Reversed-direction 2-point modelling applied to divertor conditions in DIII-D

A predictive form of the extended 2-point model known as the ‘reverse 2-point model’, Rev2PM, is applied to a range of detachment levels in the open lower divertor of DIII-D, showing that the experimentally measured electron temperature (Te ) and pressure (pe ) at the divertor entrance can be calculated within 50% from target measurements, if and only if a posteriori corrections for convective heat flux are included in the model. Unlike the standard 2-point model, the Rev2PM calculates upstream scrape-off layer (SOL) quantities (such as separatrix Te and pe ) from target conditions (such as Te and parallel heat flux), with volumetric power and momentum losses depending solely on target Te . The Rev2PM is tested against a database of DIII-D inter-ELM divertor Thomson scattering measurements, built from a series of 6 MW, 1.3 MA, LSN H-mode discharges with varied main ion density, drift direction, and nitrogen puffing rate. Measured target Te ranged from 0.4–25 eV over this database, and upstream Te ranged from 5–60 eV. Poor agreement is found between upstream measurements and Rev2PM calculations that assume purely conductive parallel heat transport. However, introducing a posteriori corrections to account for convective heat transport brings the Rev2PM calculations within 50% of the measured upstream values across the dataset. These corrections imply that up to 99% of the parallel heat flux is carried by convection in detached conditions in the DIII-D open lower divertor, though further work is required to assess any potential dependencies on device size or divertor closure.


Introduction
One of the key goals on the road towards a magnetic fusion pilot plant is core-edge integration: i.e. finding a regime capable of sustaining a robust high-performance fusion core, while simultaneously not destroying the surrounding plasmafacing components (PFCs).Even neglecting complications arising from neutron and/or helium damage, most solid PFC concepts must withstand two main plasma-driven failure modes: short-term cracking and melting due to thermal shocks (significant for deposited heat fluxes >10-15 MW m −2 ), and long-term erosion of the surface due to sputtering (significant at divertor target electron temperatures >5-10 eV) [1][2][3].Extensive experiments in tokamaks worldwide have demonstrated that detachment of the divertor targets-in which case parallel power and plasma momentum are dissipated by volumetric processes prior to reaching the target surface, thus spreading the plasma load over a much wider area of the divertor-is the most promising method for meeting these constraints [4][5][6].However, it is still an active area of research to reliably detach the divertor while still accessing highperformance core scenarios in present devices, and even more uncertainty is present when extrapolating to future devices with higher power and gradients.
The present modus operandi for designing divertor scenarios in future devices, as demonstrated in the ITER divertor physics basis [7], is to build up a large database of simulation results spanning a range of possible configurations, and use the results of this database to define the operational space in which PFC limits are not exceeded.In the case of ITER, this was performed with the SOLPS4.3code [8], and typically assumed some fixed power and particle flux leaving the core, while varying control parameters like main ion or impurity gas puff rates.A great deal of knowledge has been (and continues to be) gleaned from this method, and it works especially well when dealing with devices (like ITER) in which many of the geometric design parameters have been fixed.
However, such a workflow is less advantageous for designs with less technical maturity, such as many fusion pilot plant design efforts.These design efforts typically include optimizations over dozens of high-level parameters such as major radius, aspect ratio, and input heating power, which are carried out by so-called tokamak 'systems codes' (including the GA systems code [9], the ARIES systems code [10], PROCESS [11], and many others).This approach must make extensive use of reduced models for connecting the various aspects of tokamak physics, since it is computationally intractable to perform high-fidelity simulations for each subsystem at every potential design point encountered in the optimization algorithm.There is thus significant value-added in developing improved (and validated) reduced models that could improve the fidelity of systems code calculations, while still maintaining computational tractability.
In the context of core-edge integration, one of the key outstanding needs is a reliable model for the relationship between the plasma temperature and density at the divertor target (which is the boundary condition for the PFCs and determinant for both heat flux and sputtering processes) and the temperature and density at the upstream separatrix (which is the boundary condition for the pedestal and determinant for core performance), in the presence of strong volumetric SOL dissipation.The Reverse 2-Point Model (described in section 2) shows promise as such a reduced model.Recent analysis of a large database of SOLPS-ITER simulations for Q = 10 operation in ITER (spanning a wide range of detachment states) have shown very strong correlations between the divertor target electron temperature (and to a lesser extent the impurity seeding rate) and almost all quantities of interest for divertor physics [12].In [13], a Reverse 2-Point Model for the ITER divertor (taking the SOLPS-calculated divertor target plasma conditions as input) was developed, and it was found that the reduced model could replicate with excellent accuracy the upstream plasma parameters calculated by SOLPS.
This promising code result begs two important followup questions: (1) Does the Reverse 2-Point Model perform similiarly well for divertor geometries other than ITER?(2) Does the Reverse 2-Point Model hold up experimentally?This paper aims to address both of these questions, by assembling a relevant database of plasma measurements taken from the lower divertor discharges in DIII-D, and assessing the ability of the Reverse 2-Point Model to reproduce the upstream measurements from the target measurements.
The structure of this paper is as follows: section 2 introduces the Reverse 2-Point Model, while section 3 describes a database of DIII-D divertor measurements that will be used for model comparison.Section 4 attempts (unsuccessfully) to use the standard Reverse 2-Point Model to predict upstream database measurements from target measurements.Section 5 derives a posteriori convective corrections to the model, and shows that these greatly improve agreement of the Reverse 2-Point Model with measurements.Section 6 discusses the state of the Reverse 2-Point Model as a predictive reduced model, followed by conclusions in section 7.

The reverse 2-point model (Rev2PM)
Two-point models (2PM) refer to analytic SOL models that compare plasma conditions at the divertor target (typically denoted by the subscript 't') and at some representative upstream location (often denoted by the subscript 'u', though in this work we will use the subscript 'x' as an explicit reminder that we are using the X-point as the upstream location-see section 3.2).In its simplest form, the 2PM assumes that both parallel power flux density and plasma pressure are constant along each flux tube.However, in many cases it is necessary to make use of an 'extended' 2PM that allows for volumetric losses between the two locations (see [6] and references within).These volumetric losses are characterized by the parameters (1-f cool ) and (1-f mom ) for power and momentum dissipation, respectively: ( In these expressions, q ∥ is the parallel heat flux density, R is the major radius, and p is the total plasma pressure. Until recently, the volumetric loss terms were treated as free parameters.However, it has recently been found that in almost all cases (both experimental and computational), these can be represented by functions that depend primarily on the target electron temperature, T e,t [6,14,15].In this work, we make use of volumetric loss terms that are fit to the aforementioned large database of SOLPS-ITER simulations of the ITER divertor [12], fitting to the electron temperature in the last cell in front of the target on the computational grid used in the modeling.These fits take the form of a 4th-order log-log polynomial: The fit coefficients are as follows.(1-f cool ): 972, a 4 = −0.751.These fit functions are plotted versus T e,t in figure 1.The fit functions are clamped to their maximum for T e,t values above the range used to define these fits in [12] (about 13 eV), though this assumption has negligible effect on any of the results presented in this paper.
In the 'Forward' 2PM, the upstream condition is used as the input (independent variable), while the target condition is used as an output (dependent variable).This is the most commonlyused approach, often because measurements in the divertor have typically been more challenging to perform than doing so in the open main chamber.The aptly-named 'Reverse' 2PM (Rev2PM) takes the opposite approach, using the target condition as input to determine the upstream plasma conditions that are consistent with a certain level of volumetric dissipation.This makes it a promising candidate for a PMI-centered reduced divertor model, since one can specify the material limits for q ∥,t and T e,t (which ulitimately determine the lifetime and viability of the wall armor of the machine), and calculate the maximum tolerable upstream plasma for a given configuration.
First, let us begin with the power conservation equation, which enables a Rev2PM calculation of the upstream T e .One may begin with the component of the heat flux conducted by electrons, where κ 0e is the electron thermal conductivity and s is the parallel direction along a flux tube ( [16], section 9.6): Integrating this from target to the X-point gives the following expression, commonly used in Forward 2PM: L ∥ is the connection length between the target and the upstream location, and f cond is the fraction of total heat carried by conduction from upstream to the target.Rearranging this so that T e,x is the dependent variable, and plugging in equation ( 1) for the upstream heat flux q ∥,x , gives the following form in terms of the target heat flux: The parallel heat flux at the target can be written as a function of measurable target parameters: q ∥,t = γ sh n e,t T e,t c s,t (7) γ sh = 5.69 Here, it is assumed that T i,t is a constant multiple of T e,t (equation (10)), which modifies the sheath heat transmission coefficient γ sh (equation ( 8)) and the target sound speed c s,t (equation ( 9)).This expression for γ sh comes from analytic sheath theory ( [16], section 25.5), and assumes a deuterium plasma without impurities.
Evaluating equations ( 6)- (10), and making use of the T e,t -dependent fit for (1-f cool ) (figure 1), gives the following expression for the upstream T e calculated by the Rev2PM: 1/2 (5.69 + 3τ t + 0.5 ln (1 + τ t )) In the 'standard' Rev2PM (section 4), f cond is assumed to be equal to 1.0.In this case, every term on the right hand side is a function of T e,t and n e,t alone (plus an assumption on T i,t ).In the 'corrected' Rev2PM (section 5), f cond is allowed to vary.
Next, pressure conservation between two points along the flux tube is used to make a Rev2PM calculation of the upstream electron pressure, which, with knowledge of conditions at the target as in the power conservation approach above to determine upstream electron temperature, can be applied alongside equation (11) to estimate upstream electron density.The total plasma pressure at a point in space can be written as the sum of the static (nT) and dynamic (nmv 2 ) components, where we neglect electron dynamic pressure (due to small electron mass), assume n i ≈ n e , and again define τ ≡ T i /T e : The pressure equation has been simplified by introducing the local Mach number M, which is the ratio of the ion parallel speed to the local sound speed.
We may now combine equation (12) at the target and upstream, plug into equation (2), and make use of the T e,tdependent fit for (1-f mom ) (figure 1) to derive the following Rev2PM prediction for the upstream electron pressure: In most cases, the usual assumption of sonic flow at the target (M t = 1) is made.In the 'standard' Rev2PM (section 4), one assumes M x = 0, so that everything on the right hand side is a function of of T e,t and n e,t (plus assumptions on T i,t and T i,x ).In the 'corrected' Rev2PM (section 5), M x is allowed to vary.

Discharge conditions
An experimental database is produced from a series of DIII-D discharges operated in lower single null (LSN), with the outer strike point located on the 'shelf' of the lower divertor.All of these discharges are operated at plasma current (I p ) of 1.3 MA, toroidal field magnitude (B T ) of 2.1 T, and a nominal injected neutral beam power (P INJ ) of 6 MW.Discharges vary in main ion density (characterized in this work by pedestal electron density, n e,ped , 4.2-9.0× 10 19 m −3 ), direction of the ion grad-B drift (B × ∇B ↓ / ↑, i.e. into or out of the active divertor in LSN), and rate of nitrogen puffed into the lower divertor (0-30 Torr-L/s).The plasma remains in H-mode during the portions of the discharges in which data were taken, though the natural ELM frequency varied significantly between cases (ranging from ∼60 Hz at B × ∇B ↓ low density to ∼600 Hz at B × ∇B ↑ high density).For each plasma condition, the outer strike point is swept over a 6 cm range on the horizontal shelf, to build up radial profiles on divertor diagnostics.

Database production
The critical diagnostic for this work was the DIII-D divertor Thomson scattering (DTS) system, which at the time of this experiment consisted of 12 channels along the vertical R = 1.485 m chord above the lower outer shelf, with 50 Hz temporal resolution.As described in [17], DTS data accumulated across the strike point sweep was remapped to a single magnetic equilibrium, to produce a 2D map of Te and ne in the lower outer divertor.DTS points taken during an ELM (as determined by peaks in D α filterscope signals looking at the outer strike point) were removed, so that this database is representative of inter-ELM plasma conditions.A caveat is that a subset of this database (typically B × ∇B ↑ high density) had such high ELM frequencies that inter-versus intra-ELM has little significance; in these cases the database represents the coldest state in the constantly detaching-reattaching plasma.
For the purposes of 2-point modeling, DTS channels 0 and 1 (0.9 and 3.0 cm above target, respectively) are employed as the 'target' measurement, and DTS channels 10 and 11 (19.5 and 22.6 cm above target, respectively) slighly above the height of the X-point as the 'upstream' measurement.This is shown in figure 2, which demonstrates how the curvature of the outer divertor leg allows DTS measurements to be simultaneously taken at the divertor target and upstream of the primary volumetric dissipation region.Two DTS channels are used for each measurement to increase sample statistics and increase radial coverage, though this does have the downside of 'smoothing out' any potential steep gradients in these regions.In this database, these gradients are negligible for all but the most attached cases, where DTS channels 0 and 1 may vary by up to 50% (though this difference has little effect on the conclusions of this paper).
It is important to note that this choice of 'upstream' differs from many other applications of the 2-point model, which often take 'upstream' to mean the midplane or plasma crown.The advantage of treating the X-point/divertor entrance as 'upstream' is two-fold: (1) all power can be assumed to have entered the SOL from the core plasma by this point, reducing power balance uncertainties, and (2) all data can be taken with a single diagnostic system, reducing systematic uncertainties related to remapping plasma data relative to the separatrix (which can otherwise dominate comparisons along flux tubes [18]).The disadvantage of this choice is that there are no measurements of T i at this location, which means that T i remains a free parameter in the present analysis.
Example 2D maps of T e and n e are shown in figure 3, for both attached (a), (b) and deeply detached (d), (e) conditons.These maps demonstrate the typical spatial coverage of DTS measurements, and exemplify the transition of the divertor target plasma from high-T e /low-n e to low-T e /high-n e that is typical when moving from attached to detached regimes.
Radially continuous profiles of the target and upstream DTS data, mapped to R − R sep,mid , are produced via locally estimated scatterplot smoothing (LOESS).These profiles are shown for example attached and deeply detached cases in figures 3(c) and (f), respectively.Profiles are subsequently binned into flux tubes of width 1 mm, such that the 'R − R sep,mid = 1 mm' flux tube is the average of the LOESS profile from 0.5 to 1.5 mm.Error bars are calculated for DTSmeasured target and upstream plasma quantities by taking the square root of the sample variance of the ensemble of DTS points relative to the binned average.Error bars for Rev2PM quantities are derived by propagating uncertainties in target plasma conditions through the relevant equations defined in section 2. Target-upstream data pairs are typically available out to the 4 mm flux tube, which is approximately twice the typical near-SOL heat flux width in I p = 1.3 MA DIII-D discharges.

Database description
A database of 92 flux tubes-each containing a pair of target and upstream T e and n e measurements-was generated from 23 distinct DIII-D lower divertor plasma states.T e,t ranged from 0.5 to 25 eV in this database, and T e,x ranged from 5 to 60 eV.n e,t was closely correlated with T e,t , as shown in figure 4. For B × ∇B ↑, a continuous relationship between target temperature and density is observed.However, for B × ∇B ↓, a step-wise transition from the attached to detached state is observed, as has been reported in previous DIII-D studies [17,19,20].Additionally, at very high densities these B × ∇B ↓ discharges reach the deeply detached state (where the target particle flux density strongly decreases), which was not observed in B × ∇B ↑ discharges.In this set of discharges, very little difference in target plasma behavior was observed between deuterium-only discharges (denoted as D 2 on the plot) and deuterium discharges with nitrogen seeding (denoted as N 2 on the plot).For this reason, the nitrogen seeding rate is not used as an additional Rev2PM input parameter in this work, though the impurity seeding rate has been found to be a necessary ordering parameter in ITER simulations [12].A separate dataset spanning a greater range of impurity concentrations would be necessary to test reduced models that depend on impurity seeding rates.

Database comparison
The predictive form of the Rev2PM is tested by inputting the measured target parameters into equations ( 11) and ( 14), with f cond = 1 and M x = 0. T e,t -dependent fits for (1-f cool ) fit and (1-f mom ) fit are taken from the ITER SOLPS-ITER database (figure 1).In this database R t = R x , and we assume L ∥ = 7 m and κ 0e = 2000.Additionally, in this section we assume that T i = T e at the target (τ t = 1) and upstream (τ x = 1).Under these assumptions, the comparison between the measured and Rev2PM-predicted T e,x and p e,x are shown in figures 5 and 6, respectively.
It is clear that the standard Rev2PM massively overpredicts both temperature and pressure for all but the most attached cases in the database.Furthermore, making different assumptions for T i /T e , (1-f cool ) fit , or (1-f mom ) fit do not bring the predicted values into agreement with the measurements.Clearly there must be some physics missing from the standard formulation of the Rev2PM, in order to produce such drastic disagreement with our DIII-D dataset.As will be demonstrated in the following section, the main point of failure for the model is the assumption that all parallel heat flux is carried via conduction.Allowing some fraction of the heat to be carried via convection greatly improves model performance.

Comparison to Rev2PM with convective corrections
In this section, the upstream DTS measurements of the database are used to derive correction factors to the reverse 2point model (f cond in equation (11) and M x in equation ( 14)) to help account for convected heat fluxes.Heat can be convected to the target by a variety of processes beyond simple parallel plasma flow, including poloidal drifts, Pfirsch-Schluter currents, thermoelectric currents, and poloidal gradients in cross-field transport.Rather than calculating each of these contributions separately, these correction factors are meant to represent a reduced model amalgam of all convective processes.Since these calculations require prior knowledge of the upstream plasma conditions, they fall under the category of a posteriori knowledge (as opposed to a priori), and thus the corrected Rev2PM is no longer a fully predictive model.However, it is important to note that these correction factors are not arbitrary fitting parameters, so there is still significant value in comparing the model predictions against upstream measurements.

Calculating f cond
All parallel heat flux flowing between upstream and target can be represented in the Rev2PM by the sum of convective and conductive contributions (since volumetric radiation is already accounted for by the 1-f cool term).Since the two are directly related, the balance can be represented by a single parameter f cond , the fraction of parallel heat flux carried by conduction by both electrons and ions, which is evaluated at the upstream (X-point) location as the following:   Seeing as the Rev2PM equation for upstream temperature comes from integrating the conducted parallel heat flux, this formulation of the correction factor also slots elegantly into the existing model, equation (11).
To calculate f cond as defined in equation ( 15), the heat flux at the X-point conducted by electrons (and ions) must be approximated.This involves the local temperature T e,x (T i,x ), the local parallel temperature gradient dT e,x /ds (dT i,x /ds), and thermal conductivity κ 0e (κ 0i ).For the electrons, T e,x is taken from the upstream DTS measurements, and dT e,x /ds is approximated by linearizing the gradient between the upstream and target measurements.Previous experiments with more comprehensive DTS coverage [17] have shown that a linear dT e /ds is a reasonable approximation in attached conditions, but in detached conditions such an approximation may overestimate (underestimate) dT e /ds downstream (upstream) of the detachment front.For the ions, κ 0i is so much smaller than κ 0e (∼60 vs. ∼2000) that it is safe to neglect the conducted ion heat flux: T e,x − T e,t L ∥ (16) To calculate the denominator of equation ( 15), the total parallel heat flux is needed at the X-point for each flux tube.For this process, the global power balance is invoked, and the total power crossing the separatrix into the SOL, P SOL , is calculated as: Here, P INJ is the injected neutral beam power, P OH is the ohmic power, P rad,core is the power radiated in the core (measured by bolometry), and dW/dt is the change in plasma stored energy (estimated by a slow signal that averages over ELMs).For the time slices in our DIII-D database, P SOL ranged from 4.7-5.9MW, where most of the measured variation was due to intermittent beam drops.
Furthermore, a common assumption is made that the upstream heat flux is exponentially distributed in the radial direction, characterized by a single near-SOL heat flux width λ q : In this expression f out,x is the fraction of the total exhausted power that makes it to the outer divertor X-point.While we began with the simple assumption that f out,x = 0.5, we found that f out,x = 0.25 produced better agreement with the X-point T e and p e measurements; this will be discussed further in section 6.It is recognized that this simple prescription may not be accurate in all cases (especially with different ion B × ∇B drift directions), but it is found to be adequate for a reduced model at this stage.R OMP and (B/B p ) OMP are the radius and inverse poloidal field ratio, respectively, at the outer midplane.r is the radial flux tube location relative to the outer midplane separatrix (i.e.r ≡ R − R sep,mid ).For λ q , the multi-machine Eich scaling [21] is applied, which for this database evaluates to λ q = 1.9 mm.No change in λ q is assumed for different divertor conditions.
The resulting total parallel upstream heat fluxes for each flux tube in the database are shown in figure 7.As expected, the dominant ordering parameter for the points is the radial flux tube coordinate (each band corresponds to a specific R − R sep,mid ), with only minor variations due to other discharge conditions.The total parallel heat flux on the flux tube closest to the separatrix is approximately 120 MW m −2 in all cases.
Evaluating equations ( 16)-( 19), and plugging the results into equation ( 15) produces an estimate of f cond for each flux tube in the database, shown in figure 8. Some interesting trends are apparent: at high T e,t (the most attached cases), f cond approaches 1, meaning that most heat is carried by conduction in an attached divertor.Values of f cond > 1 imply convective heat flux away from the target (such as from plasma flow reversal [22]), which is balanced by extra conducted heat flux towards the target in the Rev2PM under the assumption that power is conserved along flux tubes.However, at lower T e,t (more detached), f cond drops below 0.01, meaning less than 1% of the parallel heat flux is carried by conduction, and over 99% is thus carried by convective processes.In the T e,t range of 1-25 eV, we find a very rough scaling of f cond ∼ T 2 e,t , shown as   a dashed line in figure 8.The trend of reduced f cond at lower T e,t is not monotonic, however, as f cond recovers to 0.01-0.1 in the most deeply detached high-n e,ped B × ∇B ↓ cases with T e,t < 1 eV.

Calculating M eff,x
The presence of convected heat flux, particularly at low T e,t , necessarily implies some level of convective plasma flow, which will carry momentum and modify the pressure balance in the SOL.Before attempting to calculate a self-consistent plasma flow, it is instructive to directly examine the electron pressure measurements in our database, since those are subject to fewer uncertainties and assumptions.Figure 9 shows the ratio of measured electron pressure at the target to that at the Xpoint, versus measured target electron temperature.Overlaid is the volumetric momentum loss function (1-f mom ) that was extracted from the ITER Q = 10 SOLPS-ITER database [12] (see also figure 1).
Two key features are evident from figure 9: first, the (1f mom ) function reproduces the shape of the measured electron pressure ratio very well.In particular, the target electron temperature at which momentum losses begin to be significant appears to match quite well.This provides evidence that the atomic and molecular physics that lead to plasma momentum loss are reasonably universal, providing hope that volumetric momentum loss functions like (1-f mom ) may extrapolate to different machine sizes and geometries.Second, it is interesting to note that p e,t /p e,x asymptotes to approximately 1, rather than 2 as would be expected for M x = 0. Looking at the total pressure at each location (equation ( 12)), this implies that Assuming M t = 1, this would be satisfied by M x = 1 and τ t = τ x = 1, or by M x = 0, τ t = 1, and τ x = 3.Given the strong collisionality in many of the cold divertor conditions in this database, high values of τ x seem unlikely, pointing again to significant convective flow at the X-point.However, a simple prescription like M x = 1 is unlikely to be appropriate for all divertor conditions, so we move on to derive a plasma flow value that is consistent with the f cond correction derived in section 5.1.
Convected heat in the divertor could be carried by a variety of mechanisms: parallel plasma flow along field lines, cross-field drifts, Pfirsch-Schluter currents, ballooning transport asymmetries, and more.Rather than differentiate between all of these mechanisms, a simplifying assumption of defining an 'effective upstream Mach number', M eff,x , is made.This quantity can be interpreted as the amount of parallel plasma flow that would be needed to carry the net convected heat flux, if all of that heat flux were carried by parallel flow.
M eff,x can be calculated from the convected component of the upstream parallel heat flux, which is simply the remainder of q ∥,x once the conducted component has been accounted for (section 5.1) A ≡ n e,x m In equation ( 20) it is assumed that n i ≈ n e , v ∥,i ≈ v ∥,e ≈ v ∥ , and higher-order electron convection is neglected due to small m e .I 0 is the atomic ionization and molecular potential (13.6 eV + 2.2 eV for deuterium).Parallel velocities are then converted to Mach numbers according to the local sound speed (equation ( 13)) with v ∥,x = M eff,x c s,x , utilizing τ x ≡ T i,x /T e,x .Finally, a convenient coefficient A is introduced (equation ( 21)), which makes clear that equation (20) simplifies to a cubic polynomial in M eff,x .This polynomial has one real root and two complex roots; the real root is used to calculate M eff,x for each flux tube.
The resulting values of M eff,x , assuming τ x = 1, are shown in figure 10. M eff,x > 0 corresponds to flow towards the divertor target, while M eff,x < 0 represents flow towards the upstream SOL (e.g.due to flow reversal).Given the phase transitions that occur in systems with parallel flow Mach numbers greater than one, it is perhaps interesting to note the large magnitudes of M eff,x , approaching 3 in the lowest T e,t cases.However, it is important to remember that M eff,x encompasses effects beyond just parallel flow; in particular, it includes convection by poloidal ExB drifts, which are believed to have an outsized role in heat and particle transport in divertors with shallow magnetic pitch angle [23].While there is a general trend of larger M eff,x at lower T e,t , the scatter in the database is large enough that attempting to fit a functional form is not meaningful.

Updated database comparison
With f cond and M eff,x in hand for each point in the database, T Rev2PM e,x and p Rev2PM e,x can be recalculated using the full form of equations ( 11) and ( 14), again utilizing T e,t -dependent fits for (1-f cool ) fit and (1-f mom ) fit (figure 1).The resulting calculated T e,x is compared to measurements in figure 11, and much better agreement is observed compared to the Rev2PM without convective corrections (figure 5).For relatively attached conditions (high T e,t ) with f cond > 0.5, there is very little difference between the standard and corrected Rev2PM T e,x , and both match the measured values well.This is to be expected, due to the T e,x ∼ f 2/7 cond dependence in the Rev2PM.However, for very small f cond this factor is still important, which leads to much smaller predictions for T e,x .Across the full dataset, the corrected Rev2PM prediction is within roughly 50% of the measured values of T e,x .The convection-corrected calculation of p e,x is compared to measurements in figure 12. Compared to the standard p e,x calculation with M x = 0 (figure 6), the corrected p e,x calculations are considerably lower, as expected since M 2 eff,x is in the denominator of equation (14).The corrected values are also in better agreement with measurements, though scatter on the order of a factor of 2 remains.Figure 12 assumes τ x = 1, and assuming a higher ion temperature further reduces the Rev2PM p e,x calculation, pushing it further from the measured values.

Discussion
A clear and immediate conclusion of this work is that the 'standard' Rev2PM, which assumes purely conductive parallel heat transport, did not have a priori predictive value for the open lower divertor of DIII-D.However, the Rev2PM could be brought into reasonable agreement with measurements by deriving a self-consistent set of a posteriori convective corrections to the model.
In light of this finding, it is interesting to examine why the standard Rev2PM found so much success in predicting the upstream plasma in the Q = 10 ITER SOLPS-ITER database [13].A closer look at the code-calculated components of q ∥,x revealed that f cond ranged from 0.74 to 1.09 for the flux tubes in that database, a much narrower (and closer to one) range than in the DIII-D database presented in this paper.Since T Rev2PM e,x ∼ f 2/7 cond , accounting for this variation in f cond would only change the model by a factor of 0.92-1.03,which is relatively negligible.
The dominance of convective transport found for the present DIII-D lower divertor database is similar to what was reported by Leonard et al in previous studies of stronglyradiating detached conditions in DIII-D [24][25][26].These works found that after accounting for radiative losses, >90% of the remaining parallel heat flux was carried to the target by convection (especially in regions with T e < 10 eV), and that parallel flows in excess of Mach 1 were necessary to carry this amount of heat.These conclusions are in broad agreement with the present work, and our database extends the previous results to a range of densities and drift directions, and bridges the gap between attached and detached conditions.The prevalance of effective upstream Mach numbers in excess of one in this work is also notable, since it suggests that convective mechanisms beyond parallel flow (which are typically limited to Mach 1) are at play.This is agreement with previous analysis that has posited that much of the convective transport in DIII-D detached regimes is due to poloidal E r xB drifts [23], since at shallow field line pitch, these drifts can 'short circuit' plasma transport along flux tubes and quickly move plasma to the divertor plates even in the absence of parallel flow.
A key lesson of the DIII-D divertor research program has been that divertor closure can be a useful knob for mediating many divertor processes.The balance of parallel conduction and convection in a divertor is one such process.The lower divertor is the most open of the DIII-D divertors (especially in the 'shelf' configuration used for this work), while the upper divertor and small-angle slot (SAS) divertors have a greater degree of geometric closure.At present the two more-closed divertors do not have the diagnostic coverage to reproduce the quantitative analysis shown in this work, but code calculations have provided some important insights in how closure affects parallel divertor transport.In particular, simulations with SOLPS [27], UEDGE [23], and OEDGE [28] have all shown that for the same core plasma conditions, more heat is carried by convection in the more open lower divertor than either of the two more-closed divertors.In addition, EDGE2D modeling of JET divertor configurations have shown that a hypothetical open divertor geometry with baffling removed exhibits much higher convective heat transport than the actual VH/VV/CC configurations with baffling intact [29].
One of the important assumptions made in deriving the convective corrections in sections 5.1 and 5.2 was setting f out,x (the fraction of P SOL that makes it to the lower outer X-point) to 0.25.This sets the total parallel heat flux at the X-point q ∥,x , and changing this modifies the calculated values of f cond and M eff,x .Assuming a larger value for f out,x leads to a smaller f cond and larger M eff,x , which propogate to a lower T Rev2PM e,x and lower p Rev2PM e,x .While a larger value for f out,x would improve the model agreement with measured T e,x in figure 11, this leads to a larger discrepancy in p e,x , and f out,x = 0.25 produces the best simultaneous agreement with the two sets of measurements.This lower value of f out,x would be consistent with some fraction of the outer divertor power being dissipated above the somewhat-arbitrary location used in this work to define 'upstream'; this process again may be a peculiarity of the unbaffled lower divertor of DIII-D, which permits more efficient transport of recycled neutrals directly into the upstream SOL compared to other divertors.However, it is also plausible that a larger f out,x is more appropriate, especially given previous power balance measurements in DIII-D that have shown that ∼60% of the power goes to the outer leg in LSN H-mode conditions [30].While T Rev2PM e,x would be well matched in such a situation, the p Rev2PM e,x calculation would significantly understimate the measured pressures, and the pressure model would need to be augmented with additional physics mechanisms to fully reproduce the p e,x measurements.
This paper has focused on convective corrections to parallel heat and momentum transport, but has neglected potentially important changes to cross-field transport at high density.
In particular, we anticipate that convective cross-field transport due to radial ExB drifts may be an important mechanism in certain regimes, especially since the derived values of M eff,x (figure 10) in excess of one may imply the influence of poloidal ExB transport.Enhanced cross-field convection would break pressure conservation along a flux tube, requiring a re-imagining of equations ( 2) and (14).It is possible that the significant residual scatter in the measured p e,x database (figure 12) is due to such cross-field transport, though deriving a model that captures these effects is beyond the scope of this paper.We also have assumed a fixed value for λ q that follows the multi-machine H-mode scaling for attached plasmas, but this neglects potential broadening effects at high density.This assumption has little qualitative effect on the results presented in this paper, as significant convective corrections must be invoked regardless of the assumed λ q .However, a more complete parametric model of λ q broadening would be useful for quantitative predictions going forward.
The lower divertor of DIII-D was chosen for our Rev2PM validation study because of the availablity of high-quality DTS data at both the target and X-point.However, one weakness of this choice is the relatively short leg length of this divertor compared to the deuterium neutral mean free path.Thus, at high density the 'divertor plasma' can extend beyond the X-point, as can be seen in figure 3.This calls into question whether the measured T e,x and p e,x values are actually representative of the pedestal boundary conditions in such a situation.This perhaps motivates a 'three-point model', incorporating the target, midplane, and an intermediate point such as the divertor entrance (or detachment front, if that is known).Such a model would allow one to capture some level of poloidal variation in f cond , most notably differentiating between the regions of strong convection between the target and Xpoint, and the more conductive conditions between the X-point and midplane, without the computational expense of a full divertor fluid code.However, deriving such a model is beyond the scope of this paper.
Given the above points, it is reasonable to ask whether the present study represents a 'fair' test of the Rev2PM.After all, if one subscribes to the belief that any next-step divertor must be closed, long-legged, and conduction-dominated (like the ITER cases), then does the failure of the model in a highly-convective open geometry even matter?So long as f cond > 0.5, then the convective corrections to the Rev2PM will be less than 20%.To address this question, it will be imperative to test the Rev2PM in a more closed divertor, even if a comprehensive database scanning many density regimes is not available.Unfortunately, in most devices this will require diagnostic development, since standard Langmuir probe analysis typically gives unreliable results in the T e,t < 5 eV regimes that are critical for volumetric dissipation processes.Making such measurements in a meaningful way will require an expansion of DTS coverage, application of advanced spectroscopic techniques (e.g.[31]), or advanced Langmuir probe analysis techniques (e.g.[32]).
While the present form of the Rev2PM with convective corrections is not a predictive model (since it requires T e,x and n e,x to define the correction factors), predictive capability could in theory be recovered if some parametric dependency for f cond and M eff,x could be found that was only a function of target parameters.This is analagous to the apparently fortuitous finding that (1-f cool ) and (1-f mom ) could be represented as functions of T e,t alone, which turned the standard Rev2PM into a predictive model.Our database for f cond exhibits a very rough scaling with T 2 e,t for cases with T e,t > 1 eV, but it is very unlikely that this holds for all levels of divertor closure or machine size.In fact, an opposite trend has already been observed in EDGE2D modeling of the more-closed and larger JET-ILW divertor [29], as they infer values of f cond = 1 at low T e,t (equivalent to 1-f conv in figure 9 of [29]) that drops to f cond < 0.35 at high T e,t .One key difference between the JET work and this paper is that they define 'upstream' as the outer midplane, and their trends are more scattered when looking at the X-point.This highlights the difficulty inherent in finding a simplified parametric dependency for our correction factors, since governing processes like flows, ionization fronts, and drifts are complicated and interconnected, and may not be able to be accurately represented by a reduced model.

Conclusion
A reduced divertor physics model known as the Reverse 2-Point Model (Rev2PM), in which upstream plasma quantities are predicted from target plasma quantities, was tested against a database of DIII-D lower divertor measurements (with conditions ranging from attached to deeply detached).It has been shown that the standard Rev2PM is insufficient to predict measured electron temperatures and pressures at the divertor entrance for DIII-D.However, it has also been shown that these measurements can be reproduced to a reasonable degree if one defines a posteriori convective correction factors to account for the fraction of parallel heat flux carried by convection (f cond ), and the effective parallel flow needed to carry that heat flux (M eff,x ).The values of these correction factors imply that up to 99% of the parallel heat flux is carried by convection in the detached cases, which is in agreement with prior measurements in DIII-D and supports the present understanding that convective heat transport is relatively more important in open divertors (such as the lower divertor of DIII-D).This motivates further tests of the Rev2PM in more closed divertor geometries (in which convective corrections are expected to be less necessary), since it is expected that next-step fusion devices will require a high level of divertor closure in order to reach adequately dissipative regimes.However, the database comparison in this paper does lend support to a key assumption of the Rev2PM: that the volumetric dissipation functions for power (1-f cool ) and momentum (1-f mom ) can be represented as near-universal functions of T e,t alone.Despite massive differences in scale and divertor closure, volumetric dissipation functions derived from ITER Q = 10 SOLPS-ITER simulations did an admirable job of reproducing (with appropriate convective corrections) measured DIII-D conditions even at T e,t < 1 eV, which bodes well for future efforts to develop more comprehensive reduced divertor models that address some of the limitations to the Rev2PM brought up in this paper.a DOE Office of Science user facility, under Awards DE-FC02-04ER54698, DE-AC05-00OR22725, and DE-AC52-07NA27344.
This report was prepared as an account of work sponsored by an agency of the United States Government.Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Figure 2 .
Figure 2. Geometry of the DIII-D lower outer divertor, with divertor Thomson scattering (DTS) measurement chords.Measurements are taken at two locations for each flux tube: near the material surface ('Target'), and just above the X-point, beyond the typical extent of the volumetric dissipation region ('Upstream').

Figure 3 .
Figure 3. 2D maps of DTS measurements of (a) Te and (b) ne for an attached plasma case.(c) 1D radial LOESS fits derived from these measurements to represent target (red) and upstream (blue) conditions.(d)-(f) Same information for a deeply detached plasma case.

Figure 5 .
Figure 5.Comparison of X-point Te calculated by the Standard Rev2PM versus measured X-point Te.Markers labeled as in figure 4. Dashed line represents 1:1 agreement.

Figure 6 .
Figure 6.Comparison of X-point pe calculated by the Standard Rev2PM versus measured X-point pe.Markers labeled as in figure 4. Dashed line represents 1:1 agreement.

Figure 7 .
Figure 7.Total parallel heat flux at the X-point (estimated from global power balance) versus measured target electron temperature.Markers labeled as in figure 4. The dominant ordering parameter for the points is the radial flux tube coordinate R − R sep,mid (see annotations).

Figure 8 .
Figure8.A posteriori estimate of f cond (the fraction of q ∥,x carried by conduction) versus measured target electron temperature.Markers labeled as in figure4.A general trend with T 2 e,t is evident for all but the most detached B × ∇B ↓ cases.

Figure 9 .
Figure 9. Ratio of measured target electron pressure to X-point electron pressure versus measured target electron temperature.Markers labeled as in figure 4. The volumetric momentum loss function (1-fmom) from [12] is overlaid.
e,x v 3 ∥,x + I 0 n e,x v ∥,x x T e,x (1 + τ x ) + I o n e,x ) M eff,x c s,x

Figure 10 .
Figure 10.A posteriori estimate of M eff,x (effective parallel flow needed to carry net convected heat flux at X-point) versus measured target electron temperature.Markers labeled as in figure 4.

Figure 11 .
Figure 11.Comparison of X-point Te calculated by the convection-corrected Rev2PM versus measured X-point Te.Markers labeled as in figure 4. Dashed line represents 1:1 agreement.

Figure 12 .
Figure 12.Comparison of X-point pe calculated by the convection-corrected Rev2PM versus measured X-point pe, assuming τx = 1.Markers labeled as in figure 4. Dashed line represents 1:1 agreement.