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ManuscriptpreparedforJ.Name withversion5.7oftheLATEXclasscopernicus2.cls. Date: 22January2015 MHD flows at astropauses and in astrotails D.H.Nickeler1,T.Wiegelmann2,M.Karlicky´1,andM.Kraus1 5 1 1AstronomicalInstitute,AVCˇR,Fricˇova298,25165Ondˇrejov,CzechRepublic 0 2Max-Planck-Institutfu¨rSonnensystemforschung,Justus-von-Liebig-Weg3,37077Go¨ttingen 2 n Correspondenceto:D.H.Nickeler a ([email protected]) J 1 2 ABSTRACT In downwind direction, a tail like structure can form, which is the astrotail. The tail is not only proposed by the ] R The geometrical shapes and the physical properties of stel- resultofsimulations,butalsobythefactthatstretchedfield S lar wind – interstellar medium interaction regions form an or streamlines minimize the corresponding tension forces, . important stage for studying stellar winds and their embed- so that the configuration is able to approach an equilibrium h p ded magnetic fields as well as cosmic ray modulation. Our state.Suchbowshocksandastrotailshavebeendirectlyob- - goal is to provide a proper representation and classification served,e.g.,aroundasymptoticgiantbranchstars(e.g.,Ueta, o ofcounter-flowconfigurationsandcounter-flowinterfacesin 2008; Sahai and Chronopoulos, 2010), and have been pro- r t theframeoffluidtheory.Inadditionwecalculateflowsand posedtoexistalsoaroundtheSun(e.g.,SchererandFichtner, s a large-scaleelectromagneticfieldsbasedonwhichthelarge- 2014). [ scaledynamicsanditsroleaspossiblebackgroundforparti- Typically, a magnetic field is embedded in the ISM. For cleacceleration,e.g.intheformofanomalouscosmicrays, stars with a strong magnetic field, the wind is magnetized 1 v canbestudied.Wefindthatforthedefinitionofthebound- as well. Hence, different null points can appear, one of the 2 aries,whicharedeterminingtheastropauseshape,thenum- flowandoneofthemagneticfield,whicharenotnecessarily 2 berandlocationofmagneticnullpointsandstagnationpoints at the same location, unless the magnetic field is frozen-in 1 isessential.Multipleseparatricescanexist,formingahighly (see,NickelerandKarlicky´,2008).Theformationofastag- 5 complexenvironmentfortheinterstellarandstellarplasma. nationpointofthefloworofamagneticneutralpointiscru- 0 . Furthermore,theformationofextendedtailstructuresoccur cial for the understanding how an interface in the form of 1 naturally, and their stretched field and streamlines provide an astropause forms between the very local ISM and a stel- 0 5 surroundings and mechanisms for the acceleration of parti- larwind.Thebestobjecttostudycounterflowconfigurations 1 clesbyfield-alignedelectricfields. observationallyistheheliosphere,wherespacecraftssuchas : Voyager 1 & 2 and IBEX perform in situ measurements of v i theplasmaparameters(e.g.,Burlagaetal.,2013;McComas X 1 Introduction etal.,2013;Fichtneretal.,2014).However,aninterpretation r of these observations is not always straight forward. While a Whenstarsmovethroughtheinterstellarmedium(ISM),the strong indications for the crossing of the termination shock material released via their winds collides and interacts with ofVoyager1exist,itisyetunclearandcontradictorywhether the ISM. This interaction produces several detectable struc- theheliopauseregionwasalreadyleft(Burlagaetal.,2013; tures,suchasstellarwindbowshockswhenstarsmovewith Burlaga and Ness, 2014; Fisk and Gloeckler, 2013; Gurnett supersonic speeds relativ to the ISM. Furthermore, a termi- etal.,2013). nation shock can form, where the supersonic stellar wind The problematics of computing counterflow configura- slowsdowntosubsonicspeed,andinbetweenthistermina- tions have been attacked from different viewpoints, such as tionshockandtheouterbowshockacontactsurfaceforms, hydrodynamics (see, e.g., Fahr and Neutsch, 1983b), kine- separating the subsonic ISM flow from the subsonic stellar maticalmagnetohydrodynamics(MHD)(see,e.g.,Suessand wind flow. This contact surface is called the astropause and Nerney, 1990; Nerney et al., 1991, 1993, 1995), and self- hasatleastonestagnationpointatwhichbothflows,thestel- consistentMHD(see,e.g.,NeutschandFahr,1982;Fahrand larwindandtheISMmaterialstopanddiverge. 2 Nickeleretal.: MHDflowsatastropausesandinastrotails Neutsch,1983a;NickelerandFahr,2001,2005,2006;Nick- tion.Suchadefinitionviathenullpointofthemagneticfield eleretal.,2006;NickelerandKarlicky´,2006,2008),focus- makes only sense as long as the star itself has a magnetic ing on different aspects and regions within the astrosphere. field, strong enough to be dynamically important. Here, we We are aware that fluid approaches like MHD are strictly focusonstarswithmagneticfields,forwhichtheastropause validonlyforcollisionalplasmas,butarefrequentlyapplied can be uniquely defined as the outermost magnetic separa- tocollisionlessconfigurationslikemagnetospheres,coronae trix. The most well-known magetic star is our Sun with its or astrospheres. The alternative approach, kinetic theory, is boundary,theheliopause. notapplicablebecauseofthedifferenceofkineticandmacro- scopicscales. It is evident that numerical simulations allow to study moreinvolvedphysicalmodels(likefullMHD),whichcan- not be solved analytically. On the other hand, the above 2.1 Definitionofaseparatrix cited analytical studies, including the present, have the advantage that they provide exact solutions, and they allow The field lines of a vector field are typically represented by us to analyse physical and mathematical effects in great the trajectories of a corresponding system of ordinary dif- detail. In particular, analytical investigations are indispens- ferential equations, the so-called phase portrait. Physically able for studying how the distribution of the magnetic null interestingphaseportraitsarethoseinwhichnullpointsex- points/stagnationpointsdeterminesboththetopologyofthe ist.Then,atopologicalclassificationofthelocalvectorfield streamlines/magneticfieldlinesandthegeometricalshapeof canbeperformedbyanalysingtheeigenvaluesofthecorre- theheliopause. spondingJacobianmatrixofthevectorfieldatthenullpoint In this paper, we provide exact mathematical definitions (see,e.g.,LauandFinn,1990;Arnol’d,1992;Parnelletal., ofastrospheresandastropauses.Wediscusspossibleshapes 1996). The configuration of the vector field in the vicinity with respect to the spatial arrangement of the stagnation of the magnetic null point depends on the dimension of the and/or null points and emphasize the role of the directions considered problem. For instance, in 3D, the magnetic null offlowandfieldofboththeISMandthestar.Furthermore, pointistheintersectionofthe1Dstableorunstablesubman- wediscusstheroleofmagneticshearflowsaspossibletrig- ifoldtermedthespine,witha2Dunstableorstablemanifold gerforparticleacceleration(e.g.,ACRs)intheheliotail. termedthefan.This2Dmanifoldistermedtheseparatrix(or pause).Incartesian2D,themagneticnullpointistheinter- section of two separatrix field lines of which one is called 2 Geometrical shapes and topological properties of stableandtheotheroneiscalledunstable.Thisnullpointin astropauses 2D is of X-point type. Trajectories, i.e., streamlines or field lines, of the vector field are called stable, when the stream- The geometrical shapes of astropauses depend on the lineorfieldlinepointstowardsthenullpoint,whiletheyare strengths and directions of the two involved flows (stellar called unstable when they point away from the null point. wind and ISM flow) and their electromagnetic fields. The Such configurations can only be obtained if the eigenvalues fact that most of the ISM flow cannot penetrate the stellar oftheJacobianmatrixofthevectorfieldatthenullpointare windregionbutisforcedtoflowarounddefinestheboundary of so-called hyperbolic or saddle point type. More specifi- calledastropause.Inthemathematicalsense,thestreamlines cally,in2Dtheeigenvaluesarerealandhaveoppositesigns, in the local ISM and in the stellar wind region are topolog- while in 3D the situation is more complex. Here, the real ically disjoint. Still, different possibilities for the definition partsoftheeigenvaluesshouldnotvanishandalwaystwoof ofthephysicalastropauseexist.Onewaytodefinetheloca- theirrealpartshavetheidenticalsign.Basically,thissepara- tionofthisboundarymightbetousethestagnationpointof trixconceptisvalidalsoconcerningtheflowfieldinHDor theflowascriterion.Thisstagnationpointistheintersection MHD. ofthestagnationlineandtheastropause(separatrix)surface. The definition of a separatrix was so far only restricted The stagnation line is this streamline along which all fluid toascenariowithasingle,isolatednullpoint.Inrealastro- elements heading towards the astropause and coming from physicalscenarios,multiplenullpointsmayexist,depending opposite directions, i.e., on the one hand from the ISM and onthecomplexityofthestellarwindanditsmagneticfield. ontheotherhandfromthestar,aredecelerateddowntozero Therefore,multipleandhighlycomplex,maybeevennested velocity.However,suchadefinitionisonlyusefulinasingle- separatrices may occur. For instance, Swisdak et al. (e.g., fluid (i.e. classical HD or MHD) theory, while in a more 2013)findthattheheliopausecanbeconsideredasaregion general multi-fluid model different pauses (i.e. one for each consisting of bundles of separatrices and magnetic islands, component)exist.Hereadefinitionviaastagnationpointof resultingfrommagneticreconnectionprocesses,whichform theflowisnotunique.Aprecisewayisthustousethenull aporous,multi-layeredstructure.Examplesofmultipleand pointofthemagneticfield,andtotransfertheconceptofthe nested separatrices originating from X-type null points are stagnationpointoftheflowtothemagneticfieldconfigura- shownanddiscussedinthefollowing. J.Name www.jn.net Nickeleretal.: MHDflowsatastropausesandinastrotails 3 2 2 1 1 y 0 y 0 -1 -1 -2 -2 -1 0 1 2 -1 0 1 2 x x 2 2 1 1 y 0 y 0 -1 -1 -2 -2 -1 0 1 2 -1 0 1 2 x x 2 2 1 1 y 0 y 0 -1 -1 -2 -2 -1 0 1 2 -1 0 1 2 x x Figure1.Shapeoftheseparatricesindimensionlessunitsforthecaseoftwonullpointslyingonthex-axis.Shownarethefieldlines(i.e., theprojectionofthecontourlinesofAintothex−y-plane).Onenullpointisfixedatu =x=−1,theotherone,u ,isatx=1(symmetric, 1 2 lefttop),x=0.8(leftmiddle),x=0.6(leftbottom),x=0.3(righttop),x=0(Parkerscenario,rightmiddle),x=−0.5(rightbottom).Inthe Parkerscenario,thesecondnullpointdisappears(coincideswiththepole)andthereforealsothedipolemoment.Theseparatricesdefinedby u areplottedwithstrongest,thosedefinedbyu withmedium,andfieldlinesareshownwithnormallinewidth. 1 2 2.2 Determination of the global shape of astrospheres: suchpatterns,westartfromthesimplestpossiblefields,the distributionofnullpointsforpurepotentialfields potential fields, and investigate how separatrices form and howtheyareshapedbydifferentspatialdistributionsofnull points. The advantage of using potential fields is given by To compute non-linear MHD flows with separatrices as we thefact thatthese fieldsobey asuperpositionprinciple, i.e., will do in Sect.3, geometrical patterns are needed for the they are linear, and every null point is automatically an X- structure of the corresponding flows and fields. To obtain www.jn.net J.Name 4 Nickeleretal.: MHDflowsatastropausesandinastrotails pointofthemagneticfield.Inaddition,potentialfieldshave origin.Here,u=x+iyisthecomplexcoordinate,B isthe S∞ nofreemagneticenergy(i.e.,theyarestable)andcaneasily asymptotical boundary condition limB=B for |u|→∞, S∞ bemappedtonon-linearfieldsbyeithergeneralizedcontact i.e., the background field, and C and C are the monopole 0 1 transformationsoralgebraictransformations.Theconceptof and dipole moments, respectively. For the case of two null contacttransformationswaspresented,e.g.,byGebhardtand points,thesemomentshavethefollowingform Kiessling (1992) and later on used and refined by Nickeler C =−B (u +u ) and C =−B u u (2) and Wiegelmann (2010, 2012); Nickeler et al. (2006, 2013) 0 S∞ 1 2 1 S∞ 1 2 todescribetheMHDfieldsofdifferentspaceplasmaenviron- whereu andu arethecomplexcoordinatesofthetwonull 1 2 ments, while the algebraic transformations were introduced points. byBogoyavlenskij(2000a,b,2001,2002). Thesimplestscenarioistheonewithtwosymmetricnull As was shown by Fahr et al. (1993) in the case of pure points (u =−u ). In this case, only the dipole moment ex- 1 2 HD and by Nickeler et al. (2006) in the case of MHD, the ists, and we can interprete this with a star with a dipole numberandspatialdistributionofthehyperbolicnullpoints magneticfieldembeddedinahomogeneousmagneticback- determinethetopologicalscaffoldofanastrosphere.This,in ground field. Such a scenario is an analogy to the classical combination with the assumption of a homogeneous back- hydrodynamical example of a cylindrical obstacle in a ho- groundfieldasasymptoticboundarycondition,allowsusto mogeneousflow.Ifweassumethestarislocatedattheori- describethegeneralshapeofitsfieldandstreamlines.Inthe ginofacartesiancoordinatesystemandtheflowisparallelto currentpaper,weaimatgivingaqualitativeoverviewofthe thex-axisandstreamsinpositivex-direction,theseparatrices different scenarios. For the detailed theoretical description formacircleinthe(x,y)-plane,wherethestagnationlineslie andtreatmentwerefertoNickeleretal.(2006). on the x-axis and intersect the circular separatrix from both If several hyperbolic null points exist, which of the sepa- sides at the two null points and pass through the pole. This ratricesdefinesthentherealpause?Whatcanbesaidabout isshownintheupperleftpanelofFig.1.TheradiusRofthe multiplenullpointsisthattheyallhavetobenon-degenerate, circular separatrix, and hence the location of the two sym- becausedoubleorhigherordernullpointsaretopologically metricnullpoints,dependsonthestrengthofthebackground unstable(e.g.,HornigandSchindler,1996). magneticfield(B )andthedipolefieldviaR2=B R2/B . S∞ 0 0 S∞ Havingdiscussedthegeneraltopologicalaspects,wenow The term B R2 is hereby the dipole moment. Such a sym- 0 0 focus on the geometrical ones, considering a simplified 2D metricscenarioisnotveryrealistic,becauseitwouldimply scenario, in analogy to typical flows in aero and fluid dy- acompletelyclosedseparatrix.Hence,noplasmacanescape namics. While in the vicinity of the star the fields are full viathestellarwind. 2D to account for a variety of multipolar field structures, Toenableatleastahalf-openastrosphere,thesecondnull asymptotically, i.e., far away from the star in downwind di- point (the one in the downwind region) must be located rection, the field converges to a tail-like (1D) structure. Let closertothepole.Theshapeoftheresultingseparatricesfor usstartwiththecaseoftwonullpointsintheframeofa2D differentpoledistancesaredepictedintheseriesofplotsin cartesian potential field. We assume that the z-direction is Fig.1.Whenmovingthesecondnullpointtowardsthepole, the invariant direction, which means that for all parameters one can notice several effects. First, the separatrix resulting ∂/∂z=0.Further, x pointstothedownwind(i.e.tail)direc- fromthefirstnullpoint‘opens’inthedownwindregion.Sec- tion, and y in perpendicular direction. The global magnetic ond,aninner,closedseparatrixisformedthatpassesthrough field,B=∇A(x,y)×e ,withtheunitvectore inz-direction, the second null point and through the pole. This separatrix z z can be described via complex analysis. As B should be a enclosesnowthedipolarfieldregion,whichbecamesmaller, potential field, it follows that ∆A=0. To solve this Laplace and the dipole field weaker, if the same background field is equation,wedefineastreamor,here,themagneticfluxfunc- present.Thisinnerseparatrixmeetsthestagnationlinefrom tionAbyA=(cid:61)(A),whereAisthecomplexstreamormag- thefirstnullpointatthepole.Ameasurablebundleoffield netic flux function. This complex flux function is obtained linescanleavetheinnerdipoleregionupstream,i.e.,between from a Laurent series of the form (see, e.g., Nickeler et al., theinnerseparatrixandthestagnationline.Thesefieldlines 2006) originfromthemonopolepartofthefield.Theyaredeflected attheouterseparatrixsothattheybendaroundtheinnersep- C A=BS∞u+C0lnu+ u1 +termsofhigherorder. (1) aratrixandextendasopenfieldlinesintotheastrotail,form- ing the inner astrosheath field lines. The closer the second In the following, we will neglect the higher order terms, so null point is located to the pole, the more opens the tail re- that the complex magnetic flux function consists purely of gion. a monopole1 and dipole. Both multipoles are located in the Assoonasthesecondnullpoint‘reaches’thepole,itvan- 1The monopole is introduced as a mathematical tool, used to ishes. Hence, only one real null point remains and the field generate radial streamlines (i.e. the stellar wind) and radial (i.e. has only a monopole moment. This scene is shown in the open)fieldlines.Withoutit,thestreamlinesandmagneticfieldlines middlerightpanelofFig.1andissimilartotheParkersce- wouldotherwisealwaysbeclosed. nario (Parker, 1961) for subsonic flows. If the second null J.Name www.jn.net Nickeleretal.: MHDflowsatastropausesandinastrotails 5 2 3 2 1 1 y 0 y 0 -1 -1 -2 -3 -2 -1 0 1 2 3 4 5 -1 0 1 2 x x Figure3.AsFig.1,butforanasymmetriccasewherethefrontnull 3 point is rotated off the x-axis by −π/4, i.e., u =−1(cos(−π/4)+ 1 isin(−π/4)),andtheotheroneisonthe x-axisatu =x=0.4.The 2 apparentnon-connectivityofsomefieldlinesaty=0and x<0is 2 notrealbutcausedbytheflipbetweenRiemannsurfaces. 1 arephysicallyconnectedbythestagnationline.Inaddition, y 0 themonopolemomentincreasessothattheastrotailbecomes evenwider. Restricting for the moment to a single null point, it is, of -1 course, not necessary that this null point is located on the x-axis.Forinstance,withrespecttotheheliopause,measure- -2 mentsfromVoyager1indicateanasymmetry(Burlagaetal., 2013). A natural way to displace the null point is provided -3 bythesolar(orstellar,ingeneral)rotation,whichresultsin awinding-upofthefieldlinesandhencetoaspiralstructure of the magnetic field. While the monopole moment in the -1 0 1 2 3 4 5 symmetric examples is a pure real number, it now becomes x complex. Hence, an azimuthal component of the outflow or field occurs. The result is a displacement of the null point offthe x-axis.ThisisdemonstratedinFig.2,whereweplot Figure 2. As Fig.1, but comparing separatrix shapes with only onenullpoint.Top:thesymmetricParkercase(u =x=−1),and theasymmetricconfiguration2incomparisontothesymmet- 1 bottom: the asymmetric case where u is rotated off the x-axis ric one. An even more complex situation is achieved when 1 by−π/4,i.e.,u =−1(cos(−π/4)+isin(−π/4)).Theapparentnon- asecondnullpointexistsintheasymmetricscene.Suchan 1 connectivity of some field lines at y=0 and x<0 is not real but exampleisshowninFig.3. causedbytheflipbetweenRiemannsurfaces. 2Considering that Voyager 1 might have passed the stagnation regionatadistanceof123AU(Krimigisetal.,2013),meaningthat pointislocatedonthesamesideasthefirstone,i.e.,inup- within our scenario u would be at roughly 123AU, our configu- 1 winddirection(lowerrightpanelofFig.1),thesecondsepa- rationshowninFig.2(bottom)andFig.3mightbeapproximately ratrixisalsolocatedinupwinddirectionandbothnullpoints scaledwith1:87AU. www.jn.net J.Name 6 Nickeleretal.: MHDflowsatastropausesandinastrotails Having multiple, nested separatrices, the real astropause Givensolutionsfor p andB oftheMHSequations S S canbeuniquelydefinedbytheoutermostmagneticseparatrix ∇p = j ×B , (12) betweenthemagnetizedinterstellarmediumandthemagne- S S S tizedstellarwind. andadditionalsolutionsforM andρofthesystems A B ·∇ρ = 0, (13) S 3 Self-consistentnon-linearMHDflows B ·∇M = 0, (14) S A Thepatternofthepotentialfields,aswecalculatedinthepre- are the parameters needed to perform the transformation, vious section, serve now as static MHD equilibria (MHS). i.e.,tocomputethegeneralsolution(see,e.g.,Nickelerand These are then mapped with the non-canonical transfor- Wiegelmann,2012)ofthesystemEq.(4)-(11) mation method to self-consistent steady-state MHD flows. B Thereby we make use of estimated or observed physical B = (cid:113) S , (15) quantities, such as density, magnetic field strength, etc., 1−M2 A withinandoutsidetheheliosphere.Thesequantitiesserveas asymptotical boundary conditions, based on which some of p = p − 1 MA2|BS|2, (16) thecoefficientsofthemappingcanbefixed. S 2µ 1−M2 0 A ObservationsfromVoyager1suggestthattheplasmaflow √ 1 M B inthevicinityoftheheliopauseandintheheliotailregionis ρv = √ (cid:113) A S , (17) µ approximately parallel to the magnetic field (Burlaga et al., 0 1−MA2 2013; Fisk and Gloeckler, 2013). In addition, the plasma M ∇M ×B j j = A A S + S . (18) withinastagnationregionisincompressible.Thiscanbeun- µ (cid:16) (cid:17)3 (cid:16) (cid:17)1 derstood in terms of the steady-state mass continuity equa- 0 1−MA2 2 1−MA2 2 tion Properties of these solutions are that the plasma density ρ, the Alfveń Mach number M , and the Bernoulli-pressure ∇·(ρv)=0 ⇔ v·∇ρ+ρ∇·v=0. (3) A Π =P+1ρv2 areconstantonfieldlines.However,thesepa- 2 Whenapproachingthestagnationpoint,i.e.v→0,theterm rameters can vary perpendicular to the field lines, which v·∇ρ in the second equation vanishes, implying that ρ∇·v, meansthatforexamplestrongshearflows,implyingastrong and in particular, as the density reaches a maximum, ∇·v gradient of the Alfveń Mach number, produce strong cur- has to vanish as well.3 The plasma flow on streamlines, rentdensities(curentsheets).ThisisobviousfromEq.(18), evenifwestartfromapotentialfield,whichimplies j =0. which originate in such stagnation point regions, transports S The occurrence of current sheets, especially around multi- thepropertyofincompressibilityfurtherintothetailregion. pleseparatrices,isknown,e.g.,insolarflarephysics,ascur- Hence, it is reasonable to investigate the heliotail and he- rentfragmentation(e.g.,Karlicky´ andBa´rta,2008a,b;Ba´rta liopause region using field-aligned, incompressible flows, etal.,2010),or,insteady-stateasfragmentedcurrents(e.g., andthebasicidealMHDequationsaregivenby Nickeleretal.,2013).Itshouldbeemphasizedthatasimilar ∇·(ρv) = 0, (4) transformation can also be performed using super-Alfveńic ρ(v·∇)v = j×B−∇P, (5) flows. In the following, we concentrate on the tail region, tak- ∇×(v×B) = 0, (6) ing the symmetric Parker-like tail (top panel of Fig.2). The ∇×B = µ0j, (7) mapping ansatz we use was described in detail in Nick- ∇·B = 0, (8) eler et al. (2006). The Alfveń Mach number is defined via ∇·v = 0, (9) MA2 =1−1/(α(cid:48)(A))2,wheretheprimedenotesthederivation with respect to A, and α is the mapped z-component of the v = ±|MA|vA (10) vector potential of A. This means that if B =∇A×e and S z vA := √Bµ ρ, (11) α=α(A),thenB=∇α(A)×ez=α(cid:48)(A)∇A×ez.Hence,α(cid:48)(A) 0 istheamplificationfactorforthemagneticfieldstrength,and itresultsto where ρ is the mass density, v is the plasma velocity, B is   tpAhlleafsvmmeńaagvpnereeltosiccsuitflryeu,,xaMnddAenµissittiyhs,etjhAeislfmvtheáengncMeutrairccehnptenrudmmenebaseibtriy,l,ivtyPAoiissf ttthhheee α(cid:48)(A) = 12(cid:113)1−1M2 − (cid:113)1−1M2  (19) 0 A∞ A,i lvaarcg3ueTuahmcirso.rsesmthaeinhselviaolpidau,seevebnouifnd∇aρryhlaapypeer.ns to become extremely ·tanh √1−MAA2d∞1B∞ −y1 −tanh √1−MAA2d∞1B∞ +y1. J.Name www.jn.net Nickeleretal.: MHDflowsatastropausesandinastrotails 7 Theansatzforα(cid:48)(A)ischoseninordertomimictwocurrent sheets located symmetrically at ±y around the heliopause 1 fieldlines,withoppositelydirectedcurrents.Tocomputethe Machnumberprofileandthecurrentsheets,weusethefol- lowing set of values (see, e.g. Frisch et al., 2012; Burlaga etal.,2013):Forthemagneticfieldoftheinterstellarmedium B =5µG, for the inner magnetic field strength in the tail ∞ B =4µG,theparticlenumberdensitiesofelectronsandions i are about equal and n ≈n =0.1cm−3, the velocity of the e i ISM plasma relative to the sun is v =25kms−1, and the ∞ MachnumbersoftheISMplasmaandoftheheliotailresult to M =0.72 and M =0.52. These values might not be A∞ A,i absolutely true, but they are used here to provide a rough ideaoftheglobalshapeofthecurrentsheets. The Alfveń Mach number profile for the chosen parameters is shown in the top panel of Fig.4. It is computed for a thickness of the current sheets in the tail of d =100AU. 1 Thisisanunrealisticscenario,andisonlyshowntohighlight thecurrentdistributioninthetail(middlepanelofFig.4).A morerealisticcaseforthecurrentsheetsrequiresmuchnar- rower widths. This is depicted in the lower panel of Fig.4 whereweuseawidthofonly10AU.Obviously,areduction in the width by a factor of ten leads to an increase of the currentstrengthbyafactoroften.Attheboundarybetween ISMandsolarwindflows,Kelvin-Helmholtz-likeorcurrent drivenreconnectioninstabilitiescanoccurduetoshearflows andextremelyhighcurrentdensities,respectively.Thesein- stability regions are thus ideal locations for plasma heating andparticleacceleration. 4 Magneticshearastriggerforparticleacceleration Theseparatrixregionsand,inparticular,theheliotailregions can serve as important particle acceleration locations. This assumptionissupportedbyobservationsofboththecosmic rayanisotropyandthebroadexcessofsub-TeVcosmicrays inthedirectionoftheheliotail.LazarianandDesiati(2010) proposethatthisexcessoriginatesfrommagneticreconnec- tion in the magnetotail. The heliotail/heliopause region was alsoconsideredbyLazarianandOpher(2009)asanimpor- tantregionforparticleacceleration.Intheirapproach,Lazar- ian and Opher (2009) use a Spitzer-like resistivity and pro- pose first-order Fermi acceleration as the dominant acceleration process of energetic particles along the magnetotail. In contrast, to approach the problematics of particle acceleration we focus on the magnetic shear generated by mag- Figure4.AlfveńMachnumberprofile(top,restrictedtoMA>0.6 neticshearflowsacrosstheheliopauseboundary.Toreduce for displaying purposes) and resulting current sheets for a sheet width of 100AU (middle) and 10AU (bottom). The current den- the complexity of the problem, we shear here the magnetic sityofthepoloidalmagneticfieldisheregiveninunitsof2.65× fieldonlyinz-direction.Thisguaranteesthatthecurrent,and, 10−17Am−2. therefore,alsotheelectricfield,arealignedwiththetaildi- rection. We investigate the generation of parallel electric fields and consider them as acceleration engines which, in solar acceleration (see, e.g., Aschwanden, 2002). The link to so- physics,istypicallyreferredtoasDirectCurrent(DC)field lar physics scenarios is obvious, as the acceleration process www.jn.net J.Name 8 Nickeleretal.: MHDflowsatastropausesandinastrotails takes place in a diluted plasma environment, i.e., in regions We use an approximate value for the magnetic shear of oflowdensity. B ≈10−11Tesla,whichisoftheorderof10%oftheheliotail z As the magnetic field jumps across the heliopause, the (or ISM) magnetic field strength and can be regarded as a magnetic shear in z-direction, B , is connected with a nar- lower bound. For the typical lengthscale of the shear layer z rowcurrentsheetlocatedaroundtheheliopauseandextend- we set l≈103km (e.g., Fahr and Neutsch, 1983a). Hence, ingintotheheliotail.Asinthetailregiontheflowisdirected wecanestimatetheresultingcurrentdensityvia along the field lines, the presence of a current density im- |B | mediatelyimpliesthateveninsuchafield-alignedflowsce- |j|≈ z =5.3×10−11Am−2. (24) µ l narioanelectricfieldcanexistduetothevalidityofresistive 0 Ohm’s law E+v×B=ηj, as long as the resistivity η(cid:44)0. For density values of 104m−3 (Fahr et al., 1986) and mag- Furthermore,aswasshownbyNickeleretal.(2014),theso- netic field strengths of 2×10−10Tesla typical for helio- lutionofnon-idealOhm’slawdecouplesfromtherestofthe tail conditions, the gyroresisitivity η is on the order of MHD equations as the flow is field-aligned. Therefore, the 105Ohmm.Consequently,theparallelelectricfieldbecomes onlyadditionalequationtobesolvedis E·B η E = ≈ |j|≈10−6Voltm−1. (25) ∇×(ηj)=0, (20) (cid:107) |B| µ0 Within the tail, the field lines are stretched, and the cur- asE=ηjand∇×E=0(stationaryapproximation). rentisconcentratedaroundtheheliopauseregion,providing Areasonableresistivityshouldbevalidinoursteady-state asufficientlyextendedenvironmentforacceleratingparticles model and account for the case of a collisionless plasma. continuouslyalongthemagneticfieldlines.Outsidethisnar- While the Spitzer resistivity is effective only in collisional row region, the current vanishes and hence also the electric plasmas,theturbulentcollisonlessresistivity(anomalousre- field,sothatthoseastrosphereregionscanbeideal. sistivityduetowave-particleinteractions)isusuallynotsta- Considering a relatively conservative case, in which the tionary. field aligned electric field extends to about 100AU, only, Theusualapproachfortheresistivityistousetheelectric meaning that the tail extends to just twice the distance than forceandtointroducesomefrictionalforceνv actingonthe D theheliopausenose,thevoltageseenbytheparticlesis chargesq,withthecollisionfrequencyνandthedriftvelocity (cid:90) vD(e.g.,1977) E ds≈E ·s≈107Volt. (26) (cid:107) (cid:107) dv q dtD = mE−νvD=0 ∧ E=ηj=ηnqvD (21) Thisvoltagecancontributetocosmicrayacceleration. mν ⇒ η= . (22) nq2 5 DiscussionandConclusions Forourcollisionlessplasma,weconsidertheinteractionbe- Wehaveshownthatthedistributionofnullpointsandstag- tween the electromagnetic field and charged particles as a nationpointsdefinestheglobaltopologyandthelarge-scale substituteforcollisions.Thisansatzismotivatedbythefact structure of an astrosphere. Multiple separatrices can exist thattheinteractiontime,whichislimitedbythetimethepar- implying jumps (tangential discontinuities) of several phys- ticleneedstocrossthecurrentsheet,i.e.the”transittime”of ical parameters, such as the magnetic field strength, parti- theparticlewithinthesystem,ismuchshorterthanthecol- cle density, etc. As the outermost separatrix defines the as- lisiontime.Hencethecollisiontime(collisionfrequencyν) tropause, its global geometrical shape is hence also deter- hastobereplacedbythegyro-time(gyro-frequencyqB/m), mined. delivering Withrespecttotheheliosphere,themultipledecreasesand 1 nq increases in the magnetic field strength as well as in other η= with σ = . (23) σ g |B| physical parameters measured by Voyager 1 (Burlaga et al., g 2013) indicates several crossings of either one or several where σ is the gyroconductivity, n and q are the parti- individual separatrices. Such a scenario is in good qualita- g cle number density and charge, respectively. This approach tiveagreementwiththemultipleseparatrixstructuresdueto was introduced by Speiser (1970) and Lyons and Speiser more than one null point as proposed here and formerly by (1985)andtheresultingresistivityiscalledinertialorgyro- Nickeleretal.(2006).Asimilarsceneconsideringmultiple, resistivity.Wewanttoemphasizethatthesubstitutionofthe nested separatrices and magnetic islands was recently sug- collisionfrequencybythegyro-frequencyautomaticallyde- gested based on detailed numerical simulations by Swisdak livershugeresistivityvaluesinthecaseofadilutedplasma, etal.(2013). which are only important for the generation of an electric Interestingly, our results for the two null point scenarios field in regions of strong current density and not in regions also agree with the recently proposed presence of a helio- wherethefieldhasthecharacterofapotentialfield. cliffregioninsidetheheliopause(FiskandGloeckler,2013). 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