Imaging radar observations and nonlocal theory of …landau.geo.cornell.edu/papers/lsw.pdfImaging radar observations and nonlocal theory of large-scale plasma waves in the equatorial

To appearin 2001

Imaging radar observations and nonlocal theory of large-scaleplasma waves in the equatorial electrojet

D. L. Hysell��

Departmentof PhysicsandAstronomy, ClemsonUniversity, Clemson,SouthCarolina

J.L. Chau

RadioObservatoriodeJicamarca,InstitutoGeofısicodelPeru, Lima

Abstract.Large-scale(

��1 km) wavesin thedaytimeandnighttimeequatorialelectrojet

arestudiedusingcoherentscatterradardatafrom Jicamarca.Imagesof plasmairregularitieswithin themainbeamof theradarareformedusinginterferometrywith multiplebaselines.Theseimagesareanalyzedaccordingto nonlocalgradientdrift instability theoryandarealsocomparedto nonlinearcomputersimulationscarriedout recentlyby Ronchi et al. [1991] andHu andBhattacharjee[1999]. Inthe daytime,the large-scalewavesassumea non-steadydynamicalequilibriumstatecharacterizedby the straininganddestructionof the wavesby shearanddiffusion followed by spontaneousregenerationaspredictedby Ronchi et al.[1991]. At night,whensteepplasmadensitygradientsemerge,slowly propagatinglarge-scalevertically extendedwavespredominate.Eikonalanalysissuggeststhatthesewavesaretrapped(absolutelyunstable)or arenearlytrapped(convectivelyunstable)andable tunnelbetweenaltituderegionswhich are locally unstable.Intermediate-scalewavesaremainlytransient(convectivelystable)but canbecomeabsolutelyunstablein narrow altitudebandsdeterminedby thebackgrounddensityprofile. Thesecharacteristicsaremainly consistentwith thesimulationspresentedby Hu and Bhattacharjee [1999]. A new classof large-scaleprimary wavesis found to occuralongbandsthat sweepwestward anddownward from highaltitudesthroughtheE regionat twilight.

1. Introduction

Large-scalewaves were first detectedin the equatorialelectrojettwo decadesagoby radarinterferometryandarenow believedto dominatethestructureanddynamicsof theregion. Thesewavesareproducedby the gradientdrift in-stability throughoutthe day and night except briefly nearsunriseand sunsetwhen the ionosphericelectric field re-verses.They propagatehorizontally in the directionoppo-�

OnleaveattheRadioScienceCenterfor SpaceandAtmosphere,KyotoUniversity, Uji, Kyoto 611-011,Japan�

Now at the Departmentof Earth and AtmosphericSciences,CornellUniversity, Ithaca,New York

sitetheelectrojetcurrentandhave wavelengthscomparableto or longer than the backgroundplasmadensitygradientscalelength. Their intricatestructureis hintedat by radarDopplerspectrograms,interferograms,andhigh-resolutionrangetime intensity(RTI) backscattermaps[Kudekiet al.,1982;Swartzand Farley, 1994]. The wavesalsodominatedensityandelectricfield irregularitiesobserved by sound-ing rockets[Pfaff et al., 1982,1987;Pfaff , 1991]. In recentyears,large-scalewaveshaveundergonenumeroustheoreti-cal andnumericalinvestigations(seeFarley [1985]; Kudekiet al. [1985], Ronchi [1990], and Hu and Bhattacharjee[1998] for reviews). The theoryof daytimewaves in par-ticular hasmaturedto the point of driving thedevelopment

1

Hysell: Large-scaleplasmawavesin theelectrojet 2

of new experimentaltechniquesto validateit.

Large-scalewavesin thenighttimeelectrojet,meanwhile,arepoorlyunderstoodby comparisonandareneitheraswellcharacterizednor asreadilyanalyzedasthedaytimewaves.Soundingrocketsare the only meansof probing the elec-trojet in situ, but themajority of therocket experimentsde-votedexplicitly to electrojetstudieswerecarriedout in day-time. Thefew nighttimeelectrojetsoundingrocket datasetsavailablerevealelectrondensityprofilesthatarejaggedandirregular (e.g. Prakashet al. [1972]; Pfaff et al. [1982]).Neutralwinds appearto substantiallyalter the polarizationelectricfield in the electrojet,particularlyaroundandaftersunset[Kudekiet al., 1987;ReddyandDevasia, 1981;Hy-sellandBurcham, 2000;Hyselletal., 2001].Thepassageofthesolarterminatorintroducessteephorizontalconductivitygradientsinto theelectrojetregionandinducesamorecom-plicatedcurrentsysteminvolving vertical currentsthat linkdirectly with theF region overhead[Haerendelet al., 1992;Haerendeland Eccles, 1992]. Numericalmodelsof night-time large-scalewavesneedto incorporatetheseshears,cur-rents,andconductivity gradientsselfconsistentlyin ordertogiveacompletetheoreticaldescriptionof thenighttimecase.

Muchof whatis knownaboutlarge-scaleelectrojetwaveswas discovered using coherentscatterradars. However,measurementsfrom conventional, monostatic,fixed-beamradarsareinherentlyambiguous;spatialandtemporalvaria-tions in the scatteringmediumgenerallycannotbe distin-guishedunlessthe velocity of the scattererstransversetotheir line of sight is known. Moreover, the scatteringvol-umeof a radarevenaslargeasJicamarcais comparableindimensionto thewavelengthof thewavesin questionhere,makingthe wavesdifficult or impossibleto resolve. Radarinterferometryusinga singlebaselinemitigatestheseprob-lemsandled to thediscoveryof large-scaleelectrojetwaves[Farley et al., 1981;Kudekiet al., 1982].

This paperpresentsobservationsof large-scalewavesintheelectrojetmadewith a new in-beamradarimagingtech-nique. The technique,introducedat Jicamarcaby KudekiandSurucu [1991] anddevelopedby Woodman[1997] andHysell and Woodman[1997], utilizes multiple interferom-etry baselinesto resolve spatial-temporalambiguityandtodiscernfine-scalestructurewithin the radarscatteringvol-ume.Weshow high-resolutionimagesof individualprimarywavesandwave packetspropagatingthroughtheequatorialelectrojetover Jicamarcawithin the radarbeam.Animatedsequencesof imageshighlight thedispersioncharacteristicsof thelarge-scalewavesthereandalsogiveevidenceof newwave phenomenain theelectrojetwhich have not beenpre-viouslydetected.Theimagesform thebasisfor directmodeldatacomparisonsandprovideamorecompletepictureof thelarge-scalewavesthanwasavailablepreviously.

Thispaperbeginswith apresentationof in-beamimagingobservationsof large-scalewavesin thedaytimeandnight-time electrojetmadeat Jicamarca.Contemporarytheoriesof large-scalewavesare then reviewed. The theoriespro-vide jumping-off pointsfor interpretingthemainfeaturesofthe images. We argue that the nonlineartheoryof Ronchietal. [1991]accuratelydescribesthedynamicalequilibriumstateevident in the electrojetin the daytime. An extensionof nonlocal,lineartheoryoffersaqualitativeexplanationforthecharacteristicsof thelarge-scalewavesobservedatnight,which appearto be mainly consistentwith the numericalsimulationsdescribedby Hu andBhattacharjee[1999].

2. Imaging radar observations

Imagesof thebackscatterbrightnessdistribution(thedis-tributionof backscatterintensityversusarrival angle)withinthebeamof a radarcanbeformedfrom interferometrydatataken with multiple baselines.It is well known that inter-ferometrywith a singlebaselineyields the first andsecondmomentsof thebrightnessdistribution [Farley et al., 1981].Morebaselinesyieldmoremoments,andasufficientnumberof momentsdefineanimage.Space-timeambiguitiesinher-ent in conventionalradarwork are resolved by the imageswhich possessspatialdegreesof freedomtransverseto theradarbeam.Furthermore,thespatialresolutionof thetech-niqueis limited by the lengthof the longestbaselineratherthan the sizeof the main radarantennaarray. The formercangenerallybe increasedreadilywhereasthe lattergener-ally cannot,andit is possiblein practiceto form very highresolutionimageseven using radarswith small main an-tennaarraysandbroadbeams.Imagesformedfrom differentDopplerspectralcomponentscanbecombinedinto compos-ite images,highlightingthespectralcharacteristicsof differ-ent regionsof the scatteringvolume. Animatedsequencesof imagesdepicthow thescatterersevolveover time. All ofthe information in the scatteredsignal is fully utilized andobjectively presentedthis way.

At Jicamarca,6 nearly collinear antennamodulesareusedfor reception. From them,the cross-correlationscor-respondingto 15 nonredundantinterferometrybaselinesarecomputed.The longestof the baselinesis 94 wavelengths.Themeasuredcross-correlationor visibility function(cross-correlationversusspatial antennadisplacement)for eachrangegateis relatedto the desiredbrightnessdistributionby a Fouriertransform(e.g. [Thompson, 1986]).Both func-tions canbe regardedasone-dimensionalsincethe scatter-ers in questionare highly field-aligned. Becausethe visi-bility distribution is incompletelyandimperfectlysampledin practice,andbecausethedirect transformationfrom vis-ibility to brightnessspacemay be ill-conditioned, regular-


izationtechniquesandstatisticalinversetheoryaretypicallyappliedto the reconstructionof the brightnessdistributionfrom the data. The particularalgorithmusedhereemploysentropy as the regularizationmetric [Ables, 1974; Jaynes,1982;Skilling and Bryan, 1984]. A detaileddescriptionofthealgorithmwasgivenby HysellandWoodman[1997].

Figure1 shows in-beamradarimagesof coherentscat-ter from plasmairregularitiesin thedaytimeelectrojetoverJicamarca.The vertical andhorizontalaxesof eachimagepanelrepresentaltitudein kilometersandzenithanglein theequatorialplanein degrees,respectively. Eachpixel in theimagesis 0.1� wideby 300to 450m high,andtheincoherentintegrationtimefor eachimageis 4 sec.At analtitudeof 100km, thezonalextentof the imagesis approximately23 km.Thelightnessof eachpixel representsthesignal-to-noisera-tio on a log scale,typically from 3–30dB. SinceJicamarcaoperatesat a frequency closeto 50 MHz, thelightnessis anindicationof theintensityof 3 m irregularitieswhichthenactastracersof intermediate-andlarge-scalewaves. Note thatthetransmittingantennamainly illuminatesthecentralpor-tion of theimagingfield of view, explainingwhy the imageperipheriesaremostlydark.Thehueof eachpixel representsthe first momentDopplervelocity, rangingtypically from -600 m/s for red to +600 m/s for blue. The color scaleiscontinuous,with greencorrespondingto zeroDopplershift.Finally, the saturationof eachpixel representsthe Dopplerspectralwidth; pure(pastel)colorsindicatenarrow (broad)spectra.

The panelson the top row of Figure1, computedfromdatataken on November6, 2000,aretypical for imagesofthe daytimeelectrojet. In them, alternatingpink andbluebandscanbe seenpropagatingwestward (toward the left).Theseare the alternatingphasesof primary gradientdriftwaveswith wavelengthsof a few km. The bandsaremostdistinctbetweenabout97 and109km in altitude.Thespec-traunderlyingtheechoesarebroadtype2. Animatedimagesreveal strongvertical shearin the apparentflow, which is150m/swestwardat 110km altitude,stationaryat 100km,andslow andeastwardbelow 100km. Shearin theelectro-jet arisesfrom altitudevariationsboth in the electrojetcur-rent and the zonalneutralwind [Hysell et al., 2001]. Thewaveformsstrainunderthe shear, twist, decreasein wave-length,andflattenover time. Eventually, they vanishevenasnew waveformsaregenerated.Thelifetimesof individualwaveformsis on the orderof tensof secondsto a minute.A shearednon-steadydynamicalequilibrium flow appearsto besustained.Similar remarkshold for the imagesin thebottomrow of Figure1, which correspondto November8,2000. This time, however, thehorizontalwavelengthof thewavesis about3 km.

Theimagepanelsin Figure2,whichcorrespondto echoes

observed a few hours earlier in the day on November7,2000, depict a qualitatively different situation. Regularwaveformswith a wavelengthof about1 km areclearlyevi-dentin the images.Thetop of thescatteringregion is moresharplydefinedthanthebottomasmightbeexpectedon thebasisof linear, local instability theory. This is due to thefactthattheverticalplasmadensitygradientreversessignintheE region topside,aneffect which is stronglystabilizing.Animation of the panelsshows that the flow hasrelativelylittle shearthis time. Thewestwardpropagationspeedof thewavesis only about20 m/s. The waveformsvisible in theimageshave long lifetimeson theorderof a few minutes.

Turning to the caseof the nighttimeelectrojet,Figure3showsimagesof large-scaleelectrojetirregularitiesobservedonApril 24,2000aftertheeveningreversalof thezonalelec-tric field. Note that the zenithanglespanof the imagesonthe top row is half that of all the other images. In the toprow, we seethedifferentphasesof a tilted large-scalewavepropagatingeastward anddownward throughthe scatteringvolume. In animations,oneseesthatall of theblueregionsthat appearto fall alongthe line from the top centerto thelower left cornerof the imagesmove togetheraspart of awavefront. This is onephaseof thewave. Opposingphasesof the wave, which appearasdisconnectedlinesof redandgreenregionsin theimages,bothprecedeandtrail theblue-shiftedphase.Many morephasesfollow continuouslythere-after (not shown). However, the large-scalewaves shownherebehave morelike wave packetsthanlike a planewave.Thewavefrontsevolvein timeandchangeorientationastheypropagate. Moreover, the amplitudeof the backscatterismodulatedin altitude,andthepatternof theamplitudemod-ulationshifts in altitudeon a timescaleshorterthanthe pe-riod of thewaves.Theeffect suggeststhatthewave packetssubtendregionsof spacethatarealternatelystableandunsta-ble. Theeastwardphasevelocity of thewave seenherewasnever greaterthanabout50 m/s andvarieddrasticallywithtime andaltitude. The wave packet extendedover a 20 kmaltituderangeandhadlifetime greaterthan its transit timethroughtheradarbeam.Differentphasesof thewave prop-agatealternatinglyupwardanddownward;in RTI diagrams,theeffect is to producetightly interleaved,crosseddiagonallines [Swartzand Farley, 1994; Farley et al., 1994]. Un-like thedaytimecase,theselarge-scalewavesdo not appearto strainanddissipateundertheactionof shearflow. Also,the wavelengthof the wavesincreasesat night, althoughitis difficult to estimatetheir wavelengthsincethey arenot somonochromaticasin thedaytime.

Notethattheblue-shiftedwavefrontin theseimageshasanearlypurebluecolor. Conventionalspectraanalysisshowsthat the echoescoincidentin altitudewith the packet havetype1 spectra.ClearlysecondaryFarley-Bunemaninstabil-


Figure 1. Radarimagesof daytimeelectrojetwaves. (Top row) November6, 2000. (Bottomrow) November8, 2000. Thelightness,hue,andsaturationof eachpixel in theimagesrepresentthebackscatterintensity, Dopplershift, andspectralwidthof theechoesasa functionof rangein km andzenithanglein degrees.


Figure 2. Imagesof daytimelarge-scalewavesobservedonNovember7, 2000.No shearis evidentin theflow here.


Figure 3. Radarimagesof nighttimeelectrojetwavesobservedon April 24, 2000. Notetherange-spreadmeteorevident intheimageson thebottomrow of thefigure.


ities areexcitedby thepolarizationelectricfield associatedwith theprimarywavepacketandtheverticalHall currentitgenerates.This secondarywave generationmechanismwasproposedby Sudanet al. [1973] andhasbeensupportedbynumerousinvestigationssince.Here,weseethemechanismillustratedvividly.

Similarremarksholdfor thewaveformsshown in thebot-tom row of Figure3. Here,upgoinganddowngoingphasesof theeastward-propagatingwavesareshown interleaved.Inaddition, therearebright, diffusebandsof backscatterjustabove and below 100 km altitude. A distinct gap lies be-tweenthe bands. Thesebandsoriginally formed at about2110LT at an altitudeof 120 km andthendescendedin aslow arcuntil about2230LT whenthey reachedanaltitudeof about96km and,soonafter, vanished.Theprimarygradi-entdrift wavesthatmakeupthesescatteringlayersevidentlyhave too smalla scaleto beresolvedhere.We canhypothe-sizethataverysteepplasmadensitylayerpresentedastablegradientoriginally at120km altitudeandunstablegradientsabove andbelow that. The layer migrateddownward overtime in a mannerreminiscentof anintermediatedescendinglayerexceptat E regionaltitudes[Mathewsetal., 1997].

Thesignatureof a range-spreadmeteorechoesis clearlyvisible in the first two framesof Figure3. Suchtrails arequite commonin the dataandoften persistfor tensof sec-onds. Coincidentally, this trail passedthrougha scatteringblob at 94 km thatappearedabruptlyin theimagesat about2155 LT at preciselythe momentand location of anotherrangespreadmeteorecho. The blob then drifted slowlywestwardandremainedvisibleuntil about2157LT. Thefirstmeteorseemsto have playedsomerole in initiating plasmainstabilitiesin the lower E region. Similar phenomenahavebeendiscussedby ChapinandKudeki[1994] andhave alsobeenobservedatmid latitudes[HysellandBurcham, 1999].

Finally, Figure4 presentsanaspectof large-scalewavesin theequatorialelectrojetnotpreviously recognizedor pre-dicted. The figure shows imagestaken aroundtwilight onApril 24, 2000(top) andNovember6, 2000(bottom). Onbothdays,linesof scatterersformedat analtitudeof about130 km and then sweptdownward and westward throughthe radarfield of view. The velocity of the scattererswasabout50 m/s. The scatteringlines were highly structuredandseemedtoexhibit thesignaturesof primarygradientdriftwaveswith wavelengthson theorderof 1 km. TheDopplershifts of the echoeswerepositive and, in someplaces,ap-proachedtheion acousticspeed.

This phenomenonis evident in several other imagingdatasetsandis frequentlydetectedby theJULIA (JicamarcaUnattendedLong-terminvestigationof the IonosphereandAtmosphere)radaratJicamarca,whereit appearsasclustersof high-altitudeE region echoesthatoccuraround1830LT.

Sometimes,severalparallellinesof scattererssweepthroughthe scatteringvolumesequentially. While the natureof theechoesis not known, we tentatively associatethemwith agradientdrift or interchangeinstability driven by the hori-zontalconductivity gradientsandvertical (upward)electro-jet currentsknown to be presentin the vicinity of the solarterminator. A detailedanalysisof thephenomenonremainsto beperformed.

In the next sectionof the paper, we evaluatecontempo-rary theoriesof large-scalewavesin theequatorialelectrojetin light of the radarimages. Theoreticalwork in the caseof gradientdrift wavesin thedaytimeis extensive but con-flicting. Ronchi et al. [1991] arguedthat intermediate-andsmall-scalewavesin the daytimeareconvectively stableinshearedflowsbut thatlarge-scalewavesarecontinuouslyre-generatedby nonlinearmodecoupling,leadingto their pre-dominance.We will argue that the daytimeradarimages,which show evidenceof a non-steadydynamicalequilib-rium statewhenshearis present,supportthis theory. In theabsenceof shearedflow, meanwhile,a linear, local theorywhich includesanomalouseffectsalonecanaccountfor thepredominanceof large-scalewavesin thedaytime.Existingtheoreticalwork in thecaseof nighttimegradientdrift wavesis, by comparisonto thedaytimecase,lessmature.We willshow with a linearnonlocalanalysisthat,at night, theshortgradientscalelengthsassociatedwith jaggeddensityprofilescauselarge-scalewavesto beabsolutelyor convectively un-stable.Thesewavesarehighlydispersiveandpropagateveryslowly. Trappedandnearly-trappedwave modescantunnelbetweenregions of spacethat are locally unstableand, inso doing, extendover a large altituderange. Suchtunnel-ing, whichwaspredictedin thecomputationalstudiesof Huand Bhattacharjee[1999] but which is an inherentlylinearphenomenon,is evidencedby the nighttime radarimages.Intermediate-scalewavescanalsobeabsolutelyunstableatnight in narrow altitudebandswheretheplasmadensitygra-dient is locally favorable. Thesebandscan sometimesbeseenin the nighttimeradarimagesandwerealsopredictedby Hu andBhattacharjee[1999].

3. Theoretical background

Large-andintermediate-scaleplasmawavesareproducedin the equatorialelectrojetdirectly by the gradientdrift in-stability [Simon, 1963;Hoh, 1963]. A linear, local disper-sion relation for the waves can be derived by combiningthe linearizedfluid equationsfor theelectronandion gasesandincorporatingappropriateequationsof state.By assum-ing planewave perturbationsfor the field quantitiesof theform exp �� , where � is thedisplacementinthe planeperpendicularto the geomagneticfield and the


Figure 4. Radarimagescorrespondingto twilight conditions.(Top row) April 24,2000.(Bottomrow) November6, 2000.


wave vector, the real and imaginarypartsof the frequency�� "!#%$ of purely field-alignedwavescan be deter-mined(see,for example,Kudekiet al. [1982]):� � � ��&�'�(*)+��'�(�,%��-+!/.0��-0!2143� 5 143�� !/ ��6' (�, (1)$ � 1 �1 7� � �8 ��' (�, �9�:1 3�;=< �?>&@�A � (2)1 �1 B --+!:. 1DC143�E F ,G , (3)

Here, ' (*) and ' (�, arethezero-ordertransverseelectronandion drift velocities,respectively. The former is associatedmainly with the Hall drift of the electronsin the electrojet,andthelatteris duemainlyto neutralwind effects.Also, @ isthedissociative recombinationrate,and . is theanisotropyfactordefinedby . = F ) F , 5 G ) G , . Thetransversediffusionisdefinedby ; < = HI�J 1 3�KL3M 5 �-N!O.P� . The F�Q termsrefer tothe ion-neutralcollision frequenciesfor speciesR , K M is theion acousticspeed,E is the vertical densitygradientscalelength, 1DC is thecomponentof the wavevectorin thedirec-tion of ' (*) , andtheothertermshave their usualmeaning.

3.1. Daytime case

Kudekiet al. [1982] performedthe first detailedexperi-mentalstudyof primarygradientdrift wavesin theelectro-jet usingboth spectralanalysisand interferometryat Jica-marca. They observed dominant,large-scalewavespropa-gatinghorizontallywith wavelengthsbetweenabout2 and6 km. Secondary, vertically propagatingtwo-streamwaveswereshown to be generatedby the electronconvectionas-sociatedwith the large-scalewavesashypothesizedby Su-danet al. [1973]. Secondarygradientdrift waveswerealsoobservedandarenow thoughtto begeneratedby nonlinearmodecouplingandplasmaturbulence[Sudan, 1983]. Per-hapsmostremarkably, thelarge-scalewaveswereobservedpropagatingat phasespeedsof only abouthalf thenominalelectrondrift velocity. Kudekiet al. [1982] notedthat thelarge-scalewavesbelongedto the S $TSVUWS � �XS or the 1YUZ1 �regime. By retainingthequadratictermin thedenominatorof (1) andconsideringthelow frequency limit of thedisper-sionrelation,they accountedfor theslow propagationspeedsof thewaves.They alsosuggestedthatthedispersivenatureof the large scalewavesinhibited their dissipationthroughmodecoupling and contributed to their long lifetimes andcoherence.Finally, they arguedthat the predominanceofwaveswith the observed wavelengthswasat leastroughlypredictedby (1) and (2) in which the recombinationtermwouldseemto imposea long-wavelengthcutoff atprecisely1=[\1 � , where 1 � E?[ 23 for typical daytimeparameters.

However, Huba and Lee [1983] showed that the growthrate predictedby the linear, local dispersionrelation actu-

ally peaksin the vicinity of 14E][ 100. Thus, local the-ory favorstheexistenceof intermediate-scalewaveswhereaslarge-scalewaves are observed to predominate. They at-tributedthe discrepancy betweentheoryandobservation tothe inapplicability of local theory to the large-scalewaves.An eigenmodeanalysiswas performedwhich allowed forverticalshearin the electrojetHall current. Shearin physi-cal spacecausesconvectionin Fourierspace(convectionofenergy to higherwavenumbers)andcausesotherwiseunsta-ble intermediate-scalewavesto bediffusively damped.Thewavesmostaffectedarethosewith wavelengthscomparableto or smallerthanthe scalesizeof the shear. Recombina-tion meanwhileestablishestheouterscalefor wave growth.Shearwas shown to stabilizethe small- and intermediate-scalemodesof the gradientdrift instability andto increasethe wavelengthof the fastestgrowing eigenmodesto 14E2U20 overabroadrangeof daytimeconditions.

Shortly thereafter, Fu et al. [1986] reexaminedthe roleof shearin E and F region plasmainstabilities. Despitethe fact that intermediate-scaleeigenmodesaredampedbyshear, one-dimensionalinitial value computersimulationsof the shearedsystempredicteda strongtransientresponseat intermediateandsmall scales. Fu et al. [1986] pointedout thattheshearedsystemis non-normalandthatthelineareigenmodesarethereforenotnecessarilyorthogonalor com-plete. They termedthe transientresponsea “quasi-local”mode,inaccessibleto eigenmodeanalysis. The predicted,strongtransientresponseat intermediatescalescontradictedradarevidenceof large-scalewaveswith 1^[_1 � dominatingthedaytimeelectrojet.

Ronchi et al. [1989] revisited theproblemof large-scaledaytimewaves,performingnonlocalanalyseswhichconsid-eredheightdependentplasmadensity, Hall current,andelec-tronandion mobility profiles.They performedaneikonalorgeometricopticsanalysisalongthelinesof Weinberg [1962]which yieldeda moretransparentinterpretationof thetran-sientresponseof Fu et al. [1986]. Themethodology, whichinvolved a WKB expansionabout the local dispersionre-lation andwhich yields informationaboutboth the steady-stateandtransientbehavior of a system,will be importantfor ouranalysisandis outlinedbelow.

Theappendixgiveslinearizedmodelequationsfor large-scalegradientdrift wavespropagatingin regionsof theelec-trojet wherethe backgroundplasmadensity, electric field,mobilities, and diffusivities too are allowed to vary withheight.ThemodelhasbeenFouriertransformedin thehori-zontaldirectionandin time; it will beevaluatedatonevalueof 1X` at a time and is consequentlyone-dimensional.As-sumethat solutionsto the equationsfor particular 1&` havetheform a %bDcd��e� a � �bfc��g ,ihDjlk�m n�o (4)


where p is the eikonal which controls rapid variationsinphasealong the ray path and

a � is a slowly varying am-plitude function. We define the local vertical wavevec-tor and frequency, respectively, by 1&kq%bDcd�� B r p 5 r b and��bfc�� B � r p 5 r � , wherethe frequency is real andobeysthelocal dispersionrelation �s�#tZ%1Xk&udv�c�� . It canthenbeshown that t playstherole of theHamiltonianandthatthetrajectoriesof theray pathsin the 1 k �/b planeareballistic(e.g.[LandauandLifshitz, 1997]):w bw � � r tr 1 kw 1 kw � � � r tr bw tw � � r tr �If thereis no explicit time variationin the local dispersionrelation, then the wave frequency � is a constantof mo-tion alongthe ray, andcontoursof constantfrequency givetheorbitsof wavepacketspropagatingthroughthemedium.Ronchi etal. [1989]extendedtheeikonalanalysisto thedis-sipative casein which thefrequency hasrealandimaginaryparts. To a goodapproximation,the trajectoriesof the raypathscontinueto follow contoursof therealpartof thefre-quency. However, the amplitudeof the wave packets canbe modifiedas they move throughregionsof physicalandFourier spacein which the local growth rate is positive ornegative.

We evaluatedthe local frequency andgrowth ratein the1XkL�2b planeaccordingto (A1) for daytimeconditionsandplottedthe resultson the top row of Figure5. This figure,calculatedfor waveswith 1&` = -=x_-�y{z}|N~�z�� , essentiallyreproducesFig. 7 of Ronchi et al. [1989]. Theconstantfre-quency contoursrepresenttrajectoriesin 1 k �Ob spaceforwave packetsoriginating from backgroundnoise. A wavepacket originatingat the point marked ‘1’ in the upperleftpanelwill propagateto theleft alongan � =constcontour, ex-periencegrowth for atime,but ultimatelydecayafterexitingtheunstableregiondelineatedby the $ =0 curvein theupperright panel. This is a convectively stable,transientwave.A packet originatingat point ‘2’, meanwhile,will follow aclosedcontourandbecometrapped.In this case,theclosedcontourlies wholly within the zonewhere $/� 0. A wavetraveling on it will be absolutelyunstableandcorrespondsto anunstableeigenmode.

Ronchi et al. [1989] producedanotherfigure similar toFigure5 exceptcalculatedfor intermediate-scalewaveswith1 ` = ��x\-�y4z 3 ~�z�� . In that case,the lack of dispersioncausedthe ray trajectoriesto be nearly straighthorizontallines. Therewereno closedcontours,andall the wave so-lutions thereforecorrespondedto transientmodes. Ronchi

et al. [1989] interpretedthe quasi-localmodeof Fu et al.[1986] in this way. Ronchi et al. [1989] also pointedoutthatmodeson opencontourscanbeeffectively trappedandabsolutelyunstableif they exist on a ridge in the 1Xk=��bplanewhere ��t vanishes.Seekinganexplanationfor whytheabsolutelyunstableintermediate-scalewavesarenotob-served in the daytime, Ronchi et al. [1990] followed St.-Maurice [1987] and consideredthe effects of small-scalegradientdrift turbulenceon electrontransport.An effectiveturbulentPedersenelectronmobility wasderivedas��< ) B -> .-+!:. � �� )� < )}� 3��%� AA �� 3X� � < )in which � � ) is theHall mobility andwherethedensityfluc-tuationsarethosedueto small-scale( �� 100m) turbulence.Thecorrectionto theelectronmobility, which canbemuchlarger thanthe classicalmobility givendensityfluctuationsof a few percent,telegraphsthroughto the dispersionrela-tion for gradientdrift wavesby increasingtheelectronPed-ersenconductivity and therefore . . Furthermore,it alterstheverticalpolarizationelectricfield profile in theelectrojet,which is well approximatedby thezonalelectricfield mul-tiplied by theradioof thelocalHall to Pedersenconductivi-ties.Themaineffecthereis to reducesignificantlytheverti-cal polarizationelectricfield, particularlyat altitudesbelowabout105km. Ronchi et al. [1990] showedthatanomalouselectronmobility causesthefastestgrowing linearlocalgra-dientdrift modesto occurat longerwavelengthsandhigheraltitudesthanwould otherwisebepredicted,independentofsheareffects. By virtue of reducingthe 1 � 5 1 ratio for thefastestgrowing mode,it also causesthe large-scalewavesto beessentiallynondispersive.This impliesthatmodecou-pling betweenthewavescanbevery effective. Finally, therelatedreductionof the local vertical polarizationelectricfield reducesthephasespeedsof thewavesto approximately1/2to2/3of thezonalelectrondrift speed,againindependentof shearor low-frequency effects.Thesefindingsareimpor-tantsincethelocaldispersionrelationultimatelydetermineswhich modes,transientandtrapped,canarise.

The bottom panelof Figure 5 shows dispersioncurvescalculatedfor waveswith 1X` = -�x?-�y4z}|+~�z�� , this time in-cluding the effectsof small-scaleturbulencewith an RMSamplitudeof 0.04. The curves indicate that the trappedmodeshavedisappeared,leaving only transientwaves.Like-wise, Ronchi et al. [1990] showed that anomalouselectrontransporteliminatedthe absolutelyunstableintermediate-scalewave modereferredto previously. Consequently, allintermediate-andlarge-scalewavesin the daytimeelectro-jet were shown to be transientmodes,convectively stablerather than absolutelyunstable. Ultimately, Ronchi et al.[1991] would perform nonlinearsimulationsof the gradi-


1

2

Figure 5. Contoursof constantfrequency andgrowth ratederivedfrom thelocaldispersionrelationdescribedin theappendixfor 1&` = -�x2-�y{z}|�~�z�� , plottedin the 1 k �2b plane. (Top) Daytimecasewith no anomalouselectronmobility. (Bottom)Daytimecasewith anomalouselectronmobility. Thezonalelectricfield for bothcaseswastakento be0.65mV/m. Modelatmosphericandionosphericparameterswerederivedfrom theInternationalReferenceIonosphere(IRI) andMassSpectrom-eterIncoherentScatter(MSIS)models[Bilitza etal., 1993;Hedinetal., 1996].Collision frequencieswerederivedaccordingto expressionfrom Richmond[1972] andGagnepainet al. [1977]andreproducedby Forbes[1981] in his equations27–30.


entdrift instability to explain thepersistenceof thetransientmodes. Shearstabilizesall waveswith wavelengthscom-parableto the sheargradientscalelengthby convectioninFourierspacewhichcausesashift from smallwavenumbersto largeandleadsto diffusivedissipation.However, thenon-linearmodecouplingthattakesplaceis suchasto transportenergy from large wavenumbersback to small wavenum-bers.A dynamicalequilibriumsaturatedstatewaspredictedinvolving theshearingandreproductionof large-scalewaveswhosecharacteristicswerepredictedby thelinear, localdis-persionrelation with anomalouseffects included. Simu-lationspreferentiallyproducedtransientwaveswith wave-lengthsof about2 km andlifetimesof about30 s.

3.2. Nighttime case

Thefactorsoutlinedabovecontinueto influencetheevo-lution of large-scalewavesatnight. Severalnew factorsalsoemerge,however, includingthedrasticreductionof thedis-sociative recombinationratewith the reducedplasmanum-ber density. This tendsto relax the long- wavelengthcut-off on the lineargradientdrift growth rateandmight beex-pectedto increasetheouterscaleof the instability. This ef-fect is balanced,however, by the emergenceof layersandsteepdensitygradientsasshown by Prakashet al. [1972]andothers.Finite gradientscalelengtheffectsareexpectedto reducethe growth rate of large-scalewaves and favorintermediate-scalewavesat night. Applying aninitial valuecalculationto nighttimerocket data,Hu and Bhattacharjee[1999]predictedthatthefastestgrowing linearnonlocalgra-dientdrift modesshouldhave wavelengthsof only about20m. This contradictsradarevidencethat the wavelengthofthedominantlarge-scalewavesactuallyincreasesfrom dayto night, however. Hu andBhattacharjee[1999] ultimatelyrecoveredthepreferencefor long-wavelengthwavesin fullynonlineartwo-dimensionalsimulationsof thenighttimegra-dientdrift instability. Wecanpredictthis resultby revisitingtheeikonalanalysis.

We have usedthe electrondensityprofile measuredbyPrakashet al. [1972] to performan eikonalanalysisrepre-sentative of the nighttimeelectrojet.The resultsareshownin Figure6. Onceagain,the effectsof anomalouselectronmobility aretakeninto account.Intermediateandlarge-scalewavesareconsideredindividually in thetopandbottompan-els, respectively. We can seethat the intermediate-scalewavescontinueto exhibit little dispersion.While it is truethattheintermediate-scalewavesgenerallyhavemuchlargergrowth ratesthanlarge-scalewaves,thefrequency contoursshown in the top row of Figure6 areall open. Most of theintermediate-scalewavesthereforearestill transientin na-ture,evenatnight.

Note that, if an intermediate-scalewave wereto emerge

at analtitudeof about107km where,in this case,��t be-comesvery small,andif it wereto fall within thesmall re-gion of 1Xk��Ob spacewherethe growth rate is positive, itwouldbeabsolutelyunstable.Whetherandwherethissitua-tion actuallyoccursin naturedependson the shapeof thebackgrounddensityand drift profiles. The emergenceofanabsolutelyunstableintermediate-scalemodein a narrowbandof altitudesseemsto be a distinct possibility at night,however.

Considernext thelarge-scaledispersioncurvesplottedinthe bottomrow of Figure6. Despitethe effectsof anoma-lous electronmobility, finite densitygradientscalelengtheffectshave reintroduceddispersionin this caseandled tothe re-emergenceof numerousclosedfrequency contours.Someof thesefall in regionswherethe growth rateis pos-itive; thesecorrespondto unstableeigenmodes.Moreover,thereareopencontourswhichspanaltitudesfromthebottomto the top of the electrojetregion andwhich wind betweentheclosedcontours.Thesepathspermitcommunicationbe-tweenthetrappedregions. Waveslaunchednearthesecon-tourswill be ableto migratein altitudebetweenthe closedcontours,growing in the unstableregionsof 1&k��_b space,tunnelingthroughthestableregions,andlingeringneartheclosedregions.Suchwavesareconvectively unstable.Notethatthefrequenciesof theclosedandnearlyclosedcontoursaresmall;thephasevelocitiesof wavesfollowing thesecon-toursareapt to be quite small, andeven wave packetsfol-lowing opencontourscanbelong lived.

The tunnelingfeatureof the large-scalewaveswasem-phasizedby Hu andBhattacharjee[1999]. Their numericalsimulationsof the nighttime gradientdrift instability pro-duceddominantkilometer-scalewavesby the time dynam-ical equilibrium wasachieved. Thesewavesspannedalti-tudesthatwerealternatelystableandunstable.Intermediate-scalewavescontinuedto bepresentthroughoutthesimula-tion andclusteredin regionsof theshearflow thatweremostfavorablefor growth. Both forwardandinversemodecou-pling weredetected.

4. Analysis

Ronchi et al. [1991] describeda non-steadydynamicalsaturatedequilibrium statein the daytimeelectrojetchar-acterizedby the dissipationof large-scaleirregularitiesbyshearand then diffusion and a regenerationof the irreg-ularities by nonlinearmode coupling from small to largescales.Their simulationswereshown to be generallycon-sistentwith radarandsoundingrocket observations,but thematchwasnot definitive. Hu andBhattacharjee[1998]pre-sentedsimulationresultspredictinga saturatedstatechar-acterizedby isotropic,large-scaledensityirregularitiesthat


Figure 6. Contoursof constantfrequency andgrowth ratederivedfrom thelocaldispersionrelationdescribedin theappendixandplottedin the 1&kT��b plane.(Top)Nighttimecasefor waveswith 100m horizontalwavelengths.(Bottom)Nighttimecasefor waveswith 2 km horizontalwavelengths.Thezonalelectricfield for bothcaseswastakento be-0.2mV/m. In bothcases,a 0.04RMSdensityfluctuationlevel wasusedto estimatetheeffectsof anomalouselectronmobility.


werealsogenerallyconsistentwith availableobservations.Discriminatingbetweenthesetheoriesrequiresinformationaboutthedetailedmorphologyanddynamicsof theelectro-jet flow.

Daytime in-beamimagingexperimentssupportthe the-ory andsimulationspresentedby Ronchi etal. [1991]. Theirfigure 16 virtually reproducesthe imagesin the top row ofFigure1, andtheir animatedsimulationsarestrikingly sim-ilar to animationsmadefrom the radar images. The pre-dicted wavelengthof the large-scalewaves (about2 km),lifetime (about30s),andpredominanceof constantlyevolv-ing, strainingstructuresexhibiting ahighdegreeof temporalvariability all agreewith observations.Thepredictedrangeof altitudesfor waveformation,about100–107km, is some-whatnarrowerthanwhatis observed.Thispredictionis con-trolled by the linear dispersionrelation, calculatedon thebasisof anelectrondensityprofile measuredduringprojectCONDOR[Pfaff et al., 1987]. Betteragreementshouldre-sultfromtheuseof aprofilemorerepresentativeof theactualconditionsin question.

Nearlyshear-freeflowscanalsoemergein theequatorialE region, asexemplifiedby the imagesin Figure2. Hyselland Burcham [2000] often observed very flat zonal phasevelocity profilesin the daytimeelectrojetusinginterferom-etry. Recently, Hysell et al. [2001] showed that the lowerthermosphericwinds predictedby the National CenterforAtmosphericResearchGeneralCirculationModel (NCARTIME-GCM) tend to producevertical polarizationelectricfields in theelectrojetthatpartially negateandtherebyflat-ten the backgroundfield profile. In the absenceof strongshear, the equationsdescribinggradientdrift waves onceagainbecomenormal, and the predominantwaves shouldbe predictedby fastestgrowing the local eigenmodes.Thelinear, local calculationsperformedby Ronchi et al. [1990],which retainthe effectsof turbulentelectronmobility, pre-dict that large-scalewaveswith wavelengthsof about500m to 1 km shouldpredominatein shear-freeflows. This es-timate agreeswith the evidenceof Figure 2, althoughtherangeof altitudespredicted,againabout100 to 107 km, isseveralkilometershigherthanthatobserved.

Contraryto intuition, radarinterferometerdatashow thatlarge-scalewavesin the electrojetcontinueto predominateat night even thoughthe densitygradientscalelengthsbe-comevery short. Hu and Bhattacharjee [1999] recoveredthis result usingnumericalsimulationsandattributed it tothe ability of the large-scalewavesto extendacrossstableregions of spaceand inhabit multiple unstableregions si-multaneously. Using an eikonal analysis,we showed thatthe large-scalewavescanexist astrappedor nearlytrappedmodesin the irregular nighttime ionospherewhile mostofthe fastergrowing intermediate-scalewavesare transients.

As thisanalysispredicted,thewavesevidentin Figure3 tendto extendover broadrangesof altitudesandundergo largeexcursionsin altitudeandorientationasthey tunnelbetweenregionsof spacethatarelocally unstable.Both thescatter-ing intensityand the Dopplershift of the echoesobservedaremodulatedasthe wavespropagate.Their zonalpropa-gationspeedsareuniformly smallcomparedto thenominalelectrondrift speed.

Finally, HuandBhattacharjee[1999]predictedthatstrongintermediate-scalewaves shouldcluster in narrow altitudebandswherethe conditionsfor instability are most favor-able.Theeikonalanalysisin section3suggestedthatabsolutely-unstableintermediate-scalemodescouldexist atnightgivenappropriatebackgroundplasmadensity and drift profiles.Theintensebandsof diffusescattershown in Figure3 likelycorrespondto thesefast-growing intermediate-scalemodes.It may be possibleto determinethis from the groundif thespatialresolutionof theimagingtechniquecanbeimproved.Otherwise,in situ dataarecrucial for investigatingthe roleof intermediate-scalewavesin thenighttimeelectrojet.

5. Discussion

Wehaveusedanew imagingtechniqueatJicamarcato in-vestigatein detail thestructureanddynamicsof large-scalewaves in the equatorialelectrojetand to evaluatenumeri-cal studiesof thewavesperformedrecently. Thenon-steadyshear-dominateddynamicalequilibrium flow simulatedbyRonchi et al. [1991] accuratelypredictsour daytimeradarobservations. That large-scalewavesare observed to pre-dominateatnight,accompaniedby intermediate-scalewavesclusteredin narrow altitudebands,wasalsopredictedby thesimulationsdescribedby Hu andBhattacharjee[1999]. Thealtitudeextentof theselarge-scalewaves,which canexceed20km, wasnotexplicitly predictedby thosesimulationsbutis consistentwith the linearnonlocaltheorypresentedhere.Thesewavesexist astrappedor nearlytrappedwave modesthatmigratebetweenaltituderegionsthatarelocally unsta-bleasthey propagateslowly eastward.Thewavesaredisper-sive at night andarevery coherentandlong-lived. Neithershearnor nonlinearmodecouplinginfluencesthedynamicsor morphologyof the wavesat night to the degreethey doduringtheday.

While shearis generallypresentduring the day and atnight, the gradientlength scaleof the shearat night tendsto be muchlongerthanit is during the day, comparabletoor longerthanthewavelengthof the large-scalewaves,andmuch longer than typical nighttime densitygradientscalelengths.Shearflow at night frequentlyexhibits amplitudes,phasing,andperiodshighly suggestiveof neutralwind forc-ing [HysellandBurcham, 2000]. Consequently, large-scale


wavesarelessinfluencedby shearat night thanduring theday. Moreover, sincethelarge-scalewavesthatexist atnightaretrappedor nearlytrappedmodesinhabitingunstablere-gionsof spacemuchof the time, they do not dependuponshear-drivennonlinearmodecouplingfor theircontinualre-generationasthey do during theday. It is possiblethat thedynamicsobservedduringthedaycould take placeat nightin regions of strongshearflow and relatively small back-groundplasmadensitygradients.Someradarimagingdatasupportsthis hypothesis.

Our imagingexperimentsidentifieda new classof large-scaleprimarywavesthatoccuraround1830LT. Thesewavesemergefirst at high altitudeswherethe ions aremarginallymagnetizedand then sweepcontinuouslydownward andwestward into the altituderegime just above that inhabitedby ordinarygradientdrift waves. Thephenomenonappearsto bevery commonbut hasnot beenstudiedquantitatively.Sucha study will requireprecisespecificationof the con-ductivity gradients,currents,winds,andelectricfieldsin thevicinity of thesolarterminator.

Theanalysespresentedherewereintendedto besugges-tive of the typesof wave phenomenathat can arisein thedaytimeandnighttimeelectrojet. More specific,quantita-tiveanalysesof in-beamradarimagesrequireconcurrentin-formation aboutthe backgroundplasmadensityand windprofilesandthebackgroundzonalelectricfield. In thepast,suchinformationhasbeenobtainableonly throughsoundingrocketexperiments.Recently, Hyselletal. [2001]andHysellandChau[2001] introducednew techniquesfor measuringwind anddensityprofilesin theelectrojetusingradarremotesensing. In the future, we hopeto couplethesenew tech-niqueswith radarimagingin orderto analyzethedispersionof thelarge-scalewavesasquantitatively asis possible.

Appendix

We derive herea linearmodelof thegradientdrift insta-bility in theequatorialelectrojetwhich takesinto considera-tion heightvariationsin theplasmadensity, mobilities,dif-fusivities, andpolarizationelectricfield. Simplified formsfor theperpendicularelectronandion velocitiesareadoptedas: '�,�� G ,F�� ¡¢� KL3M ,F ,i£ ��¤i¥{A� � , � � ; ,¦��¤i¥{A' ) � � x�§¡ 3 � F )¨£G ) �¡ � F )¨£ KL3M )G 3 ) ��¤i¥{A� � � � xY©ª � � ) � � ; ) ��¤i¥{Awhere F�«�� , G Q , and K M Q referto theion-neutralcollision fre-quency, gyrofrequency, andsoundspeedfor speciesR . In-

sertingthesevelocitiesinto thecontinuityequationsfor fluidions andelectrons,linearizing,andassumingwavesof theform

a ��Tc�� =

a %bV�q¬�4®��*%1 ` v¯�s�°�� yields the followingsystemof one-dimensionalequationsfor theperturbedden-sity andelectrostaticpotential:�\± �d� ± � 3± 3 � ± 3²3 � � A 5 A �³ � ��y (A1)

± �d� � ��¦�¯!2>´@�A � !/¨1X` � ,¦µ `�! ; , 1 3` ! � , w µ kw b! � � , µ k�� w ; ,w b?� ww b � ; , w 3w bX3± � 3 � � , � 1 3` � w 3w bV3{� � � ,E � ww b± 3 � � ��¦�¯!2>´@�A � ��¨1X` � � )*µ `�! µ�k¡ � ! ; ) 1 3`� � ) w µ kw b � � � ) µ kP! w ; )w b � µ k¡ � ww b� ; ) w 3w bV3± 3d3 � � ) � w 3w bX3 �:1 3` � ! � )E 3 ww b �8�1 ` -E ¡Here, µ ` and µ�k are the backgroundhorizontalandverti-cal electricfields in the electrojet, E is the vertical plasmadensitygradientscalelength,and E � and E 3 aredefinedas:E z�� B -A � � , ww b A � � ,E z��3 B -A � � ) ww b A � � )Nontrivial solutionsto the eigenvalueproblemrepresentedby (A1) for agivenvalueof 1 ` arethenonlocaleigenmodeswith the eigenvalues� and $ . We revert to the local solu-tionsby assumingsinusoidaleigenfunctions.Theseareeas-ily foundalgebraicallyandareplottedin the 1Xk��=b planeforgivenvaluesof 1 ` in Figure5 andFigure6.

Acknowledgments. This work was supportedby the Na-tional ScienceFoundationthroughcooperative agreementsATM-9022717and ATM-9408441to Cornell University and by NSFgrantATM-0080338to ClemsonUniversity. TheJicamarcaRadioObservatoryis operatedby theGeophysicalInstituteof Peru, Min-istry of Education,with supportfrom the NSF cooperative agree-mentsjust mentioned. The help of the staff, particularly R. F.Woodman,wasmuchappreciated.This manuscriptwaspreparedat the RadioScienceCenterfor SpaceandAtmosphereat the Ujicampusof Kyoto University. DLH is indebtedto S. FukaoandM.Yamamotofor their advice,support,andhospitality.


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