Assessing nonlocality using connection length

An analysis is instead attempted for the SOL collisionality parameter ν_SOL∗ = L/λe. Figure

7.9 shows this value calculated across the domain under the ITER base and raised/lowered density conditions. We are only interested in the upstream value of L/λe for our assessment

of nonlocality, both because it takes its minimum value here and because the nonlocality assessment can be grounded to the value of 100 again - if L/λe < 100 then the mean-

free-path of the thermal-energy-carrying electrons covers the full SOL connection length. The results show a clear trend across all three thermal transport models; the raised density scenario has a minimum L/λe value of ∼50, the ITER base scenario has a minimum of ∼35,

and the lowered density case has a minimum of ∼20. This reflects the trend observed in the simulation results in Sections 7.1 and 7.2, with the nonlocality becoming more important as

Spitzer-H¨arm or flux-limited models and then calculating the minimum L/λe value can be

used as a rough initial assessment for whether the SOL conditions for a particular tokamak would be relevant for nonlocal transport.

A simple formula can be applied for this purpose also, if theoretical, experimental or simulation estimates of upstream temperature/density conditions and connection length are known. Inserting the definition of λe = βTe

2

ne (where the constant β =

1.5×1054

√

2ln(Λ) and taking

ln(Λ) = 15) and adjusting the units in the terms, a formula for ν_SOL∗ can be written as Device Shot no L (m) Tu (eV) nu (m−3) L/λe

DIII-D 25 30 1.0x1019 ₁₅₃ JET 35703 40 59.8 1.89x1019 116 35667 40 43.6 1.49x1019 ₁₇₂ 38880 40 76.1 1.89x1019 71.8 40262 40 80.7 3.28x1019 111 40000 40 100.5 4.84x1019 ₁₀₅ 38012 40 80.2 2.81x1019 96.1 39879 40 86.0 3.75x1019 112 38013 40 42.3 4.16x1019 511 37142 40 61.4 2.18x1019 127 37134 40 72.5 3.03x1019 127 37135 40 78.5 6.25x1019 223 37136 40 77.7 7.99x1019 291 37145 40 62.9 3.48x1019 193 ITER 70 190 5x1019 53.3 70 210 4x1019 34.9 70 245 3x1019 19.2 DEMO 100 300 3.0x1019 18.3 ARC SXD 150 600 1.0x1020 22.9

Table 7.1: SOL parameter values and calculated ν_SOL∗ = L/λe from: DIII-D [176] and

JET [177] H-mode inter-ELM data; the ITER scenarios investigated in Sections 7.1 and 7.2; predicted SOL conditions for a future DEMO [178]; and parameters extracted from the ARC SXD modelling results in Chapter 5.

ν_SOL∗ = L λe = L β nu T2 u ≈ 5.5L(m) nu 1020_m−3  100eV Tu 2 (7.1)

where Tu and nu are the upstream electron temperature/density respecitvely. Using pa-

rameters representative of the FL α=0.2 model ITER cases studied in Sections 7.1 and 7.2, the L/λe values estimated from this formula are 53.3, 34.9, and 19.2 for the raised density,

ITER-base and lowered density scenarios respectively. These are consistent with Figure 7.9 and the relevance of nonlocality to the three cases observed in this study previously.

Equation 7.1 is applied to a typical DIII-D case [176] and JET shot data [177] for SOL parameters in H-mode (between ELMs), as well as to approximate SOL conditions predicted for DEMO [178]. Results are shown in Table 7.1 and Figure 7.10. Only one of the JET shots shown here has L/λe that drops notably below 100 (to 71.8), but even this case does

not reach a collisionality that is comparable to any of the ITER cases investigated here (though it is stated in Ref [177] that some of the lowest collisionality cases were excluded from their analysis). This indicates why local thermal transport models have reasonable success with the modelling of these experiments. However, experiments to study nonlocal thermal transport on existing devices may be possible for lower collisionality conditions than in the JET data shown here. Nonlocality has been observed in kinetic modelling using the KIPP code for some JET H-mode discharges [179].

The predicted DEMO conditions result in a low L/λe of 18.3, a similar value to the

ITER low density case that was studied. This similarity occurs despite DEMO having a higher upstream temperature, due to a longer connection length of a larger device which compensates. This result suggests that if nonlocal effects will have some importance for ITER conditions, as our results suggest, they will pose at least a similar level of concern for future pilot-plant reactor relevant devices like DEMO, if not greater. Nonlocal transport will almost certainly have a huge impact for modelling of ELMs/disruption events, but the results here suggest that in DEMO and lower-collisionality cases in ITER it is unlikely that nonlocal effects could be ignored even in steady-state. However, all results here were obtained under attached conditions, and whether this conclusion extends to detached divertor conditions - that future reactors will have to operate in to be feasible - is questionable.

Figure 7.10: Plot of Lnu against Tu for DIII-D [176] and JET [177] H-mode inter-ELM

data, the FL α=0.2 ITER scenarios investigated in Sections 7.1 and 7.2, and predicted SOL conditions for a future DEMO [178]. Contours of ν_SOL∗ = L/λe= 100, 50 and 30 are shown.

5, these results can be used with this formula to estimate the relevance of nonlocality for the ARC SOL as well. UEDGE also applies flux-limited Spitzer conduction as the parallel thermal transport model, and in the ARC modelling studies a flux-limiter value for α of 0.2 was used. Upstream SOL temperature and density values were extracted from the Super-X Divertor (SXD) results in Section 5.2, for the 105 MW base-case scenario with zero impurity seeding. Parameters were taken from the first grid cell ring outside the separatrix, which carried the extreme power flux density of ∼10 MW m−2. An estimate of the connection length from midplane to target was calculated from the divertor magnetic geometry. The resulting SOL parameters, also listed in Table 7.1, were found to be Tu∼ 600 eV, nu= 1x1020

and L ∼ 150 m, giving a resulting ν_SOL∗ of 22.9. This suggests a high level of nonlocality, but it is notable that it is of comparable magnitude to the values found for DEMO and the lower density ITER scenario, despite having a much higher Tu, because of the extended connection

length provided by the SXD configuration. The X-Point Target Divertor (XPTD) grids in Section 5.3 further extend the connection length up to to L ∼220, a 50% increase over the SXD, which would increase ν_SOL∗ up to 33.7 for similar upstream conditions. However, this is still a very low value, so strong nonlocal effects would still be expected to be present. The connection length would need to increase up to L∼650 in order for ν_SOL∗ > 100. These

Figure 7.11: Plot of Lnu against Tu for the FL α=0.2 ITER scenarios investigated in Sec-

tions 7.1 and 7.2, predicted SOL conditions for a future DEMO [178], and SOL parameters extracted from the ARC SXD modelling results from Chapter 5. Contours of ν_SOL∗ = L/λe

= 100, 50 and 30 are shown.

results therefore suggest that the issue of nonlocal thermal transport will be highly relevant to the ARC divertor in steady-state also, but not necessarily any greater a problem than for ITER or DEMO, despite the much smaller λq and higher Tu predicted for ARC.