phys3947 plasma physics

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Plasma Physics

Don Melrose

and Alex Samarian

Senior-level (3rd year) course

Lecture notes

Version: April 4, 2011


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Contents

Preface iii

1 Plasma: an ionized gas 1

1.1 Arc discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Solar atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Controlled fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Solar radio astronomy and space plasma physics . . . . . . . . . . . . . . . . . . . 7

1.6 Numerical values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Conversion factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8 Exercise Set 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Collective effects in plasmas 13

2.1 Electrostatic oscillations in cold plasma . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Fluid model for electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Collective response to a static eld . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Debye screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Plasma parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Exercise Set 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Wave dispersion in plasmas 21

3.1 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Expansion in plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


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vi CONTENTS

3.3 Phase and group velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Exercise Set 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Waves in isotropic plasmas 294.1 Wave equation for a plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Longitudinal response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Cold electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.5 Energetics in waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.6 Exercise Set 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Orbit theory 39

5.1 Motion of a charged particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.2 Electric drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Drift motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Adiabatic invariant: magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Appendix: formal theory of adiabatic invariants . . . . . . . . . . . . . . . . . . . 46

5.6 Exercise Set 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Cold magnetized plasma 51

6.1 Response of a cold plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.2 Dispersion equation for a cold plasma . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.3 Polarization vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.4 Polarization of cold plasma waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.5 Exercise Set 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Magnetoionic theory 59

7.1 Magnetoionic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.2 Cutoff frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.3 High-frequency limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.4 Polarization of magnetoionic waves . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.5 Exercise Set 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


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CONTENTS ix

C Assignment Sets 145

Assignment Sets 145

C.1 Assignment Set 1: Due 20 May 2011 . . . . . . . . . . . . . . . . . . . . . . . . . 145C.2 Assignment Set 2: Due 3 June 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . 147


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Chapter 1

Plasma: an ionized gas

We are familiar with three states of matter: solid, liquid and gas. Plasma is sometimes regarded

as a fourth state of matter. It is interesting that the ancient greeks considered there to be four

elements: earth, water, air and re. The rst three may be interpreted as solid, liquid and gas,

and re is one example of a plasma. In re there is a chemical reaction (usually oxidization) in

which atoms tare briey ionized, emitting optical radiation as they recombine. It is the presence

of electrons and charged ions that characterizes the plasma state.

The name plasma for an ionized gas was introduced by Langmuir 1 in the late 1920s in connec-

tion with an investigation of oscillations in an arc discharge. (I am not aware of the reason for the

name plasma.) A plasma may be dened as an ionized gas. There are also solid-state plasmas

such as metals, where the electrons in the conduction band can be regarded as an electron gas for

many purposes. In fact most of the matter in the Universe is ionized, with our environment on

the surface of the Earth being exceptional. Above us, the ionosphere is ionized, and below us, the

Earths core is a strongly conducting solid-state plasma.

In this rst lecture several examples of plasmas, and the physics of relevance to them, are

discussed, following a roughly historical sequence in the development of plasma physics as a

separate branch of physics.

1 L. Tonks, I. Langmuir, Oscillations in ionized gases, Physical Review 33 , 195 (1929)


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2 1. Plasma: an ionized gas

1.1 Arc discharges

The rst research work on what is now recognized as plasma physics was that of Langmuir in

the late 1920s on oscillations in a gas ionized by an electric arc discharge. In Langmuirs experi-ments, the plasma was created by a large electric eld driving a large current through the plasma,

such that electrons accelerated by the electric eld knock further electrons off atoms, creating a

relatively high density of electrons and positive ions. Langmuir observed oscillations in the arc

discharge at a characteristic frequency, called the plasma frequency, p, that is proportional to the

square root of the number density of electrons, ne. Dened as an angular frequency, the plasma

frequency is determined by

p = e2ne

0me1/ 2

= 56 .4 ne1 m 3

1/ 2s 1, (1.1)

where e and me are the charge and mass of the electron, respectively. The plasma-electron oscil-lations observed by Langmuir are now called Langmuir waves. Langmuir waves have frequencies

close to p, and slightly above it. Langmuir waves are longitudinal or electrostatic: they have

an electric eld along the direction of wave propagation, and no magnetic eld. This makes them

quite different from electromagnetic waves, which are referred to as transverse waves in plasma

physics.

Transverse waves are similar to electromagnetic waves in vacuo, in the sense that the electric

eld, E , in the wave is orthogonal to the direction of wave propagation. The refractive index in a

plasma is less than unity: n = (1 2 p/ 2)1/ 2, with the (angular) frequency of the wave.Langmuir actually identied two new types of wave, with the other type now being called ion

acoustic waves or ion sound waves. They are also longitudinal. Ion acoustic waves exist only

below the ion plasma frequency, pi, which is dened similar to (1.1) with the charge and mass

replaced by those of the ions: pi = ( Z 2i e2n i / 0m i)1/ 2, where the ions are assumed to have charge

Z ie, mass m i and number density, n i .

Note that a frequency can be expressed either as an angular frequency, , of as cyclic a

frequency, f = / 2. The units of are radians per second ( s 1), and the units of f are hertz

(Hz), or cycles per second in older literature.

A thermal distribution is described by its temperature. In a plasma the temperature, T e, of

the electrons can be different from the temperature, T i , of the ions. There are various heating


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1.2 Ionosphere 3

processes for plasmas, and most of these favor either the electrons or the ions. Collisions tend to

bring the electron and ion temperatures into equilibrium, but often the time required for T e to

approach T i is much longer than the heating time. More generally, collisional processes can be

quite slow in plasmas, especially for fast electrons and ions. A simple model for a plasma consists

of thermal electrons and thermal ions, with T e = T i , and various distributions of nonthermal

fast particles. Collisions can have a negligible effect on the fast particles, and the physics of

collisionless plasmas is dominated by interactions of waves and particles.

In plasma physics one often uses energy units to describe the temperature, omitting Botzmanns

constant. The temperate of an arc discharge is typically measured in electron volts (eV), with the

ionization potential for most ions being several eV. The conversion factor between kelvin (K) and

eV is given in Table 1.4. It is convenient to dene the thermal speed, V e , of electrons by writingmeV 2e = T e (note: no factor of 1/2 is included). In Langmuirs theory, T e appears through the

Debye length:

D =V e p

= 69T e1 K

1/ 2 ne1 m 3

1/ 2m. (1.2)

In modern day terminology, the plasma investigated by Langmuir is regarded as an unmag-

netized thermal plasma. Most plasmas of interest are either conned by a magnetic eld, or are

threaded by a magnetic eld that is frozen into the plasma, and such plasmas are said to be

magnetized .

1.2 Ionosphere

The Earths atmosphere is ionized by ultraviolet radiation from the Sun, and by cosmic rays. The

degree of ionization is determined by a balance between the ionization rate and the recombination

rate. The recombination rate is strongly dependent on density, and is very high near the surface of

the Earth, so that the degree of ionization is extremely small. The density decreases rapidly with

increasing height, and around a height of 100 km, the recombination rate becomes comparable

with the ionization rate. Above this height, the degree of ionization is relatively high and this

region is called the ionosphere . Initially, the electron density increases with height, due to the

rapidly increasing degree of ionization, until the plasma becomes completely ionized, and at still

greater height, the electron density decreases as the total density decreases.

The maximum plasma frequency in the ionosphere is where the electron density is a maximum.


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1.7 Conversion factors 9

Table 2: Plasma physics quantities

quantity symbol SI units gaussian units

plasma frequency p 56.4n1/ 2e s 1 5.64

104n1/ 2e s 1

f p = p/ 2 9.0n1/ 2e Hz 9.0 103n1/ 2e Hzelectron gyrofrequency e 1.76 1011B s 1 1.76 107B s 1

f B = e/ 2 2.8 1010B Hz 2.8 106B Hzproton gyrofrequency p 0.98 107B s 1 0.98 104B s 1

Alfven speed vA 2.2 1016B(n e) 1/ 2 m s 1 2.2 1011B(n e) 1/ 2 cm s 1sound speed cs 1.5 102T 1/ 2 m s 1 1.5 104T 1/ 2 cm s 1

ion sound speed vs 9 T 1/ 2e m s 1 9.1

103 T 1/ 2e cm s 1

Debye length D 69T 1/ 2e n 1/ 2e m 6.9T 1/ 2e n 1/ 2e cm

thermal e speed V e 3.9 103T 1/ 2e m s 1 3.9 105T 1/ 2e cm s 1collision frequency 0 1.37 10 5(ln)neT 3/ 2e s 1 13.7(ln)neT 3/ 2e s 1

where is the number of nucleons per electron, = 1 for a hydrogen plasma.

1.7 Conversion factors

Conversion factors between different systems of units are made most conveniently by introducing

conversion factors and regarding units as algebraic symbols. For example, consider conversion from

meters to centimeters, or vice versa. Given 10 2 cm = 1 m, one may introduce conversion factors by

writing 1 = 10 2 cm m 1 or 1 = 10 2 m cm 1, both of which follow directly from the basic relation.

Then if one is given a formula in meters, as L = x m, where x is a number, the quantity in

centimeters, that is, L = y cm where y is another number, is given by L = x m (102 cm m 1) =y cm, so that one identies y = 10 2x.

The detailed formulae in these notes are given in SI units. Another set of unit used widely

are gaussian units, also called cgs units because the basic units of length, mass and time are

centimeter, gram and second, whereas SI units are mks units (meter, kilogram second). The

electric and magnetic units are also different in SI and gaussian units, and this results in formulae,

including Maxwells equations, having different forms in the two sets of units. To convert any

formula from SI to gaussian units involves no change to the charge q , charge density , current

density J , electric eld E and electric potential . The following changes are made: magnetic


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eld B B /c , vector potential A A /c , permittivity of free space 0 1/ 4, permeabilityof free space 0 4/c 2. Other conversion factors are include in the following Table 1.3. Someother conversion factors are given in Table 1.4.

Table 1.3: Conversion factors between SI and gaussian units

quantity SI/gaussian gaussian/SI

length 10 2 m/ cm 102 cm/ m

mass 10 3 kg/ g 103 g/ kg

energy 10 7 J / erg 107 erg/ J

power 10 7 W/ ergs 1 107 ergs 1/ W

force 10 5 N/ dyne 105 dyne/ N

charge 13 10 9 statcoul / C 3 109 C/ statcoulelectric eld 3 104 V m 1/ statvolt cm 1 13 10 4 statvolt cm 1/ V m 1

current 13 10 9 A/ statamp 3 109 statamp / Acurrent density 13 10 5 A m 2/ statamp 3 105 statamp cm 2/ A m 2

magnetic induction 10 4 T/ G 104 G/ T

Table 1.4: Other conversion factors

quantity factor inverse factor

temperature 8 .6 10 5 eV/ K 1.16 104 K/ eVX-ray energy 4.1 10 15 eV/ Hz 2.4 1014 Hz/ eV

angle 2.06 105 arcsec/ rad 4.85 10 6 rad / arcsectime 3.16 107 s/ yr 3.17 10 8 yr/ s


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1.8 Exercise Set 1 11

1.8 Exercise Set 1

1.1 Calculate the plasma frequency, the electron cyclotron frequency for the following parameters:

(a) A laboratory fusion plasma: ne = 1015

m 3

, B = 1 T, T e = 10 keV.(b) The ionosphere: ne = 0 .1 m 3, B = 10 5 T, T e = 10 3 K.

(c) The solar corona: ne = 10 10 cm 3, B = 1 G, T e = 10 6 K.

(d) The interplanetary medium: ne = 1 cm 3, B = 3 10 6 G, T e = 10 6 K.1.2 Calculate the Debye length and the Debye number for the plasmas in Exercise 1.1 .

1.3 Calculate the Alfven speed for the plasmas in Exercise 1.1 .


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14 2. Collective effects in plasmas

d d

+

+ -

+ -

-

Figure 2.1: A schematic showing the electrons (dashed region) separated from the ions (solid

region) by a distance d, setting up a net charge density that is negative where there is an excess

of electrons, and positive where there is a deciency of electrons.

plasma. This induced electric eld adds to the applied eld, modifying it so that it becomes a

self-consistent eld.

Now suppose that there is no applied eld. The only electric eld is then due to free oscillations

of the electrons relative to the ions. Any displacement of the electrons from the ions sets up the

internal eld E (t) = ened(t)/ 0. This electric eld accelerates the electrons. Newtons equation

of motion implies

me d(t) = eE (t) = e2ned(t)/ 0, (2.1)

where a dot denotes a derivative with respect to time. Equation ( 2.1) is the equation for a simple

harmonic oscillator with frequency equal to the plasma frequency, p = ( e2ne/m e0)1/ 2. To see

this, suppose that d(t) = d0 cos(t) is oscillating at frequency with amplitude d0. On inserting

this into ( 2.1), one has d(t) = 2d(t), and ( 2.1) reduces to me2d(t) = e2ned(t)/ 0, whichimplies 2 = 2 p. It follows that the electron plasma frequency is the natural frequency of free

oscillations of the electrons relative to (immobile) ions.


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(2.4) with ne(x , t ) replaced by ne(x , t ), one nds that ne(x , t ) satises the oscillator equation

with a natural frequency p. Although the mathematical details are quite different, this derivation

is physically equivalent to that given above based on a slab model.

Purely temporal oscillations may be regarded as a limiting case of a wave motion in which

the wavelength is innite. Before considering the case where the wavelength is nite, we need to

consider the static limit, in which the electric eld is a function of space but is not changing as a

function of time. This corresponds to the limit of a wave with zero frequency or innite period.

2.3 Collective response to a static eld

A static distribution of charge in vacuo creates an electrostatic eld, which is a potential eld, andso may be written as minus the gradient of a potential: E = grad . The eld and the chargedensity are related by one of Maxwells equations

div E = 2 = / 0. (2.6)In particular, a point charge, q , at the origin, x = 0, creates a Coulomb eld,

(x) =q

4 0r, E (x) =

q 4 0

xr 3

. (2.7)

where r = |x| is the radial distance from the origin. To prove that ( 2.7) is the solution of (2.6) fora point charge involves introducing the Dirac -function. The charge distribution corresponding

to a point charge is described by a -function. For a charge q at x = x0, the charge density is

(x) = q 3(x x0), with 3(x x0) = (x x0) (y y0) (z z 0).A static distribution of charge in a plasma creates an electric eld which is different from that

in vacuo. Consider inserting a charge q > 0 at the origin. Through its Coulomb eld, the charge q

attracts negative charges and repels positive charges. The electrons in the plasma, moving around

like the molecules of air in this room, are attracted to the charge q , and this causes the number

density of electrons to be slightly higher near q . Similarly, the positive ions in the plasma are

repelled by q , and their number density is slightly lower near q . Thus, the presence of q induces

a charge density in the plasma. The total electric eld is the sum of the Coulomb eld and the

electric eld due to this induced charge density in the plasma.

Consider the effect of charges in a shell between r and r + dr . The number of particles in

this shell is the number density of particles times the volume, 4 r 2dr , of the shell. The force


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2.4 Debye screening 17

(and its reaction) due to each of these particles decreases 1/r 2, due to the inverse-square-

law form of the Coulomb interaction, but this is offset by the number of particles in the shell

increasing r 2. Summing over the shells (integrating over dr ) the result diverges. This leads to

the surprising conclusion that the eld associated with a test charge, q , in a plasma is modied

due to interactions with all the other charges in the plasma. This situation is quite different to the

case of an un-ionized gas. The (van der Waals) force between molecules falls off 1/r 6, and only

the interactions with nearest neighbors are important, with the net effect of other particles falling

off rapidly, 1/r 4. However, in a plasma there is no decrease in the net force, and one cannot

assume that the effect of nearest neighbors dominates. To overcome this difficulty we introduce

the concept of a self-consistent eld.

It is somewhat easier to understand the concept of a self-consistent eld in a related context:

a cluster of stars. The gravitation attraction between stars is an inverse-square-law force, and the

same problem arises: all the stars in the cluster affect the motion of any one star. We model the

cluster by a smoothed gravitational potential and a smoothed mass density, and relate these by

Poissons equation. Once we nd this self-consistent eld, we consider the motion of an individual

star in the smoothed potential. In the electrostatic case, there are charges of opposite sign, and

the self-consistent eld leads to Debye screening.

2.4 Debye screening

Debye screening may be treated in the following approximate way. The Coulomb eld ( 2.7) is the

solution of

div E =

2 = / 0, = q 3(x), (2.8)

corresponding to a point charge at the origin x = 0, with r = |x| the radial distance from theorigin. The effect of this eld on surrounding charges is to attract the charges of opposite sign to

q , and repel charges of the same sign as q . Consider the thermal electrons in the plasma. Their

number density is modied such that it becomes by

ne(r ) = ne exp[e(r )/T e], (2.9)


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2.6 Exercise Set 2

2.1 Calculate the electric eld corresponding the the screened potential ( 2.12). Show that your

result reduces to a Coulomb eld for D .2.2 Estimate N D for the following plasmas: (a) an arc discharge with ne = 10 25 m 3, T e = 1 eV,

(b) the solar corona with ne = 10 10 cm 3, T e = 10 6 K, (c) an interstellar cloud, with ne = 1 cm 3,

T e = 100 K.

2.3 Show that the Coulomb eld, = q/ 4 0r is a solution of 2 = q 3(x) by integrating 2 = q 3(x) over a sphere of radius r centered on the charge, and noting that the right handside gives q . Note that in spherical polar coordinates, r,, , one has

2 =1r 2

r

r 2 r

+1r 2

2

(cos )2+

1r 2 sin2

2

2.


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Chapter 3

Wave dispersion in plasmas

Plasmas support a rich variety of different types of waves, called wave modes. There is no system-

atic way of naming wave modes: some are historical (Langmuir waves, Alfven waves), some are

descriptive of the waves (transverse waves, cyclotron waves) and some are description of the theory

used to describe them (magnetoionic waves, MHD waves). Some wave modes can be described

using relatively simple models, but this is the case only for the simplest systems. In this lecture

and the next lecture, a general description of a wave mode in a medium is introduced, and some

properties of waves in plasmas are discussed.

3.1 Sound waves

A simple example of a waves is a sound wave in an un-ionized compressible gas. Such waves exist

only at very low frequencies in a plasma, specically at frequencies well below the collision frequen-

cies between particles. Nevertheless, the example of sound waves serves as a useful introduction

to waves in plasmas.

It is important to distinguish between a plane wave, which is a mathematical construction,

and a physical wave in a medium. A plane wave is dened to vary in time in space harmonically,

proportional to exp[i(t k x)], where is the frequency and k is the wave vector. The planewave has a period, T , and a wavelength, , with = 2 /T and k = 2 / . The wave propagates

in the wave-normal direction, described by the unit vector say, with k = k . A plane wave

is an idealization, and a real disturbance may be regarded as a superposition of plane waves.

Technically, a Fourier transform in time and space corresponds to an expansion in plane waves.


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22 3. Wave dispersion in plasmas

We describe physical waves in a medium in terms of a solution of the appropriate wave equation.

Given a model for a specic medium we can derived the wave equation from the equations for the

model.

Consider the example of a sound wave. The model in this case is a compressible uid, described

by the equations of hydrodynamics. The wave equation for sound waves is derived as follows. A

sound wave corresponds to a uctuation in the gas, and this may be described by a uctuation,

u , in the velocity of the gas. There are also uctuations in the mass density, = 0 + 1 and in

the pressure, P = P 0 + P 1, where subscript 0 implies that non-uctuating part, and subscript 1

describes the uctuating part. The adiabatic law for a perfect gas is

P = constant , (3.1)

where the adiabatic index is = 5 / 3 for a monatomic gas. This implies that the uctuations in

the density and pressure are related by

P 1 = c2s 1, c2s = P 0/ 0, (3.2)

where cs is the (adiabatic) sound speed. (The derivation of ( 3.2) involves linearizing (3.1): one

has (0 + 1) = 0 (1 1/ 0) to rst order.) The hydrodynamic equations are the continuityequation for mass

/t + div ( u ) = 0 , (3.3)

where u is the uid velocity, and the equation of uid motion

du /dt = grad P. (3.4)

One assumes that the amplitude of the uctuations, 1, P 1, u , are small, so that products of them

can be neglected. Then taking the time derivative of ( 3.3) and the divergence of (3.4), one nds

that 1, P 1, u all satisfy the same wave equation. For P 1 this is

[ 2/t 2 c2s 2]P 1 = 0 . (3.5)

A solution of (3.5) corresponds to a wave propagating at the speed cs .


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3.2 Expansion in plane waves 23

3.2 Expansion in plane waves

We look for a solution of (3.5) that varies periodically, as cos( t k x) or sin(t k x). It isconvenient to introduce complex notation by writing

cosx =12

(eix + e ix ), sin x =12i

(eix e ix ),and to choose to look for a solution that varies as exp[ i(t k x)]. Thus in (3.5) we look for asolution in which the variation of P 1 with time and position is exp[i(t k x)]. In a moreformal treatment, this step is replaced by Fourier transforming in space and time.

On expanding the uctuations in plane waves, the derivatives in ( 3.5) operate only on exp[i(tkx)]. One has

t

exp[i(t k x)] = i exp[i(t k x)],When the time derivative and the derivatives with respect to position operate only on the expo-

nential function, they are replaced according to

t i, grad = ik , div = ik, curl = ik, (3.6)

with 2 = div grad |k|2. The differential operator in ( 3.5) is replaced according to 2t 2 c

2s

2 2 + k2c2s .It follows that a plane-wave solution of ( 3.5) exists only for

2 k2c2s = 0 . (3.7)The relation = kcs is referred to as the dispersion relation for sound waves.

The simple model enables one to infer other properties of sound waves. One can show that the

uid velocity is parallel to the wave normal direction, and that there is equipartition between theaverage kinetic energy density, 12 0|u |2, and the average potential energy, 1P 1/ 20, in the wave.

Other wave modes

The uid model can be generalized in various ways, and each generalization leads to modication

of the properties of the wave modes, and to the appearance of new wave modes. Suppose one

considers the Earths atmosphere, and takes the decrease in the density with height, z , in account.


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3.3 Phase and group velocities 25

Phase velocity: particle-wave resonance

The phase velocity is the velocity at which surfaces of constant phase move. The surfaces of

constant phase correspond to t

k

x = constant. The phase velocity is dened to be along the

direction of k , that is, along the wave normal direction. The surfaces of constant phase move in

this direction at the phase speed v = /k . Thus the phase velocity for a wave in the mode M is

M (k)/k along k . There is no physical restriction on the phase velocity, which can be greater or

less than the speed of light. Waves with v > c (v < c ) are said to be superluminal (subluminal).

The phase velocity plays an important role in plasma physics through wave-particle resonance.

A particle whose velocity v is equal to the phase velocity of a wave satises the resonance condition

k v = 0 . (3.8)One can understand the importance of the resonance condition by considering what an observer

moving with the particle sees. This observer sees a particle at rest and a wave that varies periodi-

cally in space but is stationary in time. Plasma waves include an electric eld, and the particle is

systematically accelerated by this electric eld. Returning to the frame in which the particle has

velocity v , one nds that particles near resonance tend to be dragged into resonance by this effect.

Particles with velocity slightly less than that of the wave gain energy at the expense of the wave,

and particles with velocity slightly greater than that of the wave give up energy to the wave, as

they are dragged into resonance with the wave. In a thermal distribution, the number of particles

with a given velocity is determined by a Maxwellian distribution exp(mv2/ 2T ), where T isthe temperature (in energy units). There are then more particles with speed slightly less than a

given speed than particles with speed slightly greater than this speed, and the particles gain a net

energy from the wave. This causes the wave to damp, through what is called Landau damping .

A non-Maxwellian distribution may have a distribution function which is an increasing function

of velocity over some range, and in this case waves with phase velocity in the range gain energy

from the wave. This leads to growth of waves in what is called a plasma instability .

Group velocity

The energy in waves propagates at the group velocity. The group velocity for the mode M is

vgM = M (k)/ k . (3.9)


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26 3. Wave dispersion in plasmas

For transverse waves in a cold plasma, one has

vgT = 2 p + k2c2/ k = kc2/ 2 p + k2c2 = c(1 2 p/ 2)1/ 2, (3.10)where = k/k is the wave-normal direction.

In an anisotropic medium, the group velocity and the phase velocity are in different directions

in general. In terms of cartesian components, with k = ( kx , ky.kz), the cartesian components of

the group velocity are ( /k x ,/k y ,/k z )M (k).

Ray propagation

The path along which the wave energy propagates corresponds to the ray path. In a medium

whose properties change slowly with space and time, the dispersion relation changes slowly with

space and time. Writing M (k; t, x) to include this slow change, the path of a ray is determined

by the ray equations

dxdt

=

kM (k; t, x),

dkdt

=

xM (k; t, x),

ddt

= t

M (k; t, x). (3.11)

The rst of (3.11) implies that the velocity along the ray path is the group velocity. Suppose that

the properties of the medium are varying along the z direction. Then the second of ( 3.11) implies

that kz is changing along the ray path, and that the components, kx , ky, perpendicular to this

direction are constant, which is Snells law.


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3.4 Exercise Set 3

3.1 Show that the equations ( 3.1), (3.3) and (3.4), with d/dt = /t + u grad , imply t

|u |2

2+ P

1+ div |u|

2

2u + P

1u = 0 , (3.12)

which is interpreted as the equation for energy continuity for the uid. The rst two terms are the

kinetic energy density and the thermal energy density, and the second two terms are the kinetic

energy ux, and the thermal energy ux, with the latter equal to the enthalpy times u .

3.2 Show that in a sound wave, there is equipartition between the kinetic energy density, W K =12 0|u |2, and the potential energy density, W P = 12 21c2s / 0, associated with the pressure uctua-tions.

3.3 Derive the sound speed in the case where the medium is assumed isothermal rather than

adiabatic.

3.4 The dispersion relation for Langmuir waves may be approximated by 2L (k) = 2 p + 3 k2V 2e ,

where V e is the thermal speed of electrons. Show the the phase speed, v , and the group speed,

vg, for waves with this dispersion relation satisfy vvg = 3 V 2e .

3.5 Consider a model for Langmuir waves in which the electrons are treated as a compressible gaswith pressure P = nemeV 2e satisfying the adiabatic law P n e where is the adiabatic index.

(a) Assuming an equation of uid motion of the form menedu /dt = eneE grad P , show thatthe implied dispersion relation for Langmuir waves is

2 = 2p + k2V 2e . (3.13)

Hint : Use a rst order perturbation treatment with the zeroth order corresponding to P = 0.

Assume k u .(b) This model reproduces the correct form ( 3.13) for = 3. Can this value of be justied, or

is the model inadequate to describe Langmuir waves?


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30 4. Waves in isotropic plasmas

Maxwells equations (4.1) in plane waves and using (3.6), the Fourier components satisfy

k E = B , k B = 0 , k B = i0 J E /c 2, k E = i/ 0, (4.2)

respectively. One may regard the rst of the equations ( 4.2) as determining B in terms of E . The

second equation is implied by the rst. The fourth equation, combined with the third, may be

interpreted as determining the charge density in terns of the current density,

= k J . (4.3)

The third of equations ( 4.2) then becomes

k [k E ] + (2/c 2)E = i0J , (4.4)

which is one form of the wave equation.

Note what we have done: we have effectively reduced the four equations ( 4.1) to a single

equation ( 4.4). We achieve this by rst noting that div B = 0 is redundant for elds that are

varying in time, because it is implies by the rst of ( 4.1), so that our four equations are reduced

to three by expanding in plane wave, which ignores the static elds. Two of the remaining three

equations are regarded as subsidiary equations: ( 4.2) deningB in terms of

E , and (4.3) dening

in terms of J . The wave equation ( 4.4) relates the remaining quantities, E and J .

4.2 Wave equation

The next step is a particularly important one, in that it is the essential step in introducing the

self-consistent eld. First, let us separate the current into an induced (ind) part and an extraneous

(ext) part, by writingJ = J ind + J ext , (4.5)

We regard J ext as a source term and leave it on the right hand side; it is set to zero when considering

the wave modes of the plasmas. The important assumption is that the response of the medium

may be described in terms of a linear relation between the component of J ind and the components

of E . One transfers J ind to the left hand side of ( 4.4), so that all the terms proportional to the

components of E are on the same side of the equation. With this assumption, ( 4.4) becomes three


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4.4 Cold electron gas 33

The rst of these is derived from a model for Debye screening, and the second follows from a

model for a cold electron gas.

4.4 Cold electron gas

Now let us general to include both the longitudinal and transverse parts, but ignore the thermal

motions (so that there is no pressure term in the equation of motion for the electrons). This

leads to the cold plasma model, which is the simplest useful model for a plasma. The response

of a plasma tis described by the relation between the induced current J and the electric eld E ,

where we now revert to the notation used in expanding in plane waves. We wish to calculate the

elements in the matrix ij in (4.6) for a cold electron gas.

The current density in a cold electron gas is J = eneu . The only force is due to theelectromagnetic eld. The equation of motion for the uid is the same a Newtons equation

medudt

= e[E + u B ]. (4.16)

The Lorentz force, u B , is the product of two uctuating quantities, and so is neglected. Afterexpansing in plane waves, (4.16) gives

im e u = eE . (4.17)

The current density becomes

J = ie2neme

E = i02 p

E . (4.18)

On inserting ( 4.18) into (4.4) the wave equation becomes, after minor rearrangement,

(2 2 p k2c2)E + c2k k E = 0 . (4.19)

It is convenient to divide by 2, to introduce the refractive index by writing n2 = k2c2/ 2, and to

write k = k . Then ( 4.19) can be written in the matrix form

E = 0 , = n2[ 1] + 1 K (), K () = 1 2 p2

, (4.20)

where 1 is the unit matrix and K () is dielectric constant for a cold plasma.


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We are free to choose the coordinate axes such that k is along the 3-axis. Then ( 4.19) can be

written in the form ( 4.6), with the matrix being diagonal. One can write the result in the form

T

0 00 T 0

0 0 L

E 1E 2

E 3

= 0 , (4.21)

L = K L , T = K T k2c2

2, K L = K T = 1

2 p2

. (4.22)

The result ( 4.22) for K L is the second of the two simple cases written down in ( 4.15).

The dispersion equation ( 4.8) becomes

(, k) = L [T ]2 = 0 . (4.23)

The solution L = 0 corresponds to the component E 3 = 0 with E 1 = E 2 = 0, and the solution

T = 0 corresponds to the component E 3 = 0 with E 1, E 2 = 0. These are referred to as longitu-

dinal waves, with E parallel to k , and transverse waves, with E orthogonal to k , respectively.

The longitudinal waves are the electron plasma oscillations, at = p, rst identied by Lang-

muir. When thermal motions are included, the dispersion relation for Langmuir waves becomes

= L (k), with 2L (k) 2 p + 3 k2V 2e .The dispersion relation for transverse waves in a cold electron gas is T = 0. This may be

written either in term of the refractive index, n = kc/ , or in the form = T (k), with

n2 = 1 2 p2

, 2T (k) = 2

p + k2c2. (4.24)

Transverse waves do not exist for < p in a cold plasma.

4.5 Energetics in waves

The total energy in a wave may be separated into electric energy, magnetic energy and kinetic

energy associated with the perturbed motions of the particles. The ratio of magnetic to electric

energy follows from the rst of equations (4.2):

W M : W E = |B2|

20: 0|E 2|

2= k2c2 : 2. (4.25)


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4.5 Energetics in waves 35

In general it is not possible to calculate the kinetic energy in the waves directly, but the cold

plasma model is an exception. In this case, the ratio of the kinetic energy to the electric energy

follows from (4.17)

W K : W E = neme|u2

|2 : 0|E2

|2 = 2 p : 2. (4.26)Hence, for a transverse wave in a cold plasma the total energy is made up from these contributions

in the ratioW E W T

:W M W T

:W K W T

=12

:12

1 2 p2

:12

2 p2

, (4.27)

where W T = W E + W M + W K is the total energy in the waves. For Langmuir waves there is no

magnetic energy, and there is approximate equipartition between electric and kinetic energy.

The propagation of a wave implies propagation of energy at the group velocity. The phasevelocity is /k , or M (k)/k for the mode M . The group velocity, ( 3.9), viz. vgM = M (k)/ k ,

for transverse waves in a cold plasma is

vgT = 2 p + k2c2/ k = kc2/ 2 p + k2c2 = c(1 2 p/ 2)1/ 2. (4.28)In a cold plasma, the velocity of energy propagation is also given by the Poynting ux, E B / 0,divided by the total energy density in the waves. Using the rst of equations ( 4.2), the Poynting

ux becomes E B0

=kc2

0|E|2, (4.29)

and the total energy is W E + W M + W K = 2 W E = 0|E|2 for transverse waves.Langmuir waves propagate very slowly,

vgL = 2 p + 3 k2V 2e / k 3kV 2e / p. (4.30)Their group velocity reduces to zero in the cold plasma limit.


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(b) Show that there are two modes, and nd the values of n2 = k2c2/ 2 for these two modes.

(c) Show that the polarization vectors of the modes correspond to circular polarizations.

4.3 The most general form of polarization for transverse waves in an isotropic medium (or the

vacuum) consists of an unpolarized component, and a polarized component, with the latter cor-

responding to an elliptical polarization in general. Let r be the degree of polarization, with r = 1

for completely polarized radiation, and in Exercise 4.1 , and with r = 0 for unpolarized radia-

tion. Unpolarized radiation cannot be described by a polarization vector. One can describe it

by a polarization tensor, pij . The polarization tensor is hermitian, pij = p ji , and for completely

polarized radiation it reduces to the outer product of the polarization vector and its complex

conjugate: pij

eie j . We are free to choose k along the z axis, and then pij has components

that are zero for i or j equal to z . It is convenient to write it as a 2 2 matrix. We are free toimpose a normalization condition, and we require that the trace of the matrix be equal to unity:

pxx + pyy = 1. The most general form for an hermitian 2 2 matrix with this normalization is

pij = (1 r ) ij + r [ pQ (Q )ij + pU (U )ij + pV (V )ij ], (4.34)

with p2Q + p2U + p2V = 1, and with

ij = 1 00 1

, (Q )ij = 1 00 1

, (U )ij = 0 11 0

, (V )ij = 0 ii 0

, (4.35)

which are the unit matrix and the three Pauli matrices.

Show that

(a) r = 1, pQ = 1 correspond to linear polarization along the x, y axis, respectively;(b) r = 1, pV = 1 correspond to right and left hand circular polarization, respectively;(c) r = 1, pU = 0 corresponds to an elliptical polarization, with = 0 in (4.31), and express pQ ,

pV in terms of the axial ratio, T .


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Chapter 5

Orbit theory

So far we have assumed that there is no background magnetic eld: the plasma is assumed

unmagnetized. We now include the magnetic eld. We discuss three topic that involve the

magnetic eld: orbit theory, cold plasma theory and magnetohydrodynamics.

Orbit theory is concerned with the motion of charged particles in a magnetic eld. Motion in

a uniform B is a spiraling along the magnetic eld lines. Drifts across the magnetic eld occur

in the presence of an electric eld, a mechanical force, and gradients in B . Conserved quantities,

known as adiabatic invariants, are helpful in understanding the motion.

5.1 Motion of a charged particle

Consider a particle with charge q and mass m moving in a magnetic eld B and an electric eld

E . Newtons equation of motion is

dpdt

= q [E + v B ], (5.1)

where v is the velocity of the particle and where

p = m v , = mc 2, = (1 v2/c 2) 1/ 2 (5.2)are the momentum, energy and Lorentz factor of the particle, respectively.

In a uniform magnetic eld with no electric eld, E = 0, the following quantities are constants

of the motion: the energy = mc 2, and the components p = mv and p = mv of the

momentum perpendicular and parallel to the magnetic eld, respectively. The motion of the


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40 5. Orbit theory

particle may be decomposed into a motion at constant velocity along the eld lines plus a circular

motion perpendicular to the eld lines. Thus the particle exhibits a spiraling motion with the

pitch of the spiral dening the pitch angle :

v = v sin , v = v cos. (5.3)

The frequency of the circular motion is called the gyrofrequency, , and the radius is called the

radius of gyration (or sometimes the Larmor radius), R:

=0

, 0 = |q |Bm , R =v

=p

|q |B. (5.4)

The sense of gyration, which is the handedness of the circular motion in a screw sense relative to

B , depends on the sign of the charge= q/ |q |. (5.5)

Positively charged particles ( = +1) gyrate in a left hand screw sense relative to B , and negatively

charged particles ( = 1) gyrate in a right hand screw sense relative to B .The orbit of the particle is described by an equation that gives the position x of the particle

as a function of time, and can be written as x = X (t). Solving (5.1) in this case gives

X (t) = x0 + ( R sin(0 + t), R cos(0 + t), v t), (5.6)

where 0 and x0 are determined by the position of the particle at t = 0, and where the z axis is

chosen along the direction of B . The instantaneous velocity of the particle is given by

v (t) = X (t) = ( v cos(0 + t), v sin(0 + t), v ). (5.7)The sense of gyration is such that the magnetic eld produced by the spiraling charge opposes the

externally applied eld. Plasmas are diamagnetic the motions of the individual particles always

tend to reduce the applied magnetic eld.

5.2 Electric drift

Now consider the effect of inclusion of a uniform, nonzero electric eld, E in (5.1). In most

applications it is assumed that there is no parallel component, i.e. E B = 0. The reason is thatplasmas are highly electrically conducting, and charges can ow freely along magnetic eld lines


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5.2 Electric drift 41

to short out any parallel component of electric eld. However, particles do not ow freely across

eld lines and so an electric eld perpendicular to the magnetic eld is not shorted out and can

persist.

So far we choose the z axis to be along the direction of B , We are free to rotate our coordinate

system about this axis, and to choose this rotation such that E = ( E, 0, 0). For nonrelativistic

particles the different components of equation ( 5.1) are then

dvxdt

=q m

E + vy ,dvydt

= vx ,dvzdt

= 0 . (5.8)

The equation for the z -component is trivial. Differentiating the other two, using the fact that E

is constant, we obtain

d2vxdt2

= dvydt

= 2vx ,d2vydt2

= dvxdt

= 2 vy +E B

. (5.9)

The equation for vy can then be rewritten as

d2

dt2vy +

E B

= 2 vy +E B

, (5.10)

and then if we make the replacement

vy = vy + E/B, (5.11)

equation ( 5.10) reduces to the same form as the equation for vx in (5.9). These are the equations

for a simple harmonic oscillator solved for the case E = 0. By analogy with equations ( 5.6) and

(5.7) we have the following orbit equations:

X (t) = x0 + ( R sin(0 + t), R cos(0 + t) Et/B,v t), (5.12)v (t) = ( v cos(0 + t), v sin(0 + t) E/B,v ), (5.13)

for the orbit and the instantaneous velocity, respectively.

An alternative way of understanding the effect of a perpendicular electric eld is to note that

the eld may be removed by making a Lorentz transformation. The quantities E B and B 2E 2/c 2are Lorentz invariants, and provided E < B one may transform to a (primed) frame with E = 0

and B = ( B 2 E 2/c 2)1/ 2. The velocity of the transformation is in the direction perpendicularto both E and B , and is of magnitude vE = E/B . The motion of the particles in the primed

frame is a spiral around the magnetic eld B . The fact that the primed frame drifts relative


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42 5. Orbit theory

to the unprimed frame implies that the particle motion in the unprimed frame is a simple spiral

around a guiding center (or gyrocenter) which is drifting perpendicular to both the electric and

the magnetic elds. Note that in a uniform electric eld all particles drift with the same drift

velocity, vE .

5.3 Drift motions

A systematic treatment of drift motions involves assuming that the motion perpendicular to the

magnetic eld lines is circular motion about a center of gyration that is drifting. Effects that

cause drift include an electric eld, an external force, a gradient in the magnetic eld strength

and curvature of the eld lines. The following drifts are commonly identied:

electric drift : vE =E B

B 2, (5.14)

gravitational drift : vg =mg B

qB2, (5.15)

gradient drift : vB =p v2qB

B grad BB 2

, (5.16)

inertial drift : v i =B (dp /dt )0

qB2, (5.17)

curvature drift : vc = p vqB2B (B grad)BB 2 , (5.18)

polarization drift : vP =mq

EB 2

. (5.19)

A physical explanation of the electric drift ( 5.14) is given above. The gravitational drift ( 5.15)

may be derived from the electric drift by replacing q E in (5.1) by mg, and thence in ( 5.14) to

obtain ( 5.15). Note that the drift is perpendicular to both the gravitational eld and to the

magnetic eld. Also, it is in opposite senses for charges of opposite signs. Charges of opposite

signs owing in opposite directions imply an electric current. In a uid description, the current

density J implies a force per unit volume J B that opposes the gravitational force density gon the uid of mass density . The fact that the force due to the current opposes the initial force

that drives the current is an example of Lenz law. Physically, the gravitational drift may be

understood as illustrated in Figure 5.1. The gravitational force accelerates particles downward, so

that they have higher perpendicular momenta near the bottom of their orbits, and so, according

to (5.4), have larger gyroradii there than near the top of their orbits.


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5.3 Drift motions 43

Figure 5.1: The gravitational drift of a positively-charged particle is illustrated for the case where

the magnetic eld is into the page and the gravitational force is directed downward.

The gradient drift ( 5.16) is associated with a change in the strength of the magnetic eld. The

drift is perpendicular to both the direction of the magnetic eld and to the direction of grad B,

and is opposite for oppositely charged particles. In this case the implied current generates a

magnetic eld that is such as to oppose the gradient in B. The eld generated by the induced

current satises curl B = 0J which is Amperes Law in the case of constant or zero E . Physically,

the gradient drift may be understood in terms of an argument similar to that used to explain the

gravitational drift. As illustrated in Figure 5.2, if one takes an idealized case in which the magnetic

eld changes abruptly at a surface that passes through the center of gyration of the particle, then

the gyroradius is different in two halves of the orbit. On joining a sequence of semicircles with

radii that alternate between two values, one obtains the orbit illustrated in Figure 5.2, which

shows that a drift motion results.

The inertial drift ( 5.17) is attributed to the coordinate frame in which the spiraling motion is

described not being an inertial frame. For example, in a rotating plasma, the frame in which the

plasma is momentarily at rest is not an inertial frame. The quantity ( dp /dt )0, which is the time

derivative of the momentum relative to an inertial frame (e.g., the instantaneous rest frame), is

the inertial force.

The curvature drift ( 5.18) and the polarization drift ( 5.19) are both specic examples of in-

ertial drift. Curvature (or centrifugal) drift is associated with curvature of the magnetic eld.

Introducing the unit vector b = B /B , one has

B (B grad)BB 3

= b (b grad b ), b grad b =nRc

, (5.20)

where Rc is the radius of curvature of the eld lines, and where n is a unit vector along the direction


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44 5. Orbit theory

(a)(b)

Figure 5.2: (a) An idealized case which demonstrates how the gradient drift occurs: the magnetic

eld is into the page and its strength increases abruptly at a surface (dashed line) that passes

through the gyrocenter of the negatively-charged particle so that the gyroradii are different on the

two sides. The curve is drawn by joining semicircles with radii r 1 and r 2.

(b) A curved magnetic eld line may be approximated by the arc of a circle: the radius of curvature

Rc is the radius of this circle.

toward the center of gyration, as illustrated in Figure 5.2b. Polarization drift is associated with a

time-varying electric eld, in which case the inertial drift is ( dp /dt )0 = mdvE /dt .

An alternative way of writing the drift motions is in terms of the average (over the spiraling

motion) position R = X , with X given by (5.6) in the case of a uniform eld. Retaining only

the electric, gradient and curvature drift one has

R = v b +E b

B+

v p2qB

b grad BB

+v p2qB

b (b grad)b . (5.21)It is not at all obvious but it can be shown that the equation of motion for the gyrocenter has the

following components

p = q b E + 12 v p div b , p = 12 v p div b . (5.22)Magnetic elds do no work (since the force is always perpendicular to the motion) so although

the magnetic eld can induce drifts we expect any change in the particle energy to be determined

solely by the electric eld. Moreover, the perpendicular component of the electric eld can be

removed by a Lorentz transformation, and so in the simplest approximation it too does no work.


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5.4 Adiabatic invariant: magnetic trapping 45

If one writes = ( p2 + p2)/ (2m) then = v p + v p and the substitution of ( 5.22) leads to the

expected result = qE .

5.4 Adiabatic invariant: magnetic trapping

Consider a charge spiraling in a magnetic eld, B , that is changing slowly (compared to a gyro-

radius) in space. The motion is periodic is the angle known as the gyrophase , which is equal

to 0 + t in the case (5.6) of a uniform eld. The conserved quantity is the angular momentum

associated with this periodic motion, that is the -component of p times the radius, R, of gyra-

tion. The -component of p is equal to p and this gives an adiabatic invariant 2 p R, where

the 2 arises in a more formal denition, given by the integral in ( 5.27). Since R p /B , thiscontribution is proportional to p2 /B . Hence, one nds that

p2

B= constant (5.23)

is an adiabatic invariant, sometimes called the rst adiabatic invariant, and sometimes referred to

an the magnetic moment of the particle. (A charge moving in a circle corresponds to a current

loop and the magnetic moment is that associated with this current loop.)

The adiabatic invariant ( 5.25) may be shown to be an invariant using the result ( 5.22) fromthe theory of drift motions. First note the following result:

ddt

1B

= v b grad1B

=v div b

B, (5.24)

where the rst identity follows for B/t = 0, and where b = B /B and div B = 0 are used in the

second identity. Then using ( 5.22) and (5.24), one nds

d

dt

p2

B=

2 p p

B+ p2

d

dt

1

B= 0 . (5.25)

One implication of the conservation of the rst adiabatic invariant is the reection of a particle

from a magnetic compression. In the absence of any eld other than an inhomogeneous magnetic

eld, one has p = constant, and hence ( 5.25) implies sin2 /B = constant, where is the pitch

angle of the particle, cf. (5.3). Hence, as a particle propagates in a direction of increasing b grad B,sin2 increases B . As sin2 increases, |cos | decreases and so | p | decreases. If sin2 reachesunity then the particle motion is strictly circular, with p = 0, and the particle reects at that


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46 5. Orbit theory

Figure 5.3: The motion of a trapped particle inside a magnetic bottle is illustrated schematically.

point and moves back in the direction of decreasing b grad B. This leads to the concept of amagnetic bottle , which is a region of weak B between regions of stronger B such that particle

can reect at either end, as illustrated in Figure 5.3. Note that particles with sufficiently small in the center of the bottle are not reected, and escape from the ends of the bottle. The range

< 0 for which particles are not trapped is called the loss cone .

5.5 Appendix: formal theory of adiabatic invariants

Any mechanical system that has one or more nearly periodic motions has a nearly conserved

quantity corresponding to each such motion; these conserved quantities are called adiabatic in-variants . Formally, this may be seen simply in terms of Lagrangian or Hamiltonian dynamics. In

terms of Lagrangian dynamics, let us choose one of the generalized coordinates to be the angle,

, corresponding to the quasiperiodic motion, so that the Lagrangian for the system is L(, ),

where the dependence on other variables is of no interest. The Lagrangian equation of motion is

ddt

L

L

= 0 . (5.26)

Suppose one integrates ( 5.26) over one period of the motion, say over 0 < < 2. The congu-

ration of the system is the same at = 2 as at = 0 so that the nal term in ( 5.26) integrates

to zero. Thus ( 5.26) implies that the time derivative of a quantity is zero and hence that the

quantity is conserved. Thus one nds

d L = constant , dQ P = constant , (5.27)which is the desired adiabatic invariant. The second form in ( 5.27) is the corresponding form in

Hamiltonian dynamics, when the periodic motion is in an arbitrary generalized coordinate Q with


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5.5 Appendix: formal theory of adiabatic invariants 47

conjugate momentum P .

A subtle point that is that the conjugate 3-momentum (conjugate to x) is p + q A , where A

is the vector potential of the magnetic eld. This is not important in the derivation of the rst

adiabatic invariant ( 5.23), but it is important when considering the full set of adiabatic invariants

for a particle in a dipolar-like eld. There are three adiabatic invariants, often denoted M,J, .

M is the invariant ( 5.23).

A particle trapped in a magnetic bottle has a quasiperiodic motion corresponding to its bounce

motion between the reection points. There is an adiabatic invariant corresponding to this motion,

sometimes called the second adiabatic invariant. Let the distance, s, along the eld lines be a

generalized coordinate, whose conjugate momentum is p . Then ( 5.27) implies

J = ds p = constant , (5.28)where the integral is along the orbit of the gyrocenter between the reection points.

There is a third adiabatic invariant for particles trapped in a magnetic eld which is roughly

dipolar, as is the case for the Earths magnetic eld within several Earth radii, RE . The curvature

drift causes particles to drift (in magnetic longitude) around the Earth. As this drift is quasiperi-

odic there is an adiabatic invariant associated with it. This invariant is given by the integral of

the component of q A in the direction of the drift around the closed orbit, which integral involves

the radial distance r = LR E , implying that the orbit (rather the center of the bounce motion) of

the particle is conned to a given r or, as is standard jargon in this context, to a given L shell.


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48 5. Orbit theory

5.6 Exercise Set 5

5.1 Calculate the gyroradius of a particle under the following conditions.

(a) A 1 eV electron with pitch angle = / 2 in a laboratory devise where the only magnetic eldis that of the Earth, assuming B = 0 .3 G.

(b) A 2 keV electron with pitch angle = / 2 at the Earths magnetic equator at L = 3. Assume

that the Earths magnetic eld is dipolar with the magnetic eld at the pole B = 0 .3 G. Express

your answer in centimeters.

(c) A 1018 eV ion with pitch angle = / 2 in the interstellar magnetic eld with B = 3 G.

Express your answer in parsecs (1 pc = 3 1016 m).

5.2 An MHD generator is a dynamo that converts mechanical energy into electrical energy. Themechanical energy is in the form of a partially ionized gas forced (blown by a fan for example)

across a magnetic eld. The ow across the magnetic eld creates an electric eld which is such

that the electric drift is equal to the ow velocity. This electric eld is due to a (forced) charge

separation in the plasma. If one puts conducting plates on either side of the ow, surface charges

of opposite sign collect on the two plates. The dynamo operates when one connects the two plates

by a wire (outside the plasma). The voltage associated with the dynamo is found by integrating

the electric eld along a line between the two plates.Let the ow velocity, u , be along the x axis, and the magnetic eld, B , be along the z axis.

The electric eld is along the y axis.

(a) Derive a formula for the electric eld. (b) Derive a formula for the voltage assuming the plates

are a distance L apart. (c) Estimate the voltage for a ow u = 1 m s 1 between plates L = 1 m

apart in a magnetic eld B = 1 T.

5.3 The magnetic eld lines associated with a line current, I , are circles around the axis dened

by the current line. The azimuthal component of the magnetic eld is B = 0I/ 4r , where r isthe radial distance from the axis. Consider a particle with gyroradius much smaller than r moving

around the circular eld line at r , such that its center of gyration has a velocity v.

Find expressions for

(a) the gradient drift, and

(b) the curvature drift.

5.4 Assume the Earths magnetic eld is a dipole, implying that B in the equatorial plane decreases


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1/r 3, with the radial distance r = LR E , where RE = 6 .4 106 m is the radius of the Earth.Particles trapped in the eld drift in azimuthal angle around the Earth due to the grad- B drift.

Electrons and ions drift in opposite sense, and their relative drift implies a current. The drift of

particles trapped in the Earth so-called radiation or van Allen belts produce a ring current.

Estimate the time it takes for a particle to drift around the Earth at L = 4 for

(a) a 1 keV electron, and

(b) a 1 MeV ion.

5.5 A plasma consists of electrons and protons with equal number densities, n, with a uniform

magnetic eld along the x-axis with z the vertical direction.

(a) Calculate the current density, J , due to the gravitational drift of electrons and protons, where

the gravitational acceleration, g, is along the negative z -axis.

(b) Calculate the force per unit volume, J B , due to this current density.(c) Give a physical interpretation of your answer.


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50 5. Orbit theory


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Chapter 6

Cold magnetized plasma

In the absence of a magnetic eld, the response of a cold plasma can be described by a dielectric

constant K () = 1 2 p/ 2. The dispersion equation implies longitudinal waves at K () = 0,implying oscillations at = p, and transverse waves at n2 = K (). When a magnetic eld is

included, the response of a cold plasma is anisotropic, and needs to be described by a tensor.

In this lecture derive this tensor for a cold plasma consisting of electrons and various species of

positive ions. The dispersion equation becomes a quadratic equation for n2 implying that there

are two different natural wave modes of a cold plasma. At high frequencies these become themagnetoionic waves, and at low frequencies they are effectively the MHD modes for zero sound

speed. These limiting cases are discussed in later lectures. In this lecture we are concerned

primarily with describing the response of a cold plasma, and the procedure for calculating the

properties of the natural modes of the anisotropic medium.

6.1 Response of a cold plasma

The response for a cold magnetized plasma may be found by solving the equation of motion for

particles of species , with mass m and charge q :

mdvdt

= q (E + v B ). (6.1)


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52 6. Cold magnetized plasma

Expanding in plane waves and assuming that B is along the z axis, (6.1) may be written in matrix

form:

ivx

vy

vz

= q m

E x

E y

E z

+ q Bm

vy

vx0

. (6.2)

A rearrangement gives

i i 0

i 0

0 0

vx

vy

vz

=q m

E x

E y

E z

, (6.3)

with = q / |q |, = |q |B/m . Solving the matrix equation gives

vx

vy

vz

=i

q m

12 2

2 i 0

i 2 00 0 2 2

E x

E y

E z

. (6.4)

The current density for species is J = q n v . After summing over the contributions of all

species (electrons and ions), the current may be used to identify the dielectric tensor.

Cold plasma dielectric tensor

The relation between the induced current and the electric eld denes the conductivity tensor,

() say. The contribution of species to () follows by multiplying (6.4) by q n . After

summing over species this gives

() =

i

q 2 nm

12 2

2 i 0

i 2 00 0 2 2

. (6.5)

The current may be written in the form J = P /t , and the relation between P and E denes

the susceptibility tensor, () = i ()0. The dielectric tensor is identied as the unit tensor

plus ().


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6.2 Dispersion equation for a cold plasma 53

(b)

z

o

x

(a)

n 2

21.50.5

1

-2

ox

p/

Figure 6.1: Refractive index curves for the magnetoionic waves for e/ p = 0 .5. (a) For = 0

there are two curves plus a vertical line (not shown) at = p. (b) Circled portion of (a)

magnied; the dashed lines is for = 0. For = 0 the o mode and the z mode join at = p

A standard form for the resulting expression for the dielectric tensor for a cold plasma is 1

K () =

S () iD () 0iD () S () 0

0 0 P ()

, (6.6)

S () = 12 [R+ () + R ()], D() =12 [R+ () R ()],

R () = 1 2

p2 , P () = 1

2 p

2 , (6.7)

where the sum is over species, with the th species having mass m , charge q = |q |, numberdensity n , plasma frequency p = ( q 2 n / 0m )1/ 2.

6.2 Dispersion equation for a cold plasma

The wave equation can be written in the matrix form ( 4.6), that is, as E = 0, with =

n2[ 1] + K . The matrix form for , with the coordinate axes chosen such that B is along thez axis and is in the x-z plane at an angle to B , is

=

S n2 cos2 iD n 2 sin cos iD S n2 0

n2 sin cos 0 P n2 sin2 , (6.8)

1 T.H. Stix Waves in Plasmas , McGraw-Hill (1962)


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The dispersion equation is found by setting the determinant of this matrix to zero. This gives a

quadratic equation for n2:

|| = An4 Bn 2 + C = 0 , (6.9)with

A = S sin2 + P cos2 , B = ( S 2 D 2)sin2 + P S (1 + cos2 ), C = P (S 2 D 2). (6.10)The solutions may be written in the form

n2 = n2 =B F

2A, F = ( B 2 4AC )

1/ 2. (6.11)

The two solutions dene two modes. However, these correspond to propagating waves only for

n2 > 0. For n2 < 0 the solutions are said to describe evanescent waves: solutions that oscillate in

time by decay exponentially in space.

6.3 Polarization vectors

The polarization vector eM (k) for any wave mode M in a magnetized plasma may be expressed

in terms of the set of basis vectors

= (sin , 0, cos), t = (cos , 0, sin ), a = (0 , 1, 0). (6.12)These are unit vectors along the wave vector k , along the direction perpendicular to k in the Bk

plane, and along the direction orthogonal to both B and k , respectively. The component of the

electric vector along a is out of phase with the components in the Bk plane. It is convenient to

write

eM =LM + T M t + ia(L2M + T 2M + 1) 1/ 2

, (6.13)

with M = for the cold plasma modes, and M = o , x for the magnetoionic waves. The longitu-dinal part of the polarization vector is described by LM and the transverse part is described by

T M .

The transverse part corresponds to an elliptical polarization, with |T M | the axial ratio of thepolarization ellipse, as illustrated in Figure 6.2. By denition, |T M | is the ratio of the moduliof the component along t to the component along a . The triad of unit vectors ( 6.12) forms a

right hand set, and hence the sign of T M determines the handedness of the ellipse, with T M > 0

corresponding to right hand polarization and T M < 0 corresponding to left hand polarization.


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6.4 Polarization of cold plasma waves 55

D

O

B

A

C

Figure 6.2: The axial ratio |T | is equal to |AC |/ |BD |. The solid ellipse corresponds to T > 0.The wave is propagating into the page. The two ellipses correspond to orthogonal polarizations.

6.4 Polarization of cold plasma waves

So far we have only considered the condition for a solution of the wave equation to exist. When

this condition is satised, a solution of the matrix equation for E exists. The amplitude and phase

of the solution are arbitrary. It is convenient to choose them such that the solution corresponds

to a polarization vector of the form ( 6.13). This involves solving for the parameters T M , LM , with

M = here.The polarization vectors are constructed from any column of the matrix of cofactors of . One

choice gives

T M =DP cos

An2M P S , LM =

(P n2M )D sin An2M P S

. (6.14)

On inserting explicit expressions for the refractive indices into ( 6.14) one nds explicit expressions

for the polarization vectors. However, for computational and other purposes it is more convenient

to note that there is a linear relation between n2 and 1/T , and that because n2 satises a quadratic


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equation, 1 /T and T must also satisfy quadratic equations. For T this equation is

T 2 (P S S 2 + D 2)sin2

P D cosT 1 = 0. (6.15)

It is straightforward to solve the quadratic equation ( 6.15) for T = T and calculate n2 and L

in terms of T by inverting ( 6.14). This allows one to make approximations systematically: one

approximates T , and evaluates the corresponding approximations to n2 and L using (6.14).


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6.5 Exercise Set 6

6.1 At very low frequencies, the two cold modes reduce to the Alfven and magnetoacoustic waves.

(a) Show that in the limit 0 (6.6) with (6.7) implies S (0) = 1 + 2 p / 2 , D(0) = 0,P (0) = .

(b) Show that in this limit the dispersion relations for the two modes reduce to n2 = S (0)/ cos2 ,

n2 = S (0).

(c) Show that one has S (0) = 1 + c2/v 2A .

(d) Hence show that, for v2A c2, the two dispersion relations become 2 = k2v2A cos2 , 2 =

k2v2A , respectively.

6.2 Consider the response of a charge-neutral cold plasma at low frequencies. The plasma is

assumed to be composed of electrons and various species of positive ions with charge q i = Z ie,

mass m i = Aimproton and number density n i .

(a) Show that the charge neutrality condition ne = i Z in i implies

2

p

e=

i

2

pi

i. (6.16)

(b) With the Alfven speed dened by vA = B/ (M )1/ 2, where M is the mass density, show that

if the mass of an electron is neglected compared to that of an ion, then one has

i

2 pi2i

=c2

v2A. (6.17)


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Chapter 7

Magnetoionic theory

Historically, the magnetoionic theory 1 was the second important contribution (after Langmuirs

work) to what developed into modern plasma physics. The motivation was to understand radio

wave propagation in the ionosphere. In this application, the Earths magnetic eld plays an

important role, and the thermal motions of the electrons are unimportant. This corresponds

to a cold magnetized electron gas. The ions play no role, and the name magnetoionic is an

anachronism. The magnetoionic waves are important in understanding wave propagation in radio

astronomy.

7.1 Magnetoionic parameters

In the magnetoionic theory only the contribution of the electrons is retained. The plasma fre-

quency, p = ( e2ne/ 0me)1/ 2, the electron cyclotron frequency, e = eB/m e , and the wave fre-

quency, , are combined into two magnetoionic parameters:

X =2 p

2, Y =

e

. (7.1)

The dielectric tensor ( 6.6) has components

S =1 X Y 2

1 Y 2, D = XY

1 Y 2, P = 1 X. (7.2)

The coefficient (6.10) become

A = [1 X Y 2 + XY 2 cos2 ]/ (1 Y 2),1 developed by Appleton and Hartree in the early 1930s


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60 7. Magnetoionic theory

(b)

z

o

x

(a)

n 2

21.50.5

1

-2

ox

p/

Figure 7.1: Refractive index curves for the magnetoionic waves for e/ p = 0 .5. (a) For = 0

there are two curves plus a vertical line (not shown) at = p. (b) Circled portion of (a)

magnied; the dashed lines is for = 0. For = 0 the o mode and the z mode join at = p

B = [2(1 X )2 2Y 2 + XY 2(1 + cos2 )]/ (1 Y 2),C = (1 X )[(1 X )2 Y 2]/ (1 Y 2). (7.3)

For the magnetoionic modes, the two solutions ( 6.11) can be rewritten as

n2 = 1 X (1 X )

1 X 12 Y 2 sin2 + , (7.4)

with = 1 and with 2 = 14

Y 4 sin4 2 + (1 X )2Y 2 cos2 . (7.5)The two solutions are called the ordinary (o) and extraordinary (x) modes. The technical denition

of the ordinary mode is that it is the mode that has n2 1X for / 2; for > p (X < 1),this denition corresponds to n2o = n2+ and n2x = n2 .

7.2 Cutoff frequencies

Transverse waves in an isotropic plasma have n2 = 1 2 p/ 2, and so they exist as propagatingwaves only for > p, where n2 is positive. The frequency where n2 becomes zero is referred to

as the cutoff frequency.

The cutoff frequencies for the o- and x-modes are at n2+ = 0 and n2 = 0, respectively. More

generally, cutoffs occur at n2 = 0, where ( 6.9) implies C = 0, which gives

P (S 2 D 2) = 0 . (7.6)


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1 2

-10

10

20

0

w

z

x

o

n 2

p/

Figure 7.3: As for Figure 7.1 but plotted on a different scale. The whistler (w) mode branch is in

the upper left hand corner.

The handedness of observed radio emission is dened as a screw sense relative to the direction,

, of wave propagation. This is the same as the screw sense relative to b for < / 2 and the

opposite sense for > / 2. For some purposes, it is convenient to label the refractive index in

terms of right ( r ) and left ( l) hand polarizations:

n r,l 1 12 X 12 XY cos . (7.10)

The difference between the refractive indices causes Faraday rotation, which is important in

radio astronomy. Faraday rotation is the rotation of the plane of linear polarization as radiation

propagates through a magnetized medium whose waves modes are circularly polarized. One can

understand Faraday rotation qualitatively from the following idealized example. Suppose radiation

at its source, at s = 0, is linearly polarized along the 1-axis, and that it propagates along the 3-axis.

One can separate the initial linear component into right and left hand circularly components of

equal amplitude. After the radiation has propagated a distance s, the refractive index difference,

n say, implies that the two modes are out of phase with each other by k s, with k = n/c

the difference in wavenumber. On recombining the two circularly polarized components, they give

a linearly polarized component whose plane of polarization is rotated from its original direction.

The plane of linear polarization rotates at a rate k/ 2 per unit length.


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7.4 Polarization of magnetoionic waves 63

7.4 Polarization of magnetoionic waves

The polarization vectors for the magnetoionic waves are constructed from any column of the

matrix of cofactors of . One choice gives (6.14). An alternative procedure is to solve ( 6.15) forT = T and use (6.14) to nd n2 and L in terms of T . For the magnetoionic modes ( 6.15) becomes

T 2 +Y sin2

(1 X )cos T 1 = 0. (7.11)

The solutions of (7.11) are

T = T =Y (1 X )cos 12 Y 2 sin

2 =

12 Y

2 sin2 Y (1 X )cos

, (7.12)

with 2 given by (7.4), and where = 1 corresponds to the o mode and = 1 to the x mode.The two polarization ellipses are orthogonal in the sense

T + T = 1. (7.13)Approximations to the axial ratio follow by considering the ratio of the two terms in the square

root for F , cf. (6.11) or , cf. ( 7.4). For |(1 X )cos | 12 Y sin2 one ndsT

cos

|cos|1 X |1 X |

1 +Y sin2

2|(1 X )cos |+ . (7.14)

The leading terms in ( 7.14) correspond to circular polarization, with the handedness such that the

electric vector in the x mode and in the whistler mode rotate in the same sense as that in whichelectrons gyrate (right hand screw sense relative to B ) and the electric vector in the o mode and

the z mode rotate in the opposite sense. This is called the quasi-circular limit . The corresponding

approximation to the dispersion relations is, for Y |cos| 1, and X < 1,n2 = 1 X (1 Y |cos|+ ) = 1

2p2

(1 e|cos|

+ ). (7.15)

In the opposite limit |(1X )cos | 12 Y sin2 , called the quasi-planar limit or quasi-linear limit ,one has

T o , n2o 1 X, L o XY sin

1 X ; (7.16)

T x 0, n2x 1 X (1 X )

1 X Y 2 + XY 2 cos2 ,

Lx XY sin

1 X Y 2 + XY 2 cos2 . (7.17)

The transverse parts of the polarization correspond to linear polarizations along t and a for the

ordinary and extraordinary modes respectively.


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7.5 Exercise Set 7

7.1 The low-frequency branch of the magnetoionic o-mode is called the whistler mode, where the

name originates from whistling atmospherics heard in early radio receivers. (These waves areoften called helicon waves in laboratory applications.) This exercise is to derive the properties of

the whistler mode.

(a) Show that in a plasma with 2e 2 p, (7.5) implies = |(1 X )Y cos| except for a smallrange of angles about = / 2.

(b) Show that for 2 2e 2 p the solution = +1 in ( 7.4) implies

n2o 2 p

e|cos|. (7.18)(c) Show that the polarization vector for the whistler modes in this approximation is

eo =(1, i|cos|, 0)(1 + cos2 )1/ 2

. (7.19)

(d) Evaluate the partial derivatives in the expression

vgo =c

(no)/

1no

n o

t (7.20)

for the group velocity.

(e) Hence show that the group velocity is

vgo =c

no(sin , 0, cos + sec ). (7.21)

(e) Dene the ray angle by writing vgo = |vgo|(sin r , 0, cosr ), show that one hascos2 r =

(1 + cos 2 )2

1 + 3 cos2

. (7.22)

(f ) Show that there is maximum ray angle sin r = 1 / 3 corresponding to sin2 = 2 / 3.

7.2 A formal treatment of Faraday rotation involves the Stokes parameters, I ,Q,U,V , which

involve the outer produce of the wave amplitude and its complex conjugate. The degrees of

polarization are

p =(Q2 + U 2 + V 2)1/ 2

I , pl =

(Q2 + U 2)1/ 2

I , pc =

V I

. (7.23)


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The axial ratio, T , and the angle that denes the plane of linear polarization are determined by

Q pI

=T 2 1T 2 + 1

cos2,U

pI =

T 2 1T 2 + 1

sin2,V

pI =

2T T 2 + 1

. (7.24)

Show that the rate of Faraday rotation is given by

dds

=2 pe cos

2c2(7.25)

where (7.10) is used.

7.3 The plane of polarization is an observable quantity, and measurement of it at several frequen-

cies provides information of the properties of the medium along the ray path between the source

and the telescope. Assuming a homogeneous medium along the ray path, the angle through which

the plane of polarization is rotated is

=2 pe22c

D cos, (7.26)

where D is the distance to the source. Taking variations in the properties of the medium along

the ray path into account, ( 7.26) is replaced by an integral along the ray path:

=e3

20m22c

D

0ds B ne . (7.27)

The dependence on frequency implies a dependence on the square of the wavelength, = 2 c/ .

If one measures at different wavelengths one can determine the constant of proportionality,

which is called the rotation measure (RM).

Show that the rotation measure, dened by writing = RM 2, is given by

RM =e3

2(2)20m2c2 D

0ds B ne. (7.28)

The units of RM are inverse length square, usually m 2.

7.4 The polarization of transverse waves is described by the matrix

p = 121 + pQ pU ipV

pU + ipV 1 pQ. (7.29)

The degree of polarization, p, may be identied by writing p = 12 (1 p)1 + pee with

p = ( p2Q + p2U + p

2V )

1/ 2. (7.30)


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If two signals with intensities I 1 and I 2 and polarization matrices p 1 and p 2 are added incoherently,

the polarization matrix of the resulting radiation is

p =I 1p 1 + I 2p 2

I 1 + I 2 . (7.31)

Consider two signals of equal intensity being added, one that is completely linearly polarized

( pQ = 1) and the other that is completely circularly polarized ( pV = 1).

(a) What is the degree of polarization of the combined radiation?

(b) What is the axial ratio of the polarization ellipse?

(c) How does the result change if you assume pU = 1, rather than pQ = 1, for the linearly

polarized

phys3947 plasma physics

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