Plasma waves in a stochastic magnetic field
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Volume 65A, number 2 PHYSICS LETTERS 20 February 1978 PLASMA WAVES IN A STOCHASTIC MAGNETIC FIELD M.L. MITTAL and Y.S. PRAHALAD Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400076, India Received 8 March 1977 The effect of a weak stochastic magnetic field on circular polarized waves in a plasma is investigated. It is shown that the waves are damped. The damping coefficients in various cases of interest are evaluated. In a recent note [1] the authors have shown that form by noting that if = 0, they are the equations of circularly polarized plasma waves propagating along evolution of circularly polarized waves in the presence a spatially uniform but temporally stochastic magnetic of the mean field. Following Sjölund and Stenflo [31 field, are damped. But the analysis was restricted to normal modes with frequency w and wave vectors k short-range correlations and further the electron cyclo- can be introduced. These are tron frequency was chosen as the natural time scale. In — this paper, these restrictions have been removed. The a v + e (w~ + w 0) E + ~ + w0)~ B (4 present model is more realistic since the wave period ± ± m ~,2 ± — m ~,,2 w± ~‘ ‘~ is used as the natural time scale. Both short- and long- p p range correlations have been considered. The analysis where (w±, k) satisfy the dispersion relation is made using the method of normal modes. Plasma 2 2 2 2 — — — k c — w..w /(~. + ~ ) — 0. (5) fluid equations are used with immobile ions. The Co. — - P - herent waves exhibit a gaussian decay for long-range Using the relation between a± and V± and eq. (4), eqs. correlations and an exponential decay for short-range (1) to (3) can be written as correlations. The damping coefficients depend on the value of the Kubo number. aa~/at — iw~a~ + iw~K~f(t)a± 0, (6) Here following Das [2] the induced electric field where and the resulting spatial non-uniformity are neglected. — 2 2 This approximation is valid for a thin column of tenu- K~(w~) = w0w~ 2/[w~2w± + 2w~(w~ + w 0) — w~] . (7) ous plasma. The basic equations governing the evolu- Eq. (6) has the same form as the equation for the oscil- tion of circularly polarized waves, propagating along lator with random frequency modulation. This has the magnetic field oriented along the z-axis are been treated extensively by Frisch [4] . As physical ob- a V÷/at + (e/m) E~i iw (1 + f(t)) 1~ = 0, (1) servation involves time smoothing, only the moments — - of eq. (6) are of interest. Here the considerations are aB÷/at i kE~ = 0, (2) confined to coherent or mean waves only. The initial conditions can be chosen to be deterministic with val- e~ aEjat ± (k/~u0)B+ — eN0v~ = 0, (3) - — ue unity. Then as eq. (6)is formally integrable, the where B0 is the mean magnetic field, B’(t) = cB0f(t) amplitude of the coherent wave is is the fluctuating component, assumed to be ergodic r and stationary, N0 the unperturbed density and w~ (a~(t)) = e~W±t (exp iw.. K~ (wi) f f(t’) d ti). (8) eB0/m. \ L 0 -~ Eqs. (1) to (3) can be transformed into a suitable 117
Transcript of Plasma waves in a stochastic magnetic field
Volume65A, number2 PHYSICSLETTERS 20 February1978
PLASMA WAVES IN A STOCHASTIC MAGNETIC FIELD
M.L. MITTAL and Y.S.PRAHALADDepartmentofMathematics,Indian Institute of Technology,Powai, Bombay400076,India
Received8 March1977
Theeffectof a weakstochasticmagneticfield on circular polarizedwavesin a plasmais investigated.It is shownthat thewavesaredamped.Thedampingcoefficientsin various casesof interestareevaluated.
In a recentnote [1] the authorshaveshownthat form by notingthat if � = 0, theyare the equationsofcircularly polarizedplasmawavespropagatingalong evolutionof circularly polarizedwavesin the presencea spatiallyuniform but temporallystochasticmagnetic of the meanfield. Following SjölundandStenflo [31field, are damped.But the analysiswasrestrictedto normalmodeswith frequencyw and wavevectorskshort-rangecorrelationsandfurtherthe electroncyclo- canbe introduced.Thesearetron frequencywas chosenasthe naturaltime scale.In —
this paper,theserestrictionshavebeenremoved.The a v + e (w~+ w0) E + ~ +w0)~ B (4presentmodel is morerealisticsincethewaveperiod ± ± m ~,2 ±— m ~,,2 w± ~‘ ‘~
is usedasthe naturaltimescale.Both short- andlong- p prangecorrelationshavebeenconsidered.The analysis where(w±,k) satisfythedispersionrelationis madeusingthe methodof normalmodes.Plasma 2 2 2 2 — —
— k c — w..w /(~. + ~ ) — 0. (5)fluid equationsareusedwith immobile ions. The Co. — - P -
herentwavesexhibit a gaussiandecayfor long-range Usingthe relationbetweena±andV±and eq.(4), eqs.correlationsand anexponentialdecayfor short-range (1) to (3) canbe written ascorrelations.The dampingcoefficientsdependon thevalueof the Kubo number. aa~/at— iw~a~+ i�w~K~f(t)a±0, (6)
Herefollowing Das [2] the inducedelectricfield whereandtheresultingspatialnon-uniformity areneglected. — 2 2
This approximationis valid for a thin column of tenu- K~(w~)= w0w~2/[w~2w±+2w~(w~+ w
0) — w~]. (7)ousplasma.The basicequationsgoverningthe evolu- Eq. (6)has the sameform astheequationfor theoscil-tion of circularly polarizedwaves,propagatingalong lator with randomfrequencymodulation.Thishasthemagneticfield orientedalongthez-axisare beentreatedextensivelyby Frisch [4] . As physicalob-aV÷/at+ (e/m)E~i iw (1 + �f(t))1~= 0, (1) servationinvolvestime smoothing,only themoments
— - of eq.(6) areof interest.Herethe considerationsareaB÷/ati kE~= 0, (2) confinedto coherentor meanwavesonly. The initial
conditionscanbe chosento be deterministicwith val-e~aEjat ±(k/~u0)B+— eN0v~= 0, (3)- — ueunity. Thenaseq.(6)is formally integrable,the
whereB0 is the meanmagneticfield,B’(t) = cB0f(t) amplitude of thecoherentwaveisis thefluctuating component,assumedto be ergodic randstationary,N0 theunperturbeddensityandw~ (a~(t))= e~W±t(exp iw.. K~(wi) f f(t’) d ti). (8)
eB0/m. \ L 0 -~
Eqs. (1) to (3) canbe transformedinto a suitable
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Volume 65A, number2 PHYSICSLETFERS 20 February1978
If f(t) is a centeredgaussianprocesswith the correla- Case(iii). If the processis an arbitrarystationarytion function andergodicprocess,themethodof smoothingcanbe
R(t — t’) = (f(t)f(t’)>, used.Thisyieldsaa+/at— iw~(a÷)
eq.(8) yields -
(14)t
(a±(t))= e~~±texp[_~-~- w~K~(w+) = —�2w~K~f e~±(~t~R(t— t’)(a+(t’)> dt’.0
t t (9)x’ ffR(t’ — t”) dt’ dt”]. Forthecaseof white noisethis gives
0 0 (a±(t))= e~~±texp(—�2w~2K+2t).For theOrnstein—UhlenbeckprocesswithR(t — t’)
= e1tt’I/T, the integrationgives Usingeq.(6) otherlimiting cases,e.g.theKraichnan[4] approximation,canbestudied.
<a~(t))= e’~~exp[—�2w~K~T2(t/T+ e_t/T— 1)]. Fromtheanalysispresentedabove,it canbe seen(10) that no effectivecollision typeof a schemelike the
Usingeq.(10)variouscasesof physicalinterestcanbe oneproposedby the authors[1], canleadto the cor-studied. rectvaluesof the dampingcoefficients.The caseof
Case(i). Short-rangecorrelation: �w~K~T ~ 1. long-rangecorrelationyieldsan unusualdecay,ofThe problemreducesto thatof white noise gaussiantype.
Thenormal modeanalysisfor a Vlasovplasmain a(a+(t)) = e~W±texp(—e2w~K~Tt). (11) stochasticmagneticfield is underinvestigation.
Thedampingcoefficient isReferences
e2w~K~T. (12)
Case(ii). Long-rangecorrelation:ew~K~T>>1, the [1] M.L. Mittal and-Y.S.Prahalad,Phys.Lett. 58A (1976) 395.fluctuationsarevery slowandR(t) canbetreatedas a [2] K.P. Das,J. PlasmaPhys.13(1975)327.
[3] A. SjolundandL. Stenflo,Z. Physik204 (1967) 211.constant [4] U. Frisch, in: Probabilisticmethodsin appliedmathematics,
(a+(t)) = eR~±texp(— .~�2w~K~t2), (13) vol. 1, ed.A.T. Bharcha-Reid(AcademicPress,New York,1968) pp. 75—198.
showinga gaussiandecay.
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