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Volume 90A, number 3 PHYSICS LETTERS 28 June 1982 GENERAL SCHRODINGER EQUATION FOR WHISTLER WAVES PROPAGATING ALONG A MAGNETIC FIELD V.1. KARPMAN and R.N. KAUFMAN IZMIRAN, Academic City, Moscow Region, 142092, USSR Received 23 April 1982 The general Schrodinger equation (GSE) for whistler waves with their group velocity directed along an external magnetic field is derived. The “mean” wave vector of the wave beam may be parallel to or have an angle ® = arccos(24wc) with the magnetic field. Applications of GSE to the whistler propagation in density ducts are considered. The results are important for the problem of the self-focusing of whistler waves. Investigation of whistler wave propagation in an plasma density N depends only on x, we may write inhomogeneous plasma, based on the Maxwell equa- the nonvanishing components of ~ in the form tions [1—4], reveals a number of effects which are = = + ~ 2j~ not described by the Schrodinger equation (SE) de- rived in ref. [5] and used up to now. A typical ex- Ezz = —(1 + v)y2/u2 ample is the whistler trapping in a density trough at = —e~ = —i(l + v)y2/u(l u2) , (2) (~< ~i~/2 (“.‘c is the electron gyrofrequency). In this paper we show that this phenomenon as well as some where the following notations are introduced others follows from a different SE that also describes U = °~‘C’~’c’ 7 = (‘)pk’’c, P = (N(x) _N 0)/N0 a wave beam propagating along an ambient magnetic field. In the first SE, the mean (zero order) wave N0 ‘‘N(±°°), w~ = 4ire 2No/me (3) vector is parallel to the ambient magnetic field. In with ‘y ~ 1, WLH ~ ~ ~ (WLH is the lower hybrid the second one, it has an angle e = arccos(2w/wc) frequency). with the magnetic field. In both cases, however, the It would be instructive to compare some of our group velocity is parallel to the magnetic field. Both further results with those following from the WKB SEs are derived by the same approach outlined below, solutions of eq. (1), which may be written as [1,21 and they can be written in a unified form which we call the general Schrodinger equation (GSE) for the whistler waves propagating along the magnetic field. = E 0(x)exp (i ~[± f q dx + (4) The GSE may be applied to the self-focusing of whis- 0 tler waves, whistler propagation in magnetospheric where q 2 = q~ (x), m = 1, 2. ducts, etc. We start from the Maxwell equation for the wave q~(x) = (~2)_1 {(1 2u2)p2 272 [1 + electric field of the form (l/2)E(r) exp(—iwt) + c.c.: + (—1 )Jflp [p2 472 [1 + v(x)J ] l/2} (5) (I) and E 0(x) is a polarization vector, slowly changing where ~ is the dielectric tensor. Considering a cold withx. Its expression is written in refs. [1—4]. Eq. (5) plasma in the ambient magnetic field B directed along for v = 0 may also be obtained from the well-known the z axis, and assuming, at first, that B = const. and dispersion equation for whistler waves 0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland 137

General Schrödinger equation for whistler waves propagating along a magnetic field

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Page 1: General Schrödinger equation for whistler waves propagating along a magnetic field

Volume90A, number3 PHYSICSLETTERS 28 June1982

GENERAL SCHRODINGER EQUATION FOR WHISTLER WAVESPROPAGATING ALONG A MAGNETIC FIELD

V.1. KARPMAN andR.N. KAUFMANIZMIRAN,AcademicCity,MoscowRegion,142092, USSR

Received23 April 1982

ThegeneralSchrodingerequation(GSE)for whistlerwaveswith their groupvelocity directedalonganexternalmagneticfield is derived.The“mean” wavevectorof thewavebeammaybeparallelto or haveanangle® = arccos(24wc)with themagneticfield. Applicationsof GSE to thewhistler propagationin densityductsareconsidered.Theresultsareimportantfor theproblemof theself-focusingof whistlerwaves.

Investigationof whistlerwave propagationin an plasmadensityNdependsonly on x, we maywriteinhomogeneousplasma,basedon theMaxwell equa- thenonvanishingcomponentsof ~ in the form

tions [1—4],revealsa numberof effectswhich are = = + ~2j~ —

not describedby the Schrodingerequation(SE)de-rived in ref. [5] and usedup to now.A typical ex- Ezz = —(1 + v)y2/u2ampleis thewhistler trappingin a densitytrough at = —e~= —i(l + v)y2/u(l — u2) , (2)(~< ~i~/2 (“.‘c is theelectrongyrofrequency).In thispaperwe show that this phenomenonaswell assome wherethefollowing notationsare introducedothersfollows from a different SEthat alsodescribes U = °~‘C’~’c’7 = (‘)pk’’c, P = (N(x) _N

0)/N0a wave beampropagatingalongan ambientmagneticfield. In thefirst SE, themean(zeroorder)wave N0 ‘‘N(±°°), w~= 4ire

2No/me‘ (3)vectoris parallelto theambientmagneticfield. In

with ‘y ~ 1, WLH ~ ~ ~ (WLH is the lowerhybridthesecondone,it hasan anglee = arccos(2w/wc) frequency).with themagneticfield. In bothcases,however,the

It would be instructiveto comparesomeof ourgroup velocity is parallelto the magneticfield. Both

further resultswith thosefollowing from theWKBSEsarederivedby thesameapproachoutlinedbelow,

solutionsof eq.(1), whichmay be written as [1,21andthey canbe written in a unified form whichwecall thegeneralSchrodingerequation(GSE)for thewhistlerwavespropagatingalongthemagneticfield. = E

0(x)exp(i ~[±f q dx + (4)The GSEmay beappliedto theself-focusingof whis- 0

tler waves,whistlerpropagationin magnetospheric whereq2 = q~(x), m = 1, 2.

ducts,etc.We start from the Maxwell equationfor the wave q~(x)= (~2)_1{(1 — 2u2)p2 — 272 [1 +

electricfield of theform (l/2)E(r) exp(—iwt)+ c.c.:+ (—1 )Jflp [p2 472 [1 + v(x)J ] l/2} (5)

(I) —

andE0(x) is a polarizationvector,slowly changing

where~ is the dielectric tensor.Consideringa cold withx. Its expressionis written in refs. [1—4].Eq. (5)plasmain theambientmagneticfield B directedalong for v = 0 may also be obtainedfrom the well-knownthezaxis, andassuming,at first, thatB = const.and dispersionequationfor whistlerwaves

0 031-9163/82/0000—0000/$02.75© 1982North-Holland 137

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Volume90A,number3 PHYSICSLETTERS 28 June1982

= k2W~OS®/[(w~/c)2+ k2] , (6) or by

where0 is theanglebetweenthewavevectorkand P 2y, R (l +u)(1 —2u)/(1 —u)(1 + 2u), (14)theambientmagneticfield: k~= (w/c)q,k~= (w/c)p Q = 7(1 — 4u2)1/2/u . (15)= k cos B.

Now we turn to thederivationof SE. Writing the Therefore,therearetwo casesthatdiffer from eachrelativedensityvariationin theform v = v(x/a)where otherby themeanwavevectork (k~= (w/c)Q,k~a is a scalelengthof the spaceinhomogeneity,we as- = (w/c)P) andthewavepolarization.In the first case,sumethat the polarizationis circular(in zerothapproximation)

2 and thewavevectork is directedalongthemagneticc/w a~X, (7)P field at u < 1/2 aswell asat u > 1/2. In the secondwhereXis a smallparameter.It is convenientto in- case,which is realizedonly if u < 1/2, the polarizationtroducecombinations is elliptic, andthewavevectorhas a nonvanishing

= E — F + ‘8~ angle0 with themagneticfield. In zerothapproxirna-1 X -~ Y’ 2 X 1 Y “ ~ tion,

Making theansatz 2 2 1’2cos0=P/(Q +P ) / ‘‘2u, k=w~/c. (16)

F1 = E(?~x,X

2z)exp[i(w/c)(Qx +Pz)1 It is importantthat in bothcasesthegroupvelocity is

F2 = G(Xx, X

2z)exp[i(w/c)~Qx+Pz)] (9) parallelto theambientmagneticfield. This followsfrom the formula

whereP and Q areparameterswhichwill be defined 2 2 2 2 2 2 2v = w k k

1(w /c — k )/k(w /c +k ) . (17)below, andexpandingE andG m powersof X C Ii p p

Finally collectingtermswith A2 we obtainasys-

E = E0 + XE’ + X2E” +...~ tern of equationsfor E” andG”. The conditionof

G = G0 + AG’ + X2G” +..., (10) solvability of this systemgives anequationforE°thathas the form of an SE

we substitute(9) and(10) into eq.(1)andcollecttermsof equalorders,restrictingourselvesto the order 3E0 + ~ — 1N N ~ ~ = 0of A2. Omitting the detailsof the computationswhich Wg 3z 2 ~ 2 ‘~ — °‘\aNO/k_kare describedin ref. [6], we presenthereonly thefi- X (18)nal results. 2

v =(aw/ak )k=k , S=(~W/akl)kk , (19)In zerothorder,weobtaina linearhomogeneous g o 0

systemof two algebraicequationsforE0andG0. The wherethewavevectork0 is definedby thecondition

conditionof solvability of this systemleadsto expres- Vgi(ko) = 0, which is satisfiedeitherif B = 0 or if Bsion for Q that coincideswith eq. (5) if onesubstitutes = arccos2u.thereq —~Q, p -+ Pandputsv = 0. Fromthis system, Therefore,eq.(18)describesdiffraction of a waveit also follows beamin all caseswhenthemeangroup velocity of the

G0 =RE0 ‘ll~ beamis parallelto theambientmagneticfield~.For

“ ‘ the two abovementionedcases,onehaswhereR is aconstantthatdependson coefficientsof

thesystem. Vg = (2c/7)[u(l — u)3]1!2In first order in A, we havealinear inhomogeneous S = c~(1— u)(1 — ~ (B = 0); (20)

systemof equationsforE’ andG’, which is consistent c

if F, Q, andR areexpressedeitherby theformulas* Thefact that thegroupvelocity of awhistlerwaveis par-

P = 7[u(1 — u)]~!2, R = 0 , (12) allelto themagneticfield not only ate= 0 butalso ate= arccos2u is well knownfromray theory[7]. Therefore,

Q = 0 , (13) it is naturalthatwe automaticallycometo this resultwhile derivingtheSE from thefirst principles.

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Volume 90A, number3 PHYSICS LETTERS 28 June1982

ug = c/2’y, 5 ~2U(i —4u2)/72w~(cos0 = 2u). Indeed,assumetheoppositerelation: (1 — 2u)2

(21) ~ Iv(0)!. Then from eqs.(24)and(25) onehasaE/axIn both casesS (1 — 2u)at u -÷ 1/2. vi~/~,while in derivationof GSE it is assumedthat

In a similarway, onemay obtain SEif, in addition aE/ax‘-‘

1~ji!2.

to N, themagneticfield B is alsovaryingin sucha way Let usconsidernow theWKB solutionsof eq.(24):that

Cb = (B —B0)/B0~ A

2 E(x, z)~exp[i(± f p(x) ~0 -~ (27)

whereB0 B at lxi =°°. In this case,eq.(18)is re-

placedby Substitutingthis into eq.(9), we seethat it takesthesameform aseq.(4) with

~UgaE/az +~Sa2E/~x2—~wE0, (22)

p=P+~p, q~x)=Q±(c/w)p~x), (28)where

whereP andQ aredefinedeitherby expressions(12)= (N NO)(3WIaNO)k

0+ (B —BO)(aWI3BO)k0 and(13) or by (14)and (15). Ontheotherhand,sub-

= —Pw(cos0 — u)/cos 0 + w(bk0)/k0cos0. (23) stitutingp P + ~p, with P from (12)and(14), intoformula(5) andbearingin mind thecondition(26),

Here thez axis is directedalongB0,Vg andS are de- one comesin the first order of the smallparameterfinedby expressions(19), andwe writeE insteadof v(0)1(1 — 2u)

2 to expressionsfor q(x) coincidingE~1.Eq.(22) is thegeneralSchrodingerequationfor with thoseof(28). Therefore,thephasefactorsof thewhistlerspropagatingalongthemagneticfield. It is expressions(27)and(4) arein agreementwith eachvalid both for 0 = 0 andB = arccos2u. ForB = 0 it other. Onemayalso showthat thesameis true forcoincideswith theSE consideredin ref. [5] andafter- theamplitudes.wards. However,theMaxwell equationsalsohavesome

As a typical application,let usconsiderthe propa-gation of whistler wavesin a slab densityductwithb = 0. Forsimplicity, we assumethatv(x) is an evenfunctionwithoneextremumatx = 0 andv(oo) = 0.For the wavetrappedin the duct,E is proportionalto ~

exp[i(w/c)~pz]wherez~pis a constant.Thenfromeq.(22), one has theequationfor eigenmodes:

a2E/ax2+p2(x;~p)E0, p2 K[v(x)—2z~p/P](24)

where X ______

= 2w~u/(1— 2u)c2 (B = 0), ~

= —w~/(1— 4u2)c2 (cos0 = 2u). (25)

The trappingis possibleonly if ,~p>0, v~p>0, and

v(0)lP>2lz~pi.From(25)we seethatatu<1/2 ——~

—--the wavemaybe trappedbothin densitycrest(v> 0)and trough(v < 0); in thefirst caseB = 0, andin the ~1’2

secondone0 = arccos2u. At u> 1/2, the trappingis possibleonly in densitytrough with0 = 0. Fig. 1. Plots of q(x) at u. </2 for the densitycrests.Full

The GSEis valid underthecondition lines aredescribedby both theMaxwellequationsand SE.

‘~ (1 — 2u)2 . (26) Thebranchshownby thebrokenline follows fromtheMaxwellequationsonly.

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Volume90A,number3 PHYSICSLETTERS 28 June1982

It is worth to mention,that thepresenceof thecx-trabranchin fig. I makespossiblethetunnel transfor-mation q1 -~ q2 and thecorrespondingleakageof the

wavefrom the density crest[21.Finally, we note that in orderto write the GSE for

the axially symmetricwhistlerwaveoneshould re-

placein eq.(22) theoperatora2/ax2 by [61

— a2 1 D 1= ÷---

r 3r r2-xl ~a

~‘( )-~ _— Evidently, L is a particularcaseof the transversepart

of the2D laplacian

a2 ia m2

~1~It is interesting,that both for B = 0 and0 = arccos2u,- ~i’2 onehasto takem = 1, insteadof m = 0 usedin some

papers.Fig. 2. Plotsof q(x) at ~, < wJ2for thedensity troughs. In conclusion,we havederivedthe generalOvalstakeplaceat (1 — 2u)2 > lv(0)l andtheloop, shownby thebrokenline, at (1 2u)2 < ~(0)I. Theioop follows Schrodingerequationfor thewhistler wavespropagat-fromtheMaxwell equationsonly. ing alongtheambientmagneticfield B

0, i.e. with thegroupvelocity parallelto B0, andappliedthis equa-

solutionswhich cannotbe extractedfrom theGSE. tion to thewhistlerpropagationin density ducts.TheThis is seenfrom figs. 1 and2 wheretheplots of q(x) obtainedresultsform thebasis of a revisedtheory offrom eq.(5) areshown.Thoseoneswhichapproxi- theself-focusingof whistlerwaveswhichwill be de-mately coincidewith q(x) from (28)aredrawn by scribedelsewhere.solid lines,andthe “extra-solutions”by brokenlines.From fig. 1 we seethat if v>0 andu < 1/2, the GSE Referenceshasno solutionscorrespondingto q2(x).Theother

extra-solutionappearsif ~< 0 andu < 1/2 (fig. 2). [1] V.1. KarpmanandR.N. Kaufman,Zh. Eksp. Teor.Fir. 80This may be explainedasfollows. From ref. [3] it is (1981)1845.

known that solutionsof theMaxwell equationscorre- [2] V.1. KarpmanandR.N. Kaufman,Phys.Lett. 84A (1981)9-spondingto theovalsexist if (1 — 2u)

2 > I i-’(O) I. They [31 V.1. KarpmanandR.N. Kaufman,FyzikaPlazmy8 (1982)approximatelycoincidewith thosefollowing from the 319.GSEif condition(26)holds.The ovalsarefusingwhen [4] V.1. KarpmanandRN. Kaufman,J. PlasmaPhys.27

(1 — 2u)2 —~ I r’(O) I, andthey turn into one1oopif (1982), to bepublished.(1 — 2u)2 < l~(0)I. in this regiontheGSE is inappli- [5] H. Washimi,J. Phys.Soc.Japan34 (1973)1373.cable,andso theloop correspondsto an extra-solution [6] V.1. KarpmanandR.N. Kaufman,Preprintof IZMIRANno. 17 (Moscow, 1982),in Russian.which is shownby thebrokenline in fig. 2. As for [7] R.A. Helliwell, Whistlerandrelatedionosphericphenom-

u> 1/2 and ii <0, thegraphof q = q2(x) is a symmet- ena(StanfordUniv. Press,Stanford,1965).

rical oval andextra-solutionsdo not exist.

140