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# NIFS PROC 88

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```< 0 in
the case of damping)
n
\u2329
d
dt
mv2
2
\u232a
z0v0
= \u22122\u3b3W
and the growth rate \u3b3 is given by
\u3b3
\u3c9
=
\u3c0
2
(
\u3a0
\u3c9
)2 ( \u3c9
|k|
)(
v0
\u2202f(v0)
\u2202v0
)
v0=\u3c9/k
(11.11)
where \u3a02 = nq2/\ufffd0m, W \u2248 2\ufffd0E2/4,
\u222b
f(v)dv = 1.
There is a restriction on the applicability of linear Landau damping. When this phenomenon runs
its course before the particle orbit deviates from the linear-approximation solution, the reductions
leading to linear Landau damping are justi\ufb01ed. The period of oscillation in the potential well
of the electric \ufb01eld of the wave gives the time for the particle orbit to deviate from the linear
approximation ( \u3c92 \u223c eEk/m from m\u3c92x = eE). The period of oscillation is
\u3c4osc =
1
\u3c9osc
\u2248
(
m
ekE
)1/2
.
Consequently the condition for the applicability of linear Landau damping is that the Landau
damping time 1/\u3b3 is shorter than \u3c4osc or the collision time 1/\u3bdcoll is shorter than \u3c4osc.
|\u3b3\u3c4osc| > 1, (11.12)
|\u3bdcoll\u3c4osc| > 1. (11.13)
On the other hand, it was assumed that particles are collisionless. The condition that the collision
time 1/\u3bdcoll is longer than \u3bb/vrms is necessary for the asymptotic approximation of the integral
(11.9) as t \u2192 \u221e, where \u3bb is the wavelength of the wave and vrms is the spread in the velocity
distribution;
1
\u3bdcoll
>
2\u3c0
kvrms
. (11.14)
139
140 11 Landau Damping and Cyclotron Damping
11.2 Transit-Time Damping
We have already described the properties of Alfve´n waves in cold plasmas. There are compres-
sional and torsional modes. The compressional mode becomes magnetosonic in hot plasmas, as
is described in ch.5. In the low-frequency region, the magnetic moment \u3bcm is conserved and the
equation of motion along the \ufb01eld lines is
m
dvz
dt
= \u2212\u3bcm\u2202B1z
\u2202z
. (11.15)
This equation is the same as that for Landau damping if \u2212\u3bcm and \u2202B1z/\u2202z are replaced by the
electric charge and the electric \ufb01eld, respectively. The rate of change of the kinetic energy is derived
similarly, and is \u2329
d
dt
mv2
2
\u232a
z0,v0
= \u2212\u3c0\u3bc
2
m|k|
2m
|B1z|2
(
\u3c9
k
)(
\u2202f(v0)
\u2202v0
)
v0=\u3c9/k
. (11.16)
This phenomena is called transit-time damping.
11.3 Cyclotron Damping
The mechanism of cyclotron damping is di\ufb00erent from that of Landau damping. Here the electric
\ufb01eld of the wave is perpendicular to the direction of the magnetic \ufb01eld and the particle drift and
accelerates the particle perpendicularly to the drift direction. Let us consider a simple case in
which the thermal energy of particles perpendicular to the magnetic \ufb01eld is zero and the velocity
of particles parallel to the magnetic \ufb01eld B0 = B0z\u2c6 is V . The equation of motion is
m
\u2202v
\u2202t
+mV
\u2202v
\u2202z
= q(E1 + v × z\u2c6B0 + V z\u2c6 ×B1). (11.17)
As our interest is in the perpendicular acceleration we assume (E1 · z\u2c6) = 0. B1 is given by
B1 = (k×E)/\u3c9. With the de\ufb01nitions v± = vx ± ivy, E± = Ex ± iEy, the solution for the initial
condition v = 0 at t = 0 is
v± =
iqE±(\u3c9 \u2212 kV ) exp(ikz \u2212 i\u3c9t)
m\u3c9
1\u2212 exp(i\u3c9t\u2212 ikV t± i\u3a9t)
\u3c9 \u2212 kV ±\u3a9 ,
(11.18)
\u3a9 =
\u2212qB0
m
.
The macroscopic value of v\u22a5 is obtained by taking the average weighted by the distribution function
f0(V ) as follows:
\u3008v\u22a5\u3009= iq exp(ikz \u2212 i\u3c9t)2m ((c
++c\u2212)E\u22a5+ i(c+\u2212c\u2212)E\u22a5 × z\u2c6), (11.19)
c± = \u3b1± \u2212 i\u3b2±, (11.20)
\u3b1± =
\u222b \u221e
\u2212\u221e
dV
f0(V )(1\u2212 kV/\u3c9)(1\u2212 cos(\u3c9 \u2212 kV ±\u3a9)t)
\u3c9 \u2212 kV ±\u3a9 , (11.21)
\u3b2± =
\u222b \u221e
\u2212\u221e
dV
f0(V )(1\u2212 kV/\u3c9) sin(\u3c9 \u2212 kV ±\u3a9)t
\u3c9 \u2212 kV ±\u3a9 . (11.22)
As t becomes large we \ufb01nd that
\u3b1± \u2192 P
\u222b \u221e
\u2212\u221e
dV
f0(V )(1\u2212 kV/\u3c9)
\u3c9 \u2212 kV ±\u3a9 , (11.23)
\u3b2± \u2192 \u2213\u3c0\u3a9
\u3c9|k| f0
(
\u3c9 ±\u3a9
k
)
. (11.24)
140
11.3 Cyclotron Damping 141
When
t\ufffd 2\u3c0
kVrms
(11.25)
where Vrms = \u3008V 2\u30091/2 is the spread of the velocity distribution, the approximations (11.19)\u223c(11.24)
are justi\ufb01ed. The absorption of the wave energy by the plasma particles is given by
\u3008Re(qE exp(ikz \u2212 i\u3c9t))(Re\u3008v\u22a5\u3009)\u3009z
=
q2
4m
(\u3b2+|Ex + iEy|2 + \u3b2\u2212|Ex \u2212 iEy|2). (11.26)
Let us consider the case of electrons (\u3a9e > 0). As was described in sec.10.2, the wave N2 = R
propagating in the direction of magnetic \ufb01eld (\u3b8 = 0) satis\ufb01es Ex+ iEy = 0, so that the absorption
power becomes
Pe =
q2
4m
\u3b2\u2212|Ex \u2212 iEy|2.
When \u3c9 > 0, (11.24) indicates \u3b2\u2212 > 0. In the case of \u3c9 < 0, \u3b2\u2212 is nearly zero since f0 ((\u3c9 \u2212\u3a9e)/k)\ufffd
1.
Let us consider the case of ions (\u2212\u3a9i > 0). Similarly we \ufb01nd
Pi =
q2
4m
\u3b2+|Ex + iEy|2.
When \u3c9 > 0, (11.24) indicates \u3b2+ > 0. In the case of \u3c9 < 0, \u3b2+ is nearly zero, since
f0 (\u3c9 +\u3a9i/k)\ufffd 1.
The cyclotron velocity Vc is de\ufb01ned so that the Doppler shifted frequency (the frequency of wave
that a particle running with the velocity V feels) is equal to the cyclotron frequency, that is,
\u3c9 \u2212 kVc ±\u3a9 = 0,
Vc =
\u3c9
k
(
1± \u3a9
\u3c9
)
.
Accordingly particles absorb the wave energy when the absolute value of cyclotron velocity is smaller
than the absolute value of phase velocity of the wave (±\u3a9/\u3c9 < 0)(see (11.24)). This phenomena
is called cyclotron damping.
Let us consider the change in the kinetic energy of the particles in the case of cyclotron damping.
Then the equation of motion is
m
dv
dt
\u2212 q(v ×B0) = qE\u22a5 + q(v ×B1).
Since B1 = (k ×E)/\u3c9 and Ez = 0, we have
m
dvz
dt
=
qkz
\u3c9
(v\u22a5 ·E\u22a5),
m
dv\u22a5
dt
\u2212 q(v\u22a5 ×B0) = qE\u22a5
(
1\u2212 kzvz
\u3c9
)
so that
mv\u22a5 · dv\u22a5dt = q(v\u22a5 ·E\u22a5)
(
1\u2212 kzvz
\u3c9
)
.
Then
d
dt
(
mv2z
2
)
=
kzvz
\u3c9 \u2212 kzvz
d
dt
(
mv2\u22a5
2
)
,
141
142 11 Landau Damping and Cyclotron Damping
v2\u22a5 +
(
vz \u2212 \u3c9
kz
)2
= const.
In the analysis of cyclotron damping we assumed that vz = V is constant; the condition of the
validity of linearized theory is (ref.)
k2zq
2E2\u22a5|\u3c9 \u2212 kzvz|t3
24\u3c92m2
< 1.
We have discussed the case in which the perpendicular thermal energy is zero. When the per-
pendicular thermal energy is larger than the parallel thermal energy, so-called cyclotron instability
may occur. The mutual interaction between particles and wave will be discussed again in chs.12
and 13 in relation to heating and instabilities.
11.4 Quasi-Linear Theory of Evolution in the Distribution Function
It has been assumed that the perturbation is small and the zeroth-order terms do not change.
Under these assumption, the linearized equations on the perturbations are analyzed. However if
the perturbations grow, then the zeroth-order quantities may change and the growth rate of the
perturbations may change due to the evolution of the zeroth order quantities. Finally the pertur-
bations saturate (growth rate becomes zero) and shift to steady state. Let us consider a simple case
of B = 0 and one dimensional electrostatic perturbation (B1 = 0). Ions are uniformly distributed.
Then the distribution function f(x, v, t) of electrons obeys the following Vlasov equation;
\u2202f
\u2202t
+ v
\u2202f
\u2202x
\u2212 e
m
E
\u2202f
\u2202v
= 0. (11.27)
Let the distribution function f be divided into two parts
f(x, v, t) = f0(v, t) + f1(x, v, t) (11.28)
where f0 is slowly changing zeroth order term and f1 is the oscillatory 1st order term. It is assumed
that the time derivatives of f0 is the 2nd order term. When (11.28) is substituted into (11.27), the
1st and the 2nd terms satisfy following equations;
\u2202f1
\u2202t
+ v
\u2202f1
\u2202x
=
e
m
E
\u2202f0
\u2202v
, (11.29)
\u2202f0
\u2202t
=
e
m
E
\u2202f1
\u2202v
. (11.30)
f1 and E may be expressed by Fourier integrals;
f1(x, v, t) =
1
(2\u3c0)1/2
\u222b
fk(v) exp(i(kx\u2212 \u3c9(k)t))dk, (11.31)
E(x, t) =
1
(2\u3c0)1/2
\u222b
Ek exp(i(kx\u2212 \u3c9(k)t))dk. (11.32)
Since f1 and E are real, f\u2212k = f\u2217k , E\u2212k = E
\u2217
k , \u3c9(\u2212k) = \u2212\u3c9\u2217(k) (\u3c9(k) = \u3c9r(k) + i\u3b3(k)). The
substitution of (11.31) (11.32) into (11.29) yields
fk(v) =
e
m
(
i
\u3c9(k)\u2212 kv
)
Ek
\u2202f0
\u2202v
. (11.33)
If (11.32) (11.33) are substituted into (11.30), we \ufb01nd
\u2202f0(v, t)
\u2202t
=
(
e
m
)2 \u2202
\u2202v
\u2329
1
2\u3c0
\u222b
Ek\u2032 exp(i(k\u2032x\u2212 \u3c9(k\u2032)t))dk\u2032
142```