Boyd T.J.M., Sanderson J.J. The Physics of Plasmas

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6.6 Absolute and convective instabilities 247

Fig. 6.22. Dispersion curves for weakly coupled modes.

the ω(k) are complex and one of the pair has ω = ω

(k)>0 and so is unstable.

The instability is convective since for |ω|→∞the roots k(ω) are approximately

ω/v

and ω/v

and when ω

→∞, they fall in the same half k-plane. For v

> 0 they lie in the upper half k-plane. For real ω in the range (6.129) the roots

k(ω) form a complex conjugate pair. The root with k

(ω) < 0 has crossed to the

lower half-plane. Across the frequency range deﬁned by (6.129) waves propagating

in the positive x-direction will amplify.

(d) Here k(ω) is real for all real ω but ω(k) is complex across the range (6.130).

The system is therefore unstable. Since v

< 0asω →∞it follows that the

roots k(ω) fall in opposite half-planes. The roots coalesce at a point in the upper

248 Waves in unbounded homogeneous plasmas

half ω-plane for which

ω = ω

= ω

+ 2i

√

(v

)

− v



(6.131)

This corresponds to an absolute instability with growth rate ω

Exercises

6.1 Obtain expressions for the phase and group velocities of the following

modes:

(i) Alfv

en: ω

= k

/(1 + c

)

(ii) whistler: ω =−k



/ω

(iii) electron cyclotron (ω →|

|): ω

− ωω

/(ω + 

) = k

6.2 Using the data in Table 1.1, compute Alfv

en wave speeds for plasmas in

(i) interstellar space, (ii) solar corona, (iii) ionosphere and (iv) tokamak.

Assume that the positive charges are protons in (i), (ii) and (iv) and oxygen

) ions in (iii).

6.3 The energy density of a wave propagating in a plasma is the sum of con-

tributions from the oscillating electric and magnetic ﬁelds and from the

coherent particle motion induced by these ﬁelds. Suppose that the ﬁelds

show a small degree of exponential growth so that their behaviour with

time is described by exp(−iω

+ γ)t where γ  ω

. The rate of change

of the energy density averaged over a period is given by

(E

∗

· j) +

∂

∂t



|E|

|B|



where ∗ denotes the complex conjugate. Writing

= iωε

[δ

− ε

(ω)]E

where ω denotes the complex frequency ω = ω

+ iγ , show, using a

Taylor expansion along with the relation (∂/∂t)|E|

= 2γ |E|

, that the

wave energy density may be expressed in the form

W =





∂

∂ω

(ωε

|B|



Apply this result to an electromagnetic wave propagating in an isotropic

plasma (B = 0), identifying the contribution to the energy density from

coherent particle motion.

Exercises 249

Fig. 6.23. Measurement of the dependence of Alfv

en wave phase velocity on

axial magnetic ﬁeld in a hydrogen plasma compared with theory (after Wilcox et

al. (1960)).

6.4 By introducing a term into (6.10) to allow phenomenologically for the ef-

fects of electron–ion collisions through a collision frequency ν

, show that

the dispersion relation for electromagnetic waves in an isotropic plasma

becomes

= 1 −

ω(ω + i ν

)

Hence show that electromagnetic waves are damped as a result of electron–

ion collisions, with damping coefﬁcient γ = ν

(ω

/2ω

6.5 Figure 5 shows the Alfv

en wave velocity, as a function of the axial mag-

netic ﬁeld, measured in a hydrogen plasma by Wilcox, Boley and De

Silva (1960). The plasma temperature was 1 eV and the proton density

5 × 10

−3

. Using the cold plasma dispersion relation plot the Alfv

velocity as a function of magnetic ﬁeld. Verify that the cold plasma ap-

proximation is valid in this parameter range. How might the discrepancy

between the measured phase velocities and those from the simple Alfv

dispersion relation be explained?

6.6 Show that the dispersion relation for a wave propagating orthogonally to a

magnetic ﬁeld B

with its electric vector aligned with B

is ω

= ω

Explain the physical signiﬁcance of this result.

6.7 Show that the points of intersection of the plasma cut-off P = 0 with the

plasma resonance S = 0 and cyclotron cut-offs L = 0 in a two-component,

cold plasma occur at α

= 1, β

= 1 − m

and α

= 1, β

= 1 −

+ (m

)

, respectively.

250 Waves in unbounded homogeneous plasmas

6.8 Verify the topological representation of the wave normal surfaces in the

CMA diagram, Fig. 6.12.

6.9 The wave normal surfaces corresponding to fast and slow modes coincide

when the discriminant (6.34) vanishes. Find the modes propagating along

the magnetic ﬁeld for which this is possible.

For propagation orthogonal to the magnetic ﬁeld show that there is a

curve in the CMA diagram on which coincidence is possible and obtain its

equation.

6.10 Determine the group velocity v

of the ion cyclotron wave satisfying the

dispersion relation (6.71) and show that v

→ 0asω → 

. Show that the

wave is elliptically polarized, becoming LCP at the resonance frequency.

6.11 Derive (6.73) from (6.31). [Hint: Write (6.31) as n

= ( An

−C)/( An

A − B) and then substitute the biquadratic solution of (6.31) for n

on the

right-hand side.]

6.12 Determine the conditions under which the second term in the discriminant

in (6.94) is not small compared with unity.

Show that in ion acoustic waves, electron and ion velocities are of com-

parable magnitude.

6.13 Show that in a homogeneous isotropic plasma the conduction current and

displacement current in Langmuir waves cancel exactly.

6.14 Show that the dispersion relation for obliquely propagating ion cyclotron

waves in a warm plasma is

2

+ v

(

− ω

)

(

cos

θ − ω

)

Rearrange this in the form ω = ω(k) and from this show that, in the

long wavelength limit (k

 

), ω

 

+ k

sin

θ. This mode

is electrostatic as is the mode in the limit k

 

. Write down the

dispersion relation for this case.

6.15 Consider how the ion cyclotron resonance changes when a plasma contains

two ion species, as for example in the solar wind which consists of protons

with helium ions as the principal minority constituent.

Plot dispersion curves k

/ω

versus ω/ for a plasma with 80%

protons and 20% He

ions for θ = 0 and θ = π/2.

Show that as propagation switches from θ = 0 towards θ = π/2 the

resonances move from the cyclotron frequencies to the frequencies deter-

mined by S = 0, with one of the resonant frequencies shifted to

H−He



+ ω

pHe



+ ω

pHe

Exercises 251

while the other is displaced to lower frequencies.

6.16 Conﬁrm that (6.120) can be written as a three-wave interaction as

/ω

→ 1. Show that the maximum growth rate, corresponding to

k = k − ω

= 0, is given by (6.121).

Collisionless kinetic theory

7.1 Introduction

Much of plasma physics can be adequately described by ﬂuid equations, namely,

the MHD or wave equations. However, these are derivative descriptions in which

some information about the plasma has been suppressed. In situations where that

information matters it is necessary to go to a deeper level of physical description.

The information that gets lost in a ﬂuid model is that relating to the distribution

of velocities of the particles within a ﬂuid element, since the ﬂuid variables are

functions of position and time but not of velocity. Any physical properties of the

plasma that depend on this microscopic detail can be discovered only by a descrip-

tion in six-dimensional (r, v) space. Thus, instead of starting with the density of

particles, n(r, t), at position r and time t, we begin with the so-called distribution

function, f (r, v, t), which is the density of particles in (r, v) space at time t. The

evolution of the distribution function is described by kinetic theory.

With the additional information on particle velocities within a volume element

introduced by a phase space description we now have microscopic detail that we

did not have before. For that reason, kinetic and ﬂuid theories are identiﬁed as

microscopic and macroscopic, respectively.

At the most fundamental level we may deﬁne the distribution function in terms

of the individual particle positions and velocities by

(r, v, t) =



i=1

δ[r − r

(t)]δ[v − v

(t)] (7.1)

where the sum is over all particles of a given type. This is the Klimontovich dis-

tribution function which we have denoted by f

to distinguish it from f .Itis

a very spiky function being zero throughout (r, v) space except at the N points

[r = r

(t), v = v

(t)] where it is doubly inﬁnite. However, we can generate a

smoother function by integrating (7.1) over a volume element rv about the

252

7.1 Introduction 253

point (r, v) which is large enough to contain a number of particles N

(r, v, t)  1

but small enough that

f (r, v, t) =

rv



r



v

dv f

(r, v, t)

rv

(7.2)

does not change signiﬁcantly over the dimensions of the volume element. Thus,

f (r, v, t) is the number density of particles in a small volume element centred at

the point (r, v) at time t.

Provided no particles are created or destroyed, f obeys a continuity equation

in (r, v) space which is derived by exactly the same arguments used to derive

the continuity equation for the mass density ρ(r, t) in Chapter 3. There are now

two divergence terms arising from the ﬂow of particles through the surfaces of the

volume element in both r and v space. Thus, we have

∂ f

∂t

∂

∂r

· ( f v) +

∂

∂v

· ( f a) = 0 (7.3)

where a is the acceleration of the particles in the volume element. Since r and v

are independent variables in (7.3) we may bring v outside the differential operator

and, if in addition ∇

· a = 0, we get

∂ f

∂t

+ v ·

∂ f

∂r

∂ f

∂v

= 0 (7.4)

in which we have replaced a by F/m where F is the force acting on the particles of

mass m at the point (r, v) at time t. Such a partial differential equation describing

the evolution of the distribution function is known as a kinetic equation.

As a matter of fact, (7.4) is necessarily a collisionless kinetic equation since we

certainly cannot assume ∇

·a = 0 if we want to include the collisional interactions

taking place inside the volume element. A proper description of collisions is a

formidable problem as we shall see in Chapter 12. However, the transition from

(7.4) to a collisional kinetic equation can be made by a simple heuristic argument.

Assuming that F represents all the non-collisional (macroscopic) forces we note

that (7.4) states that

d f (r, v, t)

∂ f

∂t

∂ f

∂r

∂ f

∂v

∂ f

∂t

+ v ·

∂ f

∂r

∂ f

∂v

= 0

i.e. in the absence of collisions f is constant along any trajectory in (r, v) space.

Collisions, however, change this so we write

∂ f

∂t

+ v ·

∂ f

∂r

∂ f

∂v



∂ f

∂t



(7.5)

254 Collisionless kinetic theory

where (∂ f/∂t)

represents the change in f with time due to collisions. This is then

the collisional kinetic equation, though how we represent the collision term in (7.5)

is a problem we defer until the next chapter.

In this chapter we are concerned only with collisionless kinetic theory for the

very good reason that most plasmas are essentially collisionless. All of the terms

in (7.5) have the dimensions of f times a frequency. The frequency appropriate

to the right-hand side is, of course, the collision frequency ν

while that on the

left-hand side depends on the dominant macroscopic force. Since this macroscopic

force is typically the Lorentz force due to the self-consistent ﬁelds the appropriate

frequency is likely to be one of the wave frequencies encountered in the previ-

ous chapter. In particular, we have noted the dominance of the electrostatic ﬁeld

in maintaining charge neutrality and causing oscillations at the electron plasma

frequency in response to any local charge inequality. As the plasma frequency is

usually much greater than the collision frequency, unless we are speciﬁcally inter-

ested in collisional effects, we can ignore the collision term and take (7.4) as the

kinetic equation. We shall study it in order to discover some important properties

of plasmas which depend on distributions of the plasma particles in velocity space

and which, therefore, are not accessible to ﬂuid descriptions.

7.2 Vlasov equation

Vlasov ﬁrst solved the collisionless kinetic equation (7.4), now known universally

as the Vlasov equation, in the case where F = eE(r, t) and E is the self-consistent

electric ﬁeld. Interestingly he did not solve (7.4) as an initial value problem and

consequently missed its most important property! This was subsequently discov-

ered by Landau and is discussed in the next section. The collisionless kinetic

equation is sometimes referred to as the collisionless Boltzmann equation but this

is something of a contradiction in terms since the representation of collisions is at

the very heart of the Boltzmann equation.

The ﬁrst thing we shall do with the Vlasov equation is to show its formal equiv-

alence to the equation describing individual particle orbits. The latter is

r = F (7.6)

having the solution

r = r(c

, c

,...,c

, t)

v = v(c

, c

,...,c

, t)



(7.7)

where c

, c

,...,c

are the six constants of integration, which might for example

be the initial values of r and v. Inverting (7.7) gives the formal solution

= c

(r, v, t)(i = 1, 2,...,6) (7.8)

7.2 Vlasov equation 255

Now any arbitrary function of the c

f = f (c

, c

,...,c

)

is a solution of (7.4) as one can see by direct substitution:



i=1

∂ f

∂c



∂c

∂t

+ v ·

∂c

∂r

∂c

∂v





i=1

∂ f

∂c

= 0

since the c

are constants of the motion. Thus the general solution of the Vlasov

equation is an arbitrary function of the integrals of (7.6), the equation describing

orbit theory. This was demonstrated by Jeans in work on stellar dynamics and so is

generally known as the Jeans theorem.

The formal equivalence of the Vlasov equation and particle orbit theory is rig-

orous and simple but, in one sense, slightly deceptive. It should not be imagined

that solving the Vlasov equation is as easy as ﬁnding some of the orbit solutions

obtained in Chapter 2. The reason is that, in the Vlasov equation, F contains both

external ﬁelds and self-consistent ﬁelds arising from the plasma motion. In Chap-

ter 2 we assumed that the latter were negligible compared with any applied ﬁelds

and so the orbit equations solved there are only approximations to the equation of

motion (7.6). We shall see later that there is a direct relationship between solving

the linearized Vlasov equation and calculating the simple (unperturbed) orbits of

the particles in the external ﬁelds. We note, in this context, that Jeans’ theorem pro-

vides a method of obtaining zero-order, equilibrium distribution functions, namely,

any function of the constants of the motion in the (zero-order) external ﬁelds.

Illustrations of this appear in later sections of this chapter.

For the moment, we observe that the full set of equations to be solved, in the

general case, is the Vlasov equation (7.4) for the distribution function of each

species of particle together with the Maxwell equations:

∇ × E =−

∂B

∂t

(7.9)

∇ × B = ε

∂E

∂t

+ µ

j (7.10)

∇ · E = q/ε

(7.11)

∇ · B = 0 (7.12)

where

q =





dv (7.13)

j =





v f

dv (7.14)

256 Collisionless kinetic theory

and the sums are over the particle species. The fact that there are at least two

Vlasov equations to be solved (one each for ions and electrons) is a relatively minor

complication compared with the problem of solving the full Vlasov–Maxwell set

of equations self-consistently for f , when f itself is a source of the ﬁelds through

(7.13) and (7.14). (The equivalent of this latter problem in orbit theory would

be ﬁnding the orbits in the full self-consistent ﬁelds.) Nevertheless, the Vlasov–

Maxwell equations are the starting point for most calculations in plasma kinetic

theory. They include the principal effect of particle interactions, the self-consistent

ﬁeld, and in the approximation that there are a large number of particles within the

Debye sphere (nλ

 1), known as the weak coupling approximation (see Exer-

cise 7.3), they provide an adequate description unless we are speciﬁcally interested

in collisional effects. Their complexity means that one generally has to resort to

numerical methods even when a linear solution is sought. In the following sections

we present some important solutions of the linearized set of equations which are

amenable to analytic methods.

7.3 Landau damping

The most important and fundamental property of the Vlasov equation was dis-

covered by Landau (1946) who solved the linearized electron Vlasov equation for

F =−eE where E is the electric ﬁeld created when a homogeneous plasma in

equilibrium is slightly perturbed. It is assumed that the perturbation is in the elec-

tron distribution only, so that the ions remain as a steady, homogeneous, neutral-

izing backgound. This simplifying assumption avoids having to solve two Vlasov

equations. Following Landau, we solve the linearized equation as an initial value

problem, i.e. the perturbation is introduced at t = 0. The alternative in which a

perturbation is introduced at r = 0 and its spatial evolution examined is considered

in Section 7.5.

Any steady, homogeneous distribution function f

(v) satisﬁes (7.4) identically,

since the electron density is uniform and equal to the ion density and there is no

electric ﬁeld. If a small perturbation f

(r, v, t) is introduced, we may write

f (r, v, t) = f

(v) + f

(r, v, t) (7.15)

and, since the contribution of f

to E is zero, |E| is of order f

and the linearized

Vlasov equation is

∂ f

∂t

+ v ·

∂ f

∂r

−

∂ f

∂v

= 0 (7.16)

where, from (7.11) and (7.13),

∇ · E =−



dv (7.17)