MHD flows at astropauses and in astrotails
Dieter H. Nickeler, Thomas Wiegelmann, Marian Karlicky, Michaela Kraus
MManuscript prepared for J. Namewith version 5.7 of the L A TEX class copernicus2.cls.Date: 21 October 2018
MHD flows at astropauses and in astrotails
D. H. Nickeler , T. Wiegelmann , M. Karlick´y , and M. Kraus Astronomical Institute, AV ˇCR, Friˇcova 298, 25165 Ondˇrejov, Czech Republic Max-Planck-Institut f¨ur Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 G¨ottingen
Correspondence to:
D. H. Nickeler([email protected])
ABSTRACT
The geometrical shapes and the physical properties of stel-lar wind – interstellar medium interaction regions form animportant stage for studying stellar winds and their embed-ded magnetic fields as well as cosmic ray modulation. Ourgoal is to provide a proper representation and classificationof counter-flow configurations and counter-flow interfaces inthe frame of fluid theory. In addition we calculate flows andlarge-scale electromagnetic fields based on which the large-scale dynamics and its role as possible background for parti-cle acceleration, e.g. in the form of anomalous cosmic rays,can be studied. We find that for the definition of the bound-aries, which are determining the astropause shape, the num-ber and location of magnetic null points and stagnation pointsis essential. Multiple separatrices can exist, forming a highlycomplex environment for the interstellar and stellar plasma.Furthermore, the formation of extended tail structures occurnaturally, and their stretched field and streamlines providesurroundings and mechanisms for the acceleration of parti-cles by field-aligned electric fields.
When stars move through the interstellar medium (ISM), thematerial released via their winds collides and interacts withthe ISM. This interaction produces several detectable struc-tures, such as stellar wind bow shocks when stars move withsupersonic speeds relativ to the ISM. Furthermore, a termi-nation shock can form, where the supersonic stellar windslows down to subsonic speed, and in between this termina-tion shock and the outer bow shock a contact surface forms,separating the subsonic ISM flow from the subsonic stellarwind flow. This contact surface is called the astropause andhas at least one stagnation point at which both flows, the stel-lar wind and the ISM material stop and diverge. In downwind direction, a tail like structure can form,which is the astrotail. The tail is not only proposed by theresult of simulations, but also by the fact that stretched fieldor streamlines minimize the corresponding tension forces,so that the configuration is able to approach an equilibriumstate. Such bow shocks and astrotails have been directly ob-served, e.g., around asymptotic giant branch stars (e.g., Ueta,2008; Sahai and Chronopoulos, 2010), and have been pro-posed to exist also around the Sun (e.g., Scherer and Fichtner,2014).Typically, a magnetic field is embedded in the ISM. Forstars with a strong magnetic field, the wind is magnetizedas well. Hence, di ff erent null points can appear, one of theflow and one of the magnetic field, which are not necessarilyat the same location, unless the magnetic field is frozen-in(see, Nickeler and Karlick´y, 2008). The formation of a stag-nation point of the flow or of a magnetic neutral point is cru-cial for the understanding how an interface in the form ofan astropause forms between the very local ISM and a stel-lar wind. The best object to study counterflow configurationsobservationally is the heliosphere, where spacecrafts such as Voyager 1 & and IBEX perform in situ measurements ofthe plasma parameters (e.g., Burlaga et al., 2013; McComaset al., 2013; Fichtner et al., 2014). However, an interpretationof these observations is not always straight forward. Whilestrong indications for the crossing of the termination shockof
Voyager 1 exist, it is yet unclear and contradictory whetherthe heliopause region was already left (Burlaga et al., 2013;Burlaga and Ness, 2014; Fisk and Gloeckler, 2013; Gurnettet al., 2013).The problematics of computing counterflow configura-tions have been attacked from di ff erent viewpoints, such ashydrodynamics (see, e.g., Fahr and Neutsch, 1983b), kine-matical magnetohydrodynamics (MHD) (see, e.g., Suess andNerney, 1990; Nerney et al., 1991, 1993, 1995), and self-consistent MHD (see, e.g., Neutsch and Fahr, 1982; Fahr and a r X i v : . [ a s t r o - ph . S R ] J a n Nickeler et al.: MHD flows at astropauses and in astrotailsNeutsch, 1983a; Nickeler and Fahr, 2001, 2005, 2006; Nick-eler et al., 2006; Nickeler and Karlick´y, 2006, 2008), focus-ing on di ff erent aspects and regions within the astrosphere.We are aware that fluid approaches like MHD are strictlyvalid only for collisional plasmas, but are frequently appliedto collisionless configurations like magnetospheres, coronaeor astrospheres. The alternative approach, kinetic theory, isnot applicable because of the di ff erence of kinetic and macro-scopic scales.It is evident that numerical simulations allow to studymore involved physical models (like full MHD), which can-not be solved analytically. On the other hand, the abovecited analytical studies, including the present, have the ad-vantage that they provide exact solutions, and they allowus to analyse physical and mathematical e ff ects in greatdetail. In particular, analytical investigations are indispens-able for studying how the distribution of the magnetic nullpoints / stagnation points determines both the topology of thestreamlines / magnetic field lines and the geometrical shape ofthe heliopause.In this paper, we provide exact mathematical definitionsof astrospheres and astropauses. We discuss possible shapeswith respect to the spatial arrangement of the stagnationand / or null points and emphasize the role of the directionsof flow and field of both the ISM and the star. Furthermore,we discuss the role of magnetic shear flows as possible trig-ger for particle acceleration (e.g., ACRs) in the heliotail. The geometrical shapes of astropauses depend on thestrengths and directions of the two involved flows (stellarwind and ISM flow) and their electromagnetic fields. Thefact that most of the ISM flow cannot penetrate the stellarwind region but is forced to flow around defines the boundarycalled astropause. In the mathematical sense, the streamlinesin the local ISM and in the stellar wind region are topolog-ically disjoint. Still, di ff erent possibilities for the definitionof the physical astropause exist. One way to define the loca-tion of this boundary might be to use the stagnation point ofthe flow as criterion. This stagnation point is the intersectionof the stagnation line and the astropause (separatrix) surface.The stagnation line is this streamline along which all fluidelements heading towards the astropause and coming fromopposite directions, i.e., on the one hand from the ISM andon the other hand from the star, are decelerated down to zerovelocity. However, such a definition is only useful in a single-fluid (i.e. classical HD or MHD) theory, while in a moregeneral multi-fluid model di ff erent pauses (i.e. one for eachcomponent) exist. Here a definition via a stagnation point ofthe flow is not unique. A precise way is thus to use the nullpoint of the magnetic field, and to transfer the concept of thestagnation point of the flow to the magnetic field configura- tion. Such a definition via the null point of the magnetic fieldmakes only sense as long as the star itself has a magneticfield, strong enough to be dynamically important. Here, wefocus on stars with magnetic fields, for which the astropausecan be uniquely defined as the outermost magnetic separa-trix. The most well-known magetic star is our Sun with itsboundary, the heliopause. - - - - - - - - - - - - - - - - - - Figure 1.
Shape of the separatrices in dimensionless units for the case of two null points lying on the x -axis. Shown are the field lines (i.e.,the projection of the contour lines of A into the x − y -plane). One null point is fixed at u = x = −
1, the other one, u , is at x = x = . x = . x = . x = x = − . u are plotted with strongest, those defined by u with medium, and field lines are shown with normal line width. distribution of null points for pure potential fields To compute non-linear MHD flows with separatrices as wewill do in Sect. 3, geometrical patterns are needed for thestructure of the corresponding flows and fields. To obtain such patterns, we start from the simplest possible fields, thepotential fields, and investigate how separatrices form andhow they are shaped by di ff ff erent space plasma environ-ments, while the algebraic transformations were introducedby Bogoyavlenskij (2000a,b, 2001, 2002).As was shown by Fahr et al. (1993) in the case of pureHD and by Nickeler et al. (2006) in the case of MHD, thenumber and spatial distribution of the hyperbolic null pointsdetermine the topological sca ff old of an astrosphere. This, incombination with the assumption of a homogeneous back-ground field as asymptotic boundary condition, allows us todescribe the general shape of its field and streamlines. In thecurrent paper, we aim at giving a qualitative overview of thedi ff erent scenarios. For the detailed theoretical descriptionand treatment we refer to Nickeler et al. (2006).If several hyperbolic null points exist, which of the sepa-ratrices defines then the real pause? What can be said aboutmultiple null points is that they all have to be non-degenerate,because double or higher order null points are topologicallyunstable (e.g., Hornig and Schindler, 1996).Having discussed the general topological aspects, we nowfocus on the geometrical ones, considering a simplified 2Dscenario, in analogy to typical flows in aero and fluid dy-namics. While in the vicinity of the star the fields are full2D to account for a variety of multipolar field structures,asymptotically, i.e., far away from the star in downwind di-rection, the field converges to a tail-like (1D) structure. Letus start with the case of two null points in the frame of a 2Dcartesian potential field. We assume that the z -direction isthe invariant direction, which means that for all parameters ∂/∂ z =
0. Further, x points to the downwind (i.e. tail) direc-tion, and y in perpendicular direction. The global magneticfield, B = ∇ A ( x , y ) × e z , with the unit vector e z in z -direction,can be described via complex analysis. As B should be apotential field, it follows that ∆ A =
0. To solve this Laplaceequation, we define a stream or, here, the magnetic flux func-tion A by A = (cid:61) ( A ), where A is the complex stream or mag-netic flux function. This complex flux function is obtainedfrom a Laurent series of the form (see, e.g., Nickeler et al.,2006) A = B S ∞ u + C ln u + C u + terms of higher order . (1)In the following, we will neglect the higher order terms, sothat the complex magnetic flux function consists purely ofa monopole and dipole. Both multipoles are located in the The monopole is introduced as a mathematical tool, used togenerate radial streamlines (i.e. the stellar wind) and radial (i.e.open) field lines. Without it, the streamlines and magnetic field lineswould otherwise always be closed. origin. Here, u = x + iy is the complex coordinate, B S ∞ is theasymptotical boundary condition lim B = B S ∞ for | u | → ∞ ,i.e., the background field, and C and C are the monopoleand dipole moments, respectively. For the case of two nullpoints, these moments have the following form C = − B S ∞ ( u + u ) and C = − B S ∞ u u (2)where u and u are the complex coordinates of the two nullpoints.The simplest scenario is the one with two symmetric nullpoints ( u = − u ). In this case, only the dipole moment ex-ists, and we can interprete this with a star with a dipolemagnetic field embedded in a homogeneous magnetic back-ground field. Such a scenario is an analogy to the classicalhydrodynamical example of a cylindrical obstacle in a ho-mogeneous flow. If we assume the star is located at the ori-gin of a cartesian coordinate system and the flow is parallel tothe x -axis and streams in positive x -direction, the separatricesform a circle in the ( x , y )-plane, where the stagnation lines lieon the x -axis and intersect the circular separatrix from bothsides at the two null points and pass through the pole. Thisis shown in the upper left panel of Fig. 1. The radius R of thecircular separatrix, and hence the location of the two sym-metric null points, depends on the strength of the backgroundmagnetic field ( B S ∞ ) and the dipole field via R = B R / B S ∞ .The term B R is hereby the dipole moment. Such a sym-metric scenario is not very realistic, because it would implya completely closed separatrix. Hence, no plasma can escapevia the stellar wind.To enable at least a half-open astrosphere, the second nullpoint (the one in the downwind region) must be locatedcloser to the pole. The shape of the resulting separatrices fordi ff erent pole distances are depicted in the series of plots inFig. 1. When moving the second null point towards the pole,one can notice several e ff - - - - - - - - Figure 2.
As Fig. 1, but comparing separatrix shapes with onlyone null point. Top: the symmetric Parker case ( u = x = − u is rotated o ff the x -axisby − π/
4, i.e., u = − − π/ + i sin( − π/ y = x < point is located on the same side as the first one, i.e., in up-wind direction (lower right panel of Fig. 1), the second sepa-ratrix is also located in upwind direction and both null points - - - Figure 3.
As Fig. 1, but for an asymmetric case where the front nullpoint is rotated o ff the x -axis by − π/
4, i.e., u = − − π/ + i sin( − π/ x -axis at u = x = .
4. Theapparent non-connectivity of some field lines at y = x < are physically connected by the stagnation line. In addition,the monopole moment increases so that the astrotail becomeseven wider.Restricting for the moment to a single null point, it is, ofcourse, not necessary that this null point is located on the x -axis. For instance, with respect to the heliopause, measure-ments from Voyager 1 indicate an asymmetry (Burlaga et al.,2013). A natural way to displace the null point is providedby the solar (or stellar, in general) rotation, which results ina winding-up of the field lines and hence to a spiral structureof the magnetic field. While the monopole moment in thesymmetric examples is a pure real number, it now becomescomplex. Hence, an azimuthal component of the outflow orfield occurs. The result is a displacement of the null pointo ff the x -axis. This is demonstrated in Fig. 2, where we plotthe asymmetric configuration in comparison to the symmet-ric one. An even more complex situation is achieved whena second null point exists in the asymmetric scene. Such anexample is shown in Fig. 3. Considering that
Voyager 1 might have passed the stagnationregion at a distance of 123 AU (Krimigis et al., 2013), meaning thatwithin our scenario u would be at roughly 123 AU, our configu-ration shown in Fig. 2 (bottom) and Fig. 3 might be approximatelyscaled with 1:87 AU. The pattern of the potential fields, as we calculated in the pre-vious section, serve now as static MHD equilibria (MHS).These are then mapped with the non-canonical transfor-mation method to self-consistent steady-state MHD flows.Thereby we make use of estimated or observed physicalquantities, such as density, magnetic field strength, etc.,within and outside the heliosphere. These quantities serve asasymptotical boundary conditions, based on which some ofthe coe ffi cients of the mapping can be fixed.Observations from Voyager 1 suggest that the plasma flowin the vicinity of the heliopause and in the heliotail region isapproximately parallel to the magnetic field (Burlaga et al.,2013; Fisk and Gloeckler, 2013). In addition, the plasmawithin a stagnation region is incompressible. This can be un-derstood in terms of the steady-state mass continuity equa-tion ∇ · ( ρ v ) = ⇔ v · ∇ ρ + ρ ∇ · v = . (3)When approaching the stagnation point, i.e. v → , the term v · ∇ ρ in the second equation vanishes, implying that ρ ∇ · v ,and in particular, as the density reaches a maximum, ∇ · v has to vanish as well. The plasma flow on streamlines,which originate in such stagnation point regions, transportsthe property of incompressibility further into the tail region.Hence, it is reasonable to investigate the heliotail and he-liopause region using field-aligned, incompressible flows,and the basic ideal MHD equations are given by ∇ · ( ρ v ) = , (4) ρ ( v · ∇ ) v = j × B − ∇ P , (5) ∇ × ( v × B ) = , (6) ∇ × B = µ j , (7) ∇ · B = , (8) ∇ · v = , (9) v = ±| M A | v A (10) v A : = B √ µ ρ , (11)where ρ is the mass density, v is the plasma velocity, B isthe magnetic flux density, j is the current density, P is theplasma pressure, M A is the Alfv´en Mach number, v A is theAlfv´en velocity, and µ is the magnetic permeability of thevacuum. This remains valid, even if ∇ ρ happens to become extremelylarge across the heliopause boundary layer. Given solutions for p S and B S of the MHS equations ∇ p S = j S × B S , (12)and additional solutions for M A and ρ of the systems B S · ∇ ρ = , (13) B S · ∇ M A = , (14)are the parameters needed to perform the transformation,i.e., to compute the general solution (see, e.g., Nickeler andWiegelmann, 2012) of the system Eq. (4)-(11) B = B S (cid:113) − M A , (15) p = p S − µ M A | B S | − M A , (16) √ ρ v = √ µ M A B S (cid:113) − M A , (17) j = M A µ ∇ M A × B S (cid:16) − M A (cid:17) + j S (cid:16) − M A (cid:17) . (18)Properties of these solutions are that the plasma density ρ ,the Alfv´en Mach number M A , and the Bernoulli-pressure Π = P + ρ v are constant on field lines. However, these pa-rameters can vary perpendicular to the field lines, whichmeans that for example strong shear flows, implying a stronggradient of the Alfv´en Mach number, produce strong cur-rent densities (curent sheets). This is obvious from Eq. (18),even if we start from a potential field, which implies j S = .The occurrence of current sheets, especially around multi-ple separatrices, is known, e.g., in solar flare physics, as cur-rent fragmentation (e.g., Karlick´y and B´arta, 2008a,b; B´artaet al., 2010), or, in steady-state as fragmented currents (e.g.,Nickeler et al., 2013). It should be emphasized that a similartransformation can also be performed using super-Alfv´enicflows.In the following, we concentrate on the tail region, tak-ing the symmetric Parker-like tail (top panel of Fig. 2). Themapping ansatz we use was described in detail in Nick-eler et al. (2006). The Alfv´en Mach number is defined via M A = − / ( α (cid:48) ( A )) , where the prime denotes the derivationwith respect to A , and α is the mapped z -component of thevector potential of A . This means that if B S = ∇ A × e z and α = α ( A ), then B = ∇ α ( A ) × e z = α (cid:48) ( A ) ∇ A × e z . Hence, α (cid:48) ( A )is the amplification factor for the magnetic field strength, andit results to α (cid:48) ( A ) = (cid:113) − M A ∞ − (cid:113) − M A , i (19) · tanh A √ − M A ∞ B ∞ − y d − tanh A √ − M A ∞ B ∞ + y d . α (cid:48) ( A ) is chosen in order to mimic two currentsheets located symmetrically at ± y around the heliopausefield lines, with oppositely directed currents. To compute theMach number profile and the current sheets, we use the fol-lowing set of values (see, e.g. Frisch et al., 2012; Burlagaet al., 2013): For the magnetic field of the interstellar medium B ∞ = µ G, for the inner magnetic field strength in the tail B i = µ G, the particle number densities of electrons and ionsare about equal and n e ≈ n i = . − , the velocity of theISM plasma relative to the sun is v ∞ =
25 km s − , and theMach numbers of the ISM plasma and of the heliotail resultto M A ∞ = .
72 and M A , i = .
52. These values might not beabsolutely true, but they are used here to provide a roughidea of the global shape of the current sheets.The Alfv´en Mach number profile for the chosen parame-ters is shown in the top panel of Fig. 4. It is computed fora thickness of the current sheets in the tail of d =
100 AU.This is an unrealistic scenario, and is only shown to highlightthe current distribution in the tail (middle panel of Fig. 4). Amore realistic case for the current sheets requires much nar-rower widths. This is depicted in the lower panel of Fig. 4where we use a width of only 10 AU. Obviously, a reductionin the width by a factor of ten leads to an increase of thecurrent strength by a factor of ten. At the boundary betweenISM and solar wind flows, Kelvin-Helmholtz-like or currentdriven reconnection instabilities can occur due to shear flowsand extremely high current densities, respectively. These in-stability regions are thus ideal locations for plasma heatingand particle acceleration.
The separatrix regions and, in particular, the heliotail regionscan serve as important particle acceleration locations. Thisassumption is supported by observations of both the cosmicray anisotropy and the broad excess of sub-TeV cosmic raysin the direction of the heliotail. Lazarian and Desiati (2010)propose that this excess originates from magnetic reconnec-tion in the magnetotail. The heliotail / heliopause region wasalso considered by Lazarian and Opher (2009) as an impor-tant region for particle acceleration. In their approach, 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-netic shear flows across the heliopause boundary. To reducethe complexity of the problem, we shear here the magneticfield only in z -direction. This guarantees that the current, and,therefore, also the electric field, are aligned with the tail di-rection.We investigate the generation of parallel electric fieldsand consider them as acceleration engines which, in solarphysics, is typically referred to as Direct Current (DC) field Figure 4.
Alfv´en Mach number profile (top, restricted to M A > . . × − A m − . z -direction, B z , is connected with a nar-row current sheet located around the heliopause and extend-ing into the heliotail. As in the tail region the flow is directedalong the field lines, the presence of a current density im-mediately implies that even in such a field-aligned flow sce-nario an electric field can exist due to the validity of resistiveOhm’s law E + v × B = η j , as long as the resistivity η (cid:44) ∇ × ( η j ) = , (20)as E = η j and ∇ × E = (stationary approximation).A reasonable resistivity should be valid in our steady-statemodel and account for the case of a collisionless plasma.While the Spitzer resistivity is e ff ective only in collisionalplasmas, the turbulent collisonless resistivity (anomalous re-sistivity due to wave-particle interactions) is usually not sta-tionary.The usual approach for the resistivity is to use the electricforce and to introduce some frictional force ν v D acting on thecharges q , with the collision frequency ν and the drift velocity v D (e.g. , 1977) dv D dt = qm E − ν v D = ∧ E = η j = η nqv D (21) ⇒ η = m ν nq . (22)For our collisionless plasma, we consider the interaction be-tween the electromagnetic field and charged particles as asubstitute for collisions. This ansatz is motivated by the factthat the interaction time, which is limited by the time the par-ticle needs to cross the current sheet, i.e. the ”transit time” ofthe particle within the system, is much shorter than the col-lision time. Hence the collision time (collision frequency ν )has to be replaced by the gyro-time (gyro-frequency qB / m ),delivering η = σ g with σ g = nq | B | . (23)where σ g is the gyroconductivity, n and q are the parti-cle number density and charge, respectively. This approachwas introduced by Speiser (1970) and Lyons and Speiser(1985) and the resulting resistivity is called inertial or gyro-resistivity. We want to emphasize that the substitution of thecollision frequency by the gyro-frequency automatically de-livers huge resistivity values in the case of a diluted plasma,which are only important for the generation of an electricfield in regions of strong current density and not in regionswhere the field has the character of a potential field. We use an approximate value for the magnetic shear of B z ≈ − Tesla, which is of the order of 10% of the heliotail(or ISM) magnetic field strength and can be regarded as alower bound. For the typical lengthscale of the shear layerwe set l ≈ km (e.g., Fahr and Neutsch, 1983a). Hence,we can estimate the resulting current density via | j | ≈ | B z | µ l = . × − A m − . (24)For density values of 10 m − (Fahr et al., 1986) and mag-netic field strengths of 2 × − Tesla typical for helio-tail conditions, the gyroresisitivity η is on the order of10 Ohm m. Consequently, the parallel electric field becomes E (cid:107) = E · B | B | ≈ ηµ | j | ≈ − Volt m − . (25)Within the tail, the field lines are stretched, and the cur-rent is concentrated around the heliopause region, providinga su ffi ciently extended environment for accelerating particlescontinuously along the magnetic field lines. Outside this nar-row region, the current vanishes and hence also the electricfield, so that those astrosphere regions can be ideal.Considering a relatively conservative case, in which thefield aligned electric field extends to about 100 AU, only,meaning that the tail extends to just twice the distance thanthe heliopause nose, the voltage seen by the particles is (cid:90) E (cid:107) ds ≈ E (cid:107) · s ≈ Volt . (26)This voltage can contribute to cosmic ray acceleration. We have shown that the distribution of null points and stag-nation points defines the global topology and the large-scalestructure of an astrosphere. Multiple separatrices can existimplying jumps (tangential discontinuities) of several phys-ical parameters, such as the magnetic field strength, parti-cle density, etc. As the outermost separatrix defines the as-tropause, its global geometrical shape is hence also deter-mined.With respect to the heliosphere, the multiple decreases andincreases in the magnetic field strength as well as in otherphysical parameters measured by
Voyager 1 (Burlaga et al.,2013) indicates several crossings of either one or severalindividual separatrices. Such a scenario is in good qualita-tive agreement with the multiple separatrix structures due tomore than one null point as proposed here and formerly byNickeler et al. (2006). A similar scene considering multiple,nested separatrices and magnetic islands was recently sug-gested based on detailed numerical simulations by Swisdaket al. (2013).Interestingly, our results for the two null point scenariosalso agree with the recently proposed presence of a helio-cli ff ff might be interpreted as the sepa-ratrix resulting from the second null point (as shown in themiddle left panel of Fig. 1), and the streamlines originatingfrom the monopole part, which bend into the heliotail, wouldrepresent the open heliosheath as introduced by Fisk andGloeckler (2013). In the heliocli ff region, the model of Fiskand Gloeckler (2013) turns out to produce a super-Alfv´enicfield-aligned flow, while in our model the flow close to theheliopause and in the heliotail region is field-aligned but canalso be sub-Alfv´enic.Furthermore, the presence of magnetic shear flows canproduce vortex current sheets (Nickeler and Wiegelmann,2012) leading to the generation of instabilities and magneticreconnection close to separatrices. In the current work we re-strict our analysis to a maximum of two separatrices and weapply the mapping only to the heliotail with one symmetricseparatrix (top panel of Fig. 2) with two current sheets. Asmultiple separatrices can exist in the heliosphere, the pres-ence of multiple curent sheets in the vicinity of these separa-trices can lead to fragmented structures (e.g., Nickeler et al.,2013; Swisdak et al., 2013). Introducing a non-collisional re-sistivity, strong electric (DC) fields parallel to the magneticfield can be generated, which can contribute to cosmic rayacceleration as suggested by Nickeler (2005). Acknowledgements.
We thank Andreas Kopp and two more,anonymous referees for helpful comments on the paper draft. Thisresearch made use of the NASA Astrophysics Data System (ADS).D.H.N. and M.K. acknowledge financial support from GA ˇCRunder grant numbers 13-24782S and P209 / / References
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