Astron. Astrophys. 354, 697–702 (2000) ASTRONOMYAND

ASTROPHYSICS

Nonevolutionary MHD shocks in the solar windand interstellar medium interaction

N.V. Pogorelov? and T. Matsuda

Department of Earth and Planetary Sciences, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan

Received 28 May 1999 / Accepted 29 September 1999

Abstract. We present the study of the solar wind interactionwith the super-Alfvenic magnetized interstellar medium in thecontroversial case when fast magnetosonic shocks are nonevo-lutionary, while admissible switch-on shocks cannot exist dueto geometrical reasons. The flow behaviour and the accom-panying shock configuration are investigated both in the two-dimensional axisymmetric and 2.5-dimensional, that is, permit-ting rotation around the symmetry axis, statements. Generalquestions of the origin of nonevolutionary MHD shocks arediscussed.

Key words: Magnetohydrodynamics (MHD) – shock waves –Sun: solar wind – ISM: kinematics and dynamics

1. Introduction

The problem of the solar wind (SW) interaction with the localinterstellar medium (LISM) attracts close attention in the recentyears owing to a continuous stream of data obtained by Pioneer,Voyager, and Ulysses spacecrafts from various regions of the he-liosphere. Numerical calculations of Washimi & Tanaka (1996),Pogorelov & Semenov (1997), Linde et al. (1998), Pogorelov& Matsuda (1998), Ratkiewicz et al. (1998), showed the impor-tance of the LISM magnetization. These authors performed two-and three-dimensional calculations of this problem for variousangles between the LISM magnetic field and velocity vectorsB∞ andV∞. Fahr et al. (1986) also studied the influence ofthis angle on the shape of the heliopause semianalytically in theNewtonian approximation. The general scheme of the axisym-metric flow can be seen in Fig. 1, where numerical results areshown which will be discussed later. In this picture TS is theinner shock terminating the solar wind within the heliopause(HP) and BS is the bow shock resulting from the supersonicLISM interaction with the contact surface between the winds.The LISM flow is directed from the right to the left.

Matsuda & Fujimoto (1993) were the first who performednumerical modeling in the axisymmetric case withB∞ ‖ V∞

Send offprint requests to: N.V. Pogorelov? Current affiliation: Institute for Problems in Mechanics, Russian

Academy of Sciences, 101 Vernadskii Avenue, 117526 Moscow, Rus-sia (pgrlv@ipmnet.ru)

for various values of the magnetic field. Unfortunately, theirresults suffered from the lack of space resolution and interpre-tation of some of them were criticized by Baranov & Zaitsev(1995). This mainly concerned the cases with larger values ofthe LISM magnetic field. This point still requires proper clari-fication.

Magnetohydrodynamic (MHD) discontinuities are fre-quently encountered in various astrophysical and geophysicalphenomena. The variety of MHD discontinuities which satisfythe conservation-law boundary conditions of the Hugoniot type(see Kulikovskii & Lyubimov 1965) is wider than that in puregas dynamics. It is well known that this variety consists of con-tact, tangential, and Alfven (rotational) discontinuities and mag-netosonic shocks. The peculiarity of the MHD is in the fact thatnot all entropy-increasing shocks can exist in nature as station-ary structures. The condition of the entropy increase is necessarybut not sufficient. The MHD shocks must also be structurallystable, that is, stable with respect to disintegration into the se-quence of other (stable) discontinuities. Note that disintegrationof structurally unstable shocks in the ideal nondissipative state-ment occurs instantaneously under action of an infinitesimalperturbation. This is one of the reasons of difficulties arisingin numerical modeling of MHD flows. Numerical viscosity iswidely considered as a tool for selection of admissible shocks ingas dynamics. This is no longer valid for MHD. On the contrary,Barmin et al. (1996) showed that numerical viscosity, which ismuch larger than physical in a number of space physics prob-lems, can sometimes slow down the disintegration of inadmis-sible discontinuous solutions.

Since we expect the physical problem to have a unique solu-tion, it is necessary to develop the rules for selection of an admis-sible one. Disintegration of a shock is closely connected with theproblem of its evolutionarity. The shocks are called evolution-ary if the problem of their interaction with small disturbances iswell posed (Landau & Lifshitz 1960). This means that the num-ber of parameters defining an arbitrary small disturbance mustbe equal to the number of equations (the linearized boundary re-lations on the shock) they satisfy. In this case the shock can emitenough waves to adjust itself to the new boundary conditions. Ifthe amplitudes of these outgoing waves cannot be determinedunambiguously from the shock relations, then the solution isnonunique. By the choice of the coordinate system moving in the

698 N.V. Pogorelov & T. Matsuda: Nonevolutionary MHD shocks

shock plane we can always make the vectors of magnetic field,velocity, and shock normal lie in one plane ahead of the shock.The above-mentioned equations for the amplitudes of outgoingwaves split in MHD into two subsystems, one of them relat-ing the disturbance amplitudes of quantities lying in this planeand of those perpendicular to it (transverse perturbations). Thus,the evolutionary property must be checked separately for planeand transverse disturbances. As a result, some nonevolutionaryshocks turn out to be unstable only to out-of-plane disturbances.Akhiezer et al. (1958) showed that only fast and slow MHDshocks are evolutionary. These shocks propagate with respectto the unperturbed media at rest with the velocity larger than thefast and the slow magnetosonic velocities, respectively. Thereis a simple rule to determine whether the shock is evolutionaryand therefore physically realizable. The fast MHD shocks arealways super-Alfvenic, that is, the velocity component normalto the shock is larger than the velocityaA = Bn/2

√πρ of the

Alfv en wave propagation both ahead of the shock and behindit. The slow shocks are sub-Alfvenic. Nonevolutionary shocksare trans-Alfvenic. There are certain types among MHD shockswhich require a separate study. These are the cases of perpendic-ular, parallel, and singular shocks. In a perpendicular shock (itis also sometimes called transverse), the magnetic field vectoris perpendicular to the shock normal. Perpendicular shocks arealways stable. If the magnetic field vector is parallel to the shocknormal, we have a parallel shock (it is also sometimes called nor-mal). Behaviour of parallel shocks strongly depends on whetherthe Alfven velocity in front of them is smaller or larger than theacoustic speed of sound. In the former case, slow MHD shocksdo not exist, while fast shocks are evolutionary and admissiblefor all their intensities if the flow is supersonic ahead of them. Inthe latter case, on the contrary, slow shocks are admissible forall their intensities if the flow is supersonic and sub-Alfvenicahead of them, while fast shocks are admissible only in a certainrange of parameters ahead of the shock even if they correspondto a super-Alfvenic flow. Singular shocks are those for which thetangential component of magnetic field is equal to zero aheadof (behind) the shock and not zero behind (ahead of) it. Suchshocks are called switch-on (switch-off), as the tangential com-ponent of the magnetic field vector is switched on (off) at them.Switch-on shocks are always fast while switch-off shocks arealways slow, since the tangential component of the magneticfield always increases through fast and decreases through slowMHD shocks.

Let us consider what happens if we increase the value of theLISM magnetic fieldB∞ with the rest of the LISM quantitiesbeing fixed. We have two dimensionless parameters relatingthe quantities in front of the shock, namely, the Mach numberM∞ = V∞/a∞ (a∞ is the acoustic speed of sound) and theAlfv en numberA∞ = V∞/aA∞. If a∞ > aA∞ for M∞ > 1,the forward point of the bow shock corresponds to a fast parallelshock which is always realizable. If we further increaseB∞,sooner or lateraA∞ will become larger thana∞ with A∞ > 1.In this case the parallel shock, though remaining fast, will bestill evolutionary untilB∞ acquires the value corresponding to

1 < A∞ <

((γ + 1) M2

∞2 + (γ − 1)M2∞

) 12

. (1)

A similar formula can be found in Landau & Lifshitz (1960)for a particular case of the specific heat ratioγ equal to 5/3.

ForA∞ from the interval (1), the Alfven number behind theshock is smaller than 1, thus resulting in a trans-Alfenic shockwhich is inadmissible. Occasionally, a singular (fast switch-on) shock becomes admissible exactly in this range ofA∞(Lyubarskii & Polovin 1959). On a switch-on shockB ‖ nahead of the shock butB 6‖ n behind it. That is, a tangentialcomponent of the magnetic field must appear at the forwardpoint of the bow shock. On the other hand, this cannot occurdue to geometrical reasons, since there is an infinite number ofswitch-on directions and all of them are equivalent. It could befairly easily expected that the structure of the flow must be dif-ferent for the mentioned values ofB∞ from that in the regularcase.

The outlined problem is of fundamental importance in MHDmodelling of astrophysical phenomena, regardless of the factthat we discuss it only in the application to the particular prob-lem of the SW–LISM interaction. The MHD shock behaviouris not only important for interpretation of observational data inheliospheric physics, it must be quite clear to those creating nu-merical codes for MHD simulations. In contrast to pure gas dy-namic problems, numerical solution of which can generally beobtained by direct solution of the discretized Euler system, morecomplicated physical phenomena, though also governed by hy-perbolic systems, require deeper understanding of all mathemat-ical aspects accompanying them (Kulikovskii et al. 1999). Oneof the most recent views on the trans-Alfvenic shock behaviourin a magnetized plasma is given by Markovskii (1999). Discus-sion of numerical schemes that might facilitate the nonevolu-tionary shock disintegration can be found in the paper of Myong& Roe (1998).

The flow pattern originating at the forward point of the bowshock in the nonevolutionary interval was discussed by Steinolf-son & Hundhausen (1990) and Matsuda & Fujimoto (1993).Baranov & Zaitsev (1995) showed impossibility of the station-ary shock configuration presented in the latter paper but did notexplain the reason of the phenomenon. Besides, they failed toobtain the solution in the questionable range of Alfven num-bers. Myasnikov (1997) obtained the solution similar to thatof Matsuda & Fujimoto (1993) which turned out to be weaklynonstationary so that final solution could not be established. DeSterk et al. (1999) studied the plasma flow around an infinitecylinder and found out a variety of MHD shocks, some of themnonevolutionary.

In this paper we are revisiting the SW-LISM interactionproblem and perform high-resolution calculations in the two-dimensional axisymmetric and 2.5-dimensional statements. Inthe latter case we add a small axisymmetric rotational pertur-bation to the LISM velocity and magnetic field. As a result,only evolutionary shocks remain in the interaction region. Thisapproach lies in the framework of the approach of Barmin et

N.V. Pogorelov & T. Matsuda: Nonevolutionary MHD shocks 699

-600 -400 -200 0 200 400 600

-200

-400

-600

0

200

400

600

BS

TS

HP

z

x

Fig. 1.General configuration of the interaction: density (below the sym-metry axis) and total pressure logarithm isolines

al. (1996) who studied the behaviour of the nonevolutionarycompound wave under action of rotational perturbations.

2. Statement of the problem

We generally remain in the framework of the statement adoptedby Pogorelov & Semenov (1997). Although the kinetic treat-ment of the neutral component of the winds was proved to beessential in the SW–LISM interaction (Baranov et al. 1998), weconsider only the charged component and solve the MHD equa-tions for perfect, ideal, and infinitely conducting plasma. Thesolar wind is assumed spherically-symmetric. The anomalouscosmic rays and the magnetic field of the Sun are disregarded,since they are not crucial for solution of our particular problem.B∞ is supposed parallel toV∞.

The following set of determining parameters is used (Bara-nov & Zaitsev 1995):

ne = 7 cm−3, Ve = 450 km s−1, Me = 10,

n∞ = 0.07 cm−3, V∞ = 25 km s−1, M∞ = 2.

Index “e” refers to the SW at 1 AU, that is, to Earth’s distancefrom the Sun. Index “∞” corresponds to the LISM parametersat infinity. Heren andV are the number density and the velocitymagnitude, respectively, andM = V/a is the Mach number (ais the acoustic speed of sound). The adiabatic indexγ in theadopted approximation is equal to 5/3. We choose such strengthof the magnetic fieldB∞ for which A =

√2, that is, we are

within the interval (1). The dimensionless magnetic field in ourstatement is

B∞ =B∞

V∞√

ρ∞=

2√

π

A=

√2π.

This corresponds toB∞ ≈ 2.3 µGs.Calculations are performed in the half-circular region be-

tweenRmin = 24 AU andRmax = 1000 AU. The number ofcells is 501 in the radial and 504 in the angular direction. Calcu-lations are performed using the TVD second-order of accuracy

-200 0 200 400

-400

-200

0

200

400

Fig. 2.Streamlines (below the symmetry axis) and magnetic field lines

version of the Lax–Friedrichs scheme proposed and tested forMHD equations by Pogorelov (see Barmin et al. 1996). Theadvantage of the scheme is in its extraordinary robustness andcomputational efficiency, though it is obviously more viscousthan methods based on the approximate solution of the MHDRiemann problem (Brio & Wu 1988, Zachary & Colella 1992,Dai & Woodward 1994, Aslan 1996, Pogorelov & Semenov1996).

The flow parameters are fixed atR = Rmin, sinceMe > 1.The LISM parameters are fixed for the same reason at the super-sonic and super-Alfvenic inflow, whereas absorbing boundaryconditions are applied at the subsonic exit segments of the outerboundary (Pogorelov & Semenov 1997).

3. Simulation results

In this section we present the numerical results for parametersindicated in the previous section. They are subdivided into twoparts. The first one concerns a two-dimensional axisymmetricpattern. In the second part, the pattern remains axisymmetric,though the LISM velocity and magnetic field vectors are allowedto have a rotational component.

3.1. Two-dimensional modeling

In Fig. 1 we present a general view of the solar wind and inter-stellar medium interaction for the two-shock model. The pic-ture represents the chart of 19 equidistant isolines of the densitylogarithm (below the symmetry axis) and of the total pressurelogarithm located between the minimum and maximum values.If we compare this figure with the similar charts in Pogorelov &Semenov (1997), it becomes clear that by increasing the spaceresolution and by avoiding transient solutions we obtained quali-tatively different shape of the bow shock. It is now concave in thevicinity of the symmetry axis. In Fig. 2 we show the streamlines(below the axis) and the magnetic field lines. The prominent inthe streamline behaviour is that there exist a region around thesymmetry axis where they decline towards it, in contrast with

700 N.V. Pogorelov & T. Matsuda: Nonevolutionary MHD shocks

-400 -200 0 200 400

-400

-200

0

200

400

-400 -200 0 200 400

-400

-200

0

200

400

-400 -200 0 200 400

-400

-200

0

200

400

Fig. 3.Density (below the symmetry axis) and thermal pressure isolines

-0.5

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450

ρ

P10Bx

ABSHP

TS

Fig. 4. Parameter distributions along the linex = 1.296

their usual inclination for larger Alfven numbers (or weakermagnetic fields). Later they turn again in the opposite directionat the additional, or secondary, shock which is clearly seen inFig. 3. In this figure we present the thermal pressure isolines(20 isolines between the minimum and the maximum) and thedensity isolines (below the symmetry axis). In the latter case, tovisualize the region of our interest, we show only isolines be-tween 1 and 2.6 with the increment 0.04. Note that the thermalpressure suffers a jump at the heliopause everywhere except forthe stagnation point. That is why, the intensity of this jump de-creases towards the axis until it degenerates into a compressionfan. In the density isolines we can see a high-density layer. Itis also called entropy layer, since there is no jump of pressureacross this layer. It originates owing to the difference in entropybetween isolines crossing the secondary shocks at right anglesnear the axis and acute angles farther from it. This results inexistence of the elongated density wall near the heliopause.

As far as the evolutionary shocks are concerned, we wouldlike to look at the parameter distributions along the symmetryaxis. We show the profiles of several variables along the line

-400 -300 -200 -100 0 100 200 300 400

-400

-300

-200

-100

0

Fig. 5. Density isolines: perturbed LISM

-400 -300 -200 -100 0 100 200 300 400

-400

-300

-200

-100

0

Fig. 6. Streamlines andA = 1 lines: perturbed LISM

x = 1.296 in Fig. 4. We do not have points at the symmetryaxis and the above line is the closest to it available. It is clearlyseen that the Alfven numberA = 2 |V|√πρ/|B| decreasesfrom the value above 1 to the values below 1 across both thebow and the secondary shock. This means that these shocks arenonevolutionary in three-dimensions. A smallx-component ofthe magnetic field appears behind the bow shock. Later|Bx|smoothly decreases, changes its direction, then increases at thesecondary shock, and disappears at the HP. The Alfven numberprofile is abruptly interrupted, since it increases to infinity acrossthe HP. The flow pattern presented in this section is qualitativelydifferent from that obtained by Matsuda & Fujimoto (1993)and Myasnikov (1997). It also differs from the pattern obtainedby Pogorelov & Semenov (1997) for larger Alfven numbers.Thus, in contrast with pure gas dynamics, variation of the LISMparameters can cause substantial qualitative changes in the SW–LISM interaction pattern.

3.2. Rotationally perturbed flow

It is obvious from the general theory of evolutionary shocksthat the bow shock in the vicinity of the symmetry axis mustbe modified by a rotational perturbation of the flow. For thisreason, remaining in the framework of the axisymmetric state-ment, we add they-component (perpendicular to thexz-plane)to the magnetic field and velocity vectors, namely,By = 0.05andV y = 0.05/Bz∞. After that we solve the MHD systemin the 2.5-dimensional formulation, that is, taking into accountthe variation of they-components. The discontinuity pattern isshown in Fig. 5, where the same density isolines are shown as inFig. 3. Though the picture has changed quantitatively, the gen-eral structure of the flow remains similar, since the secondaryshock and the density wall still exist. The streamline pattern is

N.V. Pogorelov & T. Matsuda: Nonevolutionary MHD shocks 701

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450

ρ

A

p0

10Bx

By

BS

HP

TS

Fig. 7. Parameter distribution along the linex = 1.296: perturbedLISM

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450

ρ

A

Bx

By

BS

HP

TS

Fig. 8. Parameter distribution along the linex = 55.5: perturbed LISM

also essentially the same, as seen from Fig. 6, where they areshown together with the shocks and the isolineA = 1. We seethat the secondary shock becomes a switch-off shock, since itssurface coincides with the lineA = 1 and the velocity (andmagnetic field) vector looses its tangential component behindthis shock.

There is also a very narrow region around the symmetry axiswhereA < 1. To look closer into this region, we present theparameter distributions along the linex = 1.296 (Fig. 7). It isclear that the Alfven number becomes less than unity behind theshock. On the other hand, we see that, within a single densityjump, thex-component of the magnetic field is switched onand then switched off without any space interval. This meansthat we have effectively two merged shocks near the axis. Thesecondary shock is well seen in the total pressure profile closerto the HP (compare it with the Alfven number profile whichdrops from 1 to a smaller value behind this shock).

-0.5

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400

ρ

A

p

Bx

By

BS

HP

TS

Fig. 9. Parameter distribution along the linex = 153: perturbed LISM

If x is approximately larger than 50, the switch-off shockin the pair of merged shocks disappears, as seen from Fig. 8showing profiles along the linex = 55.5, and the bow shockbecomes a single fast MHD shock withA > 1 behind it. Aswe move fromx = 0 to x = 55.5, the intensity of the aboveswitch-off shock gradually decreases.

As we pass from the concave part of the bow shock to its con-vex part, we meet the inflection point. At this point the secondaryswitch-off shock seems to approach the bow shock, though thedensity jump across it is rather small. If we look at the parame-ter profiles along the linex = 153 (Fig. 9) which passes in thevicinity of the inflection point, we see that the bow shock againbecomes a switch-on shock, sinceA = 1 behind it. The tan-gential component is switched, however, in the direction fromthe symmetry axis in this case. As we move farther from thesecond inflection point, the bow shock transforms into a fastMHD shock. Note that they-component of the magnetic fieldacquires rather large values, though initial perturbation is verysmall. This means that the flow withinx < 200 is sensitive torotational perturbations. Farther from the symmetry axis they-component becomes much smaller behind the bow shock thanthex-component (a fraction of per cent).

4. Conclusions

We calculated the SW–LISM interaction problem for a contro-versial case of Alfven numbers close to unity. The flow structurein this case is completely different in the region around the sym-metry axis compared with that for larger values of A. Besides, ahigh-entropy layer originates, which spreads over the heliopauseup to x ≈ 300 AU. The latter effect may be important in theinterpretation of observational data.

We showed that nonevolutionary MHD shocks can be suc-cessfully avoided by adding rotational perturbations. Such kindof shocks often originate if three-dimensional MHD equationsare reduced to two-dimensional ones or even in 3D cases ifthe plasma is not ideal and/or if out-of-plane perturbations are

702 N.V. Pogorelov & T. Matsuda: Nonevolutionary MHD shocks

forcefully diminished. In this case nonevolutionary shocks canexist for a long time depending on the value of molecular andmagnetic viscosities (see Barmin et al. 1996). No need to saythat perturbations of any kind can be encountered in the LISM,since it can hardly be assumed absolutely uniform. The flow pat-tern which can be realized for other type of disturbances maybe different, of course, quantitatively, though nonevolutionaryshocks will anyway be absent. The flow pattern obtained inthe axisymmetric case can exist as a transient one. It is sta-tionary only if three-dimensional disturbances do not act at all.This pattern, however, can be rather persistent to disturbances inthe numerical treatment. This is due to the numerical viscosityand resistivity. In this case the disturbance must be either largeenough or act for a sufficiently long time (have low frequency).

Acknowledgements.The authors are grateful to V.B. Baranov,A.A. Barmin, and A.G. Kulikovskii for stimulating discussions. N.V.P.was supported, in part, by the Russian Foundation for Basic Researchgrant 98-01-00352. T.M. was supported by the Grant-in-Aid for Sci-entific Research (10147211, 10640231) of the Ministry of Education,Science and Culture of Japan.

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