magnetohydrodynamics nd plasma physics - pku · the distinction between magnetohydrodynamics and...

1 0

Magnetohydrodynamicsand Plasma Physics

10.1 Introduction and Definition s

Magnetohydrodynamics and plasma physics both deal with thebehavior of the combined system of electromagnetic fields and a con-

ducting liquid or gas . Conduction occurs when there are free or quasi-freeelectrons which can move under the action of applied fields . In a solidconductor, the electrons are actually bound, but can move considerable

distances on the atomic scale within the crystal lattice before makingcollisions . Dynamical effects such as conduction and Hall effect areobserved when fields are applied to the solid conductor, but mass motiondoes not in general occur . The effects of the applied fields on the atomsthemselves are taken up as stresses in the lattice structure . For a fluid, on

the other hand, the fields act on both electrons and ionized atoms toproduce dynamical effects, including bulk motion of the medium itself .This mass motion in turn produces modifications in the electromagnetic

fields. Consequently we must deal with a complicated coupled system ofmatter and fields .

The distinction between magnetohydrodynamics and the physics of

plasmas is not a sharp one . Nevertheless there are clearly separated

domains in which the ideas and concepts of only one or the other are

applicable. One way of seeing the distinction is to look at the way in which

the relation J = aE is established for a conducting substance . In the

simple model of Section 7 .8 the electrons are imagined to be accelerated by

the applied fields, but to be altered in direction by collisions, so that theirmotion in the direction of the field is opposed by an effective frictional

force vmv, where v is the collision frequency . Ohm's law just represents a

309

310 Classical Electrodynamics

balance between the applied force and the frictional drag . When the

frequency of the applied fields is comparable to v, the electrons have time

to accelerate and decelerate between collisions . Then inertial effects enter

and the conductivity becomes complex . Unfortunately at these samefrequencies the description of collisions in terms of a frictional force tends

to lose its validity. The whole process becomes more complicated . At

frequencies well above the collision frequency another thing happens . Theelectrons and ions are accelerated in opposite directions by electric fields

and tend to separate . Strong electrostatic restoring forces are set up bythis charge separation. Oscillations occur in the charge density . These

high-frequency oscillations are called plasma oscillations and are to bedistinguished from lower-frequency oscillations which involve motion of

the fluid, but no charge separation. These low-frequency oscillations are

called magnetohydrodynamic waves.In conducting liquids or dense ionized gases the collision frequency is

sufficiently high even for very good conductors that there is a widefrequency range where Ohm's law in its simple form is valid . Under theaction of applied fields the electrons and ions move in such a way that,apart from a high-frequency jitter, there is no separation of charge.

Electric fields arise from motion of the fluid which causes a current flow,or as a result of time-varying magnetic fields or charge distributionsexternal to the fluid . The mechanical motion of the system can then bedescribed in terms of a single conducting fluid with the usual hydro-dynamic variables of density, velocity, and pressure . At low frequenciesit is customary to neglect the displacement current in Ampere's law . Thisis then the approximation which is called magnetohydrodynamics .

In less dense ionized gases the collision frequency is smaller . Theremay still be a low-frequency domain where the magneto hydrodynamicequations are applicable to quasi-stationary processes . Frequently astro-physical applications fall in this category . At higher frequencies, however,the neglect of charge separation and ofthe displacement current is not allow-able. The separate inertial effects of the electrons and ions must be includedin the description of the motion . This is the domain which we call plasma

physics . There is here a range of physical conditions where a two-fluidmodel of electrons and ions gives an approximately correct description ofvarious phenomena . But for high temperatures and low densities, thefinite velocity spreads of the particles about their mean values must beincluded. Then the description is made in terms of the Boltzmannequation with or without short-range correlations . We will not attemptto go into such details here. At still higher temperatures and lowerdensities, the electrostatic restoring forces become so weak that the lengthscale of charge separation becomes large compared to the size of the

[Sect . 10 .2} Magnetohydrodynamics and Plasma Physics 311

volume being considered . Then the collective behavior implicit in a fluid

model is gone completely . We have left a few rapidly moving chargedparticles interacting via Coulomb collisions . A plasma is, by definition,an ionized gas in which the length which divides the small-scale individual-particle behavior from the large-scale collective behavior is small com-

pared to the characteristic lengths of interest. This length, called the

Debye screening radius, will be discussed in Section 10 .10. It is numerically

equal to 7.91 (Tln)~ cm, where T is the absolute temperature in degrees

Kelvin and n is the number of electrons per cubic centimeter . For all butthe hottest or most tenuous plasmas it is small compared to 1 cm .

10.2 Magnetohydrodynamic Equations

We first consider the behavior of an electrically neutral, conductingfluid in electromagnetic fields. For simplicity, we assume the fluid to be

nonpermeable. It is described by a matter density p(x, t), a velocity v(x, t),

a pressure p(x, t) (taken to be a scalar), and a real conductivity 6. Thehydrodynamic equations are the continuity equatio n

lapat+ 0• (pv)=0 ( 1 0 .1 )

and the force equation :

p ~ --Vp +~(J xB)+Fv+Ag ( 10.2)

In addition to the pressure and magnetic-force terms we have included

viscous and gravitational forces . For an incompressible fluid the viscous

force can be writtenFv = yJp2v (10.3)

where q is the coefficient of viscosity. It should be emphasized that the

time derivative of the ve l ocity on the left side of (10 .2) is the convective

derivative,

_~ -~v• (10.4)

which gives the total time rate of change of a quantity moving instanta-

neously with the velocity v .

312 Classical Electrodynamic s

With the neglect of the displacement current, the electromagnetic fieldsin the fluid are described by

at ~1{ 10.5}

V xB-4'TJ

C

The condition V • J = 0, equivalent to the negtect of displacementcurrents, follows from the second equation in (1 0 .5) . The two divergenceequations have been omitted in (10 .5) . It follows from Faraday's law that(7lat) V • B = 0, and the requirement V • B = 0 can be imposed as aninitial condition . With the neglect of the displacement current, it isappropriate to ignore Coulomb's law as well . The reason is that theelectric field is completely determined by the curl equations and Ohm'slaw (see below) . If the displacement current is retained in Ampere's lawand V • E = 4npe is taken into account, corrections of only the order of(U2'C2) result. For normal magneto hydrodynamic problems these arecompletely negligible .

To complete the specification of dynamical equations we must specifythe relation between the current density J and the fields E and B . For asimple conducting medium of conductivity d, Ohm's law applies, and thecurrent density is

J' = 6E' (10.6)

where J' and E' are measured in the rest frame of the medium. For amedium moving with velocity v relative to the laboratory, we must trans-form both the current density and the electric field appropriately . Thetransformation of the field is given by equation (6.10) . The current densityin the laboratory is evidently

J = J' + pev (10.7)

where p, is the electrical charge density . For a one-component conductingfluid, pe = 0 . Consequently, Ohm's law assumes the form ,

j = or I E + V x B) (10.8)~

Sometimes it is possible to assume that the conductivity of the fluid iseffectively infinite . Then under the action of fields E and B the fluid flowsin such a way that

E-}-I

(v x B)=0 (10.9)C

is satisfied .

[Sect . 1 0 .3] Magnetohydrodynamics and Plasma Physics 313

Equations (10.1), (10.2), ( 10.5), and (10.8), supplemented by an equationof state for the fluid, form the equations of magnetohydrodynamics . Inthe next section we will consider some of the simpler aspects of them andwill elaborate the basic concepts involved .

10 .3 Magnetic Diffu s ion , Viscosity, and Pressure

The behavior of a fluid in the presence of electromagnetic fields isgoverned to a large extent by the magnitude of the conductivity. Theeffects are both electromagnetic and mechanical . We first consider theelectromagnetic effects . We will see that, depending on the conductivity,quite different behaviors of the fields occur. The time dependence of themagnetic field can be written, using (10 .8) to eliminate E, in the form :

V2B (10.10)B = O x (v x B)+ 4rr6

Here it is assumed that if is constant in space. For a fluid at rest (10 .10)reduces to the diffusion equation

aB c2 p2B (10.11)

at=

4n6

This means that an initial configuration of magnetic field will decay awayin a diffusion time

T _ 4?TffL2(10 .12 )

2C

where L is a length characteristic of the spatial variation of B . The time 7,is of the order of 1 sec for a copper sphere of 1 cm radius, of the order of 104years for the molten core of the earth, and of the order of 1010 years for atypical magnetic field in the sun .

For times short compared to the diffusion time T (or, in other words,when the conductivity is so large that the second term in (10 .10) can beneglected) the temporal behavior of the magnetic field is given b y

aB = V x (v x B) (10. 1 3)at

From (6.5) it can be shown that this is equivalent to the statement that themagnetic flux through any loop moving with the local fluid velocity isconstant in time. We say that the lines of force are frozen into the fluidand are carried along with it . Since the conductivity is effectively infinite,


the velocity w of the lines of force (defined to be perpendicular to B) isgiven by (10.9) :

W = c(E B2 B

)(10.14)

This so-called "E x B drift" of both fluid and lines of force can be under-stood in terms of individual particle orbits of the electrons and ions incrossed electric and magnetic fields (see Section 12.8) .

A useful parameter to distinguish between situations in which diffusionof the field lines relative to the fluid occurs and those in which the lines offorce are frozen in is the magnetic Reynolds number Ru . If V is a velocitytypical of the problem and L is a corresponding length, then the magneticReynolds number is defined as

R~z = ~ (10.15)

where T is the diffusion time (10 .12) . Transport of the lines of force withthe fluid dominates over diffusion if R m > 1 . For liquids like mercury orsodium in the laboratory R. < 1, except for very high velocities . But ingeophysical and astrophysical applications RAT can be very large comparedto unity .

The mechanical behavior of the system can be studied with the forceequation (10.2) . Substituting for J from (10 .8), we find

2p dY _ F', - 6$ r Vl

- W ) ( 10 . 1 6)dt CZ

t

where F is the sum of all the nonelectromagnetic forces, and vi is thecomponent of velocity perpendicular to B. From (10.16) it is apparentthat flow parallel to B is governed by the nonelectromagnetic forces alone.The velocity of flow of the fluid perpendicular to B, on the other hand,decays from some initially arbitrary value in a time of the order o f

- PC2

T J -

QB2

to a value

(10.17)

2

vl = w+ C

F.L (10.18)UB 2

In the limit of infinite conductivity this result reduces to that of (10 .14),as expected. The term proportional to B2 in (10, 16) is an effective viscousor frictional force which tends to prevent flow of the fluid perpendicular tothe lines of magnetic force. Sometimes it is described as a magneticviscosity . If ordinary viscosity, here lumped into F, is comparable to the

[Sec t . 10 . 3 ] Magnetohydrodynam ics and Plasma Physics 315

magnetic viscosity, then the decay time T' is shortened by an obvious factorinvolving the ratio of the two viscosities .

The above considerat ions have shown that if the conduct ivity is largethe lines of force are frozen into the fluid and move along with it . Anydeparture from that state decays rap idly away. In considering themechanical or electromagnet ic effects we treated the opposite quantities asgiven , but the equations are , of course , coupled . In the limit of very largeconductivity it is convenient to -relate the current density J in the forceequation to the magnetic induction B via Ampere's law and to use thein finite conductivity expres sion (10.9) to eliminate E from Faraday's lawto yield (10.13) . The magnetic force term in (1 0 .2) can now be written

With the vector identity

Equation (10.19) can be transformed into

(10.19)

(10.20)

1 2

c (

J x B) -=~' ---~8

B

} + Orr 4B • V jB (1 0.21)

This equation shows that the magnetic force is equivalent to a magnetichydrostatic pressure, B 2

PM = ~(10.22)

$

plus a term which can be thought of as an additional tension along thelines of farce . The result (10 .21) can also be derived from the Maxwellstress tensor (see Section 6 .9) .

If we neglect viscous effects and assume that the gravitational force isderivable from a potential g = - VV, the force equation (10 .2) takes the form

p ~t = -D(P + P ~ + PV} + 4 (B • 0)B ( 10 .23)

In some simple geometrical situations, such as B having only one com-ponent, the additional tension vanishes . Then the static properties of thefluid are described by

p + p,U + pV = constant (10.24)

This shows that, apart from gravitational effects, any change in mechanicalpressure must be balanced by an opposite change in magnetic pressure . Ifthe fluid is to be confined within a certain region so that p falls rapidly tozero outside that region, the magnetic pressure must rise equally rapidlyin order to confine the fluid . This is the principle of the pinch effectdiscussed in Section 10 .5.


10.4 Magnetohydrodynamic Flow between Boundaries with CrossedElectric and Magnetic Fields

To illustrate the competition between freezing in of lines of force anddiffusion through them and between the E x B drift and behavior imposedby boundary conditions, we consider the simple example of an incom-pressible, but viscous, conducting fluid flowing in the x direction betweentwo nonconducting boundary surfaces at z = 0 and z = a, as shown inFig. 10.1 . The surfaces move with velocities Vi and Y2, respectively, inthe x direction. A uniform magnetic field Bo acts in the z-direction . Thesystem is infinite in the x and y directions. We will look for a steady-statesolution for flow in the x direction in which the various quantities dependonly upon Z .

If the fields do not vary in time, it is clear from Maxwell's equations(10.5) that any electric field present must be an electrostatic field derivable

from a potential and determined solely by the boundary conditions, i .e .an arbitrary external field. Expression (10 .14) for the velocity of the linesof force when a is infinite implies that there is an electric field in the ydirection. If we assume that to be the only component of E, then it mustbe a constant, E0. Because the moving fluid will tend to carry the lines offorce with it, we expect an x component Bi(z) of magnetic induction, aswell as the z component Bo .

The continuity equation (10.1) reduces to V • v = 0 for an incompressiblefluid. This is satisfied identically by a velocity in the x direction whichdepends only on z . The force equation, neglecting gravity, has the steady-state form :

Vp =I

(J x B) + r7V2v (10.25)C

z

x V1

YFig. 101 Flow of viscous con-ducting fluid in a magnetic fieldbetween two plane surfacesmoving with different velocities .

[Sect . 10.4) Magnetohydrodynamics a nd Plasma Physics

The only component of J that is nonvanish ing is J,,(z) ;

J

317

(10 . 26)

where v is the x component of velocity . When we write out the threecomponent equations in (10.25), we find

aP = ~,so B~ a 2 Uax ~ (

E° - v} +

77

sc c 2z

ap=oay

aP = 6

B

'(E

° - BO v)az c, c

(10.27)

The magnetic force in the z direction is just balanced by the pressuregradient. If we assume no pressure gradient in the x direction, the first ofthese equations can be written :

a2U U CEO-

(M\2

az

(M)2

- =2 a BO

where

~B 2~2~11

M 221 c

(10 .28)

(10.29)

is called the Hartmann number . From (10.1 7) M2 can be seen to be theratio of magnetic to normal viscosity . The solution to (10 .28), subject tothe boundary conditions v(O) = Vi and v(a) = V2, is readily found to be

v(z) = V 1 sinh [M(a - z)]+ sinh (

Mz

sinkM a sinh M a

Binh [M(a - z)]+ sigh (M')

cE+ ° 1 - - a a (10.30)

Bo sinh M

In the lim i t B., 0 , M > 0 , we obtain the standard laminar-flow resul t

zV (z) = y1 + - (V2 - V1) (10.31)

a

In the other limit of M > 1 we expect the magnetic viscosity to dominateand the flow to be determined almost entirely by the E x B drift . If weapproximate v(z) for z < a and M > 1, we obtain

v(z) cEO + yi _ cEo e-- :vlz /aBo BO

(10.32)~

318

aM

CEOBo

Classical Electrodynamics

M?? 1

V2 _. . ... - _ ... _- - ----- ---

M--'>-0

yl --------------

0 Z -~--~-~Ma

Fig. 10.2 Velocity profiles forlarge and small Hartmann

numbers M. For M- 0, lami-nar flow occurs . For M > 1, theflow is given by the E x B drift

velocity, except in the immediateneighborhood of the boundaries .

This shows that, while v(z) = V, exactly at the surface, there is a rapidtransition in a distance of order (a1M) to the E x B drift value (cEoI$o)-Near z = a, (10.32) is changed by replacing Vl by Y. and z by (a - z). Thevelocity profile in the two limits (10.31) and (10.32) is shown in Fig. 10.2 .

The magnetic field Bx(z) is determined by the equation

a$x -Orr J~= 4G~

az (Eo - G Bov (10.33)

The boundary conditions on Bx at z == 0 and z = a are indeterminateunless we know the detailed history of how the steady state was created orcan use some symmetry argument . A ll we know is that the difference inB,,; is related to the total current flowing in the y direction per unit lengthin the x direction :

a

Bx(a) - Bx(Q) =oIJ(z) dz (10.34)

c

This indeterm inacy stems from the one-dimensional nature of the problem .For simplicity we will calculate the magnetic field only for the case whenthe total current in the y direction is zero.* Then we can assume that B.vanishes at z = 0 and z = a. Using (10.30) for the velocity in (10.33), itis easy to show that then

4~ra~a2 y - ycosh

M-cosh M - a z~

B~(zl _ BO r ~ 2 1 (10.35)l

)c

22a M

sink ML 2

* This requirement means that cEQ/Bo = ~(Yj + Y?) .

[Sect . 10.4] Magnetahydradynamics and Plasma Physics 319

The dimensionless coefficient in square brackets in (10 .35) may be identifiedas the magnetic Reynolds number (10.15), since (V2- V1)/2 is a typicalvelocity in the problem and a is a typical length . In the two limits M < 1and M > 1, (10.35) reduces to

zr~ -

a

z

ll~

Bx(z) "-' 'R MBD

a ~1 _ Mz

aLM[1(e + e-Me

a

z

) )

for M < 1

for M > 1

(1 0 .36)

Figure 10.3 shows the behavior of the lines of force in the two limitingcases. Only for large R . is there appreciable transport of the lines of force .And for a given RM, the transport is less the larger the Hartmann number .

For liquid mercury at room temperature the relevant physical constantsare

or - 9 .4 X 1015 sec-1-

n = 1 .5 x 10-2 poise

p = 13.5 gm/cm-3

The diffusion time (10 .12) is r = 1 . 31 X 10-4 [L (cm)J2 sec . The Hartmannnumber ( 10.29) is M = 2 .64 x 10-2B9 (gauss) a (cm) . With L ^r a - 1 cm,this gives a magnetic Reynolds number Rm- 101V. Consequentlyunless the flow velocity is very large, there is no significant transport oflines of force for laboratory experiments with mercury . On the other hand,if the magnetic induction B0 is of the order of 1{14 gauss, then M, 250 andthe velocity flow is almost completely specified by the E x B drift (10 .14) .

RM/4

KI D

R,rr IMCo

V2 > V1

w

u M<<1

vl

V2 ? y1

M>>1

~V1

(a) (b)Fig. 10.3 (a) Axial component of magnetic induction between the boundary surfacesfor large and small Hartmann numbers . (b) Transport of lines of magnetic induction in

direction of flow ,

az - ~P-


In geomagnetic problems with the earth's core and in astrophysicalproblems the parameters (e .g., the length scale) are such that RM > 1occurs often and transport of the lines of force becomes very important .

10.5 Pinch Effect

The confinement of a plasma or conducting fluid by self-magnetic fieldsis of considerable interest in thermonuclear research, as well as in otherapplications . To illustrate the principles we consider an infinite cylinderof conducting fluid with an axial current density J. -- J(r) and a resultingazimuthal magnetic induction BO = B(r) . For simplicity, the currentdensity, magnetic field, pressure, etc ., are assumed to depend only on thedistance r from the cylinder axis, and viscous and gravitational effects areneglected. We first ask whether a steady-state condition can exist in whichthe material is mainly confined within a certain radius r = R by theaction of its own magnetic induction . For a steady state with v = 0 theequation of motion (10.23) of the fluid reduces to

0 = - `~' -d~ B2} - B2 (10.37)

dr dr 87r 47rr

Ampere's law in integral form relates &(r) to the current enclosed :

r

B(r) = 4-~` rJ(r) dr (10.38)cr o

A number of results can be obtained without specifying the form of J(r),aside from physical limitations of finiteness, etc. From Ampere's law it isevident that, if the fluid lies almost entirely inside r = R, then the mag-netic induction outside the fluid i s

B(r) =21

(10 .39)cr

whereK

I = 27rrJ(r) dr0

is the total current flowing in the cylinder . Equation (10.37) can bewritten as

dp 1 d (r2B2) (10.40)d r S7r2 dr

with the solution :

p(r) .~ po - 1fri

d(r2B2) dr (10 .41)

87r r2 dr

[Sect . 10 .5] Magnetohydrodynamics and Plasma Physics 321

Here po is the pressure of the fluid at r = 0 . If the matter is confined tor < R, the pressure drops to zero at r = R. Consequently the axialpressure P. is given by

Po =I fR1

d { f' 2B 2} dr (10.42)8rr o r2 dr

The upper limit of integration can be replaced by infinity, since the inte-grand vanishes for r ~ R, as can be seen from (10.39) . With this expression

(10.42) for po, (10.41) can be written as

p(r) =1 r

R 1 d (r 2B2) d r (10.43)87r r2 dr

The average pressure inside the cylinder can be related to the totalcurrent I and radius R without specifying the detailed radial behavior .

ThusR

(P) = R2 0 rp(r) dr ( 10 .44)

Integration by parts and use of (10 .40) give s

(p) =

12(10.45)

2rrR2c2

as the relation between average pressure, total current, and radius of thecylinder of fluid or plasma confined by its own magnetic field . Note thatthe average pressure of the matter is equal to the magnetic pressure (B 2/87r)at the surface of the cylinder . In thermonuclear work, hot plasmas withtemperatures of the order of 108°K (kT, 10 kev) and densities of theorder of 1015 particles/cm3 are envisioned. These conditions correspondto a pressure of approximately X015 x 1 08k - 1 .4 x 1Q' dynes/cm2, or 14atmospheres. A magnetic induction of approximately 19 kilogauss at thesurface, corresponding to a current of 9 x 104R (cm) amperes, is necessaryfor confinement. This shows that extremely high currents are needed toconfine very hot plasmas .

So far the radial behavior of the system has not been discussed . Twosimple examples will serve to illustrate the possibilities . One is that the

current density J(r) is constant for r < R . Then B(r) = (2Ir f cR2) forr < R. Equation (10 .43) then yields a parabolic dependence for pressureversus radius ;

P(r) = 2 2 2~ (10.46)VC R R

The axial pressure pa is then twice the average pressure ~p~ . The radialdependences of the various quantities are sketched in Fig . 10.4.


Po

Fig . 10 .4 Variation of azimuthalmagnetic induction and pressure withradius in a cylindrical plasma column

with a uniform current density J.

The other model has the current density confined to a very thin layer onthe surface, as is appropriate for a highly conducting fluid or plasma . Themagnetic induction is given by ( 10.39) for r > R, but vanishes inside thecylinder . Then the pressure p is constant inside the cylinder and equal tothe value (10 .45). This is sketched in Fig . 10.5 .

10.6 Dynamic Model of the Pinch Effect

The simple considerations of the previous section are valid for a staticor quasi-static situation . In actual practice with plasmas, such circum-stances do not arise. Generally, at some time early in the history ofcurrent flow down the plasma the pressurep is much too small to resist the

magnetic pressure outside . Consequently the radius of the cylinder ofplasma is forced inwards ; the plasma column is pinched . This has thedesirable consequence that the plasma is pulled away from its confiningwalls . If the pinched configuration were stable for a sufficiently long time,it would be possible to heat the plasma to very high temperatures withoutburning up the walls of the confining vessel .

PO

Fig. 10.5. Variation of azi-

muthal magnetic induction andpressure in a cylindrical plasmacolumn with a surface-current

density .

0 R r 3P

0 R r

[Sect . 10.6] Magnetohydrodynamics and Plasma Physics 323

A simple model, first discussed by M . Rosenbluth, exhibits the essentialdynamical features . Suppose that a plasma is created in a hollow con-ducting cylinder of radius Ro and length L. A voltage difference V isapplied between the ends of the cylinder so that a current I flows in theplasma. This produces an azimuthal magnetic induction B. which causesthe plasma to pinch inwards . The radius of the plasma column at timet > 0 is R(t) . The conductivity of the plasma is taken to be virtuallyinfinite . Then the current all flows on the surface, and the magneticinduction

BO =21

(10.47)cr

exists only between r = R(t) and r = Ro. Because of the assumption ofinfinite conductivity the electric field at the plasma surface, in the movingframe of reference in which the interface is at rest, vanishes :

E'=E+ v x B =O (10.48)C

If we now apply Faraday's law of induction to the dotted soap shown inFig. 10 . 6, the inner arm of which is moving inwards with the interface, wefind that the only contribution to the line integral of E comes from theside of the loop in the conducting wall . Thus

V= - 1 ROB, dr = - 2 d ~I In R°i~ (10.49)

L c dt Rct~ c dt R

This is the standard inductive relation between current, voltage, anddimensions (inductance). The integral of this equation i s

I In ~°~ = G EQ 11(t) dt (10.50)~Z 2 0

where Eo f {t} = VIL is the applied electric field . The function f (t) isassumed known and is normalized so that Eo is the peak value of applie d

t =o 1 = V

RFig. 10.b Plasma column inside a hollow,cylindrical conductor. L


field . In order to proceed further we must relate the current I in a dynamicway to the behavior of the plasma radius R .

The desired dynamical connection between I and R is essentially themomentum-balance equation, or Newton's second law . Some assumptionabout the plasma must be made. If the mean free path for collisions isshort compared to the radius, the dynamic behavior is characteristic ofhydrodynamic shock waves . But for a hot, tenuous plasma the mean freepath is comparable to, or larger than, the radius. Then a model withparticles moving freely inside the plasma is more appropriate . If thevelocity R of the plasma surface is large compared to thermal speeds, eachparticle approaches the interface with a velocity A in the frame of referencein which the interface is at rest . As the particle penetrates into the outerregion, it starts feeling the magnetic induction, is turned around, and leavesthe surface with velocity A . Consequently each particle colliding with theplasma surface receives a momentum transfer 2MR. The number collidingwith unit area of the surface per unit time is NA , where N is the initialnumber of particles per unit volume . Therefore the rate of transfer ofmomentum per unit area (i .e., pressure) is

p = 2NMA2 = 2pA2 (10 .51 )

where p is the initial mass density . At the surface of the plasma there is amagnetic pressure (B2/87r) due to the discontinuity in magnetic inductionfrom zero inside to the value B just outside . These pressures must balance .Consequently, using (10 .47), we find that the current is related to thevelocity by

2

P = 47rpc2R2 dR (10. 52)dt

Equation (10 .52) depends on a rather simplified model of the mechanical-momentum transfer rate in which each particle collides only once with theinterface . In fact, the velocity of the interface increases with time so thatthe surface catches up with particles which were reflected earlier and hitsthem again and again . This effect can be approximated b y the "snow-plow" model in which the interface is imagined to carry along with it allthe material which it hits as it moves in . Then the magnetic pressure andrate of change of momentum are related by

2

d hr(R)r~ = -2~R Bdt S 7 T

where M(R) is the mass carried along by the snowplow :

(10.53)

M(R) = irp(R 02 - R2) (10.54)

[Sect. 1 0 . 6] Magne tohydrodynamics and Plasma Physics

This leads to the relation

I2 = - 7rpc2R d [(R .2 - R2)dR]

ar at

325

(10.55)

between current and radius . In the initial stages when R < Ro the snow-plow model and free-particle model give the same relation between currentand radius to within a factor of 2*, and do not differ by an order ofmagnitude even at later times .

The equation of motion for R(t) is obta ined by substituting 7 2 fromeither (10.52) or (10 .55) into the inductive relation (10.50) . Choosing thefree-particle model as an illustration, we obtain

2R In(

°) dR

- -cE

o f1(t') dt' (10 .56)R dt .J4~rp

where the signs of the square root have been taken to give k < 0 . Withoutknowledge of f (t) we cannot solve this equation . Nevertheless , someidea of the solution can be obtained by introduc ing the dimensionlessvariables :

(c2Eo2r t47rp Ro

Rx=-

RoThen (10.56) becomes

r

2x In x dT o f (,r') d-r'

(10.57)

(10.58)

For the snowplow model the equivalent equation i s

d x2) dx

dT

lC dTJ

f~7 ~ ~ dT ~ ~1 2~.1 J

x( lri x)2

Without solving these equations it is evident that x changes significantlyin times such that T , 1 . This means that the scaling law for the radial

velocity of the pinch is1/4OE z

VO =_O

(10.60)47rp /

This result emerges whatever dynamic model is used, including a hydro-dynamic one. Typical experimental conditions for a fast pinch in small-scale hydrogen or deuterium plasmas involve applied electric fields of the

* The factor of 2 comes from the fact that in the one case the particles are elasticallyreflected and suffer a velocity change of 2R, while in the other the particles collideinelastically with the interface and receive a velocity change of R .


Rq

ti Fig. 10.7 Radius of plasmacolumn as a function of time

after initiation of current flow.

The characteristic velocity of

pinching is given by (10.60) .

order of 10,1 volts/cm and initial densities of the order of 10-11 gm f cm3

(- 3 x 105 deuterons/cm3). Then vo is of the order of 10' cm/sec . The

current flowing is, according to (10.52) or (10.55) ,

z~ ~ c RoEo F ~~ dx(10 .61)

vo dT

where F is a dimensionless function of the order of unity . For a tube

radius of 10 cm and the conditions described, the current I is measured inunits of 105 or 106 amperes .

The discussion of the pinching action presented so far is obviously validonly for short times after the initiation of current flow . The simplified

models indicate that in a time of the order of R,lvo the radius of the plasma

column goes to zero . It is clear, however, that before that will occur (evenapproximately) the behavior will be modified . In the hydrodynamic limit,the radial shock waves caused by the pinch will be reflected off the axisand move outwards, striking the interface and retarding its inward motionor even reversing it . This phenomenon is known as bouncing . It isevidently present also in the free-particle model . Consequently the generalbehavior of radius R as a function of time is expected to be as shown inFig. 1 0.7. Although no proper analysis has been made of the subsequentbounces, it is conjectured that there is an approach to a steady state at

some radius less than Ro.

10.7 Instabilities in a Pinched-Plasma Column

In the laboratory long-lived pinched plasmas are extremely difficult to

produce . The dynamic behavior of the previous section is found to befollowed at least qualitatively for times up to around the first bounce .

But then the plasma column is observed to break up rapidly. The reasonfor the disintegration of the column is the growth of instabilities . The

0 1 2 3T

[Sect . 1Q.7] Magnetohydrodynamics and Plasma Physics 327

(b)

column is unstable against various departures from cylindrical geometry .

Small distortions are amplified rapidly and destroy the column in a veryshort time. The detailed analysis of instabilities is sufficiently complexthat we will attempt only qualitative arguments . Two of the simplerunstable distortions will be described .

The first is the kink instability, shown in Fig. 10.$a. The lines of azimu-thal magnetic induction near the column are bunched together above, andseparated below, the column by the distortion downwards . Thus themagnetic pressure changes are in such a direction as to increase thedistortion. The distortion is unstable .

The second type of distortion is called a sausage or neck instability,

shown in Fig. lO.Sb. In the neighborhood of the constriction the azi-muthal induction increases, causing a greater inwards pressure at the neckthan elsewhere . This serves to enhance the existing distortion .

Both types of instability are hindered by axial magnetic fields withinthe plasma column. For the sausage distortion the lines of axial inductionare compressed by the constriction, causing an increased pressure insideto oppose the increased pressure of the azimuthal field, as indicated

schematically in Fig . 10 .9. It is easy to see that the fractional changes i n

Fig. 10 .9 Hindering neck instability withoutward pressure of trapped axial magnetic

fields .

------------,-(b) Sausage or neck instability .

328

Fig. 10.10 Hindering kink instability with tension of trapped axial fields .

the two magnetic pressures, assuming a sharp boundary to the plasma,are

app _ 2x a pz _ 4x(10.62)

P P R Pz R

where x is the small inwards displacement . Consequently, if

Bz2 > 1B 4' 2 (10 .63 )

the column is stable against sausage distortions .For kinks the axial magnetic field lines are stretched, rather than com-

pressed laterally together . But the result is the same ; namely the increasedtension in the field lines inside opposes the external forces and tends tostabilize the column, It is evident from Fig . 10.10 that a short-wavelength

kink of a given lateral displacement will cause the lines of force to stretchrelatively more than a long-wavelength kink . Consequently, for a givenratio of internal axial field to external azimuthal field, there will be atendency to stabilize short-wavelength kinks, but not very long-wavelengthones. If the fields are approximately equal, analysis shows that if the wave-length of the kink A < 14 R the disturbance is stabilized .

For longer-waveleng#.h kinks stabilization can be achieved by the action

of the outer conductor, provided the plasma radius is not too smallcompared to the radius of the conductor. The azimuthal field lines are

trapped between the conductor and the plasma boundary, as shown inFig. 10 .11 . If the plasma column moves too close to the walls, the linesof force are crowded together between it and the walls, causing an in-creased magnetic pressure and restoring force .

Fig. 10.11 Stabilization of

long-wavelength kinks withou ter conductor .

Classical Electrodynamics

[Sect. 14.8 Magnetahydrodynamics and Plasma Physics 329

It is clear qualitatively that it must be possible, by a combination oftrapped axial field and conducting walls, to create a stable configuration,at least in the approximation of a highly conducting plasma with a sharpboundary. Detailed analysis* confirms this qualitative conclusion andsets limits on the quantities involved. It is important to have as little axialfield outside the plasma as possible and to keep the plasma radius of theorder of one-half or one-third of the cylinder radius . If the axial fieldoutside the plasma is too large, the combined B. and B~ cause helicalinstabilities that are troublesome in toroidal geometries . If, however, theaxial field outside the plasma is made very large, the pitch of the helixbecomes so great that there is much less than one turn of the helix in aplasma column of finite length. Then it turns out that there is the possi-bility of stability again . Stabilization by means of a strong axial fieldproduced by currents external to the plasma is the basis of some fusiondevices, e.g., the Stellarator .

The idealized situation of a sharp plasma boupdary is difficult to createexperimentally, and even when created is destroyed by diffusion of theplasma through the dines of force in times of the order of 4-iraR2,1c2 (seeSection 10.3). For a hydrogen plasma of I ev energy per particle this timeis of the order of 10-4 sec for R - 10 cm, while for a 10 kev plasma it isof the order of 102 sec . Clearly the thermonuclear experimenter must tryto create initially as hot a plasma as possible in order to make the initialdiffusion time long enough to allow further heating .

10.8 Magnetohydrodynamic Waves

In ordinary hydrodynamics the only type of small-amplitude wavemotion possible is that of longitudinal, compressional (sound) waves .These propagate with a velocity s related to the derivative of pressure withrespect to density at constant entropy :

2 = (LP) (10.64)ap a

If the adiabatic law p = Kp" is assumed, s2 = ypo/Poi where y is the ratioof specific heats . In magnetohydrodynamics another type of wave motionis possible . It is associated with the transverse motion of lines of magneticforce . The tension in the lines of force tends to restore them to straight-line form, thereby causing a transverse oscillation . By analogy with

* V. D . Shafranov, Atomnaya Energy. I, 5, 38 (1956) ; R. J . Tayler, Proc, Phys. Soc .

(London), B70, 1049 (1957) ; M. Rosenbluth, Los Alamos Report LA-2030 (1956). See

also Proceedings of the Second International Conference on Peaceful Uses of AtomicEnemy, Vol. 31 (1958), papers by Braginsky and Shafranov (p . 43) and Tayler (p. ]60) .


ordinary sound waves whose velocity squared is of the order of the hydro-static pressure divided by the density, we expect that these magnetohydro-dynamic waves, called Alfuen waves, will have a velocity

B 2 V2

vA ,(

a (10.65)$7r 1Aa

where Ba21Bw is the magnetic pressure .To examine the wave motion of a conducting fluid in the presence of a

magnetic field, we consider a compressible, nonviscous, perfectly con-ducting fluid in a magnetic field in the absence of gravitational forces . Theappropriate equations governing its behavior are :

P+ 0 •(pv)- 0

Pat

+ P(v • 0)v = -VP - 4~ B x (V x B) (10 .66)at

= D x (v x B )a t

These must be supplemented by an equation of state relating the pressureto the density. We assume that the equilibrium velocity is zero, but thatthere exists a spatially uniform, static, magnetic induction Bp throughoutthe uniform fluid of constant density po . Then we imagine small-amplitudedepartures from the equilibrium values :

P = Ao + Pi(x)t} (10 .67)

V = vi(X, t )

If equations (10 .66) are linearized in the small quant i t ies , then theybecome :

ap,at +

AoO•v1 = 0

Pa ~ 1 + MP , + !10 X (V X Bl) = 0 (10. 68)at 47T

aBj - vx (vix so) =oat

where s2 is the square of the sound velocity (10.64). These equations canbe combined to yield an equation for vl alone :

°2 l - s2D(D • vl) + v_4 x V x [V x (vl x vA)] = 0 (10 .69)at,

[Sect. 10.8] 1Ylagnetohydrodynamics and Plasma Physics 331

where we have introduced a vectorial Alfven velocity :

VA - B° (10.70)~/ P

The wave equation (10 . 69) for v, is somewhat involved, but it allowssimple solutions for waves propagating parallel or perpendicular to themagnetic field direction . * With vl(x, t) a plane wave with wave vector kand frequency cv

vl(x, t) = v1eik'x iot (10.71 )

equation (10.69) becomes :

-cu2vi +(s2

+vA2)(k ' v1)k

+ TA • k[(vA • k}vi - (vA • vj)k - (k • vl)vA] = 0 (10.72)

If k is perpendicular to vA the last term vanishes . Then the solution for vl

is a longitudinal magnetosonic wave with a phase velocity :

along = S~ 2 + v~42 (10.73)

Note that this wave propagates with a velocity which depends on the sumof hydrostatic and magnetic pressures, apart from factors of the order ofunity . If k is parallel to vA, (10 .72) reduces to

2(k2271g2 - 6J2)V1 4- VJ - ) 1C2(VA • Y

j.)VA =

0 (10.74)A

There are two types of wave motion possible in this case . There is anordinary longitudinal wave (vi parallel to k and vA) with phase velocityequal to the sound velocity s. But there is also a transverse wave (vl - vA =0) with a phase velocity equal to the Alfven velocity vA . This Alfven wave

is a purely magnetohydrodynamic phenomenon which depends only onthe magnetic field (tension) and the density (inertia) .

For mercury at room temperature the Alfven velocity is [BO (gauss) f 13 .1 ])cm/sec, compared with sound speed of 1 .45 x 105 cm/sec. At all labora-tory field strengths the Alfven vclocity is much less than the speed ofsound. In astrophysical problems, on the other hand, the Alfven velocitycan become very large because of the much smaller densities . In the sun'sph.otosphere, for example, the density is of the order of 10-' gm/cm3(-b x 10111 hydrogen atoms f cm3) so that vA -- 103 Bo cm/sec. Solar

magnetic fields appear to be of the order of 1 or 2 gauss at the surface, withmuch larger values around sunspots . For comparison, the velocity ofsound is of the order of 106 cm/see in both the photosphere and thechromosphere .

* The determination of the characteristics of the waves for arbitrary direction ofpropagation is left to Problem 10.3 .


AA )

1k

(a) (b)

Fig. 1012 Magnetohydrodynamic waves .

The magnetic fields of these different waves can be found from the thirdequation in (10.68) :

kviBo(0

B 1 = 0

- ~ Boil

for k lBO

for the longitudinal k 11 Bo

for the transverse k 11 Bo

(10.75)

The magnetosonic wave moving perpendicular to Bo causes compressionsand rarefactions in the lines of force without changing their direction, asindicated in Fig. 10 .12a. The Alfven wave parallel to BO causes the lines offorce to oscillate back and forth laterally (Fig . 10.12b). In either case thelines of force are "frozen in" and move with the fluid .

If the conductivity of the fluid is not infinite or viscous effects are present,we anticipate dissipative losses and a consequent damping of oscillations .The second and third equations in (10.68) are modified by additional terms :

ail _ _ S2 0P I - 10at

-Po 4~aBz =

V x (v1 x Bo) +at

(V x sl) + ?I vwl

c2 V

4~ra 2B1

(10 .76)

where n is the viscosity* and a is the conductivity . Since both additions

cause dispersion in the phase velocity, their effects are most easily seenwhen a plane wave solution is being sought . For plane waves it is evident

* Use of the simple viscous force (10 .3) is not really allowed for a compressible fluid .But it can be expected to give the correct qualitative behavior .

[Sect . 10.8] Magnetohydrodynamics and Plasma Physics

that these equations are equivalent to

Po IV, 2 ~-S 2 VPl - l!" x (V X B l )ar k 4~(1+~~1

P ow

?B i2 2

V x (v i x Bo)

47rdCU

333

(10 . 77)

Consequently equation (10.72) relating k and co is modified by (a) multi-

plying s2 and w2 by the factor 1 + i and2

,and (b) multiplying w2 by2 4~raco1

the factor ~ 1 + i ~k ~Poo'~

For the important case of th erelation between co and k becomes

k2uA 2 = CO 2(i +

Alfven wave parallel to the field, th e

C2k

2

+ ink 2

~ (1 0.78)4~ro`~ ~ Polo

If the resistive and viscous correction terms are small, the wave number isapproximately

kW

X+ l 23 ( ,2

+

~ ~ (10.79)VA 2vA 4,r6 Po

This shows that the attenuation increases rapidly with frequency (or wavenumber), but decreases with increasing magnetic field strength . In termsof the diffusion time r of Section 10.3, the imaginary part of the wavenumber shows that, apart from viscosity effects, the wave travels for a time

T before falling to 1/e of its original intensity, where the length parameterin -r (10.12) is the wavelength of oscillation . For the opposite extreme inwhich the resistive and/or viscous terms dominate, the wave number isgiven by the vanishing of the two factors on the right-hand side of (10 .78) .

Thus k has equal real and imaginary parts and the wave is damped outrapidly, independent of the magnitude of the magnetic field .

The considerations of magnetohydrodynamic waves given above arevalid only at comparatively low frequencies, since the displacement

current was ignored in Ampere's law . It is evident that, if the frequency ishigh enough, the behavior of the fields must go over into the "ionospheric"

behavior described in Section 7 .9, where charge-separation effects play an

important role. But even when charge-separation effects are neglected in

the magnetohydrodynamic description, the displacement current modifiesthe propagation of the Alfven and magnetosonic waves . The form of


Ampere's law, including the displacement current, is :

C c , at(10.80)

where we have used the infinite conductivity approximation (10.9) ineliminating the electric field E. Thus the current to be inserted into theforce equation for fluid motion is no w

j =a

[17 x B + 1~ -a (v x B) (10 . 81)atIn the linearized set of equations (10 .68) the second one is then generalizedto read :

Po~vI +2 WA x (~vl x v

A~~ -s2Vpl - B° X (D

XBl) (10 .82)at C at ~ 47T

This means that the wave equation for v1 is altered to the form :

~ .~" v 2(VA - Yl) l "_ S2

Q(V _X1

)2 [V1 ( 1 + %2 Jat c ~ C

+vAxVxVx(v1xvA)=0 (10 .83)

Inspection shows that for vl parallel to vA (i .e., Bo) there is no change frombefore . But for transverse vl (either magnetosonic with k perpendicularto Bo, or Alfven waves with k parallel to Bo) the square of the frequency ismultiplied by a factor [I + {TJ1421c2}] . Thus the phase velocity of Alfvenwaves becomes

uA -~ cvA(10.84)

~'/C2 + vA 2

In the usual limit where vA < c, the velocity is approximately equal to vA ;the displacement current is unimportant . But, if vA > c, then the phasevelocity is equal to the velocity of light . From the point of view of electro-magnetic waves, the transverse Alfven wave can be thought of as a wavein a medium with an index of refraction given b y

cuA = -

nThu s

2 2

n~ =1 -+-v~ = t +4 8 2A ~

(10.85)

(10 .86)

Caution must be urged in using this index of refraction for the propagationof electromagnetic waves in a plasma . It is valid only at frequencies wherecharge-separation effects are unimportant .

[Sect . 10.9] Magnetohydrodynamics and Plasma Physics 335

10.9 High-Frequency Plasma Oscillation s

The magnetohydrodynamic approximation considered in the previoussections is based on the concept of a single-component, electrically neutralfluid with a scalar conductivity a to describe its interaction with theelectromagnetic field . As discussed in the introduction to this chapter aconducting fluid or plasma is, however, a multicomponent fluid withelectrons and one or more types of ions present. At low frequencies orlong wavelengths the description in terms of a single fluid is valid becausethe collision frequency v is large enough (and the mean free path shortenough) that the electrons and ions always maintain local electricalneutrality, while on the average drifting in opposite directions accordingto Ohm's law under the action of electric fields . At higher frequencies thesingle-fluid model breaks down . The electrons and ions tend to moveindependently, and charge separations occur . These charge separationsproduce strong restoring forces . Consequently oscillations of an electro-

static nature are set up . If a magnetic field is present, other effects occur .

The electrons and ions tend to move in circular or helical orbits in themagnetic field with orbital frequencies given by

(Og = eB(10 .87)

me

When the fields are strong enough or the densities low enough that theorbital frequencies are comparable to the collision frequency, the conceptof a scalar conductivity breaks down and the current flow exhibits amarked directional dependence relative to the magnetic field (see Problem

10.5) . At still higher frequencies the greater inertia of the ions implies thatthey will be unable to follow the rapid fluctuation of the fields . Only theelectrons partake in the motion . The ions merely provide a uniform back-ground of positive charge to give electrical neutrality on the average . The

idea of a uniform background of charge, and indeed the concept of anelectron fluid, is valid only when we are considering a scale of length whichis at least large compared to interparticle spacings (l > rya-!) . In fact, thereis another limit, the Debye screening length, which for plasmas at reasonabletemperatures is greater than no", and which forms the actual dividingline between small-scale individual-particle motion and collective fluidmotion (see the following section) .

To avoid undue complications we consider only the high-frequencybehavior of a plasma, ignoring the dynamical effects of the ions . We alsoignore the effects of collisions . The electrons of charge e and mass m are


described by a density n(x, t) and an average velocity v(x, t) . The equi-librium-charge density of ions and electrons is Teno. The dynamicalequations for the electron fluid are

anat

(10.88)

at m C rrin

where the effects of the thermal kinetic energy of the electrons are describedby the electron pressure p (here assumed a scalar) . The charge and

current densities are :Pe = e(n - no)

J = env

Thus Maxwell's equations can be writte n

DV . E = 47re(n - no)

v .B - o

v ~E+a~ 0~

D x B_laE_47rert ~

at

(10.89)

( .10.90)

We now assume that the static situation is the electron fluid at rest with

n = no and no fields present, and consider small departures from thatstate due to some initial disturbance_ The linearized equations of motionare

at+nov •v= 0

ayT e E+ 1 (LP } On = 0

at m mno an o

V • E --- 47ren = 0

V XB 1aE 47rena V = a

cat c

(10.91)

plus the two homogeneous Maxwell's equations . Here n(x, t) and v(x, t)represent departures from equilibrium . If an external magnetic field BDis present a [(v/e) x Bo] term must be kept in the force equation (seeProblem 10 .7), but the fluctuation field B is of first order in small quantitiesso that (v x B) is second order . The continuity equation is actually not

[Sect . 10.9] 1ylagnetohydrodynamics and Plasma Physics 337

an independent equation, but may be derived by combining the last twoequations in (10.91).

Since the force equation in (10 .91) is independent of magnetic field, wesuspect that there exist solutions of a pure ly electrostatic nature, withB = 0. The continuity and force equations can be combined to yield awave equation for the density fluctuations :

a2 2 + (4e2u\ n - 1 (a) ~2n = 0 (10.92at M I m an o ~

On the other hand, the time derivative of Ampere's law and the forceequation can be combined to give an equation for the fields :

a2E + (42hb0)E -- 1 (E) D (D - E) = cV x (10.93)at m m ano at

The structures of the left-hand sides of these two equations are essentiallyidentical. Consequently no inconsistency arises if we put aBl at = 0 .Having excluded static fields already, we conclude that B = 0 is a possi-bility. If aBlat = 0, then Faraday's law implies V x E = 0. Hence E isa longitudinal field derivable from a scalar potential. It is immediatelyevident that each component of E satisfies the same equation (10.92) as thedensity fluctuations . If the pressure term in (10.92) is neglected, we findthat the density, velocity, and electric field all oscillate with the plasmafrequency cuP

COP 2 = 47rnaea

(10.94)M

If the pressure term is included, we obtain a dispersion relation for thefrequency :

CU2 = C11 V 2 +1 (LP) k2 (10.95)m an o

The determination of the coefficient of k2 takes some care . The adiabaticlaw p = po(n f no)Y can be assumed, but the customary acoustical valuey = 3 for a gas of particles with 3 external, but no internal, degrees offreedom is not valid. The reason is that the frequency of the presentdensity oscillations is much higher than the collision frequency, contraryto the acoustical limit. Consequently the one-dimensional nature of thedensity oscillations is maintained. A value of y appropriate to I trans-lational degree of freedom must be used . Since y = (m + 2)/m, where mis the number of degrees of freedom, we have in this case y = 3 . Then

1 -^ 3 p0 (10 .96)

m(LP

) an o m no


If we use po = naKTand define the rms velocity component in one direction(parallel to the electric field),

m (0) = KT

then the dispersion equation can be writte n

w2 = w p2 + 3 (u2 )k2

(10.97)

(10.98)

This relation is an approximate one, valid for long wavelengths, and isactually just the first two terms in an expansion involving higher andhigher moments of the velocity distribution of the electrons (see Problem10 .6). In form (10.98) the dispersion equation has a validity beyond theideal gas law which was used in the derivation . For example, it applies toplasma oscillations in a degenerate Fermi gas of electrons in which all cellsin velocity space are filled inside a sphere of radius equal to the Fermivelocity VF. Then the average value of the square of a component ofvelocity is

~ U2 ) = 5 VF' (10.99)

Quantum effects appear explicitly in the dispersion equation only in higher-order terms in the expansion in powers of V.

The oscillations described above are longitudinal electrostatic oscilla-tions in which the oscillating magnetic field vanishes identically. Thismeans that they cannot give rise to radiation in an unbounded plasma .There are, however, modes of oscillation in a plasma which are transverseelectromagnetic waves . To see the various possibilities of plasma oscil-

lations we assume that all variables vary as exp (ik - x -- iwt) and look fora defining relationship between c) and k, as we did for the magnetohydro-dynamic waves in Section 10 .8 . With this assumption the linearizedequations (10.91) and the two homogeneous Maxwell's equations can bewritten :

k•vn na

co

ieE + 3(u2) nk

M60 cc) no

k • E = - i4 7 ren

k - B = O

k xB - - `~'Ec

k xE = BC

4Trenai v

c

(10 .100)

[Sect. 10.10 1 Magnetohydrodynamics and Plasma Physics

Maxwell's equations can be solved for v in terms of k and E :

V =( ie ) 1

2

E(., - e 2k2)E + C2(k - E)k]1"110 CU p

339

(10.101)

Then the force equation and the divergence of E can be used to eliminate vin order to obtain an equation for E alone :

(w2 - wp2 - C2]Z2)E + (G2 _" 3 ~Il2))(k • E)k = 0 ( 1 0 . 102)

If we write E in term s of components parallel and perpendicular to k :

E = Ell +El ~

where E ~k - E)k (10 . 13)

Ell = ~ k2

then (10.102) can be written as two equations :

(C ) 2 - (Up2 - 3 (il2)IG2)Ell = 0

(GU2 - W p2 - C2k2)E1 =

(10. 104)

The first of these results shows that the longitudinal waves satisfy thedispersion relation (10.98) already discussed, while the second shows thatthere are two transverse waves (two states of polarization) which have thedispersion relation :

0J 2 C.J p2 + C 2k2 (10 . 105)

Equation (10.105) is just the dispersion equation for the transverseelectromagnetic waves described in Section 7.9 from another point of view.In the absence of external fields the electrostatic oscillations and the trans-

verse electromagnetic oscillations are not coupled together . But in thepresence of an external magnetic induction, for example, the force equa-tion has an added term involving the magnetic field and the oscillationsare coupled (see Problem 10 .7) .

10.10 Short-Wavelength Limit for Plasma Oscillations and theDebye Screening Distanc e

In the discussion of plasma oscillations so far no mention has been madeof the range of wave numbers over which the description in terms of

collective oscillations applies . Certainly no is one upper bound on the

wave-number scale . A clue to a more relevant upper bound can beobtained by examining the dispersion relation (10.98) for the longitudinaloscillations. For long wavelengths the frequency of oscillation is very


closely co = wp. It is only for wave numbers comparable to the Dehyewave number kD,

kD2 = 2> (10.106)u

that appreciable departures of the frequency from co, occur .For wave numbers k C kD, the phase and group velocities of the

longitudinal plasma oscillations are :

VDWV

k (10.107)

V 3(U

2)

8V P

From the definition of kD we see that for such wave numbers the phasevelocity is much larger than, and the group velocity much smaller than,

the rms thermal velocity As the wave number increases towardskD, the phase velocity decreases from large values down towards (uz)%4 .Consequently for wave numbers of the order of kD the wave travels witha small enough velocity that there are appreciable numbers of electronstraveling somewhat faster than, or slower than, or at about the same speedas, the wave . The phase velocity lies in the tail of the thermal distribution .The circumstance that the wave's velocity is comparable with the electronicthermal velocities is the source of an energy-transfer mechanism whichcauses the destruction of the oscillation . The mechanism is the trappingof particles by the moving wave with a resultant transfer of energy out ofthe wave motion into the particles . The consequent damping of the waveis called Landau damping.

A detailed calculation of Landau damping is out of place here . But wecan describe qualitatively the physical mechanism. Fig. 10.13 shows a

distribution of electron velocities with a certain rms spread and aMaxwellian tail out to higher velocities . For small k the phase velocity

VPFig. 10 .13 Thermal velocity

distribution of electrons .

[Sect. 10.10] Magnetohydrodynamics and Plasma Physics 341

lies far out on the tail and negligible damping occurs . B ut as k ~ kD thephase velocity lies within the tail, as shown in Fig . 10.13, w ith a significantnumber of electrons having thermal - speeds comparable to vim. There isthen a velocity band Av around v = u, where electrons are movingsufficiently slowly re lative to the wave that they can be trapped in thepotential troughs and carried along at velocity vp by the wave . If there aremore particles in Dv moving initially slower than vP than there areparticles moving faster (as shown in the figure), the trapping process wil lcause a net increase in the energy of the particles at the expense of thewave . This is the mechanism of Landau damping . Detailed calculationsshow that the damping can be expressed in terms of an imaginary part ofthe frequency given by

e - (kD212k2 )(10 .10g)Im Lc) ' -CUp ~

(Lk3

J

8 providedk C kD. To obtain (10.10$) a Maxwellian distribution o fvelocities was assumed . For k ;z3 kD the damping constant is larger thangiven by (10.108) and rapidly becomes much larger than the real part ofthe frequency, as given by (1 0 .98) .

The Landau formula (10.108) shows that for k < kD the longitudinalplasma oscillations are virtually undamped . But the damping becomesimportant as soon as k , kD (even for k = O.SkD, Im co - -0 .7c),) .

For wave numbers larger than the Debye wave number the damping is sogreat that it is meaningless to speak of organized oscillations .

Another, rather different consideration leads to the same limiting Debyewave number as the boundary of collective oscillatory effects . We knowthat an electronic plasma is a collection of electrons with a uniform back-ground of positive charge . On a very small scale of length we mustdescribe the behavior in terms of a succession of very many two-bodyCoulomb collisions . But on a larger scale the electrons tend to cooperate .If a local surplus of positive charge appears anywhere, the electrons rushover to neutralize it. This collective response to charge fluctuations iswhat gives rise to large-scale plasma oscillations. But in addition to, or,better, because of, the collective oscillations the cooperative response ofthe electrons also tends to reduce the long-range nature of the Coulombinteraction between particles . An individual electron is, after all, a localfluctuation in the charge density . The surrounding electrons are repelledin such a way that they tend to screen out the Coulomb field of the chosen

electron, turning it into a short-range interaction . That something likethis must occur is obvious when one realizes that the only source ofelectrostatic interaction is the Coulomb force between the particles . Ifsome of it is effectively taken away to cause long-wavelength collective


plasma oscillations, the residue must be a sum of short-range interactionsbetween particles .

A nonrigorous derivation of the screening effect described above wasfirst given by Debye and Mickel in their theory of electrolytes . The basicargument is as follows. Suppose that we have a plasma with a distributionof electrons in thermal equilibrium in an electrostatic potential (D. Thenthey are distributed according to the Boltzmann factor e-'IKT where Histhe electronic Hamiltonian. The spatial density of electrons is therefor e

n(x) ~ no e- (e~JKT)(10.109)

Now we imagine a test charge Ze placed at the origin in this distributionof electrons with its uniform background of positive ions (charge density-eno). The resulting potential (D will be determined by Poisson's equatio n

V(D = -4 -aZe 8(x) - 47rena[8-[e(D/KT) - 1]~{ (10.110)

If {e(DIK7) is assumed small, the equation can be linearized :

020 - IG2DO = -47TZe fi(x)where 2

= 47rnoe2K (10.112)k~KT

is an alternative way of writing (10 .106) . Equation (10.111) has thespherically symmetric solution :

e-xDT

r

showing that the electrons move in such a way as to screen out the Coulombfield of a test charge in a distance of the order of kD 1 . The balance betweenthermal kinetic energy and electrostatic energy determines the magnitudeof the screening radius . Numerically

~,kD-~ = 591

(T~ cm (10.114)

no g

where T is in degrees Kelvin, and no is the number of electrons per cubiccentimeter. For a typical hot plasma with T = lOs°K and no = 10 15 cm --- 3 ,we find kDi ^' 2.2 x 10-1 cm.

For the degenerate electron gas at low temperatures the Debye wavenumber kD is replaced by a Fermi wave number kF :

kp -0' 13 (10.115)

VF

where VF is the velocity at the surface of the Fermi sphere . This magni-tude of screening radius can be deduced from a Fermi-Thomas generaliza-tion of the Debye-Hiicket approach . It fits in naturally with the dispersionrelation (10 .98) and the mean square velocity (10.99) .

[Probs. 101 Magnetohydrodynamics and Plasma Physics 343

The Debye-Huckel screening distance provides a natural dividing linebetween the small-scale collisions of pairs of particles and the large-scalecollective effects such as plasma oscillations. It is a happy and notfortuitous happening that plasma oscillations of shorter wavelengths canindependently be shown not to exist because of severe damping .

REFERENCES AND SUGGESTED READING

The subject of magnetohydrodynamics and plasma physics has a rapidly growingliterature . Many of the available books are collections of papers presented at confer-

ences and symposia . Although these are useful to someone who has some knowledge ofthe field, they are not suited for beginning study . Two works on magnetohydrodynamicswhich are coherent presentations of the subject ar e

Alfven,Cowling .

A short discussion of magnetohydrodynamics appears i nLandau and Lifshitz, Electrodynamics of Continuous Media, Chapter VIII .

Corresponding books devoted mainly to the physics of plasmas areChandrasekhar ,Linhart,Simon,Spitzer .

The subject of controlled thermonuclear reactions, with much material on the funda-

mental physics of plasmas, is treated thoroughly byGlasstone and Lovberg ,Rose and Clark.

PROBLEMS

10.1 An infinitely long, solid, right circular, metallic cylinder has radius (R/2)and conductivity 6. It is tightly surrounded by, but insulated from, a hollowcylinder of the same material of inner radius (R/2) and outer radius R .Equal and opposite total currents, distributed uniformly over the cross-sectional areas, flow in the inner cylinder and in the hollow outer one .At t = 0 the applied voltages are short-circuited.

(a) Find the distribution of magnetic induction inside the cylindersbefore t = 0 .

(b) Find the distribution as a function of time after t = 0, neglectingthe displacement current .

(c) What is the behavior of the magnetic induction as a function of timefor long times? Define what you mean by long times .

10 .2 A comparatively stable self-pinched column of plasma can be produced bytrapping an axial magnetic induction inside the plasma before the pinchbegins. Suppose that the plasma column initia lly fills a conducting tube ofradius Ro and that a uniform axial magnetic induction Bzo is present in the


tube. Then a voltage is applied along the tube so that axial currents flowand an azimuthal magnetic induction is built up .

(a) Show that, if quasi-equilibrium conditions apply, the pressure-balancerelation can be written :

[p(r) +8ir + $~r 2 T2 + 47r r2

Br 2 Ali" = 4rl fr ,

(h) If the plasma has a sharp boundary and such a large conductivitythat currents flow only in a thin layer on the surface, show that for aquasi-static situation the radius R(t) of the plasma column is given by theequation

R ° In R = t f (t) dt0 0

where

to = BzoRoo cE0

and Eo f (t) is the applied electric field .(c) If the initial axial field is 100 gauss, and the applied electric field has

an initial value of 1 volt/cm and falls almost linearly to zero in 1 millisecond,determine the final radius if the initial radius is 50 cm . These conditions areof the same order of magnitude as those appropriate for the British toroidalapparatus (Zeta), but external inductive effects limit the pinching effect toless than the value found here . See E . P. Butt et al., Proceedings of theSecond International Conference on Peaceful Uses of Atomic Energy, Vol. 32,p. 42 (1958) .

10.3 Magnetohydrodynamic waves can occur in a compressible, nonviscous,perfectly conducting fluid in a uniform static magnetic induction Bo. If thepropagation direction is not parallel or perpendicular to Ba, the waves arenot separated into purely longitudinal (magnetosonic) or transverse (Alfven)waves . Let the angle between the propagation direction k and the field B0be 0 .

(a) Show that there are three different waves with phase velocities given b y

u 12 ~(VA COS 6)2

2.3 - 2 52+ VA2) ±

2L(S

'2+ VA

2)2 - 4S2'UA2 COO 01'/2

where s is the sound velocity in the fluid, and VA= (Ba2/4rpo)% is theAlfven velocity .

(h) Find the velocity eigenvectors for the three different waves, andprove that the first (AIfven) wave is always transverse, while the other twoare neither longitudinal nor transverse .

(c) Evaluate the phase velocities and eigenvectors of the mixed waves inthe approximation that vA > s. Show that for one wave the only appreciablecomponent of velocity is parallel to the magnetic field, whi le for the otherthe only component is perpendicular to the field and in the plane containingk and Bo .

10.4 An incompressible, nonviscous, perfectly conducting fluid with constantdensity po is acted upon by a gravitational potential V and a uniform, static,magnetic induction Bo .

[Probs . 10] Magnetohydrodynamics and Plasma Physics 345

(a) Show that magnetohydrodynamic waves of arbitrary amplitude andform B l(x, t), v(x, t) can exist, described by the equation s

A - V)$

I \/4~F 0

aB 1

at

Bl = f ✓4Tpo v

+ +(B°

B1)2 = constantP Poi'

(b) Suppose that at t = 0 a certain disturbance Bi(x, 0) exists in thefluid such that it satisfies the above equations with the upper sign . Whatis the behavior of the disturbance at later times ?

10.5 The force equation for an electronic plasma, including a phenomenologicalcollision term, but neglecting the hydrostatic pressure (zero temperatureapproximation) is

av e(

~

where v is the collision frequency.(a) Show that in the presence of static, uniform, external, electric, and

magnetic fields, the linearized steady-state expression for Ohm's lawbecomes

where the conductivity tensor is

Wp

2

~ -

f wB

WB 1wB2 )

v2

Q 0

0

0

B2

v2(+ )w

and cup(wB) is the electronic plasma (precession) frequency . The directionof B is chosen as the z axis .

(b) Suppose that at t = 0 an external electric field E is suddenly appliedin the x direction, there being a magnetic induction B in the z direction .The current is zero at t = 0 . Find expressions for the components of thecurrent at all times, including the transient behavior .

10 .6 The effects of finite temperature on a plasma can be described approxi-mately by means of the correlationless Boltzmann (Vlasov) equation . Letf (x, it, t) be the distribution function for electrons of charge e and mass inin a one-eq*_nponent plasma. The Vlasov equation is

d~ ~ +v• Pxf+a• ~7v.f= 4

where Px and Dv are gradients with respect to coordinate and velocity, anda is the acceleration of a particle . For electrostatic oscillations of the


plasma a = eEJm, where E is the macroscopic electric field satisfyin g

D • E = 47reIfr f f (x, v, t) d3v - na

]

If fo(v) is the normalized equilibrium distribution of electron s

[no f fo(v) d-3v = no]

(a) show that the dispersion relation for small-amplitude longitudinalplasma oscillations is

k2 = 1 kvvA CI3Uc.

) P2 J k • v - co

(b) assuming that the phase velocity of the wave is large compared tothermal velocities, show that the dispersion relation gives

a) 22,,1 +2

(k~ v) +3<

(k~2

)

2 + . . .p

where ( ) means averaged over the equilibrium distribution fo(v) . Relatethis result to that obtained in the text with the electronic fluid model .

(c) What is the meaning of the singularity in the dispersion relation whenk•v = co ?

10.7 Consider the problem of waves in an electronic plasma when an externalmagnetic field B. is present . Use the fluid model, neglecting the pressureterm as well as collisions .

(a) Write down the linearized equations of motion and Maxwell'sequations, assuming all variables vary as exp (ik • x - rwt) .

(b) Show that the dispersion relation for the frequencies of the differentmodes in terms of the wave number can be written

C02(f/J 2 - C11P

2)(GJ2 - WPB - It2 C2)2

= wB2(CU 2 - k2C 2)

0002(w2 - (u V2 - k2c2) + (1Jp 2C Z (k b)2 1

where b is a unit vector in the direction of So ; co,, and wj3 are the plasmaand precession frequencies, respectively.

(c) Assuming WB < wp, solve approximately for the various roots forthe cases (i) k parallel to b, (ii) k perpendicular to b . Sketch your resultsfor w2 versus k2 in the two cases .

magnetohydrodynamics nd plasma physics - pku · the distinction between magnetohydrodynamics and...

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