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

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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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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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.

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