Hamza, A.M. and J. - Department of Physics and Engineering Physics

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Jul 1, 1993 - for analyticM purposes, lineaxize (11) in the limit of smMl. •uendes (• (yi), and wave numbers(pressure gradient term neglected), while Mso ...



VOL. 98, NO. A7, PAGES 11,587-11,599, JULY 1, 1993

A Turbulent TheoreticalFrameworkfor the Study of Current-DrivenE RegionIrregularitiesat High Latitudes: Basic Derivation and Application to Gradient-Free A.M.



Department o! Ph•tsics, Universtrotof Western Ontario, London, Ontario, Ganada

We have used a mode-couplinghypothesisto study the nonlinear evolution of E region irregu!axities at high latitudes. Conservationof energy and the identification of two distinct time scales for the problem at hand has allowedus to obtain an expressionin the fluid regime for the mean frequency and the spectral width of different types of echoesobserved by coherent radars. In this particular paper we have applied our results to a few simple cases,naxnely, to situations that are free of large-scalegradients and for which aspect anglesaxe closeto zero. Even though anomalous diffusion effects were also neglected, our theory neverthelesspredicts that in the absence of density gradients, strongly driven Farley-Buneman waves should normally saturate at a mean speed between 70% and 100% of the ion-acousticspeedof the medium. The theory also predicts that zero-frequency type 2 wavesin strongly turbulent situations should have a frequency width, which when translated to a Doppler width, should be approximately equal to the ion acoustic speed of the medium. Those results are a direct consequenceof assumingthat mode coupling is responsible for the saturation of all linearly unstable wavesin the E-region plasma. A companion paper will considerthe modificationsintroduced by large-scaledensity gradients.


cies).In addition,linear theorycannot,in principleat least,

Linear theories have been successfulin determiningthe causeof most E region irregularities seenby coherentscatter radars at low and high latitudes. Theoriesthat havesurvived the test of time include most prominently the Farley-

even producethe right value for the mean frequencyof the waves: once the waves have grown to large amplitudes and approach saturation, individual wave trains can no longer be simply described in terms of coherent fluctuations. Nonlinear


must therefore

be used in order to un-

Bunemanmechanism [Farley,1963;Buneman,1963]andthe gradient-driftinstability [Rogisterand D'Angelo,1970]. At

derstand some of the most basic properties of the turbulent plasma. A whole hierarchy of nonlinear ideas has been used high latitudes, it has alsobeenproposedthat intenseparallel in E region research and elsewhere. In order of complexcurrents carried by thermal electronscould lead to coherent ity we will classify the theories of interest for the present echoesmoving at the ion acousticspeedof the mediumeven study into the quasi-linear, resonancebroadening,weakly thoughthe local E x B drift couldbe markedlysmallerthan turbulent, and fully turbulent groups. However,it should that sameion acousticspeed[Chaturvediet al., 1987;Villain be rememberedthat other theories are possible,and that et al., 1987, 1990]. even within each one of the groupsconsideredhere a numWhile linear theorieshavebeensuccessful at providingba- ber of ingeniousideas have sometimesbeen proposed,maksicexplanationsfor the existenceof large-amplitudeplasma ing a classificationinto the basic groupssomewhatmurky at waves(for example, see the review by Fejer and Kelley times. It should neverthelessprove instructive to produce [1980]),they are unableto predict many other important at least a broad view of the subject, since we will propose properties of the observed waves. For example, it is well here yet another direction to tackle some of the challenges known that linear theory cannot be used to compute wave presentedby the abundantdata setsthat havenourished

axnplitudes (it produces a positivegrowthratefor all times) this field of study. or to computespectralwidths(the modespredictedbylinear theory are made of separateand independenteigenfrequen- 1.1. Zeroth-Order Effects

To start with a more quantitative approach,we define the unstable E region problem as the need to find a solution Copyright 1993 by the AmericanGeophysicalUnion. Paper number 92JA02836.


to an equation for the overall distribution function of the plasma. This includes finding self-consistentaverage and perturbed distributions for ions and electrons. The problem beginswith an equation for the averagevelocity distribution 11.587



f0 in the presenceof electrostaticprocesses.We



symbolicallyas Dfo Dt



turbance were weak enoughsomehow,quasi-lineareffects could produce a plateau in a distribution that had an un-

stableminimum initially. The waveamplitudewouldthen stopgrowingand the problemwouldbe well on its way to be solved.

wherefie and/•f arethefluctuatingelectricfieldanddistriIn the ionospheric E region,however,the plasmais driven bution,respectively, whileq andm arethecharge andmass unstableby large-scale electricfieldsor otherlarge-scale pheof the particles under consideration. The total derivative on nomenathat couldevenbe relatedto chemicalprocesses in theleft-handsideis obtainedby following particletrajecto-

paxt(through associated large-scale density gradients). The riesin phase space.Thismeans thatvarious forces (electric, unstablefluctuationsthereforecannotbringthe distribution

magnetic, andgravitational) contribute to modifying theve- backto its oldelectricfield-freeand/orgradient-free equilibrium,asthereis nowa "tug-of-war"betweenthequasi-linear addedto the left-handsidein E regionproblems, as they or more complicatednonlinear processesand the forcesor too play an importantrole for the determination of f0 in processes that pull the distributionawayfrom thermalequithe absence of anyinstability.Forexample,with the prob- librium. Leavingthe waveamplitudeasidefor now,weturn lem at hand,we are oftendealingwith electricfieldsthat, to two distinctquasi-linearprinciples.Theseprinciplesare in the presence of collisions with neutrals,cangiveriseto devised for a quasi-steady statesituationfor stronglydriven strongcurrentsand createin the process strongdepartures Farley-Buneman or gradient-driftwavesat highlatitudes. locity distribution. Collisionswith neutrals should also be

of the overalldistribution functionfromequilibrium (this

As a first quasi-linearsaturationprinciple,we makethe

thengivesriseto instabilities, andsoon). waveamplitudelargeenoughfor the electrontemperatureto As is well known,in the normallinear and quasi-linear shootup. The temperaturecouldthenbecomesolargethat frameworksthe fluctuatingpart of the distributionis astherelativedrift betweenionsandelectrons wouldbarelyexsumedto be smallcompared to the average. A straightfor- ceedthe newthresholdconditionsimposedby the increased waxd perturbation schemecan then be used to solve the

equationfor the fluctuations;this latter equationis well

ion acoustic speed. In the context of the formation of a

quasi-linear"plateau"this is akin to eliminatinga mini-

knownandhasnot beenwrittenhere. With thissimplest mum in the two-species distributionfunctionby broaden-

of procedures,the averagedistributionbecomesthe zeroth-

ing the ion distribution,and,moreparticularly,theelectron orderdistributionfunction;it is initiallyprescribed in terms distribution,throughwaveheating,i.e., wave-particle interof the zeroth-order forcesactingon the systemvia the left- actions. ha•d sideof (1). Sincethe perturbedquantitiesare asThe secondway to obtain quasi-linearsaturation is to

sumed,by definition,to havea negligible amplitude, they decrease the relative drift between ions and electrons un-

cannotaffectthe zeroth-orderdistribution.The right-hand til, onceagain,thresholdconditions for wavegrowthare sideof (1) must thereforebe zeroin the linear context. To put it in another way, if a disturbanceis introduced to the

system,and if that disturbanceproducesa departurefrom equilibriumstrong enoughto lead to unstablewavesafter time t = 0, one could neverthelessstill state that the linear solution will be valid for a short time of the order of a linear

growthtime. For this smallperiodof time, the right-hand

met. Wave-particleinteractionsin that casecausean effec-

tive dragbetweenthe two species.This dragcanbe coined in terms of an anomalous diffusion coefficient in the con-

text of E regionwork[e.g.,St-Maurice,1990b].It should be clear that one coulduse both quasi-linearmechanisms simultaneouslyin order to obtain saturation.

We shouldemphasize that whilethe ideaof a quasi-linear

sideof (1) hasa negligible effecton f0 (we are of course saturationprincipleis simpleand elegant,thereis onemaassuming herethat nonlineareffectsare thennegligible evjor conceptualdifficultywith usingquasi-linear principles erywhere, includingat the perturbedequation level). to saturate wavesin a strongly driven systemsuchas the

1.2. Quasi-Linear Theories

high-latitudeE region.Namely,if the wavesweretruly able to modify the temperature and drift to an extent such as

Forthe nextlittle while,onecouldusequasi-linear argu- to make the plasmamarginallystable,new wavesshouldbe mentsto describethe subsequent evolutionof the system. unableto growin sucha way as to maintainthe anomalous At that stagethe fluctuatingdistributionand the fluctu-

transport properties to the level desired. That would mean

atingfieldarestill relatedthroughthe properties givenby that if plasmaconditionswereto maintainquasi-linearsatlineartheory.However,thesetwo fluctuatingquantitiesare urationeffectsat all times,the magnitude of existing largeobviouslycorrelated.This, in principleat least,produces amplitudewaveswouldhaveto be kept quasi-constant for a nonzero average effect and affects the zeroth-order distri-

aslongasthe destabilizingdc electricfield waspresent.The

butionthroughthe right-handsideof (1) whichthen has plasmawouldconsequently necessarily be madeof coherent theformof a diffusionoperatoractingonthe average distri- wavesmoving at the ion acousticspeedof the medium. The butionfunction.At thispointwemustalreadydistinguish spectrumof these waveswould thereforebe very narrow. between two kinds of situations. With the first scenario an

This is far from what is detected when the dc electric field

initial changeis introducedin f0 and the problemis one is wellabovethenormalthreshold valueof 20mV]m,albeit for which the averagedistribution returns to the thermal with the possible exception of theso-called "type4" waves, equilibriumconfiguration.At that point if the initial dis- whichwill be discussed in St-Mauriceet al. (A studyof



the origin of non-ion-acousticcoherent radar spectra in the through resonancebroadening,are then affectedby the same high latitude E region, submitted to Journal of Geophysical wave-induced diffusion coefficient. This has made a distinction between quasi-linear effects and nonlinear effectsdue to Research,1992;referredto as paper2 from nowon). resonancebroadening rather murky at times. 1.3. Resonance Broadening Robinson's [1986]workillustrateswellhowa singlechoice We haveseenin the precedingsubsectionthat while quasi- of wave-induced transport properties could be used to take linear argumentsare the first line of battle to remedy some care of quasi-linear and nonlinear effects all at once. Robinof the problemsof linear theory suchas energyand momen- son first used the resonancebroadening theory to infer tum balance, the theory leads to unrealistic wave spectra. a wave-induced diffusion coefficient that would be large As we discussbelow, it alsoendsup overestimatingthe satu- enoughto stop wave growth. Observethat rather than use ration amplitude of the fluctuations. Consequently,one has a quasi-linear argument that would have required the relalittle choice but to move on to more complicated processes. tive drift between ions and electrons to decrease, Robinson The next logical step is to have the fluctuating electricfield ignoredthat particular quasi-lineareffect and requiredthat and the fluctuating part of the distribution function become wave-induceddiffusionbe large enoughto stop the nonlinear nonlinearlyrelated in (1). This helpsremedy a basicin- wavegrowth rate without thereforesubstantiallymodifying consistencyfrom the quasi-linear framework, namely, the the relative drift between the two species. And yet, at the quasi-linear framework assumeson the one hand that the same time, Robinson assumedthat quasi-linear effects asquadratic processesthat lead to diffusion are important for sodated with electron heating would remain important. He the average distribution function and yet that they can be thereforeuseda quasi-linearheating rate that wasconsistent neglectedat the fluctuating level. Two approachescan be with the wave-induceddiffusionrate that he had computed followed at this next simplest level: resonancebroadening at the nonlinear resonantbroadeninglevel. and weak turbulence.

In the resonancebroadeningapproachas it has been applied to the E region, one, in effect, reintroducesanomalous diffusion at the perturbed level itself. In other words, small-scale waves affect large-scale waves through shortwave-induceddiffusionin much the sameway that all waves

affectthe zeroth-orderdistribution[St-Maurice,1987a].In part this means that, just as for the quasi-linear situation,

resonance broadening[Dupree,1968]useswave-particle interactions, only to describe, this time, nonlinear effects. Thus the solution to the equation that describes the distribution function still does not depend on interactions betweenwaves. Rather, the fluctuating part of the distribution dependsonly on fluctuating electricfieldsand on the average distribution function. Physically, the diffusion comesfrom a random walk processassociatedwith the E x B drift of electronsthrough randomly distributed low frequencyelectrostatic wave packets. In the original kinetic derivation given by Dupree, the theory assumesthat the trajectory of particles in phase space is affected by encounterswith random electric field fluctuations described by a Gaussian

It is finally worth recallingthat Sudan[1983]wasfirst to proposeto use the equation for the diffusion coefficientthat resonancebroadening theory providesin order to compute a wave-induced diffusion coefficient that would ultimately

forcehigh-frequencyfluid modesin the equatorial E region

to saturate. Robinson[1986]extendedthis theory to includehigh-latitude irregularities. In the process,Robinson's anomalousdiffusioncoefficientbecamemuchlarger than the equatorial values that Sudan had in mind for his problem. 1.4.

Weak Turbulence

The next simplestway to go beyondquasi-lineareffectsis to introduce weak turbulence calculations. This brings in a new term to describethe interaction among variousmodes. The wavesare assumedto have small enoughamplitudes, however. As a result, the nonlinear coupling terms in the equation for the fluctuations can be describedadequately with the use of linear or quasi-linear results. This allows us to carry on with the time evolution of an unstable plasma for still longer time scalesthan quasi-lineartheory provides. One consequence of the weak turbulenceprocedureis that distribution. The electric fields are associated with turbuwhile wave-wavecouplingeffectsare now introducedfor the lence. This interaction leads to enhanced diffusion in the first time in the hierarchy, both the sum of the wave vectors are preservedin the fluctuatingquantitiesthemselves,not just in the background and that of their linear eigenfrequencies couplingprocess[Kadomtsev,1965; Davidson,1972,pp 31 properties. Thus, in the context of E region work, the resonance and 151]. This is an important limitation: sincethe wave it broadening theory has become equivalent to a theory of spectrum is still made of a collectionof eigenfrequencies, anomalous(wave-induced)diffusion.This has becomethe does not broaden. Consequently,if weak turbulence theory casespecifically becausethe fields used to scatter electrons is invoked as a means to saturate waves, the implication is haveimplicitly or explicitly been assumedto belongto much that when steady state turbulenceis achieved,the plasmais smaller scales than any process under consideration. An made of large-amplitude eigenmodeswhoseamplitude does implication of this has been that the use of a singleanoma- not changewith time. This shortcomingwas alsomet with lous diffusion coefficient has spread to cover all scales of quasi-lineartheory. Nevertheless,weak turbulenceshould the problem from the largest, or averagescale,down to any be more appropriate if the wave amplitudesimplied by this particular irregularity size. Although this cannot be strictly theory are smaller than those derived with quasi-lineararcorrect,the fact remainsthat both the averagedistribution guments. throughquasi-lineareffects,and the fluctuatingdistribution,



usually move at the ion acoustic speed of the medium. We

1.5. Strong Turbulence

The above argumentslead us to concludethat we need a longer time description that differs from all those given above if we wish to obtain a more dynamical description of the turbulent plasma. In this context, saturation is to be viewed as a statistical property. A turbulent steady state is achievedbecausewavesneither grow nor decay on a very long time scale. In that strong turbulence framework, frequencybroadeningthrough mode couplingplays a predominant role. We note that other theorieswould be possibleat this stage if mode coupling were to be hampered. For example, the wavescould be highly dispersive,or convective processescould be removing the energyfrom the unstable region, thereby limiting the amplitude. However,the E regioninstabilities that we are dealing with here are essentially nondispersiveat small aspectangles,while the growthrates

are usuallytoo largefor convectiveprocesses to matter [StMaurice,1985].This meansthat modecouplingshouldtake place relatively easily and should be consideredfirst, since large-amplitudewaveshave plenty of time to be createdand to interact


each other.

1.6. E Region Theories Currently in Use

In spite of the generalumbrella givenhere to variouslevels of theoreticalcomplexities,many E regiontheoriesdo not fit the mold too easily,as they usea mixture of ideasto produce various results. The reasonfor this lack of clarity at times is that



are more difficult

to obtain


linear ones. Nevertheless,some nonlinear ideas are easier to

understandthan others while at the same time being able to make useful predictions. Their relative simplicity may have made some of these ideas more popular than others, with researcherspushingthem perhapsa little too far in the process.

One particularly elegant nonlinear mechanismin E re-

have also mentioned

that the enhanced diffusion is then as-

sofiated with enhancedamountsof heating by plasmawaves

[Robinson, 1986].The plasmawaveheatingphenomenon is clearly observedwhenever the ambient electric field exceeds

50 mV/m [e.g., Schlegeland St-Maurice,1981; Wickwaret al., 1981;St-Mauriceet al., 1990,and references therein]. In termsof the magnitudes involved,St-Maurice[1990a,b] used Robinson's principle to estimate that the broadband wave amplitude that would be required in order to slow the phase velocity of large-amplitude waves down to their thresholdvaluesat 110 km at high latitudes. The rms wave amplitude implied by this kind of calculation is of the order of 20% or more. The inferred density fluctuation level

is thereforetoo large (4 timesgreaterthan valuesnormally measured).This raisesdoubtsaboutthe capabilityof quasilinear expressionsof the type used by Robinsonto solvethe problem beyond the realm of simple estimates.

Note that the applicationof Sudan's[1983]work to the high-latitude case was originally suggestedby Nielsen and

Schlegel [1985]when they foundthat the phasevelocityof E region irregularities was much closerto the ambient ion acousticspeedof the medium than to the E x B drift of the plasma. The latter would have been expected from linear theory. We should also observethat other authors have used anomalousdiffusion in one form or another to explore other

consequences (see Fejer and Kelley [1980]for a thorough review). In particular, Weinstockand Sleeper[1972],Sato [1977],and St-Maurice[1990b]haveusedstronganomalous diffusion as means to modify the ambient medium, for example, by changing appreciably the vertical profile of the equatorial electrojet or by creating fine structuresin the roral electrojet. Our main point about anomalousdiffusionis that we see two major difficulties with using this mechanismin order to make large-amplitude wavesmove at the ion acousticspeed

gion work was broughtin by Sudanet al. [1973]. They

of the medium

proposeda two-step secondarywave generationmechanism

these theories ignore the fact that the observedspectra are usually quite broad. The secondproblem is that the wave amplitude levels inferred from that model would have to be very large, namely, more than five times greater than normally observed. We conclude that even just with the 5% amplitude levels that are measured,seriousconsideration has to be given to mode coupling as a meansto saturate waves. In fact, even "type 1" or "primary wave" radar spectra are relatively wide. This means that the turbulent modescannot be describedin terms of a simple shift in the eigenvalue,the way people using anomalousdiffusiontheories would have it. The primary aim of the presentpaper, of

that waslater confirmedby observations [e.g.,Kudekiet al., 1982].Accordingto thismechanism, large-scale wavesgrowing from a gradient-drift mechanismcan generatesecondary instabilitiesif their amplitude is so large that the waveelec-

tric fieldsand/or gradientsthemselves exceedthe threshold for Farley-Bunemanor gradient-drift growth. We also mentioned another relatively simple nonlinear idea that has received widespread attention particularly in high-latitude research. The mechanismwas originally pro-

posedby Sudan [1983]for equatorialwork and relied on anomalous (wave-induced) diffusionto saturatefast-growing

near 110 km altitude.

The first one is that

modes. The mechanism,which we may be reinterpreting Harnzaand St-Maurice[thisissue],and of St-Mauriceet al. somewhat here, works as follows. As the wave amplitude [paper2] will be to offeran alternativetheoryin an attempt of a particular oscillation increases,it generatesa turbulent to produce more realistic spectral descriptionswith smaller fieldthat createssmaller-scale irregularities. Thesein turn wave amplitude levels. Finally, others have dealt beforeus with the more complienhance the overall amount of local diffusion through the anomalous diffusion process described above. The nonlin- cated mode-couplingphysics. This kind of work was again ear phase velocities then become so small that waves can pioneered by Sudan and his coworkersin E region irreguno longer extract "free energy" from the ambient electrical larity research.Most notably,Sudan[1983]synthesized this currents. As a result, the largest-amplitude wavesshould kind of work under a unified treatment in which he stressed



that mode coupling between slowly growing gradient-drift hasbeensuggested by thoseusingRobinson's [1986]theory modesleads to two-dimensionalturbulence in the magne- [e.g., Haldoupis,1989, and references therein]. For clarity, tized E region plasma. Sudan's work was basedon the so- we also limit our analysis for now to the two-dimensional called"directinteractionapproximation"(DIA) and "mix- caseand do not get into the debate as to whether large asing length" theories of turbulence and their relationship. pect angle echoesare produced by turbulence or could be The work was used in particular to predict the spectral explained through a bending of radar rays by refraction and shape of the irregularities and to provide a basic explana- other processes [e.g., Prikryl et al., 1992; Uspensky et al., tion as to why the spectral energy peaks at the lowest un- 1993].The workshownherewill, in otherwords,be strictly stable frequenciesin weakly unstable E region systems. A wlid only for wavesgrowing at anglesnearly perpendicular noticeablepoint about this work is that the nonlinear eigen- to the geomagneticfield. Becauseof the large amount of information involved and frequencieswere assumedto be equal to the linear valuesto first order. This meant that one would look at the linear exof the different points of emphasis,our presentationhas been pressionsfor the phase velocity of "type 2" irregularities in divided into three separate papers. In the current paper we order to interpret the data. Experimentalists were further coverthe basic theoretical framework and draw generalconenticed to follow this assumptionfollowing numerical simu- clusionson spectral widths and mean Doppler shifts as they lationsby Keskinenet al. [1979]who obtainedmeanphase pertain to so-called type I and 2 waves, namely, wavesthat velocitiesfor linearly stable modes that were consistentwith are produced in the absenceof any significantlarge-scale the linear eigenfrequencyapproximation. Another idea that gradient.In a secondpaper [St-Mauriceet al., paper2], we may need to be explored further was proposedby Keskinen will apply our new theoretical framework to stronggradient

[1981],whointroducedmodecouplingfroma differentpoint situations. As a result, we will offer a mechanism for the of view, by having high-frequency Farley-Buneman waves normal occurrenceof so-calledtype 3 and 4 wavesas well as beat with the ambient lower-frequency turbulence. an alternative explanation for some of the 10 MHz observa-

tionsmadeby Villain et al. [1987,1990]. Our mechanism 1.7. Contents of Current Work

doesnot, in the latter case, require large currents along the

In the work presentedin this paper and its companions, magneticfield. Finally, in a third paper [Hamza and Stwe focus on the mode-couplingidea introduced by Sudan Maurice,this issue](referredto as paper 3 from hereon), [1983,andreferences therein].We will, however,modifythe we present our self-consistentclosureschemefor the theooriginaltheory by removingthe assumptionthat the phase retical determination of the relation between mean Doppler velocity has to be closeto the linear value in any way, even shiftsand the spectral widths in E regionsituations. We also though this possibility will remain for particular situations. relate these quantities to specificwave amplitude levels. The current paper itself is organizedas follows. In section We will also show that for the normal E regioninstability models,coupling terms are not affectingthe overall energy 2 we usethe standard assumptionsleading to the production balanceof the waves;couplingonly strictly redistributesthe of Farley-Buneman and gradient-drift wavesto derive nonenergybetweenmodes. We will use a closureprocedurethat linear expressionsin terms of explicit time derivatives in the differsfrom the one proposedby Sudan. In it we will derive fluid limit. In section 3 we show how conservationof energy a general equation that, in a quasi-steadyturbulent state, can be usedin this caseto derive important wave properties relatesthe mean Doppler shift and the spectralwidth to the when nonlinear processesare dominated by mode coupling paxametersthat drive the instability. We will show in the and we obtain an expressionfor the nonlineargrowth rates processthat the linear expressionfor the eigenfrequencyis of the turbulent waves. This leads us to apply our results to usefulonly as a necessaryconditionfor growth;it cannotbe a steady state turbulence model in section 4. Consequences usedin general to predict nonlinear phasevelocities. Indeed, for ordinary type I and 2 spectra are also discussedin that we will prove that the nonlinearphasevelocitiespredictedby section. Our main conclusionsare presentedin section 5. our model have to be smaller than their linear counterparts. Moreover, we will find that in the strong turbulencelimit, 2. BASIC DERIVA•ON the mean phase velocity and the spectral broadening are The equations that we use contain the usual assumpsuch as to maintain the waves at a zero nonlinear growth rate, for which we will have derived an approximate fluid tions about E region physics for the Farley-Buneman and expression. For Farley-Buneman waves in fully turbulent gradient-drift instabilities in the fluid regime. Our derivaconditions this means average phase velocities that could tion is similar to what has been donein the past, particularly evenbe detectably smaller than the ion acousticspeeditself. the presentations made by Sudan[1983]and Sahr [1990]. In this part of our theoretical work we stress that we The ions are treated as collisional,unmagnetized,and nonwill, for simplicity, completelyneglect anomalousdiffusion convective.The electronsare taken as strongly magnetized effects.This, however,may turn out to haveits advantages; and weakly collisional, and with negligibleinertia. We use for example, St-Maurice [1990b]showedthat the Doppler quasi-neutrality and neglect any production or loss of ionshift of secondarywaves(wavesthat can only be produced ization. However,we includea backgrounddensitygradient. throughnonlineareffects)shouldnot havea phasevelocity We adopt Sabresnotation and use the followingdecompogiven by the linear results if anomalousdiffusion were to be sitionsfor the electrostatic potential •, the density N, and 20 to 100 times greater than the normal value. Such a value the pressure P•:




= Eo - Vd


•v = •Vo(•+.)

tiplying by the parameter •, we obtain a secondequation relating n to •, namely,

Po= NoTo(1+ where Non and d represent the density and electrostatic potential fluctuations. We adopt a two-fluid model based on the continuity and momentum equations for both the ionsand electrons,alongwith the quasi-neutralitycondition. Our staxtingset of equationsis thereforegivenby ON

+ v. (•v,) = 0



(11) where

(12) Equations(9) and(11) cannowbe combined. In the process the • terms cancel,and we get 2

o-•+ v. (Nv•)= o



vi •-•-C, c Vx c n+(l+•b)-•-+Vr.Vn (i x,V•) 0 (13) +K. v4• + •V.

3_.(Nvi ) =_ miqiNV•- VPi_viNvi (5) wherethe ion acousticspeedC• • Ot

0= q'N(V•-ve ) -•



given by

c]= •Z +%T•






The latter expressionactuMly representsa •nearization

isothermM(7• = 1). It is convenient to manipulate the electron momentum u•ess the speciescan be considered The equation still involves two field variables,namely,the equationfirst. To leadingorderin the ratio v,/O, we get

v, = m,O•, A x O,



density and electrostatic potentiM fluctuations, through the ambient density gradient term and through a nonlinearexch•ge term. Both terms comefrom the electronequations. B•ause of them, we need a second equation to close the


system. FollowingSudan[1983]and Sahr [1990]we can, for analyticMpurposes,lineaxize(11) in the limit of smMl •uendes (• (yi), and wavenumbers(pressure gradient term neglected),while Msothro•ng awayany othergr•i-


A = v(q,))+ n

As is customary, we have separated ß into its uniform gradient and fluctuating parts, but only for the first of the

two termson the right-handsideof (7). This procedureis followedby an evaluation of the divergenceof Nv,, which is then inserted back into the electron continuity equation. Usingthe standard procedure,we then considerzeroth-order gradients in the perpendicular plane to be important only if they have a component that is either paxallelor antiparallel to the ambient electricfield. The resultingequationcan be

ent effect on the ions. These •sumptions axe reasonably co•istent with our use of the fluid description, which a• pliesbetter to lowerfrequencies.This givesthe appro•mate bMance


0W •

q• yi mi



We nowFouriertransformequation(13) in spaceanduse (15) to replaceM1 potentiMfluctuationterms by density fluctuation terms. This gives


(16) (9) where

wherethe nonlinearcouplingcoefficientis givenby

c xi- •--•y• cO,(E.i) i Vr = •E K=

VNo x i No

n, (VNo. i)i •'•


Mkklk• = Mkk•k• =-M_k,_kl,_k• = (10)

Noticethat parallel (i.e., z component)electricfieldsand density gradients will not be discussedany further here. With similar manipulations for the ions and after mul-

I + •b k• k• i20,v'i'k•xk2ñ (Wk•


Sincethe couplingcoefficientwasderivedby linearizingthe ion equations,it is not inconsistentat this stageto use the linear expressionsfor the eigenfrequencies in the equation



for the couplingcoefficient (moreon thisin paper3). This We then take the density,or n moment,of (13), whichleads is obtainedby droppingthe nonlinearcouplingterm itself

to the conservation


in (16). In that case,weusea Laplacetransform in timeto recoverthe well-knownlinearexpressions [e.g.,Fejer et al., 1986] •

dzdynZ•n =0 (2a) In termsof the Fouriercomponents, (23) can be rewritten

k. Vr


(1+•b) {•---(a•k - k•C•)-t•//k} The expression for the lineareigenfrequency cannowbe used in (17) to yield

0 • I"k a-• k

-2Zk l +g,v•

Ot •

Yi •.' kl.L X k•.L

Mkk•k• = i--

(1 +

, (kl•l•VEk2•2•VE )


In (20), wecanoftenidentifytwodistincttimesc•esfor

the problemat hand. The two time sc•es are related to an oscillation•equency and to a decay/growthrate, respecWe can nowrewriteequation(16) in the followinguseful tively. Starting with the linear regime,the decay/growth


rate term is usually sm•l comparedto the eigen•equency

ank k+ Z Mkk,k•nk•nk• OtPiw•n = k•,k•=k-ki

--(1-1-•) .-•'•+k•C•

and modecouplingtermsbecause,first • is typicallymuch le• than 1 and, second,K