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Nonlinear oscillations in a cold plasma

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1968 Nucl. Fusion 8 183

(http://iopscience.iop.org/0029-5515/8/3/006)

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NUCLEAR FUSION 8 (1968)

NONLINEAR OSCILLATIONS IN A COLD PLASMA*

R.W.C. DAVIDSON** AND P.P.J .M. SCHRAM***

DEPARTMENT OF PHYSICS, UNIVERSITY OF CALIFORNIA,

BERKELEY, CALIFORNIA, UNITED STATES OF AMERICA

ABSTRACT. In a Maxwell-fluid description large-amplitude electrostatic and electromagnetic oscillations in a coldplasma are analysed in situations where the spatial variations are one-dimensional and the ions form a fixed neutralizingbackground. In the electrostatic approximation a Lagrangian description following the electron motion is adopted, and exactsolutions obtained in these variables in situations where multistream flow does not develop. The inversion to Eulerian co-ordinates is carried out explicitly for the particular example of an initial sinusoidal perturbation in density. A model describ-ing the dispersive modifications of (weak) thermal effects is analysed showing the phase-mixing of (moderately) large-amplitudeinitial disturbances. After sufficient time, the electron fluid becomes stationary and a low level of electric field remainsbalancing the force due to pressure variations. In addition, for the case of nonlinear electrostatic oscillations perpendicularto a static magnetic field, the exact solution in Lagrangian variables shows that for a cold plasma coherent oscillations at theupper hybrid frequency are maintained indefinitely over the regions of initial excitation. Electromagnetic oscillations are con-sidered as developing from small initial values on the background of the large-amplitude longitudinal oscillations already cal-culated. The resulting wave equation is analysed for several limiting cases. It is shown that for sufficiently short wavelengthsof the transverse fields there exists an infinite number of narrow ranges of the wave number in which the transverse oscillationsare unstable. The growth is ultimately limited by the magnetic force which was neglected in the description of the longi-tudinal motion. Finally, stationary solutions are studied. In the electrostatic approximation a very special class of periodicBernstein-Greene-Kruskal waves without trapped particles is obtained. Again the electromagnetic solutions are unstable (inspace), the growth being limited by the same effect as in the time-dependent problem. A simple class of special stationarysolutions is obtained for the complete problem including all magnetic force terms.

1. INTRODUCTION

It is the purpose of the present article to con-sider in some detail the problem of large-amplitudeelectrostatic and electromagnetic oscillations in acold plasma in situations where the spatial varia-tions are one-dimensional, and the ions form afixed neutralizing background. The nonlinearelectrostatic problem of Dawson [l] and Kalman[2] is briefly formulated in 2.1 introducingLagrangian variables following the electron fluid,and the exact solutions for relevant physical quan-tities are obtained in those variables. Through-out the present paper, analysis is restricted toinitial-value problems for which multistream flowdoes not develop. In 2. 2 a particular example isconsidered in which the electron fluid is initiallyat rest, and the electron density has the form

n(xo,O) = no(l + A coskx0) (1.1)

where no is the uniform ion background density.The inversion to Eulerian variables is carried outexplicitly and exact nonlinear expressions for the

* Research supported in part by the National Aeronautics andSpace Administration, Grant No.NGR05-003-220; by Euratom, NationalScience Foundation Grant GP4454; and by the Air Force Office ofScientific Research, Office of Aerospace Research, under Grant No.AF-AFOSR-130-66.

* * Present address: Department of Physics and Astronomy,University of Maryland, College Park, Maryland 20740, United Statesof America.

* * * Present address: Department of Mechanical and AerospaceSciences, University of Rochester, Rochester, New York, 14627, UnitedStates of America.

velocity, electric field, and density in the labora-tory frame are obtained. The relation betweenthese solutions and the solution to the Vlasov equa-tion with appropriate (cold) initial conditions isbriefly discussed.

Various extensions of the electrostatic problem,and the inclusion of electromagnetic effects, areconsidered in sections 3 and 4, respectively. In3. 1 the analysis of 2. 1 is extended to describelarge-amplitude electrostatic oscillations perpen-dicular to a static magnetic field, and the exactsolution is obtained in Lagrangian variables. Inthis case, coherent oscillations at the upper hybridfrequency are maintained indefinitely over the re-gion of initial excitation and remain local to thatregion. A model relevant to describing the dis-persive effects of small, but finite, plasma tem-perature is analysed in 3.2. Coherent oscillationsat the plasma frequency are no longer maintainedfor all time as in the cold plasma considerationsof 2. 1. It is shown that for absolutely integrableinitial disturbances the electron velocity phasemixes to zero; however, a low level of electricfield, balancing the force due to pressure varia-tions, remains in the time-asymptotic limit. In3. 3 the analysis of 2. 1 is modified by the inclu-sion of a collisional drag term in the equation ofmotion for the electron fluid. For arbitrary,large-amplitude, initial disturbances (which donot lead to multistream flow), this dissipationleads to a uniform, field-free, stationary, equi-librium. For clarity, the various generalizationsof the electrostatic problem considered in section 3have been treated separately; it should be noted,however, that nothing restricts the effects of sta-tic magnetic field,, dispersion and dissipation frombeing included in a single Lagrangian analysis.

183

DAVIDSON and SCHRAM

In section 4 the development of transverse fieldsis considered in detail. The solutions of the elec-trostatic problem enter as known coefficients de-pending on space and time in the wave equation.Several limiting cases are treated. In one of these,the solutions are unstable for an infinite numberof narrow ranges of the transverse wave number.The growth-rate is calculated explicitly for a spe-cific example. This nonlinear instability is limit-ed by the influence of the magnetic force whichwas neglected in the electrostatic approximationof the preceding sections. The total energy in thetransverse oscillations is bounded by the totalamount of energy in the initial electrostatic andelectromagnetic perturbations.

In section 5 the stationary solutions are studied.In the electrostatic approximation the cold plasmaform of the Bernstein-Greene-Kruskal waves isderived. These are periodic in space and the re -quirement of uniqueness of the Lagrangian t rans-formation is shown to imply the absence of trappedparticles. It is also shown that the oscillations ofsection 2 never take the form, in space, of thesestationary solutions.

The transverse fields are unstable in space.The growth is again limited by the magnetic forcein the longitudinal problem. Finally, an exactclass of solutions is obtained for the completeproblem including all magnetic force terms, anda specific example is considered in detail.

2. ONE-DIMENSIONAL ELECTROSTATICOSCILLATIONS IN A COLD PLASMA

Introducing the Lagrangian variables, (XO,T), asnew independent variables [3-5]

T = t

T

rX0 = X- / v(xo,T')dT'

d0

(2.5)

Equations (2. 1) - (2. 3) may be rewritten in the newvariables as

(2.6)

(2.7)— E(xo,T)=47renov(xo,T)

and

(2.8)

where Poisson1 s equation has been used in ob-taining Eq. (2. 7). The transformation, Eq. (2. 5),has the effect of replacing the convective deriva-tive, 3/3t + v9/3x, by the local time derivative,9 / 9 T . From Eqs (2.7) and (2.8), V(XO,T) has themotion of a simple harmonic oscillator, oscillat-ing at the plasma frequency, u0, i . e .

(j (2.9)

2 . 1 . SOLUTION IN LAGRANGIAN VARIABLES

Assuming that the ions form a fixed, uniformbackground, and that the plasma is cold, the one-dimensional Maxwell-fluid equations for the elec-tron density, n(x, t), electric field, E(x,t), andelectron velocity, v(x, t), read in the electrostaticapproximation:

|fn+|;(nv)=0

9t

_9_at

8x

E - 4?renv = 0

9 e TP

-r—V = E

9x mand

— = -47re(n-n0)

(2.1)

(2.2)

(2.3)

(2.4)

where no is the uniform ion density. Poisson1 sequation may be thought of as an initial value toEq. (2. 2). By virtue of the continuity equation andEq. (2. 2), Eq. (2.4) remains true for all times iftrue initially.

The general solutions to the system (2. 6) - (2. 8)are then simply

V ( X O , T ) = V(x0)cosuoT+tJoX(x0)sinu)0T (2. 10)

E(x0 , T) = —u0V(x0) sinw0T - — U6 vJ

COSU0T

(2.11)and

n(xo,r) =n(xo,O)

— T—V(xo)sinu>oT+^—X(xo)(l -IJQ CJXQ O X O

(2.12)

The functional dependence of V and X on x0 is re -lated to the initial velocity and electric field pro-files, v(x0,0) and E(xo,O), through

V(x o )=v(x o ,O) , and X(x0) = ^ E ( x 0 , 0) (2.13)0

In addition, X(x0) is related to the initial density,n(x0, 0), through Poisson1 s equation at T =0, viz.

184

NONLINEAR OSCILLATIONS

The co-ordinate transformation, Eq.(2.5), maynow be written as

T = t,

x = x o + ^ ^ s i n u 0 T + X ( x 0 ) ( l - c o s w 0 T ) (2. 15)

The transformation from Lagrangian to Eulerianco-ordinates, i . e . the determination of XQ as afunction of x and t from Eq. (2. 15), requires anexplicit specification of the initial conditions,V(xo) and X(x0), and in general entails the alge-braic solution to a transcendental equation. It isevident from the solutions, (2.10) - (2.12), how-ever, that coherent oscillations at the plasmafrequency, w0, are maintained indefinitely overthe region of initial excitation.

In addition, there is a restriction of the classof initial-value problems which may be treatedby the procedure just outlined. The condition thatthe density be non-negative initially, requires onlythat

n(xo,O) 5 0 (2.16)

However, asking that the solution for the densityas given by Eq. (2. 12) remain non-negative andfinite for all times gives the more restrictive con-dition

n(x o ,O)>n o /2 (2.17)

as well as

(2.18)

Equation (2. 14) has been used in deriving inequali-ties (2.17) and (2.18). Consequently, a lowerbound of no/2, not zero, is placed on the ampli-tude of the initial density perturbation. Also, therate of change of V(x0) with x0 is restr icted.These do not represent physical limitations on thecold plasma initial conditions, but rather mathe-matical limitations on the Lagrangian formalismthat has been used. Mathematically if condition(2.17) or (2.18) is violated for some rangeof XQ , the transformation from Lagrangian toEulerian co-ordinates, as determined from Eq.(2. 15), does not remain unique for all x and t .Physically, circumstances in which these condi-tions are violated for some range(s) of x0 lead tothe development of multistream flow within halfthe period of a plasma oscillation [ l ] . Considera-tions here, however, will be restricted to initial-value problems for which inequalities (2. 17) and(2.18) are satisfied.

2.2. EXAMPLE WITH INVERSION TOEULERIAN CO-ORDINATES

As a particular non-trivial example for whichthe inversion to Eulerian co-ordinates may becarried out explicitly, let us consider initial con-

ditions specified by a sinusoidal perturbation indensity

n(x 0 ,0)=n 0 ( l + Acoskx0), | A | < 1/2 (2.19)

and zero velocity

V(x0) = 0 (2.20)

The solutions (2. 10) -(2.12) may then be written

V(XO,T) = —2- AsinkxosinwoT (2.21)

E ( X O , T ) = UQ-T-si (2.22)

and

n(xo,T) = n01+ Acoskxp

1+Acoskx0(l - cosu>oT)(2.23)

In addition, the co-ordinate transformation, (2.15),becomes

T =t

kx =

where= 2Asin2 ^ 1

(2.24)

(2.25)

The condition that | A | < 1/2 ensures that | f2(T)| < 1and that the solution, xo(x, t), to Eq. (2. 24) issingle-valued.

The solution may be determined numericallyfrom Eq. (2. 24) at different times, and the cor-responding forms of n, v, and E, in Eulerian vari-ables deduced from Eqs (2. 21) - (2. 23). Forexample, taking A= 0.45, the density is shownas a function of x in Fig. 1, at successive quarter-

n(x,t)

= o,2ir

FIG.l. Density profiles with A = 0.45.

185

DAVIDSON and SCHRAM

periods of a plasma oscillation. The x and t de-pendences are periodic, with periods 2TT/1C and2TT/U0, respectively. The dense regions fill in therare regions in the first quarter-period (wot = 7r/2).As time goes on, the electrostatic forces are suchas to cause a "bunching" of electrons around

= (2n+l)7r; n = 0, ±1, ±2, (2.26)

reaching a maximum density of 5. 5n0 at uot = n.The dense peaks fill in the rare regions and thesystem reverts to its initial state. The corres-ponding velocity and electric-field profiles (notpresented here) exhibit a steepening of the initialwave form sinkx, without change in maximumamplitude.

We return to the problem of determining explicitanalytical expressions for n(x, t), v(x, t), andE(x,t), using the general procedure for inversionfrom Lagrangian to Eulerian co-ordinates dis-cussed by Jackson [6], It is evident from Eq.(2.24) that sin kxo and coskx0 are periodic func-tions of x with period 27r/k. Consequently, thedensity, electric field, and velocity as expressedin Eulerian co-ordinates, possess Fourier-seriesrepresentations in x. For example, v and E eachcontain the factor sinkxo(x, t), which may bewritten as

sinkx

where

o(x, t) = ) an(t)sinnkx (2.27)

n = l

2ir

k fn(t) = — / dx sinnkx sin kxo (x, t) (2.28)IT {J

The integration may be carried out using the relation between x and x0 prescribed by Eq. (2. 24),giving

where the Jn are Bessel functions of the first kind.The electron fluid velocity and electric field inEulerian variables are thus given by

v(x,t) =

and

(nfi(t)) sinnkx sinuot

(2.30)

X sinnkx cosuct

From Eqs (2.4) and (2.31) we obtain for thedensity

(2.3.1)

n(x,t) = ) ) c o s n k x

Equations (2.30) - (2.32) are thus exact solutionsto the initial-value problem (2.19) and (2. 20), anddemonstrate quite explicitly the distortion of waveforms as manifested through the generation ofhigher harmonic dependence on kx. In addition,the form of Eqs (2. 30) - (2. 32) is useful to providea convergent series representation of n, v and Ein powers of the amplitude, A, of the initial den-sity perturbation. The reader will also have notedthat the method of inversion used above is not lim-ited to the case in which n(x0, 0) has a simplesinusoidal dependence on x0, but may be general-ized to treat any initial density perturbation whichitself has a Fourier series representation in x0.

In addition, we remind the reader that the pre-ceding analysis determines the solution to theVlasov equation with self-consistent electric field,and initial value for the distribution function ofthe form

f(x, i/J0)=n0(l+ Acoskx)6(i/), (2.33)

The solution for all times may be written in theform

f(x, v, t) = n(x, t) d(y-v(x, t)) (2.34)

where n(x,t) and v(x, t) are given by Eqs (2.30)and (2. 32). The plasma which is initially cold,remains cold for all times in the sense that norandom motion relative to v(x,t) develops. Formore general initial conditions for the density,n(x, 0), and mean velocity v(x, 0), the solution forf(x, v, t) in the cold plasma problem may still bewritten in the form given by Eq. (2. 34), providedinequalities (2. 17) and (2.18) are satisfied andconsequently multistream flow does not develop.

3. GENERALIZATIONS OF THEELECTROSTATIC PROBLEM

3.1. NONLINEAR ZERO-TEMPERATUREBERNSTEIN MODES [7]

The analysis of section 2 may be generalized todescribe the nonlinear behaviour of large-amplitudeelectrostatic oscillations perpendicular to a uni-form, static magnetic field, So. With unit Car-tesian vectors (e\, %2> ^s)> w e take So along e3,and as before consider spatial variations in thex-direction (e*i). The electron velocity, v"(x, t),may be written as,

v(x,t)=v(x,t)?1+v2(x,t)ev2+v3(x,t)eV3 (3.1)

and the electric field, which is along the directionof spatial variations in the electrostatic approxi-mation, becomes

E(x,t) =E(x,t)e1 (3.2)

(2.32)The plasma is again assumed cold. Equations(2.1) and (2. 2) describing the evolution of E(x, t)

186

NONLINEAR OSCILLATIONS

andn(x.t), remain unchanged. However, theelectron velocity in the ^-direct ion now obeys

0 2 ; om ° l ° me

and the two additional degrees of motion satisfy

( 3 p 3 )

in Eulerian variables depends on the details of theinitial conditions chosen for the problem. How-ever, it is clear that coherent oscillations at theupper hybrid frequency are maintained for alltime in the region of initial excitation. In termsof the initial conditions, V(x0), v2(x0,0) andn(xo,0), the conditions that the transformationfrom Lagrangian to Eulerian co-ordinates beunique and that multistream flow does not develop,now become

and

_9_9t

(3 .5)

Defining Lagrangian co-ordinates following the x-motion by Eq. (2.5), Eqs (3.3)- (3.5) read in thenew variables

and

97V2 (XO,T) =nov(xo,

(3.6)

(3.7)

(3.8)

The system of equations to be solved in Lagrangianvariables now consists of (2. 6), (2. 7) and (3.6) -(3.8). From Eqs (2.7), (3.6) and (3.7), V(XOJT)is seen to have the motion of a simple harmonicoscillator, oscillating at the upper hybrid fre-quency, u ^ , i - e -

(3.9)

Consequently, the only modifications of the solu-tions (2. 10) and (2. 12) for V(XO,T) and n(xo,r) ,and the co-ordinate transformation (2. 15), is thatu0 is replaced by wUH. The quantities, v2(x0, T),E (X O ,T) and v3(x0, T), as determined from Eqs(3. 6) - (3. 8), are then given by

= v2(x0,0) + —W U H

sinwUHT

wUHX(x0)(l-cosuUHT) (3.10)

E(x0, T) = ̂ "T2- V(x0) sinwUHr - ^-u\ X(x0)cos WUHTe UH

(3.11)

and

V 3 ( X 0 , T ) = V 3 ( X 0 , 0 ) (3.12)

thus giving a complete description of the problemin Lagrangian variables. The explicit behaviour

and

(3.14)

Conditions (3. 13) and (3.14) reduce to the inequa-lities (2. 17) and (2. 18) for n0 - 0, as they should.If there is no initial shear in the electron velocity(8v2 (x0, 0)/3x0 = 0), conditions (3. 13) and (3. 14)become less and less restrictive with increasingmagnetic field strength.

3.2. THERMAL EFFECTS

Modifications due to dispersion resulting fromfinite plasma temperature may be described byincluding the force due to pressure variations inEq.(2.3), i . e .

m nm

The evolution of the electron pressure, P, as de-termined from the appropriate moments of theVlasov equation, is given by

(3.16)

in circumstances where the electron heat flowmay be neglected [8]. This corresponds to theapproximation, (uo/k) » vTH, where l /k is the(typical) length scale of the disturbance beingstudied, and vTH is the electron thermal speed.In Lagrangian variables defined by Eq. (2.5),Eqs (2.6) and (2.7) remain unchanged with inclu-sion of pressure effects; however, the equationfor V(XO,T) is now given by

— V(XO,T) = E(XO,T) ; — P(XO, T)

9 T \ 0' / m v ° ' n(x0,0)m 9x0 v ° '

(3.17)where

(3.18)

187

DAVIDSON and SCHRAM

Differentiating Eq. (3. 17) with respect to T, wethen have that

V(XO,T)(3.19)

where

(3.23)

The quantity, V(k0), is the Fourier transform ofthe initial velocity perturbation, v(x0, 0), andX(k0) is related to the transform of the initialelectric field by

u2(k0)X(k0) = - - (3.24)

The solution to Eq. (3.19) is not mathematicallytractable except within some additional approxi-mation scheme; we discuss only one of these here.It should be noted that in an order of magnitudeestimate the thermal effects are smaller by a fac-tor k2v2,H/u2 relative to the other terms in Eq.(3.19). For purposes of a qualitative description,the model we adopt is one in which the final termin Eq. (3. 19) is approximated by its linearizedversion. In particular, for the case of an (initial-ly) uniform pressure PQ, Eq.(3. 19) becomes

^ V ( X O . r) (3.20)

where

>2 - (3.21)

The remaining part of 3. 2 will be directed towardsthe use of Eqs (3. 20), and (2. 5) - (2. 7) as a modeldescribing the evolution of electrostatic oscilla-tions in a plasma including thermal effects. Anyconclusions drawn must be interpreted within thismodel. Strictly speaking, Eq.(3.20) is applicableonly in a small-amplitude analysis. However, itmay be expected to give qualitatively correct be-haviour even for moderately large initial ampli-tudes provided steep spatial gradients do not de-velop in the course of time. For initial values ofthe form of a single monochromatic wave of wavenumber k', the analysis goes through as in 2. 2,the essential modification being a slight frequencyshift from u0 to uo(l + Zk2\'%} )

1/2 . However, forinitial conditions which are absolutely integrable,the long time solution to Eq. (3. 20) has quite dif-ferent behaviour than for the case of a cold plas-ma. The analysis of section 2 indicated that co-herent plasma oscillations are maintained inde-finitely in the region of initial excitation, where-as, in relation to Eq. (3.20) the inclusion of thermaleffects leads to a dispersion of the disturbancethroughout the plasma. The solution to Eq.(3. 20)may be written in terms of its Fourier integralrepresentation as

V(XO,T)• / •

dkoelk°X° {V(k0)cosw(k0)T

A straightforward stationary phase analysis ofEq. (3. 22) indicates that the oscillatory termsphase-mix to zero as (1/T1/2) for large r [9], Forexample, the first term decays as

• ) V(k0 = 0) X (oscillation at frequency u 0)

(3.25)

for large T. Estimating V(k0 = 0) = VA where Vand A are the characteristic magnitude and (spa-tial) width, respectively, of the initial disturb-ance v(xo,O), we see that V(XO,T) decays to anegligible level compared to ? in a time t, where

WQX SS A /A.J-J \5. &o)

It should be noted that the time integrations overv(x0, T) involved in the transformation (2.5), andthe solution for n(xo,r) as determined from Eq.(2.6), have non-oscillatory portions which do notphase-mix to zero. The time asymptotic be-haviour is conveniently written in terms ofX(x0) and may be summarized as follows:

v(xo ,T -• oo) -* 0

n(xo,T -*oo) -> n 0 - 3n0x2) -

and

(3.27)- l

where xo(x, r — oo) is determined from

x = xo+X(xo)

( 3 . 28)

(3.29)

(3.30)

u(ko)X(ko)sinw(ko)T} (3.22)

That is to say, a low-level, stationary electricfield remains, together with its associated densi-ty profile. This represents a balance betweenelectrostatic force and the (long time) force due topressure variations in the model that has beenused.

It is possible to compare the asymptotic results(3. 27) - (3. 30) with the exact asymptotic statefollowing from Eqs (3.17) - (3.19) and (2.6) - (2.8).

188

NONLINEAR OSCILLATIONS

We assume P(x0, 0) = Po to be uniform and writeEq.(3. 19) in the form

92v(x0/ 3 9

-3(3.31)

We integrate this equation with respect to r from

= 0 and usingzero to infinity assuming -r—

9v8T r = 0

= E(XQ, 0). This leads to the equationm

- E ( x 0 . 0) + h)«I(x0) = -m x ° ' o v o/X

m

where

N(xo,O) dx0 dxf

-3

(3.32)

I(x0)

CO

= / v ( x 0 , T) dr, N(x0> 0) a

Note that is of the order where k0 isthe characteristic wave number of the initial per-turbations. We require k^X^ « 1. In that caseI(x0) - I0(x0) - X(x0). The results reduce to theasymptotic state (3. 27) - (3.30) if we require inaddition k0X « 1.

3.3. DISSIPATION DUE TO COLLISIONAL DRAG

The effect of dissipation in the nonlinear ana-lysis of section 2 may be simulated by includinga collisional drag term, - vv, on the right-handside of the equation of motion, Eq. (2. 3). Theonly modification of the system (2.5) - (2.8) occursfor Eq. (2. 8) which now reads

rv

— V(XO,T) = -—E(X O ,T) - I /V(X O ,T) (3.38)

The velocity in Lagrangian variables then satisfies9

-V(X O ,T)+U)2 V (X O ,T)=O (3.39)

We have assumed that the integral defining I(x0)exists. This implies v(x0, oo) =0. Integrating(2.7)and using (3. 32) we obtain

E(x0,oo) = E(x0,0) + 47ren0I(x0)

o D i o '

Poisson1 s equation together with

~4

~ - ) (3.33)d x / x '

9x T1 +

dl V ddx 0 / dx0

leads to

N(x0,oo)-1, 3 X ^ 1 + -^-J - | -

Furthermore for T -• oo

I(x0)

(3.34)

(3.35)

Equations (3. 33) - (3. 35) are analogous to the set(3.29), (3.28), (3.30). Instead of X(x0) we haveI(x0) and this function can be found by solvingEq. (3. 32). We do this by an expansion

= I O + X 2 I 1 (3.36)

and find

(3.37)

[, = - N'^xn, 0)-J2_ {N'^X-,, 0)}1 v ° 'dx0 * 0' "

where v has been assumed constant. The motionthus exhibits damped oscillations, with a dampingfactor exp(-vT/2), and oscillation frequency(U2, - z/2/4)1/2, where u0 > v/2 by assumption. Con-sequently, V(XO,T) tends to zero for large r, asdoes the electric field, E(XO,T) . Similarly, thedensity may be shown to damp to the value of the*uniform background density, n0. This asymptotictime behaviour of course persists in the Eulerianframe, and is valid for any initial conditions thatdo not lead to multistream flow. For the case

u 0 (3.40)

this restriction is still given (approximately) byinequalities (2. 17) and (2. 18).

4. ELECTROMAGNETIC OSCILLATIONS

The complete set of cold plasma equations insituations where all gradients are in the x-directioncan be conveniently split up into three subsets.

Subset A

3t

9t

= -47re(n-n0)

= 47renvv

9xp I 1 1

= - ^ - (E»+-vwB,-rv«B,

(4.1)

(4.2)

(4.3)

(4.4)

This subset describes the longitudinal oscillationstreated in the preceding sections if the contribu-

189

DAVIDSON and SCHRAM

tions to the Lorentz force in Eq.(4.4) due to self-consistent magnetic field are neglected. Thisneglect and the use of non-relativistic mechanicsare justified if the fluid velocities are muchsmaller than the velocity of light and if the fieldsBy, Bz arising from electromagnetic oscillationsare not much larger than the electrostatic fieldE x . These conditions can always be met bychoosing suitable initial conditions.

Subset B

9E

at

9Vy

at

9BZ—

8E

9vy e (_ 1 _—-L = IE - - v B,9x m V y c x '

(4.5)

(4.6)

(4.7)

This set describes transverse oscillations. Wehave put B (satisfying 9Bx/9x = 0) equal to zero.

Subset C

9E2

9t

8BV

at

at

= 4?renv

= c9EZ

3x

9v9x

e / 1= (E_+-vY\ z x B

(4.8)

(4.9)

(4.10)

The system (4. 8) - (4. 10) describes the additionaldegree of freedom for transverse oscillations andis identical to subset B with the interchange of yand z, and the replacement of c by -c .

We now treat the case without external magne-tic field, and in addition neglect the magneticforce in Eq. (4.4). The solutions of subset A arethen known from section 2. Consequently, thequantities n(x, t) and vx (x, t) enter the equations(4. 5) - (4. 10) as known functions. We solve theequations of subset B keeping in mind that thesolutions of Eqs (4. 8) - (4. 10) may be written downby analogy, and introduce the vector potential Ayrelated to the fields by

c at B , =• (4.11)

This satisfies Eq.(4.6), and (4.7) reduces to

0 0 \ I

Jt+V* 9x7v (4.12)

expressing conservation of canonical momentumin a reference system moving with vx . Equation(4. 5) becomes

a2Ay i a2AvIx*"" c2 at2

47re— nvy (4.13)

From (4. 12) we obtain vy in terms of A

e e

where x0 is the solution of

t

x0 =x-Jvx(x0,f)df (4.15)o

Substituting (4. 14) into (4. 13) it follows that

92A,

with

and

(4.17)

The function N(x, t) is periodic in t. ThereforeEq. (4. 16) suggests the possibility of parametricresonance. In section 2 expressions for N(x, t)and vx(x, t) were obtained for the case of an initial-ly sinusoidal density perturbation and vx(x,t =0)=0.We use these results in order to investigate theproperties of Eq.(4. 16) explicitly. However,generalizations to other initial conditions for thelongitudinal problem are straightforward.

We denote the characteristic wave numbers ofthe longitudinal and transverse oscillations by k 0and k respectively and the characteristic frequen-cy of the transverse oscillations by u. SinceN(x, t) and F(x, t) in Eq.(4. 16) vary in space andtime with characteristic wave number kQ andfrequency u0, considerations will be restrictedto solutions, Ay, with k ;>k0 and u k. u0 . We dis-cuss four different ordering schemes.

(1) u) , kc

Eq.(4. 16) reduces to

= F(x,t) (4.18)

describing the direct generation of the magneticfield Bz = 3Ay/9x by the electric current density,cF(x, t).

(2) to ~ kc » u0

Eq. (4.16) reduces to

190

NONLINEAR OSCILLATIONS

The solutions of the homogeneous part of theequation are plane electromagnetic waves. Theinhomogeneous equation (4.19) is easily solved bytransformation to the new variables, x± =x±ct .The general solution of Eq. (4. 19) in terms of theinitial values of the fields is given by

E y = - | [E y (x-c t , t = 0) + Ey(x + ct, t = 0)

+ B z (x -c t , t = 0) -B z (x + ct, t=0)]

x + ct

, t = 0)

dx'F x', t -

Bz = |[Bz(x-ct, t =

Ey(x-ct, t = 0)-Ey(x + ct,

(4.20)

where the appearance of retardation is clear. Thedifference from the analogous three-dimensionalexpressions for the particular solution of the in-homogeneous wave equation should be noted.Equation (4. 20) describes the generation of highfrequency short wavelength transverse fields.

(3) k » k0, u ~u 0 ~ kc

We now consider situations where the transverseperturbations are short wavelength in comparisonwith the longitudinal perturbation. This limitingcase has the effect of introducing two space scalesin Eq. (4. 16). Reserving x for the fast variationsin space, and introducing £ = kox for the slow va-riations, we see that 3/9x is replaced by 9/9x+ ko9/9| . In lowest order, Eq. (4.16) thenbecomes

(4.21)

We start by studying the solutions, Ay, of thehomogeneous part of Eq. (4.21). Introducing theFourier transform

rAk(?,t)= / A"(x,?,t)e-ikx dx (4. 22)

it follows that

»2

+ u2N(?,t)}Ak = 0 (4.23)

Since N(£,t) is periodic in t this is a Hill equation.In order to discuss Eq.(4.23) and make us of well-known results of the theory of Mathieu and Hillequations [10, 11], we write N(f, t) as a doubleFourier series for the case of an initially sinusoidal

density perturbation with vx (x, t = 0) = 0, (the example discussed in detail in section 2), i. e.

(4.24)

where T=wot. Then with f0 =koxo and Eqs (2.23)-(2.25) we find

2ir 2TT

N,

X exp[iiT-im?(?0JT)]2ir 2ir

= ( w / d T / d ? o ( 1 + AC°S?o)

0 0

X exp[iiT-im?0-imA(l - cosT) sin?0]

Using standard trigonometric identities and seriesexpansions with Bessel functions the integrals canbe evaluated with the result

<i = - » (4.25)

The following properties are readily established

No,o

Ne.o = 0

Therefore

N(C, t) = 1 + 2Y(?) + 4 ^ Nc (?) cos J0T (4. 26)

with

and

),m cosmC (4.27)

cosmf (4.28)

If A « 1 we may expand in powers of A and obtainafter considerable algebra

9 1 1N = 1 + A cos £ cos T + A cos 2? - - cos T +- cos 2T

+ A° cos £JI..L\8 32

cos T +- cos 2T - — cos 3To o6

+ cos 3? i -| + || cos T -|cos 2 T + ^ cos 3T

+ O(A4) (4.29)

191

DAVIDSON and SCHRAM

Equation (4. 23) may be written in the standardform [10, 11]

e2Jcos2iT ]Ak=0 (4.30)

where T = T/2. The 02j follow immediately fromEqs (4. 25) - (4. 28). For small A we get fromEq.(4.29)

+ O( A4)

• * I H - #

O(A4

= A2 \j

32

- | Acos3? +O(A4)

A 4

= — (- cos § + 9 cos 3?) + O(A4)

08 = O(A4) (4.31)

It is well known [10, 11] that the solutions of theHill equation, Eq. (4. 30), have the form

Ak = a exp(/uT) <£(T) +b exp(-/ur) <£(- T) (4. 32)

where a and b are integration constants, and $(T)is a periodic function. Depending on the coef-ficients 028> M is either purely imaginary imply-ing stability, or of the form, ri+/uR, where r isan integer or zero and/uR is real, implying in-stability. It is the latter possibility which is ofspecial interest here.

The quantity \x is given by [10]

cosh 7TJU = 1 - 2S(0) sin2-^

where S(0) is the infinite determinant

(4.33)

S(0) =

1

02

02 0 060ft-16 0n-16

- 4

£4* o

0 0 - 4

0 o - 4

- 4

0 O - 1 6

0 o - 4

00

-16 0n-16 - 16

- 4

(4.34)

It has been assumed that 60 ^4r2, r = 0, ±1 etc.

192

If all terms of O(A2) are neglected, then Eq.(4.30) is a Mathieu equation with stability dia-gram [10, 11] as given in Fig. 2. We see fromEq. (4.31) that 0n >4 and the first unstable regionis near 0O = 9. Therefore we consider Eq. (4. 30)with

/ k 2 r 2 \eo=4 1 + ^ = 9 , 02= 2Acos?,

(4.35)72C = 0 for i > 1

FIG.2. Stability diagram for Mathieu equation according to Refs [6, 7].Stable regions are shaded.

In this case we use the results of Ref. [10], page17, describing the boundary curves of the stableregions in Fig. 2 in order to find the region ofinstability. Unstable waves develop for k in therange5+-L^ ^

, 2 2

k c

cos?|: (4.36)

The width of this range is clearly of order A3 .The maximal growth-rate follows from Ref. [10],page 101, and is given by

(4.37)

The growth-rate takes this maximal value whenk2c2/u§ is approximately in the middle of theinterval (4. 36). On both sides of the middle /uRdecreases until it vanishes at the boundaries ofthe interval. In a similar way results may beobtained near 0o = 16, 25 . . . with growth-ratesand unstable k ranges of order A4, A5 etc.

NONLINEAR OSCILLATIONS

If A is not small (but still A < 1/2) it can beshown that the unstable regions and the growth-rates are much larger . This is suggested byFig. 2. This stability diagram, however, is nolonger valid because 62a, $> 1 will also play arole . It should also be realized that sincevx ~ Au o /k o ~ Ack/k0 the present non-relativistictreatment is only justified if

Eq. (4.4) it is easily seen that

A « M «k

(4.38)

Therefore larger values of A together with ko«krequire a relativistic treatment or restriction tospatial regions for kox ~ mr (n is an integer) sincethere vx is small according to Eq. (2. 21). It mustalso be recognized that Eq. (4. 23) breaks downafter a sufficiently long time. This can be seenas follows. If we take a derivative with respectto x in Eq. (4. 32), a factor t appears since JU de-pends on kox (see Eq.(4.37)). Since such termswere neglected in writing Eq. (4. 23) we have therequirement

wotA3ko « k (4.39)

Consequently the explicit calculations are onlyvalid for a restricted time; however, considerablegrowth is possible in Eq.(4.32) since the growthrate is of order UQ A3 and ko « k . Secularitiesmight be avoided by means of a generalizedW.K. B.-method and it is therefore not to be ex-pected that the restriction (4. 39) represents aserious limitation of the theory.

Questions may be raised regarding the energysource of the instability. It is clear from totalconservation of energy that the amount of energyin the unstable transverse oscillations can neverexceed the energy content of the initial longitudi-nal and transverse perturbations. It is of someinterest to examine the energy balance for thelongitudinal and transverse oscillations separate-ly. It can be easily shown that

a n 2 + E =x 8%

3 ) -x = - K

where

(4.40)

(4.41)

(4.42)

The right hand side of Eq. (4.40) is due to themagnetic force - e/mc vyBz in Eq.(4.4) whichwas neglected in preceding calculations. Thequantity K in Eq. (4.41), however, was not neg-lected in the treatment of the transverse oscil-lations. It is this quantity which makes the trans-fer of energy from the longitudinal into the trans-verse oscillations possible. If we start initiallywith Ey + B2 « Ex then ultimately E2, +&\ is of thesame order of magnitude as E2. Comparing themagnetic force with the electrostatic force in

(4.43)

Therefore in view of Eq. (4. 38) the magnetic forceis still relatively small when the energy of thetransverse oscillations has reached its ultimatelevel. Nevertheless it is this magnetic force thatis responsible for turning off the instability.

Thus far, only the solution to the homogeneouspart of Eq. (4. 21) has been discussed. This cor-responds to initial conditions with zero canonicalmomentum in the y-direction as is seen fromEq.(4.17). In general, ifF(x,0)/0, we must adda particular solution of the complete inhomoge-neous equation. For instance instead of Eq.(4. 23)we have

(4.44)

where Ak(?,t) and Fk(?,t) are the Fourier transforms of A(x, £, t) and F(x, £, t), respectively. Asolution of Eq. (4. 44) is

(4.45)

where Ak(§,t) is a solution of the homogeneouspart of Eq.(4.44). The zeros of Ak(C,t') inEq. (4. 45) can be accommodated by suitablechoice of integration contour in the complex t1 -plane.

(4) k~k0, kc

In this case Eq. (4. 16) reduces to

2

^ | 2N(xt)A =c2u2N(x,t)Ay =c2F(x,t) (4.46)

and we recover Eq. (4. 30) and Eq. (4. 31) wherenow 0o = 4 + O(A2). Instability seems possibleagain since 6$ = 4 is a point where an unstableregion extends to the 90-axis in the stability dia-gram, Fig. 2. However, we remind the readerthat the width of the unstable region is of orderA2 and the neglect of k2c2/u§ in 0O as given byEq. (4. 31) is only correct if kc <w0 A. Thereforewith k ~ ko we find in general that vx ~ UoA/ko> cwhich is inconsistent. Consequently, the unstablesolution is possible only for kox ~ nir (n being aninteger) where vx is small according to Eq.(2.21).

5. STATIONARY SOLUTIONS

We now investigate time-independent solutionsof the Eqs (4.1) - (4. 7). Again solutions of Eqs(4. 8) - (4. 10) are quite analogous to those of Eqs(4. 5) - (4. 7). We will t reat the purely electro-

193

DAVIDSON and SCHRAM

static problem leading to cold plasma version ofthe Bernstein-Greene-Kruskal (B.G.K.) waves[12], the electromagnetic problem using the electro-static results, and a special case of the completeproblem including the magnetic force in Eq.(4.4).

5. 1. ELECTROSTATIC PROBLEM

Allowing for a constant velocity, v0, of the uni-form background, the relevant equations are

nvv =

vv -7-*- = E,x dx m '

(5.1)

(5.2)

- ^ = -47re(n-n0) (5.3)

We introduce the Lagrangian co-ordinate n by

implying

rx=Jvx(rj')dr)'

o

x dx drj

(5.4)

The equation for vx becomes

d v

Consequently

vx = v0 +a sinuor)

Ex = u0 (b cos uor] - a sinu0

(5.5)

(5.6)

(5.7)

(5.8)

where a and b are .integration constants.The transformation given by Eq.(5.4), becomes

w0- cosw0r))} (5.

If we choose the origin such that Ex = 0 at x = rj = 0,then b = 0 and Eq. (5. 9) is the same as the secondline of Eq. (2. 24) if we replace x0 by v0 r\, k byuo/vo and £1 by a/v0 . We can then use the inver-sion formulae, Eqs (2.27) and (2.29), and obtainfor E,,

Ex (x) = i - i. 10)

Substituting this into Eq.(5.3) it follows that

Writing

where

.cosnx1

n = 0

2JT

an = (x1) cosnx'dx1

2ir

i r

= - Jdx*

v (ri'Jcosnx'fr]1)— dr)', n

217

we find after some algebra

vv(x)=v 2vn U n v ' n Vvft. cos

n = i (5 .12)

where J^ is the derivative of Jn with respect toits argument.

Some interesting results are implied by theabove analysis. In section 2 the calculations wereonly valid for Cl < 1. In a similar way the validityof the present analysis requires a<VQ. It thenfollows from Eq. (5. 6) and b = 0 that vx nowherechanges sign. Consequently, no electrons aretrapped. The condition a<v 0 , which is necessaryto ensure the uniqueness of the Lagrangian trans-formation, excludes trapped electrons from theformalism. This is not surprising since the co-existence of trapped and free electrons cannot bedescribed by the one-stream cold plasma model.

It should also be noted that the solutions (5.6),(5. 7) and (5. 8) cannot be obtained by "freezing"the solutions of the time-dependent problem insection 2, i . e . at no instant of time do the solu-tions in section 2, properly generalized to includethe background flow velocity, take the spatial formof a cold B.G.K. wave. Furthermore, it is re-markable that in a cold plasma only periodicB.G.K. waves of a very special form, given inEqs (5. 10), (5. 11) and (5. 12), are possible. Thisis in sharp contradistinction with the situation ina hot plasma [12], and is due to the absence oftrapped particles. In addition, it should be notedthat the spatially averaged electron velocity doesnot equal v0 but rather v0 +a2/2v0 as is seen fromEq.(5. 12).

194

NONLINEAR OSCILLATIONS

5.2. ELECTROMAGNETIC PROBLEM

Having obtained the solutions to the electrostaticproblem we now consider the stationary form ofthe Eqs (4. 5) - (4. 7) treating n(x) and vx(x) asknown quantities through Eqs (5. 11) and (5. 12).It follows that

dEv

dx= 0 (5.13)

(5.14)

(5.15)

where Ay is the y-component of the vector poten-tial. Choosing Ey=0 and vy=(e/mc)Ay inorder tosatisfy Eqs (5.13) and (5.14), we obtain (n = n0N)

(5. 16). o - T - N(x) Av = 0dx2 cl v ' y

i . e . a Hill equation in space. In general the so-lutions of Eq.(5. 16) are growing in space becauseof the minus sign. For special values of theFourier coefficients of N(x), however, the solu-tions can be "stable"; see Fig. 2 in section 4. Itmay be expected, as in section 4 that growingsolutions are limited by the magnetic force whichwas neglected in the longitudinal part of theproblem.

or Bz(x) is prescribed, the others can be calcu-lated. As an example we consider

n = n0(l + Acoskx),

which gives

_ 47ren0A . ,EY = -—:—"—smkxx k

and

B, =2 2 2

r ^ n

(5.21)

(5.22)

(5.23)

where B? enters as an integration constant thatmust obviously satisfy the inequality

47reno A . . [^2r*—sinkx Bo -vy = c ^ — sinkx Bo

X (4coskx+ A cos 2kx) (5.24)

These results remain valid in special relativity.It is easily proved that (Vy/c^ ^ A/(l + A).

A C K N O W L E D G E M E N T

5.3. SPECIAL CASE OF THE COMPLETEPROBLEM

We thank Dr. A.N. Kaufman for valuable dis-cussions concerning this work.

If we consider the background to be at rest,i . e . v0 = 0, but retain all magnetic force terms,it follows from Eqs (4. 1) - (4. 7) that

and

v x = 0 ,

dEx _dx

* c

47renv

Ey =0

47re(n- n0)

ryBz =0

dR= C"dx"

(5.17)

(5.18)

(5.19)

(5.20)

[2][3][4][5]

[6][7][8][9]

[10]

T i l l

Eqs (5. 18) - (5. 20) permit a non-trivial class ofsolutions. If one of the functions n(x), Ex(x), vy(x)

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STURROCK, P . A . , Proc. R. Soc . A242 (1957) 277 .

KONYUKOV, M . V . , Soviet Phys. JETP 37 (1960) 570 .

COURANT, R., FRIEDERICHS, K . O . , Supersonic Flow and Shock

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JACKSON, E . A . , Phys. Fluids J3 (1960) 8 3 1 .

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BERNSTEIN, I . B . , TREHAN. S . K . , Nuc l . Fus ion^ (1960) 3 .

LIGHTHILL. M . J . , J. Inst. Math. Appl. 1 (1965) 1 .

McLACHLAN, N . W . , Theory and Application of Mathieu

Functions, Oxford University Press, 1947.

ARSCOTT, F . M . , Periodic Differential Equations, Pergamon Press,

1964.

[ 1 2 ] BERNSTEIN, I . B . , GREENE. J . M . , KRUSKAL, M . D . , Phys. Rev.

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195