Rfct-fc.Kt.NOc:IC/66/64

INTERNATIONAL ATOMIC ENERGY AGENCY

' • * :

INTERNATIONAL CENTRE FOR THEORETICALPHYSICS

LECTURESON THE NON-LINEAR THEORY

OF PLASMA

R. Z. SAGDEEV

AND

A. A. GALEEV

1966

PIAZZA OBERDAN

TRIESTE

IC/66/64

I n t e r n a t i o n a l Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEOEETICAL PHYSICS

LECTURES OH" TEE NON-LIUEAR THEORY OP PLASMA

H.Z. Sagdeev*

and

A.A. Galeev *

TRIESTE

September 1966

* Permanent address: Institute of Nuclear Physics,Novosibirsk, USSR

tii'iSh&:* * ! •«••

LIST OF CONTENTS

Page No.

PART I: THE WAVE-WAVE INTERACTION IF PLASM

1. Resonant non-linear coupling "between plasma waves ............ 1

2. Interaction of the three waves of finite amplitude .*,,,,,,,,, 8

3. Many-wave interaction in random phase approximation 20

4. Plasma turbulence in terms of the kinetic equation for

waves ...................,,.*.,« , , 28

5. Coupling "between different modes in a turbulent plasma ...... 35

References «... , 40

PART II: PARTICLE-WAVE INTERACTION

1. Particle interaction with a single wave 42

2. The many-waves case , 51

3. Quasi-linear theory of the electromagnetic modes 62

4. Quasi-linear theory of the Post-Rosenbluth "loss-cone"

instability .....*......«*. 70

5. Hon—resonant adiabatic wave-particle interaction 8l

6. ^uasi-linear theory of drift instability 90

References 10<?

PART III; WAVE-PARTICLE NON-LINEAR INTERACTION

1. Electron plasma oscillation turbulence **..*»...*...«•*.«... 112

2. Current-driven ion sound turbulence *•*•*... 118

3. Non-linear theory of the drift instability ....*....... 131

4. The general scheme of weak turbulence theory 140

Referencea , * * 145

PART IV: NON-LINEAR PLASMA DYNAMICS IN COLLISIONLESS SHOCKS

1. .Non-linear dispersive wave trains and solitons ............. 147

2. The onset of instability and turbulence 150

Simplified schematic table 159

References 160

Note: Equations and references are numbered separately for each part.

PART I \ THE WAVE-WAVE IHTERACTI01T I F PLASMA

1. Resonant Non-linear Coupling Between PlaBma Waves

We would like to start with the linear theory of plasma waves.

In a uniform plasma we can expand the equations of the linear wave

theory in series of Fourier harmonics* The main problem of this

chapter will "be to apply a systematic approach of this kind to non-

linear phenomena..

The simplest approach to nonlinear dynamics is to look first

of all for very small amplitude osoillations in fluid-type equations}

then to look at finite amplitudes, where non-fluid-type phenomena

depending on the Vlasov equation are now included.

If we start from the analogy with non-linear gas kinetics and

assume we have monochromatic waves of frequency' Wv» then the main

nonlinear effect will "be steepening of these waves»

Fig. 1

We can describe this process as the generation of higher'harmonics*

If the plasma wave satisfies a linear equation, then this can

be written as

L (<*>»*)

» linear operator

Taking into account the non-linear terms in real plasma oscillations,

we get in the simplest approximation a quadratic term (containing ID )

with a small operator I *

If we apply perturbation theory to obtain the solution we have

f ~ fo + fx

where

and f<

If the relation -

then 2 ldK and

•o (K/has no dispersion ( / = constant),

also satisfy it. Then.the driving term on the

r.h.s» resonates with a normal mode of the unperturbed system, caus-

ing (secular) growth. If, on the other hand, the relation £ 0 = O ( ^

is far from linear (the modes are characterized by large dispersion)-

then ^ is driven only to finite amplitude. "This is generally the

case with plasma waves, in contrast with acoustic phenomena in fluid

dynamics.

Qualitatively, we see that if we start with an initially mono-

chromatic wave of frequency £0, a substantial amount of energy will

be transferred to the 2CJ mode, a small amount to 3 ^ , and higher

modes will be almost unexoited. This qualitative illustration demon-

strates that periodic waves in a plasma, in the region of frequencies

where there is an essential deviation from a linear law of dispersion,

may propagate without considerable non-linear shape distortion, moreover,

such one-dimensional periodic wares are sometimes the exact steady-state

non-linear solutions

* This problem will be considered in a second set of lecturest

"Collision-free Shooks in Plasma"

- 2 -

From these results we may drawiiie conclusion that noniinear

periodic one-dimensional waves in a plasma can exist for a long time,

at least in a case of no damping. However, we must also ascertain

if they are stable with regard to various random distortions. If

they turn out to be unstable, this implies a transition of their

energy to some other type of motion of the plasma, possibly to random

turbulent motion.

We will illustrate the physical mechanism of one of these in-

stabilities by means of the example of fhe simplest

nonlinear plasma wave, that isT the flute oscillation (the Alfven

mode in the ICED limit)C?J, For this mode, noniinear effects never

change the wave sh&pe even for the linear dispersion law 0) - K V^ ,

because of transversal polarization.

We will take an initial undisturbed flow as a monochromatic wave

described by

«. 2 %M4 Cos

\£ , t , etc. can be related to n ^ by the fluid and Maxwell equations,

and so are also specified by implication. Of course, any other

periodic dependence (e.g., exponential) on C^o^ "(J o'tJ could have

heen taken, but this is convenient to work with.

In order to treat the stability of this wave as usual, we in-

vestigate the time dependence of the small perturbations. We write

the fluid velocity V , field H and number density // in terms of

perturbed and unperturbed quantitiesi

w}jere small letters designate perturbed quantities and U, is the velocity

with which the Alfven wave propagates in the z-direction, Tor the sake

of simplicity we will restrict ourselves to consider one-dimensional

perturbations.

First we discuss the equation for perturbed longitudinal motion

(i.e., |j ). Time derivatives of perturbed quantities are treated by

assuming dependence like &Xp (-W>t) . The pressure term on the r.h.s.

is rewritten using the adiabatio law connecting pressure and density,

and the lowest order magnetic force term is retained!

Here C s is the sound velocity, and H ^ is taken as a first-order

quantity.

The linearized continuity equation is

Eqs. (l) and (2) just describe the longitudinal behaviour of the system,

and>with the deletion of the last term in (l), yield longitudinal sound

waves.

For the third equation, we take the linearized induction equation

which gives an equation generating ll^ . The fourth equation describes

the change of the transverse linearized velocity, driven by the magnetic

force, expanded for small density perturbation

U

Here & - KoVl is "the unperturbed mass density and H O is the uniform

magnetic field along which the Alfven waves propagate. Eq,s. (3) and (4)

describe the tiansverse motion and if the last terms (the nonlinearity)

are dropped, yield the dispersion relation for Alfven waves.

Let us make an additional simplifying assumption, namely,ft* ^

We can estimate the effectiveness of the nonlinear terms in (3) and (4)

by comparing them with linear terms in their respective equations.

and

Prom (l) we can estimate (neglecting the non-linear term)

implies V* ^ C ^ , or since UJ -x k^C^ tC '•' >> - f-Kj/tlL* ^ai-

we see, on substituting for \H , that this implies T S ^ ^ T *j thus

the noniinear term of (3) may be dropped, and we are left with a very

symmetrical system of equations. They almost split into longitudinal

and transverse systems, but do not because of the crossing terms in the

equations

Let us calculate the effect of these crossing terras by means of per-

turbation theoryf3j. As we know from quantum mechanics, the "energy

shift" (properly, the frequency shift here) ia caloulated in first order

by means of the diagonal matrix element

If we take for Vo either of the two eigenmodes of tins unperturbed

system, the a-integration gives zero*

- 5 -

because of the periodic "behaviour of the perturbed "potential". If,

however, we start with a pure state (consisting, in this case, of the,

Alfven wave) and include a small amount of the other mode in the per-

turbation, the result in second order is

provided K>^ i k . o ~ K.^-0 . Thus, if there are waves in the excited

spectrum which satisfy this selection rule, the matrix elements can be

non-zero, in constrast to the non-degenerate situation.

In the wavo frame, the argument of the cosine in "^becomes simply

Kot , i.e., the frequency U)oof the wave in its own rest frame vanishes,

so that the seleotion rule

holds trivially, stating that both modes propagate at the same frequency.

We can now write the z-dependence of a l l longitudinal perturbed

quanti t ies as Q & and of a l l transverse perturbed quanti t ies as 6

and use the momentum selection rule ( i . e . , KA - K.e-tK5 ) to cancel

th is z-dependence everywhere, ffow (l) to (4) take the form

(5)

iKsun + i/cs riov;, - o (6)

- 6 -

-ico

iAjtUL nMH.iK.iHj.

Note that in (5) and in (8) only one of the exponentials in C©£ (Ksatisfies the selection rale, and the other may thus be dropped. Comhin

(5) and (6) we get

(9)

Likewise, combining (7) and (8) and substituting fOT tx from (6) we find

+

Eqs. (9) and (10) without the crossing terras on the r.h.s« are

the Doppler--ohifted dispersion relations for sound and Alfven waves,

respectively. In order to (iscuss stability we put ft) +^s^if * (0$ + I*

cO 1 * i ^ U) V " ^A" V*2") a n d calculate the determinant of this system

of equations* The result is

The selection rules used in Eq.s. (5) to (8) are

(13)- 7 -

Because of the form of the dispersion relations, al l terms in the a"bove

equations can be written with either plus or minus sign. For convenience,

let us specify that CO o •» Ko V^ >O t JVom the assumption 9* *"** ~

i t follows that Ct, « VA . Now, if COA «. KA V^ , (12) and (13)

imply that ICS "=• O ; therefore, we must have LO& s-— K*V/\ * Then

K/v'Ss ~K£6 and Kj, "X -2-K o <O . Thus, Eq. ( l l ) implies

(instability) only for CO5 <-0 • The energy of a "quantum" of the

in i t ia l wave must "be larger than the energy of a "quantum" of the per-

turbed waves ( i . e . , h)o ">

2. Interaction of the three waves of finite amplitude

The problem of the stability of Alfven waves of small amplitude was

considered in Section 1. For disturbances in the form of a sum of sound

and Alfven waves, we can write two coupled equations for the perturbed

longitudinal velocity and transverse magnetic field. These equations

describe the coupling between sound and Alfven waves, We have found

the frequency rule for the decay instability, which corresponds to the

usual energy conservation law of quantum mechanics. Therefore, it can

be seen more directly if we write the wave equations in the Hamiltonian

form using quantum parameters, such as the energy of a wave quantum

and the number of waves.

To do this, let us rewrite the wave equations. \fe will replace UJJ

and Kx by £ VI (O} h± [£)] CKp (-id 4 +i«2) where Vu (£) andh^l-tr) a r e slowly varying and 60 and l . satisfy the sound dispersion

relation in the longitudinal equation and the Alfven dispersion relation

in the transverse equation. If we go to the laboratory coordinate system

and label transverse quantities, with an index (2) and longitudinal ones

with an index (l),

- 8 -

Eqs. (9) and (10) beoome

where ire have'now taken into aocount the phase of the finite amplitude

Alfven wave d H *=• I* H \ € * . Let us define the number of quanta W ^

as the total energy in a mode divided by the frequency of the mode.

f

Then we can write Eqis, (14) and (15) in a symmetrical form by introducing

probability amplitudes ( i . e . ,

amplitudes are the following!

probability amplitudes (i#e«, IC; | =• V\ ' )• For the present case these

C.W - - =__ , c, W =

In terras of these variables, Eqs. (14) and (15) can be written in a form

similar to the Schrodinger equation in the interaction representation^ '

- 9 -

7\ JL = Yk (19)

where

Lft-'• " W

We see that the "matrix elements" of the interaction operators differ only

"by a sign for the two modes. This is due to the Hamiltonian form of the

hydromagnetic equations. Starting from this point we would like to in-

troduce an important generalization: For any kind of plasma waves (not

only for Alfven ones), if it is possible to represent three-wave inter-

action in a Hamiltonian form (as was done here for MHD Alfven waves) the

sane symmetry rules should exist also for decay-type instability. Actually,

this holds for any fluid*approximation. Moreover, in such arbitrary cases

one may expect to have for the probability ampHude variables the same

type of equations as (18) and (19)» Of course, the matrix elements and

normalizations should be specified in a different way for the various

canes. Besides, in each case it is necessary to fulfil the "decay"

condtions (12) and Q-3).

In Fig, 2 various forms of the spectra are represented for clarity.

As it is easy to demonstrate, "decays" of the oscillations may occur for

Fig. 2

spectra 1 and 4. Oscillations having spectra similar to spectra 2 and 3

are stable with relationship to ''decay". However, in the presence of

Including multi-fluid cases.

- 1 0 -

mk *

several branohes in the spectrum of the oscillations, that is, oscillations

characterized by the spectra similar to spectra 2., may be unstable with

relationship to the "decays" to oscillations out of which at least one does

not belong to the given branch. More accurately, "decays" are possible

when we may draw a curve similar to either curve 1 or curve 3 through the

three points corresponding to oscillations W e , Cd4 ,U) j_ (these three

points, generally speaking, may lie on different branches). (For differ—1

, ent branches of the oscillations, however, "prohibitions" associated with

the polarization of the waves may arise.) The fulfilment of "decay"

conditions, however, in itself still does not signify instability.

By solving Eqs, (l8) and (19) we find an exponential behaviour (i.e.,1

\Q. I1*- _ €^tp£2."$"n)* •tn h e Seneral case the growth rate of the per-

turbing waves: is given by

From this expression we find that the decay-type instability occurs only when

the frequencies of the perturbing waves are both smaller than the large

amplitude wavesj^J , i.e., j td±\ ,|&>gj < | 0^o | •

In the: case when the resonance conditions for the frequencies and

wave vectors- cannot be satisfied between three waves only, we may include

in our consideration the fourth wave. In order to have the finite growth

rate, this fourth wave must obviously be the finite amplitude wave.

Therefore, we must,, in. factr consider the stability of .th® second harmonic

of finite amplitude waves. The resonance conditions have now the form

U3infftho decay rule ( i .e . , 2LU>C>Ol >.» ° C K 0 U^W\) andthese resonance conditions, i t may be shown that in the second order the

spectra 3 of Fig. 2 is unstable and spectra 2 is stable. The instability

of the gravity waves on the ocean surface and ion sound waves in the second

order was considered as an example of this kind of instability [5j, We

can expect that the diagram of the unstable regions in the frequency ware

amplitude plane qualitatively looks like the same kind of diagram as for

the parametric resonanoe problem (Fig. 3)»

- 11 -

-:.• 3

The.width of the unstable region near the nth harmonic Is of the order of

the growth rate and proportional to the nth power of the amplitude. Of

course, we must satisfy the decay conditions

n, K, *. K

One of the interesting examples of higher-order decay of this type is the

instability of the periodical Alfven waves with a saw-toothed wave form

of a period " CL M with respect to perturbations with biA.it \.*04 (0t » \k /Q

frequencies^ .

( Another question is the relaxation of the initially steady waves

under the influence of the disturbances, To deal with this question we must

consider the case of finite amplitudes of the disturbances,and the nonlinear

terms in the equation for decaying, say, Alfven waves must be taken into

account. When Co , Ci and C^ are all of the same order, one would expect

to find the following equation;

i& - K -K(20)

One can derive Eq. (20) from the same MED equation as we have used to derive

Bq. (18).

- 12 -

Let

In the case where Q)9rules of the V ' s imply

and <»)x < Q J *b© symmetry

.

Taking the real and imagittayy parts of (l8)-(20) arid, using the variables

) and ©•= ^ 4 ~ & - 0 o ^ i v e s u s the

1 = # a. a, to*ff

/^ * M' - Sgt

(22)

By integrating:the last of these equations, ye. find

- 13 -

By using the frequency rule, we can easily find a first integral

of the remaining equations

By integrating Qo ;gv* •+

constants of the motion :

m,-& n

%. h<

const * (24)

we oan also find the following

const.

const *

const,

(25)

They are well known in the theory of the so-called "parametric

amplifiers" f8j as the vectorial Manley-Rowe relations taken in the

direction of propagation, and may be understood in terms of the diagram

for the three-wave decay process

"by saying that when a quantum disappears from the (0KO mode, quanta

appear in each of &)<, and W K L f so Aft,--/ , A ^ - £ ^Aft,

By combining the relations (23) and (25) we have

no =

The three roots of

-n.) -r*J (a6)

are labelled as

Eq, (26) can be transformed into

This can be transformed to^an elliptic integral "by tb.e change of variables

(27)

If we define the time "te in such a way that *| vTo J^ 0- , then

(26)

Therefore,

5fand, by definition of 5f » we haire the general solution

(29)

Let us consider two simple, oaaes *7.]}t

Case Ai At time "t - 0 , n.o(e>)=0 t WA(o) » O ^ . Withoutloss of generality we can put f-G in Eq. (26), Then the three roots

of Eq. (26) are simply

, =. h, lo ) = »it o) - o -

We can simplify the solution by neglecting

this case V << 1 • Thus

in B(l» , sinoe in

n.W- ^

= Mo )-

The time variation of the occupation numher is shown in Fig. 4

Fig. 4

The situation described "by this figure is the case in which the frequency

of the finite amplitude wave is smaller than the frequency of the per-

turbations, and, hence, it is stable to the deoay-type instability. The

small periodic variation in its amplitude io due to the small amount of

- 16 -

• • * • ! • ( * * * & •

•*$}•_ X- J » I

energy initially in the perturbations.

Case Bt Let us nov consider the decay of finite amplitude waves, when at

time £ -0 , H j > ) - 0 , no(t>) > > n ^ o ) > 0 « Putting P = 0again, we find the constants

Thus from (29) and (25) we obtain

= 11,(0)

n t It) =

where i_

(31)

Since at •£ — 0 I2Z -0 , we have

Therefore, ffl I e ffQ is £ of the period of

V <z, fj-V2' >With V <z, fj-V2' > we have

- 17 . -

Therefore, from (31)-(33)

n,to) (34)

During this time the amplitude of the initial steady wave Ho decreases

so we can say that it is the time of decay. It differs from the linear

growth rate "p — *W Ytl+ only by logarithm factor. This factor we

need when the amplitudes of disturbances are comparable with the amplitude

of the initial wave.

The behaviour' of the relative number of wave quanta are shown in

Pig. 5.

- 18 -

"T"

It turns out that the amplitude of the two perturbing modes for long times

drop down to zero again. Thus, the "decay" found in this problem is not

re al ly " ixrevers i"bl e ".

This type of problem was considered first in 1954 "by PIERCE (YJ. He

discussed the use of such resonant three-wave interaction in high frequency

electronics. Now it is a highly developed part of radiofrequency electronics

where these nonlinear methods are used for the production of mixing

frequencies and parametric amplification.

Another illustration of these methods is noniinear optics. Here the

nonlinearity comes usually in terms of the ourrent dependence on electric

field [?].

Also, all optical media have dispersive properties.

In the simplest approximation the dielectric function is given in

terms of the resonant frequencies - at least far from these resonances -

"L.

Such a dispersion makes it possible to satisfy the resonance

relations C00 - U)*+^2. > K o = ^i "*" ^ 2 - • Thus, if a ruby laser beam

is transmitted through quartz, a small amount of the "second harmonic" is

generated. However, if two lasers are passed through quartz at the

appropriate angle with respect to one another, in order to satisfy the

conditions CO,-tu) i~ 5 > *• v* x. - **•*> > a considerable amount of

energy can be converted into harmonics. In general, in nonlinear optics

there is a fairly high variety of different kinds of noniinear effects.

As an example of another sort, one might mention the effects which are

seen by reflecting laser beams off the' surface of a metal, and so on.

1 - 19 -

3* 13any-wave interaction in random phase approximation

In plasmas, we do not deal very often with highly ordered phenomena

since many modes are available to satisfy the conservation relations.

Instead of the Manley-Rowe type of behaviour (Pig. &.)

Fig* 6a

there are many modes present, and the recurrent character of the evolution

is destroyed (for incommensurable frequencies) (see Fig, 6b)

Fig, 6b

He can treat this situation using the random phase approximation (EPA) and

look at the modulus of the amplitude, averaging over phases, JTow there is

no way to distinguish, the "main wave" from the others, and this is reflected

in the notation in the equation for the wave amplitude

K'(35)

- 20 -

1

With a particular choice of normalization the amplitudes CKplay the

role of the probability amplitudej so, the mod© occupation number is

equal to the square of this amplitude*

Within this normalization the "matrix elements" have the following

symmetry properties (compare with the ones for the interaction of the

Alfven and sound waves);

(37)

Since we are interested in the time variation of the occupation number

f\/£ only for the eigenoseillations, we can involve only them in this

consideration. Then the equation for the amplitudes of the eigen-

oscillations can be treated with the methods of time-dependent perturb

ation theory from quantum mechanic s[,4-j. On this basis we expand

in powers of the interaction y% •£, ic-x1 a s

and calculate the change in the occupation number after time

\ CffCOV1- A C!*5 \Z • F r o i a (35) we have

- 21 -

o o

V:

K

Or* *, ~

(38)

The quantity L.;f is time—independent and corresponds to the solution in

the absence of interaction between modes. It can be written as a positive

amplitude times a phase factor of the form 6 ^A t w6 postulate that

although the Jt > are fixed by initial conditions at time £ ~ 0 > the

correlation between the phases C^ vanishes completely in a time small

compared with the time required for the change • in \Q-*\ owing to the

non-linear interaction between modes. Therefore, there is no correlation

at all between <h-> &&& ©-*/unless they correspond to the same mode, i.e.,

we can average over an ensemble of states with completely chaotic phases

in the wave amplitudes. Generally,application of the random phase average

(indicated "by a bar over the quantity being averaged) takes the form of an

identity such as

- 22 -

W,i. • * ' ( 3 9 )

Thus, the lowest order

(40)

Substituting (38) in (39) we obtain -

(41)

Co)

The product of four C;» on the r .h#s v after averaging over chaotic

phases in the wave amplitude, can be reduced to the product of tvo occupation

numbers* The two possibilities of combination of C? ar© shown in

(41) by dashed and straight lines. In the first term the C* combine

as \ C& 1* It®, \% = &4» or in the last two t*c*s ^or /r '°>/£ j r***!^

- 23 -

Application of the symmetry relations among the matrix elements

V?,K'tK" shows that the product of two of them can always be writtena s ( S ^ K ' i ? " $)$) times a sign depending on the signs of W ? , ^ ^

For the times longer than a wave period, the time integrals can "be

evaluated approximately as

Thus, the change in occupation number is proportional to time} for times

shorter than the characteristic time required for A/j> to change, ^

is «™P/dt approximately. In this manner, we obtain the kinetic

equations fio, 11, 14]

V?,, .. J N V,. V,, - Si« («»»

(42)

"ITe can go from Eq, (35) directly to this result, written in terms of

positive frequencies only, byusing time-dependent perturbation theory and

the "golden rale". Really, let us consider the part of the nonlinear

processes with tO^* CO j£', U^jJ'^Oj which corresponds to decay prooess

of the high frequency mode COj* and the combination process of the low

frequency modes w^ , (•Oj * « Then we can write the changing of the

occupation number A^f due to these processes

- 24 -

J^ y lCy lC

(43)

In the classical limit, A ^ ^ i , we recover the part of the previous

result. From this derivation it follows that the "collisional terra" for

the four—wave interaction is proportional to the third power of the

occupation number. In the absence of the resonance interaction between

three waves (as, for example, for the electron plasma wavesplj or

shallow water waves [}T>~\) the mode coupling then is the third order in

the wave energy,

The kinetic equation in the form (43) is used in the physics of

solids for the description of the phonon interaction due to lattice

irregularities, which can be important (e;g., heat transfer)[9J. But

there is a qualitative difference between applications of this equation to

the phonon and plasma turbulence. In solids we usually deal with a state

not far from thermal equilibrium. Thus, nonlinear phenomena produce

small corrections only. On the contrary, in a plasma they are very

important. The mean free path for a wave in1 a turbulent plasma can "be

ouite short and a behaviour like that described in the Rayleigh-

Jeans law does not hold for, plasma. As to particles, we can follow the

familiar treatment of the Boltzmannequation by expanding around a

Ilaxwelli&n.

In concluding this section we would like to make the following

notation. In the previous considerations we used the time-dependent

perturbation theory from quantum mechanics, which operates in terms of

the eigenoscillations only. But very often it is more convenient to use

the Fourier transform for the description of turbulent plasma motion.

Then we cannot exclude from our consideration the beats with the

frequency different from the eigenfrequenoy. The equation for the wave,

disturbance,

- 25 .-

« 2.where V^"^^^^, is the operator with respect to the time variable,

can be written now in the Fourier representation in a more general formi

= 2 - Vr-.--, Lfi\a^)- (44)K ** V

where V? is the growth rate of the eigenoscillations, <n*i Q^Q are the

Fourier transform of oJ?\fc/• The application of the Fourier

transform is rigorous for the stationary amplitude of the disturbances

We will work within the same normalization of the amplitude C £

a3 before; so the matrix elements have the same properties of symmetry

as (37) if we put all frequencies equal to the eigenfrequencies.

In the first approximation we omit the terms of order VC ? •v(Y;T£0;rJ C

and obtain from Eq. (44) the usual result: •

(45)

Tfe have again supposed that the phases of L-g* are completely chaotic.

Then, multiplying Eq.. (44) by C^*cO a n^ averaging the result over the

chaotic phases, we obtain that the non-Linear tern cancels in the first

approximation (45)« Therefore, we must take into account the beats with

the amplitude

(46)

Here we suppose that the beats damp very fast, so the damping rate is

higher than the growth rate of the eigenoscillations. We need

- 26 -

this condition in the Fourier representation in order to get the same

result as in the time-dependent perturbation theory and we -will use it

in the consideration of the wave interaction in the kinetic description

of the collisionlees plasma. Substituting (46) in (44) we obtain the

stationary state equation very similar to Eq. (42)

a)

, ^ ^ £ l ( - ^ ^

co-w',^.- G>to'-£Or,-cO^'

(47)

Using the definition

V -> Oand the symmetry properties (37) we can reduce this equation to

Eq. (42) only if the beats damp rapidly enough • ( - Y ^ < 0 )

- 27 -

4* Plasma turbulence in terms of the kinetic equation for waves

In the previous section we derived a kinetic equation for waves

using the random phase approximation (EPA). Mow we will investigate

some of the preperties of this equation, and how to apply it to several

problems.

Investigation of plasma stability in magnetic confinement shows

that often, under the influence of various small disturbances, the

plasma arrives at a state of disordered motion. In general we must

describe this motion by means of all of the plasma characteristics:

velocity, temperature, etc. (in the fluid-type description of the

plasma) at each point in space and time. If the deviation from steady

state is slight (or the total turbulent energy is small), we can re-

present this turbulent motion by a set of collective oscillations in

a linear approximation.

% , etc. (4o)

where the frequencies are determined from the dispersion relation

CO * <Ot")I'Tow the state may be described in terhs of the amplitudes of the eigen-

modes. The problem is to determine these amplitudes in phase space

as a function of K ,

The distribution of energy between different scales of turbulence

can be found on the basis of wave-wave interactions. These interactions

are easily treated within the framework of the EPA, which is probably

valid for the case of many interaction waves. This kind of approach

was called "weak turbulence theory".

If we wish, we can write the results of this treatment either in

terms of mode energy or the number of waves ti,-* in the K th mode. In

the latter representation, the wave kinetic equation becomes

(49)

where the form of the collision term, J*- "L "Mtj *•»?[, was derived in

the previous lecture. In problems of nonlinear stability, it is

- 28 -

.v"*. * 4 ""*"

necessary to consider the sources of instability and dissipation. This

accounts f or the inclusion of the first term on the r.h.s.' In steady-

state, we caji drop

equated to zero.

and just consider the r»h,s, of Eq, (49)

Let us begin a preliminary examination of the relationship of weak

plasma turbulence to Kolraogoroff's kind of treatment in the conventional

hy&rodynamic turbulence. It is very difficult to find a rigorous de-

scription of the energy transfer among different scales of hydrodynamic

turbulence, because we cannot ezpress!lstrong turbulence" in terms of

eigenoscillations (waves) and we have no rigorous statistical equivalent

of the equation corresponding to the wave kinetic equation above. In

conventional hydrodynamics the most reliable procedure is the use of

dimensional arguments, ,

•CO

t. 7.

Imagine a situation wherenthe source of large-scale (small k) turbulent

motion (region l) is separated from the region where the turbulent motion

damps rapidly due to viscosity in small-scale motion (region 2), Energy

passes from large scales to small continuously in K -space and in the

niddle region in phase space, practically no ..dissipation occurs • In the

intermediate region (between (l) and(2)) the well-known dimensional

arguments lead to the spectrum

(Kolmogoroff-Obukhov law)

where tj* is the energy per wave number per unit volume.

- 29 -

in making this it was assumed that the turbulence is isotropic and that

it may be described in terms of local characteristics. Energy is trans-

ferred from small ;K to large in a resonant manner, passing through each

decreasing scale in succession (see, for example, Ref. (16)).

It is interesting to find an analogy with Kolmogoroff'e spectrum for a

weakly turtmlent plasma since in this case we have the equation for the

spectral density derived from the previous principles. In the pls.sna,

however, we can imagine a great variety of different waves, so probably

there is no universal-type spectrum with some power dependence H K -v \£ .

However, for any given type ol" plasma waves one hopes to have a simple

spectrum.

Let us consider an idealized model investigated by ZAREAROV [j7j.

Tfe will deal with the noniinear dispersive (and therefore, plasraa-like)

media described by the wave equation

( 5 0 )

2.where V i s the Laplacian,When we Fourier-transform, we find

with

(52)

which corresponds to a decay-type spectrum. We can introduce the mode

.-.rrplitudes C^- defined by

(53)

In terras of (,-»we have 3q. (51) in a canonical form:

I 2S? ._ f 1/ c-n)r

(54)where

Let us note here that the wave kinetic ecuation was derived in.

Section 3 for the case of a set of discrete nodes. In the continuum

limit, "V -/(.l<,*CK-b) (instead of *~/(C ±)) and the kinetic

equation seens to lack a dimensionality factor of K in the r«h*s.

Therefore, in the derivation* instead of O^if we will have o -

functions! crfi* ~* —~f}f£-£j t where v is a normalization volume.

1 XThus we should really dei'ine IT.£» - T ^ 1 1 , whereupon everything is

correct.

\Je will take 00- K for all modes (i.e., £<*.O ) except in

evaluating resonant denominators} this imposes a restriction on the

magnitude of the amplitudes for which the analysis applies, namely,

l^«?) c<* t&U* • A 1 1 modes would interact strongly if € «• O

exactly.

In terms of h.* we have

(55)

- 31 -

To avoid difficulty with signs, we have legislated all frequencies to

be positive f^ - / = (it/ • The "terra £ j ^ /££• has "been added to

represent a source and sink of wave energy.

Since we are interested in isotropic solutions, we will average

the wave equation over angles. First we integrate out the Independence,

replacing K' &u - K-K according to the frequency rule. The^we

write

measured fron K, direction)

(in the first term)

and integrate with respect to cos 0 from -1 to +1. As a result the

$ -function is replaced "by

r 1 1r K.H

Integration with respect to \i> gives a factor of 2.77 . The second

term can he treated similarly.

In the first term, the condition fcLt<0umeans that the integration

over the modulus of R1 is only over the interval O ~* K, . In the

second term, however, this restriction is lifted and the upper limit

is oo .

We can put the wave kinetic equation in a more convenient form Toy

introducing the new variable. If we define

K = n.K-k*the result is

- 4- £(57)

- 32 -

One solution in the form of a power law oan "be found (for V^-0

"by inspection, in the form • • ..

i . e . , "K.A*/yjr« Fnis is just the Rayleigb-Jeans law for equipartitionof energy among a l l modes. However, i t is clear that for the- tfcrinproblem this law is useless, and some of the integrals taken separately

in I3GL. (16) diverge at large K ,

I t is clear that we must seek a solution of tha. ij&rra Vr ^K. (

to avoid this "ultraviolet catastrophe". The proport i'onal ity constant

will be determined by requiring that the result be connected with the

regions of growth and dissipation. •

It might appear that a similar divergence now occtirs at K-*0*

That this is not so may be seen by combining terms ia the previous

equation; -

2

llote that the first term diverges in the same way at both K = O and

K'-K • The result is convergent if A^ *** ^-** whete %<3 *

AHow substitute ^x.~Z$ ^^ the collision term and look for •

solutions with 2. <S <. 3 . Using the relation

K •• "'••• '" • ' ' \

- 33 -

we find

Rs) =r(a-2.s)

(53)

From the behaviour of the 1 -functions and the last two terms, it

is clear that FtS)•»*<*>at S*o and'— °o at S~2. « and so one expects

to find a root near the middle of this interval. In actual fact, £ = £.5*

is a solution, as may be verified by substitution.

Let us apply this analysis to the problem in which a source of

instability and damping are present. Ordinarily we can estimate A*.

simply by

I/. (59)

where K.o is a characteristic magnitude in K -space over which V varies.

This is incorrect if regions of damping and growth are separated by a

transparent region. Let the damping be described by

and the instability by Eq, (59) with V*7.

For sufficiently large R. , the damping must end. The inter-

mediate region ig described (for small enough Y ) by

*. *Z & K-2"5 • (61)

The boundary of the damping region is given by k A , where

- 34 -

-*, or ic

By integration of the steady state equation, we obtain the

conservation law

From this follows **

where the lower bound of the integration on the r.h.s. is taken at some

K » K O . We use Bq# (62) to solve for B

Thus Eq. (61) is approximately valid if the transparent region is

sufficiently "broad and the damping sets in sufficiently abruptly. Then

a spectrum similar in appearance to the Kolmogoroff spectrum holds, al-

though the mechanics involved are quite different. In our case, this

spectrum is a result of wave-wave hierarchy^' in the Kolmogoroff's case,

the modes are the vortices whose statistical interaction we cannot

deduce from the previous principles.

5. Coupling "between different modes in a turbulent plasma

This simple form of the turbulence spectrum is due to the fact

that the kernel of the wave-wave collision integral is very simple

for the idealized model described in Eq, (50)« In a more realistic

situation, this kernel usually has a complicated form. Even the

procedure for its derivation is quite cumbersome, especially in the

- 35 -

cases of interaction of different modes. It is, therefore, very important

to find some additional simplifications of this problem.

Let us show how it can "be simplified if the interacting modes have

very different dispersive properties. If we have two modes (characterized

"by frequencies COjrand £1*) such that CGg*»Q.-*, we may treat the problem

adiabatically* For example, let us consider the interaction between high

frequency short wave longitudinal plasma oscillations

and low frequency long wave ion acoustic vibrations

with K ^ q . This problem was first considered by VEDEKTOV and RUDACOV [I^J.

We start from the Liouville equation for the number of electron plasma

oscillations (plasmons) in the six-dimensional space of coordinates and

wave vectors:

(64)

In a uniform plasma, changes in the frequency t> as a function of position

arise only from the ion-acoustic vibration which we describe by the hydro-

magnetic equations;.

(2.

(65)

«_ (67)

where the last tenn on the r.h.e. of (65) is obtained by averaging the

: - 36 -

/""* Air-

electron term -a/m2T< yfi) over many plasma oscillation periods, andjust represents a gradient of the radiation pressure:

€'"T

(67) is just the adiabatic equation of state*

Using the definition of the plasma frequency

(68)

and linearizing Eq. (66), we can write Eq. (64) in the form

(69)

Since s

VJhile (65) "becomes

- 37 -

i? (70)

In (69) and (70), f is the fluid displacement.

In order to see that this set of equations is quite acceptable to work

with, let us apply it to some specific problems.

(a) Damping of an ion acoustic wave in a gas of Lan^muir plasmon.

Let us assume a dependence of Ng and on T and "t , of the form

p — iSLk "*"t "'*. Linearizing Eq.. (69), we obtain the correction to

the 'plasraon distribution function:

Substituting in (73) and dotting with <j we find the dispersion relation

connecting the frequency_Q,and Wave vector G of an ion acoustic wave:

For Z having a small imaginary part I -p. .ic. j we find

\-Ie can see a direct analogy with the usual Landau damping (growth) of waves

on particles. Here, ion sound waves damp on quasi-particles with velocity

*• and distribution function Av*. And, of course, under the reciprocal

influence of the ion sound wave spectrum, this plasmon distribution will

relax.

For this relaxation process we have from {69) and (73)

- 33 -

a i \t

-•&%£ <*>

where ( > means an average over the fast (plasma) oscillations.

(b) Instability of a plasmon gas.

Another interesting problem which we can solve within this framework

is the instability of a gas of plasmons due to the coupling with the

sound waves.

Under suitable conditions, a narrow spectrum of Langmuir oscillations

is formed, which we can regard as excitation of a single mode:

( 7 5 )

After substitution in .(72) and integration by parts, we obtain

where 1*7*-= \u) /V^w^K-Aft^wis the total, energy in the spectrum and the

dispersion equation (68) has been used again in calculating ^'S^T" 5

If Ct • -2-z >> C- » "this equation has a solution7 9<T S

Prom which it is clear that the criterion for instability is

- 39 -

The growth rate is

It can also be shown that the "plasraon gas" is unstable with re-

spect to the decay mode discussed in the first lecture in connection

with Alfven waves'. A "steady state" plasma oscillation (6)0, K.o>decays

into another plasmon (G), K ) and an ion sound wave (£1,*?) \j>> '9j*

. , REFERENCES

1 . VEDENOV, A . A . , VELIKHOV, I ' ] . P . , a n d SAGDEEV, R . Z . , H u c l . F u s i o n 1.,

8 2 ( 1 9 6 1 )

2 . GALEEV, A.A., and ORAEVSKII, V.IT,, Soviet Phys.-Doklady 2 , 983

(1962-1963) (nhokl. Akad. ITauk SSR _1£7, 71 (1962))

3 . ORAEVSKII, V.K., and SAGDEEV, R.Z., Soviet Phys.-Tech. Phys. £, 955

(1963) (Zh. Teohn. P i e . ^32, 1291 (1962))

4 . GALEEV, A.A., and KARPLIM, T.1T., Soviet Phys.-JETP ]J_, 403 (1963)

• (Zh. Eksperim. i Teor. F i z . ^4_, 592 (1963)

5. ZAKAROV, V.E. , Dokl. Akad. Hauk SSR, in press

6. GALEEV, A.A. and ORAEVSKII, V.IT., Soviet Phys. Doklady £ , 154 (1964)

,(Dokl. Akad. Nauk SSR 1^, 1069 (1964))

7. AKISTROITG, I.A., BLOEIIBERGEU, N. et al., Phys. Rev. 122, 1918 (1962)

8. PIERCE, J.R.,"Travelling Wave Tubes" (Van ITostrand, U.Y. (1950))

9. BL03HBBRGEIT, ET., "Nonlinear Optics", (¥.T. Benjamin (1965))

10. PEIERLS, R., "Qiantum Theory of Solids, Oxford (1965)

11. CAT/IAC, 1.1., KAHTROWITZ, A.R., LITVAK, M.H., PATRICK, R.H., and

PETSCHEK, H.E., Uucl. Fusion, Suppl. Pt. 2, 423 (1962)

- 40 - • . .

12. DRUMJIOHD, W.E., and PI IKS, D. , Huol. Fusion, Suppl. Pt . 3, 1049 '(1962)

13. ZAKHAROV, V.ff., FIL0HEM0, n . , - D o k l . Akad Nauk SSR, in press

14. KADOHTSEV, B.B. t and PEPVIASHVILY, V . I . , Soviet Phys.-JBTP l ^ , 1578

(1963) (Zh. Skeperim. i Teor. Piz. 41, 2234 (1962)) '

15. Ki\DOIlTSBV, B.B., "Plasma Turbulence", Acad. Press , London (1965)

16. LABDAU, L.D. and LIFSHITZ, E.M.," "Fluid Mechanics", Pergamon Press (1963)

17. ZAKEAROV, V.E., "Prikl. Hekh. i Tekh. Fiz" £, 35 (1965)

18. VSDE1T0V, A.A., RUDAKOV, L.I., Soviet Phys. -Doklady % 1073 (l965)

(Dokl. Akad. Hauk SSSR ,152, 767 (1964)

19. ORAEVSKII, ViH., llupl. Fusion 4., 263 (1965)

- 41 -

PART II. PARTICLE-WAVE INTERACTION

1. Particle interaction with a single wave

"We have finished consideration of non-linear wave theory in

fluid-type media. Let us now start to take into account a second type

of interaction, namely that between waves and particles*

It is convenient to start with the simplest problem, the

interaction between small amplitude monoohromatic waves and partioles.

For simplicity we assume that there is no magnetic field, and we take

the wave potential in the form

-I (cot'f * % eThe main contribution to the interaction arises from the group of

resonant particles, those with velocity close to the wave phase

velocity W / K . In linear theory it gives as a result the so-

called Landau damping. The non-linear problem has not been solved

in its entirety, but qualitative arguments show that the distribution

function is reconstructed for the velocities close to /K» The

simplest type of such reshaping is expected to "be in the form of a

"plateau".

As an indication of suoh non-linear distortion we would remind you of

the BERWSTEIN-GREEUE-KRUSKAL (BCK) [l] solution which is an exact

non-linear steady-state solution. According to this, there is no

Landau damping in steady state. It is tempting to treat suoh steady

state waves as an asymptotio limit (•£-><»=>) — Landau's initial value

problem.

Let us see in a very crude qualitative fashion how to approach

the BGK-type solution from an initial ohoice of the distribution

function. Consider a wave propagating in a plasma with phase velocityW/K. . Looking in the frame of reference moving with the wave, we see

that there are two groups of particles! if I IT 1 > iTit - A

particles pass over the peaks in the periodic potential without

trappings if \-vr\ <• \^^c ln » t h ey aJce trapped. If the number

of particles moving in one direction is not balanced by the number

moving the opposite way, as indicated by the arrows in Fig. 1, the

distribution fttnotion changes with time.

- 42 -

Fig. 1

Let us define energy-angle (similar to action-angle) variables

and Q and write the distribution funotion for the trapped particles

at^Ot and for short times,-fc>0 ,'•/ is anisotropic (depends on 0 ) in

contrast to the steady-state BGK solutions, whioh depend only on £ .

Every partiole has some frequency of oscillation <!)&/• POT partioles

near to the bottom of the potential well, CO is independent of £ . For

two particles with fi^E^and $<- 9^ we have, slightly different

frequencies

CO., - = CO

After a time ^ > f , where . "CAU)*!* the partioles will be separated

by a phase A9 "• 1 • T h u s ^Q phases of the particles become randomized,

and 1 ceases to depend on Q . This phase-mixing process is similar

to that which affects partioles in a non*«niform magnetic field (where

the frequency depends on particle energy) and has been treated many

times. '

We can ask about the cause of the apparent ohange in entropy

as a result of phase mixing (since a phase-independent function has

more entropy than the original l^fcQ))*. This question can also be

* Such an entropy production was discussed in [2^3j as adissipation mechanism in oollisionless plasma shooks*

answered qualitatively for the case of trapped particles in such a

monochromatic wave.

The distribution function must conserve the characteristics

of the oollisionless kinetic equation. These characteristics are the

particle orbits. How, on the "basis of period differences we can

conclude that the distribution function at any fixed OC will be

modulated invelooity space (oscillations appear in *f(v))i and this

modulation will grow with time £3] 1 a e e Fig. 2).

0 IT IT 0

Pig,

There is a limit in which one can easily solve the problem of

such peculiar behaviour of the distribution function. Suppose ^/^ is

so large that the resonant particle energy is small compared with

the initial wave energy. Since the Landau damping is small in such an

approximation the particle motion occurs in an almost oonstant

periodic field.

We can easily find the particle orbits in such a field and

immediately write a general solution for the distribution function;

^le'e%

[kn

where Xb {xJ£~J tj is the point from which X evolved and£= sign V j at

the ini t ia l moment -i'O we negleoted here the oorreotion to the

- 44 -

distribution funotion due to the presence of a sinusoidal W ? » in

plasma ^jt Sin kx and had used the initial condition, in %h$

expanded near the resonance velooityi '.."'" ..,

The distribution function (l) is normalised to th» energy variable"BV. Introducing the new variable

for trapped particles with 96 2 > 1, we oan express the partiole

trajectory in terms of an elliptio integral of the first kind and

ae takes the, sign of " V " '

Therefore Eq.. ( 1 ) for trapped, partioles takes a foann £ 3

Analogously, for the untrapped particles we obtain

(S;nIf we now go baok to the variables X, &9 ~t, we oan describe quantitative-

ly the modulation of the distribution in the velocity spaoe.

It becomes much easier if we introduce the less realiatio but

simpler model with a square-tooth wave { 3J,(see Pig. 3 ) . It is known

that this model permits one to derive linear Landau damping correctly.

As with a sinusoidal wave, CO ~ U)(E), How; we can obtain an

- 45 -

analytic* .Istfitttion for the time development of oscillations in tTspaoe,

and find -that as £ -> CK>t the wavelength \ (t) ~> Q

and l(v) iB strongly modulated.

° Fig. 3

Tt is interesting to compare this with the results obtained "by

Landau in his original treatment of damping. He found two types of

damping^ for a monochromatic wave, there are terms in -f like £ L

arising from the initial value of j- , in addition to the exponential

damping. Although Landau did not discuss the point in the paper, he

used this result to show how entropy changes. For long enough times,

the wavelength of the modulation "becomes so small that it is no

longer justified to neglect the " J/Sy^" '&e:riIIls i1* "the collision

term} so even for a "collisionless" plasma, collisions will act to

smooth out 4 , The time required for this will be finite and quite

insensitive to the collision frequency S> . The same argument applies

here, and thus we get relaxation to one particular steady state

trapped particle distribution (out of the entire set of BGK solutions).

We now have a situation similar to that in MSD turbulence

theory. There, in the usual co-ordinate space, turbulence first

develops on the scale of large wavelengths, then degrades into smaller

scales. For sufficiently short lengths, real damping occurs. In

the usual picture, the spectrum is independent of viscosity. Here we

have a similar situation in \T-spaces the steady-state distribution

is independent of collisions (viscosity).

How we oan calculate the Landau damping as1• • )

*' We can now see that this kind of approximation is actually thefirst term in the series off-- f[£_ .

- 46 -

• ~ ~£I-CO

c.

0

Pig. 4

The decrement starts off with the value found by Landau, but after a

time i/tO(£) ohanges sign. This ooours because after on© wave period

the direction of the imbalanoe in trapped particle motion is approx-

imately reversed [33. For the exact sinusoidal wave treated by

O'HEIL [ 4], the same qualitative behaviour exists.

Assuming the slow change of the wave amplitude O'Neil calculated

the time-dependent growth rate of the oscillation. The result is

- 47 -

where P * F (* ^ 4 ) , P ' r W f t f ) ^ 4 ]

YL - ? U) —fe •-&•/ is the damping coefficient found

by Landau. For the time less than the period of motion of trapped

partioles and larger than the phase-mixing time -jf s- -/^ jr. for

untrapped ones* we obtain the usual formula of Landau:

In the limit of large time the growth rate goes to zero

A more complicated problem, taking the collisions into account,

is now solved only for the asymptotic limit of the steady state [5],

since finding the distribution function is not so easy as in

the case of equations(2).

The kinetic equation for this function in the system of the

resting wave may be written down in view of collisions as

where all values are reduced to the dimensionless form:

where fflx.) is the wave potential in the system at rest, k and V=A.

are its wave number and phase velooity respectively, "£"" is the

effeotive time of electron collisions whose velooity coincides with

the wave phaae velocity, L are Coulomb logarithms and *P(y), y, °l and

\J_ are the dimensionless potential energy, co-ordinate, phase velocity

and collision frequency, respectively.

* As was shown by Landau, a large damping of the wave appears at the

initial time -fc. *< /KTT a n^ ^e wave", amplitude .cannot *be treated as a

constant. Therefore that formula for this time should not be believed.

-48 -

s: is • •«• . wlui«*> ... ....•»•.«*».

Let us introduce one restriction which, makes the problem

soluble (in praotioe)i

. -U, « '% « I . (4)X . •

This corresponds to a wave of finite but small amplitude and low

frequenoy of oollisione.

How it will be more convenient to deal with new independent

variables for the distribution funotions

B

where .£" is the dimensionless total particle energy in the wave field.

Then Eq. (3) takes the form

(5)

where the i signs before the root correspond to the different

directions of electron velocities (plus sign is taken for particles

overtaking the wave).

In solving Bq. (5) we must oonsider two oasesi £• > (external

region) and £ <*po (internal region). ¥e find the solution of Eq.. (5)

as an expansion in series»

By substituting Eq.. (6) into Eq* (3) we get

where fo(g}(zBTO approximation) is defined (in the external region) by the

oonditionthat ••f1(y) is periodic, and by the boundary condition that

at E » ^p0 the value -f&(£) should asymptotioally approach the

Maxwell distribution for the plasma moving at the velooity- c relative

to the wave. Thus the equation for X, (E) at E" > ^o will be

77 r.ttiAare some elliptic integrals.

i:ow we can find -to (£7 and substitute it in Eq.'(7). Finally we must

have /,€A and W ^ - - jfiffat whioh, as one can see, will beexpressed in terms of some elliptic integrals.

Let us write this, omitting a calculation and using some

expansions of elliptic integrals for,the case

nt - o.i The same procedure must be used also in the internal region ( E

However, the distribution function here is quite symmetrical X+where •f-t-(^i^) and 4-(Ety) correspond to electrons with positive and

negative velocities relative to the wave. This means that its

contribution to wave damping W is quite small, and we can therefore

neglect it.

The major problems arise when we want to find distribution

functions inside the singular domain lying between external and

internal regions. Fortunately, as shown by ZACHAROV and KARFMAU [~5_],

we do not need to know in detail the behaviour of this function to

calculate W$j.no t it is enough to know its value at both sides of the

singular region. " In fact,

where 6 is the width of the singular region*

Now, by using integration "by parts and taking into aooount

only the lowest derivatives of the distribution funotion inside a

singular region, we obtain

If we substitute here the values of p/^ F » which we find from

external (E-^ o+ 0) and internal ( E- (f0- 0) regions, we finally

have

- c *

2. The many-waves oase

Let us now turn to a problem with two waves. If the waves

have well-separated phase velocities ( W/ K}^then they will not

interact and we oan 3ust superpose the results for the trapped particles

of a single wave. However, if the waves are close together,

At)where ^f ' is the order of magnitude of the amplitude, then the situation

becomes completely different. We ezpeot to find overlapping or "collect-

ivization of the trapped particles.

-51-

We oan continue by looking for solutions with 3, 4, etc., waves

present, but the analysis is hopelessly complicated and only rough

qualitative conclusions can be drawn. However, with a very large

number of waves present, we can introduce the random phase approximation

and the statistical approach used "before in these leoturea.

0Pig. 5

Suppose that there is a domain in velooity space

CJO < \r <K /

with waves present having phase velocities throughout this interval,

such that between any two neighbouring waves there is collectivization

of the resonant particles. If the phases of these waves are random,

particles undergo Brownian motion in velooity space and we derive the

familiar quasi—linear theory L6,7j» We will not go into the derivation

in detail herd, as it is well known to most people and discussed

critically in a number of books (e.g., that of TIEM-MT and MONTGOMERY

[8]). The most rigorous way of obtaining the finite quasi-linear

equations, as well as the criterion of overlapping between neighbouring

monochromatic waves, was given by AL'TSHUL1 and KAEFMAK

> fit] mm-• 4*1-M* 3E5OTII

We oontent ourselves here with a simple non-rigorous derivation,

Starting from the Vlasov equation

in one dimension, we write the distribution function as the sum of a

slowly varying and rapidly varying part

For the rapidly varying part, we have just the usual linearized form

of (10):

During a single oscillation period, L changes only slightly, and we

can use a WKB approximation in time to solve for •£, . For / we have

2J'+ y-21 _'J. vu> 2k = o in)

The only non-linear effect of which quasi-linear theory takes account

is the average of the product of two rapidly oscillating functions,

i . e . , the last term in (11).

The analysis is mathematically similar to that of the familiar

van der Pol method, where we have two kinds of motion, one fast and

one slow. The same kind of averaging in time is used there. The

analogy with the van der Pol method can be useful in justifying

quasi-linear theory for many waves. Here we have a partial

differential equation in ( Xjl^t ) instead of an ordinary differential

equation. If just one monochromatic wave were present, we could

transform the frequency to zero by going to a frame of reference

moving with the phase velocity of the wave, so that an average over

the fast osoillations becomes meaningless* But if many waves are

present, no suoh oo-ordinate system exists, and the average can be

employed. We take the fast time "2"^ ~ ^-/j^ , where AtO-Ct) - k!) .

and compare it with the time "£l on which 4 ohangesj to get the cri-

terion for the applicability of the quasi-linear theory, ^t <<'^~2..

This oriterion does not depend on the group velooity.

Although the discussion has concerned itself only with the one-

dimensional problem, we will write down the results for electrostatic

waves in three dimensions:

Jo

where *YK is related to r by Landau's formula.

Usually, Eq. (13) is written with %(u)-Kir) instead of the last

factor} this is done because the condition that the wave amplitude be

small requires J( K <<- u ) * • However, then 1 ) , a contains only the

interaction with the resonant particles; the form written above

also contains the adiabatic interaction with the non-resonant

particles. The resonant par;t of (13) gives the reconstructions of IT

in the resonant region? the adiabatic part desoribes the ordered

motion of the plasma fluid under the influence of the wave field.

In the one-dimensional case we get a steady state solution

after some time with a plateau for a restricted region in IT space.

We can estimate the time ^ T D required for relaxation from the initial

velocity distribution to this plateau from Eq. (12):

L ^ -X—J-— r~ —-j—

- 5 4 -

t

*

where ^ \T* (w ) m Vj- (K/mAV t]aa width of the wave p»ok«t ia

ivelocity spaoe, and A CO ~ K ^\T . Then the condition

just yields

AIT

or, equivalently, -'/ACO i a muoh smaller than the trapping time. It ia

the case opposite to the condition which we obtained "before for two

neighbouring raonochromatio waves to overlap, A IT <

The main result we find is that the plasma rapidly evolves to

a state with a plateau on the distribution function and a finite

amplitude wave packet which does not interaot with the plasma. Of

course, this ie not really a steady-state solution: we know already

from considering fluid-type phenomena that the wave spectrum will

interact with itself and change in time. However, for small amplitude

waves we may hope that the time required for establishing the plateau

is much shorter than that for wave-wave interactionsj

But it turns out that both "Z~ and fwware inversely proportional

to the first power of the wave energy or wave number for three-wave

interactions* In order to justify the ordering we need to take

into account the other parameters of the system* 2"^ is very short

for narrow wave packets, so we make a new assumption,

1 (15)

vwhere IT is a typical phase velocity.

He can invoke a second argument. The criterion (14) comes from

wave-7-wave interactions whioh obey the decay or resonance conditions

(see Fart i); for Langmuir oscillations we cannot conserve quasi-

momentum and energy in a three-wave interaction. For a four-wave

interaction, however, we find that the typical interaction times go

inversely as the square of the wave energy. So, one way or the other,

we can sometimes justify neglect of non-linear wave-wave interaction

on the scale t"_ .

- 55 .

Now our task is to find some examples of situations where quasilinear theory is applicable, just to see what kinds of phenomena can

arise.

For Langmuir plasma oscillations we have the system of Eq. (12)

and (for the wave energy)

together comprising the quasi-linear approximation. The first problem

considered was that of the relaxation of an unstable distribution with

a gentle bump, where it is possible to find the wave energy (f^l (where

<${o)^O ) as a function of the bump parameters and of time for

-£.** O and •{•*• cxs . This solution is, of course, well known from

many papers [Sfl'Jt so ye shall not discuss it again here. The

second problem treated using (12) and (l6) was that of the quasi-

linear Landau damping of a wave packet (,^/K )bvUi ^ ^/K ^ ( Jmcxjk

with £(o)£0 inside this interval [6,7].

These are two typioal examples of one-dimensional quasi-linear

problems. In two or three dimensions, it is easy to find oomplioations

and additional properties of the theory, because the one-dimensional

problem is in a sense degenerate: the resonant particles occupy only

a restricted region in velocity space. In higher-dimensional cases,

even for a wave packet which is localized in K -space, we find a-

broadening of the resonance region.

Let us consider a two-dimensional problem. In Pig. 6 are

plotted the curves of the initially constant 4 , which are, say,

oircles centered on the origin. Let us introduoe a fairly narrow

wave packet propagating in the IT -direction. Because of the formation

of a quasi-linear plateau, there will be established inside this

narrow band, \TX ~uVy<; > a n e w system of equipotentials (character-

istics) parallel to \5^. These, of course, oonneot with the circles

in the part of velooity space outside the resonant band,. It is easy

to see that only a finite amount of energy is needed to reoonstruot

in this way.

Now suppose we have many suoh wave packets present propagating

in different directions. In Fig. 6 each will be traversed by its own

I I

Pig. 6

set of equipotentials* Since these equipotentiala will intersect

some domain extending to infinity, an infinite amount of energy is

now needed to make 4- be constant along all these equipotentials*.

If,1 for example, all directions of propagation are present, this

domain fills all of velocity apace outside of the circle

•>_ 2 ^-t •_. / t O / K ) ^ because every part of this region is

common to at least two different resonance "bands* Since j is

constant out to infinity, it is obvious taat an infinite amount of

energy is involved. This means that any steady state, corresponding

to finite wave packet energy, is impossible, and the wave speotrum

must damp to zero

* Since the quasi-linear plateau is no longer in exactly steady statein the two-dimensional case (even for quite narrow wave packets) it isneoessary to undexetand its meaning by considering it as an approxi-mation, i.e., a "quasi-plateau".

- 5? -

To see what will actually happen to the distribution function,

let us suppose that we have a two-dimensional wave packet which has

cylindrical symmetry in K -space. Then 4 will "be isotropio

Fand on substitution in (12) we find

...2. / .?

(17)

To get this, we replace the summation over ft by an integral, and take

a very narrow spectrum, / ^ / — /<f/ h( K-Ka)* (Quasi-linear

theory is still valid because this distribution gets smeared out when

the spread in angles is taken into account.) For W / K > IT we have

Together with (15), Eq.. (17) has an exact solution if we make one

additional simplification. If initially there is a loss of energy

in the wave packet, then we obtain a reconstruction of j in the form

of an outward expansion, so that finally for most of the distribution

function we can neglect ®/y\ in comparison with IT. Now it is

quite easy to solve the equations. As an example of a case where it

is valid to neglect ~/K without restrictions, we can point to the

interaction between eleotrons and ion sound waves, since

Kf\j

For Langmuir osoillations this will be valid only after a considerable

time, when the distribution has spread a long way.

How let us introduoe a new variable in place of time

tie have

where J^ is a constant. .

To solve (Id) for a given ini t ia l distribution •£> we oari

Laplace—transform and obtain the solution in terras of the Green's

function for £ using modified Bessel functions of order -"7^»:

This is not a very oonvenient solution to work with, "but we can

go to -t **510 > where $ no longer depends on I T and write the

similarity solution of (18) as pLl]

To find V , we haveV

For the asymptotic form (19)» we calculate ,jllj

- 5 9 -

Note that ^ V ^ drops out of the expression (20) for • Y* because the

main contribution to VK comes from ^T»t^/K.«

The original system of equations has now been reduced to (20)

alone, which oan easily be transformed to a seoond order non-linear

differential equation, which can be solved by a not quite convenient

quadrature, though this is a cumbersome pxooedurej the qualitative

behaviour is obvious. Starting with some initial value, the energy

will damp and V will go to zero.

Initially we have Landau damping; after some finite time,

C • There is no energy available for further reconstruction

of i , and y tends to a constant value* The results are clearly

different from the one-dimensional case in which a plateau is formed.

The picture we have obtained has obvious applications to so-

called turbulent heating. Suppose we have a system in which turbulent

heating is being employed; usually this means that the plasma is

carrying a current whioh drives some instability, so we have relative

motion of the electrons and ions. If this drift velocity is slightly

greater than **VK. ic>n sound waves are unstable*. As mentioned above,

/y^ drops out of the problem and the results can be applied direct.

So even without knowing the wave spectrum we know that,after heating,

the eleotron spectrum will not be Maxwellian but will have the form

just seen. The actual form, of the tail of the distribution will come

not from quasi-linear theory but from other considerations, perhaps

wave-wave interactions•

However, it must be remarked that in the oaloulation just

completed, we assumed an isotropic wave spectrum; for ion sound

turbulent heating this is oertainly not satisfied. Suppose a ourrent

to be directed perpendicular to the magnetio field (the usual situation):

Non-linear theory of this kind of instability will be discussed

later (see Part III).

- 60 -

is the y. -direotion, W is in the ? -direction.

For a small but finite value of the magnetic field, this problem

is exactly the one just considered, since partioles gyrate about the

direction of the field, so that after a time of the order of a few

lanaor periods, particles are mixed in the TK^ -\5L plane. We oan

regard this as a rotation of the wave packet instead of a rotationof particles, and the distribution function will depend in V^ITJ.,..rough XTx* .•+ u only» even for a one-dimensional wave packet.

a

We thus find that there is also a difference between the one-

di;aansional case with and without a magnetic field. Of course, H

crust not be too large, because the simple picture of plasma dynamics

used here»then gets drastioally modified. H is used only to contain

the partioles. We oan negleot it in considering longitudinal wave

properties because ^ p t ^ ^ ^ t , I this is always the case in turbulent

heating situations.

- 61 -

3. Quasi-linear theory of the electromagnetic nodes.

In this section we will apply quasi-linear theory to electromagnetic

nodes propagating through a plasma immersed in a uniform magnetic field H

We will consider the simplest cases of these modes, namely- ion arid

electron whistlers propagating parallel to the field n, i- -», The

limitation to parallel propagation simplifies the algebra "but does not

significantly change the results.

The basic equations for the problem are the kinetic equation

and Maxwell's equations. If we divide the distribution into a slowly

i varying and rapidly varying part, the equation for the slowly varyingi

•part takes the form

( 2 0 a )

where we have introduced cylindrical co-ordinates in velocity spao©, and

the "** sign in the resonant denominator refers to righi)* and Icft-fcand

circularly polarized waves. In the resonant region this equation be-

come

• ••'• . - ; '*•'•' i i ••--•' :

The resonanoe condition now picks out partioles with velocity

VI rK.

The frea.ucncy of the field as seen in a co-ordinate system moving with

this velocity has been Itoppler-3hiftecL to the gyro frequency. Consequently,

the resonant particles rotate around the field H» at the sane rate as

the electric vector E"^ and they are accelerated very effectively fay

this field.

A3 a first application of Eq, (21), we assume a large amplitude

•wave packet is impressed on a Maxwellian plasma. The resonant region

.or the wave packet will be as shown in Fig* 1. It lies in the left-

hand plane because iO-u\«j is negative for the whistler modes* The

colid circles in this figure are the level curves for the Maxwellian*

~ 63 "

•1

As in the quasi-linear theory of Langmuir oscillations, the resonant

particles diffuse in velocity space until a quasi-static state is

reaohed. From Eq. (21) we can easily see that this state will be such

that

-T. - OV

This condition is completely equivalent to the condition that <-T.e V

in the quasi-linear theory of Langmuir oscillations. The solution

of Eq. (2?) is

I'll?'- (23)

as can be checked by substituting solution (23) in Eq. (22) Thus, it

follows that the level curves of the quasi-static distribution are given by

(For a sufficiently narrow wave packet A — ^ —• it will appear

-p-f - — Vj = const.) Therefore, these curves are almost circles but

their origin is displaced to the right by /^ . Consequently, they will

look like the dashed curves in Fig, 1, Since the marginal stability

condition for the whistler mode is just Eq. (22) integrated over IT, ,

~ 0 (24)

K.

CO

we find that the damping coefficient goes to zero in the time-asymptotic

limit, just as it did in the analogous case for Langmuir oscillations.

" 64 "

In order to continue this analogy, one can reduce the two-dimensional

quasi-linear diffusion operator in Eq. (2l)to the one-dimensional form.

Indeed, if we introduce

K(25)

as one of the new variablesyJ*~. derivatives will cancel, and Eq. (21)

with the new variables -jjj- and, say, IT=17 will have the form

•at-

Also the integral (24), which determines the imaginary part of the

frequency, can easily "be written in a one-diraensional fora

mlO

(27)

* In the original paperf^J the one-dimensional form of the quasi-linear.'equation for whistlers was written inaccurately. It was corrected inHef, Tl23 * But the one-dimensional whistler case is also degenerate in srespectt there is no exact plateau in a non-epne-dimensional case. Soactually, we can have only a quasi-plateau Fl3J.

ome

- 65 "

How we already have the full analogy with longitudinal oscillations.

?he integral (27) might be negative, giving an instability, in

plasma with non-isotropic distribution of particle velocities. As one

of the most interesting exanples of this distribution one should mention

the distribution with the so-called "loss cone". This situation will

have some bearing on laboratory plasmas and on space plasmas. The loss-

cone distribution is of the form

(28)

where

and

=1

Fig. .8

For a given Ot the level curves appear as follows

- 66 "

ft* afflibft ;i*si

Substituting this*-distribution into the stability oriterion for -whistlersg i v e s • . • : • • . . - . . - • • : . • . . • • . - ' . , • • . . . '

(29)

where p ~ ft* tfL

,. ,..v.Integrating the first term "by p&rts and carrying out the differentiation

on the other two terms gives

ooI 1 • f.

L <Q

The second term in this equation vanishes identically, and,'provided the

range over which 6 changes is small compared to T , we can replace

.t}i(*') by S(<) in the third term. Consequently we find [i3]J

-•••(30)

»^~*~ n — '— '. .' • Since W is negative,this criteriori gives

instability for large enough \XT^ or .small enough *

At the end of a mirror machine, < . goes to' zero so there is always

a small unstable region.

- 61 -

However, the violation of the looal stabil i ty criterion in a smallregion A2 does not mean that the plasma i s unstable as a whole.

* This would require the wave to grow through many Q -foldings beforei t left the unstable region

Carrying out this integration gives the following non-looal criterion

for instabilityi

] (31)

where &\ H. —~. 'J and F ( S " ) S o e s t o i n f i n i t y as Sj goes t o

zero. In present maohines this condition cannot be satisfied by ionwhistlers, but i t can be satisfied by electron whistlers, if & isnot too small.

According to Eq. (30) the plasma will also be unstable when theresonance velocity is much larger tlian the thermal velocity. In any

•i

' thermalized laboratory plasma there will, of course, be very fewpartioles so far out on the t a i l of the distribution, and the instab-

'< i l i ty will be slow. So the point is that partioles are swept into theloss cone (Fig. 10) when the plasma t r ies to form the quasi-linearplateau, but very slowly. C.F.KENHSL and H .E, PHPSCHEK [ (4]successfully applied this basic diffusion into the loss-conemechanism to the plasma in the magnetosphere where thenecessary high-energy particles exist in large enough numbers

- 68 -

to: make tills in stability an iniportsrit mechanism fo^ particle'untrapping.

Fig. 10

The importance-of ..wbis^ler-type lofes-cone instalnility isoft,en de-

creased by the competition of another instability due"^o the loss cone:

namely, the Post-Rosenbluth one. This instability is a pure electrostatic

mode with a more complicated kind of polarization. Sino© this is very

important we will discuss it in detail.

- 69 "

4» Quasi; linear, theory of the Post-Ro.senbluth. "loss-oone" instability

We: should: lilt© to consider in more detail the Pbst-Rosenbluth in-

stability £l;1fj:in the case where tfie volume of the "loss cone" in the velocity-

space is very-small (for example, it nay correspond to the large mirror ratio).

In this case, we will use expansion on the small parameter (the relative

volume of the loss cone). Further, in real traps of finite length \_ , the

time of escape through the magnetic mirrors is finite and is of order /V.

i where V is the ion velocity along magnetic field lines. Here we willi S

consider the highly idealized situation where this time is much greater than

] the time of the quasi-linear diffusion into the loss cone ^

; L >i If we neglect the particle escape through the mirrors, the loss cone will "be

filled during the tine T^ and then instability will disappear. The relax-

ation of the ion distribution function can be described in the frame of the

quasi-linear theory., In the case of the large mirror ratio it turns out

that the level of the tur&ulence is sufficiently small, to be able to neglect

\ mode-mode coupling.

As in Ref.Fl5|'we assume that:

(a) unstable oscillations are electrostatic;

(b) perturbation scale is sufficiently small for the plasna to be

regarded as homogeneous,, so. that the electric field potential Cf> can be

< expanded in a sum over the fields, of the individual harmonic oscillations:

where K^ and «.. are the components of the wave vector,respectively^along

and transverse to the unperturbed magnetic field VAe - \O(O,

(c) electrons are coldf

(d) the frequency and wavelength of the oscillations are within the

interval,

- TO -

(33)

where £0H] is the gyrofrequency of the species, fo =. *Kj /tGKj i s

the Larmor radius of the particle with thermal velocity JJ - . Develop-

ment of the instability within this interval of frequency is possible

only in a dense plasma

(34)

Under the assumptions (33) we negleot the influence of the magnetic

field on the ion motion and use the drift approximation for the descrip-

tion of the electron motion. Then the kinetip equations for the ions and

electrons respectively can "be written in the form

J2(35)

n

where M is the magnetic moment of the electron (i.e., JA — *"* -*• /$_\\ V V S L ) ,

The complete system of equations contains, "besides Eqs. (35) snd (36), the

equation for the electric field potential

- 71

In the linear approximation we reduce Eqs, (35)-(37) to the dispersion

equation

In a very dense plasma £!,«. >>lOue we must take into account in 3q, (36)

the inertial drift of the electrons which corresponds to the additional

term 6)pt /tOji in Bq. (33). We omit this term in the following

calculations because it changes only the definition' of the plasma

frequencies:.

, etc*

Insofar aa K+ *><Rjmiyte neglect the ion motion along magnetic field

lines. Then the unperturbed ion distribution does not depend on the

azimuthal angle in the velocity space, and we can easily integrate over

this angle:

(39)

Here ¥(X-J is the ion distribution with respect to the dimentionless

- 72 -

. j.. M »..»• a*,... _j -"I" —iii—iw- _

velocities W - Vl/vU> » and it satisfies the normalization condition.

The root of the integrand of 'F(H) corresponds to the taking of the in-

tegrals in (39) for CO in the upper half plane) so that for ^ = yi-+^

W >jL . (40)

Finally, we neglect- the electron thermal motion according to the third

assumption and rewrite the dispersion equation in the form

For the plasma near the marginally stable state one may expand Eq. (41)

over JM. Co = V <-< C0r

= . + *)„£ 11+^ 3 /

The solution with 3 W v O ^ 0 , corresponding to growing disturbances,

appears only if

oo

Vy* > o

The particle escape from the loss cone through the magnetio mirrors leaves

this condition unfulfilled,

- 73 "

Averaging the kinetic equation over the period of the rapid oscilla-

tions in the co-ordinate space we derive the'quasi—linear equation for the

averaged distribution function 4o\.

MK 60 CO

(44)

Under our first assumption L ( is the diffusion time into

the loss cone) no loss of the particles from the trap appears during the

process of the relaxation of the ion distribution. Then, due to the

small volume of the loss cone (our second assumption) the wave energy

is small. Hence, we can neglect mode-mode coupling and take the rapidly

oscillating part of the distribution function -fpQ and the electric field

potential ID^^ in the linear approximation

jitw7-

J

where ^ 0 * is the frequenoy of the eigenosoillation with wave vector K and

amplitude

Within this approximation.we reduce Eq. (44) to the "quasi-linear equation"

in the usual form to "be vaiid now even for strong instability

if j P « K IT^L

- 74 -

f \

9* - 'M*H;We used here the axial symmetry of the distribution function in the

velocity space to average Eq. (44) over the azimuthal and longitudinal

components of the ion velocity* We can do this even for the wave

spectrum without axial symmetry in the K—space since the gyration

of the particles in the strong magnetic field symmetrizes the velocity

distribution.

The quasi-linear kinetic equation (45) and the equation

(46)

where V ^ is determined in terms of "V'otiVA'y e expression (42), are

the "ba ic system, which we have now to solve to find the time evolution

of plasma with the loss cone for -fc^O * This is actually a very

complicated system* However, we can simplify Eq, (45) i*1 e same way

as we did once in Section 1 of Part II, with Eq. (12). This procedure is

right if the main contribution in the integral (43)

comes from that domain of the velocity space where lr>> /^ . If

we take the initial distribution function in the form Qj) shown

qualitatively in Fig# 11,

the condition V»£ would mean that $- «1L where \TP is the

plateau size. Actually, any kind of initial distribution, even, say,

of the type t) very soon will have a form oS) , since the quasi-

linear diffusion is very high for smaller IT" ,

Thus, Eq. (15) might be rewritten in the approximate form £

W.w -y.

(47)

where

This equation can be easily solved using Laplace transform. The

result is [\J~J

(46)

where is the initial ion distribution

is the Green function of Eq, (46)j <k. =

Now we can find the growth rate V^ in terras of

i t in Sq. (46):

])J, and substitute

e»o

^ - ^

- 76 "

Thon wo will have [ 17*1

(50)

where M u <) is the degenerate hypergeometric functionflfij.

Let us consider the idealized ion distribution for which we can take

the integrals of Eq.s. (48) and (50) exactly:

= Aw

where As <? pn o r m a li z a' ti o n constant;

^ ^ ^ >. Q ( A*£ 1. corresponds to the case.of empty loss cone: ^/bJ-0 )•

Then the solution for the distribution function and the equation for wave

amplitude can "be written in the form

- A (•<•

(52)

•-'f.

(53)

- 11 -

From tho last expression we see that, if initially A > , the growing

oscillations cause' the turbulent diffusion of ions into the loss cone.

As we can see from expression (52), for the particular1choice of the initial

ion distribution (51) the quasi-linear relaxation of the turbulent spectrum

in tine can be described by changing in time the parameters A(-t) , £ (-fcO

and the effective "temperature" of the rcain body of the ion distribution

only. This change can be represented by using the, formula (42) :

= e

How it is easy to describe the solution of Eq. (53) qualitatively.

At some moment *td we reach the marginally stable point £ £-t«, ) ~ \ £.C"ko3'

and the oscillations stop growing. However, the ions continue to diffuse

into the loss cone (see Eq. (47)) and the oscillations become damping.

It is obvious that the relaxation process stops only when the oscillation

amplitude reaches the zero level, when

The final ion distribution has a margin of stability with respect

to the considered perturbations (i.e., JJ «* < 0 for arbitrary wave

veotor K. ). The parameters of this distribution depend on £) only

and can be found from the energy conservation law*

«*» CO

where £ _. ^ w J)

* In ?Lof.ri7jit was proposed to use-for this aim the equation % Cpoo)-0because after the first state of relaxation to-the quasi-ateady,state (55)the particle loss through the mirrors supports the instability on a verylow level. But in that case we must add at the r .h.s , of Eq, (47) theterm describing this particle loss and change the expression for the growthrate (50) (This was not done in Ref . j l ] )

- 78 -

From (52) and (55) we immediately obtain

In the limit of large mirror ratio the parameter £ is small

and the amplitudes of the oscillations remain small during the relax-

ation process (i.e., D <<• 1 ). On the other hand, by reducing the

quasi-linear equation (45) to the form (47) we suppose that the width

of the sink on the distribution function essentially increases due to

the quasi-linear diffusion. Eenoe, we can apply the result (56) only

to the case of initially strong instability

(57)

Sxpanding (56) on small parameters £. , D U D , ^ / A Jwe obtain

(58)

The ion distribution in the large time limit, as it follows from Eqs. (52)

and (58), is equal to

~ JLrfc) t f* J

-(IE) %

(59)

- 79 -

Here we can see directly that the sink on the ion distribution under

condition (57) becomes wider and less deep, after relaxation, and we

can justify the approximation (47). The main part of the distribution

function is only slightly disturbed ~ay "quasi-linear diffusion".

In order to write the equation for the waves in terms of one

variable J) only, we need to suppose, as usual in the quasi-linear theory,

that the wave spectrum has a sharp maximum at the point K = K* >

V ^ ^ } ^*£f(.K*) • Now we neglect the change of frequency for

I>[t)>> B an<* integrate Eq. (53) up to the quadrature with

conditions dD/ji \ -=_ Q T} (<xf\ •=. "T)

£•/£**- (60)

where

For the distribution (51) the maximal growth rate is achieved at point

(61)

Parameter JJ in the saturation regime can be easily found from Sq. (60)

as well as from Eq, (55)

3 n ,

[ * e(62)

- 80 "

W f l W M l If I •"*< •

By doing this , we introduced a lot of idealizations in order to getsoluble equations. Of course, in 0L real is t ic situation, the develop-ment and non—linear relaxation of tlie loss oone instability have a morecomplicated nature, due, for example, to the continuous loss of part-ioles through the mirrors, which we did not take into acoount, and soon. In these circumstances, the non-linear mode-mode ooupling becomesvery important, as was shown in Ref. [l7j .

5* Non-raaonant adiabatio wave-particle interaction

Up t i l l now we have considered only the resonant interaction ofparticles and waves. For their non-resonant (or adiabatic) interactionwe oan no longer approximate the resonance term in the diffusion equa-tion by a o -funotion. For example, in the oase of Langmuir oscillations,the diffusion equation takes the form

21 - si 2. y

- U/. 9

where we have approximated )td* ~KV~' by ^ ^ OJ*, in thelast step. Using the equation for wave growth, we oan rewrite this as

- J/e )2_X l£) IE T?t ~ 2 &/ cop \dt 4 - ' sl n in . (63)

- 81 -

Multiplying T*oth sides of Eq,. (63) to C~? and integrating over allIf, we have

Prom Eq.. (64) it ia obvious that non-resonant interaction in quasi-

linear theory has a trivial meaning: i t describes the participation of the

main body of the plasma distribution m thaoscillations. Therefore,

such interaction should "be taken into aooount in order to have an

energy conservation law, and so on.

Let us come back to Eq, (63). Changing co-ordinates

from *to r - H 2 |Es(x)|Z/4T,n.

If we choose the init ial conditions -f (v)- o(v) and 7? —0 then3<1» (65) b-as the solution

Consequently, the non-resonant interaction heats the bulk of the

plasma to an effective temperature '£*'=• 13 I E^J / LTTTl .

We should note in passing that the Gaussian shape of the final

distribution is.another demonstration that quasi-linear theory oannot

be applied to a single wave. For a single monoohroinatio wave generated

in an init ially oold plasma, the distribution would be of the form

y/v;= < >>(v-v(V)> = -J—===; (6T)

A lot of instabilities in plasma are due to the non-resonant

interaction of plasma and waves. It is obvious that the quasi-linear

relaxation in such oases should be described in terms of adiabatio

- 82 -

in te rac t ion . Lot us s t a r t with the firehose ins tab i l i ty [19 , 20I whioh gives thesimplest example of adiabatic wave-partiole interaction [21] . In th i sin s t ab i l i t y the Rc(u)) i s zero and the 3*1 fed) i s given by

when »i

If the plasma is not far from marginal stability (i.e., -r-— « ]_ )

then ivv\(tO) will be muoh less than K V*hl , and we can apply

quasi-linear theory. ITote that quasi-linear theory does not always

require Xwi (GO) << f je.(u)| } as mentioned, above, the KJI(U)) is in

fact zero for this instability. Since KIT^: «OL) H J for the f ire-

hose instability.there will be no resonant partiolesj the instability

itself is algebraic in nature and does not require resonant particles.

To get the diffusion equation, we replaoe Oi in Bq, (2C&) by i Ivrv (tO

and use the reality condition Im. (ULKW ITM(U)IS) • T 3 a i s procedure

gives

2H = _ ^ 1 — -

(69)

where we have used

(TO)

We can solve .Bq.. (69) by the following approximate procedure

Since the plasma is near marginal stability (i.e., _£L! « [. )» theTdistribution can be expressed as

where 4 M is a Maxwellian and j 1 is a oorreotive term produoing the

-83 -

anisotropy. If we negleot this small oorreotive term when evaluating

the r.h.s. of Eq. (69) we obtain

£-J

The solution of this equation is

* / •

e

(71)

-JMLIHKI (72)

This funotion oan now "be used to calculate the growth rate

2^cdh*CcXIHJ

(73)

The maximum growth rate in the linear theory is found for K "jT'.lfrIf we are willing to use this value of K[ for the coeffioient in Eq.(73), we obtain a tractable non-linear equation for the growth of thewaves

(74)

The solution to this equation is of the form

where

- 84 -

-fc

Fig. 23

One can give a simple physical interpretation for the removal of

the temperature anisotropy. The instability is loir frequency so the

integral

3 -- (75)

should be constant. At first the field lines are straight, but, as the

amplitude of the wave "builds up, the field lines beoome very wiggly.

no wave

with wave

Fig. 24

Since the path of integration in Eq. (75) beoomes longer

oome smaller, and this presumably reduces l u.

must be-

Although the adiabatio quasi-linear diffusion describes the wave

interaction with all particles, nevertheless the strength of this

interaction might be quite different for different domains in velooity

space. In such situations the quasi-linear distribution function might,

in the process of relaxation, have a quite peouliar form.

An example of such quasi-linear theory is the case of the "whistler"

mode with no (initial) magnetio field [23,24). If we have a non-isotropio

distribution function, it is usually unstable against whistler type

perturbations.

Fig. 25

Let us consider suoh an anisotropio distribution j [ 2 > x. I •

As indicated in Pig. 25 , the effective temperature in the x-direction

is greater than that in the y-direction. We can easily show that a

pure transverse perturbation ( F J. _K ) will be unstable for any arbitra-

rily small anisotropy* To do this, we write the linearized Vlasov

equation, where the first order quantities contain the faotor Q i {

(76)

and

- 86 -

(T7)

7 (78)

So we have four equations for ft , f€ , Ay, A*. We get (expressing

the fields in terms of Hu)

(79)

Suppose now that the ion distribution is anisotropio. (if 4Qi were

isotropio, the magnetic field terms in the Lorentz foroe would oanoel.)

Substituting in the expression (77) for the current, we get

_ 1£ /• / + the similar electron contribution

from the current in the x-direotion.

The dispersion relation now has a simple form

in*/ Uv ASiIf the plasma is anisotropio, this becomes the usual dispersion

relation for waves in a plasma in the absenoe of a magnetic field*

When wo have an anisotropio distribution, we oan look for a new root

- 87 -

of this equation* It arises as very low frequenoy mode. Let us suppose

that J- ^ €. 2.T* 2-Ti , Taking ^ « fc^lT. t we o a n derivefrom (81) . '

Now we oan recognize that this wave is unstable for sufficiently

small K if i_ — x/-y < O (in terras of a nor© general distribution

function ^ (v*^^ 7 - ) , it would be ^ ^ ^kdlT + 1 < 0 )•There is no real part in ( , and therefore"it describes an aperiodic

instability. The imaginary part is very small if the electron

distribution is anisotropic.

Suppose we have the opposite situation, t DL <. i j . . Then instabil-

ity results if the direction of propagation is rotated ( i . e . , by inter-

changing the roles of x and z) by 90 degrees. So this situation is

absolutely unstable, even for very small anisotropy.

If we look at the imaginary part of ft) , we see that i t increases

with increasing wave numberj but we cannot take K, very large because

the additional term beoomes significant and gives stabilization. Thus

there exists a oritioal wave number below which waves are unstablet

Let us see how quasi-linear theory may be applied to such a wave.

To find i t we oonsider the quasi-linear theory for this kind of pertur-

bation. The equation for X has the form

n i r lme ^ -^ J 1V

where *->- denotes rapidly varying functions. Bert we use the expression

found in the linear theory for the rapidly varying functions, as is

customary. The quadratic expression is averaged over many cycles, and

we obtain the final result

- 88 -

= Jm

( KC ~ ~ J

where the bar indioatea the average.

)2J-°

4 t , LWe can represent ^ ^ K ^ as _ - _ — and neglect to

5 & £in the denominator since i O l <t *C ITU'

•t«v

How we can easily go through the same sort of oaloulation as thatjust completed for the firehose instability* Finally, the quasi—linearequation becomes

The physioal meaning of this quasi-linear diffusion in the velooityspaoe is clear. The instabili ty creates magnetic flutes, producing achaotic magnetic f ield. This chaotio field influences the motion of theparticles* Thus we have scattering of particles due to small-scalefluctuations in the magnetic f ield.

Although this case is en adiabatio interaction of modes with a l lparticles, we oan see from Eq. (84) that the quasi-linear diffusion forthe particles with small \T^ is much greater than for the main body of thedistribution.

Therefore one oan expect the considerable reconstruction ofpartiole distribution to ocour mostly for very small region, where

. 8 9 -

6. Quasi-linear theory of drift instability

A. Linear theory of drift waves

One of the most important types of plasma instability is the drift

instability of non-uniform plasma, which leads to increasing heat

and partiole loss due to the turbulent transport process in the plasma

Therefore, more careful attention should be paid to the influence of

this type of instability on plasma confinement in traps and to the

non-linear stage of their development. Let us consider here in detail

the low-frequency ( i . e . , <*> << ^>ni) "universal drift instability" of

a plane slab of low ft plasma in a strong magnetic field H =^0,()>rtr \25~2%J

% Jji

We take the plasma to be inhomogeneous in the x-direction, with

V \ ^ p = 0 and small density gradient

r ^ <(86)

For simplicity we put \i~ U and ^pi ^ni. , so the perturbation

can be considered as quasi-neutral. Then the plasma is unstable in

respect to the electrostatic perturbation with the phase velocity within

the interval

(87)

The electric field potential ^f is written here as the sum of propagating

i plane waves

(88)

Generally speaking, the equation for the x-dependence of the wave

disturbance has a form of the integro-differential equationJ29-3ljwhich

has in the HKB approximation a solution of the usual type:

- 90 -

(89)

Here A^ is the complex function, but with small imaginary part for theinstabil i ty with small growth ra te . Then the square of this function— Kx (zCj to j K) - Uo K fo )-ri J/^ . cy piays a role analogous to

the complex potential of the Shrodinger equation. The turning points2.

of the real part of " - Ax M restr ic ts the region of co-ordinatespaoe, where the wave packet in the form (88) can move. The descriptionof the unstable perturbation by wave packets (88) is justified if theygrow in the time between reflection from the turning points up to thelevel at which we cannot neglect the non-linear mode oouplingJ32] .Therefore, by choosing perturbation in the form (4) we suppose that* [17J

- (90)

where % ^a "t]ie amP l i ' tu<i-e °f "fclle wave in the quasi-stationary turbulentstate and ^Kih i s "the amplitude of the electric field fluctuation attime t = 0 (for the quiet plasma i t is the amplitude of the thermalfluctuation).

We start by deriving the linear dispersion relation for the driftinstabili ty with the help of the integration of the Boltzmannequation over the unperturbed particle trajectory [34I . The equilibriumdistribution function depends on the constants of motion 1£} TT,,t j \only,

(91)

* Convection of the wave packets in the non-uniform plasma was con-sidered in \}i\' The instabili ty condition found there is similar to(90).

- 91 -

In the linear approximation we can write the Boltzmann equation with

an electrostatic potential perturbation in the form

Sh

. Tir.2

On the l.h.a. we have a derivative taken along the unperturbed trajectory

of particles in the magnetic field

j(t) 'W«j [V-1)) - Sin

(93)

UTvHl-t)

where U^ and )J[| are the velocities of the particle across and along the

magnetic field, and 0j is the angle the velooity vector makes with

the U^axis at the instant t ="t.

Ve will write these in vector form using the unit vector rL - ~ A\'

H \

Carrying out the integration over trajectories in (92) using (91)

and (93), we obtain the first order correction to Xj in the form

(94)

where all quantities which do not depend on the time t1 following the

trajeotory have lift en carried outside the integral.

Now we use1 the fact that the total time derivative along the

particle tTajeotory ia

< * - • • - » - > • • • • : ( 9 5 )

and use the Bessel function expansion

(96)

where

Then we oan do the time integration in (94) and obtain the Fourier

component of the perturbed distribution function

(97)

Here the small additional term t £• has been inserted to make the

integral convergent* and corresponds to the adiabatio switching on of

the perturbation at -£ =• — °o .

Let us now consider the stability of the local Maxwellian

distribution

- 93 -

where r\o^t 1] are the density and temperature of the jth species. For

low frequency waves (Co <^OHjJ, we negleot in Eq. (97) al l terms with

XT 0. Also, by (87) we may drop the K-j.\J term for ions,which

describes resonant wave-ion interaction. Then on integrating with

respect to velocity we find the perturbed ion density

Here the function

CM = L UVs>) e.

describes the weakening of the effective electric field experienced

by the ions, averaged over their circular Larmor orbits. In the limits

of small and large Larmor radius, respectively,

/ « ^ Of**)]

The Larmor radius of the electrons is vexy small and for them Eq. (97)

corresponds to the usual drift approximation. Expanding the eleotron

term in <*> / K^\J^ we obtain

In the limit of quasi-neutrality ( A?4 \j, < c £ , \j)

the frequenoy andv the growth rate are given by

' /*•

- 94 -

Kv/(r.t-i) (101)

We see that the instabili ty arises as a result of finite ion Larmorradius effect and that the growth rate goes to zero simultaneously withthe ion Larmor radius. In order to see more olearly what oan happennear this marginally stable case we will write the expression for thegrowth rate through the local characteristics of the distribution ofthe resonant electrons:

In the approximation of the small ion Larmor radius, two terms on ther . h . s . almost oancel accidentally for the Maxwellian electron d i s t r i -bution at one point IQ. - ^/^ and instabili ty becomes very weak.

(This cancellation takes place also for the arbitrary temperatures ofthe species ~7^7c )• Therefore, even small deviations from the lineartheory may strongly change the stabil i ty in respect of the finiteamplitude waves. There are two effects of this type:

1) If the amplitude of the drift wave is large enough, theresonanoe region will be broadened by a considerable amount ( i . e . ,AU|1 •"— y *^if ) . We will now take this broadening into account

and show that a wave of finite amplitude can be stable in the non-linearregime.

2) The relaxation of the eleotron distribution under the influ-ence of the oscillations of finite amplitude, as we know already, alsoleads to stabilization of the instability'.

Let ua f i rs t consider case l ) whioh takes place in the stabil i typroblem of the monochromatic drift wave.

B. Kon-linear stability, of the monoohromatio drift wave

We choose the following form for the eleotrio potential:

where we have set tf » 0 for simplicity. If we work in a co-ordinate

system moving with the wave, the rate of increase in resonant eleotron

kinetic energy, which must equal the rate of decrease in wave energy,

can be written as 1 /

dT _ rz_

In the drift approximation, the general solution for the distri

bution function can "be written as

where L [J£,\J^o,o] is the initial distribution and (jo o ) is

the initial position of the particle. We can divide the distribution

into two parts

,1] = fi (105)

where the f irst part is the local Maarwellian and the second part is a

< perturbation due to the wave. The second part only makes a contribution

! to the harmonic generation and we can neglect i t while we evaluate the

I rate of change of kinetic energy. Consequently, we find

16 -2t

0 o] !£•

( 1 0 6 )

The particle equations of motion in the drift approximation,

o(107)

have a solution in the form of e l l ip t io integrals. First we Introducea new variable, KgLf +• Mt £ « Z p , and rewrite the equation for

conservation of energy,

2.in the form

(108)

2.e<where

For the case where 5£?< 1, we can write the solution of Eq. (108) as

When 2£^> 1, it is convenient to introduce the new variable

defined as

D-This equation has the solution

. (no)

To get the time dependence of the growth rate we need only put

these solutions into Eqs, (104)-and (106), This problem was solved by

O'HEIL fSl in the approximation which reduces to the linear growth rate

at t - 0. We would now like to take into account non-linear effects

that are present init ially, so we must include more terms than O'ffeil

did. After substituting Eq. (106) into Eq. (104), we ootain

i

- 97 -

whore we have dropped even functions of £ by symmetry arguments and have

neglected the time dependence of 3L .

Carrying out the integration in Eq. (ill) gives us the following

expression for the growth rate ;

T <

**&3&

ftrrrtt 1

In the limit where .

growth rate oomes from the

<<c 1» the main contribution to the

1 term in the f i r s t sum) so we find

(113)

Consequently, a wave with finite amplitude can be stable in the non-

linear regime if the linear growth rate ia small enough.. Here we took

into account only finite Larmor radius effects, but the same idea can

also "be applied to the current- or gravity-driven drift instabilities

in an inhomogeneous plasma. Therefore, we can use this expression to

estimate the amplitude of the separate drift waves observed by H",S,

Buchelnilcova in a potassium plasma.

Since the amplitude we obtained from Eq.. (113),

is very small, we can obtain the level of harmonic generation by expand-

ing the Vlasov equation in the small parameter

the amplitude of the second harmonic is given by

e

For example,

where we have chosen

this oan be written as

In the limit of small Larmor radius

T, *One can expect that the amplitude of the nth. harmonic will be; propor-

tional to the nth power of tke main harmonio

- 99 -

TConsequently, we find that the amplitude of the hannonicB decreases

exponentially as a function of the frequency.

The picture drawn here is valid only for the narrow wave packet

in whioh the phase velocities of the different waves are very close,

A /,<- < >J __Lf . But during experiment the instability can

arise in a wide region of the phase spaoe and this condition is

violated. Then,the main stabilization effect comes from relaxation of

the electron distribution.

C. Quasi-linear relaxation of the particle distribution and

transport processes [/3Jj 3GJ

As usual, we express the distribution as the sum of a slowly and

rapidly varying part (i»e»» -T• =• jC • -t- X* ) and obtain an equation for

the slowly varying part by averaging the Vlasov equation over the rapid

oscillations

Expanding $ and o L\ in Fourier components allows us to express the

r .h . s . of Eq. (114) as

( jC * $ ,x v) (us)

where the cross terms have vanished in the averaging process and we

have explicitly made allowance for a slow x-dependenoe in (^ (x) and

In accord with the philosophy of quasi-linear theory, we assume

that the distribution can be written in the form "

- 100 -

».

where the dependence on the last two variables is slow enough to be

ignored while evaluating o -f x j • With this point in" mind, we

oan easily generalize the expression (97) for b /•£ /

I,K r/-l± - 1 1

We negleot here the contribution to the integral from the seoond term

of the expansion on slow oo-ordinate. • ,

%\x(4- f

Substituting this expression into Eq., (115) and averaging the result

over the azimuthal angle in velocity space, we find

Since Ki^th' ^ (O < K Uiku f ^e irL^erao>''ion °^ *^e waves with

the ions will be primarily adiabatio and the interaction with the

eleotrons primarily resonant. Also, we may drop finite Larmor radius

effects in the eleotron equation but not in the ion equation. Conse-

quently* the electron and ion equations become

- 101 -

(117)

and

coHi

(118)

Of course, the eleotron equation could have been derived more simply

from the drift approximation.

Bext we investigate the formation of the quasi-linear "plateau".

For this purpose it will he convenient to evaluate the K. -dependence

of the eleotron diffusion coefficient at that value of K whioh is only

unstable (i.e., K-=K , where K^ -~ **** /yy and K. can he found by

taking into acoount the competition between quasi-linear plateau fornn-

at ion and collisions, as will he shown later. The waves with KTa < K y

a <Jare damping.) If we introduce the new variables

,r- i. yr 2 Ct)~ (One

we can reduoe the differential operators, in Eq. (11?) as

- 102 -

Consequently, the level curves of the "plateau" are described "by the

equations

Since }?^. <<• lT^£ in the resonant region, the level curves of the

original Maxwellian oan he described by the equations

X const•

where \r/ ^ - Si ^ Since «? -no : * <> / -2-ro 'it follows that the two sets of curves differ only by finite Larmor

radius effects (see Pig.26). The energy gain which we receive after

relaxation of the electron distribution, for this reason is also small

for the small ion Larmor radius.

resonant region

quasi—linear "plateau"

Maxvellian

- 103 -

If we plot the distribution as a function of IT^ (see Fig.2.T), we

find that the slope in the resonant region must beoome very steep, so

that the enhanced Landau damping oan stabilize the waves. During this

process the electrons lose the energy of the motion along field lines.

Therefore, we can say that the energy goes to the unstable waves from,

the longitudinal motion of the eleotrons.

Fig. Z%

We can also use the oonstanoy of B to estimate the change in

an electron's position /35_/

(120)

Since i>l% £ 1TJ\ the displacement of the resonance electrons is

muoh smaller than plasma radius /"*• —- .

n

Thus the instability inhibits itself rapidly and the change in eleotron

density is small.

- 104 -

One oan easily see that the electrons and ions diffuse across

the magnetio field at the same rate. Integrating Eq,. (117) overvelocity cof-ordinates gives the diffusion equation for the electrons

n 'HP aSW j*

= Is

where we have introduced the groirth rate YK. and multiplied and

divided by ^ x * Using Eq. (ll8), we find a similar equation for

the ions

One oan easily see from the dispersion relation (i.e., ^ £ ~

) ^a° /%- r1 ) "fciia"b ^e cLiff^si011 coefficients

in these two equations are identioal.

Since the oollisionless theory gave only negligible diffusionacross the magnetio field, we will modify the previous work by addinga small collision term to the r .h . s . of Eq. (114). The collisions willtry to make the distribution take the form of a local Maxwellian andwill thereby prohibit the formation of the quasi-linear'plateau11.Consequently, we should find enhanced diffusion in this oase. Theelectron equation will now be of the form

where

- 105 -

S i n c e we a r e t r e a t i n g t h e c a s e w h e r e " S £ - . .{fe f <•<

we oan solve Eq. (117) by successive approximations /36y. We express

the distr ibution^ J~ r T^o>-t •£'*** and then demand thatJc je je

and that

Eq. (123) has the solution

( 1 2 5 )

To solve Eq. (124), we integrate i t with reepeot to C

) 7J

Since | will be more strongly dependent on UV than onX , we can

neglect the derivative with respect to JC in the first bracket in

Eq. (126). If we also use Eq. (125) to evaluate the second term on

the r .h .a . of Eq. (126), we find

- 106 -

Integrating this expression over If. gives

(i)where we have introduced the actual growth rate, $* ,, and the

growth rate for a Maxwollian plasma, )(i (see (102) )•

Although this expression cannot he used to evaluate either the

growth rate or the field amplitude separately, it can "be used to

evaluate the diffusion co-efficient, which depends on the produot of

these two quantities. From Eq. (l2l) we can see that the diffusion

coefficient is given by

10

T H e n ' T & ® 5 • ' " ' ( 129 )

By using E-q. (128) we can express th i s as

Evaluating this expression at K^ ~ -/J/t and .2 «t

gives / 7

- 107 -

Of course, this formula is only valid when ^"t.Qu ifnef >>- "^^Zl^e (

In a later lecture we will investigate the wave amplitude and rewrite

this condition in a concrete way.

- 108 -

REFERENCES

1. BERNSTEIN, I.B., GREEN, JJI., and KRUSKAL, MJ)., Phys. Rev.

108/ 546 (1957).

2. BUNGEY, J., J. Fluid Mech. (1963).

3- MAZITOV, RJC.,Zh.Prikl. Mekh. i Tekhn. Fie . ' (1965).

4. O'HEIL, T.f PhyB. Fluids, 8,2255 (1965).

5. ZAKHAROV, TJ!.» and KARPMAJT, V . I . , Soviet Phys . - JETP 16, 351

(1963) (Zh . Bkspexim. i Teor . F i z . , ^ 3 , 490 (1962) ) .

6. VEDEJTOV, AJi.., TTBLIKHOV, B J ? . , and SAGEEEV, R.Z. , Uuol. Fusion

_1, 82 (1961)* Nuol. Fusion, Suppl. Pa r t I I , 465 (1962) .

7. DRUMMOSD, WJS., and PIKES, D . , Nuol. Fusion Suppl. Pa r t I I I ,

1049 (1962) .

8- TIDHAJT, D., and H01TTGCMERY, D. , "Plasma Kine t i c Theory",

MoGraw H i l l , If .T,, . (19.64).

9. AL'TSHDL1, L J i . , and KARPMAN, V J , , Soviet Phys.—JETP ^20,

1043 (1965) (Zh. Eksper inui Teor . F i z . 81, 515 (1965)).

10. KADOMTSSV, B^B., "Plasma Turbulence", Acad. Press, London (1965).

11. GALEEV, A.A., KEKHEL,. C .F., and SAGUEEV, R.Z., IOTP, Trieste,

Preprint. IC/66/83,

12. ROWLAHDS, J., SHAPIRO, VJ)., and SHBVCEBHKO, V.I. Zh. Eksperim,

i Teor. Fiz. JjO, 979 (1966)

13. KEHBEL, C.F., ENGEIMAN1T, F.,"ICTP, Trieste, preprint ic/66/90.

14. KSHHBL, .C.F., and PBTSCHEK, H.JE., J. Geophys. Res. 71, 1, 1

(1966).

- 109 -

15- ROSENHWTH, M .N., and POST, R.F . , Fhys. F lu ids £ , 547 (1965) .

16. GALEEV, A .A., Zh. Eksperim. i Teor, Piay; . , J

4 9 , 672 (1965))

17. GAL3EV, A.A., "Plasma Physics and Controlled Thermonuclear

Research", Vol I , p . 393 (IAEA, Vienna, 1966)

18. GRADSTEIIsr, I . S . , and RYZHIK, I J I . , "The t a b l e s of the i n t e g r a l s ,

. . . . " Goatekhisdat (1962) .

19. VEDENOV, A.A., and SAGDEEV, R.Z. , C o l l . Plasma Physics and the

Problem of Controlled Thermonuclear React ions" 3_ ( 1 9 5 8 ) ^ 0 o i l .

P i z . plasmy i p r o b l . : u p r a v l . temayad. r e a k t s i i , _3» 2 7 8 (I958)j.

20. ROSEHBLUTH, MJST., '.2nd Geneva Conference on the

Peaceful Uses of Atomic Energy (1958) .

21 . SHAPIRO, VJ>., and SHBVCHEUKO, V . I . , Soviet Phys. -JETP l8t

1109 (1964)' (Zh. Eksperim. i Teor . P i z . £fc 1612 (1963)j.

22. ZASLAVEKII, G M., and MOISEEV, S .S. , P r i k h . Mekh. i Tekhn. P i z .

(1962) .

23. MOISEEV, S .S . , and SAGDEEV, R . 2 . , Plasma Phys. ( j . Uucl .

Energy, Pa r t C) £ , 43 (1963 ) .

24. HJRTH, H J . , Phys.. Fluids _6, 48 (1963).

25. RUDAK0VtL.I.,anl SAGDBEV, R.Z. , Soviet Phys . -Doklady (Dokl.

Akad. Uauk. SSSRj 138, 581 (1961 )j ITuol. Fusion, Suppl. Par t I I ,

481 (1961) .

26. KADOMTSEV, B.B., and TBIOPEEV, A.V., Soviet Phys. -Doklady J , 826

(1963) (Dokl. Akad. ffauk SSSR 146,581 (1962) .

27. GALEEV, A.A., ORAEVSKII, VJff., and SAGDEEV, R.Z. , Soviet P h y s . -

JETP 1J,615 (1963) (Zh. Eksperim. i Teor . P i z . 44,903 (1963) .

- 110 -

28. MKH^ILOVSKII, A.B., and RIIDAKOV, L . I . , Soviet Phys. -JETP 17,62T (1963) (zh. Eksperim. i Teor. F i z . ,4J;i912 (1963)).

29- GALEEV, A.A., Soviet Phys. -Doklady jS , 444 (1963) (Dokl. Akad.

Nauk SSSR 3^0, 503

30. SILIff, V.P. , Soviet Thys. -JETP OJ, 857 (1963) [ zh . Eksperim.

i Teor. Fiz. 44,1271 (1963)).

31 . OALBBV, A.A., Soviet Phys. -JETP 17, 1292 (1963) (Zh. Eksperim.

i Teor. F iz . ^ , 1 9 2 0 (1963)).

32. KADOMTSEV, 3.B., Soviet Phys. -JETP 18 , 847 (1964)

Eksperim. i Teor. F iz . 4^ j 1231 (1963)).

33 . COPPI, B. , LAVAL, G., PELLAT, R., and ROSENBLUTH, MJST., ICTP,

T r i e s t e , p repr in t i c / 6 5 / 8 8 .

34. ROSENBLUTH, M.N., KRALL, IS,, 'and ROSTOKER, JT., ITuol.. Fusion,

Suppl. Par t I , 143 (1962).

35- GALEEV, A.A., and HDLAKOV, L . I . , Soviet Pliys. -JETP 28

(1964). (Zh. Eksperim, i Teor. F i z . , & , 647 (1963)) .

36. ORAEVSKII, VJT., and SAGDEEV, R.Z., Soviet Phys. -Doklady

568 (1964) (Dokl. Akad. Nauk SSSR 150 , 775 (1963)) .

- I l l -

PART I I I . WAVE-PARTICLE NON-LIEEAR IBTERACTIQN"

1. Electron plasma oscillation turbulence -

Let us now turn to tlte last principal meolianism for non-linear

interaction between the waves and the plasma, in the case whe"re there

is no magnetic field. If we impress two waves on a plasma, then these

waves will beat with the mixed frequency ( £j, t 0J2, ) and mixed wave-

length ( Kf ± K} ) . We have already considered the resonance of

this beating with a third wave ( C()3 , K3 ) in the decay-type inter-

action. But, in analogy with the linear theory, this beating can also

resonate with particles moving at the velocity

(1)

This type of process' has been included for the f irst time in the.theory of

weak turbulence by ¥. DBUMOUD and D. PIKES [ l j for a one-dimensional

wave packet, and by KADQMTSEV and PETVIASHVTU [2] for the general case

(see also / 3J). In our derivation we will follow the article [4] .

Of course, the rate of energy-increase (or decrease) due to this process

will be proportional to the product of the energy in the two primary

waves. Consequently, i t will be a higher order correction in the

expansion on wave amplitude. But very often the linear growth rate is

small, because only a few particles can resonate with the wave, and

the non-linear oorrection to the damping coefficient can be important.

To illustrate this effect, we oonsider a wave packet of random-

phased Langrauir oscillations. The frequency for Langauir oscillations

is essentially constant ( i . e . , CO1 = u)pt(£ + \ K^Aj,) where

^ \ * i< 1 ) . Consequently, i t takes at least four waves to

satisfy the frequency resonance condition

and wave-wave scat ter ing cannot enter the problem un t i l third order in

the -wave energy. On the other hand, in seoond order in the wave

energy, we oan sat isfy the resonance oondition for non-linear i n t e r -

action with the par t ic les

- 112 -

(2)

As usual, we expand the distribution in powers of the wave

amplitude by using the iteration formula

K'+K"*! JoO

Substituting this expression into Poissonfs equation gives the dynamic

equation for the waves

(3)

where

(4)

etc.

- 113 -

• 4 . - , , * . •;••.

We solve Eq. (3) by expanding in powers of the field amplitude. The

lowest order solution just defines the field eigenfunotions

(5)

where 60 (K^ is the solution of RQ. \ £ K (**)) J ? 0 . One can

easily see that the second order solution is

To derive the wave kinet ic equation, we f i r s t multiply Eq. (3)

Toy <p* (K LS ) P *>'*** ~ )t and integrate over did du) • The

first term in the resulting expression will be of the form

In • * / *-v \

% C (to) f(i,«) f *(jyo j

Sinoe / (K ((O) is peaked around k\c , we can rewrite the imaginary part

of this term as

U i y«V»£%)f{!S^^C!i8)€

C 8 )

If we use Sqs. (5), and (6) to evaluate the remaining two terms andthen average over random i n i t i a l phases ( i . e . , < ipj<* ipK, J> ~

£ ; ) we find the well-known kinetio wave equation[2,4,5]

- 114 -

4 £?.,

(3 )

•where we have dropped terms which are higher than fourth order in thefield amplitude. The f i r s t term on the r . h . s . of th i s equation i s justthe quasi-l inear growth r a t e . The second term gives the rate at whichthe modes ^Ki and *fV' decay into f* . The third term gives therate at which tpK and U^i decay into ^K" » but i t also gives acontribution due to the non-linear interaction of the waves and pa r t i c l e s .The third term i s also due to the non-linear wave-particle in teract ion.

For Langmuir osci l la t ions the decay-type interactions make no

contribution, since we oannot satisfy the frequency condition

t r t ^ * ~*^&! * 1^ we r e s t r i c t our considerations to the caseof a narrow spherical wave paoket with width (&k) ^j>e

<c 0 ^ / 4 . \ *$ >)^

then we can also drop the non-linear interaction of the waves with theelectrons. With these points in mind, one can easi ly see that the maincontribution to the r . h . s . of Eq.. (9) wil l come from the third term andthat in th i s term £ z^ wil l be determined by the electrons and £ ^ bythe ions . To evaluate <5 ,.i /u)* _<0K') w e expand -the integral inEq. (4) in terms of the small parameter £ (ui t -u)K iJ //y._vi.' \\r ~\

* Singulari t ies of type X/L (co,lCj near 4j -6>(K)must be treated in Eq,. (9)

according to the rule (See Part l ) fe^ = ^_ v , 0 + lV

- 115 -

vCHi - f t '

7> )(^\\( \

(10)

In a similar way, we find that

T(11)

and that

. IT" **•

where (12)

Consequently, Eq. (9) becomes

(13)

- 116 -

We can see from Eq,. (13) that the number of waves is conserved

in this process:

This is easy to understand if we look at the resonance condition as

an equation determining the energy a partiole receives when wave <pK

scatters from it into wave (PK .

Since this quantity is much smaller than the energy in either wave

( i . e . , {K^-KiVy^"31 Ul^-tOo ^ ^iWjj the number of waves must be

conserved. Let us note that in the limit of large phase velocity of

waves the wave number conservation law can be found for arbitrary waves

using the symmetry properties of the coefficients £ and C in

Eq. (9) [ 6 ] .

Eq. (13) oan easily be solved in the case where the spread of

the wave packet is much larger than the ion thermal velooity

» I— . CH)

In this limit, Eq. (13) oan be reduced to the following differential

equation:

3f

where

- 117 -

(15)

The r.h.s. of this equation describes decrease in energy due to the

scattering of waves to longer wavelengths. The l.h.s. of the equation

describes the steepening of the wave packet in k -space. The general

solution of the equation was given in ("4],

where &- G J^~, P { f>'L U " ^J is the initial energy

distribution in the wave packet.

In the time asymptotic limit the solution can be written in the

form [ 7

2. Current-driven ion sound turbulence

Now we will discuss current-driven ion sound turbulence in the

limit where ~T* »r"TT and Ho '— 0 . fceoausie the instability

is electrostatic in nature, we Can use the wave kinetic equation

derived in the previous section ( i . e . , Bq,. (9))«

For waves with phase velocities between the ion and electron

thermal velocities the linear dispersion relation takes the form

.- 118 -

f" •'. *" 1

where C-s ~ j /rn«v is the sound velocity, U is the electron drift

velocity, and B is the angle between U and K . In the long wave-

length limit, the phase velocity of the oscillation is equal to the

ion sound velooity CO =1^1^- S » an<^ *^e short wavelength limit

the frequency is equal to the ion plasma frequency (see Pig. 1 ).

IssFig. 1

We need

a > c (17)

for the existence of the instabili ty in respect of the long wavelengthperturbation. For the short waves the cr i t ical current is less for smallion temperature "T ^ <<hT^ .

Since the ion sound instability is the resonance type weakinstabili ty, we may use the quasi-linear theory for the description ofthe ourrent relaxation in the collisionless plasma. For this aim weconsider the particle distribution in a plasma with a current.

- 119 -

\

\

Pig. ;.2, depicts -the equipotentials (characteristics) of the

electron and ion velocity distribution functions. The two speoies are

displaced "by the electron drift velocity U. .

Let us take a narrow wave packet propagating parallel to VX as

shown in Fig. 2. The interval /between the resonant region and the

origin is of the same order as the ion sound velocity, "because the ion

sound velocity is calculated in the ion rest frame. We can expect a

plateau to arise in the resonant region with electron equipotentials

(Pig. 2 ) . If we look for waves with k not parallel to the current

velocity \it , then for U, >> C we can expect these waves also

to he unstable. So we will have'"a resonant region in the form of a

cone as shown in Fig. 2 by dashed lines, inside whioh all waves

are unstable.

The angle of the vertex of this cone of course depends on the

ratio of the ordered velocity to the oritioal velocity. If the

ordered velocity is much larger than ^ $ , almost all angles of

propagation will be unstable; only waves within a v^xy small angle

with respect to \L will be stable, and this angle-^U>will approximately

satisfy .

- 120 -

So only electrons moving within a small angle — Vp of \f are not in

resonance with thp wave spectrum. Some small part of the electrons are

always out of resonance with the wave spectrum because the projections

of their velocities in the direction of the current are much larger than

If we now have a magnetic field in the \f, direction, then,as was

mentioned in Part II, it will play the role of a mixer} all electrons

will rotate around the 1? . axis, and in this case the small cone of

stable electrons has no important effect. So we can consider this

problem as if all electrons are resonant with waves.

But if we have no magnetic field of this sort (i.e., if we

have H !\ \T ) ( then the electrons in the stable cone will not be

in resonance. The only possibility is that they can resonate with

waves (ion sound or other) arising through wave-wave interactions.

If we now look for the simplest experiment, we have a-magnetic

field in the u -direction and apply an electrio field parallel to H ,

producing a current velocity which satisfies (l7)» We will find a

quasi-linear interaction between the ion sound waves and the electrons

outside the stable cone* In other words, all electrons outside the

stable cone take energy from the electric field, aocelerate and then

give up energy to the ion sound waves due to the quasi-linear inter-

aotion. So we will have some kind of balance between the electric

field acceleration and the retarding force due to radiation of ion

sound phonons.

A fraction of the electrons will be accelerated just as in the

usual runaway processes when this happens. But if (P<^I (i.e.,

the electrio field is much above the critical value.) only a small

number of electrons will run away. The simplest experimental evidence

of such phenomena can be expected to arise as follows: first, if

the number of electrons in the stable cone is small, then for such

high electrio fields we will find thai the electrical conductivity

is muoh lower than classical values. Secondly, runaway electrons will

be observed. Both of these effects have been observed already in

discharges with high electric fields.

In the opposite case, if we have a magnetic field perpendicular

to f , all the partioles rotate and we will not have runaways. JIow

- 121 -

we expect to observe only the first effect above, i.e., an anomalously low

6" , This also has "been confirmed by experiment. In this case, the

anomalous conductivity should be even smaller, because when E waB

parallel to H j some ourxent was carried by the runaway electrons.•• ~ * * • . • • • • , , • "

Every eleotron is resonant at some phase of its rotation about the

magnetio field. This kind of geometry has been produced in the fast

v -pinoh experiments at Novosibirsk.

&

\\

\

Fig. 3

Electrical conductivities are found up to five orders of magnitude

lower than the classical conductivity.

Let us look again at the first case, where W » ~ 0 . For

strong electric fields we can consider almost all electrons resonant,

and we find runaway phenomena which aj?e closely analogous to the run-

aways for the case of a Lorentz gas in which electron-ion interactions

are retained. For a Lorentz gas, the oollision integral depends dimen-

sionally on velocity as the inverse fifth power for large velocities.

We know that if a Lorentz gas is subjected to an electron field, the

acceleration due to the field beooraes greater than the collisional

retarding force for sufficiently high velocities. This problem was

solved by Kruskal and Beaaastein. (The Lorentz gas is an exactly soluble

model for the runaway problem in & collisional plasma.)

We now recall that the quasi-linear oollision term whioh was

- 122 -

written down in Part II has the form

" * / /te &>and has the same power dependence on velocity as the Lorentz collisional

; term. Thus the rather unrealistic Lorentz gas model of Kruskal and

Bernstein can now be given the physical interpretation of collisions

between electrons and electrostatic waves. (This analogy was suggested

by HUDAKOV S} t wll° applied the results of Kruskal and Bernstein to

1 this problem and found that for sufficiently large electric fields,

even neglecting the cone of stable electrons, runaways result; in

' other words, over a sufficiently long time, the ion sound instability

cannot prevent the runaway process from occurring.) "We may inquire what

happens after these electrons run away. There exist other instabilities,

stronger than this onej for example, two-stream instabilities with a

critioal velocity on the order of the electron thermal velocity may

play a role.

Thus this is the situation in two- or three-dimensional quasi-

• i linear theory. To calculate any kind of concrete expressions (for

example, electrical conductivity) we can use the quasi-linear

collision term, following the usual procedure starting with thei

•< transport equation. To do this, we need to know the quasi-linear

diffusion coefficient which is proportional to the square of the

electron field. Sometimes this can be done within the framework of

pure quasi-linear theory, as in the example of a spherically symmetrio

problem considered in Part I I ; but for the ion wave instability, i t

"\ is necessary to consider higher order processes like wave-wave

scattering, etc. So we will now turn to the consideration of

mode coupling for this problem. In Part I , i t was shown that the1 dispersion relation of the type drawn in Fig. .1 ( i . e . , concave downward)

cannot satisfy the resonance Condition for the decay-type interaction.

Consequently, the main contribution to the mode ooupling comes from

the non-linear wave-particle interaction, which is described by the

last two terms in Eq. (9)» The electron contribution to these terms

^ is negligible, because only a few electrons can satisfy the resonanoe

condition ( i . e . , oVl ~ ft o ( O r CO1) /lj< _ K% V\T <<L °^ ' s i n o e t h e

wave phase velocity is much larger than the ion thermal velocity, the

waves arc only scattered by the ions, and the number of waves will be

conserved.

- 123 -

If we set U>*- CO* - COz> ~k-lkL and

then to lowest order in the small parameter we find

(18)

However, if we substitute these expressions into the wave kineticequation we see that the non«-linear terms cancel. Consequently, wemust take into account terns of higher order in the small parameter

" Hkl . For CWwe find

Jrn

(19)

To perform the integration we separate ft'?£" into two parts:

i l l K"'(20)

The value of the f i rs t part is obvious from the argument of the deltafunction and the second part is independent of the delta funotion.Consequently, we find

-124-

[1 USL CO

(21)

ft)

In an analogous way we find

_ = e

"'OJ'pMM. I -

(22)

By using EqS.(2l) and (22) we can write the kinetio equation in

the form £~9,10_J

(23)

-125-

As expected, the mode coupling term oonserves the number of waves

Consequently, the mode coupling- cannot stop ths increase in the numberof waves produced by the linear instability, and we need to take intoaccount some additional process if we want to find a stationary spectrum.The energy flows from the high-frequency short-wavelength oscillationsto the low-frequency long-wavelength osoillationsj so we assume, asKADOMTSKV did /~10_y, that ion-ion collisions produce a turbulence sinkin the long-wavelength region. So we can cut off the spectrum in thisxegida and construct a stationary solution in the rest of phase spaceby balanoing the linear growth rate with the noniinear flow of energyto the long wavelength.

Aa an example of this kind of solution, we f irs t assume that the

turbulence is limited to two line's in K -space and consider the long-

wavelength turbulence

where ( K , S ,VP ) are spherical oo-ordinates in ^ -space With the

polar axis along the direction of the current.

When U. is only slightly larger than C s ,»-*ke linear instabilityproduces waves propagating at small angles with respect to the current( i . e . , D c<- t ) . In this case, we can reduoe Eq.(23) to the form

(24)

where we put

" 1= - k" 1 1

-126-

and replaced the summation over K by

2The general solution of Eq.(24) is

where we have demanded that I ( (0 - 0 at some very long wavelength J

This solution i s , of course, not unique. Moreover, the solution

is unstable, because any wave propagating along the direotion of the

current has a larger linear growth rate and smaller mode ooupling term

than the wave propagating at the angle Q o . From this argument

we conclude that the spectrum tends to collapse to a line parallel to

the current.

I .A, JLKHIE2SR JC^JT ^ i a s s-1 0 w n that i t is possible to construot

a self-similar osoillatory solution in the long-wavelength region ( i . e . ,

K X-Q «. J_ ) . Here we only follow his arguments qualitatively. For

an axially symmetric solution ( i . e . , l^*)2"- X(.V. ^}4;))» we can write

Eq.( 23) in the form

kCs Hr ' SL

One can see that this equation is invariant under the transformation

-127-

where cL ie an a rb i t rary parameter. Therefore, we oan reduce thenumber of variables by introducing ( X-U ) and X. F /5~pjT, / z X*>£ i

whioh are invariant under the above transformations. In the l inearstage, the function I K depends only on 3£ l ike •-^ *f "{ X " -Jwhere . y 4 - N X • Consequently,, J.K •-.•should depend only.on X

in the l a t e r s tages . With regard to the angular depend©npe of the :turbulent spectrum, Akhiezer found that the energy osc i l la tes betweenthe Cherenkov cone ( i . e . , I {&) - O L^?^ ^ ~" & /VJL)/

and the current l ine ( i . e . , X. (-9 ^ """ ^ t- V— &>$> ^ ) ) , with a.period of osc i l la t ion proportional to the energy in the spectrum.Consequently, most of the time the ener-gy wil l be distr ibuted through-out the Cherenkov oone (see P ig .4 ) .

F i g . 4 '* • • •.

In the short-wavelength limit ( i . e . , K A^ » 1. ) again we can

reduoe 3q..( 23) for the wave amplitude to the iri*egro-diffexential equa-

tion of typo ( 26) ; <

- ^ \ I (k0

(28)

, ^ ) fG* e Gse1 tSinB Ue'G& If -f)] j i -

-128-

There is no 'rigorous'solution of type (2?) for this equation. '

However, we can find, as we did for the wave-wave interaction in 'Fart I,

some power-type solutio^ at least f or .-the k-dependexicSe. The' natural sink

of wave in the short-wavelength region is due to the linear ion

Landau damping. Prom Eq.(28) we find that the spectral energy of waves

very rapidly deoreases to the short scales

The power of-Wssolution is completely different from that found:in Part I I .

Therefore there is no universal power solution for the plasma turbulence

like the Kolmogarof spectrum in hydrodynamio turbulence.

Sow with the help of the expression founds for the spectral energy of

waves an the limits of both large (25) and short (28) scales, we estimate

the turbulent resistivity. Multiplying the quasi-linear equation for

the electrons

-it

integrating the result over the velooity space, w«t obtain

the electron momentum loss due to the radiation of the ion sound waves

Introducing the effective electron-ion collision frequency as

with the help of Bqs.(l6) we obtain from this expression

(30

-129-

- .i:-,ip-m-:B "-••ww

On the. r.h.s. of Eq.(3O) is the wave momentum density radiated h^-

electrons per unit time (i.e., — ' t^J /- By Eq..(3O) we supposed

• that all the waves radiated by electrons are absorbed by ions. The

main contribution to the integral on the r.h.s. of Eq.(30) corresponds

to the wavelength of order Debye length K * r> •'"•* £ ,

For the case of large ourrent velocity M >> Cj, we find fro«.Sqs.(25)

and (29) •.- ;

u • ~ t i •• • •

Therefore the effective collision frequenoy ia larger than the growthrate. , '.

As in the usual Joule heating,the turbulent heating changes mostly the

electron temperature fl2] if'-\A,ie greater than the ion thermal

velocity. The heating rate of the ions is always nuoh Smaller than the

heating rate of the electrons. Really, the rate of the energy changes

due to the scattering of the electrons by ion sound waves is equal to (see

31.(30))

where £ - O £ ^ \ €1*,^) "> ' ^ ^ ' i s the wave

Then, even if we suppose that the total energy of the waires is

absorbed by ions, the ratio of the heating rate for both species is limited

by the inequality < ~12_7" : • ' ;. ,

^ ^ (32)

fc-U

This ratio is small if the current velocity is not so close-to the

oritioal one.

-130-

3. Non-linear theory: of the 'drift instability ~ ' ' ' ;: ' ' * ' " '

In' order to ' complete consideration of: the 'anomalous transport '"'" '

processes in non-uniform plasma, we will try to" descriW in

the non-linear stage of "the low-frequency "universal drift

The' nbn lineal E| {9} for the :< spectral energy of ;t^e" ele^^statid^waves'1 is°

written in terms of the expansion coefficient of the particle distribution

in powers of the wave amplitude,J These coefficients are defined by the

iteration formula V- ., '•""'' - ; '

We can d e s c r i b e th© mode c o u p l i n g of i ^ r i f t i w a T e s i i n a; ii:on»-xmif orm

p l a s i a a . w i t h the , h e l p . o f , t h i s -equat ion $o l o n g .aa .: • r • ;- - •;. o [yjs,rz ••;•..;;:;',::.,

(34)^ *

This inequality just says that" the difference between "the eigenfre^uencies

in a non-uniform plasma is much less than the growth rate. In other

words, the difference between the energy levels is much smaller than the

energy of interaction. •-.'.• • ' . ' .• ,-, .- ''-' • .. .'> :

The dispersion relation in the linear approximation was found in Part II ,

How let us proceed to the next ste^p: the derivation of the rioh-lin6^r* ceiiua-

tion for the wave energy. Substituting in (33) the-2in©J&r approximation -

( 1 1 - 9 7 ), we get2.

-131-

In (3) the velocity derivative gives several terms? however, we recall

that in the first order calculation, the contribution from differentia-

tion of the argument of the Marwellian cancelled up to a small term, and

the same thing happens here; also, if we assume K x ^> /ift, » "tb-6

term arising from differentiation of the density in ^ is negligible.

So the main contribution comes from the rapid velocity dependence of the

exponential exp £X*^> j\ /&«»"] in "blie seoond term of S £ j .Carrying out the time integration and symmetrizing with respect to Ky

and K__ , we have

(36)

Using this expression in the same approximations, we obtain the third

order correction to the distribution function.

-132- (37)

In Eq. (9) for the wave amplitude, the main non-linear contribution comes

from the ion terms, until K^f^ *• N \T/yr\ • We also use the fact

that the phase velocity along the lines is much larger than the thermal

velocity for most unstable waves, so the dielectric constant for the beats

can be written in a form*t

% (38)

If we keep only the ion contribution to

K,

, Eq.(9) becomes

rK"

* The approximation (38) breaks down if ^^ ^C < \<p> - B u t f o r s u c h

a long wave the interaction through the beats is very small and we canneglect i t .

-133-

•

On the l . h . s . we take into account a weak dependence on TT , DC and

in W , in addition to tho oscillatory factor already obtained in^

the linear approximation. The way to do this is very similar to what

we did-to obtain the dependence on x in ^ K (see Eq.(8)). But we

may obtain the I .h . s . of Eq.(39) by a more formal method if we expand

the frequency, x-component of wave vector and x-co-ordinate near the

real position of the wave packet in the configuration space ( (S>^ ltx X.).

In aocordanoe with Eq.(8) we find

This we easily-reduoe to the substantial derivative in configuration

space by the help of relations

Three terms on the r . h . s . of Eq.(39) desoribe respectively the linear

growth of instability, node coupling due to wave scattering by ions

and decay type interaction* which can be written in symmetrical form with

the particular choice of normalization (See Part l ) . ,

Let us f i rs t consider the long-wavelength drift turbulence

( i . e . , Kj5 r ;1" ^•<-. i- )• . 1^ this limit we neglect the wave

scattering by ions,, whioh is small, (as xj- r .2<&J, J in CPEtpaxison with

the decay type interaction, .

As we go to the limit of zero ion Laaanox radius, the phase

velocity of the wave becomes very close to the electron drift velooity

CC (40)

Therefore, the waves are propagating almost in phase with each other for

a long time A\. <** t^) " , and interaction becomes stronger.

-134-

This is reflected by the fact that the integrand of the non-linear

decay type term goes to zero as oU)K , But the argument of the

£ -function in (39) vanishes also. Thus

Then the non-linear term is of order

By balancing growth rate and the rate of energy output from the mode

due to-the decay interaction we obtain

For the truncation of the wave kinetic equation (39) in the lowest non-linear

order to oe valid, we need X*. A LGK ^ ^ i j thus (4l) implies that

the E * V drift is always smaller than the initial velocity due to

the pressure gradient,since &-/ku)K ^ ^- i s needed for the deriv-

ation of (39). In a plasma with P> > v w * / m . , as was shown in Part I I ,

the electrostatio waves satisfy

IS (42)^ i

For some instabilities, the perturbation theory leading to| (39) breaks down because ^K > oVwj< . For example, let us consider

bad curvature effects on the long-wavelength drift instability 15} •

expressions (11-98) and (II-1OO) hold in the drift frames of the respeot( ive species. In the co-ordinate frames moving with the respective gravi

' tatio.ial drift velocities, we have instead

n

135-- 1 3

where Vlo is "the principal normal to the curved field lines andis the radius of curvature. Here we n«gleot the ion L&rmor radiusin comparison •with the wavelength.

Row the dispersion relation yields*

If we go to the co-ordinate system moving with velocity

the real part of the frequency vanishes and we have pure growingmodes, i . e . , a strong instability.

This is related to the fact that the non-linear termin (39) iraniahes for o cO -*•• O , Up to now we have asaumed th&tthe electrons (which can move freely along the lines of force) areestablished in a Boltzmann distribution. However, this is not trueof the resonant electrons. As ion non-linear effects become small,i t is necessary to consider electron terms which were not includedin the foregoing treatment. When )(*~ " *^K , (41) impliesthat the eleotron ,E«-V4 drift is as large as the drift arisingfrom the pressure gradient. Moreover, sometimes we can treat thedrift wave as monochromatic in the zero Larmor radius limit, Inthat case the amplitude limitation due to the nott-linear motion ofthe resonance electrons is more restrictive (see Part II ) .

Up to now wo have considered only long-wavelength drift

oscillation. But if we start from the thermal level of fluctuation,

the wave grows more rapidly with the largest growth rate.

* Let us note that in the trap with fly f 0 the wave packets with k x » K.can be growing even in the f ie ld with minimum B-geometry ty x ^~ > 0 "(the plasma stabi l i ty in the general case dtT, a ^ O disouBsect in^ ««tail in [l7] and [iS - l$] respeotiv«lyj. •

The linear growth rate for the drift wave (sec 11-101)

increases as a function of K^Y^- , until K^c ~ \-li- or until

finite plasma pressure effects come into play £~1-§J* The finite

plasma pressure effects add a term due to.the ion diamagnetic drift,

H HI n.

Putting this drift into the linear dispersion relation gives us the

following marginal stability condition;

Consequently, finite plasma pressure stabilizes the drift wave when

(43)

Thus, the maximum linear growth rate will occur when

j u s t b e l o w t h e s m a l l e r o f ( ? Ti a 1 ) " 1/z- . o r { P ^

Prom Eq., (39) we can' see that the rate of the wave scatter-

ing by ions is the rapidly increasing function of Kj_ and that this

process makes the main contribution to Eq,.(39) for short wavelengths.

As usual, the short waves scatter by ions to the long-wavelength region

of K -space and vexy soon limit themselves to the level (41) (of course,

we now estimate- Yv_ , U) *, in the short-wavelength l imit) .

From Eq.(4l) i t also follows that the boundary of the turbu-

lent region moves from small wavelengths to large wavelengths. However,

this movement must stop when Kx "HL ~D9Coroes of order unity, because

the scattering process becomes less effective there and decay processes

-137-

start to return energy to the short wavelengths. In. the time-asymptotic

-limit, the spectral energy of waves roust be equal to zero for ^.r^- >^

If we use the estimate for j IP^I2" given in Eft»(41}, wefind the following value for the diffusion coefficient (see Part I I ) .

2.

As was shown, in the time-asymptotic l imi t , we maximise Ji)± with respect

to the long wavelength Kj_ ]THV ^ 1 •

In th i s expression X x appears as the basic unit in the"random walk". I t i s tempting to choose i t as large as the radius ofthe plasma; th i s would correspond to a.quasi-mode. But Eq. (41) showsthat the • amplitude of such a quasd-niode would be enormous-, and, conse-quentlyr it& s t a b i l i t y i*s questionable. Since i t i s d i f f icul t to checkthe s t a b i l i t y of such a mode, we gjve another argument against i t s ex i s -tence. Since the l inear growth rate i s Independent of' ^ ^ , a l l thenormal modes up to Kx "- Ky would have to grow to a large amplitudewith the quasi-mode, and these modes would suppress the amplitude of thequasi-mode. Therefore, we put ^ ^ ~ '")?•„ ** 1^. and find

(45)

When collisions are not strong enough to maintain the Maxwel

lian form of the electron dis-tribution, quasi-linear relaxation will

reduce the growth rate of the oscillations (see Part II). If * e is the.

collision frequency and I ^ K | is the amplitude- of the turbulence,

then the growth rate will be reduced when

-138-.

Using Eq.(4l) to estimate the level of turbulence shows that the v,

oscillations with K, rv.; ~ 1 will be suppressed when V* < lO... { *' ) L-iP \ .Kv y\ / wtf J

jjut this does not mean that the instability will be completely 4

suppressed, because the rate of energy flow from the short-wavelength

region to the long-wavelength region will also be reduced. Consequently,

tlic level of turbulence in the short-wavelength region will be given by

3q.(4l) with ^i. T C ^ J- a n ^ "*i'ie '^i'fu.sion coefficient by Eqs.(44)

and (45).

The diffusion rate i^ decreased when the relaxation of the

distribution affects the short wavelengths ( i . e . , Kxf^ ^ ( ,"5 &ZJ" ^ )

Then the diffusion coefficient can be estimated with the help of Part II

*y^fwhere the condition O ^ O ^ *J^ is the condition for the applicabil-

ity of Eq.(46).

-139-

4. The general scheme of weak turbulence theory

In order to establish the relationship "between the different parts

of weak turbulence approach, it is useful now to summarize what we did

in the form of a symbolic tree;

Vlasov equation(for each, species of

oharged particles)\

Usingstatisticalapproach

Maxwell equation

Particle kineticequation

Quasi—particle (wave) kineticequation

(f or each: "species-" of plasma wares)

These kinetic equations have the form:

where the Stoss-tenns represent:

in the' 1st approximation

This i s a quasi-linear approximation ( Q L A I I o f

these lectures). Im. U)K M l ) } : means the imaginary part of U)< as a

functional of. (froin ' t l a e l i n e a r dispersion relation).

This approximation takes into account only cross-interaction

particles + waves, aocoxding to the resonance condition

-140-

o) - o

in the 2nd approximation ( a )

Wave, + wave; (3-waves) interaction (part i ) ,

according to the "decay11 condi !.ons

cO, + '2. ~

The collisional term in the

particle ' kinetic equation

describes the adiabatic inter-

action of particles with waves,

since the particles participate

in the oscillations.

in order to

character of

Stoss (part

A, A ,

- IV • hiSymbolic

stress the

notation

quadratic

3-wave: interaction

I)

in the 2nd approximation (b)

Tlave - particle - wave interaction (part IIl)»

The contribution of this process

to Stoss-term is too cumbersome

and we did not' consider i t in the

lectures (see /~4» 10_7)

A A A

N hi in order to show that the particles

also participate, see the details in

Part III.

in the 3rd approximation

4-wave interaction

K, +•

-141-

1 . . —Again only adiabatic interactionof particles and waves.

A/ A/ A/Since this Stoss-terra is

cubic, i t can be important

only in the undecayable

cases. ¥e did not consider

i t in the lectures.

+ etc., which -we also did not consider.

In the case of strong turbulence such approximate procedure

breaks down, since higher numbers of wave, interactions have the same

order of magnitude. In other words, in the strong coupling limit there

is no closure. Here we do not discuss the various attempts to make

non-linear estimations in the strong turbulence cases /f*10, 20 , 21 _/ .

In the case of weak turbulence, the statement of concrete problems

in terms of, generally speaking, quite complicated non-linear integro-

different'ial kinetic equations for particles and waves does not lead,of course,

to easy interpratation. , But, at least, there exist? 'a.finite

number of equations. < '•In the different limiting cases one could

solve i t using various realistic and idealized approximations, as we

were trying toi demonstrate in these lectures..

Feak plasma turbulence theory permits fit simple generaliza-

tion in order to take into account the spontaneous (thermal) fluctuations

in plasma. One can expect these fluctuations to be the additional source

of wave generation in.the kinetic equation for the waves and,.corre-

sponding to it, the recoil effect (friction term) in the-particle kinetic

equations £~IO, 22 J\

Let us treat the fluctuating . part of the distribution function

^ r ^. \J*,r \) > • which satisfies the ideal gas correlation law

in a plasma without magnetic field

(46)

-142-

lie will take 0 L into account for simplicity in tlie case of the

longitudinal oscillations

Since o L represents an additional source in the- equation for

the electric field.

But we are interested in the. quadratic terms-1 ^

Multiplying Eq. (47) "by obtain

or

Using the sane symbolic approximation as we did in part III

according to the correlation law(46)', we easily derive*

(48)

(50)

As in most of our lectures^we consider here the discrete set ofwaves in the finite system of volume equal to unity.

-143-

XMk

This is an additional term, representing the spontaneous generation

of the plasma waves, which we should now add to the r .h . s . of

the wave kinetic equation.

The recoil effect in the quasi-linear equation for the distribu-tion function is derived in the following way:

where oVD is the contribution to the electric field from, the sponta-

neous fluctuations. Averaging c>*Px> (- i using Eq.s. (46)

)'ve derive this additional term in the kinetic equation;

(52)

These additional terms (50)(in the wave kinetic equation) and (52)

(in the quasi-linear equation) are not significant in the case of unstable

plasma. In stable plasma the competition between the term (50)

and "Landau damping" determines the equilibrium level of the thermal

field fluctuation. This equilibrium background of plasma waves,after

the substitution into the quasi-linear Stoss, will give

"IT -Combining the terms (52) and (53) we get the Lenard-Balesou equation.

-144-

REFERENCES

1. / DRUMMOND, W.E., and PINES, D., Nuol. Fusion Suppl.. .part III,

1049 (1962).

2. KADOMTSEV, fl.B., and PETVIASHVILI, V . I . , Soviet Phys. - JETP

16. , 1578 (1963)! (Zh.Eksperim. i Teor. F i z . 4 } » 2234 (1962)).

3 . KARPMAN, V . I . , Soviet Phys. - Doklady 8, , 919 (1964)

(Dokl. Akad. Nauk SSSR 1^2 , p37 (1963)) .

4 . OALEEV, A.A., KARPMAN, V . I . , and SAGDEEV, R.Z., Soviet Phys.

- Doklady 9_ , 681 (1965)

(Dokl.Akad. Nauk SSSR 1^]_ , 1088 (1964)) .

5. SILUST, V.P. , Zh. P r i k l . Mech. i Teohn. F i z . 1_ , (1964) 3 1 .

6. AL'TSHUL', A.M., and KARPMAN, V . I . , Soviet Phya.-JETP 20 ,

1043 (1965); (Zh. Eksperira. i Teor .Fiz . 4J. , 15.52 (1964)) .

7« ZABUSKY, K.J . , p r i v a t e communication.'

8 . RUDAKOV, L . I . , KORABLEV, A.B., Zh. Eksperim. i Teor .F iz . jQ t

220 (1966).

9. PETVIASHVILI, V . I . , Soviet Phys.-Doklady 8_ , 12l8 (1964)

(Dokl.Akad. Nauk SSSR 1 ^ , 1295 (1963).

10. KADOMTSEV, B.B., "Plasma Turbulence", Aoad. Press , London,

1965.

11.' AKHIEZER, I.A., Soviet Phys.-JETP 20 , 617 (1965); 2£ , 1519

(1965); (Zh. Experim. i Teor. Fiz. 4J. , 952 (1964)$ 4J. , 2269

' (1964).-

12. SAGDSEV, R.Z., Congress on Applied Me thematica, New York, April

,1965 ( to be pub l i shed) .

13. GALEEV, A.A., RU3AK0V, L . I . , Soviet Phys.^JETP l8_ , 444 (1964)

(Zh. Eksperim. i Teor. F i z . 4Ji , 647 (1963)) .

14. KADOMTSEV, B.B., Soviet Phys.-JETP 18 , 847 (1964) (Zh.Eksperira,

i Teor. F i z . 45 , 1230 (1963)) .

- 1 4 5 -

15. GALEEV, A.A., Soviet Phys. - JETP 1£ , 1292 (1963)

(Zh. Bksperim. i Teor. Piz. 44 , 1920 (1963)).

16. MIKHAILOVSKAYA, L.V., and 1CKHAIL0VSKII, A.B., Soviet Phys.

- JETP, JL8_ , 1077 (1964)5 (2h. Eksperim. i Teor. Fiz. _4j£ ,

1566 (1963)).

17. KOGAN, E. Ya., MOISEEV, S.S., and OBAEVSKII, V.K., "Plasma

stability in complex magnetic field configurations", Prikl.

Mekh. i Tekhn. Fiz. (in press).

18. COPPI, B., ICTP, Trieste, preprint 1C/66/63, (I96fe).

19. GREENE J., JOHNSON, J., private communication (1966) to be

. ' published.

20. GALEEV, A.Av MOISEEV, S.S., and SAGDSEV, B.2., Plasma Physics,

J. Nuol. Energy, Part C, 6, , 645 £1964).

21. GALEEV, A.A-, ICTP, Trieste, preprint lC/66/82, (1966).

(Submitted to Phys. Fluids); see also Prikl. Mekh. i Tekhn.

fiz-. 1 , 39 (1965)22 VEJ)MOV, A.A., VELIKHOV, E.P., SAGDEEV, E.Z., Usp. Fiz. Nauk

, 701 (1961).

-146-

PAST IV. ffOff-LIHEAB PLASMA DYNAMICS Iff COLLI3I0ILESS SHOCKS*

1. Ifon—linear dispersive wave t r a ins and solitons. -

One of the most in teres t ing f ie lds wh&rB one oan apply the

whole arsenal of the plasma non-linoar theory methods i s the theory

of col l is ionless shocks.

Actually, what we laiov Jibout the wide variety of these problems

are some l imi t ing oases where the plasma behaviour can be described

almost in terms of moments approximation (as we did in Fart I ) . ' Such

approximations can be applied to col l i s ionless plasmas only in some

degenerate cases, for example, when we have guiding centre motion

across a magnetic f i e l d . Another re la t ive ly well developed case i s

the low & ( & = ~TT7— ^ 1 ) l imi t , where the moments approxi-

mation appears simply as the equations of motion for each specie,

since the thermal spread i s almost negl ig ib le .

The cases whBn the f lu id- l ike description breaks down are not

yet understood sat isfactor i ly? therefore we have not too much to say

about i t .

The logical way to find the features of the shock in co l l i s ion-

less plasma i s to follow the time evolution of an i n i t i a l Rieniannian-

type disturbance. In a f lu id- l ike approximation, the development of

such profi les can also be described in terms of Eieittann's solution

(unless dispersion does not play a role) where any point of profile

oc i s moving with velocity

• — - Cbr) V i r (i)a i • . . . . . . . .

where C (xrj is a local value of the sound speed and Vis the local

velocity amplitude. According to solution (l), the non-linear steepen-

ing will create sufficiently strong gradients in the front of such a

* This part of the lecture notes represents only a short review. Forthe omitted details we will refer to the reviews £l» 2_/ (See also

-147-

wave. Finally, the dispersion phenomena become very important.

They are represented in a fluid-like plasma model in termsof thehighest derivatives. Unfortunately nobody has yet found thegeneralization of a Riemannian-type solution for the case with dis-persion. But for a simple analysis one oan expect the formula

n -T(2 )

to "be qualitatively right. Very often the linear dispersion law

has a behaviour similar to that of the two different cases shown in

Pig. 1.

K

Fig. 1.

So one can write it down as

"= C K * ( (3)

where dL > O for the case (a) and

expression (2) means

< O . for the case (b)

Now using (2) and (4) one can easily see that the competition betweennon-linear and dispersive effects will lead qualitatively to the timeevolution of non-linear vrave profile, shown in Fig. 2.

>r -t=o @

\r

Fig. 2.

-148-

In other words\:f the initially monotbnic profile becomes intensively

modulated, reminding us finally of a non-linoar wave "train.Obvious-

ly, in the case (a) this wave train is going far ahead and in-the case

© it is going 'backwards.

The next step in order to see iHe asymptotic "time "behaviour"

( t -> oo ) of such wave trains is to look for the steady-state

solutions Q- =. Q (or the solutions depending on (x-t tcO.

One can expect these solutions to "be the asymptotic form of any

init ial one-dimensional profile, as happens in conventional fluid-

dynamics, where ( X - LfC- )-solutions represent the shock structure.

Since in our case dispersion plays a determinant role, these steady-

state non-linear solutions are reversible (in the case of no dissi-

pation). They appear to be non-linear periodic waves. In order to

find their form in a moments approximation, one must solve the system

of non-linear ordinary differential equations. There is no difficulty

in doing i t , since the coefficients do not depend on the argument (x - i

The characteristic space scale for these steady-state waves is the

order of characteristic dispersion length ( -v, ifdL ^ *^ie case of

dispersion relation of the type (3J). As a degenerate limiting

case of these non-linear solutions,the solitary waves (solitons)

appear. For sufficiently high amplitudes such steady^state waves

do not exist .This happens when the dispersion, at last,cannot compete

with non-linear steepening. The same thinf* happens to shallow

water solitary waves, as is well known. This breaking of non-linear

steady state waves means the breakdown of the moments.approximation: the

plasma motion becom«s multistreaming ahd very likely unstable /~2»4-_/*

But, as we know, some types- of instabilities/iii-hon-linear waves can

arise even for smaller amplitudes (much below breaking conditions).

Before considering the instability problems, vre should

discuss briefly the influence o£ specific Qollisionless.damping mech-

anism on theae steady-state waves. • The "resonant" particles (with

velocities close to the wave velocity) are', contributing to the damping

of this kind of wave. They are also responsible for the. entropy

production mechani»m: similar to what we discussed in Part ,11,when consider

ing - the non-linear Landau damping, Ma^ietio fie^d;plays, a. special

role in this case £~$Jf* Also the thermal motion of the partioles

-149-

with velocity muoh less than "resonant" gives rise to an additional

randomization, since magnetic moments of ions in such waves are no

longer the adiabatic invariants.

2. The onset of instability and .turbulence' -

However, the most effective mechanisms of "collisionless"

dissipation are the various types of plasma instability which lead

to the transformation of the energy of regular non-linear motion

into the energy of chaotic oscillations (and finally to random-

ization). ¥e start "by discussion of the decay instability (Part i )

of the almost periodic wave trains in laminar shock waves. Let us•

take the wave train propagating exaotly across the magnetic field

in a low G> case. The dispersion law cO(.t ^ for this wave is

shown in Pig. 1 (case (b) ) and belongs to the non-decayable type

(see Part i ) . Therefore this wave can be unstable only in respect

of the perturbation which-includes some other type of wave.

Analysis of stability £"sj" shows that the non-linear oscillations

with frequency

are unstable with respect to decay to a fast magnetohydrodynamic

wave and fast Tahistler-type oscillation with the following frequencies;

The growth rate of the instability is of order

^ U)o fjj^l « U)o (7)

As a consequence of decay instability, non-linear regular oscillations

are damped much earlier than according to the theory of laminar shocks,

since their energy is transferred to the energy of a whole spectrum of

noises. The length of damping A obtained in suoh a manner may be

-150-

identified with the thickness of the front of a shock wave. Promthe dimensional point of view we may expect that Z^ must be of

an order of several lengths of oscillation (the rough estimate for

A was given in reference £"lj* But if the amplitude of the

shock is large enough a number of faster instabilities come into

play and are much more important than those of decay-type.

The most obvious type of instability for non-linear waves

in a magnetic field is "beam" instability, when the mean regular velo-

city of the electrons relative to the ions exceeds the mean thermal

velocity ( i£ y* T~^-/tY\ )• Using some results of soliton

theory to find the current velocity, we observe that this condition

begins to be fulfilled for waves with a Mach number exceeding

M ~

Physically, this instability signifies that the electrons moving

relative to the ions are losing momentum and energy not only because

of ordinary collisions, but also because of some new effective friction

of a collective nature - coherent "radiation" of plasma oscillation as

a consequence of instability. Therefore,upon fulfillment of the condi-

tion of instability "\T >• M ~^e/™ a n anomalous electric resistance

should appear, leading to an anomalous dissipation. We expect, of

course, that the main part of the ordered energy of the eleotrons due

to this collective friction between the electrons and ions goes into

the disordered energy of the electrons (as in usual joule heating by

high density electron current). Therefore the effective "temperature1*

of the electrons increases in comparison with that of the ions .

"T, »~T (8)

Starting from this moment tie "beam" instability becomes less important

because the ion sound wave instability comes into play. Let us show

now that the electron ourrent in the turbulent shock front becomes very

low since the anomalous electric resistance in the shock front is very

large. In order to describe rigorously the shock front structure and

compute this current we must at least be able to solve the equations for

-151-

t h e m o m e n t s o f q u a s i - l i n e a r d i s t r i b u t i o n f u n c t i o n l - i f V 4")j v K "*- > ~ s *

(taking into account tho effective eleotrio resistivity) together with

the Maxwell equations for the electric and magnetic fields.

For the steady-state case these equations in a moments approxi

mation will have a form

(I)

In such an approach* the only difference from conventional magnetogas-

dynamics lies in a very complicated form of 6"~ which is now some

functional of the distribution function,and so on (See Part III, where

we discussed the electric conductivity in the presence of the ion sound

instability).

The system (i) can be easily reduced to the one equation for H(x)

AT,ff

^ $ JBut since o is now very complicated, there is no reason to try to

solve Eq. ( a. ) exactly. ¥e will give an estimate of the shock thick-

ness using the effective collision frequency between ions and electrons

due to the radiation of ion sound waves by eleotrons, estimated in Part III

* It is possible to justify this approach, at least, for low Mach numbers•1 jl|_ _ < 1 » using weak turbulence theoiy.

; -152-i

'J'l IIB—ILII I

Ji

Substituting Eq.(io) into the well-known expression for the thickness

of the shock wave in the magnetohydrodynamios of a plasma for a case

of a weak wave being propagated across a magnetio field £~ 8

g- vxe2- • (11)

we obtain the thickness of the turbulent shock front

\LM-1)¥e estimate temperature ratio in this expression assuming that the ion

sound wave energy i? absorbed by ions. Using the results of Part III

we have

JL ~ JL (13)

Finally, we express the current velooity CL through the

thickness of the shock front using a Maxwell equation for the magnetic

field

Prom Bqs.(12) - (14,) w e immediately obtain

] 1 A

It can also be found from Bq.

-153-

¥e assume that the'turbulent heating in the shock front is sufficiently

effective1 t-Iiat

Then from ( 15 ) we find . the final result,

This estimate is close to the experimental data £~S, 10_/ in order

of magnitude in a fairly wide range of density.

Since the model used is quite similar to what we have in

conventional gas dynamios, we expect the existence of so-called iso-

magnetic jump when the Mach number becomes greater than critical.

This critical Mach number is quite independent of conductivity. It can

be found from the left-hand side of Eq.. (9) if we take into account that

the condition of the singularity is that the expression under the square

root be zero at H — H ^ (magnetic field behind shock).

=; r\

- u.H. -

For the case of V• =; r\ <, this critical Hach number coming from

Eqs. (17) and (iB ) is close to jU[ c « 2,8 . Z"8_7*

In our case an isomagnetic jump means the breaking up of the

momenta approximation for the ions. It probably leads to the multi-

streaming picture of ion motion and to spreading the effective shock

front.

Let us discuss now the case of the shook wave propagating

net exactly across a magnetic field (dispersion law type 1- (£) ).

-154-

[ M 'iii il' • * • . - *

If the angle 6 between the plane of the shook front and the magnetic

field lines is much larger than

V

the turbulent spectrum can be completely different from the case of the

perpendicular propagation. The oscillatory structure of the shock (non-

linear wave train) is still unstable(in respect of the decay to two

Alfven waves,with growth rate

But development of the beam instability can now be impossible if the

angle of propagation 9 is large enough*

Therefore we may expect that in the initially isothermal plasma the ion

sound instability cannot also arise. But in a strong magnetic field

even small eleotron current U, <-«- y ^ leads to the development of

some other instability, discovered by 0 .BUDEMiUJ /T*11-/ • T]ae

oscillations have the frequency and wavelength in the intervals;

(Generally speaking, higher oyclotron harmonics can also exist).

The dispersion relation for this kind of wave was found in Part II,

where we disoussed the-"loss-cone" instability.

(22)

is is the case 0

-155-

Tfe expand the integrals in the case of phase velocities within the inter-vals:

CO (23)

ihe.

and take a plasma- with the particle distribution in a form

h - e(24)

Then from Eq.(22) we obtain

= 0

Further, for simplicity, we consider the case of a -dense plasma:

(25)

(26)

In this case the growth rate of the instability is given by the equation

and has a maximum

< L

Jf i u)(28)

-156-

4H.W !«.»,»-*,.»,-,.J»» T

It is difficult to desoribe rigorously the non-linear stage of the insta-

bility, but we can expect the instability to be Baturated when the effect

of turbulent motion of the electrons* in the random electric field stabi-

lizes the instability £~ 12 J

\The turbulent Ohm's law can easily be found from the quasi-linear equation

for eleotron distribution and has the form

•cllTx (30)

It follows from Sqs. (29) and (30) that

U- (31)

Using this estimation of the effective frequency of collisions one can

find the order of magnitude of the shock thickness in the usual manner

A completely different olass of instabilities is becoming

veiy important in the collisionless shock waves in high ft -plasmas.

These are mostly the pitch-angle instabilities. Por detailed considera

tion we will refer to the preprints /~13,

Turbulent motion of ions oan be negleoted because of |j£ .= > ' n__X_-0

h

Since the whole problem of plasma oollisionless shook

•waves is divided into a wide variety of separate pieces, it is useful

to represent the entire picture of the different existing limiting

cases in the form of a simplified schematic table.(See p.159)

ACKNCWLSDGI.IEM'S

The authors are indebted to the Plasma Physics Group

at the ICTP, and particularly to Professor M JTJtosenbluth, for much

discussion and encouragement. Dr. D .Book and Dr. T.O'Neil made an

important contribution in the preparation of this manuscript. We are

grateful to Miss. LI .Lewis for heroic.attempts to improve the English

of the final version of the lecture notes* It is a pleasure to ack-

nowledge the hospitality of Professor Abdus Salara and the IAEA at the

International Centre for Theoretical Physios, Trieste.

-158-

V *r> r*-

1

-H1 1 5

\f

•1- fa:

3i

a J

o

t—A

a)

6

41*

ia) - ? •

0> o

H T

-159-

REFERENCES

1. VEDEHOV, A.A., VELIKHOV,- E .P ., and SAGDEEV, R.Z., Nuclear Fusion

1_ , 82 (1961).

2. SAGDEEV, R.Z., "Problems in Plasma Theory" edited by Leontovich,

Atomizdat, Hosoow 1964, Vol. IV, 20-80.

3. POLOVIN, R.V., Nuclear Fusion ± , 10 (1964).

4. SAGDEEV, R.Z., Zh. Tekhn. Fiz., 31 , 1955 (l96l) (transl. Soviet

Phys. - Tech.Phys. (1962)).

5. SAGDBEV, R.Z., Shook Waves in a Harified Plasma, Uppsala Conference

on the Ionized Gases, 1959.

6. KARPMAJT, V.I., and SAGDSEV, R.Z., Zh. Tekhn. Fiz. ,33 (1963)

(transl. Soviet Phys. - Tech.Phys. (1964)).

7. GALEEV, 7L.A., and KAEPMAU, V.I., Zh. Eksperim. i Teor. Fiz. _4J. ,

592 (1963) (transl. Soviet Phys. -JETP, r7 , 403 (1963)).

3. MARSHALL, ¥., Proc. Roy. Soc. (London) A233 , 367 (1956).

9. KURTHULLJIEV, RJC., MB3EBRIKHIN, YuJS., PILSKY, V.I, and SAGDEEV,H.Z.,

"Plasma Heating by Collisionless Shock Waves" , Plasma Physics and

Controlled Nuclear Fusion Research, Vol. II,IAEA, Vienna, 367-387,

(1966).

10. HIETZ, E., private communication,

11. EUHEMAN, 0., Preprint, Stanford University (l96l).

12. GALEEV, A.A., Preprint ICTP IC/66/82, Trieste, 1966 (submitted to

Phys. Fluids).

13. KEMEL, C.F., and SAGDBEV, R.Z., ICTP, Trieste, preprint, Ic/66/68,

(1966).

14. KENEBL, C.F., and SAGDEEV, R.Z., ICTP, Trieste, preprint, IC/66/88,

(1966).

15. ADLAI-1, J.H., and ALLEIT, J JE., "Hydromagnetio Disturbances of Large

Amplitude in a Plasma", Proo. 2nd UJJ. Int. Conf. on Peaceful Uses

of Atomic Energy, 221-224 (1958).

-160-

16. DAVIS, L., LTJST, R., and SCHLUTER, A., ZJlaturf orsoh. X3a, 916,

1958.

17. SAGDEEV, R.Z,, "Non-Linear Motion of a Raxified Plasma in a

Magnetio Field","Plasma Physios and the Problem of Controlled

Thermonuclear Research" £ ,454-460, Mosoow, .(1958).

18. GARDUER, C.S., GOERTZEL, II., GRAD, H.f MORAWETZ, C.SM ROSS, M.H.,

and RUBINS, H., Proo. 2nd UJf. Int.Conf.on Peaoefxa Uses of Atomic

Energy, ^ 1 , 230-237, (l958).

19- SAGDEEV, R.Z., "liagnetodynamios Shook Waves in Low Density IonizedGaB% Proceedings, Symposium on Eleotromagnetio and Fluid Dynamicsof Gaseous Plasmas, 3-6 April 1961, pp. 443-450, Polytechnic Press,Brooklyn, USA, 1962.

-161-

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