Chapter 1

General Properties of High

Temperature Plasmas

1.1 Introduction

The �rst systematic studies on the properties of ionized gases were undertaken by Langmuir and

Tonks in the 1920�s. Although mercury vapor plasmas used then were of low density and low

temperature, and thus not fully ionized, several fundamental properties of plasmas were revealed.

These include the concept of Debye screening (originally predicted for dielectric �uids, but equally

applicable for plasmas), capability of a plasma to support both high frequency (of the order of

electron plasma frequency) and low frequency (of the order of sound frequency) waves, and exper-

imental observation that plasma electrons obey a nearly Maxwellian velocity distribution function

even though they are not expected to do so because of insu¢ cient collisions. The last �nding

is often referred to as Langmuir�s paradox and was later given plausible explanation in terms of

plasma turbulence. The existence of both high-frequency and low-frequency waves is due to certain

destabilization mechanisms, and the capability of a plasma to self-excite various waves was already

apparent in the series of experiments carried out by Langmuir and Tonks. Experimental investiga-

tion of these plasma properties was made possible by a small probe (Langmuir probe) which is still

widely used to diagnose low-temperature plasmas. By varying the probe bias voltage, Langmuir

was able to measure the energy distribution function of electrons. This was already an important

indication that, because of many body nature of a plasma, statistical mechanics (Boltzmann equa-

tion, Vlasov equation) must be employed to describe plasma phenomena in a satisfactory manner.

1

Simple, rare�ed mercury discharge plasmas used by Langmuir and Tonks opened the door to a new

physics discipline, namely, plasma physics. (Mercury vapor plasma is easy to produce because of

the relatively low ionization potential. Electronic devices based on mercury vapor plasmas are still

commercially available, e.g., Ignitron, a high voltage, high current switching device.)

In astro- and space physics, the importance of plasma interaction with a magnetic �eld was

�rst recognized. Magnetohydrodynamics (MHD) is a useful approximation to describe macroscopic

plasma dynamics in a magnetic �eld and is still widely used to predict MHD equilibria and stability

of magnetically-con�ned plasmas in nuclear fusion research. To achieve self-sustained fusion reac-

tions, a high temperature (T & 108 K), high density (n & 1014 cm�3) deuterium-tritium plasma

must be stably con�ned with a con�nement time � & 1 sec. The early experimental results in var-

ious con�nement schemes (mirrors, pinches, stellarators and tokamaks) were rather disappointing.

It was in the late 1960�s that the superiority of tokamaks to other devices was recognized, although

the tokamak con�nement times were (still are) anomalously short compared with the theoretical

prediction based on collisional transport.

Today, fully-ionized plasmas having an extremely high temperature (exceeding 10 keV), high

density (n & 1014 cm�3), and long con�nement time (& 0:1 sec) are realized in tokamaks. (The

idea of tokamak was conceived by Sakharov and Tamm in the 1950�s and experimental research was

initiated by Artsimovich in the early 1960�s. Since then, tokamak research has undergone extensive

development worldwide. At present, it is the most successful approach to controlled nuclear fusion

reactors.) These parameters, or combination of them, are not very far from those required for self-

sustained fusion reactions, and in some tokamaks (e.g., JET, Joint European Torus and JT-60 in

Japan), the conditions of the breakeven have been achieved. (Self-sustained fusion reaction means

that fusion products heat the plasma to maintain a high temperature. In the case of D-T fuels, �

particles carrying an energy of 3.5 MeV are born through fusion reactions. They collide with the

background plasma and deposit energy mainly on electrons.

Although tokamaks exhibit con�nement times longer than any other types of devices (e.g.,

stellarators, mirrors), the con�nement times are still much shorter than expected from the clas-

sical (or neoclassical) theory. It is believed that plasma turbulence, grown from various plasma

instabilities, is responsible for the anomalously short con�nement times. Ion thermal di¤usivity

of the order of the neoclassical value has been achieved in present tokamaks with strong plasma

rotation. It is believed that the anomalous ion thermal transport caused by the ion temperature

gradient (ITG) instability is suppressed by nonuniform plasma rotation. However, electron ther-

2

mal transport continues to be anomalous (approximately two orders of magnitude higher than the

neoclassical transport). at present, it is still not clear which instabilities play dominant roles. One

of the problems is that there are so many instabilities expected in the tokamak magnetic con�gu-

ration. Identifying instabilities does not automatically mean that they are actually responsible for

enhancing particle or thermal losses, for particle and thermal transport processes are inevitably due

to nonlinear e¤ects (or at least quasilinear e¤ects). Although signi�cant progresses have recently

been made in how to predict the saturation level of an instability and consequent anomalous par-

ticle and thermal transport, the origin of the anomalous electron thermal conduction in tokamaks

still remains largely unresolved.

Besides the active research on controlled fusion currently going on worldwide, space plasma

physics has been making its own steady progresses. In this discipline, too, studies on wave prop-

agation and instabilities have been very active. Laboratory plasma physics greatly bene�ts from

the knowledge gained in space plasma physics and vice versa. Mutual interaction between the two

disciplines is becoming increasingly bene�cial.

Plasmas of our interest can thus have an extremely wide range of parameters (temperature,

density, etc.) and di¤erent con�gurations (torus, mirrors, etc.). Often, plasmas without particular

con�nement e¤orts are also the subject of current research (e.g., plasmas in inertial con�nement

fusion research). The purpose of this introductory chapter is to give a brief summary of basic prop-

erties of magnetically con�ned, high temperature plasmas, with a slight bias toward the tokamak

magnetic con�guration.

1.2 Dielectric Properties of Plasmas

A plasma consists of free electrons and ions, both randomly moving because of their thermal motion.

It is due to the mobility of electrons and ions that any excess charge created in a plasma tends to

be rapidly neutralized and thus a plasma can maintain its gross charge neutrality. This, however,

does not mean that a plasma cannot support static electric �elds in itself. Laboratory plasmas are

all con�ned in a �nite volume, and they are bound to di¤use as long as collisions among charged

particles exist. (Collisional encounter between charged particles and consequent particle di¤usion

will be discussed later.) Whenever ions and electrons tend to di¤use at di¤erent rates, an electric

�eld (so-called ambipolar electric �eld) will be set up to ensure that both ions and electrons di¤use

at the same rate. Otherwise, charge neutrality cannot be realized.

3

An important consequence of the mobility of electrons and ions is the e¤ective screening of the

electric �eld due to an excess charge placed in a plasma. One of the Maxwell�s equations (Gauss�

law) is

r �E = 4��t = 4��free + 4��ind; (1.1)

where �t is the total charge density, consisting of the excess, free charge �free and the induced

charge, �ind. In vacuum, we obviously do not have �ind, and the electric �eld will be the Coulomb

�eld. In a plasma, however, both electrons and ions will react to the Coulomb �eld, and they create

additional charge density, �ind, through polarization. The resultant electric �eld will not be the

Coulomb �eld, but a screened �eld.

In equilibrium, the induced charge density may be found from Boltzmann distribution. Noting

that a charge e placed in the electrostatic potential � has a potential energy e�, we may assume

that the spatial distribution of the particle density is governed by

n(r) = n0 exp(�e�=T ); (1.2)

where T (in erg) is the temperature of the particle species of concern, and n0 is the particle density

in the absence of the excess charge. Consider an excess, point charge q placed in a plasma. Then,

�free = q�(r) where �(r) is the three dimensional delta function. Noting E = �r�, we may rewrite

Eq. (1.1) as

r2� = �4�q�(r) + 4�n0e�exp

�e�

Te

�� exp

��e�Ti

��; (1.3)

where Te; Ti are the electron and ion temperatures, respectively. For simplicity, we have assumed

singly-ionized ions. As it is, Eq. (1.3) is a nonlinear equation for the potential �(r). If e�� Te; Ti;

however, Eq. (1.3) can be linearized as

�r2 � k2D

�� = �4�q�(r); (1.4)

where kD is de�ned by

k2D = 4�n0e2

�1

Te+1

Ti

�= k2De + k

2Di: (1.5)

kD is called the inverse Debye length (or Debye wavenumber). Eq. (1.3) can be readily solved with

a solution

�(r) =q

rexp (�kDr) : (1.6)

This is valid in the region where r & r0, with r0 de�ned by

r0 ' ejqj=min(Te; Ti): (1.7)

4

At r ' r0, the potential energy becomes of the same order as the thermal energy. The linearized

solution for the potential holds only for r � r0.

A collection of electrons and ions is appropriately de�ned as a plasma when the characteristic

length r0 is much smaller than the Debye length rD = 1=kD. We may restate this requirement

as follows: A �uid of charged particles is de�ned as a plasma when there are a su¢ ciently large

number of particles in the Debye sphere, i.e.,

n0r3D � 1: (1.8)

When this condition is well satis�ed, an ionized gas behaves collectively as a dielectric. In usual

laboratory plasmas, the condition is well satis�ed. Exceptions are extremely high density, low-

temperature plasmas such as solid state plasma composed of electron-hole pairs. In dusty plasmas,

negative ions charged with a large number of electrons exist and often exhibit phenomena due to

strong coupling such as �crystal formation�and plasma condensate.

The solution for the potential, Eq. (1.6), which is known as the Debye (or Yukawa) potential,

clearly indicates the screening action of a plasma. At a distance larger than the Debye length, the

potential decays much more rapidly than the usual Coulomb potential. This also indicates that the

e¤ective dielectric constant of a plasma for static (no time variation involved) electric �eld is given

by

�(! = 0) = 1 +k2Dk2; (B = 0; no magnetic �eld); (1.9)

as immediately follows from the Fourier transform of the potential. The frequency ! does not have

to be exactly zero for Eq. (1.9) to be valid. As long as both ions and electrons are able to establish

Boltzmann distribution in a potential �; Eq. (1.9) holds. Since density adjustment occurs with a

time scale of the order of the transit time over one wavelength, � ' 1=kvT , where vT is the thermal

velocity, and the ion thermal velocity is much smaller than the electron thermal velocity, we obtain

the following condition for the validity range of Eq. (1.9),

! � kvTi (� kvTe)

where vTi(e) is the ion (electron) thermal velocity, vTi =p2Ti=M; vTe =

p2Te=m:

Most laboratory plasmas are produced in relatively strong magnetic �elds for the purpose of

con�ning charged particles. A plasma placed in a magnetic �eld is a typical anisotropic medium for

electromagnetic �elds, and the anisotropy manifests itself particularly in the dielectric properties.

The dielectric constant, Eq. (1.9), is still applicable for electric �elds parallel to the magnetic �eld.

5

Figure 1.1: Cycloid ion trajectory when a dc electric �eld E in y direction is suddenly applied

perpendicular to the magnetic �eld B in z direction: �yi indicates the average shift of the ion from

the original location.

For electric �elds purely perpendicular to the magnetic �elds, the expression is signi�cantly altered

and given by

�?(! = 0) = 1 +Xs

!2ps2s

; (E ? B) (1.10)

where !ps is the plasma frequency de�ned by

!ps =

�4�n0se

2s

ms

�1=2; (1.11)

and s is the cyclotron frequency

s =esB

msc; (charge sensitive). (1.12)

In conventional plasmas of thermonuclear interest, the ion contribution dominates since

�?(! = 0) '!2pi2i

=M

m

!2pe2e

� max

!2pe2e

; 1

!; (1.13)

where M is the ion mass and m is the electron mass.

The cross-�eld dielectric constant, Eq. (1.13), may be derived from the following intuitive

argument. Let an electron and ion be at the origin x = 0; y = 0 in a plane perpendicular to a

6

magnetic �eld B0 in the z direction, as shown in Fig. ??. When steady electric �eld Ey is suddenly

applied, both electron and ion start moving along cycloids. The motion of the ion is described by

Mdv

dt= e

�E(t) +

1

cv �B

�; v(0) = 0 (initial condition) (1.14)

where

E(t) = EH(t)ey;

with H(t) the unit step function, H(t) = 0; t < 0; H(t) = 1; t > 0. The solution is

8<: vx = VE(1� cosit)

vy = VE sinit(1.15)

where

VE = cE

B; or in vectorial form, VE = c

E�BB2

; (1.16)

is the so-called E�B drift velocity. Further integration of Eq. (1.15) with initial condition

x(0) = y(0) = 0 yields the location of the ion,8><>:x(t) = VEt�

VEisinit;

y(t) =VEi(1� cosit);

(1.17)

which describe a cycloid characterized by a circle having a radius VE=i. The average displacement

of the ion in the direction of the cross �eld electric �eld Ey is thus

4yi =VEi: (1.18)

The calculation can be repeated for an electron with the result

4ye = �VEjej

: (1.19)

Then, the charge separation yields the electric dipole moment density

Py = 4�n0e(4yi �4ye) = !2pi2i

+!2pe2e

!Ey; (1.20)

and the corresponding dielectric constant is given by

�?(! = 0) = 1 +!2pi2i

+!2pe2e

: (1.21)

(A more rigorous derivation will be given later when we discuss the full dielectric tensor of a

magnetized plasma.) As mentioned earlier, in conventional laboratory plasmas, !2pi=2i is large.

7

Consequently, an electric �eld externally applied to a plasma in the direction perpendicular to the

magnetic �eld is signi�cantly reduced in the plasma.

In toroidal axisymmetric devices such as tokamaks, the permittivity is modi�ed as

�? 'q2

"2!2pi2i

=!2pi2ip

; (1.22)

where ip is the ion cyclotron frequency in the poloidal magnetic �eld Bp; ip = eBp=Mc; " = r=R

is the inverse aspect ratio and q is the safety factor,

q =rB�RBp

: (1.23)

The enhancement of �? in toroidal plasmas is due to the fact that the E �B drift due to a radial

electric �eld Er is in the toroidal direction with a magnitude

V� ' cErBp

:

1.3 Diamagnetic Properties

Both electrons and ions placed in a magnetic �eld undergo cyclotron motion. The direction of

cyclotron motion is such that the secondary magnetic �eld produced by cyclotron motion itself

opposes the primary (external) magnetic �eld. Thus, a plasma tends to be diamagnetic. (In a

tokamak, the external toroidal magnetic �eld is enhanced by the self-transformer action of the

plasma, unless the poloidal beta, �p = 8�p=B2� , exceeds unity. (The poloidal beta will be explained

in Chapter 2.) However, the resultant, total magnetic �eld, toroidal plus poloidal, should be

regarded as the e¤ective external �eld. The statement that a plasma is a diamagnetic medium still

holds.)

Let us consider a charged particle having a charge e, a mass m, and a perpendicular energy

12mv

2?. The cyclotron radius of the particle is

rc = v?=; or vectorially, rc =� v2

; (1.24)

with =e

mcB; and the corresponding magnetic dipole moment (per particle) is

� =1

2c

Zr� JdV

=e

2c

Zr� v?� (r � rc) dV

= �12mv2?

B

B: (1.25)

8

The minus sign evidently indicates that � is opposite to the magnetic �eld B (diamagnetism). In

Section 1.5, we will see that the magnetic moment � is a constant of motion for slowly varying

electromagnetic �elds. The magnetic dipole moment density is, by de�nition,

M = n0� = �12mv

2?n0

B0eB; (1.26)

where n0 is the particle density, and eB is the unit vector in the direction of B. Recalling the

Maxwell�s equation (Ampere�s law),

r�B = 4�

c(Jc + cr�M); (1.27)

with Jc being the conduction current density, and summing the contributions from electrons and

ions to the magnetic dipole density in Eq. (1.26), we �nd

r���1 +

�

2

�B

�=4�

cJc; (1.28)

where

� =8�n0(Te + Ti)

B2; (1.29)

is the ratio between the plasma thermal pressure and the magnetic pressure. (Note we have replaced

the energy 12mv

2? by the temperature T for both electrons and ions.) The presence of a plasma

thus reduces the magnetic �eld for a given external current.

The � factor is an important parameter both in plasma physics and in fusion reactor engineering.

In MHD stability analyses, one of the major objectives is to �nd to what extent � can be increased

before a plasma becomes unstable. Much theoretical and computational e¤ort has been devoted

to this important problem. It appears that the � factor in tokamaks is limited by the MHD

instabilities, such as the ballooning and internal kink instabilities. In fusion reactor engineering, a

higher � indicates a more economical fusion reactor, for fusion power output is proportional to �.

The maximum average � so far achieving in tokamaks is less than 10%, which may be su¢ cient for

deuterium-tritium based fusion reactors. However, for other fusion fuels having smaller reaction

cross sections (e.g. D-D, D-3He), � values of tens of % will be required. (The reaction D + 3He !

p + � + 18.3 MeV is aneutronic and thus environmentally attractive. However, the reaction cross-

section is roughly 10 times smaller than that of D-T reaction, and reactors based on D-3He will

have to be operated at higher temperature and density than those in D-T reactors. Furthermore,

3He is not abundant on the earth.)

9

The term cr�M in Eq. (1.27) may be de�ned as the magnetization current density,

JM = �cr�� p

B2B�; (1.30)

where p = n0(Te + Ti) is the thermal pressure of the plasma. The magnetic �eld B is in general

nonuniform because of its gradient and curvature. Then,

JM = �cr�� p

B2B�

= cB�rpB2

� 2cpB3B�rB � c p

B2r�B: (1.31)

The �rst term is the diamagnetic current due to plasma pressure gradient. In a low � plasma, the

last term may be ignored. The second term is due to the nonuniformity of the magnetic �eld, and

in a low � plasma, it cancels the guiding center current due to the magnetic drift,

Jc? =cp?B3

B�rB +cpkB4B� (B �rB) '2cp

B3B�rB; (1.32)

which will be discussed in Section 1.4. Here, p?(pk) is the plasma pressure perpendicular (parallel)

to the magnetic �eld, and in the case of isotropic pressure, p = pk = p?: Also, it is noted that in a

low � plasma, the gradient and curvature of the magnetic �eld are almost identical,

B �rB ' BrB: (1.33)

This can be seen from the equilibrium condition,

rp = 1

cJ�B: (1.34)

Substituting J =c

4�r�B, the force balance condition reduces to

r�p+

B2

8�

�=1

4�B �rB: (1.35)

If the plasma pressure is small compared with the magnetic pressure, B �rB ' BrB follows.

Therefore, the dominant cross-�eld (perpendicular to the magnetic �eld) current in a low �

plasma with an isotropic pressure is the diamagnetic current due to the pressure gradient,

J? = cB�rpB2

; (1.36)

which alternatively follows more directly from the force balance equation in equilibrium

rp = 1

cJ�B: (1.37)

10

(To obtain Eq. (1.36) from Eq. (1.37), multiply B� (vectorially) on both sides of Eq. (1.37).)

The fact that the diamagnetic current given in Eq. (1.36) is the only current in a low � plasma

with an isotropic pressure does not mean that a simple torus with a toroidal �eld alone can con�ne

such a plasma. We note that the divergence of the magnetization current, Eq. (1.30), identically

vanishes because r �r�A � 0 for an arbitrary vector A. Therefore, the magnetization current

does not contribute to charge separation. However, the divergence of the magnetic drift current,

Eq. (1.32), is not zero,

r � Jc? =2c

B3rp � (B�rB) 6= 0;

and charge separation will take place unless the resultant excess charges are short-circuited along the

magnetic �eld. This condition (the absence of charge accumulation) can be expressed by r �J = 0,

or

r? � Jc? +rk � Jk = 0; (1.38)

which is the equilibrium form of the general charge conservation law,

@�

@t+r � J = 0:

The concept of the rotational transform, which will be elaborated later, is just to provide the

required short-circuiting mechanism by adding an appropriate poloidal magnetic �eld to the simple

toroidal magnetic �eld.

The diamagnetic drift velocities of electrons (charge �e) and ions (charge e) are

VeDM =

cTeeBr(ln pe)� eB; (1.39)

ViDM = �cTi

eBr(ln pi)� eB; (1.40)

where pe and pi are the electron and ion thermal pressure, respectively. These drift velocities are

not associated with translational motion of charges, but is due to the nonuniformity in the (dia-

)magnetization current associated with Larmor motion of charges. In order to see the origin of

the diamagnetic current more clearly, let us compute the velocity moment of a local Maxwellian

distribution function with zero mean velocity,

F (v2; r) = n0(r)fM�v2; T (r)

�; (1.41)

where

fM�v2; T (r)

�=

�m

2�T (r)

�3=2exp

�� mv2

2T (r)

�; (1.42)

11

is the Maxwellian velocity distribution function with a space dependent temperature. We assume

a local rectangular coordinate system in which a uniform magnetic �eld is in z direction, B = Bez,

and density and temperature nonuniformities are in y direction, as illustrated in Fig. 1.2. A charged

particle with a charge e and mass m follows the trajectory determined by8>>>><>>>>:dv

dt=

e

mcv �B;

v =dr

dt;

(1.43)

where r(t) and v(t) are instantaneous particle coordinate and velocity. From these equations, we

�ndd

dt

�v � e

mcr�B

�= 0 (1.44)

or

v � e

mcr�B = const. (1.45)

although both velocity and particle position are evidently time dependent. This conservative quan-

tity can be identi�ed as the canonical momentum,

p = mv � e

cA:

Multiplying Eq. (1.45) by B� (vectorially), we also �nd (recall that a uniform magnetic �eld has

been assumed)

r� eB � v

= const. (1.46)

where again eB is the unit vector along B, and is the cyclotron frequency. The y component of

the above equation reads

y(t)� vx(t)

= const. (1.47)

Therefore, the density and temperature may be arbitrary functions of this constant of motion,

n(r) = n�y � vx

�;

T (r) = T�y � vx

�:

If the characteristic length, over which the density and temperature vary appreciably, is much larger

than the cyclotron radius vx=, the velocity distribution function may be expanded as

F�v2; y � vx

�' F (v2; y)� vx

dF

dy: (1.48)

12

Figure 1.2: Diamagnetic current is created when a density gradient across the megnetic �eld exists.

The varying thickness of ion and electron gyro circles indicates local density. The diamagnetic drift

velocities of ions and electrons are in opposite directions but the currents are in the same direction.

Therefore, even though the velocity distribution function is locally Maxwellian, and thus contains

no guiding center drifts, the velocity moment of the nonuniform distribution function yields a �nite

particle �ux,

�x =

ZvxF (v

2; r)d3v

= � 1

d

dy

Zn0v

2xfM (v

2)d3v

= � 1

m

dp

dy: (1.49)

The macroscopic (or �uid) �ow velocity associated with this �ux is identical to the diamagnetic

drift velocities calculated earlier.

1.4 Guiding Center Drift in a Nonuniform Magnetic Field

We have already seen that a static (or low frequency, ! � i the ion cyclotron frequency) electric

�eld perpendicular to the magnetic �eld creates the E�B drift of both electrons and ions,

VE = cE�BB2

: (1.50)

13

This drift velocity is that of guiding centers of electron and ions, that is, translational motion of

charged particles is involved in the E�B drift. In this section, some other mechanisms that cause

cross �eld guiding center drifts of charged particles will be discussed. Cross �eld (perpendicular to

B) drifts of charged particles are important from con�nement point of view, because they lead to

deviation of particle motion from a given magnetic �eld line (or from a magnetic surface).

First, we introduce an e¤ective gravitational acceleration g in the equation of motion for a

particle with charge e and mass m;

dv

dt=

e

mcv �B+ g: (1.51)

g is assumed to be perpendicular to the magnetic �eld. If g is time independent, the drift velocity

can be immediately found by taking time average of the above equation over one cyclotron period,

V = mcg �BeB2

=1

Bg �B: (1.52)

The case of E�B drift can be recovered by choosing g = e

mE?. The charge e and mass m cancel

in Eq. (1.49), and the E�B drift is thus independent of particle species.

The e¤ective g �eld can simulate another important case, namely, the curvature of the magnetic

�eld. The centrifugal force to act on a particle moving along a curved magnetic �eld with a velocity

vk is

F =mv2kR2

R;

where R is the local curvature radius. Then the e¤ective g may be de�ned by

g =v2kR2R; (1.53)

and the corresponding guiding center drift velocity is

Vc = mcv2kR�BeB2R2

= mcv2k1

eB4B� (B �r B); (1.54)

where the de�nition for the curvature

R=R2 = �B �r B=B2; (1.55)

has been substituted. Note that the curvature drift is charge dependent. Ions and electrons drift

in opposite directions, thus creating a guiding center current. This is schematically illustrated in

Fig. 1.3.

14

Figure 1.3: Ion and electron magnetic drifts due to a nonuniformity in the magnetic �eld intensity.

In a nonuniform magnetic �eld, gyro motion is non-circular.

A similar guiding center drift is caused by the gradient in the magnetic �eld intensity. To

calculate the drift velocity due to the gradient of B (r), it will be su¢ cient to �nd the average

Lorentz force acting on the particle,e

chv �Bi:

Here h� � �i indicates averaging over one cyclotron period. As before, we assume B = B ez, but its

magnitude B now depends on y; B = B(y). As we did earlier for the diamagnetic drifts, we may

expand B(y) as

B(y) = B(y0)�vx

@B

@y; (1.56)

where y0 is the y�coordinate of the guiding center. Therefore, the average force is in y direction

and given by

Fy = � e

c

v2x� @B@y

= � e

c

v2?2

@B

@y: (1.57)

Consequent e¤ective g is

g = �v2?2

1

B

@B

@yey; (1.58)

15

which may be generalized as

g = �v2?2

rBB

: (1.59)

Then, the drift velocity due to the gradient of the magnetic �eld becomes

Vg =mv2?2

cB�rBeB3

: (1.60)

Combining Eqs. (1.54) and (1.60), the total guiding center drift velocity caused by the nonunifor-

mity in the magnetic �eld (curvature and gradient) becomes

VD = mcv2k1

eB4B� (B �rB) +mcv

2?2

1

eB3B�rB: (1.61)

In a low � plasma, the diamagnetic current does not strongly modify the magnetic �eld. There-

fore, in a low � plasma, we may assume r�B '0: Consequently, the curvature and gradient

become identical, B �rB ' BrB; and the guiding center drift may be approximated by

VD ' mc

�v2k +

v2?2

�1

eB3B�rB; (low �): (1.62)

Averaging over a Maxwellian velocity distribution function for electrons and ions, we �nd the

average (or thermal) drift velocity of each species,

VDe ' �2cTeeB3

B�rB; VDi '2cTieB3

B�rB: (1.63)

Since electrons and ions drift in opposite directions, the curvature-gradient of a magnetic �eld

creates a current associated with the guiding center drifts. However, as discussed in the preceding

section, this current is cancelled by a part of the magnetization current. (See Eq. (1.31).) The

cancellation is not surprising, because a magnetic �eld does not do any work on charged particles,

and thus is unable to modify the velocity distribution functions.

We now discuss a guiding center drift velocity of ions due to a perpendicular electric �eld which

slowly varies with time. �Slow�here means that the characteristic frequency ! of the time varying

electric �eld is much smaller than the ion cyclotron frequency, ! � i. Earlier, we saw that such a

static electric �eld perpendicular to a magnetic �eld creates E�B drift of both ions and electrons.

Since both electrons and ions move in phase in E�B motion, the E�B drift does not create a

current. However, if the electric �eld is varying with time, a current in the same direction as the

electric �eld can be induced due to the di¤erence in ion and electron masses. In Sec. 1.2, we saw

that the ion displacement in the direction of E? is much larger than that of electrons, by a factor

M=m, the ion to electron mass ratio.

16

The equation of motion for the ion is

dv

dt=

e

M

�E?e

�i!t +v

c�B

�; (1.64)

where M is the ion mass. Assuming B = Bez; E? = E?ex, we write down each component as

dvxdt

=e

M

�E?e

�i!t +1

cvyB

�; (1.65)

dvydt

= � eB

cMvx: (1.66)

Eliminating vy, we obtain

d2vxdt2

=e

M

��i!E?e�i!t �

eB2

Mc2vx

�: (1.67)

Since ! < i; d2vx=dt2 can be neglected compared with 2i vx. We thus obtain the time averaged

ion drift velocity,

Vx =e

M2i

@E?@t

: (1.68)

This may be generalized as

Vp =e

M2i

@E?@t

; (! � i): (1.69)

This drift velocity is called polarization drift and has the same physical origin as the cross-�eld

dielectric constant discussed in Section 1.2, since the displacement of ions in the direction of the

cross-�eld electric �eld is given by

4r =ZVpdt =

e

M2iE?:

(Electron displacement is smaller by a factor of the electron to ion mass ratio.) The polarization

drift thus creates a current

Jp = n0eVp =n0e

2

M2i

@E?@t

: (1.70)

Therefore, the Maxwell�s equation

r�B = 4�

cJp +

1

c

@E?@t

=

1 +

!2pi2i

!1

c

@E?@t

;

readily de�nes an e¤ective dielectric constant

�? ' 1 +!2pi2i

;

for a slowly (! � i) time varying electric �eld perpendicular to the magnetic �eld.

17

It is noted that the ion polarization drift in nonuniform plasma and/or electric �eld should be

given in the form of substantive derivative,

Vp =e

M2i

dE?dt

=e

M2i

�@

@t+Vi �r

�E?; (1.71)

where Vi is the total ion �uid velocity, including the E � B drift, magnetic curvature-gradient

drift, and diamagnetic drift velocities. The ion polarization drift in this form will play important

roles in analyzing drift type, low frequency modes in toroidal devices.

1.5 Formal Derivation of the Drifts

In this Section, a general formulation is given for the various guiding center drift velocities we have

encountered. The practice to be presented not only summarizes the drift velocities, but also yields

higher order drifts. We start o¤ with the equation of motion for a charged particle with mass m

and charge e;

mdv

dt= e

�E+

1

cv �B

�; (1.72)

where the magnetic �eld B is assumed to be nonuniform B = B(r): Expanding B to the �rst order

of the Larmor radius,

B(rg + rc) ' B(rg) + rc �rB (1.73)

where rg is the guiding center coordinate, and rc is the particle coordinate measured from rg, we

obtain an equation for the time averaged velocity u;

mdu

dt= e

�E+

1

cu? �B+

1

chv � rci �rB

�; (1.74)

where h� � �i indicates averaging over one cyclotron period. Noting

hv � rci = �1

2

v2?

B

B; (1.75)

and using the low � approximation, B �rB =BrB; we may rewrite Eq. (1.74) as

mdu

dt= e

E+

1

cu? �B�

1

e

12mv

2?

BrB

!: (1.76)

Multiplying B� (vectorially) and denoting the drift velocity along B by vk, we �nd for the total

guiding center velocity,

u = cE�BB2

+B

B2��mc

e

du

dt+c

e�rB

�+ vk; (1.77)

18

where � = 12mv

2?=B is the magnetic moment de�ned earlier. The time derivative d=dt must be

evaluated along with the guiding center motion,

d

dt=

@

@t+ u �r:

In the lowest order approximation, we have u = VE +vk where VE is the E�B drift velocity.

Then,

du

dt' @VE

@t+�VE + vk

�� r�VE + vk

�+@vk@t

' @VE@t

+�VE + vk

�� rVE +VE � rvk +

v2kB2B � rB+vk

@eB@t

+ eB@vk@t

; (1.78)

and the cross-�eld guiding center drift velocity becomes

u? = VE +mc

e

�v2?2+ v2k

�B�rBB2

+mc

e

B

B��@VE@t

+VE � rVE +vkB(B � rVE +VE � rB) + vk

@

@t

�B

B

��: (1.79)

Most of the terms have already been discussed earlier. The �rst term in the RHS is the E�B

drift, the second term is the gradient and curvature drift, and the third term,

mc

e

B

B2� @VE

@t=

e

m2@E?@t

;

is the polarization drift. There are four new terms, however. The term containing VE �rVE is

the nonlinear polarization drift term. It is due to the convective derivative u �r. The last three

terms are all proportional to vk, and they are usually negligible when averaged over the velocity

with Maxwellian weighting.

The equation for the parallel velocity vk can be found from Eq. (1.76). Taking the scalar

product between the magnetic �eld B and Eq. (1.73), and noting

dvkdt

' du

dt� eB +VE �

deBdt

; (1.80)

we �nddvkdt

=e

mEk �

v2?2B2

B �rB + VEB� dBdt; (1.81)

where it is understood thatd

dt' @

@t+ (VE + vk) � r:

The �rst term in the RHS of Eq. (1.81) is the acceleration by the parallel electric �eld Ek. The

second term indicates a force due to the nonuniformity of the magnetic �eld. In a static, nonuniform

19

magnetic �eld, a charged particle approaching the region of higher magnetic �eld experiences a

repulsive force in the direction of the magnetic �eld, and unless the particle has a su¢ cient initial

parallel velocity, it will be re�ected back. The mirror device exploits this property to con�ne a

plasma. In a time varying magnetic �eld superposed on a static magnetic �eld, the nonuniformity

in the wave magnetic �eld causes energy exchange with charged particles. Plasma heating based

on this process is called magnetic pumping. The last term in Eq. (1.78) is of higher order in 1=B

and can be ignored under usual circumstances.

Finally, we note that in a low � plasma (B �rB ' BrB), the magnetic drift velocity

mc

eB3

�v2?2+ v2k

�B�rB; (1.82)

can alternatively be written in terms of the parallel velocity vk alone as

mcvkeB

r��vkB

B

�: (1.83)

The identity between these expressions can be established readily through the following manipula-

tion,

r��vkB

B

�= rvk �

B

B� vk

1

B2rB �B: (1.84)

rvk = rr2

m(E � �B) = � �

mvkrB; (1.85)

where E = 12mv

2 is the total kinetic energy and � is the magnetic moment, both constants of

motion. Eq. (1.85) is convenient for analyzing motion of trapped particle orbits (such as �banana�

in tokamaks and stellarators) as we will see in the following section.

1.6 Some Constants of Motion

If collisions among charged particles are rare, as in a high temperature plasma, several useful

constants of motion can be identi�ed. Finding these constants facilitates a kinetic description of

charged particles, since the particle distribution function in an equilibrium should be a function of

these constants of motion.

We assume that there are no static electric �eld along the magnetic �eld. This assumption is

required to avoid runaway problems of electrons. The equation of motion for a charged particle is

thendv

dt=

e

m

�E? +

1

cv �B

�: (1.86)

20

But the cross-�eld electric �eld will simply cause E�B drift, and if we ride on the frame moving

with this velocity, the electric �eld disappears. Then, the equations to be analyzed are

dv

dt=

e

mcv �B ; v =

dr

dt: (1.87)

Since a magnetic �eld does not do any work on the charged particle, the energy

1

2mv2 =

m

2

�v2? + v

2k

�; (1.88)

is a constant of motion. Here, v? and vk are velocity components perpendicular and parallel to the

magnetic �eld.

Another important constant of motion is the magnetic (dipole) moment,

� =12mv

2?

B: (1.89)

This follows from the parallel equation of motion in Eq. (1.78). When E = 0, it reduces to

mdvkdt

= ��dBds; (1.90)

where s is the coordinate along the magnetic �eld. Noting vk = ds=dt, we obtain

mvkdvkds

= ��dBds;

ord

ds

�mv2

2� mv2?

2

�= ��dB

ds: (1.91)

However, the total energy 12mv

2 is constant. Then,

d

dsln

12mv

2?

B

!= 0; (1.92)

which indicates that � = 12mv

2?=B is a constant of motion. Although we have derived the invariant

for the case of a steady, nonuniform magnetic �eld, it also holds for a slowly time varying magnetic

�eld. This has an important implication for plasma heating by adiabatic compression. By slowly

(! � i) increasing the magnetic �eld, particles further acquire perpendicular energy. The particle

density also increases, and the thermal energy density (n0T ) can be signi�cantly increased through

adiabatic compression. Of course, the compression must be done fast enough for the process to be

adiabatic, that is, compression must be achieved well before the loss due to thermal and/or particle

di¤usion becomes signi�cant.

21

The invariance of the magnetic moment indicates that the magnetic �ux enclosed by the particle

remains constant, since the �ux enclosed by a Larmor circle is given by

' �r2cB /v2?B;

where rc = v?= is the Larmor radius.

The concept of plasma con�nement by magnetic mirrors makes full use of the two constants of

motion, 12mv2 and �. The magnetic �eld intensity in a mirror device is strongly nonuniform having

maximum intensities at the mirror throats. (Modern tandem mirrors have extremely complicated

�eld con�gurations in order to improve stability against MHD instabilities, and also to reduce

particle loss through mirror throats. However, the basic concept remains essentially unchanged.)

The two constants of motion require

1

2mv2(0) =

1

2mv2(s); (1.93)

�(0) = �(s); (1.94)

where again s is the coordinate along the magnetic �eld, and s = 0 is a reference point which is

chosen at the �eld minimum. Solving for v2k(s), we �nd

v2k(s) = v2k(0)�2

m[B(s)�B(0)]�: (1.95)

Therefore, the particle cannot exist in the region

B(s)�B(0) >mv2k(0)

2�; (1.96)

and it must be re�ected back at the turning point determined from

B(s) = B(0) +mv2k(0)

2�: (1.97)

If the maximum and minimum intensities of the magnetic �eld are denoted by Bmax and Bmin ,

we see that particles satisfying the condition

v2kv2?

<Bmax �Bmin

Bmin; (1.98)

are trapped by the mirror �eld. Since particles not satisfying the above condition are inevitably lost

through the mirror throats, plasma particles in mirror machines usually do not establish Maxwellian

distribution functions, and plasmas con�ned in mirror devices are susceptible to rapidly growing

22

plasma instabilities, such as the loss-cone mode. One method to close the mirror throat and reduce

the axial loss is to implement electrostatic con�nement by creating the so-called thermal barrier.

Similar particle trapping can occur in toroidal devices as well. In tokamaks, for example,

particles follow (in the lowest order approximation) a helical magnetic �eld, and thus alternately

experience strong (in the inner major radius region) and weak (in the outer major radius region)

magnetic �elds. The presence of trapped particles can cause so-called trapped particle instabilities.

When collisions are infrequent, trapped particles are in a sense isolated from the rest of particles,

and they are not able to establish Boltzmann distributions in an electrostatic potential, even for

relatively low frequency perturbations. Since the average velocity of trapped particles along the

magnetic �eld is small (though not quite zero), short circuiting of charge accumulation becomes

ine¤ective for trapped particles, and �ute type instabilities may occur. Also, the average drift

motion of trapped particles is quite di¤erent from that of untrapped (or transit) particles. The

following Section describes particle motion in the tokamak magnetic geometry. Knowledge of drift

motion in a given magnetic con�guration allows us to estimate particle and/or thermal di¤usivities.

1.7 Magnetic Geometry in Cuurent Carrying Toroidal Plasma

(Tokamak)

In this section, plasma equilibrium in an axisymmetric, current carrying toroidal plasma is ana-

lyzed. Tokamaks, Reversed Filed Pinch (RFP), and spheromaks (or Compact Torus) belong to this

category. Toroidal plasma equilibrium cannot be achieved by simply bending a straight current

carrying plasma con�ned by a solenoidal magnetic �eld. Current carrying toroidal plasma is subject

to a radially expanding magnetic force and also the ballooning force due to plasma pressure and

equilibrium requires an additional magnetic �eld (vertical magnetic �eld).

In tokamaks, plasma equilibrium,

rp = 1

cJ�B;

is realized by combination of three essential components of magnetic �elds, the toroidal �eld B�,

poloidal �eld B�; and vertical �eld B?: The toroidal magnetic �eld is to maintain the Larmor

radius of ions at a su¢ ciently small level so that ions do not hit the wall of vacuum chamber. The

toroidal �eld alone is unable to con�ne a plasma in toroidal devices because the magnetic surfaces

of purely toroidal �eld are not closed and intersect with the vacuum chamber. (Magnetic surfaces

of purely toroidal magnetic �eld are described by R = constant, where R is the radial distance from

23

Figure 1.4: In toroidal con�nement devices, the poloidal magnetic �eld B� is required to form closed

magnetic surfaces and rotational transform. In tokamaks, B� is created by the toroidal current,

while in stellarators, it is applied externally.

the axis.) Ions and electrons drift vertically in opposite directions along the magnetic surface and

are lost. The function of the poloidal magnetic �eld B� is to form closed magnetic surfaces of the

total �eld B� +B�. The vertical magnetic �eld B? is required to counterbalance various radially

outward forces to act on a current carrying high pressure plasma as will be shown in Chapter 2.

In the geometry shown Fig. 1.4, the toroidal magnetic �eld is approximately given by

B� 'B0

1 +r

Rcos �

' B0

�1� r

Rcos �

�; (1.99)

where B0 is the �eld at the center r = 0 and R is the major radius. In tokamaks, the plasma pressure

is negligible compared with the magnetic pressure and the toroidal �eld may be approximated by

the vacuum �eld. The gradient of the toroidal �eld is

rB� ' �B0R(cos �er � sin �e�) = �

B0ReR; (1.100)

which coincides with the curvature of the �eld,

� =1

B2�B� � rB� = �

B0ReR: (1.101)

Note that@

@�e� = �eR:

24

The poloidal magnetic �eld B� depends on the radial distribution of the toroidal current density

J�(r) and plasma pressure p(r): In the lowest order, the following form may be assumed,

B�(r; �) = B�0(r) (1 + � cos �) ; (1.102)

where the factor � is of order r=R and depends on the plasma pressure and internal inductance of

the discharge.

Figure 1.5 illustrates qualitatively how the magnetic �eld lines look like as the safety factor

q =rB�RB�

;

is varied (increased). In pure toroidal �eld without B� (q = 1) as shown in Fig. 1.5 (a), the

magnetic �eld lines are all circular centered about the axis. As q decreases (or the toroidal current

I� increases), the �eld lines become helical. If q is rational number, q = m=n where m and n

are integers, magnetic �elds lines trace on themselves. Fig 1.5 (b) shows the case q = 5=2: The

magnetic �eld line comes back to the original starting position after 5 rotations in the toroidal

direction accompanied by two rotations in the poloidal direction. If q is irrational, the �eld lines do

not trace on themselves and form tightly nested ergodic surfaces as shown in (c) for q =p5 and in

(d) for q =p2: In actual tokamaks, the safety factor q is a continuous function of the minor radius

r; and there exist many radial positions where q takes rational number. If q is rational, ergodic

magnetic surfaces are not formed and plasma con�nement is endangered.

­1.0­1.0

0.50.5

0.2

0.00.0­0.2

­0.5 ­0.5z

y x1.01.0

(a)

1.0 1.00.50.5

z

y x­0.2

­1.0­1.0­0.5­0.5

0.00.0

0.2

(b)

25

0.00.0z

y x

0.2­0.5 ­0.5

­1.0 ­1.0

­0.20.5 0.5

1.01.0

(c)

­0.5­0.5­1.0­1.0

­0.2

0.2

0.0 0.0z

y x0.50.51.01.0

(d)

Figure 1.5: Magnetic �eld lines in tokamak when q = (a) 1 (no toroidal current), (b) 5/2, (c)p5;

and (d)p2:

1.8 Collisional Transport

As is well known, the collision frequency among charged particles in a fully ionized plasma is

inversely proportional to T 3=2 with T being the kinetic temperature. For example, the average

collision frequency of electron-proton collisions for a Maxwellian electron distribution is given by

�ei =4p2� n0e

4

3pmT

3=2e

ln�; (1.103)

where m is the electron mass, Te (in erg) is the electron temperature, and ln� is the Coulomb

logarithm with � given by

� = 12�n0�3D: (1.104)

Note that � is essentially the number of particles in a Debye sphere which appeared in Section 1.1.

For typical laboratory plasmas, ln� ' 15 may be assumed. It is noted �ei in Eq. (1.103) is the

momentum transfer collision frequency, or the so-called 90� scattering collision frequency. In some

cases, the topology of charged particle trajectory can change signi�cantly at a scattering angle much

smaller than 90�: For example, the e¤ective detrapping collision frequency for trapped electrons

in a tokamak is approximately given by �eff ' �ei=�; where � = r=R is the inverse aspect ratio

with r being the minor radius and R the major radius in the toroidal geometry. The enhancement

26

in the e¤ective collision frequency is due to the fact that a scattering angle of order 4� 'p�

(rather than 4� ' 90�) is su¢ cient to change the status from trapped to untrapped electrons. In

a tokamak, particles having small parallel velocity, v2k < "v2?; are trapped in the region of smaller

toroidal magnetic �eld as will be discussed shortly.

In a high temperature plasma, the collision frequency is small but still �nite, and various trans-

port processes (e.g., di¤usion across a con�ning magnetic �eld) are ultimately determined by colli-

sional e¤ects. Often, however, experimentally-observed plasma con�nement times are much shorter

than that expected from collisional processes. It is believed that electrostatic or electromagnetic

instabilities are responsible for con�nement deterioration. This so-called anomalous transport was

noted as early as 1945 by Bohm, and the Bohm di¤usion time once prevailed as an insurmountable

limit in various devices.

Studies on plasma instabilities have been clearly motivated by attempts to explain various

anomalies in transport phenomena. Perhaps the most outstanding anomaly widely observed in

tokamaks is the anomalous electron thermal di¤usivity which is nearly two orders of magnitude

larger than the neoclassical prediction. A high temperature plasma con�ned by a magnetic �eld is

necessarily in thermal nonequilibrium with the environment even if plasma particles may acquire

Maxwellian velocity distribution in the velocity space. In coordinate space, a con�ned plasma

necessarily has nonuniformities in the density and temperature. Such nonuniformities can cause

various plasma instabilities, typi�ed by the drift instability, which will be discussed in Chapter 3.

In some experiments, positive aspects of anomalous transport phenomena are utilized. One

example is turbulent heating which plays key roles in theta-pinch and some toroidal experiments.

In turbulent heating, the collision frequency between electrons and ions is enhanced by deliberately

making a plasma unstable against short wavelength plasma waves. In Reversed Field Pinch (RFP),

the dynamo mechanism induced by plasma turbulence seems to be of fundamental importance in

reversing magnetic �eld and maintaining the equilibrium for a period much longer than the �eld

di¤usion time.

The classical di¤usion coe¢ cient across a straight magnetic �eld can be qualitatively estimated

from

D? = �r2c ; (1.105)

where � is the momentum transfer collision frequency and rc =pT=m= is the average Larmor

27

radius. Note that this expression is in the form

D? = (collision frequency)� (random walk distance)2; (1.106)

in which the random walk distance is taken to be the Larmor radius. For electrons,

De? = �eir

2ce; (1.107)

where �ei is the electron-ion collision frequency given in Eq. (1.103). It is noted that the collision

frequency in Eq. (1.105) should be a momentum transfer collision frequency between di¤erent

particle species, and collisions among like particles (e.g., electron-electron, ion-ion) should not

make signi�cant contributions to particle di¤usion. This is because collisions between like particles

do not involve a net momentum transfer, as can be seen from the following simple argument. The

vectorial form for the guiding center (negative of the Larmor radius) is

rg = cp�BqB2

; (1.108)

where p =mv is the particle momentum. The change in the guiding center position due to a change

in the momentum 4p is

4rg = c4p�BqB2

: (1.109)

After a collision between two particles, one particle loses a momentum, and the other gains, so that

the net momentum change is zero,

4p1 +4p2 = 0: (1.110)

Therefore, for a collision between like particles (e.g., electron-electron), there is no net change in

the particle position before and after the collision,

4rg1 +4rg2 = 0 (for like particle), (1.111)

which indicates there is no net di¤usion. For a collision between unlike particles (e.g., electron-ion),

it is evident that

4rge �4rgi = 0 (for unlike particles), (1.112)

which asserts that electrons and ions di¤use together.

The simple di¤usion coe¢ cient given by Eq. (1.105) is applicable only for straight magnetic

�elds without curvature and gradient. In a nonuniform magnetic �eld, the di¤usion coe¢ cient is

28

in general signi�cantly enhanced. In a tokamak, for example, the di¤usion coe¢ cient is enhanced

by a factor

1 + 2�?�kq2; (1.113)

where q = rB�=RB� is the safety factor against the kink instability with R=r being the aspect

ratio, and B� and B� the toroidal and poloidal magnetic �elds, respectively. (�? and �k are the

components of the plasma conductivity tensor in the direction perpendicular and parallel to the

magnetic �eld. For a hydrogen plasma, �k ' 2�?:) This enhancement factor is due to P�rsch and

Schlüter, and originates from an e¤ective increase in the random walk distance.

The appearance of P�rsch-Schlüter factor may be understood from the following qualitative

argument. A particle follows the helical magnetic �eld in a tokamak in the lowest order. However,

the curvature and gradient in the toroidal magnetic �eld causes guiding center drift (see Eq. (1.63)),

VD =2cT

eBRez; (1.114)

in the con�guration shown in Fig. 1.4. Therefore, the equation of motion for the guiding center is

dr

dt= VD + vk

B�Be�; (1.115)

where vk is the velocity component parallel to the total magnetic �eld, B�+B�. Note that vkB�=B is

the projection of vk on the poloidal plane. Also vk can be positive or negative depending on parallel

or antiparallel motion with respect to the magnetic �eld. Decomposing into � and r components,

we have

rd�

dt= VD cos � + vk

B�B; (1.116)

dr

dt= VD sin �: (1.117)

Since the magnitude of vk is of the order of the thermal velocity which is much larger than the drift

velocity, VD, we obtain1

r

dr

d�=VD sin �

vk

B

B�: (1.118)

Integrating with respect to �,

r = r0 � qRVDvkcos �; (1.119)

where r0 is the radius of the reference magnetic surface chosen, and q = r0B=RB� is the safety

factor on the magnetic surface. Equation (1.119) indicates that the shift from the magnetic surface

depends on the sign of vk. The maximum deviation from the magnetic surface is qRVD=vk when

29

cos� = 1, or � = 0. When the particle su¤ers a collision, vk changes its sign, and the shift calculated

above can be regarded as the random walk distance. Then, the e¤ective di¤usion coe¢ cient is

D? ' �(qR)2V 2Dv2T

= q2�r2c ;

�for � >

vTqR

�(1.120)

where vT = (T=m)1=2 is the thermal velocity.

In a fully-ionized plasma, �? and �k are determined by Coulomb scattering and �k=�? ' 2 for

a hydrogen plasma. Then the P�rsch-Schlüter factor may be approximated by 1 + q2.

The P�rsch-Schlüter di¤usion is valid when the plasma is su¢ ciently collisional. The criterion

for this condition is that the mean free path along the magnetic �eld is shorter than the system

size. In tokamaks, the helical magnetic �eld lines close on themselves (or reconnect) over a distance

2�qR along the �led line. Taking this quantity as the system size, we may restate the collisionality

condition as

� >vTqR

: (1.121)

(The factor 2� disappears because the wavenumber kk corresponding to the length 2�qR is 1=qR,

and thus the transit frequency is !T = kkvT = vT =qR.)

As the plasma temperature increases, the collisionality condition becomes violated. (Recall that

� / T�3=2.) When � < �3=2vT =qR, the plasma enters the collisionless (or banana) regime. The

appearance of the factor �3=2 is due to the condition that su¢ ciently large number of particles be

trapped in the local mirror �eld. As we saw in Section 1.6, particles can be trapped in a mirror

�eld ifv2kv2?

<4BB

; (1.122)

where in tokamaks, the di¤erence between the maximum and minimum magnetic �elds 4B is

approximately

4B ' 2�B:

Then, the fraction of trapped particles is approximatelyp�, which is not a small number even

though � itself may be small.

The important aspect of the trapped particles is that their e¤ective detrapping collision fre-

quency is enhanced by a factor 1=�; as brie�y explained earlier in this section. This is because

the scattering angle can be much smaller than 90� to cause topological change of trapped particle

orbits. (The Coulomb collision frequency in Eq. (1.103) is de�ned as the 90� scattering collision

frequency.) Note that the condition for particle trapping is

v2k < 2�v2?: (1.123)

30

This means that a loss in the perpendicular energy by an amount �v2? is su¢ cient to cause particle

detrapping.

To clarify this point, let us review the fundamental mechanism behind Coulomb collisions. Let

an electron having a velocity v collide with a proton at an impact parameter r. The loss in the

velocity in the original direction is

4v = e2

mr24t ' e2

mrv; (1.124)

where 4t = r=v is the approximate interaction interval. The order of magnitude estimate for the

collision cross section is thus

� ' �r2 = �e4

m2v2j4vj2 : (1.125)

For 90� scattering, j4vj = v, and we essentially recover the ordinary cross section for the Coulomb

collisions,

� ' �e2

(mv2)2; (1.126)

except for numerical factors of order unity. However, for smaller scattering angle, the cross section

becomes large. For trapped particles, j4vj2 ' �v2 is su¢ cient to cause topological change in the

trajectory. Then,

�Tr '1

��; (1.127)

and the e¤ective collision frequency becomes

�eff '1

��c; (1.128)

where �c is the ordinary 90� Coulomb collision frequency given in Eq. (1.103).

To analyze the motion of trapped particle, we employ the equation of motion,

mdv

dt= e

�E+

1

cv �B

�; (1.129)

where the electric �eld E has both toroidal and radial components,

E = Er + E�:

Averaging over the cyclotron period yields the magnetic drift velocity, Eq. (1.114), and averaging

over the bounce period of trapped particle yields a radially inward E�B drift,

Vr = �cE�B�

(radially inward), (1.130)

31

which is known as Galeev-Ware drift of trapped particles, and another E�B drift along the �

direction,

V� = �cErB�; (1.131)

which is often referred to as return plasma �ow. In general, a plasma in axisymmetric devices such

as tokamaks rotates in the toroidal (�) direction at a velocity

V� =c

enB�

d

dr(2pi + pe + en�) ;

independent of collisionality where pi = nTi; pe = nTe are the ion and electron pressure and � is the

electrostatic potential. The poloidal �ow velocity is proportional to the ion temperature gradient,

V� =c

eB

dTidr

:

If the toroidal electric �eld is ignored, the energy is conserved and the trajectory of the trapped

particle is described by Eqs. (1.117) and (1.118), except vk is now replaced by

vk ! �qv2k � �v

2? (1� cos �): (1.132)

Integratingdr

d�= � r

B

B�

VD sin �qv2k � �v

2? + ��

2? cos �

; (1.133)

we obtain

r = r0 � r0B

B�

2VD�v2?

qv2k � �v

2? + �v

2? cos �; (1.134)

where the sign � corresponds to the sign of vk. Mirror re�ection occurs at angles ��0 where �0 is

determined from

cos �0 = 1�1

�

v2kv2?

= 1� �; (1.135)

where

� =1

�

v2kv2?: (1.136)

Trapped particles bounce back and forth between these poloidal angles. Particles are trapped if

� < 2: Particles falling in the range � > 2 are untrapped (or circulating along the toroidal direction).

Fig. 1.6 shows particle trajectories projected on a poloidal plane (� = const.) for di¤erent values

of �: The dashed circle in each �gure indicates the magnetic surface.

1 + :1p0:5� 1 + cos �

32

­1.0 ­0.5 0.5 1.0

­1.0

­0.5

0.5

1.0

x

y

1 + :1p1� 1 + cos �

­1.0 ­0.5 0.5 1.0

­1.0

­0.5

0.5

1.0

x

y

1 + :1p2� 1 + cos �

­1.0 ­0.5 0.5 1.0

­1.0

­0.5

0.5

1.0

x

y

1 + :1p2:5� 1 + cos �

33

­1.0 ­0.5 0.5 1.0

­1.0

­0.5

0.5

1.0

x

y

Figure 1.6: Particle trajectories projected on a poloidal plane (� = const.) when, from top,

� = 0:5; 1, 2, and 2.5. Particles with � < 2 are trapped in the local mirror �eld.

The oscillation or bounce period can be found by integrating

rd�

dt=B�B

qv2k � �v

2? + �v

2? cos �;

or

T =4B

B�

r

vk

Z �0=2

0

d�q1� 2�(v?=vk)2 sin 2�

: (1.137)

For well-trapped particles, v2? � �v2k, the integral reduces to

T =4B

B�

r

vk

�02

Z 1

0

d�p1� �2

=p2�

qRp"vT

; (1.138)

where vT is the thermal velocity. Then, the bounce frequency !b may be de�ned by

!b =2�

T=

p2�vTqR

: (1.139)

The condition for existence of trapped particles is therefore given by demanding that the e¤ective

collision frequency �eff be less than the bounce frequency, since collisions will destroy bounce

motion. This yields � < �3=2vT =qR. The ratio

�� =�

�3=2vT =qR; (1.140)

is called the collisionality parameter. A collisionless tokamak is characterized by �� � 1. The

collisional P�rsch-Schlüter regime corresponds to

�3=2�� =�

vT =qR> 1: (1.141)

34

The intermediate regime

1 < �� <1

�3=2; (1.142)

is peculiar in that the di¤usivity is relatively insensitive to the collision frequency. For this reason,

this regime is called the plateau regime.

The bounce motion of trapped particles is not strictly along a given magnetic �eld line, but

deviated from the magnetic surface by the amount

�r = r0B

B�

2VDvkv2?

'p"

vTeB�=mc

=p"rc�; (1.143)

where rc� is the cyclotron radius with the poloidal magnetic �eld. The trajectory, which is qual-

itatively shown in Fig. 1.7, resembles a banana. (Also, the banana progressively moves along

the toroidal direction thus creating a toroidal current. The toroidal drift velocity of bananas will

be calculated later.) Taking this deviation as the random walk distance, and using the e¤ective

collision frequency, we obtain a rough estimate for the di¤usion coe¢ cient of trapped particles,

Dt 'p"�eff (�r)

2

=1

"3=2q2D?c; (1.144)

where the factorp� is the fraction of trapped particles and D?c is the classical di¤usivity given in

Eq. (1.105). The di¤usion coe¢ cient of untrapped particles is still given by the P�rsch-Schlüter

formula, and thus smaller than that of trapped particles by a factor �3=2.

Let us return to the problem of the toroidal drift of banana. In Section 1.3, we saw that the

magnetic drift due to the curvature and gradient can alternatively be written as

VD =mcvkeB

r��vkB

B

�: (1.145)

If we apply this to the coordinates (r; �; �) in Fig. 1.4, we obtain

V� = �mc

eBvk@vk@r

; (1.146)

where the axisymmetry in tokamaks, @=@� = 0; is noted. Since V� is the projection of the toroidal

drift V� on the poloidal plane, we �nd

V� = �mc

eB�vk@vk@r

: (1.147)

35

Denoting the coordinate along the trapped particle orbit by s, we then �nd

V�

Ids

vk= � mc

eB�

@

@r

Ivkds;

or

V� = �c

eB�

@Jk=@r

@E=@r (1.148)

where E = 12mv

2 is the total energy, Jk =Ivkds and use has been made of

@vk@E =

1

mvk: (1.149)

The quantity Jk =Ivkds is the action along the magnetic �eld and is another constant of motion.

The integralIis to be done over one complete bounce motion.

The toroidal current carried by trapped electrons is therefore given by

Jt = �nteV�

= "3=2c

B�

dpedr

; (1.150)

where nt 'p�n0 is the trapped electron density. The bootstrap current has its origin in the trapped

particle current calculated above. Trapped electrons collide with untrapped, transit electrons at

a rate �ee=� where �ee is the electron-electron collision frequency which is approximately equal to

�ei, the electron-ion collision frequency. Therefore, the momentum lost by the trapped electrons

�ee�Jt = �ee�

1=2 c

B�

dpedr

; (1.151)

should eventually be transferred to ions through collisions between transit electrons and ions, and

the bootstrap current JBS can be estimated from

�eiJBS = �ee�1=2 c

B�

dpedr

: (1.152)

Noting �ei ' �ee, we �nd

JBS = �1=2c

B�

dpedr

: (1.153)

For the bootstrap current to be signi�cant, a tokamak should have a su¢ ciently high temperature

and density, close to those required in fusion reactors so that a su¢ ciently large number of trapped

particles exist. The bootstrap current has been experimentally con�rmed in several tokamaks,

and in some cases, more than half of the toroidal current is due to the bootstrap current. The

36

steady state operation of a tokamak entirely based on the bootstrap current has been proposed

by Bickerton et al. If this scheme is viable, one of the fundamental engineering di¢ culties of the

tokamak fusion reactor will be eliminated.

However, it should be noted that operation with the bootstrap current alone does not allow

high �. From the Maxwell�s equation (Ampere�s law), we �nd

1

r

d

dr(rB�) =

4�

cJBS ' 4�

p"

B�

dP

dr; (1.154)

and thus the poloidal beta is of the order of

�P =8�p

B2�' O

�1p�

�: (1.155)

This is signi�cantly smaller than that imposed from equilibrium limitation and MHD instabilities.

(Both impose a poloidal � limit �p . 1=�.)

We now turn to the problem of the di¤usion coe¢ cient in the plateau regime 1 < �� < 1=�3=2.

Detailed analysis for the di¤usion process in this regime is complicated, but a qualitative derivation

goes as follows. In equilibrium, the velocity distribution function f(v; r) should obey

V � @f@r

= ��c(f � fM ); (1.156)

where fM is the Maxwellian distribution, and � is the momentum transfer collision frequency.

This simple approximation for the collision term is due to Krook and is su¢ cient for qualitative

analyses. Also, note that the e¤ects of con�ning magnetic �eld are included in V, the guiding

center drift. This approximation is called drift kinetic model, and signi�cantly simpli�es analyses

of low frequency (! � i) electromagnetic waves in a plasma as well. The drift (or gyro) kinetic

equation will be discussed in detail in later chapters.

Our purpose here is to solve Eq. (1.156) for f , that is, to �nd possible deviation from Maxwellian

due to toroidicity. As the drift velocity, V, we have vk, the velocity parallel to the magnetic �eld

which is approximately a constant of motion (recall that we are in the intermediate regime where

trapped particles are not important), and of course VD, the magnetic drift. Noting vk � VD, we

may linearize the equation as

vk1

qR

@f1@�

+ VD sin �@fM@r

= ��cf1; (1.157)

where f1 = f � fM . Note that1

qR

@

@�is the derivative along the magnetic �eld. Integrating over �,

we obtain

f1(vk; r) = �VD

@fM@r�

vk=qR�2+ �2c

��c sin � �

vkqR

cos �

�: (1.158)

37

Therefore, the radial particle �ux is given by

n0Vr =

Z 2�

0d�

Zd3vVD sin �f1

�vk; r

�(1.159)

= �12V 2D

dn0dr

Z�c�

vk=qR�2+ �2c

g�vk�dvk; (1.160)

where

g(vk) =

rm

2�Texp

�mv2k2T

!; (1.161)

is the one dimensional Maxwellian distribution function. Since � < vT =qR for the regime of interest,

we may make the following approximation,

�c(vk=qR)2 + �2c

' ��

�vkqR

�= �qR�(vk); (1.162)

where �(x) is the delta function. Then,

n0Vr = �p�

2V 2D

@n0@r

rm

2T; (1.163)

which is independent of the collision frequency. De�ning the di¤usion coe¢ cient by

n0Vr = �D@n0@r

;

we thus �nd

D ' vTqR

q2r2c ;

��3=2

vTqR

< � <vTqR

�; (1.164)

More accurate analysis shows that no clear plateau for the di¤usion coe¢ cient exists, and the

transition from trapped to P�rsch-Schlüter regime occurs in a rather gradual manner. The reason is

that the boundary between di¤erent regimes (such as � = �3=2vT =qR) is not a well-de�ned quantity.

For a single particle, such boundary is well de�ned, but when many particles are statistically

involved, exact averaging over distribution should be performed. Our treatment is too crude typi�ed

by replacing, whenever appropriate, the velocity by the thermal velocity, etc. By the same token,

the radial displacement from the magnetic surface and bounce frequency must be calculated by

averaging over particle distribution. The simpli�ed approach, however, does give us a clear picture

of physical mechanisms behind the particle di¤usivity in the tokamak magnetic geometry which is

signi�cantly enhanced from that in a uniform magnetic �eld. The overall di¤usivity in a tokamak

discharge is summarized in Fig. 1.8 as a function of the collisionality parameter,

�� =1

�3=2�c

vT =qR: (1.165)

The dashed line shows the result of more rigorous kinetic analysis of the neoclassical transport.

38

Figure 1.7: Particle di¤usivity D in a tokamak as a function of the collision frequency �c: " = r=R

(inverse aspect ratio) and v is the thermal velocity. In the collisionless regime, di¤usivity is

dominated by trapped particles. Dotted line shows the di¤usivity predicted by kinetic theory.

39