Chapter 6

Waves in a Uniform Plasma

6.1 Introduction

Although we seldom encounter uniform unbounded plasmas in practice, studying wave phe-

nomena in such an idealized case reveals numerous fundamental waves that can be excited in

a plasma. Also, when the characteristic lengths of plasma nonuniformities are much longer

than wavelengths of concerned waves, wave propagation can still be treated by the method

essentially identical to those developed for uniform plasmas (e.g. WKB approximation).

A magnetized plasma is a typical anisotropic medium for electromagnetic waves and can

support various kinds of waves. Since a plasma consists of light electrons and heavy ions,

characteristic frequencies range from low frequency ion cyclotron frequency to high frequency

electron cyclotron frequency. In general, particle dynamics of both electrons and ions must

be incorporated in rigorous analysis of plasma waves.

Another characteristic feature of waves in a plasma is that they are subject to damping

even in the absence of particle collisions. Well known examples are Landau and cyclotron

damping. The collisionless wave damping plays important roles in plasma heating (and

current drive) which can be e¤ectively used in further raising the temperature of a plasma

already having a temperature high enough so that collisional Joule heating is ine¤ective.

In this Chapter, a general kinetic method based on particle velocity distribution function

will be presented for waves in a collisionless, magnetized, uniform plasma. Contribution of

1

both electrons and ions to the conduction current will be found �rst. The resultant current

J = !� � E

with !� being the conductivity tensor will then be substituted into Maxwell�s equations

which yield a general dispersion relation.

6.2 Perturbed Particle Distribution Function

Although �uid or hydrodynamic approximation is much less involved than kinetic description

and provides more transparent physical pictures, it overlooks collisionless damping such as

Landau and cyclotron damping. Also, when wave phase velocities approach thermal velocity

of electrons or ions, �uid approximation tends to break down. For these reasons, satisfactory

description of plasma waves requires kinetic theory.

The starting equation is the collisionless Boltzmann equation or Vlasov equation for a

particle species having a charge e and mass m,

@f

@t+ v � @f

@x+e

m

�E+

1

cv �B

�� @f@v

= 0: (6.1)

Both electric and magnetic �elds, E and B contain in general external �elds E0 and B0, and

those associated with plasma waves E1 and B1. Since the distribution function f implicitly

depends on the �elds E and B, Eq. (6.1) is a nonlinear di¤erential equation. Throughout

this chapter, we assume that perturbed �eld quantities are small, so that Eq. (6.1) may be

linearized asdf1dt+e

m

�E1 +

1

cv �B1

�� @f0@v

= 0; (6.2)

whered

dt=@

@t+ v � r+ e

mc(v �B0) �

@

@v; (6.3)

is the substantive derivative along the particle unperturbed trajectory. For simplicity, we

have assumed that there is no external electric �eld, E0 = 0: In magnetically con�ned

plasmas, electric �elds usually exist. In tokamaks, for example, the toroidal current is

driven by an inductive toroidal electric �eld. This �eld is much smaller than the runaway

2

(Dreicer�s) �eld in most tokamaks. Except for a small number of runaway electrons, deviation

of the electron velocity distribution from Maxwellian may be ignored. In the direction

perpendicular to the magnetic �eld, an ambipolar electric �eld tends to develop near the

edge of a magnetically con�ned plasma. In general, a plasma self-consistently induces an

ambipolar �eld to maintain charge neutrality. The assumption E0 = 0 should therefore be

made with some caution. In axisymmetric devices such as tokamaks, the radial electric �eld

Er causes E �B plasma rotation in the toroidal direction,

V� = �cErB�;

where B� is the poloidal magnetic �eld. The toroidal rotation velocity may approach the

ion thermal speed vTi: However, transport in tokamaks is fairly insensitive to the toroidal

rotation and in the lowest order the radial electric �eld may be ignored. It should be noted

that the radial electric �eld does not cause plasma rotation in the poloidal direction which

is proportional to the ion temperature gradient, dTi=dr:

There are several known methods to solve Eq. (6.1). The method most frequently used

is to integrate it along the particle unperturbed trajectory (the method of characteristic) as

brie�y outlined in Chapter 3. Here, we directly integrate Eq. (6.1) over the gyro angle �,

tan� = vy=vx; (6.4)

in the geometry shown in Fig. 6.1. A uniform (thus straight) magnetic �eld B0 is assumed in

the z direction, and the wavevector k in the x-z plane without loss of generality in a uniform

plasma. Assuming that all perturbed quantities, f1; E1 and B1 are proportional to

exp [i(k � x�!t)] ;

and eliminating the perturbed magnetic �eld B1 via Faraday�s law,

r� E1 = �1

c

@B1@t

or k� E1 =!

cB1; (6.5)

we rewrite Eq. (6.1) as

i(k � v�!)f1(v)� @f1@�

= � em

��1� k � v

!

�E1 �

@f0@v

+v � E1!

k � @f0@v

�; (6.6)

3

Figure 6.1: Geometry assumed in the analysis. The external magnetic �eld is in the z

direction. The wavevector k is assumed to be in the x� z plane, k = k?ex + kkez:

where

=eB0mc

; (6.7)

is the cyclotron frequency (e < 0 for electrons, and i > 0 for ions). Also, note the identity,

v �B0 �@

@v= �B0

@

@�: (6.8)

Introducing

U =1

!

�(! � kkvk)

@f0@v?

+ kkv?@f0@vz

�; (6.9)

W =@f0@vk� k? cos�

!

�v?@f0@vk� vk

@f0@v?

�; (6.10)

where

v2? = v2x + v

2y ; vk

4

are constants of motion in a uniform magnetic �eld (v2? = const. from the constant magnetic

moment and vk = const. from the assumption E0 = 0) and characterize the unperturbed

distribution function f0(v);

f0(v) = f0(v2?; vk); (6.11)

we can rewrite Eq. (6.6) in the form

@f1@�� ik?v? cos�+ kkvz � !

f1 =

e

m[U(Ex cos�+ Ey sin�) +WEz] ; (6.12)

which can be readily integrated as

f1(v;k; !) =e

mexp

�i

Z �

0

(k?v? cos�0 + kkvz � !) d�0

��Z �

0

exp

"� i

Z �0

0

(k?v? cos�00 + kkvk � !) d�00

#� [U(Ex cos�0 + Ey sin�0) +WEz] d�0: (6.13)

Note that f1 must be a periodic function of �; f1(� + 2�n) = f1(�) with n an integer. For

this reason, the general solution of Eq. (6.12) has been discarded, and only the particular

solution retained.

The integral over � in Eq. (6.13) can be performed if the following expansion is exploited,

exp(ix sin�) =1X

n=�1Jn(x)e

in�;

where Jn(x) is the n-th order Bessel function. The result is

f1(v;k; !) = ie

m

Xm

Xn

Jm(�)ei(m�n)�

n

�Jn(�)UEx + iJ

0n(�)UEy + Jn(�)WEz

kkvk � ! + n; (6.14)

where

� =k?v?

; J 0n(�) =dJn(�)

d�; (6.15)

W is now modi�ed as

W =! � n!

@f0@vk

+nvk!v?

@f0@v?

; (6.16)

and we have used the following recurrence formulae of the Bessel functions,

Jn+1(x) + Jn�1(x) =2n

xJn(x); (6.17)

5

Jn�1(x)� Jn+1(x) = 2J 0n(x): (6.18)

Equation (6.14) is our desired expression for the perturbed distribution function. Various

moments of f1(v) can be calculated from f1. For our purpose, it su¢ ces to �nd the perturbed

current density given by

J1 =Xs

esn0s

Zv f1s(v) d

3v; (6.19)

where s indicates particle species (electrons and ions).

If f0(v) is isotropic,

f0(v) = f0(v2);

U and W are simpli�ed as

U =@f0@v?

; W =@f0@vk

: (6.20)

In this case, the perturbed magnetic �eld B1 disappears from the original Eq. (6.6), since

v �B1 �@f0@v

= 0; (6.21)

identically, provided f0 is isotropic. However, this does not mean that waves are electrostatic,

since the electric �eld is given by

E1 = �r��1

c

@A

@t; (6.22)

where � is the scalar potential and A is the magnetic vector potential. Electrostatic waves

require that the electric �eld E1 be written in terms of a scalar potential alone,

E1 = �r�; (6.23)

and so far we have not made any such assumptions.

6.3 Dispersion Relation

The perturbed velocity distribution function calculated in the preceding Section yields the

perturbed current density through

J1 = n0e

Z(f1i � f1e)v d3v; (6.24)

6

where for simplicity we assume a plasma composed of electrons and singly ionized ions,

both having the same particle density n0. Since f1 consists of terms proportional to each

component of the electric �eld E; Eq. (6.24) can be cast into the form

J = !� � E; (6.25)

where !� de�nes the conductivity tensor, which can be evaluated once the unperturbed

distribution function f0(v) is prescribed. (For brevity, we hereafter omit the subscript �1�

in the perturbed quantities.) The dispersion relation of electromagnetic waves can then be

found from Maxwell�s equations,

r� E = �1c

@B

@t; (6.26)

r�B = 1

c

�@E

@t+ 4�J

�=1

c

�@E

@t+ 4� !� � E

�; (6.27)

by eliminating the perturbed magnetic �eld B (or electric �eld E) between these equations.

It is convenient to introduce a dielectric tensor de�ned by

!" = 1+ i4�

! !� : (6.28)

Eliminating B between Eqs. (6.26) and (6.27), we obtain

k2E� (k � E)k�!2

c2 !" � E = 0; (6.29)

or in the tensorial form �k2�ij � kikj �

!2

c2�ij

�Ej = 0; (6.30)

where �ij is the Kronecker�s delta. The dispersion relation is thus given by

det

�k2�ij � kikj �

!2

c2�ij

�= 0: (6.31)

The tensor !" has nine components. Only under special circumstances, the tensor be-

comes Hermitian, �ij = ��ji, which is the condition for the absence of wave energy dissipation

by the plasma. The expression for �ij is shown below without specifying the velocity distri-

bution function f0(v) contained in U and W ,

� = 1+Xs

!2ps!

1Xn=�1

Zd3v

!S

! � kkvk � n; (6.32)

7

where the tensor !S is given by

!S =

266666666664

v?

�n�

�2J2nU iv?

n

�JnJ

0nU v?

n

�J2nW

�iv?n

�JnJ

0nU v?(J

0n)2U �iv?JnJ 0nW

vkn

�J2nU ivkJnJ

0nU vkJ

2nW

377777777775; (6.33)

!ps is the plasma frequency de�ned by

!2ps =4�n0e

2s

ms

; (6.34)

and Zd3v = 2�

Z 1

0

v?dv?

Z 1

�1dvk:

To proceed further, the unperturbed velocity distribution function f0(v2?; vz) must be spec-

i�ed.

If the velocity distribution function f0(v2?; vz) is isotropic Maxwellian

f0(v2) =

� m

2�T

�3=2exp

��mv

2

2T

�; (6.35)

with v2 = v2? + v2k; �ij can be simpli�ed somewhat. For example, �xx becomes

�xx = 1 +Xs

!2ps!

Z Xn

v?(n=�)2J2n(�)

! � kkvk � ns

��msv?

Ts

��ms

2�Ts

�3=2exp

��msv

2

2Ts

�d3v

= 1 +Xs

!2ps!2

Xn

2n2

�s

Z 1

0

xe�x2

J2n

�p2�s x

�dx� !

kkvTs

1p�

Z 1

�1

e�t2

t� �sndt; (6.36)

where x = �?=vTs; vTs =p2Ts=ms is the thermal velocity, t = �k=vTs; �sn = (! �

ns)=kkvTs, and

�s =1

2

k2?v2Ts

2=k2?Ts=ms

2= (k?�s)

2; (6.37)

with

�s =

pTs=ms

s;

being the thermal Larmor radius. The integral over x reduces toZ 1

0

xe�x2

J2n(p2�x)dx =

1

2exp (��) In (�) ; (6.38)

8

where In(�) is the modi�ed Bessel function of the �rst kind. The integral over t can be

written in terms of the plasma dispersion function de�ned by

Z(�) =1p�

Z 1

�1

e�t2

t� � dt: (6.39)

Therefore, Eq. (6.36) becomes

�xx = 1 +Xs

!2ps!2

Xn

n2

�se��sIn(�s)�0Z(�sn): (6.40)

The other components can be calculated in a similar manner:

�xy = ��yx = �iXs

!2p!2

Xn

ne��(In � I 0n)�0Z(�n); (6.41)

�xz = "zx =Xs

!2p!

Xn

nm

k?kkTe��In(�)�nZ(�n); (6.42)

�yy = 1 +Xs

!2p!2

Xn

e���n2

�In + 2�(In � I 0n)

��0Z(�n)

= "xx + 2Xs

!2p!2

Xn

�e��(In � I 0n)�0Z(�n); (6.43)

"yz = �"zy = iXs

!2p!2

Xn

r�

2e��(In � I 0n)�0Z 0(�n) (6.44)

�zz = 1�Xs

!2p!2

Xn

e��In(�)�0�nZ0(�n): (6.45)

The subscript �s�has been omitted for brevity.

In deriving these expressions, use has been made of the following identities:

e��1X

n=�1In (�) = 1; (6.46)

Z 1

0

e�x2

xJn(ax)Jn(bx)dx =1

2exp

��a

2 + b2

4

�In

�ab

2

�: (6.47)

Di¤erentiating by a yieldsZ 1

0

x2J 0n(ax)Jn(bx)e�x2dx =

1

4

�bI 0n

�ab

2

�� aIn

�ab

2

��exp

��a

2 + b2

4

�: (6.48)

Since a = b =p2�; we obtainZ 1

0

x2J 0n(p2�x)Jn(

p2�x)e�x

2

dx =

p�

2p2[I 0n (�)� In (�)] e��; (6.49a)

9

which has been used in calculation of "xy = �"yx and "yz = �"zy: Di¤erentiating Eq.(6.49a)

further with respect to b and substituting a = b =p2�; we also obtainZ 1

0

x3hJ 0n(p2�x)

i2e�x

2

dx =

�n2

4�In (�) +

�

2(In � I 0n)

�e��; (6.50)

where use is made of the di¤erential equation satis�ed by In (�) ;

I 00n +I 0n���1 +

n2

�2

�In = 0: (6.51)

This is needed in calculating "yy: Also, it is noted that the plasma dispersion function Z(�)

satis�es the following di¤erential equation,

Z 0(�) + 2[1 + �Z(�)] = 0; (6.52)

since

Z 0 (�) =1p�

d

d�

Z 1

�1

e�x2

x� � dx

= � 1p�

Z 1

�1

�d

dx

1

x� �

�e�x

2

dx

= � 2p�

Z 1

�1

xe�x2

x� � dx = �2 [1 + �Z (�)] : (6.53)

6.4 Plasma Dispersion Function Z(�)

When the particle velocity distribution function f0(v2?; vz) is characterized by isotropic

Maxwellian or bi-Maxwellian

f(v2?; vz) =m3=2

2�T?(2�Tk)1=2exp

"�mv

2?

2T?�mv2k2Tk

#;

the dielectric tensor �ij contains the plasma dispersion function

Z(�) =1p�

Z 1

�1

e�x2

x� � dx; (6.54)

and its derivative, Z 0(�). Since this important function so often appears in analyzing plasma

waves, it may be appropriate to devote one Section. Fried and Conte have tabulated numer-

ical values of Z(�). Mathematically speaking, the plasma dispersion function is the Hilbert

transform of the function 1=(x� �):

10

Approximate series and asymptotic expansions of Z(�) can be found in two limiting cases,

j�j � 1 and j�j � 1. We �rst note that Z(�) satis�es the following di¤erential equation

dZ

d�= �2 [1 + �Z(�)] ; (6.55)

as shown in the preceding section. This can be integrated as

Z(�) = Z(0)e��2 � 2e��2

Z �

0

ex2

dx; (6.56)

with Z(0)e��2being the general solution and the last term the particular solution. To �nd

the �initial value�Z(0), we evaluate

Z(0) =1p�lim&!0

Z 1

�1

e�x2

x� � dx: (6.57)

Letting � = �+ i�, we �nd

Z(0) =1p�lim

�; �!0

Z 1

�1

e�x2

x� �� i� dx

=1p�lim�!0

Z 1

�1

x+ i�

x2 + �2e�x

2

dx

= ip� (6.58)

Therefore,

Z(�) = ip�e��

2 � 2e��2Z �

0

ex2

dx; (6.59)

which can be used for any values of the complex quantity �. As we have seen in the preceding

Section, � is given by

� =! � nkkvT

;

and in general a complex number since ! and/or kk can be complex depending on growing

(!i > 0) or damped (!i < 0) wave. Even if Im � is small, the plasma dispersion function is

intrinsically complex as indicated in Eq. (6.59). Physically, this means that plasma waves

in general su¤er damping through wave-particle interaction. Under certain circumstances,

the intrinsic dissipation can be the source of plasma instability as we have already seen in

Chapter 3.

11

When j�j � 1, Eq. (6.59) yields the following series expansion for Z(�),

Z(�) ' ip�e��

2 � 2�1� �2 + 1

2�4 � � � �

�Z �

0

�1 + x2 +

1

2x4 + � � �

�dx

= �2� + 43�3 � � � �+ i

p�e��

2

; (j�j � 1) : (6.60)

In the opposite limit j�j � 1, we directly expand the de�ning equation for Z(�),

Z(�) =1p�

Z 1

�1

e�x2

x� � dx

' � 1p��

Z 1

�1

�1� x

�

��1e�x

2

dx

= �1�� 1

2�3� � � �+ i

p�e��

2

; (j�j � 1) : (6.61)

The plasma dispersion function Z(x+ iy) satis�es the following symmetry properties,

Re Z(�x+ iy) = �Re Z(x+ iy); (6.62)

Im Z(�x+ iy) = Im Z(x+ iy); (6.63)

or

Z(�x+ iy) = �Z�(x+ iy); (6.64)

which can be seen from the de�nition of Z(�);

Z(x+ iy) =1p�

Z 1

�1

e�t2

t� (x+ iy) dt =1p�

Z 1

�1

t� x+ iy(t� x)2 + y2 e

�t2dt: (6.65)

Also, for y > 0,

Z(x� iy) = Z�(x+ iy) + 2ip�e�(x�iy)

2

: (6.66)

Therefore, knowing Z(x+ iy) for x; y > 0 is su¢ cient to evaluate Z(x+ iy) for arbitrary x

and y.

6.5 Unmagnetized Plasma

In the absence of external magnetic �eld, the perturbed distribution function satis�es

i(k � v�!)f1(v) = �e

m

��1� k � v

!

�E � @f0

@v+v � E!k � @f0

@v

�: (6.67)

12

If the velocity distribution is isotropic, Eq. (6.67) simpli�es as

i(k � v�!)f1(v) = �e

mE � @f0

@v: (6.68)

The wave vector may be assumed to be in arbitrary direction, say, k = kez: The electric

�eld may be assumed to be

E = Ex + Ez: (6.69)

Evidently, Ex is associated with the transverse wave while Ez is associated with the longi-

tudinal wave. We further assume that f0 is Maxwellian,

fM (v) =� m

2�T

�3=2exp

��mv

2

2T

�: (6.70)

The current density is

J = ne

Zvf1d

3v

= �ine2

T

Zv

kvz�!E � vfM (v)

= �i ne2

mkvT

1p�

Z 1

�1

e�t2

t� � dtEx � 2ine2

mkvT

1p�

Z 1

�1

t2e�t2

t� � dtEz

= �i ne2

mkvTZ (�)Ex � 2i

ne2

mkvT� [1 + �Z (�)]Ez: (6.71)

The permittivity pertinent to the transverse wave Ex is

"T = 1 + i4�

!

��i ne

2

mkvTZ (�)

�= 1 +

!2p!kvT

Z (�) : (6.72)

The dispersion relation is given by�!k

�2=c2

�T=

c2

1 +!2pe!kvTe

Z (�e)

; (6.73)

where the ion contribution has been ignored. When ! � kvTe;

Z (�e) ' �1

�e;

13

and �!k

�2=

c2

1� (!pe=!)2;

!2 = !2pe + (ck)2 : (6.74)

In the opposite limit, ! � kvTe;using

Z (�e) ' �2�e + ip�e��

2e ' i

p�;

we �nd

k3 = ip�!2pe!

c2vTe; k =

p3 + i

2�1=6

�!2pe!

c2vTe

�1=3: (6.75)

The wavenumber becomes complex which indicates spatial damping or evanescence. The

inverse of the damping factor

� =1

ki=

2

�1=6

�c2vTe!2pe!

�1=3;

is often called anomalous skin depth since it can be larger than the classical skin depth

�c =c

!pe; (6.76)

provided

! <vTec!pe:

The longitudinal wave Ez is characterized by the dispersion relation

"L = 1 + 24�ne2

mkvT� [1 + �Z (�)] = 0; (6.77)

or

1 +

�kDk

�2[1 + �Z (�)] = 0; (6.78)

where

kD =

r4�ne2

T;

is the inverse Debye length. For an electron plasma,

1 +

�kDek

�2[1 + �eZ (�e)] = 0; (6.79)

and for electron-ion plasma,

1 +

�kDek

�2[1 + �eZ (�e)] +

�kDik

�2[1 + � iZ (� i)] = 0: (6.80)

These dispersion relations will be discussed in detail in Chapter 7.

14

6.6 Waves in a Cold Plasma

In Section 6.2, we have seen that all components of the dielectric tensor �ij contain harmonics

of the cyclotron frequency in the argument of the plasma dispersion function

�n = (! � n)=kkvT :

The appearance of harmonics is due to deviation of particle orbit from complete circular

motion when acted by wave electric �eld perpendicular to the external magnetic �eld. When

the Larmor radius is small (cold plasma), such deviation becomes ignorable, and harmonics

are expected to disappear. Only the fundamental cyclotron frequency will enter the

dispersion relation.

Another manifestation of particle thermal motion is collisionless wave damping. Both

Landau and cyclotron damping require that waves �nd particles which travel along the

magnetic �eld with the speed corresponding to the phase velocity (or Doppler shifted phase

velocity) of the waves. In cold plasmas, the number of these resonant particles is very small,

and all damping mechanisms are expected to disappear. The dielectric tensor in this case

should become Hermitian, �ij = ��ji since no absorption of wave energy by plasma is involved.

Let us �rst assume that both electrons and ions are characterized by delta function

distribution,

f(v) = �(v) =�(v?)

2�v?�(vk); (6.81)

where �(v) is the three-dimensional delta function. This assumption is equivalent to the con-

dition that the phase velocity of concerned wave be much larger than the thermal velocities

of both electrons and ions. (For high frequency waves, such assumption is often appropriate.

However, for low-frequency waves such as Alfven wave, the condition can easily be violated.

In Chapter 3, we have in fact assumed that !=kk � vTe for Alfven waves when analyzing the

MHD ballooning instability.) Then, the dielectric tensor can be evaluated from Eq. (6.32).

15

For example, �xx becomes

�xx = 1 +Xs

!2p!

1Xn=�1

2�

Z 1

0

v?dv?

Z 1

�1dvz

v?�n�

�2J2n(�)

! � kkvk � n@

@v?

��(v?)

2�v?�(vk)

�= 1� 1

2

Xs

!2p!

�1

! � +1

! +

�= 1�

Xs

!2p!2 � 2

= 1�!2pe

!2 � 2e�

!2pi!2 � 2i

; (6.82)

where only n = �1 terms remain �nite because of the delta function distribution. Also note

that J1(x) ' x=2 for x� 1. Similarly, we �nd

�xy = ��yx = �iXs

!2p!(!2 � 2) ; (6.83)

�xz = �zx = �yz = �zy = 0; (6.84)

�yy = �xx = 1�Xs

!2p!2 � 2 ; (6.85)

�zz = 1�Xs

!2p!2: (6.86)

Substituting these �ij in Eq. (6.31), we obtain the following dispersion relation,���������������������

k2k �!2

c2

1�

Xs

!2p!2 � 2

!i!2

c2

Xs

!2p!(!2 � 2) �kkk?

�i!2

c2

Xs

!2p!(!2 � 2) k2 � !

2

c2

1�

Xs

!2p!2 � 2

!0

�k?kk 0 k2? �!2

c2

1�

Xs

!2p!2

!

���������������������

= 0:

(6.87)

As expected, the dielectric tensor is Hermitian, �ij = ��ji; indicating no absorption of wave

energy in a cold plasma.

The components

�xx = �yy = 1�Xs

!2ps!2 � 2s

; (6.88)

16

very much resemble the dielectric constant � (scalar) of an isotropic dielectric medium if the

cyclotron frequency s is replaced with the frequency of bound harmonic electron motion,

!0. As we will see shortly, the dispersion relation of electromagnetic waves propagating along

the magnetic �eld is given by �ckk!

�2= �xx; (6.89)

with electric �elds perpendicular to the magnetic �eld. This also resembles the dispersion

relation of electromagnetic waves in an isotropic dielectric medium,�ck

!

�2= � = 1�

Xj

!2pj!2 � !20j

; (6.90)

where !0j is the frequency of harmonic motion at the j�th bound state. Therefore, cyclotron

motion of charged free particles in a plasma can be considered as bound harmonic motion,

and this analogy holds particularly well in a cold plasma.

The dispersion relation, Eq. (6.87), is a sixth order algebraic equation for ! and in

principle can be solved with the propagation angle

� = tan�1�k?=kk

�;

with respect to the external magnetic �eld as a parameter. However, fundamental modes

can be revealed by considering two particular angles, parallel propagation � = 0 (k? = 0)

and perpendicular propagation � = �=2 (kk = 0). Propagation at arbitrary angles can be

regarded as linear combination of the fundamental modes.

6.7 Parallel Propagation (k? = 0)

If k? = 0 in Eq. (6.87), we obtain two independent solutions, c2k2k!2� �xx

!2+ �2xy = 0; (6.91)

�zz = 0: (6.92)

In terms of electric �eld components Ei, these solutions correspond, respectively, to c2k2k!2� �xx

!Ex � �xyEy = 0; (6.93)

17

�xyEx +

c2k2k!2� �xx

!Ey = 0; (6.94)

�zzEz = 0: (6.95)

Equation (6.91) yieldsc2k2k!2� �xx = �i�xy: (6.96)

In this case, the electric �elds Ex and Ey are related through

Ex = �iEy;

that is, the wave is circularly polarized with either positive (Ex = �iEy) or negative (Ex = +iEy)

helicity. Superposition of these two modes yields a plane wave having Ex only. The dispersion

relation of the plane wave is given by

c2k2k!2

= �xx = 1�!2pe

!2 � 2e�

!2pi!2 � 2i

:

6.7.1 Modes with Positive Helicity

Electron Cyclotron and Whistler Modes

We �rst consider the mode described by

c2k2k!2� �xx = +i�xy: (6.97)

The two components of the electric �eld Ex and Ey are related through Ex = �iEy; that

is, the �eld is circularly polarized with positive helicity (circulation in the same sense as the

electron gyration motion). Substituting

�xx = 1�Xs

!2ps!2 � 2s

= 1�!2pe

!2 � 2e�

!2pi!2 � 2i

; (6.98)

�xy = �iXs

!2ps!(!2 � 2s)

= ijej!2pe

!(!2 � 2e)�

i!2pe

!(!2 � 2i ); (6.99)

we obtainc2k2k!2

= 1�!2pe

! (! � jej)�

!2pi! (! + i)

; (6.100)

18

where subscripts e and i refer to electrons and (singly ionized) ions. As ! approaches jej

from below, the second term in RHS becomes large. The wavenumber kk correspondingly

becomes large, too, and such phenomenon on the !�kk space is called resonance. Resonance

indicates possibility of strong absorption, and in the present case, absorption at the electron

cyclotron frequency is suggested. The wave damping rate will be calculated in Section 6.9

after removing the cold plasma assumption.

When kk ! 0, Eq. (6.100) gives

! = 0 and1

2

�jej �

q2e + 4!

2pe

�; (6.101)

provided !2pe � !2pi, jej � i as in conventional plasmas. The solution with the negative

sign must be discarded since for kk ! +0, it corresponds to a wave having negative helicity

thus violating the assumption made earlier. The positive solution,

!c1 =1

2

�jej+

q2e + 4!

2pe

�; (6.102)

is called a cut-o¤ frequency since waves become evanescent below this frequency (but above

jej). The solution ! ! 0 with kk = 0 corresponds to the Alfven wave branch, ! ' kkVAwhere VA is the Alfven velocity. This mode exists in very low frequency regime, even lower

than the ion cyclotron frequency, ! � i: The Alfven mode will be discussed in more detail

in the section to follow.

In the intermediate frequency range i � ! � jej, Eq. (6.100) yields

! ' jej!2pe

c2k2k; (6.103)

which is known as the whistler or helicon wave. This wave is strongly dispersive. Both phase

and group velocities are proportional top!. As we will see later, the whistler wave can

become unstable when the electron distribution function is characterized by bi-Maxwellian

having T? > Tk. Such anisotropic distribution function is believed to prevail in the space

plasma trapped by the earth magnetic �eld. The name �whistler�is coined after whistling

tones accompanying electromagnetic radiation emitted from the ionospheric plasma. The

whistler wave can be excited by lightning and propagate ove a long distance in the ionosphere

along the earth magnetic �eld.

19

Kinetic Alfven Mode

In the low frequency limit, ! � i, Eq. (6.100) yields

c2k2k!2

' 1 +!2pe! jej

�1 +

!

jej

��!2pi!i

�1� !

i

�= 1 +

!2pe2e

+!2pi2i'!2pi2i; (6.104)

in typical laboratory plasmas with !2pi � 2i : Solving for !, we obtain

! = kkVA; (6.105)

where

VA = ci!pi

=B0p4�Mn0

; (6.106)

is the Alfven velocity with M the ion mass and Mn0 the mass density of the plasma. The

Alfven wave described by Eq. (6.105) is nondispersive. The Alfven mode with negative

helicity is also allowed as will be shown in the following section. If two Alfven modes with

positive and negative helicities are superposed, a plane Alfven wave is realized provided they

have the same amplitude.

In order to �nd �nite ion Larmor radius e¤ects on the Alfven mode, the assumption

k? = 0 must be removed. Let us go back to the starting equation, Eq. (6.97). When

! � i, �xy ' 0 if summation over electron and ion is taken. Also, �xx may be approximated

by !2pi=2i . Therefore, the Alfven wave is essentially described by

c2k2k!2' �xx; (6.107)

and �xx ' !2pi=2i plays the role of plasma permittivity perpendicular to the external magnetic

�eld as discussed in Chapter 1. The dispersion relation Eq. (6.107) holds even when the

condition k? = 0 is relaxed. In Section 6.2, we have derived an expression for �xx in the

case of isotropic Maxwellian distribution. If we neglect electron contribution, and assume

j� inj � 1 for ions in Eq. (6.36), we obtain

�xx '!2pi2i

1

�i

�1� e��iI0(�i)

�; (6.108)

20

where use has been made of the following identity

2

1Xn=1

e��In(�) =1X

n=�1e��In � e��I0(�) = 1� e��I0(�):

Since I0(�) ' 1 + 14�2 for � � 1, we �nd the correction to �xx due to the �nite ion Larmor

radius,

�xx =!2pi2i

�1� 3

4�i

�: (6.109)

Physically, the correction term �i is due to change in the e¤ective electric �eld experienced

by the ion as discussed in Chapter 3,

E? e¤ ' [1�O(�i)]E?: (6.110)

The ion polarization drift corrected for the ion �nite Larmor radius is

vpi =e

M2i

@

@t([1�O(�i)]E?) : (6.111)

Therefore, the cross-�eld ion permittivity is subject to a correction of order �i;

�? '!2pi2i[1�O(�i)]: (6.112)

The dispersion relation of Alfven wave based on the permittivity in Eq. (6.109) is

!2 ' k2kV 2A�1 +

3

4�i

�; �i � 1 (6.113)

which can hold even when k? � kk as long as the condition �i � 1 is satis�ed. In a plasma

with an electron temperature comparable with the ion temperature, this should be modi�ed

as

!2 ' k2kV 2A�1 +

3

4�i + �i

TeTi

�= k2kV

2A

�1 + k2?�

2s

�1 +

3

4

TiTe

��; (6.114)

where �2s = (Te=M)=2i (ion Larmor radius with the electron temperature) which often

appeared in Chapters 3 and 4. Alfven waves can therefore have long parallel, but short

perpendicular, wavelengths. The Alfven wave described by Eq. (6.114) is often called kinetic

Alfven wave.

The correction k2?�2s in the kinetic Alfven mode is due to the deviation from the ideal

MHD. In ideal MHD, the parallel electric �eld is vanishingly small,

Ek = �ikk�+ i!

cAk = 0; (6.115)

21

and the parallel Ampere�s law combined with the charge neutrality conditionr�(J?+Jk) = 0;

rkr2Ak =4�

cr � J?; (6.116)

readily yields the dispersion relation !2 = (kkVA)2 if for the perpendicular current the lowest

order ion polarization current

J? =n0e

2

M2i

@

@tE?; (6.117)

is substituted. The charge neutrality condition itself is not useful in ideal MHD because

both electron and ion density perturbations are vanishing. In order to �nd corrections of

order (k?�s)2; the ideal MHD assumption should be removed and one has to use two-�uid

approximation or kinetic analysis as done for the ballooning mode in Chapter 4. In a uniform

plasma, the ion density perturbation in the frequency regime kkvTi � ! � i is (cf. Eq. (4.

21))

ni = �e�

Tin0 + e

��iI0(�i)e�

Tin0 = ��i

�1� 3

4�i

�e�

Tin0; (6.118)

and that of electrons in the low frequency limit ! � kkvTe is

ne =

��� !

ckkAk

�e

Ten0: (6.119)

From the charge neutrality condition ni = ne and Ampere�s law

r2Ak = �4�

cJk;

where the parallel current is largely carried by the electrons,

Jk ' Jke =n0e

2

kkTe(�!)

��� !

ckkAk

�; (6.120)

one readily �nds the dispersion relation in Eq. (6.114). The parallel electric �eld associated

with the Alfven mode is of order (k?�)2kk� (� kk�):

The kinetic Alfven mode can of course be recovered from the general dispersion relation

in Eq. (6.30) provided the following assumption is made, Ey = 0 because the cross-�eld

electric �eld is essentially curl-free (electrostatic) in a low � plasma. (Recall that we have

assumed k? = k?ex:) Then, the dispersion relation in Eq. (6.114) readily follows from�k2k �

!2

c2�xx

��k2? �

!2

c2�zz

�� (k?kk)2 = 0; (6.121)

22

where

�xx '!2pi2i

�1� 3

4�i

�; �zz '

k2Dek2k�!2pi!2' k

2De

k2k: (6.122)

In �zz; the ion term is ignorable for the Alfven mode in a low � plasma since !=kk ' VA � cs

(the ion acoustic velocity).

Figure 6.2: Parallel modes with positive helicity. Electron cyclotron resonance�kk !1

�occurs at ! . jej : The non-dispersive low frequency mode ! � i (� jej) is the Alfven

mode described by ! = kkVA:

6.7.2 Modes with Negative Helicity

Ion Cyclotron Mode

The mode described byc2k2k!2� �xx = �i�xy; (6.123)

has negative helicity (Ex = +iEy). Again in the cold plasma approximation, Eq. (6.123)

becomesc2k2k!2

= 1�!2pe

! (! + jej)�

!2pi! (! � i)

; (6.124)

23

This exhibits a resonance (kk ! 1) at the ion cyclotron frequency i. For ! � i, Eq.

(6.124) also reduces to Alfven mode,

c2k2k!2' 1 +

!2pe2e

+!2pi2i'!2pi2i;

!2 = V 2Ak2k:

Therefore, in the low frequency limit, Alfven wave can have either helicity, negative or

positive, and a plane, linearly polarized Alfven wave can thus be constructed.

Figure 6.3: Parallel modes with negative helicity. The Alfven mode ! = VAkk with negative

helicity exists in the low frequency region ! � i. The ion cyclotron resonance occurs at

! . i: The cuto¤ frequency !2 is given by !2 =�p4!2pe +

2e � jej

�=2:

The resonance at the ion cyclotron frequency is approached from below as in the case of

electron cyclotron resonance. Usually in plasma heating by cyclotron resonance, waves are

launched from the region where the magnetic �eld is strong so that ! < i: As they propagate

along the magnetic �eld toward weaker �eld region, waves eventually hit the resonance point

(or resonance plane) ! = i where strong absorption takes place. Absorption mechanism

24

(cyclotron damping) is due to thermal e¤ects and will be discussed later. It should be noted

that the cuto¤ frequency ! ! 0 in the limit kk ! 0 corresponds to the Alfven mode.

Electron Mode

In the regime ! � !pi (� i) ; Eq. (6.124) yields

c2k2k!2' 1�

!2pe! (! + jej)

:

The cuto¤ frequency of this mode is

!2 =1

2

�q2e + 4!

2pe � jej

�:

There exist no modes with negative helicity in the frequency domain

i < ! <1

2

�q2e + 4!

2pe � jej

�:

6.7.3 Plasma Oscillation

Finally, the mode described by

�zz = 0; (6.125)

is purely electrostatic, since E k k both along the external magnetic �eld. In a cold plasma,

�zz = 1�!2pe!2�!2pi!2' 1�

!2pe!2

= 0; (6.126)

which indicates electron plasma oscillation having arbitrary wavenumber kk. Electrostatic

waves in a magnetized plasma will be discussed in detail in Chapter 7.

6.8 Perpendicular Propagation (kk = 0)

When kk = 0; Eq. (6.31) yields26664��xx ��xy

�xyk2?c

2

!2� �yy

3777526664Ex

Ey

37775 = 0; (6.127)

25

and �k2?c

2

!2� �zz

�Ez = 0: (6.128)

Corresponding dispersion relations are�ck?!

�2= �yy +

�2xy�xx; (6.129)

andk2?c

2

!2= �zz: (6.130)

The mode described by Eq. (6.129) is called extraordinary mode while the mode described

by Eq. (6.130) is called ordinary mode in analogy to optical waves in anisotropic crystals

which exhibit double refraction.

If thermal e¤ects are negligible, the expression obtained earlier for cold plasma may be

substituted into the dielectric components, �ij. Eq. (6.129) reduces to

�ck?!

�2= 1�

!2pe!2 � 2e

�!2pi

!2 � 2i�

�e!

2pe

!2 � 2e+

i!2pi

!2 � 2i

�2!2�1�

!2pe!2 � 2pe

�!2pi

!2 � 2i

� : (6.131)

This does not exhibit resonance at ! = jej and i. (Showing this is tedious but straight-

forward. Please try.) Resonance occurs when �xx = 0, or

1�!2pe

!2 � 2e�

!2pi!2 � 2i

= 0; (6.132)

which yields

!2 = !2pe + 2e = !

2UH ; (6.133)

and

!2 =!2pi

1 +!2pe2e

= !2LH ; (6.134)

where !UH is the upper hybrid frequency, and !LH is the lower hybrid frequency. At these

resonance frequencies, the waves essentially become electrostatic. To see this, we take the

�rst equation in Eq. (6.127),

�xxEx + �xyEy = 0: (6.135)

26

When �xx = 0, we must have Ey = 0, since �xy remains �nite at the resonance frequencies.

Recalling that we have assumed k? = k?ex, we observe that

k k E; and r� E = 0;

which indicates that upper and lower hybrid waves are electrostatic.

The cuto¤ frequencies are the same as those in the parallel modes,

!1;2 =1

2

�q4!2pe +

2e � jej

�:

This is because when k? = 0 (cuto¤ condition), Eq. (6.127) requires that

�2xx + �2xy = 0;

or

�xx = �i�xy:

These relationships are identical to Eq. (6.96) corresponding to the cuto¤ condition of the

parallel modes.

At very low frequencies such that ! � i, Eq. (6.131) becomes�ck?!

�2= 1 +

!2pi2i� 2

!2pi!2(k?�i)

2 � 2!2pe!2(k?�e)

2 ; (6.136)

!2 =V 2A + 2

Ti + TeM

1 + (VA=c)2 k2?: (6.137)

This mode is called the magnetosonic mode or compressional Alfven mode corrected for the

sound speed Vs;

V 2s? = 2Ti + TeM

; kk = 0: (6.138)

It is noted that the adiabatic coe¢ cients are i = e = 2 for waves propagating strictly

normal to the magnetic �eld kk = 0. In the case ! � kkvTe; which may occurs for samll but

�nite kk (peopagation slightly tilted from � = 90�); e = 1; and the sound speed for k? > kk

becomes

V 2s? =2Ti + TeM

; ! � kkvTe:

27

Except for the sound speed, the dispersion relation is formally identical to that of the shear

Alfven wave propagating along the magnetic �eld. However, �eld polarization is entirely

di¤erent. In order to see the characteristic di¤erence between shear Alfven wave propagating

along the magnetic �eld and compressional Alfven wave propagating perpendicular to the

magnetic �eld, we go back to the �eld equation,

r� E = �1c

@B

@t; (6.139)

or

k� E = !

cB: (6.140)

For shear Alfven wave, kk = kkez; both the electric and magnetic �eld are perpendicular

to the unperturbed magnetic �eld B0. Physically, such �eld con�guration corresponds to

bending of the magnetic �eld lines. On the other hand, the magnetosonic wave is associated

with k = k?, and E?, with k? and E? being normal to each other. Then, from Eq. (6.140),

we observe that the perturbed magnetic �eld associated with compressional Alfven wave is

parallel to the unperturbed �eld B0;

B k B0;

which creates compression and rarefaction of magnetic �eld lines.

In Chapter 2, we have seen that the most dangerous MHDmodes are incompressible char-

acterized by r � v = 0. Compressional Alfven (magnetosonic) mode is obviously accompa-

nied by �eld line and thus plasma compression, andr � v remains �nite. The corresponding

plasma density perturbation may be found from the continuity equation,

@n

@t+r � (nv) = 0; (6.141)

where for ! � i, the dominant cross-�eld velocity is the E �B drift,

vE = cE�B0B20

; (6.142)

where E is the induction (rather than electrostatic) electric �eld related to the magnetic

perturbation through

r� E = �1c

@B

@t: (6.143)

28

If the density n0 and the magnetic �eld B0 are uniform, the perturbed density n1 can be

found as@n1@t

= �n0r � v =n0B20B0 �

@B

@t; (6.144)

orn1n0=BkB0: (6.145)

This indicates that plasma density and magnetic perturbations are in phase and equal in rel-

ative magnitude which is another manifestation of frozen-in nature of plasma to the magnetic

�eld lines. The appearance of sound speed in the dispersion relation of the magnetosonic

mode is thus understandable.

The mode given in Eq. (6.130) is the ordinary electromagnetic wave propagating across

the magnetic �eld. If thermal e¤ects are neglected, �zz ' 1�!2pe!2, and Eq. (6.130) becomes

!2 = !2pe + c2k2?: (6.146)

This mode is una¤ected by the external magnetic �eld because of the fact that the electric

�eld assocaited with the mode is along the magnetic �eldB0 and the corresponding perturbed

current is also along B0. The Lorentz v �B0 force vanishes in this case, and particle motion

remains una¤ected by the magnetic �eld. That the phase velocity is independent of the

magnetic �eld makes this particular mode an extremely convenient probe in measuring the

electron density. In typical fusion devices, fpe = !pe=2� is of order of tens of GHz, and

millimeter wave or shorter wavelength infrared laser is required in interferometric plasma

density measurements.

Figure 6.4 summarizes the perpendicular modes.

6.9 Propagation at Arbitrary Angle

When both kk and k? are nonzero, the dispersion relation, Eq. (6.87), must be analyzed as

it is. One of the important applications of wave analysis is in radio frequency (rf) heating of

plasma, and the discussion presented here will be developed having this particular application

in mind. In rf heating, the wave frequency ! is given. (It is the frequency determined by

29

Figure 6.4: Perpendicular modes. Resonance (k? !1) occurs at the upperhybrid frequency

!UH =p!2pe +

2e and the lowerhybrid frequency !LH = !pi=

p1 + (!pe=e)2 which are slow

electrostatic modes and allow e¢ cient wave absorption. The cuto¤ frequencies !1 and !2

are identical to those found in parallel propagation.

wave sources.) Therefore, rather than solving Eq. (6.87) for !, we attempt to �nd kk

and k? for a given frequency !. Wave propagation is allowed if real solution for k exists.

Otherwise, waves become evanescent being cut o¤ by the plasma itself. However, in several

practical applications, waves must go through an evanescent region before fully penetrating

deep into plasma core. If the spatial damping due to evanescent region is tolerably small,

such excitation mechanism is still a useful method in plasma heating. A typical example is

the lower hybrid wave which has successfully been developed for plasma heating and current

drive.

Let us introduce the parallel and perpendicular indices of refraction,

nk =ckk!

; n? =ck?!: (6.147)

30

Also, to save subscripts, we let8>>>>>>>><>>>>>>>>:

�? = �xx = �yy; (in cold plasma)

�k = �zz;

�X = �xy = ��yx:

(6.148)

Expanding the determinant in Eq.(6.87), we obtain

�?n4? +

��?n

2k + �kn

2k � �?�k � �2? � �2X

�n2? + �k

�n2k � �?

�2+ �k�

2X = 0: (6.149)

The direction of wave injection in most heating applications is perpendicular to the magnetic

�eld or in the radial direction. For this reason, we assume that nk as well as ! is prescribed

(kk is usually determined by wave excitation mechanisms, such as grill structure in lower

hybrid wave exciters. Even if kk is not well de�ned, one can always Fourier analyze along the

direction of magnetic �eld.) Eq. (6.149) is an algebraic equation for n2?, and the condition

that n2? be real is given by��?n

2k + �kn

2k � �?�k � �2? � �2X

�2 � 4�? h�k �n2k � �?�2 + �k�2Xi > 0: (6.150)

n2? may become negative. In this case, wave becomes evanescent. However, as explained

earlier, in a nonuniform plasma, evanescence does not necessarily mean complete cuto¤,

and if evanescence region is con�ned at the plasma-vacuum boundary, waves can still tunnel

through and penetrate into a plasma. If n2? becomes complex, on the other hand, wave acces-

sibility is usually much reduced. Also, two otherwise independent modes become degenerate

when n2? is complex. For these reasons, Eq. (6.150) may be regarded as the accessibility

condition. In general, the condition is necessary, but not always su¢ cient, and accessibility

of a particular mode should be examined carefully for given experimental conditions, such

as the plasma density pro�le. In the following, accessibility problems of some typical modes

of practical interest will be discussed.

31

6.9.1 Electron Cyclotron Mode

In Section 6.2, we have seen that electron cyclotron wave propagating along the magnetic

�eld is described by the dispersion relation,

n2k =c2k2k!2

= 1�!2pe

! (! � jej): (6.151)

The wave has positive helicity with circular polarization. When the propagation angle � is

slightly titled from the magnetic �eld, the dispersion relation is modi�ed as

n2 = n2k + n2? = 1�

!2pe! (! � jej cos �)

: (6.152)

To see this modi�cation, we directly solve the dispersion relation which is rewritten in terms

of the total index of refraction n and propagation angle � as

n4��? sin

2 � + �k cos2 ��� n2

���2? + �

2X

�sin2 � + �?�k

�1 + cos2 �

��+ �k

��2? + �

2X

�= 0:

(6.153)

This can readily be solved for n2,

n2 =B �

pB2 � 4AC2A

; (6.154)

where 8>>>>>>>><>>>>>>>>:

A = �? sin2 � + �k cos

2 �

B = (�2? + �2x) sin

2 � + �?�k(1 + cos2 �)

C = �k (�2? + �

2x)

(6.155)

Note that in Figs. 6.2 and 6.3, there exist at most two solutions for kk or k? for a given

frequency !. The two solutions given in Eq. (6.154) indicate that at arbitrary propagation

angle �, we still have two propagation modes.

It is convenient to rewrite Eq. (6.154) as

n2 = 1� 2(A�B + C)2A�B �

pB2 � 4AC

: (6.156)

32

For the electron cyclotron wave, we may ignore ion dynamics since ! � !pi (� i). Then,

the dielectric components assume

�? = 1�!2pe

!2 � 2e; (6.157)

�X = ijej!2pe

! (!2 � 2e); (6.158)

and

�k = 1�!2pe!2: (6.159)

Substituting these into A; B and C, we �nd

n2 = 1�2!2pe

�1� !2pe=2e

�2!2

�1� !2pe=2e

�� 2e sin2 � � jej

pD; (6.160)

where

D = 2e sin4 � + 4!2

�1�

!2pe!2

�2cos2 �: (6.161)

If � = 0 (parallel propagation), we recover the familiar result for the parallel electron cy-

clotron mode,

n2k =

�ckk!

�2= 1�

!2pe! (! � jej)

: (6.162)

Also, if � = �=2;

n2? =

�ck?!

�2= 1�

!2p�!2 � !2p

�!2�!2 � !2p � 2e

� ; (6.163)

and

n2? = 1�!2pe!2; (6.164)

which are also consistent with those found in Section 6.3.

The dispersion relation, Eq. (6.152) derives from Eq. (6.160) if sin2 � is su¢ ciently small.

Since D is positive de�nite, the accessibility condition for electron cyclotron wave may be

imposed by

n2 > 0; (6.165)

everywhere along the wave trajectory, that is, from the plasma edge where !p = 0 to the

plasma core where !p is maximum, and cyclotron resonance is aimed at. In toroidal devices

such as tokamaks and stellarators, the toroidal magnetic �eld varies being proportional to

33

1=R, where R is the radius from the toroidal axis. Therefore, if the wave frequency ! is so

chosen that cyclotron resonance is to take place at the plasma minor center, accessibility

can be achieved only from the inner side. Often this requirement causes technical di¢ culties

since the inner side of toroidal devices is crowded with mechanical structures (toroidal coils,

poloidal windings, etc.).

Electron cyclotron resonance heating has become feasible rather recently after devel-

opment of high power (& 100 kW); high frequency (' 30 GHz) microwave sources (e.g.

gyrotron). Although heating reactor scale devices will require a large total power at much

higher frequencies (because of higher magnetic �elds), ECR heating is one of the promising

auxiliary heating methods to achieve ignition temperatures. One restriction of ECR heating

is its density limit at the region when cyclotron absorption takes place. The frequency !

must be above the lower cuto¤ frequency

! > !c1 =

p4!2pe +

2e � jej

2

everywhere along the wave trajectory. (We note that the cuto¤ frequencies to make n2 = 0 in

Eq. (6.160) is independent of the propagation angle, �.) Therefore, the maximum allowable

density in terms of the plasma frequency !pe is

!2pe (at resonance) � 22e:

6.9.2 Lowerhybrid Wave

As seen in Section 6.6, the lower hybrid resonance frequency

!LH =!pip

1 + (!pe=e)2; (6.166)

is essentially proportional to the ion plasma frequency and resonance region is a point rather

than a plane as in the case of cyclotron resonance. Fortunately, in the sense of geometrical

optics, any waves tend to refract toward lower phase velocity region. At the resonance region,

the wavenumber k increases and thus waves tend to converge.

In the case of lower hybrid wave, it is impossible to completely avoid evanescent region.

However, as we will see shortly, the evanescent region is limited to the plasma edge where the

34

plasma density is low. Waves can easily tunnel through the evanescent region and penetrate

into the plasma core.

Accessibility of lower hybrid waves has been �rst analyzed by Golant, and we follow the

analysis developed by him. If we assume jej � ! � i appropriate for the lower hybrid

wave, the dielectric components become8>>>>>>>>>>><>>>>>>>>>>>:

�? = 1 +!2pe2e�!2pi!2

(� 0)

�X = i!2pe! jej

� ii!

2pi

!3' i

!2pe! jej

�k = 1�!2pe!2'!2pe!2

(< 0)

(6.167)

As the accessibility condition, we adopt Eq. (6.150) which only ensures that n2? be real,

but not necessarily n2? be positive. The inequality, Eq. (6.150), can easily be solved for the

range of n2k,

n2k >

�?��? � �k

�2+��? � �k

��2xy + 2

r�?�k�

2X

h��? � �k

�2+ �2X

i��? � �k

�2 : (6.168)

Since ���k��� j�?j ; j�X jwe may approximate Eq. (6.168) as

n2k > �? +�2X�k+ 2

r�?�k�X =

p�? +

j�X jpj�kj

!2: (6.169)

The RHS of the above equation becomes maximum at a distance xm from the plasma edge

where the plasma density n0(x) satis�es

!2pe(xm) =!4r

2e + !2r

: (6.170)

Here, !r (= const.) is given by

!2r =!22e

m

M2e � !2

; (6.171)

35

which is the electron plasma frequency at the resonance point, x = xr,

!r = !pe(xr) ; xm < xr: (6.172)

The maximum value of RHS takes a simple value and is given by

n2k > 1 +!2pe(xr)

2e: (6.173)

This is the condition to avoid complex solutions of n2?, and can be regarded as the accessibility

condition for lower hybrid waves.

At the plasma edge where the plasma density is zero, we have the vacuum solution

n2? + n2k = 1:

Therefore, when the accessibility condition is satis�ed (n2k > 1), the wave at the edge region

should be evanescent with respect to transverse propagation,

n2? = 1� n2k < 0:

The distance from the plasma edge beyond which the wave becomes transparent n2? > 0

can readily be found from the dispersion relation, Eq. (6.149). In the edge region, �? ' 1

can still be assumed, but �k, �x can quickly become large. When, Eq. (6.149) is written in

the form

n4? +Bn2? + C = 0; (6.174)

the condition for n2? to have a positive real solution is simply

C < 0: (6.175)

Note that the lower hybrid wave corresponds to the �slow wave�solution,

n2? =1

2

h�B +

pB2 � 4C

i: (6.176)

Therefore, the transition position becomes

!2 = !2pe(xt): (6.177)

Since ! is of the order of ion plasma frequency at the plasma core where the density is

maximum, we �nd that the transition distance from the plasma edge is extremely small.

The inevitable existence of evanescent region in launching lower hybrid waves into a plasma

is not expected to be problematic as successfully demonstrated in several recent experiments.

36

6.9.3 Ion Cyclotron Wave

RF heating with the frequency in the range of ion cyclotron frequency is one of the oldest

methods in fusion research. This is due to the obvious reason that high power rf sources

at typically tens of MHz are readily available. ICR heating is still in active use and can

well compete with neutral beam injection in attaining high ion temperatures. Since the

frequency is considerably lower than electron cyclotron and lower hybrid schemes, wave

excitation based on waveguides cannot be used. Ion cyclotron waves are usually excited by

antennas located near the plasma edge. Coupling e¢ ciency is expected to be higher, the

closer the antenna is to the plasma. This may cause impurity problems, for it is practically

impossible to completely insulate the antenna from the plasma. It is commonly observed

the impurity level signi�cantly increases when attempts are made to increase rf power. Of

course, impurities (high Z ions) directly contribute to plasma energy loss through radiation,

and deteriorate energy con�nement times.

Analysis on accessibility of ion cyclotron resonance is less involved than ECR and LHR.

n2? remains real in the approximation to be used. As in the case of LHR, mild evanescence

exists, but it can be tolerated.

Ion cyclotron wave is characterized by the frequency

! . i � jej

Therefore, �?; �X ; and �k in Eq. (6.149) may be assumed to be8>>>>>><>>>>>>:

�? ' 1 +!2pe2e�

!2pi!2 � 2i

' �!2pi

!2 � 2i;

�X ' �ii!

2pi

! (!2 � 2i )� i

!2pi!i

;

�k ' �!2pe!2;

(6.178)

and the solutions for n2? are given by

n2? =1

2�?

!2pe!2�n2k � �?

�� 1

2�?

�!4pe!4�n2k � �?

�2 � 4!2pi!2 � 2i

!2pe!2

�n4k +

2!2pi!2 � 2i

n2k �!4pi

2i (!2 � 2i )

��1=2:(6.179)

37

In the square root, the �rst term dominates over the rest. Then, the two solutions are

n2? '1

�?

!2pe!2�n2k � �?

�; (6.180)

and

n2? 'n4k (

2i � !2)� 2n2k!2pi + !4pi=2in2k (!

2 � 2i ) + !2pi: (6.181)

The �rst solution is evanescent unless n2k is very large. The second solution corresponds to

ion cyclotron wave propagating at an angle with respect to the magnetic �eld. When n? = 0

(parallel propagation), we recover the solution previously found,

n2k '!2pi

i(i � !); (6.182)

which is in the form of

n2k = �? � i�X : (6.183)

From Eq. (6.181), it can be seen that cuto¤ (n2? = 0) occurs when

!2pi(xc) = !(i � !)n2k; (6.184)

and the resonance when

!2pi(xr) =�2i � !2

�n2k: (6.185)

At the plasma edge where !pi = 0, the wave is evanescent, n2? = �n2k < 0. (We do not

recover the vacuum solution n2 = 1 since in the expression for �? and �k, the vacuum term

1 has been omitted.) Wave propagation is allowed (n2? > 0) in the region

! (i � !)n2k < !2pi(x) <�2i � !2

�n2k: (6.186)

When ! <�i, the ratio between the density at the resonance and that at the cuto¤ is

approximately equal to 2. This should be compared withpM=m (M=m being the ion-

electron mass ratio) found for the lower hybrid wave. The evanescent region in ion cyclotron

wave is substantially larger than that in lower hybrid wave. However, this does not necessarily

mean that the wave damping due to evanescence is correspondingly strong. In the evanescent

region, n2? may be approximated by

n2? ' �n2k +3!2pi

2i � !2(< 0): (6.187)

38

Thus, the spatial damping factor jk?j in exp (� jk?jx) is substantially reduced from the

vacuum value (jkyj = kk).

Ion cyclotron resonance heating based on the mechanism described above is best suited

to cylindrical plasmas, such as those in tandem mirror experiments. In addition to transverse

resonance, longitudinal resonance (n2k ! 1) can be achieved if the magnetic �eld itself is

nonuniform along the plasma column (concept of magnetic beach).

In toroidal plasmas, the toroidal magnetic �eld is strongly nonuniform and accessibility

of ion cyclotron waves encounters similar di¢ culties as electron cyclotron waves. For this

reason, ICR heating based on slow wave (the branch of conventional ion cyclotron mode) is

seldom used particularly in tokamak research. Instead, fast wave (magnetosonic wave) with

! ' 2i is commonly used. Since fast wave heating requires knowledge of wave propagation

in a plasma with �nite ion temperature, we defer this important topic until a later chapter.

6.10 Kinetic E¤ects

The cold plasma approximation employed in the preceding Sections is applicable when the

phase velocity (or the Doppler shifted phase velocity) is su¢ ciently remote from the thermal

velocities of electrons and ions. The components of the dielectric tensor �ij all contain the

plasma dispersion function Z(�n) or its derivative where

�n =! � nkkvT

:

Cold plasma approximation pertains to the limit j�nj � 1 which allows us to use the asymp-

totic expansion of the plasma dispersion function,

Z(�n) ' �1

�n� 1

2�3n� � �+ i

p�e��

2n :

The imaginary residue term, though small, indicates wave damping through Landau (n = 0)

and cyclotron (n 6= 0) resonance.

39

6.10.1 Cyclotron Damping

In this Section, we analyze the dispersion relation of the electron cyclotron mode�ckk!

�2= �xx + i�xy; (6.188)

by retaining kinetic resonance. Ion contributions to �xx and �xy can be ignored and we

approximate them by

�xx ' 1 +1

2

�!pe!

�2 !

kkvTe

�Z(�e;+1) + Z(�e;�1)

�;

�xy ' i1

2

�!pe!

�2 !

kkvTe

�Z(�e;+1)� Z(�e;�1)

�:

Substitution into Eq. (6.188) yields�ckk!

�2= 1 +

!2pe!kkvTe

Z(�e;�1); (6.189)

where

�e;�1 =! � jejkkvTe

:

If j�e;�1j � 1; Eq. (6.189) reduces to�ckk!

�2' 1�

!2pe!(! � jej)

+ ip�!2pe

!kkvTeexp

"��! � jejkkvTe

�2#: (6.190)

Solutions for ! or kk must be complex which indicates wave damping. (Recall that we have

assumed a propagation function ei(k�r�!t):) In steady state plasma heating experiments, ! is

real and Im kk (> 0) becomes the spatial damping factor. In the lowest order, the solution

for kk is given by

kk = kk0 + iki;

where

kk0 =!

c

s1�

!2pe!(! � jej)

; (6.191)

ki =

p�

2

!!2pec2k2k0vTe

exp

"��! � jejkkvTe

�2#: (6.192)

40

At the electron cyclotron resonance, ! = jej ; the plasma dispersion function takes a

simple value Z(0) = ip�: Then Eq. (6.189) becomes�

ckk!

�2' 1 + i

p�

!2pejejkkvTe

: (6.193)

In magnetic fusion plasmas, jej ' !pe: Also, kk � kDe =p2!pe=vTe: Then the unity can

be ignored and we have

kk = �1=6ei�=6

(jej!2pe)1=3

(c2vTe)1=3: (6.194)

Im kk is comparable with the real part indicating strong spatial damping.

In the case of second harmonic electron cyclotron resonance heating, ! ' 2jej; the

term Z(�e;�2) contained in �xx and �xy plays the dominant role. The damping mechanism is

the same as in the fundamental mode except that a target plasma should have appreciable

electron temperature for the second harmonic heating to be e¤ective. This is because for a

cold plasma, the �nite electron Larmor radius e¤ect is vanishingly small, I2(�e)=�e / �e:

The ion cyclotron resonance can be analyzed in a similar manner.

6.10.2 Whistler Instability due to Temperature Anisotropy

TheWhistler mode has been �rst observed as naturally occurring radiation from the ionospheric

plasma. The mechanism of self-excitation (instability) of the mode is generally attributed to

the anisotropic electron temperature Te? > Tek which is plausible in the ionospheric plasma

con�ned by the earth mirror magnetic �eld. In a mirror �eld, particles can freely escape

through the mirror throats and the velocity distribution is in general non Maxwellian. Here

we assume that the electron velocity distribution is bi Maxwellian characterized by two

temperatures, Te? and Tek ;

fe(v2?; vk) =

m

2�Te?exp

��mv

2?

2Te?

�rm

2�Tekexp

�mv2k2Tek

!: (6.195)

The dielectric components �xx and �xy in Eqs. (6.31) and (6.32) were for Maxwellian distrib-

ution and cannot be used for the present purpose. Instead, �xx and �xy have to be calculated

41

from Eq. (6.26),

�xx = 1 +!2pe!2

Xn

Zd3v

v?

�n

�e

�2J2n(�e)

! � kkvk � ne

�(! � kkvk)

@fe@v?

+ kkv?@fe@vk

�; (6.196)

where ion contributions have been ignored. For k? = 0; we have

�xx = 1��!pe!

�2+1

2

�!pe!

�2 " jejkkvTke

Z(�e;�1)�Te?2Tek

Z 0(�e;�1)

#

�12

�!pe!

�2 " jejkkvTke

Z(�e;1) +Te?2Tek

Z 0(�e;1)

#; (6.197)

where vTke =q2Tek=m and Z 0(�) = �2[1+ �Z(�)] is the derivative of the plasma dispersion

function. Similarly,

�xy = i1

2

�!pe!

�2 "� jejkkvTke

Z(�e;1)�Te?2Tek

Z 0(�e;1)�jejkkvTke

Z(�e;�1) +Te?2Tek

Z 0(�e;�1)

#:

(6.198)

Substituting these into the dispersion relation�ckk!

�2= �xx + i�xy;

yields �ckk!

�2= 1 +

�!pe!

�2 " !

kkvTkeZ(�e;�1) +

1

2

�1� Te?

Tek

�Z 0(�e;�1)

#: (6.199)

Assuming j�e;�1j � 1; we thus obtain�ckk!

�2= 1�

!2pe!(! � jej)

+ ip��!pe!

�2 �jej

�1� Te?

Tek

�+ !

Te?Tek

�e��

2e;�1

kkvTke: (6.200)

If Te? > Tek; the last resonance term can change its sign from positive to negative and

a whistler instability (Im ! > 0) occurs. Since the frequency ! is much smaller than

the electron cyclotron frequency, a slight temperature anisotropy is su¢ cient to excite the

whistler mode.

An anisotropic distribution function is thermodynamically unstable in the sense that

it has freedom to relax to more stable isotropic Maxwellian distribution. A plasma can

42

approach thermodynamic equilibrium through particle collisions. However, if the growth

rate of plasma instability is much larger than the collision frequency, the relaxation process

is accelerated by the instability. In the case of the whistler instability, it enhances the

temperature equilibriation by transferring the perpendicular energy (Te?) to parallel energy

(Tek) at a rate much faster than expected from the collisional process.

6.10.3 Weibel Instability

Another well known instability caused by temperature anisotropy is the Weibel instability.

This is an electromagnetic instability in unmagnetized plasma and may occur when a plasma

is heated anisotropically as in strong rf heating. To analyze the instability, we assume the

geometry shown in Fig. 6.4. An electromagnetic wave propagates in the z direction with

a wavevector k and electric �eld in the x direction, E = Exex: The unperturbed electron

distribution function integrated over the ignorable vy is assumed to be bi Maxwellian,

f0(vx; vz) =

rm

2�T?exp

��mv

2x

2T?

�rm

2�Tkexp

��mv

2z

2Tk

�; (6.201)

where ? and k are with respect to the direction of wave propagation. After eliminating the

perturbed magnetic �eld from the linearized Vlasov equation for the electron, we have

�i(! � kvz)f1 �e

m

�E

�1� k � v

!

�+k

!(E � v)

�� @f0@v

= 0; (6.202)

where@f0@v

= ��m

T?vx +

m

Tkvz

�f0: (6.203)

Solving Eq. (6.202) for f1;

f1 = �ie

T?

Exvx! � kvz

�1 +

�T?Tk� 1�kvz!

�f0: (6.204)

Only the current density in the x direction is non vanishing and given by

Jx = �en0Rvxf1dvxdvz

= �i n0e2

mkvTe

�T?TkZ(�e) +

1

�e

�T?Tk� 1��Ex; (6.205)

43

where vTe =p2Tk=m and �e = !=kvTe: This de�nes a scalar conductivity

� =JxEx

= �i n0e2

mkvTe

�T?TkZ(�e) +

1

�e

�T?Tk� 1��; (6.206)

and the dielectric constant,

� = 1 + i4�

!� = 1 +

!2pe!kvTe

�T?TkZ(�e) +

1

�e

�T?Tk� 1��: (6.207)

Then the dispersion relation for electromagnetic mode is given by�ck

!

�2= � = 1 +

!2pe!kvTe

�T?TkZ(�e) +

1

�e

�T?Tk� 1��: (6.208)

Let us �rst consider isotropic case, T? = Tk;�ck

!

�2= 1 +

!2pe!kvTe

Z(�e): (6.209)

This describes electromagnetic modes in an unmagnetized plasma with kinetic corrections.

Evidently, it formally agrees with the dispersion relation of the ordinary mode in a magne-

tized plasma in Eq. (6.146). For �e � 1; we indeed recover the familiar form,�ck

!

�2= 1�

�!pe!

�2or !2 = !2pe + (ck)

2:

In the opposite limit �e � 1; we have�ck

!

�2= 1 + i

p�!2pe!kvTe

' ip�!2pe!kvTe

;

which yields

k = �1=6ei�=6�!!2pec2vTe

�1=3: (6.210)

The quantity

�a =

�c2vTe!!2pe

�1=3; (6.211)

is known as the anomalous (or kinetic) skin depth. It can exceed the conventional collisionless

skin depth

� =c

!pe; (! � !pe) (6.212)

44

provided ! < vTe!pe=c: (Of course, at extremely low frequency ! < �c (the electron collision

frequency), the collisional skin depth becomes dominant,

�c =c

!pe

r�c!; (6.213)

which emerges from the dispersion relation�ck

!

�2= 1 + i

4�

!�c = 1 + i

!2pe!�c

; (! � �c)

where �c = n0e2=m�c is the collisional conductivity.)

We now return to the Weibel instability. When �e � 1, Eq. (6.208) reduces to�ck

!

�2' 1�

�!pe!

�2�!2pek

2v2Te!4

T?Tk: (6.214)

There always exists a negative solution for !2 which indicates a purely growing instability.

The maximum growth rate is of the order of

max 'pT?=m

c!pe: (6.215)

The distribution function with anisotropic temperatures may be replaced by two cold

electron clouds drifting in opposite directions along the x axis,

f0(v) =1

2[�(vx � V ) + �(vx + V )] �(vy)�(vz): (6.216)

(The factor 1/2 is for the normalization,Zf0d

3v = 1:) In this case, the growth rate is given

by

max 'V

c!pe: (6.217)

However, as we will see in Chapter 8, such two-stream distribution function is unstable

against rapidly growing electrostatic instability with a growth rate far exceeding that of the

Weibel instability.

The physical mechanism of the Weibel instability is in the magnetic Lorentz force. In the

geometry assumed, the perturbed magnetic �eld is in the y direction and the Lorentz force

v �B directed in the z direction causes the �lamentation of the plasma. If the unperturbed

distribution is isotropic, the magnetic force term

(v �B) � @f0@v; (6.218)

45

in the Vlasov equation identically vanishes. For the anisotropic distribution assumed, this

term remains �nite,

m

�1

T?� 1

Tk

�vxvzByf0; (6.219)

and contributes to the perturbation in the distribution function.

46