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GEOPHYSICAL RESEARCH LETTERS, VOL. 22, NO. 23, PAGES 3275-3278, DECEMBER 1, 1995

Magnetosheath flow near the subsolar - Zwan-Wolf and Southwood-Kivelson theories reconciled

DavidJ. Southwood I and Margaret G. Kivelson2 Instituteof Geophysicsand PlanetaryPhysics. University of California.Los Angeles

Abstract. We reconcilethe superficiallycontradictory theories the flow upstreamof the magnetopause,the magneticfield should of flow immediately upstreamof the subsolarmagnetopause. decreaseand the gaspressure and plasmadensity should increase We show that the formation of a layer of enhancedfield and whilst Zwan and Wolf [1976] predict a region where the field depresseddensity immediately adjacent to the magnetopauseas increasesand the densityand pressuredecrease. Southwoodand predictedby Zwan and Wolf appearsto requirethe presence Kivelson[1992] acknowledgedthe differencebetween the Zwanand upstreamof a regionof enhancedpressure and field rarel•tction Wo/f[1976]predictions and their own theory. It is thepurpose of aspredicted by Southwoodand Kivelson. We thusargue that the thispaper to link thesetwo apparentlycontradictory notions and to detachedregions of enhanceddensity and the depletionlayer arguethat not only do botheffects occur in practicebut they arepart adjacentto the magnetopauseare actually manifestationsof a of one and the same effect. singlephenomenon. The models of the solar flow around the Earth's magnetospherein the gas dynamic limit include the field only Introduction kinematically [Spreiter and Alksne, 1969; 1970]. The approach would be entirely adequateif the gas pressurebehind the shock The structureof the magnetosheathfield and flow patternnear everywheregreatly exceededthe magneticpressure. As Zwan and the magnetopauseis of greatinterest in solarterrestrial physics, the Wo/f[1976] point out, it is implicit in Alksne's[1967] calculations region marking as it does the interface between and that there alwayswill be a regionnear the magnetopausewhere field terrestrialplasma regimes. The problemis complexbecause of the pressureis important. The kinematiccalculations based on the gas anisotropicnature of magnetohydrodynamicbody forces and the dynamic code predict that the field grows without limit as one allied fact that there are three separatewave modes.Gas dynamic approachesthe magnetopause[see also, Sonnerup,1980]. This fact studiesofthe flow around the weredone with great on its own does not mean that the field necessarily hasto grow large successbySpreiter and co-workers [Alksne, 1967: Spreiter and nearthe boundary; it simply means that an approach thatignores the Alksne,1969: 1970] but these treatments inevitably broke down in fieldforce gives an inadequate description. Recently, some full the immediate vicinity of the subsolarmagnetopause as the MHD simulations[Wu, 1992] have shownboth a densitydepletion magneticfield wastreated kinematically but not self-consistently. and upstreamof that a densityenhancement which may representa In 1976, Zwan and Wolf [1976] analyzed the flow near the combination of Southwood-Kivelson and Zwan and Wolf effects stagnation streamline. They showed that near the subsolar appearingin the results. magnetopausea depletionlayer would form in which the density The net magneticbody forcej x B is alwaysperpendicular to the would decreaseand the magneticfield increase.This was one of the magneticfield and so the stressesexerted by the magneticfield on earliestworks attemptingto model the flow near the magnetopause the are fundamentally anisotropic. Furthermore, small includingthe field forces. Subsequently,a variety of observations disturbances in the medium generate three distinct indicatedthe presenceof a layerjust outsideof the magnetopause magnetohydrodynamic(MHD) wave modeswhose propagationis in which the field was enhancedand the plasmadensity depressed as also stronglyaffected by the field direction. Both the phenomena Zwan and Wolf had predicted[Crooker et al., 1979]. discussedby Zwan and Wolf [1976] and those discussedby Motivated to explain the regions of compresseddensity and Southwoodand Kivelson [1992] are associatedwith the slow MHD rarified magneticfield detectedby the ISEE I and 2 spacecraftby mode. Although only the latter work relies explicitly on Song et al. [1990; 1992] in the ,Southwoodand considerationsof its properties, the slow mode gives rise to Kivelson [1992] adopteda different approachto look at the flow compressionand rarefactionof the field and plasmapressure with immediately upstreamof the magnetopause.They discussedthe the property that the respective pressureschange in antiphase. formationof slow modedisturbances in the magnetosheathflow Indeed for the caseof propagationat a large angle to the field, the upstreamof the magnetopause.The Zwan andWolf effectpredicts presshrechanges in the wavemore or lessbalance. The group exactlythe oppositephenomena to thosepredicted by Southwood velocity of the wave mode is stronglyguided by the magneticfield andKivelson [1992]. Acrossa compressionalslow front standing in undermost conditions. Dungey [1968] likens the mode to a wave in a pipe (aligned along the field) althougha wave in an elastictube might be a betteranalogy. •Department of Physics, Imperial College of Science, To understandthe dynamicalfunction of the slow mode, let us Technologyand Medicine,London SW7 2BZ, U.K. 2Departmentof Earth and Space Sciences,University of startby noting that an ideal planar slow MHD mode wave satisfies California,Los Angeles,CA 90024-1567 the coplanaritytheorem. In other words, the backgroundmagnetic field, the wave velocitychange, the magneticfield changeand wave normalall lie in the sameplane. It followsthat a standingwave can Copyright1995 by the American Geophysical Union. modify the velocity of the flow in the plane of the upstreamfield and flow. If we take the plane containingthe upstreammagnetic Papernumber 95GL03131 field and velocityto be the X, Y plane, we concludethat, in steady 0094-8534/95/95GL-03131503.00 state, slow waves standingin the flow will serve to modify the

3275 3276 SOUTHWOOD AND KIVELSON: ZWAN-WOLF & SOUTHWOOD-KIVELSON MERGED

velocity only in this plane. One can note that the diversionin the Z Zwan and WoWs 'thin tube approximation'is an analogyand as direction would introduce intermediate mode disturbances. suchis not fully rigorous. An individualtube of the magnetosheath doesnot actually retain its identity and so there are elementsof the Review of Zwan and Wolf [1976] scenariowhich are not completelyexplained, as we now discuss. As the tube moves towards the magnetopause,one expects Zwan and Wo/f[ 1976] considerthe evolutionof a singlethin transversepressure balance to be more or less maintained. The tubemoving in fro.m the solarwind as illustratedin Figure1. The compressionof the tube increasesthe magneticfield magnitudebut incident solar wind field is assumedto lie in the Y direction, the case the corresponding enhancement in magnetic pressure cannot we shalluse throughout the remainderof the paper.As the plasmain actually act to push plasmaalong the field (the "squeezing"effect the tube crossesthe ,it receivesa tangentialacceleration describedby Zwan and Wolf [1976]). Only a gradient in gas that sets up motion parallel or antiparallelto the field in order to pressurealong the field can push the plasma in such a direction. deflect the plasmaaway from the nose of the magnetosphere.This Both the assumedoutflow along the field from the stagnation motion cannot but decreasethe number of particles per unit flux streamlineimposed at the bow shockand the increasingcontribution near the part of the tube incident on the nose and this is the first of the magneticcomponent to pressurebalance seem to be working source of the depletion effect. One can envisagethat as the flux to make the gaspressure near the constrictiondecrease. This is the tube moves towardsthe magnetopausethe number density on the oppositeto what is requiredto sustainthe outflow itself. tube near Y= 0 decreasesand the magneticfield rises.It would seem We proposea resolutionof the paradoxthat the Zwan and Wolf that to sustainthis motion behind and away from the shock, the and Southwoodand Kivelsontheories predict different behavior by shock must be strongestnear the nose so that there is a pressure linking the two theories. Already in Zwan and Wolf [1976], the gradientalong the field directioneverywhere to maintaina net force authorspointed out their solutioncorresponds to a slow mode wave along the field direction. The pressureon the stagnationstreamline expandingalong the flux tube. However,implicit in their pictureis (i.e. the streamline at Y = 0) would decrease towards the the paradoxalluded to above. They describea slow modesignal in magnetospherenose and to maintainperpendicular pressure balance which the field increasesand the pressuredecreases behind the the pressurewould rise, thus creating a region wave as the wave propagatesalong the flux tube away from the nose adjacentto the magnetopausein which the field is enhancedand the region.Yet, the pressuregradient in sucha wave pulseis exactlyas plasmadensity is depleted. we have describedin the previousparagraph; it would exert a force The secondsource of depletion that Zwan and Wolf identi•j is to pushmaterial back towardsthe nose. Sucha motion in isolation associated with compressional stress near the nose of the is opposite to what we have concludedoccurs behind the bow magnetosphere. As the flux tube is pushed up against the shock, opposite to the direction of the force squeezingat the magnetopauseit can be squeezednear the Y = 0 plane. In other boundaryand generally oppositeto expectations. Without some words, Zwan and Wolf picture there being a pressuregradient furtherelement, the "squeezing"of the sheathflux as it approaches external to the flux tube which exerts a force that also moves , the magnetopauseintuitively describedby Zwan and Wolf is not material in the flux tube away from the flow stagnationpoint. For achieved.We believethat a more complexsystem of stressmust lie this pictureto make full sense,the flux tubemust be very thin and in behindthe effectthan Zwan and Wolf envisage. someway detachedfrom the surroundingmedium. Zwan and Wolf [1976] specify the external pressureas an ansatzbased on a form Review of Southwood and Kivelson [1992] taken from Spreiter et al.'s [1966] gas dynamiccalculations. This appearsto introducea kinematicelement into the calculation.If one Southwoodand Kivelson [1992] come at the problem of flow regardsthe magnetopauseitself as a rigid boundary,then the force it near magnetopausefrom a different direction to Zwan and Wolf. exerts is external to the magnetosheathand acts only at the They startby referenceto Kantrowitzand Petschek's[1966] analysis boundaryof the sheath. If the plasmaaway from the boundaryis of the fbrm of the threeMHD modesand point out that in generalan alsoto be subjectto the force theremust be someinternal means in obstaclein a plasma flow shouldexcite all three modesstanding the sheathitself to achievethe transferof stress. Zwan and Wolf upstreamfrom the obstacle.The fastmode forms the bow shock,as [1976] allude to the external (i.e. external to the flux tube itself) is well known. The other two modes should form fronts well inside nature of the force in their conclusionssection. Here they compare the magnetosheathat the point where their phasevelocity matches differencesbetween their work and the earlier work of Lees [1964] the componentof magnetosheathvelocity normal to the front. who also consideredmotion along the stagnationstreamline in the Retainingthe model geometrydiscussed earlier, the slow mode magnetosheath. forms to adjust the level of flow diversion around the magnetosphericobstacle in the X, Y plane. If the flow is to be diverted away from the stagnationstreamline at Y = 0 then the pressuremust rise acrossthe slow front and it couldsteepen to form a shockalthough shock formationis not an essentialfeature. Now the field mustdrop if the pressurerises and it followsthat the field I geometryin the vicinity of the front implied by the Southwoodand Kivelson[1992] modelhas the form illustratedin Figure2. As Figure 2 shows,the Southwoodand Kivelson[1992] proposal has the unwelcomeprediction that the front bendsthe field towards the magnetopausenose. This is an unavoidablefeature of requiring thatthe front deflectmaterial away from the nose. As they point out Figure 1. Figure 1 is taken from Zwan and Wolf [1976]. The some of the flux can be bent out of the plane by a standing sketchesshow the idealized squeezingof a solar wind flux tube intermediatemode but the overall effect nonethelessstands, namely at successivetimes T 1, T2, T3 as it is draped over the that the slow compressionalfront bendsthe field towardsthe nose. magnetosphericnose region. However, in this respect,by enhancingthe field near the noseof the SOUTHWOODAND KIVELSON:ZWAN-WOLF & SOUTHWOOD-KIVELSONMERGED 3277

magnetopauseitself but the deflectionitself cannotbe achieved withoutsome plasma compression region near the nose.However, thereis no a priori reasonthat the plasma compression region need lie immediatelyagainst the magnetopause.Indeed the insightwe offerhere is thatit mayappear detached as, downstream of it, must lie a regionof depletedplasma density, a resultof theevacuation of the tubeby the motioninduced away from the nose. On pressure balance grounds,the evacuationof plasma leads to field enhancement downstream of the front and thus immediately adjacentto the magnetopause. We have sketcheda symmetricalsituation where the upstream field is in the Y directionin Figure3. The structureillustrated is envisagedas standingimmediately upstream of the magnetopause. The threeregions we identifyare (i) an upstreamregion where the flow in the (X, t) planeis insufficientlydeflected to movearound the boundary,(ii) a frontalregion where deflection of the flow is imposed,(iii) a regionof depletionagainst the magnetopause.In generala fluxtube embedded in theflow moves from being entirely in region(i) to threading(i) and(ii) andsubsequently to threading all threeregions. Putative Usingthe sketchin Figure3, onecan outline the 'history'of a source region flux tubeas it movesnear the subsolarmagnetopause. Consider the labeled 1. The tube is embeddedin the magnetosheath flow and the sectionclosest to the nose has just encounteredthe standingstructure. The field is bentwhere it passesthrough the Figure 2. Figure2 is Figure4 of Southwoodand Kivelson standingslow shock/t¾ont which depresses the field and compresses [1992]. The sketchillustrates the field bendingas it passes the plasma. The dissipationcritical to a shockdoes not seeman through a slow mode front immediatelyupstream of the essentialelement of the structureand accordingto Songet al. [1990: magnetopause.The slowmode structure stands in theflow and 1992] theremay be little heatingassociated with the observed performsthe dynamicalfunction of deflectingthe flow in the densityenhancements. The plasmacompression results in a net planecontaining the upstream field around the magnetopause.

magnetopause,such a frontprovides a further mechanism for the formationof a depletionlayer.

A Unified View In Figure3 we showa sketchwhich provides a rationalized scenariofor the flow structureand resolvesthe ambiguitiesof the earlierdescriptions. In essence,we envisagea field compression regionwith roughly the properties of Zwan and Wolfs flux tube immediatelyadjacent to the magnetopause. However the field is not alignedwith the outer boundary of theregion. Field lines threading the compressionregion bend towards the Sunand enter a field rarefactionregion which is immediatelybehind a slowMHD wave • Flow shock/t¾ont. Outside the t¾ont,the field threadsthe incoming ---• Field magnetosheathflow. In essenceone pictures a slow mode front in r'-] UpstreamRegion whichfirst the plasma compresses and then there is an expansion "'•*•Compressed Density wave in which the field compresses. Field Rarefaction Thescenario we put forward here not only fits very well with the • CompressedField Density Rarefaction observationsof bothcompression and rarefaction regions but also introducesinto the Zwan and Wolf picture a mechanismfor 'squeezing'the plasmaalong the field whilstat the sametime Figure3. Figure3 showsa sketchwith a rationalizedscenario squeezingthe dimensions of the flux tubes and thereby increasing for the flow structureupstream of the subsolarmagnetopause. the field. Constrictionof the flux tube itself althoughintuitively Thereis a field compressionregion with roughlythe properties appealingas a squeezingmechanism is not compatiblewith the of Zwan and Wolf,s flux tube immediately adjacent to the continuumMHD equationsin whichmagnetic forces act solely magnetopause.However the field is notaligned with the outer perpendiculartoB. In reality,stress must be communicated through boundaryof theregion. Fieldlines threading the compression themedium such that material approaching the magnetopause in or regionbend towards the and enter a fieldrarefaction region nearthe Z = 0 planeis deflectedin theY directionas it approacheswhich is immediatelybehind a slow MHD wave shock/front themagnetopause. The deflection will giverise to a depletionof [Southwoodand Kivelson,1992]. Outsidethe frontthe field materialeventually on flux tubeswhich are pushed up againstthe threadsthe incomingmagnetosheath flow. 3278 SOUTHWOOD AND KIVELSON: ZWAN-WOLF & SOUTHWOOD-KIVELSON MERGED force exertedon the plasmaat the front which causesthe deflection upstreammagnetosheath flow in the plane containingthe field is an of the flow away from the nose of the magnetosphere.On those integral part of the reconnectionstructure when reconnectionis partsof the field line which have crossedthe front, thereis motion occurring. away from the nose. This flow, inducedby the slow front,gives rise to a depletionnear the nose. Acknowledgments. We thank Richard A. Wolf, who identified himself to us as a referee,for his very constructivecomments. This At a later time we envisagethe flux tube in position2. Note that work was supportedby the Divisionof AtmosphericSciences of the sectionsof the tubeat largeenough [Y] are still in the upstreamflow. National ScienceFoundation under grant ATM 93-14239. The tubecrosses the frontbut closeto the noseit now is pressedup againstthe magnetopause.Here combinationof the evacuationof References the tube resultingfrom the deflectioninitiated upstream by the plasmacompression region and the normalpressure exerted by the Alksne, A.Y., The steady-statemagnetic field in the transition magnetopauseresult in an enhancedfield and a depresseddensity region betweenthe magnetosphereand the bow shock,. regionjust as describedby Zwan and Wolf [1976]. Evidently,the SpaceSci., 15, 239, 1967. depletionwould be detectedmost readily on trajectorieswhich pass Crooker, N. U., T. E. Eastman, and G. S. Stiles, Observationsof nearthe subsolarpoint or otherwisenear the Z = 0 plane. plasma depletion in the magnetosheathat the dayside magnetopause,d. Geophys.Res., 84, 869, 1979 Dungey, J.W. Hydromagneticwaves, in Physicsof Geomagnetic Conclusion Phenomena,edited by W.H. Campbelland S. Matsushita,p. 913, Academic Press,New York, 1968. In this paperwe have describeda scenariofor flow immediately Kantrowitz, A., and H. E. Petschek,MHD characteristicsand shock upstream of the subsolar magnetopausewhich combines the waves, in Plasma Physicsin Theoryand Application,edited by superficiallycontradictory notions advanced in the theoriesof Zwan W. B. Kunkel, McGraw Hill, N.Y., 1966. and Wolf[ 1976] and Southwoodand Kivelson[ 1992]. The earlierof Lees, L., Interaction between the solar plasma wind and the the two theorieswas remarkablyuseful in not only identifyingthe geomagneticcavity, AIAA d. 2, 1576, 1964. existenceof a depletionlayer with compressedmagnetic field close PetschekH.E. Magneticfield annihilation,AAS-NASA Symposium on the Physicsof SolarFlares, NASA Special Publication, SP-50, to the magnetopausebut alsoin organizingideas of what went on in 425, 1964. the region in question. However, as we pointed out, the theory Rijnbeek, R.P., H.K. Biemat, M.F. Heyn, V.S. Semenov,C.J. contained some paradoxical elements. We have resolved the Farrugia,D.J. Southwood,G. Paschmann,N. Sckopke,C.T. paradoxesby pointing out that both theoriesconcern a slow mode Russell, The structureof the reconnectionlayer observedby disturbance set up near the stagnation stream line and which ISEE I on September8 1978,Ann. Geophys., 7, 297, 1989. propagatesin the plasmaframe along the field and away from the Song,P., C. T. Russell,J.T. Gosling,M. Thomsen,and R. C. Elphic, magnetospherenose. The magnetic field threading the field Observationsof the densityprofile in the magnetosheathnear the compressionregion immediatelyadjacent to the magnetopause stagnationstreamline, Geophys. Res. Lett., 17, 2035, 1990. connectsto the upstreamflow region at large [Y]. The slow front Song,P., C. T. Russell,J.T. Gosling,and M. Thomsen,Slow mode transitionin the frontsidemagnetosheath, d. Geophys.Res., 97. structure, which locally rarefies the field, lies between the 8295.1992. compressedfield region and the upstreamflow. Its presenceis Sonnerup,B.U.O, Transport mechanisms at the magnetopause,in critical in initiating flow deflectionaway from the magnetopause Dynamicsof the Magnetosphere,edited by S.-I. Akasofu,p. 77, nose. Close enough to the magnetopausethis flow tums into a D. Reidel, Dordrecht, Holland, 1980. roughlyfield alignedflow of the kind envisagedby Zwan and Wolf Southwood,D. J., and M. G. Kivelson, On the form of the flow in [1976]. the magnetosheath,d. Geoph.vs. Res., 97, 2873, 1992. Our discussion here has been restricted to circumstances where Southwood,D. J., and M. G. Kivelson, The formation of slow mode the magnetopauseis regarded as impervious and magnetic frontsin the magnetosheath.in ThePhysics of the Magnetopause editedby P. Songand B.U.O.Sonnerup (in press),American reconnectionis not occurring.Slow shockstructure is also a feature Geophys.Un., Washington,1995. of scenariosfor reconnectionflow as was originally proposedby Spreiter, J. R., and A.Y. Alksne, Plasma flow around the Petschek[1964]. Detailed comparisonof the magnetopausedata magnetosphere,Rev. Geophys.,7, 11, 1969. with the Petschektheory has been made tbr an individual event Spreiter,J.R., andA.Y. Alksne,Solar wind flow pastobjects in the [Ro'nbeeket al., 1989] although the slow shock identification solarsystem, Ann. Rev.Fluid Mech., 2, 313, 1970. remainscontroversial [B. U. O. Sonnerup,private comm. 1994]. Walthour,D.W., J.T. Gosling,B.U.O. Sonnerupand C.T. Russell, Walthour et al. [1995] have produced a similar more detailed Observations of anomalous slow-mode shock and reconnection comparisonof data and theory that makes allowancesfor the layerin the daysidemagnetopause, J. Geophys.Res., 99, 23,705, 1995. pressureanisotropy inherent in collision-freeplasmas which again Zwan, B. J., and R. A. Wolf, Depletionof solarwind plasmanear a givesevidence of a slow modestructure in the reconnectionregion. planetaryboundary, J. Geophys.Res., 81, 1636, 1976. As we pointed out in Southwoodand Kivelson [1995], the implicationof the Petschektheory and subsequentelaborations of it (Received:May 1, 1995;revised: August 30, 1995;accepted: is that the physicalprocesses associated with the deflectionof the September18, 1995)