ⊥ is reduced from 1.9 MW to just under 1 MW, and heat flux to the primary outer target increases from virtually nothing to 2 MW. Similarly, the secondary outer divertor (not shown in the figure) also reattaches, despite receiving considerably less heat flux than the primary divertor; the heat flux entering the secondary outer divertor leg increases from 3.1 MW to 3.7 MW and the Q ⊥ to the secondary divertor side walls reduces from 1.5 MW to 0.8 MW, while the heat flux to the secondary outer target increases from virtually nothing to 0.8 MW.

Main chamber heat flux.
While detachment processes proceed 'as expected' with similar behavior and scaling criteria to standard divertors, perpendicular transport also plays a significant role in the reduction of heat fluxes entering each divertor leg, which ultimately lowers the incoming power in the legs to a tolerable level for the 'standard' dissipative processes to become effective and the 'expected' detachment behaviors to proceed. There are significant perpendicular losses to the side walls as particle count increases, shown in figure 15, which eventually saturate and stop increasing with higher densities.
At low plasma densities, power exhaust from the core enters the SOL and travels straight to the divertor legs. As the particle count is increased, the heat flux leaving the plasma and impacting the main chamber walls increases, observed in equation (2), and the remaining heat flux that enters the divertor legs decreases, observed in equation (1). This effect occurs until the outer divertor reaches its peak in the ion flux rollover curve, at which point the synergistic effect of losses to the side walls and ionizations in the divertor volume reduce the temperature in the divertor. At this low temperature, volumetric recombination processes begin in the outer leg, and upstream pressure saturates. A significant increase in volumetric recombination processes in the primary outer divertor, shown in figure 9(a), with a saturation of the total ionization events in the primary outer divertor, shown in figure 9(b), correspond to At this point, radial transport towards the main chamber walls and divertor legs stays fairly high, but stops increasing, while the heat flux to the outer primary divertor leg walls starts to decrease, shown in figure 11.
This effect is the result of the upstream pressure saturation; density increases, temperature lowers, and as the pressure saturates, the radial gradient in the temperature relaxes, which nearly eliminates the conductive component of the cross-field heat fluxes, shown in figure 16. There is strong Figure 16. Midplane ion temperature for a lower particle count simulation and a higher particle count simulation. variation in the radial temperature profile at lower number of particles N edge D , which flattens out after recombination processes begin in the outer leg. All versions of the SOLPS code use a diffusive ansatz for radial transport, which is an approximation of turbulence that emulates a spatiotemporal average of cross-field transport through spatial gradient with diffusion coefficients. This follows an equation of the form where D and χ are the user-specified diffusion and thermal diffusivity coefficients that articulate the convective and conductive cross-field heat fluxes, respectively. The saturation of the upstream pressure and the relaxation of the temperature gradient mean that essentially the only contribution to the crossfield transport is the convective heat flux. This convective contribution is significant, which is typical and expected for edge plasmas, but does not increase with increasing particle count because of the saturation in pressure. The consequence of this is a synergistic beneficial effect of the flux rollover and transition to the detached state: as recombination begins, the conductive component of the cross-field transport is eliminated, limiting the cross-field transport to a fixed level and stabilizing conditions upstream.

Trace impurity radiation
A trace neon impurity consisting of a fixed inventory of 1 × 10 17 particles is present across the entire simulation set. All charge states of neon are fully resolved, meaning neutral neon atoms (and the associated atomic reactions) are treated by the EIRENE code and continuity, momentum, and energy balance equations are solved for all ten charge states in the B2 fluid code. The neon is primarily localized next to the core in the intermediate region between the primary and secondary separatrices, rather than radiating in the divertor legs and away from the core plasma. Since the overall contribution of neon to the overall energy balance in the transition to plasma detachment is small, we do not anticipate that the impact of this trace impurity was significant on the analysis presented here. However, it is noted that the localization of the neon close to the core, even in this trace amount, is not ideal for future reactor-relevant situations. While neon is often considered to be a favorable radiator because it is inert and will not introduce complexities with molecule formation, strong localization to the core plasma can decrease core performance, which should be avoided, and further study is required to understand the distribution and spread of impurity radiation in these long-leg configurations.

Conclusions
A scan of plasma particle count is performed with the SOLPS 4.3 code package to assess the physics of detachment in a longleg divertor configuration, which is a promising mitigative feature of the magnetic topology used in many alternative divertor configurations. Up-down and in-out divertor asymmetries are observed, with plasma behavior in the primary divertor that is consistent with experiments and modeling of other long leg divertor geometries. Across this scan, which displays the typical 'rollover' of ion flux to the material surface as the plasma detaches, it is observed that the physics-based scaling relationship between upstream plasma pressure and incoming divertor heat flux during plasma detachment corresponds well to the existing physical understanding of divertor detachment in 'standard' divertor configurations. However, in the absence of impurities, significant spreading of the heat flux and losses to the side walls from perpendicular transport are required for the 'normal' atomic and molecular detachment processes to begin, and a new energy balance is suggested for long leg configurations to address the perpendicular energy losses, as equation (2). The influence of losses to the side walls, characterized here as Q ⊥ , was not initially identified as an important component of the SOL energy balance, but appears to be significant in long leg geometries [11,12]. Once detachment proceeds, a synergistic effect between the limitation of cross-field transport and divertor plasma recombination and detachment is observed. Overall, the existing theory and many features of divertor plasma detachment in standard divertors appear to translate to long leg divertor geometries. With this foundation, several other key aspects of the physics of long leg divertors can be studied in future simulations, including the spread of impurity radiation, since this study indicates that localization of impurity radiation to the core plasma might be a major concern.