Mhd Wave Breaking in the Outer Plasmasphere

GEOPHYSICALRESEARCH LETTERS, VOL. 14, NO. 10, PAGES1007-1010, OCTOBER1987

MHD WAVE BREAKING IN THE OUTER PLASMASPHERE

T. E. Moore, D. L. Gallagher • SpaceScience Laboratory, NASA Marshall Space Flight Center, Huntsville, Alabama 35812.

J. L. Horwitz, and R. H. Comfort Departmentof Physics,The Universityof Alabama in Huntsville,Huntsville, Alabama 35899.

Abstract.Empirical models of the average magnetospheric Plasma Parameters magneticfield, plasmadensity, and temperaturedistributions are used to construct a model of the distribution of MHD wave mode Obtaininga crediblemodel of the plasmamass density and effec- speedswithin the magnetosphere.Although the MHD wavespeeds tive temperature(required for the ion acousticspeed) is a more in generalhave a smallerdynamic range than the magneticfield difficult matter. The plasmaelectron density can be measuredfairly intensityor the plasmaproperties, considerable structure and vari- reliably by noting the frequencyof the upper hybrid resonancein abilityis foundwhich will leadto interesting"optical" effects on the passivelysensed plasma waves (Persoonet al., 1983), or by an propagationof low frequencywaves. A persistentfeature of the active plasma wave soundingtechnique (Higel and Wu, 1984). derivedoptical structure, which is qualitativelyinsensitive to known However, measurementof the ion masscomposition and tempera- variabilityof the field or plasma,is a pronouncedminimum of the ture, andelectron temperature, is dependentat presentupon accurate wave speedsin the outer plasmasphere,i.e., a magnetospheric measurementsof the low-energy"core" plasmaions and electrons. "shoal."This featuredoes not mapalong magnetic field lines,but is These measurementsare very difficult to make at the low densities confinedto theequatorial region, leading to a positiveradial gradient typicalat high altitudesdue to photoelectronemission by spacecraft of wave speedsnear synchronous orbit. The breakingof earthward surfacesand the consequentpositive floating potential relative to the propagatingdisturbances in this region may play an essential role in plasma. At present,good informationon theseparameters exists the formationof the substorminjection boundary and in the creation onlyat densitiesgreater than approximately 100 cm -3, i.e., in the of equatoriallytrapped warm ion distributions. plasmasphereand plasmapauseregions. Even with this restriction, statistical information on electron temperature is not routinely Introduction available. In orderto proceed,a plasmadistribution model was constructed The propagationof magnetohydrodynamic(MHD) waveswithin usingas muchempirical information as possible,with extrapolation spaceplasmas has long been recognized as the meansby whichthe or guessworkheld to a minimum, as describedbelow. The resulting energy and momentumassociated with transientphenomena are densityand temperaturemodels for quiet conditionsare shownin transported(Southwood and Hughes, 1983, and references therein). Figure 1 as equatorialplane and midnightmeridian cross sections. Within a systemlike the terrestrialmagnetosphere, the plasmaand The plasmaelectron density is tiedto the observationsfrom GEOS magneticfield are highlynonuniform, and this leadsin generalto a 2 at geosynchronousorbit reportedby Higel and Wu (1984). These veryinhomogeneous medium for propagationof suchwaves. Burton observations have been reduced to mean states for three levels of et al. (1970) reportedobservations of magneticfield and plasma activity: quiet, average, and disturbed. Results shown here densitiesfor four passesof OGO 5 throughthe plasmapause region, correspondto the averageconditions. The radial distributionof calculatingAlfv•n speedprofiles. We report here an attemptto equatorialnumber density uses the observationsof Berchemand constructa three-dimensionalmodel of the typical magnetospheric Etcheto(1981) from ISEE, indicatinga power law dependenceon distributionof Alfven and fast mode wave speeds,based on the radius close to Earth. We have taken the power law in the form extensivestatistical data bases now available.Our goal is to identify R-[3.5+ 0.5sin(2'n'MLT/24)] soas to reflectthe closer approach to persistentor characteristicfeatures of thesedistributions which have diffusiveequilibrium which prevails in the duskbulge region. This potentiallyimportant effects on the dissipation of suchwaves or their R dependenceis modified by an exponentialdropoff beyond a accessto various regionsof space.This approachimplies an variableR value. The scalelength of the exponentialincreases with emphasison propagationat wavelengthswhich are smaller than the the R at which the exponentialdrop begins.The exponentialdrop magnetosphericsystem, in contrastto the emphasison long continuesuntil an asymptoticnumber density is reachedwhich is wavelengthsand standing modes found in muchof the literatureon representativeof near-Earthplasma sheet values at midnight,i.e., magneticpulsations. 1.0 cm'3 (Strangewayand Kaye, 1986),and is somewhathigher at noon with a smooth variation over intervening local times. This MHD Wave Opticsof the Magnetosphere asymptoticplasma sheet density falls slowly with radius (as R-•). The positionat which the exponentialdrop beginsis locatedfor a particularlocal time so as to producethe correctnumber density at Magnetic Field geosynchronousorbit as specifiedby the GEOS 2 dataset. A crude indicationof the magnetopauseposition was introduced by invoking We have usedthe Mead and Fairfield (1975) empiricalmagnetic a typical magnetosheathdensity outside the boundaryindicated in field model, which providesfour differentstates of magnetotail Figure 1, panelsA and C. stretching:superquiet, quiet, disturbed, and superdisturbed. The plasmacomposition is takenfrom Horwitz et al. (1986), i.e., Referenceis madeto the originalpublication for field line maps, approximately75% H +, 20%He +, and5% O +. In thepresent model, intensitydistributions, and other documentationconcerning this evidencefor heavyion enhancements relative to thesenumbers in the model.Though we haveused all fouractivity levels, here we only plasmapauseregion (Roberts et al., unpublishedmanuscript) and in showresults for the quietmodel. Other magnetic field modelscan the near-Earthplasma sheet (Strangeway and Kaye, 1986)has not easilybe substituted,but this model was considered to be adequate been factored in. The net result is a mean ionic mass of 2 amu. It is withinthe region of interest,taken to be thatregion within a radial anticipatedthat more sophistication in this parameter will be incor- distanceof 12 Re geocentric. poratedin futureversions of thismodel, when statistical data bases canprovide the properjustification. Such adjustments will tendto reducethe wave speedsderived for theseregions. Copyright1987 by the AmericanGeophysical Union. The "effectivetemperature" plays two roles in thisstudy. It is used to definethe soundspeed contribution to theMHD fastmode speed. Papernumber 7L6609. As well, it could be usedto controlthe plasmascale height along 0094-8276/87/007L-6609503.00 magneticfield lines, along which good thermalconduction is

1007 1008 Mooreet al ßMHD Wave Breaking

A PLASMA DENSITY 12 B PLASMADENSITY

MODELN3 ' ,, • • 3 MODELN3 CM-3 "'".••.,,, • .,••'• • ..2 LT=00CM-3 30000.0 30O00.0

18 lO 12 RE 3000.0 - 3000.0 1000.0 0 •-' 1000.0 300.0 300.0 100.0 •-• I 100.0 10000.030.0 I I I 10000.0 3O.O 10.0 10.0 3.0 ' 2 3.0 1.0 • 1.0 0.3 ' -- 3 0.3 0.1 • '• :•:'•-- 0.1 0.0 4 0.0

0LT .. 0 I 2 3 4 5 6 7 8 g 10 11 125 RE

PLASMA TEMPERATURE c D PLASMA TEMPERATURE 12

-.-' -- MODEL N3 3 MODEL N3 LT = 00 ELECTRON ,. 2: ELECTRON VOLTS VOLTS 30000.0 1OOO0.0 1 30O00.0 '2 4 6: 8

1000.0 0 3000.01000.0 1012RE 3000.0300.0 ) t 10000.0300.0 ! 00.0 -• 1 100.0 30.0 30.0 10.0 3.0 •' 2 10.03.0 1.0 1.0 0.3 3 0.3 0.1 0.1 0.0 4 0.0 0LT 5 0 1 2 3 4 5 6 7 8 9 10 11 12 RE

Fig. !. Contourplots of theequatorial and midnight meridian distributions of plasmaelectron density and effective temperature accordingtothe model used in this work. assumedto maintain isothermalconditions. In practice, the com- boundaryof the plasmasheet, including the entireplasmasphere, the binationof low meanionic massand centrifugal reduction of gravity equatoria!!yassigned temperature and density values are simply in the nearly corotating frame of the low-energy plasma in the mappedalong magneticfield lines to low altitudewhere the topside plasmasphereleads to very small gradientsof plasmadensity along ionosphereis not modeled.This procedureleads to very low densi- magneticfield lines (Gallagher eta!., 1987). This assertionis of ties at low altitudes in the region conjugateto the plasma sheet, coursesuspect in the topsideionosphere at the feet of the field lines, producingan "auroral plasmacavity" consistentwith the observa- wherea largedensity gradient exists. However, this regionis outside tionsof Persoonet al. (! 987). In the polar regionoutside the plasma the regionof presentinterest (above 0.5 R•. altitude) and will not be sheetboundary field line, the resultsof Persoonet al. (1983) are used dealt with here. to provide densitiesaccording to a simple expressionwith a power Sinceelectron temperatures representing the full energydensity of law in radius.Clearly the detailedstructure of the F layer ionosphere the plasmaare not readily availablethroughout the magnetosphere, is not addressedby thesetechniques. However, the agreementwith we assign"effective" temperatureequal to the ion temperaturewith auroral plasma cavity values observedby Persoonet al. (1987) guidancefrom the work of Comfort (1986), Gallagheret al. (1987), confirmsthe plausibilityof this procedure.As a furtherdevelopment and Garrett and DeForest(1979). The temperaturerises gradually of this work, we plan to incorporateeither fully three-dimensional with radiusfrom a baseof 2500 K deep in the plasmasphere,then empirical data or a physical model of the field-aligned plasma rises inversely with the exponentialdensity drop describedabove distribution. until an asymptotic value is reached which is representativeof plasmasheet temperatures, i.e., !. 16 x l0 s K. No attempthas been madeto enforcepressure equilibrium with theJ x B forceimplied by Wave Velocities the magnetic field model; however, a realistic pressuregradient resultsfrom this uniform temperatureand the assumedweak radial Theresults for A!fv6nwaves [computed as V^ = B/(4,rrNM) ø's, dependenceof the densityoutside the plasmasphere. with M the meanionic mass]and magnetosonicor fast modewaves Number densitiesaway from the equatorialplane are derived as [computedas VF•a = (V^2 + Vs2)O.S,where Vs = (kTeff/M)ø's] are follows: At locations earthward of a field line taken to mark the outer shown in Figure 2. Note that we have used the ion acousticspeed Moore et al.' MHD Wave Breaking 1009

ALFVEN WAVE SPEED ALFVEN WAVE SPEED

A 12 B

MODEL B2N3 3 MODEL B2N3 ß LT=00 KM/S ' KM/S

30000,0 30000,0 20000.0 20000.0 10 12 RE 10000.0 10000.0 2 4 6 8 ,0 ''•. 700O.0 :-:i 7000.0 • :': 3000.0 •.: 3000.0 .: :."2000.0 1 "' 2.000.0 ß ::' 1000.0 •" 1000.0 700.0 700.0 300.0 300.0 200.0 200.0 100.0 100.0 70.0 70.0 30.0 30.0

0LT 11 125 RE

FAST MODE WAVE SPEED FAST MODE WAVE SPEED c 12 D

MODEL B2N3 .., 3 MODEL B2N3 LT=00 KM/S !2 KM/S ! 30000.0 1 30000.0 .20000.0 20000.0 .. 10000.0 10000.0 -o . '.• 7000.0 • .' 7000.0

. . • 3000.0 3000.0 2000.0 1 ' 2000.0 ,::"lOOO.O 1000.0 700.0 -2 700.0 300.0 300.0 200.0 200.0 3 100.0 lOO.O 70.0 70.0 30.0 4 30.0

125

Fig. 2. Contourplots of the equatorialand midnight meridiandistributions of Alfv(•n speedand magnetosonicspeed.

correspondingto an assumptionof isothermalconditions or unity mean •tate. We wish to draw particular attention to the deep ratio of specific heats. Both equatorial and midnight meridional minimum of the wave propagationspeeds found roughly between sectionsare shown. The only region of significant difference 4-6 R•:• in the outer plasmasphereor inner plasma sheet rather betweenA!fv(•n and fast mode speedsis found in the plasmasheet, independentof local time. For the particular model plotted, the wherethe effective temperature is very high and the magnetic field 'is deepestminimum is found in the evening near 5-6 Rye,though the depressedby tail stretching.Examining first the equatorialsection existenceof this featureis relatively independentof the activity level we find a deepminimum in bothwave speeds which extends around chosenfor the plasmaand field models.The basicvelocity minimum all local times, but is deepestbetween 5 and 6 Ru at early evening featureis simplya consequenceof the approximateL-3 magnetic localtimes. In this regionthe fastmode speed drops as low as 150km fieldvariation and the L -3 to L-4 variation of theplasma density in the s-!, whereastypical valuesthroughout the rest of the equatorial inner plasmasphere,coupled with the rapid drop in plasmadensity, magnetospherearenear 1000 km s-!. drop in magneticfield intensity,and rise in plasmatemperature in the Examiningthe correspondingmeridional distribution, we find plasma sheet. thatthe deep minimum does not extend very far outof theequatorial Acousticwaves in a gasobey a wave equationwhich is essentially plane alonggeomagnetic field lines, being confinedto a region identicalto that describingshallow water waves. Therefore, we can withinabout ___ 20 ø of theequator. In factthe Alfv(•nspeed rises very make use of our experienceand intuition with water waves in antici- rapidlypassing out of the plasmasphereat higherlatitudes, and the pating the behaviorof magnetosonicwaves. Of course,the index of "horns"of the plasmasheet form regionsof extremelyhigh A!fv•,n refraction can be very anisotropic, and the fast mode should be speed,much higher than values found in the centralplasma sheet. coupledto the Alfvfin mode, so the situationis more complex. How- ever, we expectaccelerated steepening and breakingof waveswhen they propagate into regions of decreasingwave phase speed, as Discussion would be the case for compressionalwaves propagatingearthward throughthe synchronousorbit region at any local time. By analogy The purposeof this studyhas been to computethe distributionof with water wave behavior, we proposeto refer to the wave speed MHD wave speedsthroughout the near-Earthmagnetosphere. To do minimum region as the magnetospheric"shoal." The term "beach" this we havedefined a crudeempirical model of the plasmadistribu- might be appropriate,as well, and has been used to describethe tion within the magnetospherewhich is neverthelessplausible as a propagationof plasmawaves into regionsof slowerphase speed. In 1010 Moore et al.: MHD Wave Breaking

the presentcase the regionof low speedis embeddedwithin regions the Data SystemTechnology program at MSFC. JLH andRHC were of higherspeed, so that the analogy with a "shoal"seems preferable. supportedin partby NASA grantNAG8-058 with The Universityof Breakingof an acousticwave correspondsto the transientforma- Alabama in Huntsville. tion of a shock,with the implicationthat bulk flow energyflux is convertedinto thermalenergy of the gas as it is processedby the References shock.Since the acousticwave would be occurringin a collisionless Berchem,J., andJ. Etcheto,Experimental study of magnetospheric plasma, all the phenomenaassociated with collisionlessastro- convection,Adv. SpaceRes., 1, 179, 1981. physicalshocks are expectedto occur, includingthe creationof Burton, R. K., C. T. Russell, and C. R. Chappell,The Alfv•n highly unstableion distributionsand very energeticparticles. The velocity in the magnetosphereand its relationshipto ELF wave breakingand shockformation would be expectedjust as the emissions,J. Geephys.Res., 75, 5582, 1970. wave enters the region of increasingcore plasma density and Comfort, R. H., Plasmaspherethermal structureas measuredby decreasinghot plasma, i.e., just asit beginsto compressand displace ISEE-1 and DE-l, Adv. SpaceRes., 6(3), 31, 1986. earthwardthe plasmasheet inner boundary. Though this situation is Gallagher,D. L., P. D. Craven,and D. A. Gurnett,A newmodel of clearlyvery complex,the physics of it is a logicalextension of recent the plasmadensity in the Earth'sinner magnetosphere (abstract), advancesin our understandingof steadycollisionless shocks. The Eos Trans. AGU, 28, 385, 1987. net effect on the plasma would include some combinationof Garrett, H. B., and S. E. DeForest,An analyticalsimulation of the boundary displacementand local heating along the displaced geesynchronousplasma environment,Planet. Space Sci., 27, boundary.Since the outerplasmasphere is knownto consistof iono- 1101, 1979. sphericplasma, we shouldexpect the plasmaheated by wavebreak- Higel, B., and Wu Lei, Electron density and plasmapause ing to be primarily ionospheric. characteristicsof 6.6 RE:A statisticalstudy of the GEeS 2 relaxa- Evidencefor theseeffects in substermevents is summarizedby tion sounderdata, J. Geephys.Res., 89, 1583, 1984. Moore (1986) andreferences cited therein. It seemsplausible as well Horwitz, J. L., R. H. Comfort, and C. R. Chappell,Plasmasphere that the concentrationof wave heatingof the low-energyplasma and plasmapausecharacteristics measured by DE-l, Adv. Space observednear the equatorin the outer plasmasphere(Olsen et al., Res., 6(3), 21, 1986. 1987)is associatedwith thebreaking of wavedisturbances propagat- Mcllwain, C. E., Substerminjection boundaries, in Magnetospheric ing from the outer magnetosphere. Physics,edited by B. M. McCormac, p. 143, D. Reidel Publ. Co., Dordrecht, Holland, 1974. Mead, G. D., and D. H. Fairfield, A quantitativemagnetospheric Conclusions model derived from spacecraftmagnetometer data, J. Geephys. Res., 80, 523, 1975. We have constructedan empirically-basedmodel of the plasma Moore, T. E., Accelerationof low energymagnetospheric plasma, and magneticfield parametersfor the purposeof specifyingthe Adv. SpaceRes., 6(3), 103, 1986. distributionof MHD wave phasespeeds throughout the magneto- Olsen, R. C., S. D. Shawhan,D. L. Gallagher,J. L. Green, C. R. sphere,exclusive of the magnetotail.In thispaper we havepointed Chappell,and R. R. Anderson,Plasma observations at theEarth's out a majorfeature of the resultingwave speed distribution, a torus- magneticequator, J. Geephys.Res., 92, 2385, 1987. shapedminimum or magnetospheric"shoal." This feature is a Persoon,A.M., D. A. Gurnett, and S. D. Shawhan, Polar cap characteristicof the statisticallyaveraged configuration of the electron densities from DE 1 plasma wave observations,J. magnetosphere,though it maybe moreor lesspronounced for plaus- Geephys.Res., 88, 10,123, 1983. ible instantaneousstates of the magnetosphere.The featureis not Persoon, A. M., D. A. Gurnett, W. K. Peterson,J. H. Waite, Jr., J. entirelysymmetric and leadsto equatorialcontours of equalwave L. Burch, and J. L. Green, Electron densitydepletions in the speedwhich spiral outwardin the eveningsector similar to the nightsideauroral zone, J. Geephys.Res., in press, 1987. substerm injection boundary inferred from spacecraftdata Southwood,D. J., and W. J. Hughes, Theory of hydromagnetic (Mcllwain, 1974). We assertthat the existenceof sucha feature waves in the magnetosphere,Space Sci. Rev., 35, 301, 1983. impliesthat magnetosphericwaves will be subjectto well-known Strangeway,R. J., and S. M, Kaye, Quiettime masscomposition at wave phenomenasuch as refraction,reflection, and breaking, near-geosynchronousaltitudes, J. Geephys. Res., 91, 7105, dependingupon their wavelengths in relationto thescale of thefea- 1986. ture. We suggestthat these wave phenomena are of generalimport- anceto magnetosphericdynamics, specifically in the formationof substorm-injectedplasma populations which produce bright diffuse aurora and the ring current.

Acknowledgments.This work benefitted from the efforts of J. Foy (Received June 17, 1987; and B. L. Giles. Supportwas providedby NASA Headquarters revised August 6, 1987; underRTOP 442-20-10, by theDynamics Explorer program, and by accepted August 11, 1987.)