MHD flows at astropauses and in astrotails

8.2 MB·English

by  Dieter H. Nickeler

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 ad-
vantage 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
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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.
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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 Alfve´n 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 Alfve´n 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-Alfve´nic
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 Alfve´n 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
 
tpAhlleafsvmme´naagvpnereeltosiccsuitflryeu,,xaMnddAenµissittiyhs,etjhAeislfmvthe´aengncMeutrairccehnptenrudmmenebaseibtriy,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.
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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 Alfve´n Mach number profile for the chosen parame-
ters 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 acceler-
ation process of energetic particles along the magnetotail.
In contrast, to approach the problematics of particle accel-
eration we focus on the magnetic shear generated by mag- Figure4.Alfve´nMachnumberprofile(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).
J.Name www.jn.net
Nickeleretal.: MHDflowsatastropausesandinastrotails 9
Inparticular, theheliocliff mightbe interpretedas thesepa- Fahr, H. J. and Neutsch, W.: Stationary plasma-field equilibrium
ratrix resulting from the second null point (as shown in the states in astropause boundary layers. I - General theory, Mon.
middle left panel of Fig.1), and the streamlines originating Not.Roy.Astron.Soc.,205,839–857,1983.
fromthemonopolepart,whichbendintotheheliotail,would Fahr, H. J. and Neutsch, W.: Pressure distribution at the inner
boundaryofanastropausecausedbyacompressiblestellarwind,
represent the open heliosheath as introduced by Fisk and
Gloeckler(2013).Intheheliocliffregion,themodelofFisk Astron.Astrophys.,118,57–65,1983.
Fahr, H. J., Neutsch, W., Grzedzielski, S., Macek, W., and
and Gloeckler (2013) turns out to produce a super-Alfve´nic
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Fahr,H.-J.,Fichtner,H.,andScherer,K.:Determinationofthehe-
alsobesub-Alfve´nic. liosphericshockandofthesupersonicsolarwindgeometryby
Furthermore, the presence of magnetic shear flows can meansoftheinterstellarwindparameters, Astron.Astrophys.,
produce vortex current sheets (Nickeler and Wiegelmann, 277,249,1993.
2012)leadingtothegenerationofinstabilitiesandmagnetic Fichtner,H.,Scherer,K.,Effenberger,F.,Zo¨nnchen,J.,Schwadron,
reconnectionclosetoseparatrices.Inthecurrentworkwere- N.,andMcComas,D.J.:TheIBEXribbonasasignatureofthe
strictouranalysistoamaximumoftwoseparatricesandwe inhomogeneityofthelocalinterstellarmedium, Astron.Astro-
phys.,561,A74,2014.
apply the mapping only to the heliotail with one symmetric
Frisch,P.C.,Andersson,B.-G.,Berdyugin,A.,Piirola,V.,DeMa-
separatrix (top panel of Fig.2) with two current sheets. As
jistre, R., Funsten, H. O., Magalhaes, A. M., Seriacopi, D. B.,
multiple separatrices can exist in the heliosphere, the pres-
McComas,D.J.,Schwadron,N.A.,Slavin,J.D.andWiktorow-
enceofmultiplecurentsheetsinthevicinityofthesesepara-
icz,S.J.:TheInterstellarMagneticFieldClosetotheSun.II.,
tricescanleadtofragmentedstructures(e.g.,Nickeleretal.,
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anonymousrefereesforhelpfulcommentsonthepaperdraft.This SituObservationsofInterstellarPlasmawithVoyager1,Science,
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