Radio Frequency Wave Dissipation by Electron Landau Damping in Tokamaks with Solov’ev Equilibrium
Kharkiv National University, Department of Physics and Technology, Kurchatov str., 31, Kharkiv, UKRAINE
Abstract. Longitudinal permittivity elements are derived for RF waves in a toroidal plasma with Solov’ev type equilibrium
by solving the drift-kinetic equation for untrapped and trapped particles. Our dielectric characteristics are suitable to estimate
the wave dissipation by electron Landau damping in the frequency range of the Alfvén and fast magnetosonic waves, for both
the Large and Low Aspect Ratio Tokamaks with circular, elliptic and D-shaped magnetic surfaces. Contributions of the
untrapped and usual trapped electrons to the imaginary part of the longitudinal permittivity are computed for a spherical
tokamak plasma in the wide range of wave frequencies at the different magnetic surfaces.
Keywords: D-shaped tokamak, arbitrary aspect ratio, Solov’ev equilibrium, electron Landau damping
PACS: 52.35.-g, 52.50.Qt, 52.55Ez
Kinetic wave theory in any toroidal plasma should be based on the solution of Vlasov-Maxwell’s
equations [1-7]. However, this problem is not simple even in the scope of the linear theory since to solve the
differential wave equations one should use the complicated integral dielectric characteristics valid in the given
frequency range for realistic two- or three-dimensional plasma models. The form of the dielectric tensor
components depends substantially on the geometry of an equilibrium magnetic field.
In this paper, we analyze the contributions of the untrapped and usual trapped particles to the imaginary
part of the longitudinal permittivity elements for RF waves in an axisymmetric D-shaped toroidal plasma with
Solov’ev equilibrium under the condition when the additional groups of the so-called d-trapped particles are
absent in the plasma volume. In the general case, these dielectric characteristics were derived analytically in Ref.
 by solving the drift-kinetic equation for untrapped and three groups of the trapped particles as a boundaryvalue
problem, using an approach developed for low aspect ratio tokamaks with concentric circular, elliptic and
D-shaped  magnetic surfaces.
2. PLASMA MODEL
The D-shaped transverse magnetic surface cross-sections corresponding to a Solov’ev type equilibrium
 are not concentric and can be plotted.
4. NUMERICAL RESULTS
As some application, let us consider the contributions of the untrapped and usual t-trapped
particles/electrons to m m
Imε||, and m m
Imε ||, in the spherical tokamak with a = 50cm, b = 90cm, d =15cm,
1 70cm R = , 0 86cm R = ; 0 0.6T = φ H , N(0)=3 ⋅1013 cm-3, T(0) =2000 eV. To have a parabolic profile of plasma
pressure the radial structures of plasma density and temperature are presented as N(ρ) = N(0) 1− ρ 2 and
T (ρ) = T (0) 1− ρ 2 , respectively. Under these conditions [small elongation b/a=1.8, and small central beta
= 0.067 o β ], the additional groups of the d-trapped particles are absent in the plasma volume. Computations of
the diagonal elements of the longitudinal permittivity elements are carried out for waves with poloidal and
toroidal mode numbers m=m´=1 and n=2.
The dependence m m
Imε||, and m m
||, Imε versus ω are presented in Fig. 1a, Fig. 1b and Fig. 1c for RF
waves in our plasma model at the magnetic surfaces: a) ρ = 0.2, b) ρ =0.5 and c) ρ =0.8, respectively. As shown
in these plots, the waves (usually, the low-frequency waves) interact effectively with the trapped electrons at the
external magnetic surfaces, see e.g. Fig. 1c, where the fraction of the trapped particles increases and the fraction
of untrapped particles decreases (and tends to zero in the spherical tokamaks if / 1 1 a R → ). By this reason, the
effective heating of the trapped electrons is possible in tokamaks using the Alfvén waves, when the local Alfvén
resonance condition is realized near the plasma boundary or in the region of the moderate radii.
By comparing m m
Imε||, and m m
Imε ||, in Fig. 1, we see that the fast waves with a high phase velocity,
[ ] v ph = R0qoTu / 2 (m + nq) > vT ω π , dissipate mainly due to their transit-time resonant interaction with untrapped
electrons. It means, using the fast waves, the favorable conditions can be created to transform the wave
momentum into the momentum of the untrapped electrons leading the non-inductive current drive.
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