A reduced model for the ITER divertor based on SOLPS solutions for ITER Q = 10 baseline conditions: B. A reduced model based on reversed-direction two point modeling^*

Divertor target survival is a paramount requirement for fusion reactors. Controlled nuclear fusion by definition generates net energy without destroying the container. For tokamaks, the divertor targets are potentially the components most at risk of being seriously damaged by plasma contact. Too high deposited power flux density on the target, q_t, will cause almost immediate target destruction. Too high T_e,t will result in slower, but still unacceptably rapid, destruction of the target by sputter–erosion–wear.

For the ITER Q_DT = 10 baseline, the present analysis of the SOLPS 4.3 simulation database converted to SOLPS-ITER (S-I) results shows that the total power flux density deposited on the target, q_tot,t (W m⁻²), is a function of $\left({T}_{\text{e},\mathrm{t}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$. It is therefore potentially useful to consider reversing the usual view where target conditions are considered to be governed by the boundary conditions of the confined plasma—which are the 'upstream' ('u') quantities for the edge plasma—i.e. q_||,u $\left({P}_{\text{SOL}}\right)$ and p_tot,u (n_e,u, T_e,u, T_i,u)—and to instead consider the boundary conditions of the confined plasma to be governed by the target conditions required for target survival, e.g. by specifying the particular set of values of $\left({T}_{\text{e},\mathrm{t}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ that are known to be compatible with target survival.

Clearly, the implications of this 'reversed perspective' are potentially of major consequence since the values of q_||,u $\left({P}_{\text{SOL}}\right)$ and p_tot,u (n_e,u, T_e,u, T_i,u), must also satisfy critically important requirements for the confined plasma including achieving ignition and Lawson conditions and staying under Greenwald density and above H–L back-transition limits. All this, of course, is just giving quantitative expression to the requirement that a controlled fusion device has to provide net power whilst maintaining integrity of its plasma-facing components, but here the latter constraint is made the starting point

The divertors of DT tokamaks will have to satisfy a number of other critical constraints, including: efficient pumping needed for density control and He ash removal; (de-) enrichment/compression of He; strong radiative cooling without instability or degradation of fusion performance; etc. These additional constraints, however, differ in a fundamental way from the constraint of not destroying solid targets: they are constraints on the plasma state, specifically on the plasma state of highly dissipative divertors. Unfortunately, present understanding of that plasma state is still significantly incomplete making it difficult to make quantitative predictions with high confidence; the models currently used for prediction typically only match present measurements qualitatively [1]. By contrast, the constraint of not destroying solid targets involves the first state of matter, the solid state, almost exclusively. Fortunately, thermo-mechanical engineering of solid structures is a relatively mature subject, as is also the field of sputtering of solids, making the requirement of target survival a very robust starting point. Divertor plasma physics, with its various uncertainties, is not directly involved in quantifying the constraint of divertor target survival; that constraint is the same for a limiter tokamak—and in fact is the same for any solid device containing plasma, whether for fusion or any other application.

Section 2 describes a reduced model for the Q = 10 ITER divertor based on reversed-direction two point modeling, Rev2PM. Section 3 presents tests of the predictive capability of the Rev2PM for the Q = 10 ITER cases analyzed in the part A paper. Section 4 contains a summary, discussion and conclusions.

For a detailed description and discussion of standard, i.e. forward-direction, two point modeling, 2PM, see e.g. section 4 of [2]; sections 5.2 and 5.4 of [3]; also [4, 5]; etc. The basic 2PM assumes that q_|| and p_tot are constant along SOL field-lines, i.e. there are no volumetric losses within the plasma. We are particularly interested in ${T}_{\text{e},\mathrm{t}}^{2\mathrm{P}\mathrm{M}}$:

$$\begin{equation}{T}_{\text{e},\mathrm{t}}^{\text{Basic}2\mathrm{P}\mathrm{M}}=\left[\frac{8{m}_{\text{i}}}{e{\gamma }_{\text{sh}}^{2}}\right]\left[\frac{{q}_{\Vert}^{2}}{{p}_{\text{tot}}^{2}}\right].\end{equation} \tag{ 1a }$$

In forward-direction 2PM, the independent variables, i.e. the inputs, are the upstream quantities, q_|| and p_tot, i.e. q_||,u and p_tot,u, and the dependent variables, i.e. the outputs, are downstream target quantities such as T_e,t,gc. The extended 2PM includes the effect of volumetric losses, and sometimes additional features, see e.g. equation (15) of [2]:

$$\begin{align}\hfill {T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{Ext}2\mathrm{P}\mathrm{M}}& =\left[\frac{8{m}_{\text{i}}}{e{\gamma }_{\text{sh}}^{2}}\right]\left[\frac{{q}_{\Vert,\mathrm{u}}^{2}}{{p}_{\text{tot},\mathrm{u}}^{2}}\right]\left[\frac{{\left(1-{f}_{\text{cool}}\right)}^{2}}{{\left(1-{f}_{\text{mom}}\right)}^{2}}\right]\left[\frac{\left(1+{\tau }_{\text{t}}/{Z}_{\text{t}}\right)}{2}\right]\hfill \\ \hfill & \quad \times \left[\frac{{\left({1+\mathcal{M}}_{\text{t}}^{2}\right)}^{2}}{{4\mathcal{M}}_{\text{t}}^{2}}\right]\left[{\left(\frac{{R}_{\text{u}}}{{R}_{\text{t}}}\right)}^{2}\right].\hfill \end{align} \tag{ 1b }$$

The average charge of the ions at the target is Z_t; the Mach number at the target is ${\mathcal{M}}_{\text{t}}$; R_u (R_t) is the major radius at the divertor-entrance (target); the other quantities in equation (1b) have been defined in part A. It should be noted that ${f}_{\text{pwr}}^{\text{TK}}\equiv {f}_{\text{cool}}$, see appendix A. The upstream (u) location here continues to be at the poloidal height of the X-point on the low field side. For the present set of ITER simulation cases, there is little difference between ${q}_{\Vert,\mathrm{u}}^{\text{TK}}$ and ${q}_{\Vert,\mathrm{u}}^{\text{TKP}}$, less than 1% on average; T, K and P represent thermal, kinetic and (recombination) potential energy, and so q_||,u can be used to represent either. On the other hand, the difference between ${q}_{\Vert,\mathrm{t}}^{\text{TK}}$ and ${q}_{\Vert,\mathrm{t}}^{\text{TKP}}$ becomes large at low T_e,t, about an order of magnitude, see figure 8(a) of part A and figure 3(a) of this paper.

The form of the 2PM expressions for T_e,t,gc used here, equations (1a) and (1b), hold independently of whether q_|| is carried by conduction or convection, see discussion following equation (5.8) in [3].

The 2PM is virtually always used in the forward direction, but algebraically it is obviously a simple matter to move a target quantity e.g. T_e,t,gc to the rhs of equation (2) and $\left[\frac{{q}_{\Vert,\mathrm{u}}^{2}}{{p}_{\text{tot},\mathrm{u}}^{2}}\right]$ to the lhs, then treat T_e,t,gc as a dependent variable. In reversed-direction 2PM, Rev2PM, the independent variables, i.e. the inputs, are target quantities, e.g. T_e,t, and/or specified global quantities, e.g. Inj_Ne, while the dependent variables, i.e. the outputs, are upstream, divertor-entrance (X-point) quantities, e.g. q_||,u and p_tot,u. We therefore solve the 2PM, equation (1b), for upstream quantities to obtain the most basic relation of the Rev2PM, which is for the now dependent variable ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M}}$ as a function of the now independent variables $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$, i.e. the quantities that will now be specified as input:

$$\begin{align}\hfill {\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M}}& ={\gamma }_{\text{sh}}{\left[\mathrm{e}/4{m}_{\text{i}}\left(1+{\tau }_{\text{t}}\right)\right]}^{1/2}{T}_{\text{e},\mathrm{t}}^{1/2}\left[\frac{\left(1-{f}_{\text{mom}}\right)}{\left(1-{f}_{\text{cool}}\right)}\right]\hfill \\ \hfill & \quad \times \left[\frac{{R}_{\text{t}}}{{R}_{\text{u}}}\right].\hfill \end{align} \tag{ 2 }$$

${T}_{\text{e,t,gc}}^{2\text{PM}}$ is the key quantity in standard (forward) 2PM and ${\left[\frac{{q}_{\Vert,\text{u}}}{{p}_{\text{tot,u}}}\right]}^{\text{Rev}2\text{PM}}$ is the equivalent in Rev2PM. In each case, they are the starting point for obtaining the other divertor quantities of interest, see next section.

Mathematically, the difference between standard (forward) 2PM and Rev2PM is clearly insignificant. The motivation to use Rev2PM for modeling the ITER divertor was discussed in the introduction: this anchors the model in very solidly established properties of the first state of matter, the solid state, rather than in properties of the plasma state which is less well understood generally, and is particularly so for detached divertor conditions in tokamaks.

It will be noted that all of the quantities on the rhs of equation (2) are known and fitted functions of T_e,t,gc, see table 1.

Table 1. Quantities used on the rhs of equation (2) are known and fitted functions of T_e,t,gc in part A.

Quantity in equation (2)	Figures in part A
γsh	Figure 8(f)
$\left(1-{f}_{\text{mom}}\right)$	Figure 10(a)
$\left(1-{f}_{\text{cool}}\right)$	Figure 9(a)
τt	Figure 8(e)

In the case of $\left(1-{f}_{\text{cool}}\right)$, only, there is also a significant secondary dependence on Inj_Ne. It is therefore straightforward to calculate ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M}}$ as a function of input parameters T_e,t,gc and Inj_Ne, see figure 1. In equation (2) the approximations z_t = 1 and ${\mathcal{M}}_{\text{t}}=1$ have been made since for the 3rd flux-tube, ft3, $\left\langle {z}_{\text{t}}\right\rangle =1.0058$ (last-cell) and $\left\langle {\mathcal{M}}_{\text{t}}\right\rangle =1.015$ (guard-cell), where the averages are for the 41 cases. The effect of impurities on the ion mass at the target has also been neglected and m_i = m_D has been assumed since for the ft3 data, $\left\langle {\left({m}_{\text{i}}/{m}_{\text{D}}\right)}_{\text{t}}\right\rangle =1.026$.

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As can be seen, the Rev2PM matches the code values for $\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}$ fairly well. There is a non-negligible dependence on Inj_Ne for low T_e,t, which disappears for high T_e,t; this behavior is due to the secondary dependence of $\left(1-{f}_{\text{cool}}\right)$ on Inj_Ne, since there is little/no Inj_Ne—dependence for the other inputs to the Rev2PM.

The fits used to calculate ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M}}$ are for cases with T_e,t < 15 eV only, and so are only valid for low T_e,t. As was noted in part A, hotter target conditions are likely to cause unacceptable sputtering rates for steady-state conditions—i.e. for the Q = 10 baseline conditions that are the focus of these papers; however, they are of potential interest regarding transients, such as brief loss of detachment. The code values of $\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}$ for the hotter cases with 15 < T_e t < 41 eV are also shown in figure 1(a) as open symbols. T_e,t,gc—fits were not made for these hot cases for the quantities used on the rhs of equation (2); instead, simple averages of the values for the T_e,t > 15 eV cases were used to calculate the dashed curve shown in figure 1(a): $\left\langle {\gamma }_{\text{sh}}\right\rangle$ = 6.1; $\left\langle {\tau }_{\text{t}}\right\rangle$ = 0.29; $\left\langle 1-{f}_{\text{mom}-\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\right\rangle$ = 1.089; $\left\langle 1-{f}_{\text{cool}}\right\rangle$ = 0.860 for ft3. As noted, there was no significant dependence on Inj_Ne for the hot cases for any of the quantities involved. For ft3, R_t = 5.554 m and R_u = 5.264 m. As can be seen, ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M}}$ fairly well matches the code values of $\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}$ for the hot cases as well, confirming the expected variation for well attached conditions, $\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\propto {T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{1/2}$, which is the basic 2PM prediction ${T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{2\mathrm{P}\mathrm{M}}\propto \left[\frac{{q}_{\Vert,\mathrm{u}}^{2}}{{p}_{\text{tot},\mathrm{u}}^{2}}\right]$ for hot target conditions where volumetric losses are small. It is evident from figures 1(a) and (b) that there is also a significant correlation between $\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}$ and Inj_Ne; we will return to this below.

Thus two important steps have now been taken toward the development of a reduced model for the ITER divertor based on reversed-direction two point modeling, Rev2PM:

• (a)  
  In part A, a pair of reduced model control parameters were identified as promising, [T_e,t,gc, Inj_Ne].
• (b)  
  In part B, Rev2PM has been identified as a possible reduced model that can logically and easily use [T_e,t,gc, Inj_Ne], since these are either downstream (T_e,t,gc) or global (Inj_Ne) control parameters.

Any useful reduced model must be capable of making reliable predictions, eventually to be compared with ITER experimental results, but for now to be compared with the results for cases whose code output was not used as input to the reduced model. A limited test of predictive capability was carried out within the context of this study: the present set of 41 ITER cases was divided manually into two subsets: (a) a subset of 20 database (D) cases, and (b) a subset of 21 test (T; not to be confused with T for thermal) cases. The division was based on the values of T_e,t such that the database cases spanned the full range of T_e,t values, leaving the remainder as the test cases. The database cases (only, which is in contrast with part A, where the fits were for all 41 cases) were then used to make the fits needed to quantitatively characterize the inputs to the Rev2PM, which was then applied to the test cases to assess the reliability of the Rev2PM predictions. The database and test cases spanned essentially the same range of T_e,t values.

3.1. Testing the predictive capability of the Rev2PM for $\frac{{\boldsymbol{q}}_{\Vert,\mathbf{u}}}{{\boldsymbol{p}}_{\mathbf{tot},\mathbf{u}}}$

The most important term in equation (2) is the combined volumetric loss term which appears as the ratio $\left(1-{f}_{\text{mom}}\right)/\left(1-{f}_{\text{cool}}\right)$, therefore a fit using the database cases was made for (the reciprocal of) this ratio, i.e. ${\left[\frac{\left(1-{f}_{\text{cool}}\right)}{\left(1-{f}_{\text{mom}}\right)}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$, see figure 2(a).

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Of course, the fit could just as well be done for $\left[\frac{\left(1-{f}_{\text{mom}}\right)}{\left(1-{f}_{\text{cool}}\right)}\right]$ rather than the reciprocal; however, use of the latter facilitates comparison of this lumped loss term with the individual volumetric power loss term, figure 9 of part A. As can be seen from equation (1b), the individual loss terms are in competition with regard to their effect on T_e,t. In principle these terms might have cancelled or even reversed the direction of the T_e,t—dependence on the lumped loss term; however, as can be seen from comparing figures 2(a) and 9 of part A, although it weakens it, the effect of momentum-loss does not overwhelm the effect of power-loss.

The only other quantities that require fits are γ_sh and τ_t; as can be seen from figures 8(e) and (f) of part A, there is very little scatter in these quantities and so it is not worthwhile to make separate fits specifically for the database cases: for γ_sh (τ_t) the average of the ratio of (the code values for the database cases) to (the fit for all the cases) = 0.997 (0.994), with standard deviation = 0.010 (0.024), respectively; therefore, for these quantities the fits for all the cases were used, ${\gamma }_{\text{sh}}^{\text{fit},\mathrm{a}\mathrm{l}\mathrm{l}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}\right)$ and ${\tau }_{\text{t}}^{\text{fit},\mathrm{a}\mathrm{l}\mathrm{l}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}\right)$. There is no significant Inj_Ne—dependence for these quantities.

From equation (2) we then obtain:

$$\begin{align}\hfill & {\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}={\gamma }_{\text{sh}}^{\text{fit},\mathrm{a}\mathrm{l}\mathrm{l}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}}\right){\left[{T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}}\right]}^{1/2}\hfill \\ \hfill & \qquad \quad \times {\left[\mathrm{e}/4{m}_{\text{i}}\left(1+{\tau }_{\text{t}}^{\text{fit},\mathrm{a}\mathrm{l}\mathrm{l}}\left({T}_{\text{e},t}^{\text{code},\mathrm{T}}\right)\right)\right]}^{1/2}\left[\frac{{R}_{\text{t}}}{{R}_{\text{u}}}\right]/\hfill \end{align}$$

$$\begin{equation}{\left[\frac{\left(1-{f}_{\text{cool}}\right)}{\left(1-{f}_{\text{mom}}\right)}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right).\end{equation} \tag{ 3 }$$

The Rev2PM predicted values for the 21 test cases, ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, are shown in figure 2(b). As can be seen, ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ reproduces fairly well the code values for the test cases i.e., ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{totu}}}\right]}^{\text{code},\mathrm{T}}$, without using any code data from the code test cases as input. From the viewpoint of understanding and scaling, this is the most important prediction of Rev2PM since, as noted earlier, $\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]$ is the equivalent for Rev2PM to T_e,t,gc for the 2PM.

3.2. Testing the predictive capability of the Rev2PM for q_||,u

We consider next the prediction for the upstream power flux density, q_||,u, i.e. ${{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, where we will use as input the target power flux density, ${q}_{\Vert,\mathrm{t}}^{\text{TK}}$. It should be noted that we want to avoid using as input the code values for the test cases i.e.${\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{code},\mathrm{T}}$, since we want the only required inputs to be $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right)$. Thus we must first construct ${\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ and then check that it does adequately predict ${\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{code},\mathrm{T}}$. Figure 3(a) shows ${\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ and figure 3(b) shows that ${\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{code},\mathrm{T}}$ is in fact fairly well predicted by ${\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t}\cdot \mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right)$.

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We can now proceed to calculate ${{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$:

$$\begin{equation*}{{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}=\end{equation*}$$

$$\begin{align}\hfill & {\left[{q}_{\Vert,\mathrm{t}}^{\text{TK}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right)/{\left[\left(1-{f}_{\text{cool}}\right)\right]}^{\text{fit},\mathrm{D}}\hfill \\ \hfill & \qquad \left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right).\hfill \end{align} \tag{ 4 }$$

Equation (4) also uses the fit ${\left[\left(1-{f}_{\text{cool}}\right)\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$, which is shown in figure 4.

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Figure 5 shows that ${{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ predicts fairly well the code values of ${{q}_{\Vert,\mathrm{u}}}^{\text{code},\mathrm{T}}$. The agreement is in fact rather better than for ${\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, see figure 2(b).

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From the practical viewpoint, the Rev2PM prediction for q_||u is possibly more important than $\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]$. The most critical boundary values for the confined plasma are typically considered to be n_e,u and P_SOL. The latter is directly related to q_||,u, although (experimentally) also involving the somewhat uncertain value of the SOL parallel power width.

3.3. Testing the predictive capability of the Rev2PM for p_tot,u

We turn next to the Rev2PM prediction for the upstream total pressure, p_tot,u, i.e. ${{p}_{\text{tot},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, which is given by the mathematical identity:

$$\begin{equation}{{p}_{\text{tot},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}={{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}/{\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}.\end{equation} \tag{ 5 }$$

Figure 6 shows that ${{p}_{\text{tot},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ predicts fairly well the code values of ${{p}_{\text{tot},\mathrm{u}}}^{\text{code},\mathrm{T}}$. The quality of the match is about the same as for $\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]$, since the match for q_||u was much better than for $\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]$. Results, not shown here, are quite similar for ${{p}_{\text{tot},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}={\left[{p}_{\text{tot},\mathrm{t}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right)/{\left[\left(1-{f}_{\text{mom}}\right)\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right)$.

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3.4. Testing the predictive capability of the Rev2PM for T_e,u

We turn next to the Rev2PM prediction for the upstream electron temperature, T_e,u, i.e. ${{T}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, which we will take here to be given by:

$$\begin{equation}{{T}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}={\left(7{{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}{L}_{\Vert}/2{\kappa }_{\text{e},\Vert,0}\right)}^{2/7}.\end{equation} \tag{ 6 }$$

The basis for this simple estimate, which assumes the approximation that q_|| is carried entirely by Spitzer parallel electron heat conduction, is discussed, e.g. in section 4.10 of [3]. For ft3, L_||,div = 20.5 m. Here κ_e,||,0 = 2597 has been used without any Z_eff correction [6]. Results are shown in figure 7(a). As can be seen, ${{T}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ predicts fairly well the code values of ${{T}_{\text{e},\mathrm{u}}}^{\text{code},\mathrm{T}}$, although there is very little variation among the cases, which is to be expected because of (i) the 2/7th power involved, and (ii) the fact that ${{q}_{\Vert,\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ varies little, figure 5, for these specific ITER cases all of which have P_SOL = 100 MW.

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For the Rev2PM prediction for the upstream ion temperature, T_i,u, i.e. ${{T}_{\text{i},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, we will use the strong correlation that has been found in the present code results between ${{T}_{\text{i},\mathrm{u}}}^{\text{code}}/{{T}_{\text{e},\mathrm{u}}}^{\text{code}}\equiv {\tau }_{\text{u}}^{\text{code}}$ and Inj_Ne, see figure 7(b) which is for the database cases only. This provides the functional relation $\left[{\tau }_{\text{u}}^{\text{code},\mathrm{D}}\right]\left(\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$. The values of ${{T}_{\text{i},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ are then given by:

$$\begin{equation}{{T}_{\text{i},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}={{T}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}\left[{\tau }_{\text{u}}^{\text{code},\mathrm{D}}\right]\left(\mathrm{I}\mathrm{n}{{\mathrm{j}}_{\text{Ne}}}^{\text{code},\mathrm{T}}\right).\end{equation} \tag{ 7 }$$

Results are also shown in figure 7(a).

3.5. Testing the predictive capability of the Rev2PM for n_e,u

Finally we obtain the Rev2PM prediction for the upstream electron density, n_e,u, i.e. ${{n}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$, which is given by:

$$\begin{equation*}{{n}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}=\end{equation*}$$

$$\begin{equation}{{p}_{\text{tot},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}/\left[\mathrm{e}\left({{T}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}+{{T}_{\text{i},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}\right)\right]\end{equation} \tag{ 8 }$$

where the upstream Mach number ${\mathcal{M}}_{\text{u}}$ is assumed to be small enough that ${\mathcal{M}}_{\text{u}}^{2}\ll 1$. Figure 8(a) shows that ${{n}_{\text{e},\mathrm{u}}}^{\text{Rev}2\mathrm{P}\mathrm{M},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ fairly well predicts the code values of ${{n}_{\text{e},\mathrm{u}}}^{\text{code},\mathrm{T}}$, and as for the other upstream quantities, this is done without using any code data from the code test cases as input. The quality of the match is about the same as for $\left[\frac{{q}_{\Vert,\mathrm{u}}}{{p}_{\text{tot},\mathrm{u}}}\right]$ and p_tot,u.

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Equations (1) and (2) are the 2PM expressions that are appropriate when the energy loss to the electrons due to ionization of the recycling neutral deuterium is not explicitly taken into account, but is instead directly included with the other electron power losses. The expressions are somewhat different when the recycle power loss is included explicitly, see e.g. section 5.5 of [3], equation (4) of [5], etc. The above Rev2PM analysis requires some small modifications in order to take recycle power loss explicitly into account, resulting in small changes in the figures starting with figure 1. Details will be reported elsewhere. Here we only report the result for n_e,u, figure 8(b). As can be seen, the effect of taking recycle energy loss explicitly into account is small; however, it does provide a somewhat better match to the code.

As an aside: it is evident from figure 8 that there is a strong correlation between n_e,u and Inj_Ne. The related correlation between n_e,u and ⟨c_z⟩_sep was pointed out in [7], see figure 15 there and accompanying text, (${\left\langle {c}_{\text{z}}\right\rangle }_{\text{sep}}\equiv \sum {n}_{\text{z}}/{n}_{\text{e}}$ (where the sum is over the charge states of species Z) is the impurity concentration averaged along the separatrix upstream of the X-point) where it is noted that this may have important implications for ITER baseline operation:

'Returning to the question of maximum tolerable n_e,sep, the highest values in figure 15(a) ... are found at the lowest c_Ne and peak at n_e,sep ∼ 6 × 10¹⁹ m⁻³* ⁴ . This is 0.5 n_GW for the 15 MA ITER baseline burning plasma.... The analysis in [93 of [1]], finds the observed H-mode operational boundary to be roughly limited by n_e,sep/n_GW < 0.4–0.5 in JET and ASDEX Upgrade.... If the same limit also applies to ITER, then the lowest seeding values in the SOLPS-4.3 database would be marginal, .... Since the lower c_Ne are those at which Q_DT is maximized [3 of [1]], a compromise will need to be sought for the appropriate seeding level, assuming Ne as extrinsic radiator and depending on the real λ_q which is found on ITER. What is clear is that more simulations are required, particularly at reduced transport in the W/Be environment with impurity seeding (Ne and N)'.

In summary, it is sufficient to specify the value of the variable pair $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}\right.$, $\left.\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ alone, for the Rev2PM to be able to predict the values of all upstream (divertor-entrance) quantities of practical interest. The specified values of $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}\right.$, $\left.\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ may be any values within the ranges (for the database cases) of the variables used to make the numerical fits needed to quantitatively characterize the Rev2PM. Further, since all the target quantities of practical interest are strongly correlated with the same variable pair, $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}\right.$, $\left.\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$, database fits, e.g. for ${q}_{\Vert,\mathrm{t}}^{\text{TK}}$, figure 3(a), can also be used to predict the values for all the target quantities of practical interest as well. Therefore it is sufficient to specify the value of $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}\right.$, $\left.\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ to be able to reproduce the values of all the major target and divertor-entrance quantities calculated by S-I for ITER baseline conditions.

As noted earlier, only a limited test of predictive capability could be carried out within the context of this report, based on dividing the set of 41 cases into two subsets. All the cases used in this report are for ITER baseline burning plasma conditions, Q_DT = 10, q₉₅ = 3, P_SOL = 100 MW, metallic walls, and Ne seeding; they do not include different values of P_SOL or pumping speed, nor other impurity species, nor classical currents or drifts, nor different ad hoc radial transport coefficients, etc. Extensions of the present study will therefore be required to be able to draw more definitive conclusions.

As discussed in section 1, it would be valuable to have, if possible, a comprehensive Rev2PM reduced model for the ITER divertor, i.e. a single model that would cover a range of P_SOL values and of other critical inputs such as the assumed values of the radial transport coefficients, χ_⊥, D_⊥, or the power width, λ_q,||, etc, that is a reduced model whose sole inputs would be $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}},{P}_{\text{SOL}},{\lambda }_{\text{q},\Vert},{S}_{\text{pump}},\dots \right)$. It remains to be seen if this will be possible. If it is not, then an alternative approach to the Rev2PM is to use a reduced model based purely on numerical fits for the upstream quantities, similar to the ones in sections 3–5 of part A for the target quantities. Here we will call this the fully fit-based reduced model, to distinguish it from the Rev2pm reduced model. Of course, the latter also uses numerical fits, but the former is purely fit-based and does not use Rev2PM.

Consider, for example, the key quantity n_e,u. The lines in figure 9(a) are ${\left[{n}_{\text{e},\mathrm{u}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ and were constructed in the same way as the fits in figures 2(a), 3(a) and 4, i.e. using the database cases only. Figure 9(b) shows that ${\left[{n}_{\text{e},\mathrm{u}}\right]}^{\text{code},\mathrm{T}}$ is about as well predicted by ${\left[{n}_{\text{e},\mathrm{u}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}^{\text{code},\mathrm{T}},{\mathrm{I}\mathrm{n}\mathrm{j}}_{\text{Ne}}^{\text{code},\mathrm{T}}\right)$ as by the Rev2PM reduced model, compare figure 8. It seems less likely, however, that it will be possible to develop a comprehensive fully fit-based reduced model than a comprehensive Rev2PM reduced model: for the former it would seem necessary to calculate separate ${\left[{n}_{\text{e},\mathrm{u}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ for each value of P_SOL, although this remains to be seen.

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What has been achieved so far then is that (a) two limited-scope reduced models for the ITER divertor have been developed, and (b) a general procedure has been identified for constructing a Rev2PM reduced model as well a fully fit-based reduced model for a tokamak divertor based on a set of code results for the tokamak. It remains to be seen if comprehensive reduced models can be found, even just for ITER.

Finally, it should be noted that, so long as these reduced models are based on phenomenological, numerical correlations, e.g. for $\left(1-{f}_{\text{cool}}\right)-{T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}$, then, strictly, they can only be used to make projections to lower values of T_e,t,gc than were used to make the numerical fits used in the reduced models, and not true predictions: the latter would require a physics-based model for the functional relationships, e.g. $\left[1-{f}_{\text{cool}}\right]\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$, valid for the required range of $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ values.

Here in part B of the study, reversed-direction two point modeling, Rev2PM, was identified as a limited-scope (because of a single value of P_SOL, primarily) reduced model that is able to directly and readily take advantage of using $\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$ for its reduced model control parameters, i.e. its input, since these are either downstream or global quantities. In standard, i.e. forward-direction 2PM, the independent variables, i.e. the inputs, are the upstream quantities, q_||,u and p_tot,u, and the dependent variables, i.e. the outputs, are target quantities such as T_e,t,gc. In reversed-direction 2PM, Rev2PM, the independent variables, i.e. the inputs, are target quantities, e.g. T_e,t,gc, and/or specified global quantities, e.g. Inj_Ne, while the dependent variables, i.e. the outputs, are upstream quantities, e.g. q_||,u and p_tot,u.

In the present study, three steps have thus been taken toward the development of a (comprehensive) reduced model for the ITER divertor based on reversed-direction two point modeling, Rev2PM:

• (a)  
  In part A, a pair of promising reduced model control parameters have been identified, [T_e,t,gc, Inj_Ne].
• (b)  
  In part B, the Rev2PM has been identified as a (limited-scope, at least) reduced model that can logically and easily use [T_e,t,gc, Inj_Ne] as inputs, since these are downstream or global control parameters.
• (c)  
  In part B, a first, limited test of the predictive capability of the reduced model was successfully carried out: the present set of 41 ITER cases was divided into two subsets: (a) a subset of 20 database cases, and (b) a subset of 21 test cases. The database cases (only) were used to make the fits needed to apply the Rev2PM to the test cases, and demonstrated reasonable reliability for the Rev2PM predictions.

At this point it is not known if the Rev2PM can provide the basis for a comprehensive reduced model for the ITER divertor since the database of cases used in the present study is too narrow. Extensions of the present study will therefore be required to be able to draw more definitive conclusions.

As was shown in figure 8, values of n_e,u calculated using the fully fit-based reduced model, FFBRM, reproduce the individual code values about as well as the Rev2PM reduced model, Rev2PMRM. This is as to be expected since, strictly 'Rev2PM' should be designated as 'Rev2PMF', where the 'F' stands for formatting, see section 11 'two point model formatting of EDGE2D, SOLPS, SONIC, UEDGE, etc code output' of [2]: all the 2PM and Rev2PM expressions used here are identically true if the individual values of T_e,t,gc, (1 − f_mom), etc for each case are used in the expressions, e.g. equations (1b) and (2). The first step toward a true model involves using numerical fits to e.g. (1 − f_mom)[T_e,t,gc], etc, as has been done in the present work. As just noted, the next step will be to develop more comprehensive reduced models based on code cases for a range of values of P_SOL, etc, if that is possible. However, a model with full predictive capability will require that quantities like (1 − f_mom)[T_e,t,gc] be obtained using information from other sources than codes, for example, from first-principle physics models or from experimental measurements. A potentially useful step short of full predictive capability will be possible if it is found that a variety of codes give similar results for key quantities like (1 − f_mom)[T_e,t,gc].

This still leaves the question that if, for example, the values of n_e,u calculated using the FFBRM reproduce the individual code values about as well as the Rev2PMRM does, then why bother with the latter? As noted earlier, it seems less likely that it will be possible to develop a comprehensive FFBRM than a comprehensive Rev2PMRM: for the former it would seem likely that it will be necessary to calculate ${\left[{n}_{\text{e},\mathrm{u}}\right]}^{\text{fit},\mathrm{D}}\left({T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}},\mathrm{I}\mathrm{n}{\mathrm{j}}_{\text{Ne}}\right)$, etc, for each value of P_SOL, etc. It also seems likely that physical understanding will be better aided by Rev2PMRM than by FFBRM, but this remains to be seen.

High quality experimental results are now emerging from present day tokamak studies for a number of the quantities reported in this study, and can be compared with the above code results for ITER to assess their general plausibility. It will also be informative to apply the general procedure developed in this study to construct a reduced model of the divertor based on a set of code results for a presently operating tokamak, thus making it possible to test predictions of the reduced model with experimental measurements in the near term.

The methodology presented here is a step on the path toward the development of reduced models which could be transferred, eventually, to real tokamak operational control, or the implementation of simple descriptions of the complex processes at work in the divertor which could be used in pulse design simulators. Such simulators need to be fast so that they can be used to turn around a pulse design in a matter of minutes for use in planning discharges on ITER.

The data set used in this report is restricted to baseline ITER conditions, i.e. P_SOL = 100 MW, etc, an important limitation of the scope and potential application of this study that has to be kept in mind. With regard to absolute values, and possibly even to the trends, the findings here may not apply generally for ITER, e.g. to other values of P_SOL, let alone to other tokamaks.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the US Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). The views and opinions expressed herein do not necessarily reflect those of the ITER Organization.

For simplicity, the subscript 'gc' is usually dropped in appendix A, and it is to be understood that T_e,t ≡ T_e,t,gc.

The following is an extension of [2]. There are four components of q_||, the parallel power flux density carried by the plasma, i.e. by the charged particles_|: (i) thermal energy, kT, (T) (ii) kinetic energy, mnv², (K) (iii) potential energy of ionization and molecular dissociation, ${\varepsilon }_{\text{H}}^{\text{pot}}$, (P), and (iv) −jV electrostatic potential energy, (J). A derivation of the −jV term is given in appendix B; however, it should be noted that for the cases used here, plasma currents, as well as drifts, were turned off, i.e. j = 0. As discussed e.g. in [2], the target sheath causes plasma cooling by removing the plasma's thermal and kinetic energy. The sheath heat transmission coefficient for plasma cooling, ${\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$, is therefore defined by:

$$\begin{equation}{\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\equiv {q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}/\left[\mathrm{e}{T}_{\text{e},\mathrm{t},\mathrm{g}\mathrm{c}}{{\Gamma}}_{\Vert,\mathrm{e},\mathrm{t}}\right].\end{equation} \tag{ A1 }$$

It can be shown that, approximately ${\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\approx 7.5$ [3], and thus with several further approximations the result is:

$$\begin{equation}{q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\equiv {\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}\mathrm{e}{T}_{\text{e},\mathrm{t}}{{\Gamma}}_{\Vert,\mathrm{e},\mathrm{t}}\approx 7.5{n}_{\text{e},\mathrm{t}}e{c}_{\text{st}}\end{equation} \tag{ A2 }$$

where the further approximations are to assume: (i) ${c}_{\text{st}}=\sqrt{2\mathrm{e}{T}_{\text{e},\mathrm{t}}/{m}_{\text{i}}}$, the isothermal plasma sound speed at the target, (ii) T_i,t = T_e,t, (iii) no plasma impurities, and (iv) Mach 1 flow at the target. It should be noted, however, that in this paper the full expression, equation (A1), is used to evaluate ${\gamma }_{\text{sh}}\equiv {{\gamma}}_{\text{sheath}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$ using the S-I output values of ${q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}$, T_e,t,gc and Γ_||,e,t.

Here, with the aim of being more explicit, compact and hopefully clearer, we will define:

$$\begin{equation}{q}_{\Vert,\mathrm{t}}^{\text{TK}}\equiv {q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}.\end{equation} \tag{ A3 }$$

As discussed in section 11 of [2] on two point model formatting, the plasma cooling factor, $\left(1-{f}_{\text{cool}}\right)$, is defined by:

$$\begin{equation}\left(1-{f}_{\text{cool}}\right){q}_{\Vert,\mathrm{u}}{R}_{\text{u}}={q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}{R}_{\text{t}}\end{equation} \tag{ A4 }$$

see equation (64) of [2]. The volumetric momentum loss factor, $\left(1-{f}_{\text{mom}}\right)$, is defined similarly, equation (65) of [2]. Here, to be more explicit and clear we will use:

$$\begin{equation}{f}_{\text{pwr}}^{\text{TK}}\equiv {f}_{\text{cool}}\end{equation} \tag{ A5 }$$

$$\begin{equation}\text{Thus}\,\,:\left(1-{f}_{\text{pwr}}^{\text{TK}}\right){q}_{\Vert,\mathrm{u}}^{\text{TK}}{R}_{\text{u}}={q}_{\Vert,\mathrm{t}}^{\text{TK}}{R}_{\text{t}}.\qquad \end{equation} \tag{ A6 }$$

As discussed, e.g. in [2], the power flux density deposited on the target by the plasma includes three energy components (T, K and P), minus the power reflected as neutrals by ions incident on the target, thus:

$$\begin{align}\hfill {q}_{\text{dep},\mathrm{t}}^{\text{plasma}}-{q}_{\text{reflect},\mathrm{t}}^{\text{neut}}& ={q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}-\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}\,\sin \,{\theta }_{\perp }\hfill \\ \hfill & ={\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}-\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}{n}_{\text{e},\mathrm{t}}\mathrm{e}{T}_{\text{e},\mathrm{t}}{c}_{\text{st}}\,\sin \,{\theta }_{\perp }\hfill \end{align} \tag{ A7 }$$

where θ_⊥ is the incidence angle between B and the target surface and:

$$\begin{equation}{\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}-\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}={\gamma }_{\text{sheath}}^{\text{plasma}-\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}}+\frac{{\varepsilon }_{\text{H}}^{\text{pot}}}{\mathrm{e}{T}_{\text{e},\mathrm{t}}}\end{equation} \tag{ A8 }$$

where the (recombination) potential energy ${\varepsilon }_{\text{H}}^{\text{pot}}\equiv {E}_{\text{e}-\mathrm{i}}^{\text{surface}-\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}+{E}_{\text{atom}-\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}}^{\text{surface}-\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}\approx$ 15 eV. Figure A1 compares ${q}_{\Vert,\mathrm{t}}^{\text{P}}$ and $e{\varepsilon }_{\text{H}}^{\text{pot}}{{\Gamma}}_{\Vert,\mathrm{e},\mathrm{t}}$ where, as can be seen, simply assuming a constant ${\varepsilon }_{\text{H}}^{\text{pot}}$ = 14.2 eV gives a quite close match.

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Note that ions that are reflected as atoms do not deposit ${E}_{\text{atom}-\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}}^{\text{surface}-\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{b}}$, and in principle this should be allowed for. For refractory metals, the fraction of ions reflected as neutrals, as well as the fraction of incident energy carried by the reflected atom can be quite high. Figure A2 shows ${q}_{\Vert,\mathrm{t}}^{\text{TKP}}$ and ${q}_{\text{reflect},\mathrm{t}}^{\text{neut}}$ vs T_e,t.

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As can be seen for these cases, ${q}_{\text{reflect},\mathrm{t}}^{\text{neut}}$ is small relative to ${q}_{\Vert,\mathrm{t}}^{\text{TKP}}$ but not negligible. Since the primary purpose of the present report is to identify control parameters, ${q}_{\Vert,\mathrm{t}}^{\text{TKP}}$ is used rather than ${q}_{\Vert,\mathrm{t}}^{\text{TKP}}-{q}_{\text{reflect},\mathrm{t}}^{\text{neut}}$, etc; however, when comparisons with experiment are involved, the difference should be used.

Here we will use: ${q}_{\Vert,\mathrm{t}}^{\text{TKP}}\equiv {q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}-\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}$

In [2] the power loss factor $\left(1-{f}_{\text{pwr}-\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\right)$ is defined by:

$$\begin{equation}\left(1-{f}_{\text{pwr}-\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\right){q}_{\Vert,\mathrm{u}}{R}_{\text{u}}={q}_{\Vert,\mathrm{t}}^{\text{plasma}-\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}-\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}{R}_{\text{t}}.\end{equation} \tag{ A9 }$$

$$\begin{equation}\text{Here}\;\;\text{we}\;\;\text{will}\;\;\text{use}\,\,\,:{f}_{\text{pwr}}^{\text{TKP}}\equiv {f}_{\text{pwr}-\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}\end{equation} \tag{ A10 }$$

$$\begin{equation}\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}:\left(1-{f}_{\text{pwr}}^{\text{TKP}}\right){q}_{\Vert,u}^{\text{TKP}}{R}_{u}={q}_{\Vert,\mathrm{t}}^{\text{TKP}}{R}_{\text{t}}\end{equation} \tag{ A11 }$$

In section 2 it was noted that there is little difference between ${q}_{\Vert,u}^{\text{TK}}$ and${q}_{\Vert,u}^{\text{TKP}}$, and we can simply use q_||,u. Since ${q}_{\Vert,\mathrm{t}}^{\text{TKP}} > {q}_{\Vert,\mathrm{t}}^{\text{TK}}$, then ${f}_{\text{pwr}}^{\text{TKP}}< {f}_{\text{pwr}}^{\text{TK}}$, and so$\left(1-{f}_{\text{pwr}}^{\text{TKP}}\right) > \left(1-{f}_{\text{pwr}}^{\text{TK}}\right)$, which is illustrated in figure A3. From the above it is readily shown that:

$$\begin{equation}\left(1-{f}_{\text{pwr}}^{\text{TKP}}\right)=\left(1-{f}_{\text{pwr}}^{\text{TK}}\right)+\left(e{\varepsilon }_{\text{H}}^{\text{pot}}{{\Gamma}}_{\Vert,\mathrm{e},\mathrm{t}}/{q}_{\Vert,\mathrm{u}}\right)\left({R}_{\text{t}}/{R}_{\text{u}}\right).\end{equation} \tag{ A12 }$$

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The total power flux density deposited on the target by the plasma plus neutrals & photons is:

$$\begin{equation}{q}_{\Vert,\mathrm{t}}^{\text{TKP}}\,\sin \,{\theta }_{\perp }+{q}_{\text{dep},\mathrm{n}\mathrm{e}\mathrm{u}\mathrm{t},\mathrm{t}}+{q}_{\text{dep},\mathrm{r}\mathrm{a}\mathrm{d},\mathrm{t}}\equiv {q}_{\text{tot},\mathrm{t},{\theta }_{\perp }}\end{equation} \tag{ A13 }$$

where θ_⊥ is the incidence angle between B and the target surface. For the ITER outer target, the toroidally symmetric value is θ_⊥ = 2.7° in the strike point vicinity [7]; with target toroidal tilting and divertor tungsten monoblock surface beveling (to avoid exposed leading edges), the angle between B and the toroidally asymmetric outer target surface is increased to θ_⊥ = 4.2°.

The term q_dep,neut,t allows for the power carried by the reflected neutrals, and thus may even be negative in principle at certain spatial locations, although of course the power flux density ${q}_{\text{dep},\mathrm{t}}^{\text{total}}$ integrated over the entire plasma facing surface must be equal to the power entering the simulation through the core or via 'external' volumetric sources.

The physical validity of the −jV component of q_||,t has sometimes been questioned; it is discussed in appendix B. When included then the variable is ${q}_{\Vert,\mathrm{t}}^{\text{TKPV}}$, where 'V' is for electrostatic potential energy, to be distinguished from recombination potential energy which is indicated by 'P'. As noted, for the present cases j = 0.

The following extends the analysis in section 25.5 of [3] 'Expressions for the Floating Potential, Particle and Heat Flux Densities Through the Sheath', to now include a derivation of the −jV term. In [3] it was simply stated without proof on page 652 that 'The power removed from the plasma as a function of V is q_se(V) = q_surface(V) − j_TOT(V)|V|. It is important to distinguish between the power to biased and floating surfaces, i.e. equation (25.54) versus equation (25.46), [25.30, 25.31]'. In the following the −jV term is derived.

The current density j(≡j_TOT) from the plasma to the target is given by:

$$\begin{equation}j=\mathrm{e}\left({{\Gamma}}_{\text{i}}-{{\Gamma}}_{\text{e}}\right)\end{equation} \tag{ B1 }$$

where Γ_i(e) is the ion (electron) particle flux density.

At the sheath edge, se, the thermal energy removed from the plasma electrons (thereby cooling them) by the target sheath is:

$$\begin{equation}{q}_{\text{e},\mathrm{s}\mathrm{e},\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}}={{\Gamma}}_{\text{e}}\left(2\mathrm{e}{T}_{\text{e}}-\mathrm{e}\mathrm{V}\right)\end{equation} \tag{ B2 }$$

where V is the electrostatic potential of the target; the reference potential is taken here to be at the se, i.e. V_se = 0. For an electrically floating target V < 0. Targets are almost never biased such that V > 0, owing to the extremely large (damaging) current of electrons that would result. Thus, usually −eV is positive, and −eV $\approx$ 3eT_e for hydrogenic plasmas and floating targets.

At the sheath edge, se, the thermal energy removed from the plasma ions (thus cooling them) by the target sheath is:

$$\begin{equation}{q}_{\text{i},\mathrm{s}\mathrm{e},\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}}={{\Gamma}}_{\text{i}}\left(\sim 2.5\,\,\mathrm{t}\mathrm{o}\sim 3.5\mathrm{e}{T}_{\text{e}}\right)\end{equation} \tag{ B3 }$$

where the value of the numerical coefficient depends on assumptions made about the amount of thermal energy convected by the ions in the pre-sheath, i.e. the plasma upstream of the sheath entrance, see section 25.1 of [3]. In SOLPS the factor is 3.5, and we will use that value in the following for concreteness .

Table A1. Table of variable names. Concentrations are defined as $\left\langle {c}_{\text{z}}\right\rangle =\frac{\sum {N}_{\text{z}}^{i,j}}{\sum {N}_{\text{e}}^{i,j}}$, where i and j are radial and poloidal cell indices of the numerical simulation grid. ${N}_{\text{e}}^{i,j}={V}^{i,j}{n}_{\text{e}}^{i,j}$,where V is the cell volume. ${N}_{\text{z}}^{i,j}=\sum {V}^{i,j}{n}_{\text{z,k}}^{i,j}$, where k indicates charge state. Simply connected lower single null, LSN, grids are assumed, where the poloidal cell index of the outer target guard cell is nx, and the cell just 'in front' of the target has the poloidal index nx − 1. Radial index i_sep + 1 is the first set of cells outside the separatrix. Note that ${j}_{\text{x}\;-\;\text{point}}^{\text{outer}}$ is used as the upstream index, essentially where the peak in q_|| occurs.

Description	Symbol	S-I var. name(s)	Source	Indices
Density of ion	na	NA	b2fstate	i, j
Volume of cell	V	VOL	b2fgmtry	i, j
Puff strength D2	InjD	USERFLUXPARM	b2.neutrals.parameters	D2 gas puff stratum
Puff strength neon	InjNe	USERFLUXPARM	b2.neutrals.parameters	Neon gas puff stratum
Flux tube neon concentration	${\left\langle {c}_{\text{z}}\right\rangle }_{\text{div}}$	—	Post-processed	${j}_{\text{x}\;-\;\text{point}}^{\text{outer}}\leqslant j\leqslant nx$
Flux tube neon concentration	${\left\langle {c}_{\text{z}}\right\rangle }_{\text{Xpt}-\mathrm{X}\mathrm{p}\mathrm{t}}$	—	Post-processed	${j}_{\text{x}\;-\;\text{point}}^{\text{outer}}\leqslant j\leqslant {j}_{\text{x}\;-\;\text{point}}^{\text{inner}}$,
				i ∈ {jsep, jsep + 1} {Thi
Deuterium atom	nD,t	DAB2	fort.44	j = nx − 1
Deuterium molecular	nD2,t	DMB2	fort.44	j = nx − 1
Guard cell electron density	ne,t,gc	NE	b2fstate	j = nx
Average neutral pressure	${\left\langle \mathrm{p}\mathrm{n}\right\rangle }_{\text{div}}$	DAB2 X TAB2 + DMB2 X TMB2	Post-processed from	See figure 1
			B2PLOT USER routine, input
			ig = ny + 1
Total plasma pressure at target	ptot,t	—	Postprocessed from b2fstate	j = nx − 1
Total plasma pressure upstream	ptot,u	—	Postprocessed from b2fstate	j = ${j}_{\text{x}\;-\;\text{point}}^{\text{outer}}$
Parallel heat flux density at the target	${q}_{\Vert,\mathrm{t}}^{\text{TKP}}$	(FHTX − FHJX)/PBSX	Postprocessed from B2PLOT a	j = nx
Parallel heat flux density upstream	${q}_{\Vert,\mathrm{u}}^{\text{TKP}}$	(FHTX − FHJX)/PBSX	Postprocessed from B2PLOT a	j = ${j}_{\text{x}\;-\;\text{point}}^{\text{outer}}$
Potential energy flux density at target	${q}_{\Vert,\mathrm{t}}^{\text{P}}$	FHPX/PBSX	Postprocessed from B2PLOT a	j = nx
Potential energy flux density upstream	${q}_{\Vert,\mathrm{u}}^{\text{P}}$	FHPX/PBSX	Postprocessed from B2PLOT a	j = ${j}_{\text{x}\;-\;\text{point}}^{\text{outer}}$
Electrostatic energy flux at target	${q}_{\Vert,\mathrm{t}}^{\text{J}}$	FHJX/PBSX	Postprocessed from B2PLOT a	j = nx
Electron temperature at target	Te,t,lc	TE	b2fstate	j = nx − 1
Electron temperature in guard cell	Te,t,gc ≡ Te,t	TE	b2fstate	j = nx
Ion temperature at target	Ti,t,lc	TI	b2fstate	j = nx − 1
Ion temperature in guard cell	Ti,t,gc	TI	b2fstate	j = nx
Power to target from neutral particles	qneut,t	WNEUT		j = nx
Power to target from radiation	qrad,t	WRAD	Post-processed from B2PLOT a 100 WLLD	j = nx
Total neutral flux to target	ΓD,sum,t	D_FLUX_ATM + 2x a 2_FLUX_MOL	Post-processed from B2PLOT a 100 WLLD	j = nx
Parallel electron particle	Γ||,e,t	SUM(FNA/PBSX)	B2PLOT a , note: neglecting poloidal	j = nx

^aB2PLOT is a postprocessing program packaged as part of the S-I code. In the table above the arguments to the routine, variable names, and indices are given where appropriate.

As the electrons pass through the sheath they each lose an amount of kinetic energy equal to −eV in the retarding electric field of the sheath by the point they reach the solid surface, where they are deposited. Thus the thermal energy deposited on the surface by the electrons is:

$$\begin{equation}{q}_{\text{e},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}={{\Gamma}}_{\text{e}}\left(2\mathrm{e}{T}_{\text{e}}\right).\end{equation} \tag{ B4 }$$

As the ions pass through the sheath they each gain an amount of kinetic energy equal to −eV in the attracting electric field of the sheath by the point they reach the solid surface, where they also deposit. Thus the thermal plus kinetic energy deposited on the surface by the ions is:

$$\begin{equation}{q}_{\text{i},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}={{\Gamma}}_{\text{i}}\left(3.5\mathrm{e}{T}_{\text{e}}-\mathrm{e}\mathrm{V}\right).\qquad \end{equation} \tag{ B5 }$$

The totals are therefore:

$$\begin{equation}{q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{e},\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}}={{\Gamma}}_{\text{e}}\left(2\mathrm{e}{T}_{\text{e}}-\mathrm{e}\mathrm{V}\right)+{{\Gamma}}_{\text{i}}\left(3.5\mathrm{e}{T}_{\text{e}}\right)\,\end{equation} \tag{ B6 }$$

$$\begin{equation}\text{and}\,\,:{q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}={{\Gamma}}_{\text{e}}\left(2\mathrm{e}{T}_{\text{e}}\right)+{{\Gamma}}_{\text{i}}\left(3.5\mathrm{e}{T}_{\text{e}}-\mathrm{e}\mathrm{V}\right).\end{equation} \tag{ B7 }$$

$$\begin{align}\hfill \mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}:\,\,{q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}-{q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{e},\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}}& =-{{\Gamma}}_{\text{i}}\mathrm{e}\mathrm{V}+{{\Gamma}}_{\text{e}}\mathrm{e}\mathrm{V}\hfill \\ \hfill & =-\mathrm{e}\mathrm{V}\left({{\Gamma}}_{\text{i}}-{{\Gamma}}_{\text{e}}\right)=-\mathrm{j}\mathrm{V}\hfill \end{align} \tag{ B8 }$$

$$\begin{align}\hfill \mathrm{a}\mathrm{n}\mathrm{d}\,\,\mathrm{s}\mathrm{o}:\,\,{q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{e},\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}}& ={q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}+jV={q}_{\text{e}+\mathrm{i},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}-j\left\vert V\right\vert \qquad \hfill \end{align} \tag{ B9 }$$

which is the result stated without proof on page 652 of [3].

Such a mathematical exercise may leave one still feeling unclear about what is happening physically. Consider a thought experiment where V becomes more negative than V_floating, i.e. |V| becomes larger and so Γ_e decreases. Thus $j=\mathrm{e}\left({{\Gamma}}_{\text{i}}-{{\Gamma}}_{\text{e}}\right)$ increases, becoming positive. Thus from equation (B8) q_e+i,surf > q_e+i,se,cool and also q_i,surf − q_i,se,cool = −Γ_ieV, which is positive, i.e. the impact energy of the ions on the target is increased. We may wonder how it is possible for more power to be deposited on the target than entered the sheath. The explanation is that we have not as yet considered the electrostatic potential energy of the ions and electrons at the two different locations. At the se each ion has (positive) electrostatic potential energy ɛ_i,pot = e|V| relative to the target, while each electron has (negative) electrostatic potential energy ɛ_e,pot = −e|V|. For V = V_floating, Γ_i = Γ_e and thus there is no net electrostatic potential energy deposited on the target when the e–i pair deposit there (there is, of course, the e–i recombination potential energy, but that is included separately, as discussed in appendix A). However, when V is more negative than V_floating, then q_i,pot,surf = Γ_ie|V| (W m⁻²), q_e,pot,surf = −Γ_ee|V| (W m⁻²), and ${q}_{\text{e}+\mathrm{i},\mathrm{p}\mathrm{o}\mathrm{t},\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}=-{{\Gamma}}_{\text{e}}\mathrm{e}\left\vert V\right\vert \left({{\Gamma}}_{\text{i}}-{{\Gamma}}_{\text{e}}\right)=\mathrm{j}\left\vert V\right\vert$ (W m⁻²), which is positive since j > 0. This then explains where the apparently missing energy/power came from. An equivalent argument applies to the situation where V becomes less negative than V_floating.

The fitting procedure used .excel polynomial fits. It was found convenient to first generate the fits using logarithmic values of the quantities, and then convert back to the values themselves. The numerical values of the log–log fit parameters are given in table C1.

Table C1. Polynomial fit coefficients for the log–log fits. y = ax⁴ + bx³ + cx² + dx + e, where y = log(dependent variable) and x = log(independent variable).

Figure	dependent variable	independent variable	a	b	c	d	e	R2
2a.4 a	(1 − fc)/(1 − fm)	Te,t,gc	−0.144 461 667	0.458 138 567	−0.428 428 825	0.502 895 328	−0.693 627 469	0.917
3a.1 a	${q}_{\Vert,\mathrm{t}}^{\text{TK}(\mathrm{D}\mathrm{B})}$	Te,t,gc	0.035 465 3727	0.141 084 131	−0.651 254 906	1.009 724 27	7.694 892 95	0.999
3a.2 a	${q}_{\Vert,\mathrm{t}}^{\text{TK}(\mathrm{D}\mathrm{B})}$	Te,t,gc	−0.641 012 006	1.296 714 93	−0.933 316 251	0.845 008 876	7.674 263 93	0.999
3a.3 a	${q}_{\Vert,\mathrm{t}}^{\text{TK}(\mathrm{D}\mathrm{B})}$	Te,t,gc	−1.247 390 13	2.026 465 77	−1.071 506 99	0.690 673 584	7.713 269 44	0.999
3a.4 a	${q}_{\Vert,\mathrm{t}}^{\text{TK}(\mathrm{D}\mathrm{B})}$	Te,t,gc	−1.161 351 36	1.957 950 57	−0.787 572 771	0.632 895 878	7.583 794 18	0.999
4.1	1 − ${f}_{\text{pwr}}^{\text{TK}}$	Te,t,gc	0.077 998 1122	0.096 756 0884	−0.724 103 668	1.071 330 96	−0.696 036 483	0.999
4.2	1 − ${f}_{\text{pwr}}^{\text{TK}}$	Te,t,gc	−0.536 138 518	1.176 515 19	−1.019 099 07	0.937 207 54	−0.728 959 293	0.999
4.3	1 − ${f}_{\text{pwr}}^{\text{TK}}$	Te,t,gc	−1.431 211 31	2.140 399 82	−1.063 060 56	0.726 311 26	−0.708 501 836	0.999
4.4	1 − ${f}_{\text{pwr}}^{\text{TK}}$	Te,t,gc	−1.020 846 45	1.834 714 23	−0.921 557 828	0.726 005 873	−0.825 420 658	0.999
9a.1 a	ne,Xpt	Te,t,gc	−0.227 456 696	0.351 570 479	−0.268 267 958	0.060 971 8348	19.922 4712	0.999
9a.2 a	ne,Xpt	Te,t,gc	0	−0.041 252 9932	−0.052 880 6776	0.079 120 0878	19.801 269	0.839
9a.3 a	ne,Xpt	Te,t,gc	0	0.001 612 23456	0.005 588 895 517	−0.029 945 7665	19.749 3639	0.209
9a.4 a	ne,Xpt	Te,t,gc	0	0.240 516 625	−0.332 525 131	0.007 918 862 69	19.719 0914	0.916

^aThe numbers 1, 2, 3, 4 after the period indicate ⟨Inj_Ne⟩ = 0.54, 1.1, 1.8, 3.9 (e²⁰ atoms s⁻¹) respectively.