Wave dissipation by electron Landau damping in low aspect ratio tokamaks with elliptic magnetic surfaces

Wave dissipation by electron Landau damping in low aspect ratio tokamaks with elliptic magnetic surfaces

N. I. Grishanov

Laborato´rio Nacional de Computac¸a˜o Cientı´fica, 25651-070 Petro´polis, Brazil G. O. Ludwig

Instituto Nacional de Pesquisas Espaciais, 12221-010 Sa˜o Jose´ dos Campos, Brazil C. A. de Azevedo and J. P. Neto

Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, Brazil 共Received 25 March 2002; accepted 19 June 2002兲

Longitudinal dielectric permittivity elements are derived for radio-frequency waves in an axisymmetric tokamak with elliptic magnetic surfaces, for arbitrary elongation and inverse aspect ratio. A collisionless plasma model is considered. Drift-kinetic equation is solved separately for untrapped 共passing or circulating兲 and three groups of the trapped particles as a boundary-value problem. Bounce resonances are taken into account. A coordinate system with the ‘‘straight’’

magnetic field lines is used. Permittivity elements, evaluated in the paper, are suitable to estimate the wave dissipation by electron Landau damping共e.g., during the plasma heating and current drive generation兲in the frequency range of Alfve´n, fast magnetosonic, and lower hybrid waves, for both the large and low aspect ratio tokamaks. The dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and nondiagonal elements of the longitudinal permittivity. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1499953兴

Low aspect ratio tokamaks共or spherical tokamaks兲rep- resent a promising alternative route to magnetic thermo- nuclear fusion.^1–5 In order to achieve fusion conditions in these devices additional plasma heating must be employed.

Effective schemes of heating and current drive in tokamak plasmas can be realized using rf waves. As is well known, the kinetic wave theory of any toroidal plasma should be based on the solution of Vlasov–Maxwell’s equations. How- ever, this problem is not simple even in the scope of linear theory since to solve the wave共or Maxwell’s兲equations it is necessary to use the correct dielectric共or wave conductivity兲 tensor valid in the given frequency range for the concrete two- or three-dimensional共2D or 3D兲plasma model. In this paper, the longitudinal permittivity elements are derived for rf waves in a 2D axisymmetric tokamak with elliptic mag- netic surfaces under the arbitrary elongation and arbitrary aspect ratio. The drift-kinetic equation is solved separately for untrapped particles, usual t-trapped particles, and two additional groups of the so-called d-trapped particles as a boundary-value problem, using an approach developed for low aspect ratio tokamak with circular magnetic surfaces⁶ and for large aspect ratio tokamaks with elliptic magnetic surfaces.^7,8

To describe a low aspect ratio tokamak with elliptic magnetic surfaces we use the new variables (r,␪⬘^,␾^{) instead} of quasitoroidal coordinates共␳^,␪^,␾兲as

r⫽␳

冑

共a²/b²兲sin²␪⫹cos²␪^,

共1兲

␪⬘⫽2 arctan

冋 ^冑

^{11⫺␧⫹␧tan}

冉

^12arctan

冉

^abtan␪

冊冊 册

^{, ␾⫽␾,}

transforming the initial elliptic cross sections of the magnetic surfaces to circles with radius a of the external magnetic

surface, where the Cartesian coordinates are x⫽(R

⫹␳^cos␪^)cos␾^{, y}⫽(R⫹␳^cos␪^)sin␾^{, z}⫽⫺␳^sin␪^{. In the} (r,␪⬘⁾ coordinates, the magnetic field lines become

‘‘straight,’’ and the module of an equilibrium magnetic field, H⫽兩H兩, is

H共r,␪⬘^兲⫽

冑

^H_␾0

2 ⫹H_␪²₀g共r,␪⬘兲,

共2兲 g共r,␪⬘兲⫽

冑

共1⫺␧cos␪⬘兲²⫹␭共␧⫺cos␪⬘兲²/共1⫺␧²兲, where

␧⫽r/R , ␭⫽h_␪²

共

^b2/a2⫺1

兲

^,

共3兲 h_␾⫽H_␾0/

冑

^H_␾0

2 ⫹H_␪²₀, h_␪⫽H_␪0/

冑

^H_␾0 2 ⫹H_␪²₀. H_␸0(r) and H_␪0(r) are the toroidal and poloidal projections of H for a given magnetic surface at the points␪⫽⫾␲^{/2; R} is the major tokamak radius; and b and a are the major and minor semiaxes of the elliptic cross section of the external magnetic surface. In this model, all magnetic surfaces are similar to each other with the same elongation equal to b/a.

To solve the drift-kinetic equation for plasma particles we use the standard method^9–14 of switching to new vari- ables associated with conservation integrals of energy and magnetic moment. Introducing the variables ␯ 共particle en- ergy兲and␮共nondimensional magnetic moment兲in velocity space instead of ␯储 and␯⬜:

␯²⫽␯储2⫹␯_⬜^2, ␮⫽ ␯_⬜²

␯储2⫹␯_⬜²

冑

^H_␾0 2 ⫹H_␪²₀

H共r,␪⬘兲 , 共4兲 the perturbed distribution function of plasma particles 共any kind of ions and electrons兲can be found as

PHYSICS OF PLASMAS VOLUME 9, NUMBER 9 SEPTEMBER 2002

4089

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f共t,r,␪⬘^,␾^,␯储,␯_⬜兲⫽_s⫽⫾

兺

₁^{fs共r,␪⬘,␯,␮兲exp共⫺i␻t⫹in␾兲, 共5兲}

where we have determined that the problem is uniform in both time t and toroidal angle ␾. In the zeroth order over magnetization parameters 共i.e., neglecting the finite Larmor radius effects and assuming that the wave frequency is much larger than both the drift frequency and the precession fre- quency兲, the linearized drift-kinetic equation for harmonics f_s can be written as

共1⫺␧cos␪⬘兲² 共1⫺␧²兲^1.5

冑

¹⫺␮^g共r,␪⬘兲

g共r,␪⬘兲

冉

^{⳵␪⳵fs⬘⫹inq fs}

冊

^⫺isrh␪^␻␯ ^fs

⫽2共erE_储F/ M h_␪␯T

2兲

冑

¹⫺␮•g共r,␪⬘兲, 共6兲 where

F⫽ N

␲^1.5␯T

3exp

冉

^⫺␯␯T22

冊

^{, ␯T2⫽2TM , q⫽h}␪

冑

^␧1h⫺^␾␧². 共7兲 E_储⫽E"h is the parallel 共to h⫽H/H) electric field compo- nent; th steady-state distribution function F is given as Max- wellian with the particle density N, temperature T, charge e, and mass M . The index of particles species 共ions and elec- trons兲is omitted in Eq.共6兲. By the indexes s⫽⫾1 for f_s, we distinguish the perturbed distribution functions with positive and negative values of the parallel velocity ␯储

⫽s␯

冑

¹⫺␮^g(r,␪⬘) relative to H. Thus, the initial drift- kinetic equation is reduced to the first-order differential equation with respect to the poloidal angle ␪⬘, where the variables r, ␯^,␮共as well as R, a, b, q, N, T兲appear as the parameters. Note, using the smallness of␧Ⰶ1 and␭Ⰶ1, that Eq. 共6兲 can be readily reduced to the initial drift-kinetic equation in Ref. 7 for plasma particles in the large aspect ratio tokamaks with small elongation.

After solving Eq.共6兲, the longitudinal component of the current density j_储⫽j"h can be expressed as

j_储共r,␪⬘兲⫽␲^eg共r,␪⬘兲

兺

^⫾_s^{1 s}

⫻

冕

^0⬁␯3

冕

^{01/g(r,␪⬘)fs共r,␪⬘,␯,␮兲d␮d␯. 共8兲}

Depending on ␮ ^and ␪⬘, the phase volume of plasma particles should be split in the phase volumes of untrapped, t-, and d-trapped particles by the following inequalities:

0⭐␮⭐␮u, ⫺␲⭐␪⬘^⭐␲ for untrapped particles, 共9兲

␮u⭐␮⭐␮t, ⫺␪t⭐␪⬘^⭐␪t for t-particles, 共10兲

␮t⭐␮⭐␮d, ⫺␪t⭐␪⬘^⭐⫺␪d for d-particles, 共11兲

␮t⭐␮⭐␮d, ␪d⭐␪⬘^⭐␪t for d-particles, 共12兲 where关analyzing the condition␯储(␮^,␪⬘⁾⫽0兴

␮u⫽ 1⫺␧

冑

¹⫹␭, ␮t⫽ 1⫹␧

冑

¹⫹␭, ␮d⫽

冑

^1⫹␧␭², 共13兲 and the reflection points ⫾␪t and⫾␪d for t- and d-trapped particles, respectively, are

⫾␪t⫽⫾arccos

再

^{␧共1⫹␭兲/共␭⫹␧2兲}

⫺

冑

^␧共␭⫹^{2共1⫹␭兲}␧²兲²²⫺ 1

␭⫹␧²

冋

^{1⫹␧2␭⫺}

冉

^1⫺␮␧2

冊

²

册 冎

^,

共14兲

⫾␪d⫽⫾arccos

再

^{␧共1⫹␭兲/共␭⫹␧2兲}

⫹

冑

^␧共␭⫹^2共1⫹␧^2␭兲兲²²⫺ 1

␭⫹␧²

冋

^{1⫹␧2␭⫺}

冉

^1⫺␮␧2

冊

²

册 冎

共15兲 for the given magnetic field structure by g(r,␪⬘^{) in Eq.}共2兲. Further, we solve Eq. 共6兲, in the general case, for un- trapped, t-trapped, and two groups of d-trapped particles, i.e., under the condition when the tokamak magnetic field configuration can be considered as a system with two local minimums of H(r,␪⬘). In this case, the existence criterion of the d-trapped particles is ␧⬍␭or

b/a⬎

冑

¹⫹␧⫹q²共1⫺␧²兲/␧. 共16兲 Otherwise, if␧⬎␭, the equilibrium magnetic field has only one minimum, and the d-trapped particles are absent at the given magnetic surface 共as it is in tokamaks with circular magnetic surfaces where b⫽a,␭⫽0 and accordingly␭⬍␧兲. Of course, the d-trapped particles are characteristic only of elongated tokamaks. Moreover, in the D-shaped tokamaks, in the general case, the additional groups of trapped particles can appear. However, the description of the new groups of the trapped particles there will be more complicated.

The solution of Eq.共6兲should be found by the specific boundary conditions of the trapped and untrapped particles.

For untrapped particles, we use the periodicity of f_sover␪⬘^, whereas the boundary condition for the t- and d-trapped par- ticles is the continuity of f_sat the corresponding stop points, Eqs.共14兲and共15兲. As a result, we seek the perturbed distri- bution functions of untrapped, f_s^u, t-trapped, f_s^t, and d-trapped, f_s^d, particles as

f_s^u⫽

兺

^⫾⬁_p ^{fs, pu exp}

冋

^{i2␲共p⫹nq兲␶共T␪u⬘兲⫺inq␪⬘}

册

^{, 共17兲}

f_s^t⫽

兺

p

⫿⬁

f_{s, p}^t exp

冋

^{i2␲p␶共T␪t⬘兲⫺inq␪⬘}

册

^{, 共18兲}

f_s^d⫽

兺

^⫿⬁_p ^{fs, pd exp}

冋

^{i2␲p␶共␪⬘兲⫺Td␶共␪d兲⫺inq␪⬘}

册

^{, 共19兲}

where p is the共integer兲number of the bounce resonances for trapped particles, and the transit resonances for untrapped particles,

␶共␪⬘兲⫽

冕

0

␪⬘ 共1⫺␧²兲g共r,␩兲d␩

共1⫺␧cos␩兲²

冑

¹⫺␮^g共r,␩兲, 共20兲 is the new time-like variable 共instead of␪⬘ to describe the bounce-periodic motion of untrapped, t-, and d-trapped par-

4090 Phys. Plasmas, Vol. 9, No. 9, September 2002 Grishanovet al.

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ticles along the magnetic field line with the corresponding periods T_u⫽2␶⁽␲^{), T}t⫽4␶⁽␪t), and T_d⫽2关␶⁽␪t)

⫺␶⁽␪d)兴. The Fourier harmonics f_{s, p}^u , f_{s, p}^t , and f_{s, p}^d for un- trapped, t-, and d-trapped particles can be readily derived after the corresponding bounce-averaging.

To evaluate the dielectric tensor elements we use the Fourier expansions of the current density and electric field over the poloidal angle ␪⬘^:

j_储共␪⬘兲

共1⫺␧²兲g共r,␪⬘兲⫽

兺

^⫾⬁_m ^{jm储 exp共im␪⬘兲, 共21兲}

E_储共␪⬘兲共1⫺␧²兲g共r,␪⬘兲 共1⫺␧cos␪⬘兲² ⫽

兺

m⬘

⫾⬁

E_储^m⬘exp共im⬘␪⬘兲. 共22兲 As a result, the whole spectrum of the electric field, E_储^m⬘, is present in the given mth harmonic j_储^mof the current density:

4␲ⁱ

␻ ^j储m⫽

兺

^⫾⬁_m⬘ ^{␧储m,m⬘E储m⬘⫽}

兺

^⫾⬁_m⬘ ^{共␧储m,m,u ⬘⫹␧储m,m,t ⬘⫹␧储m,m,d ⬘兲E储m⬘,}

共23兲 where␧储,u

m,m⬘,␧储,t

m,m⬘and␧储,d

m,m⬘are, respectively, the separate contributions of any kind of the untrapped, t-, and d-trapped particles, to the longitudinal共parallel兲permittivity elements:

␧储.u

m,m⬘⫽ ␻p 2r²

2h_␪²␯T

2␲³_p⫽⫺⬁

兺

^⬁

冕

0

␮_uT_uA^m_pA_p^m⬘

共p⫹nq兲²关1⫹2u_p²

⫹2i

冑

␲^up

3W共u_p兲兴d␮^, 共24兲

␧储.t

m,m⬘⫽ ␻p 2r²

2h_␪²␯T

2␲³_p

兺

_⫽^⬁₁

冕

␮u

␮tT_t

p²B_p^mB_p^m⬘关1⫹2␯p 2

⫹2i

冑

␲␯p

3W共␯p兲兴d␮^, 共25兲

␧储.d

m,m⬘⫽ ␻p 2r²

h_␪²␯T

2␲³_p

兺

_⫽^⬁₁

冕

␮t

␮_dT_d

p²共C_p^mC_p^m⬘⫹D_p^mD_p^m⬘兲

⫻关1⫹2z_p²⫹2i

冑

␲^zp

3W共z_p兲兴d␮^. 共26兲 Here we have used the following definitions:

W共z兲⫽exp共⫺z²兲

冉

^1⫹

^冑

^2i␲

^冕

^{0zexp共t2兲dt}

冊

^{, 共27兲}

u_p⫽ r␻

冑

¹⫺␧²T_u 2␲^h␪兩p⫹nq兩␯T

, ␯p⫽r␻

冑

¹⫺␧²T_t 2␲^h␪p␯T

,

共28兲 z_p⫽r␻

冑

¹⫺␧²T_d/2␲^h_␪^p␯T, ␻p

2⫽4␲^{Ne2/ M ,}

A_p^m⫽

冕

0

␲

cos

冋

^{共m⫹nq兲␩⫺共p⫹nq兲␲␶␶共共␲␩兲兲}

册

^{d␩, 共29兲}

B_p^m⫽

冕

0

␪_t

cos

冋

^{共m⫹nq兲␩⫺p␲␶2␶共共␪␩t兲兲}

册

^d␩

⫹共⫺1兲^p⫺1

冕

0

␪t

cos

冋

^{共m⫹nq兲␩⫹p␲␶2␶共共␪␩t兲兲}

册

^{d␩, 共30兲}

C_p^m⫽

冕

␪d

␪_t

sin关共m⫹nq兲␩兴cos

冋

^{p␲␶␶共共␪␩t兲⫺兲⫺␶␶共共␪␪dd兲兲}

册

^{d␩, 共31兲}

D^m_p⫽

冕

␪d

␪_t

sin关共m⫹nq兲␩兴sin

冋

^{p␲␶␶共共␪␩t兲⫺兲⫺␶␶共共␪␪dd兲兲}

册

^{d␩. 共32兲}

As was mentioned previously, Eqs. 共24兲–共26兲 describe the contribution of any kind of untrapped, t-, and d-trapped particles to the dielectric elements. The corresponding ex- pressions for plasma electrons and ions can be obtained from 共24兲–共26兲replacing T, N, M , e by the electron T_e, N_e, m_e, e_eand ion T_i, N_i, M_i, e_iparameters, respectively. To obtain the total expressions of the permittivity elements, as usual, it is necessary to carry out the summation over all species of plasma particles. It must be pointed out that the dielectric characteristics, Eqs. 共24兲–共26兲, are derived neglecting drift effects and the finite particle-orbit widths. These effects 共as well as the finite pressure and Larmor radius corrections兲can be accounted for in the next order共s兲 of perturbations over the magnetization parameter.

Note that Eqs.共24兲–共32兲have been written in the quite general form where the ellipticity is accounted implicitly by the functions␶^(r,␪^{), g(r,}␪^{), and}␭(r) defined, respectively, in Eqs. 共20兲, 共2兲, and 共3兲. In particular, for tokamaks with circular magnetic surfaces, where ␭⫽0 and ␧储,d

m,m⬘⬅0, the expressions ␧储,u

m,m⬘ and ␧储,t

m,m⬘ 共and the corresponding phase coefficients A_p^m, B^m_p兲can be simplified substantially⁶because the ␶^(r,␪) functions for trapped and untrapped particles can be reduced to 共i兲 the third kind elliptic integrals in low (␧

⬍1) aspect ratio tokamaks, or共ii兲the first kind elliptic inte- grals in large (␧Ⰶ1) aspect ratio tokamaks. An important feature of the dielectric characteristics of Eqs. 共24兲–共26兲 is the fact that, since the phase coefficients A_p^m, B_p^m, C_p^m and D^m_p are independent of the wave frequency␻and the particle energy ␯, the analytical Landau integration of the perturbed distribution functions of both the trapped and untrapped par- ticles in velocity space is possible. As a result, the longitu- dinal permittivity elements are written by summation of bounce-resonant terms including the well-known plasma dis- persion function W(z), i.e., by the probability integral of the complex argument, Eq. 共27兲. After this, the numerical esti- mations of both the real and imaginary parts of the longitu- dinal permittivity elements become simpler, and their depen- dence of the wave frequency is defined only by the arguments u_p,␯p, and z_pof the plasma dispersion functions W(u_p), W(v_p), and W(z_p).

One of the main mechanisms of the radio frequency plasma heating is the electron Landau damping of waves due to the Cherenkov resonance interaction of E_储 with untrapped, t-, and d-trapped electrons. Here, it should be taken into account that the Cherenkov resonance conditions are differ- ent for trapped and untrapped particles in the considered plasma model, see the arguments of the plasma dispersion functions共28兲, and have nothing in common with the wave–

particle resonance condition in the cylindrical magnetized plasmas. Another important feature of tokamak plasmas is the contribution of all E_储^m⬘ harmonics to the given j_储^m har- monic of the current density, Eq.共23兲. As a result, after av-

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eraging in time and poloidal angle, the wave power ab- sorbed, P⫽Re(E_储j_储*), due to the untrapped, t-trapped, and d- trapped electrons can be estimated by

P⫽ ␻

8␲

兺

^⫾⬁_m

兺

^⫾⬁_m⬘ ^{共Im␧储m,m,u ⬘⫹Im␧储m,m,t ⬘⫹Im␧储m,m,d ⬘兲}

⫻共Re E_储^mRe E_储^m⬘⫹Im E_储^mIm E_储^m⬘兲, 共33兲 where Im␧储,u

m,m⬘, Im␧储,t

m,m⬘, and Im␧储,d

m,m⬘are the separate con- tributions of untrapped, t-, and d-trapped electrons to the imaginary part of the longitudinal permittivity elements:

Im␧储m,m⬘⫽Im␧储,u

m,m⬘⫹Im␧储,t

m,m⬘⫹Im␧储,d m,m⬘,

Im␧储,u m,m⬘⫽ ␻p

2r⁵␻³ 8h_␪⁵␯T

5␲^{5.5共1⫺␧2兲1.5}_p⫽⫺⬁

兺

^⬁

冕

0

␮_u T_u⁴ 兩p⫹nq兩⁵

⫻A_p^mA_p^m⬘exp共⫺u²_p兲d␮^, 共34兲

Im␧储.t

m,m⬘⫽ ␻p 2r⁵␻³ 8h_␪⁵␯T

5␲^{5.5共1⫺␧2兲1.5}_p

兺

_⫽^⬁₁

冕

␮u

␮_tT_t⁴ p⁵B_p^mB^m_p^⬘

⫻exp共⫺␯p

2兲d␮^, 共35兲

Im␧储.d m,m⬘⫽ ␻p

2r⁵␻³

4h_␪⁵v_T⁵␲^{5.5共1⫺␧2兲1.5}_p

兺

_⫽^⬁₁

冕

␮t

␮dT_d⁴

p⁵共C_p^mC^m_p^⬘

⫹D_p^mD_p^m⬘兲exp共⫺z_p²兲d␮^. 共36兲 In the simplest case of toroidicity-induced Alfve´n eigenmodes¹⁵ 共TAEs兲, describing the coupling of only two harmonics with m₀ and m₀⫺1, the terms with m₀, m₀⫺1 also should be accounted for in Eq. 共23兲 to estimate the TAEs’ absorption by the trapped and untrapped electrons. As a result, the dissipated power of TAEs by the electron Lan- dau damping is expressed as

P⫽ ␻ 8␲_m⫽

兺

_m

0⫺1 m₀

Im␧储m,m⬘兩E_储^m兩²⫹ ␻ 4␲^Im␧储

m₀,m₀⫺1

⫻共Re E_储^m0Re E_储^m0⫺1⫹Im E^m_储⁰Im E_储^m0⫺1兲, 共37兲 where 兩E_储^m兩²⫽(Re E_储^m)²⫹(Im E_储^m)² is the squared module of the mth electric field harmonic. Of course, our dielectric characteristics, Eqs. 共24兲–共26兲 and Eqs. 共34兲–共36兲, can be applied as well to study, e.g., the excitation/dissipation of the kinetic Alfve´n waves in toroidal plasmas as was made in Ref.

9 for a large aspect ratio tokamak with circular magnetic surfaces, and the ellipticity-induced Alfve´n eigenmodes¹⁶ in elongated tokamaks when the (m₀⫾2) harmonics of the electric field and the permittivity elements Im␧_储^m_,u^0,m0⫾2, Im␧_储^m_,t^0,m0⫾2, Im␧_储^m_,d^0,m0⫾2should be involved.

Note that the nondiagonal elements␧m,m储 ⬘兩m⫽m⬘are char- acteristic only for toroidal plasmas. For the one-mode共cylin- drical兲approximations, when m⫽m⬘^⫽m0, the nondiagonal elements vanish, i.e., Im␧储

m,m⬘兩m⫽m⬘⫽0, and Eqs. 共33兲 and 共37兲can be reduced to the well-known expression

P⫽共␻^/8␲兲Im␧储m,m兩E_储^m兩². 共38兲

The longitudinal permittivity elements evaluated in the paper are suitable for both the large and low aspect ratio tokamaks with elliptic magnetic surfaces and valid in the wide range of wave frequencies, mode numbers, and plasma parameters. Expressions共24兲–共26兲,共34兲–共36兲have a natural limit to the corresponding results⁶for tokamak plasmas with circular magnetic surfaces, if b⫽a and␭→0. Since the drift- kinetic equation is solved as a boundary-value problem, the longitudinal permittivity elements 共24兲–共26兲can be applied to study the wave processes with a regular frequency such as the wave propagation and wave dissipation during the plasma heating and current drive generation, when the wave frequency has being given, e.g., by the antenna-generator system. Of course, the best application of our dielectric char- acteristics is to develop a numerical code to solve the two- dimensional Maxwell’s equations in elongated tokamaks for electromagnetic fields in the frequency range of Alfve´n, fast magnetosonic, and lower hybrid waves. On the other hand, they can be analyzed independently of the solution of Max- well’s equation, by analogy with the computations in Refs.

6 – 8.

Note that in analyzing the collisionless wave dissipation by the plasma electrons one should remember other kinetic mechanisms of the wave–particle interactions, such as the transit time magnetic pumping and/or the cyclotron reso- nance damping, which can be described by the transverse and cross-off dielectric permittivity elements. Recently, a comprehensive theoretical analysis on the Cherenkov ab- sorption of the magnetohydrodynamic waves 共Alfve´n and fast magnetosonic waves兲 by electrons has been done¹⁷ for large aspect ratio tokamaks with circular magnetic surfaces.

The corresponding approach can be used as well for elon- gated spherical tokamaks. However, this is a topic for addi- tional investigation.

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