Abstract
Runaway electron distributions are strongly anisotropic in velocity space. This anisotropy is a source of free energy that may destabilize electromagnetic waves through a resonant interaction between the waves and the energetic electrons. In this work, we investigate the highfrequency electromagnetic waves that are destabilized by runaway electron beams when the electric field is close to the critical field for runaway acceleration. Using a runaway electron distribution appropriate for the nearcritical case, we calculate the linear instability growth rate of these waves and conclude that the obliquely propagating whistler waves are most unstable. We show that the frequencies, wave numbers, and propagation angles of the most unstable waves depend strongly on the magnetic field. Taking into account collisional and convective damping of the waves, we determine the number density of runaways that is required to destabilize the waves and show its parametric dependences.
This work, supported by the European Communities under the contract of association between EURATOM, Vetenskapsrådet and the Hungarian Academy of Sciences, was carried out within the framework of the European Fusion Development Agreement. The authors are grateful to H. Smith, G. Papp, and P. Helander for fruitful discussions. One of the authors acknowledges the financial support from the FUSENET Association. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
I. INTRODUCTION
II. DISPERSION RELATION
A. Electronwhistler wave
B. Magnetosonicwhistler wave
III. RUNAWAY CONTRIBUTION
A. Distribution of the runaway electrons
B. Resonance condition
1. Anomalous Doppler resonance
2. Cherenkov resonance
IV. UNSTABLE WAVES
A. Most unstable wave
B. Stability diagram
V. CONCLUSIONS
Key Topics
 Plasma waves
 74.0
 Runaway electrons
 41.0
 Electric fields
 29.0
 Whistler waves
 29.0
 Magnetic fields
 18.0
Figures
(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 electronwhistler 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) 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 electronwhistler 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 magnetosonicwhistler 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.
(a) Comparison of the lowest frequency solution of Eq. (5) (blue solid) with the magnetosonicwhistler 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.
C_{s} 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.
C_{s} 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 nearcritical field, plotted with respect to the parallel and perpendicular momentum normalized to , for Z = 1 and α = 1.3.
Normalized runaway electron distribution function in nearcritical 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 nearcritical, (blue solid) and avalanche, (red dashed) distribution functions. For the nearcritical 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) 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 nearcritical, (blue solid) and avalanche, (red dashed) distribution functions. For the nearcritical 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 nearcritical, (blue solid) and avalanche, (red dashed) distribution functions. For the nearcritical 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).
Comparison between the nearcritical, (blue solid) and avalanche, (red dashed) distribution functions. For the nearcritical 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 electronwhistler wave (a,b) and themagnetosonicwhistler wave (c,d). Both in (a,b) and (c,d), the black line is , the electronwhistler approximation is valid in the region above it. In (c,d), the dashed line denotes , the magnetosonicwhistler 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.
Normalized growth rate for the electronwhistler wave (a,b) and themagnetosonicwhistler wave (c,d). Both in (a,b) and (c,d), the black line is , the electronwhistler approximation is valid in the region above it. In (c,d), the dashed line denotes , the magnetosonicwhistler 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 nearcritical 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.
Most unstable wave in the nearcritical 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.
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 nearcritical electric field, for electron temperature . (a,b) Stability threshold as function of magnetic field for the electronwhistler wave and magnetosonicwhistler waves, respectively. The runawaybeam 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 electronwhistler 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 thresholds for the most unstable wave in nearcritical electric field, for electron temperature . (a,b) Stability threshold as function of magnetic field for the electronwhistler wave and magnetosonicwhistler waves, respectively. The runawaybeam 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 electronwhistler 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 electronwhistler wave in nearcritical electric field, for the experimental parameters of the T10 tokamak. The parameters are , Z = 3, .
Stability threshold for the most unstable electronwhistler wave in nearcritical electric field, for the experimental parameters of the T10 tokamak. The parameters are , Z = 3, .
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