(a) Solution of the analytical approximation of the dispersion relation from Eq. (5) (solid) together with the numerical solution using the hot plasma susceptibilities, for plasma temperature T = 20 keV, density , magnetic field B = 2 T, and propagation angle . Dashed line shows . For the electron-whistler wave, . (b) The lowest frequency solution of the analytical approximation of the dispersion relation Eq. (5) (blue solid) together with the whistler approximation from Eq. (6) (red dashed), for propagation angles θ = 0 (thick lines) and (thin lines).
(a) Comparison of the lowest frequency solution of Eq. (5) (blue solid) with the magnetosonic-whistler wave of Eq. (10) (red dashed). The parameters are the same as in Figure 1 . (b) Contour plot of the lowest frequency solution of Eq. (5) and the solution of Eq. (6) . The values plotted are on both figures.
Cs as function of α and Z. The distribution function is valid in the region . Solid black line shows and dashed black line is . The region between the solid and dashed lines gives the combinations of α and Z for which the condition is fulfilled.
Normalized runaway electron distribution function in near-critical field, plotted with respect to the parallel and perpendicular momentum normalized to , for Z = 1 and α = 1.3.
(a) Contour plot of the distribution function, for (solid, corresponding to E = 0.06 V/m) and (dashed, E = 0.069 V/m). The effective charge is Z = 1.5. (b) Comparison between the near-critical, (blue solid) and avalanche, (red dashed) distribution functions. For the near-critical distribution, we used Z = 1 and α = 1.3. For the avalanche distribution, we used , Z = 1, and E = 40 V/m (corresponding to α = 865).
Comparison between the near-critical, (blue solid) and avalanche, (red dashed) distribution functions. For the near-critical distribution, we used Z = 1 and α = 1.3. For the avalanche distribution, we used , Z = 1, and E = 40 V/m (corresponding to α = 865). (a) The distribution function as a function of for (thin lines) and (thick lines). (b) The distribution function as a function of for (thick lines) and (thin lines).
Normalized growth rate for the electron-whistler wave (a,b) and themagnetosonic-whistler wave (c,d). Both in (a,b) and (c,d), the black line is , the electron-whistler approximation is valid in the region above it. In (c,d), the dashed line denotes , the magnetosonic-whistler approximation is valid in the region above it. The rest oftheparameters is , T, and . (a,c) Ultrarelativistic resonance condition for . (b,d) General resonance condition, sum of the cases and m = 0.
Most unstable wave in the near-critical case: maximum of the growth rate ( , contour lines) on the line corresponding to the maximum runaway energy (2.6 MeV, white dots). The parameters are , T.
The value of wave number (blue dashed) and propagation angle (red dotted) (a) and frequency (b) of the most unstable wave as function of maximum runaway energy.
Stability thresholds for the most unstable wave in near-critical electric field, for electron temperature . (a,b) Stability threshold as function of magnetic field for the electron-whistler wave and magnetosonic-whistler waves, respectively. The runaway-beam radius is m (dashed) and m (solid). In (a), we assume and in (b) . (c,d) Sensitivity of the stability threshold to the normalized electric field α for the electron-whistler wave. The runaway beam radius is , and the maximum runaway energy is , corresponding to . In (c) Z = 1 and in (d) Z = 1.5.
Stability threshold for the most unstable electron-whistler wave in near-critical electric field, for the experimental parameters of the T-10 tokamak. The parameters are , Z = 3, .
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