Brazilian Journal of Physics, vol. 26, no. 4, december, 1996
731
Short Wavelength Oscillations of a
Magnetized Current-Carrying Plasma
N.I. Grishanov, A.G. Elmov+, C.A. de Azevedo, A.S. de Assis
Universidade do Estado do Rio de Janeiro, 20550-013, RJ, Brazil
+ Instituto de Fsica, USP, C.P. 66318, 05315-970, SP, Brazil
Universidade Federal Fluminense, Niter
oi, 24020-000, RJ, Brazil
Received March 28, 1996
The linear wave propagation is analyzed for inhomogeneous collisionless cylindrical plasmas
in a helical magnetic eld. Using the geometric optics approximation the electron Landau
damping of the kinetic Alfven, fast Alfven, fast magnetosonic and atmospheric whistler
waves are studied in a current-carrying plasma with \hot" electrons and one kind of ions.
The expressions obtained for real and imaginary parts of the radial refractive indexes can be
used in the wide range of the wave phase velocities relative to the electron thermal velocity.
The ion-cyclotron damping of fast magnetosonic waves is estimated taking into account the
magnetic shear corrections.
I. Introduction
The problems of plasma heating and current drive
by the Alfven and ion-cyclotron waves in tokamaks have
stimulated an interest renewed in a study of the waves
excited in magnetized current-carrying plasmas. The
theory of magnetohydrodynamic (MHD) waves, in homogeneous magnetic eld plasmas, has been basically
completed (see, for example, Ref. [1, 2] and the references therein). In the last years, using the kinetic models of high-temperature cylindrical and toroidal plasmas in a helical magnetic eld, the MHD waves were
intensively studied[3;7] in connection with laboratory
experiments. It was shown that the properties of MHD
waves depend strongly on the magnetic eld structure,
on the ratio of the wave phase velocity to the thermal
velocity of particles, and on the density and temperature gradients of the background plasma.
As is well known, the theory of linear waves is based
on the solution of Maxwell's equations for the components of the perturbed electromagnetic elds E; H and
current density j. The set of Maxwell's equations will
be closed if we know the relation between of j and E.
Usually, this connection is dened via the wave conductivity tensor ^ik : ji = ^ik Ek , or via the dielectric
tensor ^ik : ^ik = ik +i4^ik =!, where ik are the Kroneker constants. Depending on the various plasma parameters, the elements of ^ik can have dierent forms,
which is related to the nature of the wave phenomena
observed in the plasma, in particular, by its oscillation
spectra and by the magnetic eld conguration. On the
other hand, every plasma model needs to justify the dielectric tensor components valid at a given frequency
range. For collisional plasmas, to derive the tensor ^ik ,
it is possible to use the ideal MHD plasma model or twouids MHD equations[8] . Usually, the laboratory fusion
plasma is collisionless, it means that the wave frequency
! is larger than the electron-ion collision frequency ei
and the mean free path of electrons is bigger than the
wavelength along the magnetic eld lines. The corresponding expressions for ^ik of a collisionless plasma
can be derived solving the linearized Vlasov equation
for the perturbed distribution functions of plasma particles. The solution of the Vlasov equation and the evaluation of the dielectric tensor components are shown in
Ref. [9], which are applied for Alfven heating and current drive problems in a current-carrying plasma.
Our purpose in this paper is to nd the real and
imaginary parts of the radial refractive index for the
basic eigenmodes, using the geometric optics approximation in cylindrical current-carrying plasmas for the
frequency range of Alfven and ion cyclotron waves. In
this range of the frequencies, it is possible the excitation
of kinetic Alfven waves[10] , the fast Alfven (see Refs.
[1; 2], called sometimes as sheared Alfven[4] ) waves, and
732
N.I. Grishanov et al.
fast magnetosonic (called sometimes as compressional
Alfven[4] ) waves. Analyzing the dissipation characteristics of these waves in collisionless plasmas, we should
take into account two kinds of collisionless damping:
the electron Landau damping (Alfven heating problem); and the ion-cyclotron damping, when the resonant ions absorb the wave energy via the Doppler effect (ion-cyclotron resonant heating). In this paper we
prolong the analysis of those phenomena taking into account a plasma inhomogeneity and extending results in
Refs. [4-6],[9] .
The present paper is organized as follows. In section II, we describe the plasma model. The dispersion characteristics of the Alfven and fast magnetosonic
waves we analyze in section III, taking into account the
Landau damping by the plasma electrons. In section
IV, we present the contribution of the resonant ions,
to the dielectric tensor, for wave frequencies near the
ion-cyclotron frequency, and evaluate the ion-cyclotron
damping of the fast magnetosonic wave in currentcarrying plasmas.
II. Cylindrical plasma model
II.1. Equilibrium current and helical magnetic eld
Here, we use a simple model of a cylindrical currentcarrying plasma, when the equilibrium current j0 is parallel to the helical magnetic eld B0 . Under the equilibrium conditions, we assume that the ions have no
directed current velocity, so that only electrons are the
source of the current j0 in the plasma. This longitudinal
current induces the poloidal magnetic eld B0 (r) . In
this case, the magnetic surfaces may be represented by
the circular and concentric cylinders. For this model we
use cylindrical coordinates (r; ; z), and the same type
of a stationary magnetic eld, as in Ref. [4]:
c
B0r = 0;
B0 = h B0 ;
B0z = hz B0 ;
q
B0 = B02 + B02z ;
(1)
d
where h = B0 =B0 and hz = B0z =B0 are the cylindrical projections of the unit vector, h = B0 =B0. The
magnetic eld conguration (1) is dened by the parameters 1 and 2:
c
@ h = ; hz h dq ;
1 = rhz 2 @r
rh
q dr
z
2
@ rh = 2 hz h 1 ; r dq :
2 = hrz @r
hz
r
2q dr
(2)
d
Here, q = rhz =Rh corresponds to the tokamak safety
factor, where R is the tokamak major radius. Keeping in the mind that many of our results are valid for
toroidal systems, we shall consider the plasma cylinder of length 2R as an approximation of a large
aspect ratio torus where the trapped particle eect
will be neglected. Furthermore, we assume that the
poloidal magnetic eld is much smaller than the axial magnetic eld, B0 << B0z . In the used plasma
model, we also take into account the radial inhomo-
geneity of the plasma density n0 (r) via the parameter,
n = @ ln n0 =@r. This model is reasonable for plasma
1, where cs =
devices
with low pressure, = c2s =c2A p
p
T0e =Mi is sound velocity and cA = B02 =4n0 Mi
is Alfven velocity. The current velocity v0 is assumed
to be much smaller
than the electron thermal velocity,
p
v0 << vTe = T0e =me , where T0e is the plasma (electron) temperature and Mi and me are mass of ions and
electrons, respectively.
Using the Maxwell's equation, for equilibrium cur-
Brazilian Journal of Physics, vol. 26, no. 4, december, 1996
rent density, and magnetic eld,
4 en v ;
(rotB0 )k = 4
j
0 = ;
c
c 00
(3)
we can nd the following expression for the current velocity v0 :
2
(4)
v0 = ; 2!cA
ci
where !ci = eB0 =Mi c is the ion cyclotron frequency.
This velocity should be taken into account (see Ref.
[9]) to describe correctly the steady-state distribution
function of plasma electrons.
Using this model, we can consider three dierent
limits ("submodels"):
a) when the current velocity is equal to zero, the
poloidal magnetic eld is absent, and we have a plasma
733
conned in a straight magnetic eld;
b) when the plasma current is uniform, j0 = const, the
safety factor q is constant too, B0 r; dq=dr = 0 ,
and we have a helical magnetic eld without shear;
c) when the equilibrium current is nonuniform, the radial derivative of q is not equal to zero, dq=dr 6= 0,
which is the characteristic of a sheared magnetic eld,
and we have general model.
II.2. Maxwell's equations in the currentcarrying plasma
We use the normal A1 , binormal A2 and parallel A3
components, relative to the B0 eld line, for the set
of vector values A = f E, H, jg, which are related to
the cylindrical components Ar ; A , and Az , through the
following relationships:
c
A1 = Ar ;
A2 = A hz ; Az h ;
A3 = Az hz + A h :
(5)
Since the equilibrium parameters are independent of time t, angle and variable z, the dependence of oscillating
values on these variables may be represented by exp(;i!t + im + ikz z) (one plane wave approximation). Here, !
is the wave frequency, kz is the axial projection of the wave vector, m is the azimuthal wave number. It means
that we deal with waves (m and kz are arbitrary), propagating in the nonuniform current-carrying low plasma.
For such waves, the Maxwell's equations can be rewritten as
H1 = !c (kbE3 ; kk E2);
h2 E );
3
H2 = !c (kk E1 ; i1 E2 + i @E
+
i
@r
r 3
2
@ rE + i h E ; i E );
H3 = !c (;kb E1 ; i r1 @r
2
2 3
r 2
(6)
kbH3 ; kkH2 = ; !c ^1j Ej ;
h2 H = ; ! ^ E ;
3
kkH1 ; i1 H2 + i @H
+
i
@r
r 3
c 2j j
@ rH ; i h2 H + i H = ; ! ^ E ;
kbH1 + ri @r
2
2 3
r 2
c 3j j
(7)
d
where kk =kz hz + h m=r and kb = hz m=r ; kz h are
the parallel and binormal components of the wave vector relative to B0 . Here, we have taken into account
the magnetic shear eect via the dependence of kk on
r, which is the principal eect in our model. If the
expressions for dielectric tensor elements are known,
the equations (6) and (7) are basic to study the eigenmodes of current-carrying plasmas in the helical magnetic eld. In this paper, we use these equations to evaluate the dispersion characteristics of fast magnetosonic,
734
N.I. Grishanov et al.
lar to B0 , in a nonuniform current-carrying plasma for
the frequency range, !dr << ! << !ci, where !dr kbnc2s =!ci is the diamagnetic (or drift) frequency. The
assumption of a small scale wavelength permits us to
use the geometric optics approximation to describe analytically the waves in a broad frequency range for an
arbitrary plasma density inhomogeneity. Here, we use
the general expressions ^ij given in Ref. [9] and the ap2
proximationof cold plasma ions, kk2 vTi
=!ci2 << 1. After
the summation over electrons and ions, we can obtain
the following expressions for the operator representing
the dielectric tensor of current-carrying plasmas:
fast Alfven, kinetic Alfven, atmospheric whistler waves.
III. Electron Landau damping of short wavelength oscillations in a current-carrying plasma
III.1. Dielectric tensor-operator for MHD waves
In a rst step, we are going to analyze the waves of
a small scale wavelength in the direction perpendicu-
c
2 2
^11 = 1 + i~ k!b c2 ;
2
@ (r:::);
^12 = ig ; ~kr!b c2 @r
^13 = i kkb ? ;
k
@ 1 ::: ; ^ = ; i~ c @ r @ 1 ::: ; ^ = ; ? @ + ;
^21 = ;ig ; ~k!bc2 r @r
22
2
23
r
!2 @r @r r
kk @r n
@ (r:::);
^32 = k?r @r
^33 = 3 + ? kn2kb ;
^31 = ;i kkb ? ;
k
k
k
2
where
2
(8)
p
2
2
1 = cc2 1 + !!2 ;
A
ci
c
2 1 v0 !ci
2 = 1 + c2 !2 ;
~ = 2 Ze We ;
2
kv! g = cc2 !! + k !02 ci ;
ci
A
2
;
p
? = cc2!!ci 1 + i Ze We :
A
A
2
2
p
3 = cc2 k!2cic2 (1 + i Ze We );
A k S
In (8), We is the plasma dispersion function of electrons
"
#
Z Ze
2i
We = We (Ze ) = exp(;Ze ) 1 + p
exp(t2 )dt ;
0
2
!;k v
Ze = p k 0 :
2kk vTe
(9)
d
Note that, in the expressions ^ik , we keep only the
main terms, which are proportional to the electron current velocity v0 and the radial gradient of plasma density n. The terms in ^ik , which are proportional to
the 1 and 2 parameters, are of the same order as the
shear induced terms in the Maxwell's equations (6) and
(7). Furthermore, we take into account the Doppler effect via the term, kk v0 , in the argument of the plasma
dispersion function (9). Supposing that v0 = 0 (or
h = 0), we get the known result for the dielectric permeability, of a nonuniform cylindrical plasma, in the
straight magnetic eld (see Ref. [11]).
Using (8), we can see that the o-diagonal elements
^13;^23; ^31, ^32 are not small, and allow to evaluate the
so called "transit-time magnetic pumping" dissipation
for MHD waves[1;2] . Moreover, these elements of ^ik are
important and give a contribution to the Alfven current
drive via ponderomotive forces[9] .
III.2. Dispersion equation
To obtain the dispersion equations for Alfven and
fast magnetosonic waves, we should solve the equations
Brazilian Journal of Physics, vol. 26, no. 4, december, 1996
(6) and (7). The corresponding solutions can be represented in the form:
Z r
E1;2;3(r) exp i kr ()d ;
0
where E1;2;3(r) are the slowly varying amplitudes, and
kr is the normal (radial) component of the wave vec-
735
tor, which satises the condition kr (r)=n >> 1. In
this paper, we do not consider the mode conversion effects because the geometric optics approximation is not
valid[2] at the conversion zones where 1 kk2 c2 =!2.
Using expressions (8) and (6), the equations (7) can
be written as
c
h
i
1 ; Nk2 ; Nb2 (1 ; i~) E1 + Nb Nr (1 ; i~) + ig + iNk N1 E2
+ Nk Nr + iNb N? ; N2 E3 = 0;
k
h
i
2
2
Nb Nr (1 ; i~) ; ig ; iNk N1 + iNk N2 E1 + 2 ; Nk ; Nr (1 ; i~) E2
N
?
n
+ Nk Nb ; iNr N ; N1 ; N2 ; N ? E3 = 0;
k
k
?
?
Nk Nr ; iNb N ; N2 + N1 E1 + Nk Nb + iNr N ; N1 ; N2 E2
k
k
;
+ 3 ; Nr2 ; Nb2 E3 = 0;
(10)
where Nr;b;k = c kr;b;k =! are corresponding to the normal (radial), binormal and parallel refractive index components
relatively to B0, Nn = c n =!; N1;2 = c 1;2=!. Using the condition when the determinant of (10) is equal to zero,
we obtain the dispersion relation for Alfven and fast magnetosonic waves. These two wave branches are coupled by
the !=!ci terms and by the eects connected with equilibrium current, magnetic shear, and density gradient. To
simplify this equation, we assume that ? >> Nk N1 ; Nk N2 . In this case, we can derive the following equation, for
the complex radial refractive index:
"
NNNN
3(1 ; Nk ) 2 ; Nk2 ; (Nr2 + Nb2 )(1 ; i~) ; i b ;r Nk 2 2
1
k
#
2
Nb N1 N2 ; (g + Nk N1 )(g + Nk N1 ; Nk N2 )
+
1 ; Nk2
h
;
+ Nr2 + Nb2 1(Nr2 + Nb2) ; 2 (1 ; Nk2 ) + (g + Nk N1 )(g + Nk N1 ; Nk N2 )
#
Nr2Nb2 + Nb4 ; Nk2Nr2
+ iNr Nb Nk N2
; N1 N2
Nr2 + Nb2
"
#
2
2
Nk2(g + Nk N1 )
2
2 ?
2
?
2
;(Nr + Nb ) N 2 (1 ; Nk ) + iNn Nr N 2 (1 ; Nk ) 1 ; ( ; N 2) = 0:
? 1
k
k
k
2
Using (11), it is possible to analyze some interesting
limits for the wave vector in parallel and perpendicular directions to the ambient magnetic eld (the cold
plasma limit, nal Larmor radius eects and so on).
(11)
d
III.3. Fast waves
Retaining in zero approximation the largest terms
proportional to 3 (i.e., assuming that 3 ! 1), we can
get from (11) the equation for the real part of the radial
refractive index, ReNr for the fast waves. In the next
736
N.I. Grishanov et al.
approximation (by ;3 1 ), we obtain the damping coecient = ImNr =ReNr , where Nr2 = (ReNr )2(1 + 2i)
and << 1. As a result, for the fast waves we have:
c
N 2 N N ; (g + Nk N1 )(g + Nk N1 ; Nk N2 )
(ReNr )2FW = 2 ; Nk2 ; Nb2 + b 1 2
;
; N2
(
1
k
p
p
Nb2
1 + Nb2 =(ReNr )2 Ze exp(;Ze2 )
2
1 + (ReN
Z
exp(
;
Z
)
;
e
e
2
3(1 ; Nk2) j 1 + ipZe We j2
r)
h ;
1 (ReNr )2 + Nb2 ; 2(1 ; Nk2) + (g + Nk N1 )(g + Nk N1 ; Nk N2 )
#
"
#)
Nb2 + Nb4=(ReNr )2 ; Nk2
kk2c2A (g + Nk N1 )
Nb Nk N2
N
n
; N1N2 1 + N 2 =(ReN )2
; ReN ( ; N 2 ) + ReN 1 ; !! ( ; N 2 )
:
r
r 1
r
ci 1
b
k
k
FW = 21
It should be noted that the dispersion equation (12) is
corresponding to the well known dispersion relations derived in Ref. [4] for the compressional and shear Alfven
waves if we omit the terms, which are proportional
to the current density gradients in equation (12) (i.e.,
N1 ; N2 ! 0). Using equations (12) and (13), we can es-
(12)
(13)
timate the radial refractive index, and the damping coecient of the fast magnetosonic (FMS) and fast Alfven
(FA) waves. In the frequency range ! << !ci, for the
fast magnetosonic wave with !2 (kr2 + kb2 + kk2)c2A ,
these expressions can be simplied to:
c
(ReNr )2FMS = 1 ; Nk2 ; Nb2 ; N1 N2 ;
(14)
2
pZ exp(;Z 2 ) ; Nb Nk N2 + Nn :
Nb
FMS = 1 + (ReN
(15)
e
e
2
)
2
2ReNr (1 ; Nk2) 2ReNr
r
For the fast Alfven wave with ! kk cA , we have
N 2 N N ; (g + Nk N1 )(g + Nk N1 ; Nk N2 )
; Nb2 ; N1 N2 ;
(16)
(ReNr )2FA = b 1 2
1 ; Nk2
#
"
2
kk2 c2A pZe exp ;;Ze2 ;
p
N
2
b
FA = 1 + (ReN )2 2 Ze exp ;Ze ; !2 ( ; N 2)
p
2
r
ci 1
k j 1 + i Ze We j
!
Nb Nk N2
Nn 1 ; !(g + Nk N1 ) :
+ 2ReN
(17)
; 2ReN
2
(
;
N
)
!ci (1 ; Nk2)
r 1
r
k
The dispersion characteristics of the fast Alfven waves in the current-carrying plasma depend on the relation between
two small parameters kk v0=! and !2 =!ci2 . In particular, for kkv0 =! << !2 =!ci2 , we have the following expression of
(ReNr )2FA , to consider the FA-wave radial structure:
!2 21
(ReNr )2FA = !2 (N
; Nb2 ; N1 N2 :
(18)
2
ci k ; 1)
For kk v0 =! >> !2 =!ci2 for (ReNr )2FA , we have
N 2N1 N2 ; Nk2(2N22 ; 3N2N1 + N12)
; Nb2 ; N1 N2 :
(19)
(ReNr )2FA = b
1 ; Nk2
Brazilian Journal of Physics, vol. 26, no. 4, december, 1996
From equation (18), assuming v0 ! 0, we obtain the
well-known condition of the FA-wave propagation in
the plasma with a straight magnetic eld, Nk2 > 1 .
In the case, v0 =cA >> !2 =!ci2 , the FA-waves can be
excited in the plasma with a strong nonuniform current (dq=dr 6= 0) under the condition, Nk2 < 1 . For
the waves with kb = 0 (that usually corresponds to the
wave excitation with a poloidal wave number m = 0),
the propagation condition becomes the usual one. Note
that the radial wavelength of the fast Alfven waves, in
the current-carrying plasma, can be smaller than it is
in the plasma in the straight magnetic eld under the
same values of !; m; and kz . These values can be given,
for example, by the antenna/generator system.
The resonant conditions of the fast wave excitation
substantially depend on the distribution of the equilibrium current. If we assume that the current velocity is
equal to zero in equations (12) and (13), the real and
imaginary parts of the radial refractive index, respectively, will be the same as the well-known results for the
homogeneous magnetic eld [1,4,7]. For plasmas with
an uniform equilibrium current (a nonsheared magnetic
eld, N1 = 0) it is necessary to take into account the
current velocity in the dielectric tensor elements ^12 and
^21, which are important to derive the dispersion rela-
737
tion for the fast waves. For plasmas with a nonuniform
current (N1 6= 0 and dq=dr 6= 0) the frequencies of the
fast magnetosonic and fast Alfven waves will depend
also on the terms driven by shear in ^22 and g.
The correction related to the density gradient in
(15) and (17) is important to evaluate the damping
of the fast magnetosonic and fast Alfven waves in the
nonuniform plasma. Moreover, the third term in equation (15), which depends on the sign of ReNr under
the given sign of n, can cause either decrease or increase of the wave dissipation. In laboratory plasmas,
the density usually decreases to the boundary; it means
that in this case n < 0. Using (15), we nd that the
waves propagating into the plasma are absorbed more
eectively due to the density gradient than it is in an
uniform plasma. On the other hand, the waves propagating
direction ( ReNr > 0; n < 0, and
p in opposite
p
j 2n Nk vTe = ReNr ! j> 1) may become unstable.
III.4. Kinetic Alfven wave
Another solution of the dispersion equation (11) corresponds to the so called kinetic Alfven (KA) wave. In
this case, the refractive index Nr2KA of KA waves is a
large value proportional to 3 and we have respectively:
c
(ReNr )2KA + Nb2 = 3 1 ; Nk2 ;
1
p
2
Nb
Z exp ;;Z 2 KA = 1 + (ReN
e
2
2 e
r)
!
!(g
+
N
N
N
N
N
N
1)
b
2
k
k
n
+ 2ReN ( ; N 2 ) ; 2ReN 1 ; ! ( ; N 2 ) :
r 1
r
ci 1
k
k
(20)
(21)
d
Analyzing these expressions, we see that the equilibrium current does not make a substantial eect on
the dispersion characteristics of the kinetic Alfven wave
in the weakly nonuniform low beta plasma, if <<
NrKA cA =Nn vTe . As is in the uniform plasma[10], the
damping rate of these waves is dened by the Im3 with
a shifted argument
p for the plasma dispersion function,
Ze = (! ; kkv0 )= 2kk vTe .
Note that the inuence of the current on the radial
attenuation of the kinetic Alfven, fast Alfven and fast
magnetosonic waves with kb = 0 (or m = 0) will be
smaller by a factor than the corresponding damping
rates, , in equations (15), (17) and (21). The eect of
the shear terms in these expressions shows us that FA
and FMS waves, which have the small damping coecients, FW Ze , can be unstable in the nonuniform
current-carrying plasma.
The exact solutions for Ez ; Hz ; and @(rEr )=@r +
imE components of the perturbed electromagnetic
eld are proportional to Jm (k?r)exp(;i!t + im +
ikz z) (see Ref. [11]) in homogeneous plasmas conned
in the homogeneous magnetic eld. Here, Jm (k? r)
738
N.I. Grishanov et al.
is the Bessel function. It means that the expressions
obtained for real and imaginary parts of the considered waves become corresponding to the expressions for
cylindrical waves[11] in the uniform plasma, under the
following conditions: k?2 = kr2 + kb2 and kr2 >> kb2 .
III.5. Atmospheric whistlers
Now we are going to analyze the high-frequency fast
magnetosonic waves in the frequency range, !ci <<
! << !ce = eB0 =me c. These waves are called as
atmospheric whistlers (AtW). To obtain the real and
imaginary parts of the AtW's radial refractive index,
we should take into account that for these frequencies
the values of 1 and g in (11) are given by expressions:
c
!pi2 !ci2 1 = ; !2 1 + !2 ;
p
2
2
kv
g = ; cc2!!ci 1 ; k! 0 + !!ci2 ;
A
(22)
where !pi = 4n0e2 =Mi is the plasma ion frequency, and the other components of the dielectric tensor will be
dened by the electron contributions to ^ik in (8). In this case, the expressions for dispersion characteristics of
2
atmospheric whistlers, !AtW
(kr2 + kb2 + kk2 )c2A (!ci2 + kk2c2A )=!ci2 , are given by:
(ReNr )2AtW = 1 ; Nk2 ; Nb2 + N 2g; ;
2
Nb2
AtW = 1 + (ReN
r )2
k ;
#
p
4 4
kk cA Ze exp ;Ze2
pZ exp ;;Z 2 +
e
e
2
(!ci2 + kk2c2A )2 j 1 + ipZe We j2
1
"
N k k c2
Nn
!ci2
+ 2ReN 2(!b2 k+Ak2c2 ) + 2ReN
2
+ kk2c2A ) :
r ci
r (!ci
kA
(23)
d
One can see above that the radial damping rate of atmospheric whistlers have the order of the magnitude
as higher as the damping rate (15) for FMS waves
in the low-frequency range, !; kkcA << !ci . When
!ci2 << kk2c2A (or kk kr and j Ze j<< 1), the damping rate of the high-frequency fast magnetosonic wave
is two times higher than the damping rate of the lowfrequency magnetosonic waves.
IV. Ion-cyclotron damping of the fast magnetosonic waves in a magnetic shear plasma
IV.1. Contribution of ion-cyclotron resonance
to the dielectric tensor
In the previous section, we have studied the wave
dissipation via the electron Landau damping. In magnetized plasmas, besides of the electron damping, it is
possible to obtain the ion cyclotron damping. The ion
cyclotron damping occurs, for example, when the wave
interacts with plasma ions in the resonant condition
! ; kk vk !ci. As is well known (see Ref. [7]), the
presence of a small group of the resonant particles modies strongly the plasma dielectric properties. To calculate the contribution of the resonant ions to the plasma
conductivity, it is necessary to solve the Vlasov equations for an arbitrary parameter kk vTi =(! ; !ci). In
this case we obtain the following expressions for contributions of the resonant ions to ^(jki) , at the fundamental
(rst) harmonic of the ion cyclotron frequency:
c
!pi2
!
+
!
ci
;
= ; 2!(! + ! ) 1 ; ! ; ! (i ; 1) ;
ci
ci
2
i!
!ci (; ; 1) ;
^(12i) = ;^(21i) = 2!(! +pi ! ) 1 + !! +
; !ci i
ci
^ i
( )
11
= ^ i
( )
22
Brazilian Journal of Physics, vol. 26, no. 4, december, 1996
739
i!pi2 ;i @
i!pi2 ;i 1 @
(i)
= ; 2!2 k + @r ; kb ; ^31 = ; 2!2 k + r @r (r:::) + kb ;
k
k
2
2
;
;
!
!
!2
@
1
@
^(23i) = ; 2!pi2 k +i @r ; kb ; ^(32i) = 2!pi2 k +i r @r (r:::) + kb ; ^(33i) = ; !pi2 ;
k
k
^ i
( )
13
where
p
;i = 1 + i Zi; Wi (Zi; ); Zi; =
(24)
!
;
!
h
h
r
dq
ci
z
p
; = 2 r 1 ; 4q dr :
2 j kk + j vTi
d
In these expressions we have omitted the gradient terms
of the ion density, assuming that it will be not important for excitation of high-frequency waves in the frequency range, ! !ci . As can be seen in (25), the
characteristic peculiarity of ^(jki) is the gradient shift of
the kk wave number, in the plasma dispersion functions
;i . This shift can be related to the dierence between
the conditions of the cyclotron resonances in the plasmas with the helical and straight equilibrium magnetic
elds. When plasmas are in straight magnetic eld, the
cyclotron resonance condition is ! ; kz vk = l!c , and
the corresponding condition for current-carrying plasmas (in the helical magnetic eld) is given by
c
! ; kk vk = l !c + 2 hzrh 1 ; 4qr dq
dr ;
(25)
d
where = e; i is a kind of the plasma particles,
l = 1; 2; ::: is the number of resonant cyclotron harmonic. The similar current eect on the cyclotron resonance condition has been found for the passing particles
in the toroidal plasmas (see Ref. [12]).
analyze the ion cyclotron resonance (ICR) dissipation
of eigenmodes in the current-carrying plasma.
Putting together the expressions (25) and (8), it is
possible to obtain the dielectric tensor-operator, which
can be used to study the wave phenomena, in the frequency range !dr << ! << !ce; that allows us to
To estimate the ion cyclotron dissipation of the fast
magnetosonic wave, at the fundamental cyclotron frequency we use the conditions, ! kr cA !ci, and
Zi; << 1. In this case, we have
IV.2. Ion-cyclotron damping of FMS waves in
the current-carrying plasma
c
r
2
(11i) = (22i) = ; 4cc 2 1 ; i 2 j k !++!jciv ;
Ti
k
A r
2
ic
!
+
!
(i)
(i)
ci
12 = ;21 = ig = ; 4c2 3 ; i 2 j k + j v ;
and using the conditions
k
A
Nk2 << Nr2 c2 =c2A <<j 11 j;
kkv0 << !ci;
Ti
Nk N1 <<j 12 j;
the following dispersion equation for fast magnetosonic waves can be obtained
;
Nr2FMS = 211 + 212 =11:
(26)
740
N.I. Grishanov et al.
As a result, the real and imaginary parts of the radial refractive index of these waves are given by
2
vTi :
p
(ReNr )2FMS = cc2 ;
FMS =j kk + 2 hrhz 1 ; 4qr dq
j
dr
8!ci
A
In this case, the ion cyclotron damping of the fast magnetosonic waves (28) and analogous damping for plasmas in an uniform magnetic eld[1] are the same order
of magnitude if the wavelength along the magnetic eld
is much smaller than the screw step of the helical magnetic eld line, kk = (m + nq)h =r >> h =r. The main
dierence of the obtained result from the well-known[1]
is: in the uniform magnetized plasma, the FMS waves,
with ! = !ci and propagating exactly perpendicular to
a straight magnetic eld (kz = 0), are not absorbed by
the plasma ions; in the helical magnetic eld case, the
absorption of FMS waves, with kk = 0 and ! = !ci ,
may be substantial due to the magnetic shear correction in the resonant condition (26). In this case, the
ion-cyclotron damping rate of FMS waves propagating
exactly across to the magnetic surfaces in the currentcarrying plasmas is
r dq j p vTi : (28)
FMS jkk =0 =j hr 1 ; 4q
dr
2!ci
This feature of the ion cyclotron absorption, of the fast
magnetosonic waves, should be taken into account to
analyze the wave dissipations and stability questions in
the current-carrying plasmas near the so-called rational
magnetic surfaces, where the kk (r) change the sign.
V. Conclusions
To analyze the wave dispersion for eigenmodes,
propagating in cylindrical current-carrying plasmas, we
use the dielectric tensor elements obtained through the
solution of the linearized Vlasov equation for plasmas
in a helical magnetic eld, taking into account the effects of the equilibrium current, magnetic shear, and
the density gradient. To evaluate the dispersion equation for eigenmodes in a magnetized current-carrying
plasma, we used all nine dielectric tensor components.
Account of these components is necessary for the correct estimation of the damping rate of the basic MHD
waves. This eect is related especially to the transit
time magnetic pumping[1;2] dissipation of waves in the
frequency range much smaller than the ion-cyclotron
frequency, ! << !ci .
(27)
Using the geometric optics approximation, for small
amplitude waves, we derived the analytical expressions
for the electron Landau damping of fast magnetosonic
waves (see the equations (14) and (15)), fast Alfven
waves (see the equations (16)- (19)), kinetic Alfven
waves (see the equations (20) and (21)), and atmospheric whistlers (see the equations (23) and (24)), in
the plasma with "hot" electrons and one kind of ions,
taking into account current, shear, and density inhomogeneity. It is shown that the radial gradients of plasma
density and equilibrium magnetic eld inuence substantially on the damping rate of the eigenmodes. The
damping rate of these modes contains three independent terms (see the equations (15), (17), (21) and (24)),
which are associated to three "submodels" of plasmas in
the helical magnetic eld. The rst term corresponds to
the damping rate of eigenmodes in uniform magnetized
plasmas without the equilibrium current; these expressions are corresponding to the well known results (see,
for example, Refs. [1] and [7]). The second and third
terms describe the inuence of the radial gradients of
the safety factor (by dq=dr) and the plasma density (by
dn0=dr) on the damping rate of eigenmodes in nonuniform current-carrying plasmas.
The expressions for real and imaginary parts of the
radial refractive indexes of the considered waves, when
the wave frequency, azimuthal, and longitudinal wave
numbers are given, are valid in a wide range of the wave
phase velocities relative to the electron thermal velocity in the argument
p of the dispersion plasma function,
Ze = (! ; kk v0)= 2kkvTe ). It allows us to analyze the
"hot" and "cold" limits related to wave propagation,
Ze << 1 and Ze >> 1, respectively.
The contributions of resonant ions to the dielectric
tensor elements are presented in equation (25). The
ion-cyclotron damping of the fast magnetosonic waves,
in the frequency range ! !ci, is evaluated taking into
account the magnetic shear correction (see the equations (28) and (29)). It is shown that the ion-cyclotron
damping of FMS waves, propagating perpendicular to
the ambient helical magnetic eld, is not equal to zero in
contrast to the uniform magnetic eld case. The magnetic shear corrections in the argument of the plasma
Brazilian Journal of Physics, vol. 26, no. 4, december, 1996
p
dispersion function, Zi; = (! ; !ci )= 2 j kk + j vTi ,
are also important to analyze the drift cyclotron instabilities. These instabilities can be excited around the
rational magnetic surfaces where the longitudinal projection of the wave vector is equal to zero.
The results of this paper can be applied to study the
Alfven wave heating and current drive phenomena in
tokamak plasmas with circular magnetic surfaces and
a large aspect ratio. However, in a general case, the
one-mode approximation for waves (typical for cylindrical plasmas) is not suitable for plasmas in toroidal
geometry. Here, it is necessary to take into account the
"non-local" eects, connected with the poloidal modecoupling eect and the inuence of trapped and untrapped particles on the wave dissipation [13] .
Acknowledgments
We are grateful to Dr. F.M. Nekrasov for many
useful discussions. This work was supported by Rio de
Janeiro Research Foundation (FAPERJ) and Brazilian
Council of Research (CNPq).
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