Interactions between magnetohydrodynamical discontinuitiesWenlong Dai and Paul R. Woodward Citation: Physics of Plasmas (1994-present) 1, 3662 (1994); doi: 10.1063/1.870901 View online: http://dx.doi.org/10.1063/1.870901 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/1/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hall magnetohydrodynamics model of double discontinuities Phys. Plasmas 9, 4905 (2002); 10.1063/1.1521420 Nonevolutionary discontinuous magnetohydrodynamic flows in a dissipative medium Phys. Plasmas 5, 2596 (1998); 10.1063/1.872946 Interaction between perpendicular magnetohydrodynamic shocks Phys. Fluids B 5, 732 (1993); 10.1063/1.860516 Propagation of discontinuities in magnetohydrodynamics Phys. Fluids 24, 1386 (1981); 10.1063/1.863544 Propagation of discontinuities along bicharacteristics in magnetohydrodynamics Phys. Fluids 23, 648 (1980); 10.1063/1.863024

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Interactions between magnetohydrodynamical discontinuities Wen long Dai and Paul R. Woodward University of Minnesota, School of Physics and Astronomy, Army High Peiformance Computing Research Center. Supercomputer Institute, noo Washington Avenue South, Minneapolis, Minnesota 55415

(Received l3 May 1994; accepted 3 August 1994)

Interactions between magnetohydrodynarnical (MHD) discontinuities are studied through numerical simulations for the set of one-dimensional MHD equations. The interactions include the impact of a shock on a contact discontinuity, the collision of two shocks, and the catchup of a shock over another shock. The shocks involved in the interactions may be very strong. Each shock in an interaction may be either a fast or a slow shock.

I. INTRODUCTION

Interactions between magnetohydrodynamical (MHD) discontinuities are important for understanding some natural phenomena, such as the interaction between supernova blast waves and interstellar clouds,I-3 the interaction between so­lar wind irregularities and magnetosphere,4 and the shock interactions in the out of the heliosphere.s Efforts have been paid in modeling such phenomena through the interactions. Qualitative and quantitative descriptions of the interactions involving strong MHD shocks are often desired.

In this paper the interactions between MHD discontinui­ties are investigated through numerical simulations for the set of one-dimensional MHD equations. The discontinuities may be very strong. The simulations give quantitative de­scriptions for the interactions, and some of them have never been realized before. Since one-dimensional MHD equations are used in our investigation, our model is highly idealized when compared with what may be occurring in the nature or a laboratory, and we do not intend to be modeling a portion of any real phenomena. However, our simple treatments of the interactions may serve to refine our intuition of some real phenomena.

The plan of this paper is as follows. Basic equations and the numerical scheme we used will be introduced in Sec. II. Section III is the simulation results for various interactions between MHD discontinuities. The summary and a brief dis­cussion of this paper will be put in Sec. IV.

II. BASIC EQUATIONS AND NUMERICAL SCHEME

Our investigation will be based on one-dimensional MHD equations. The influence of the internal structure of a shock on an interaction is supposed to be negligible. Thus, ideal MHD equations plus numerical viscosity/resistivity are used in our investigation. Under the one-dimensional ap­proximation (ay = a z = 0), ideal MHD equations are in the form6

atP+aJpUx)=O,

at(pux) + Bx(pu; + P) = 0,

a/cpu\,) + axC pUxu" + Ay)=O,

a/(puz) + aJpuxuz + A z) = 0,

at(pE) + ax(puxE+ uxP+ uyAy+ uzA z) = 0,

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Here p is the mass density, (u x ,uy ,uz) and (Bx ,By ,Bz) are the three components of the flow velocity and the magnetic field, respectively, E is the specific total energy. and P, A y ,

and A z are the diagonal and off-diagonal total pressures. We see that B x is a constant under the one-dimensional approxi­mation. E, P, A y , and A z are defined as

_ 12221222 E=I:+ 2" (Ux+uy+uzl+ 81TP (Bx+By+B z )·

Here I: is the internal energy and p is the thermal pressure. The thermal pressure is related to the internal energy through the gamma-law equation of state p = (y- 1 )Pl:, with y the ratio of the specific heat capacities. Although the electric field E and current j do not explicitly appear in Eqs. (1)-(7), they may be obtained through the magnetic field and flow velocity: E=Bxu/c and j=cVXB/41T under the approxima­tion of ideal MHD.

For the simplicity of the formulations in our numerical scheme, we write the basic equations in a Lagrangian mass coordinate defined by dm == p dx:

v Ux

u y

U== UZ

VBy VB z E

Here V== lip.

-ux P Ay

F(U)== A z -Bxuy -Bxuz

PUx + Ayuy + Azuz

(8)

3662 Phys. Plasmas 1 (11), November 1994 1 070-664Xi94/1 (11 )/3662/14/$6.00 © 1994 American Institute of Physics

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Our scheme is of the Godunov type for Eqs. (1)-(7), which has second-order accuracy in both space and time. Considering a numerical zone Xi<X<xi+ I> we write Eq. (8) in a different form:

(9)

Here At is the time step and Ami is the mass contained in the ~one, (U(t»i is the average of U over the zone at time t, and Fi is the time-averaged flux at the interface Xi during the time step, i.e.,

I !Xi+1 (U(At)i=;r-: U(x,t)dx,

X, Xi

- I f Fi= A F[U(Xi ,t)]dt, u.t !!'t

(10)

with AXi the width of the zone. The discretized equation (9) may be obtained by integrating Eq. (8) over the rectangular Xi<X<xi+1 and O<t<A.t in the (x-t) space. The dis­cretized form, Eq. (9), is exact if the time-averaged fluxes F; may be exactly found. Thus, one of the key issues for this type of schemes is to obtain the set of time-averaged fluxes at the interfaces of numerical zones. The fluxes needed in our scheme are approximately calculated through a nonlinear Riemann solver/ which solves a Riemann problem in ideal MHD equations.

A Riemann problem in ideal MHD equations is an initial value problem, Eqs. (1)-(7), subject to a specific initial con­dition:

if x<O,

if X>O.

Here U(/) and U(r) are any two constant states. It is well known that the basic equations (1)-(7) allow

three kinds of waves: fast, Alfven, and slow waves, besides entropy waves. The entropy wave is stationary in the La­grangian coordinate, and the fast, slow, and Alfven waves have their wave speeds Cf , Cs' and Ca , respectively:

C}.s= M( C~+ C;+ C;)::t ~( d+ C~+ C;)2-4dC~].

Here the plus (or minus) sign is for a fast (or slow) wave. Here Co, Ca , and Ct are defined as Co=(ypp)I12, Ca= (B;pI47T) 112, and Ct=[(B~+B;)pI47T]1f2. Note that the wave speeds are defined in the mass coordinate. Their equivalences in a x coordinate may be obtained through di­viding them by the mass density p.

An outline of the Riemann solver we used is given here, and a more detailed description for it may be found in Ref. 7. The Riemann solver is based on jump conditions for any discontinuity in ideal MHD equations. The jump conditions may be derived directly from the conservations of mass, mo­mentum, energy, and magnetic flux, and may also be ob­tained simply by integrating Eq. (8) over a discontinuity:

W[V]= -[ux],

W[ux]=[P],

Phys. Plasmas, Vol. 1, No. 11, November 1994

(11)

(12)

slow wave.

rotation .... R3 fast wave R2····· ..

".

slow wave rotation

fast wave

__ ~L~ __ ~~~~~ ___ x

FIG. I. All possible waves generated in a MHD Riemann problem and eight possibly different states divided by the wave fronts.

W[Uy] = [Ay], (13)

W[uzJ=[A z], (14)

W[VBy]= - Bx[uy], (15)

W[VBz] = -Bx[uzJ, (16)

WEE] = [uxP] + [uyAy] + [uzAz]' (17)

Here W is the speed of the discontinuity propagating in the mass coordinate. The bracket [X] stands for the difference between the states on the two sides of the surface, i.e., [Xl = XI - Xo. Physically, W is the mass flux across a dis­continuity, i.e., W=(s-uxo)Po, with s being a usual shock speed and u xO being the x component of the flow velocity in the prewave state. Here W is negative when the discontinuity propagates in the negative x direction. This set of jump con­ditions is equivalent to that derived from the Eulerian equa­tions in the book by Landau and Lifshitz.6 In our Riemann solver, we approximate rarefaction waves by rarefaction shocks obeying the jump conditions given above (and in­volving decreases in the entropy). This approximation limits the Riemann solver itself to weak rarefaction waves. We use the Riemann solver only for the approximate calculation of the fluxes needed in the scheme.

The solution of a general Riemann problem in ideal MHD equations contains six waves moving leftward and rightward, which are separated by a contact surface. Each wave may be discontinuous. The appearance and strength of each wave depend on specific data in the initial discontinuity. The wave fronts of the six waves in both directions and a contact surface separate whole (x-t) plane into eight possi­bly different regions. We label them L, R2, R3, R4, R5, R6, R7, and R, respectively, as shown in Fig. l.

To find a Riemann solver, we find a formula for the shock speed Wf or Ws in terms of the prewave state, and one transverse component of the magnetic field in the post-wave state:

W},s 122

2(l+Ao) [(Cs+Cf+A 1)

::t V( C; + C}+ A 1)2- 4(1 + Ao)( C;C}- A 2)].

(I8)

Here the plus (or minus) sign is for fast (or slow) shocks. Here AO,I,2 are defined as

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A2==~A.p2[A}Y + (y+ 2)A.p2 Ay[Ay] + (y+ l)C~C;

4 2 2}[Ay] +( y+ 1)Ca-2COCa A'

y

Here A.=1+(B/By)2, and all the variables in A O•I ,2 are evaluated at the prewave state, except for the jump [Ayl. For a given set of left and right states, the Riemann solution is obtained by the following procedures:

(1) Guess one transverse component (e.g., y component) By2, By4 , and By7 of the magnetic field in regions R2, R4, and R7, and the orientation 1{1 of the transverse part of the magnetic field in regions R3, R4, R5, and R6. Here tan 1{1= B z3/ By 3'

(2) Consider the right' and the left states as two prewave states, calculate two fast shock speeds using Eq. (I8), one for the wave moving rightward and the other for the wave moving leftward; then from the jump conditions, Eqs. (11)-(17), with these two shock speeds and the field component By in the two post-wave states, find the complete states in regions R2 and R7, respectively.

(3) Perform the rotations with the earlier guess 1{1, then use the jump conditions with the speed W equal to the Alfven speeds in regions R3 and R6 to obtain the com­plete states in regions R3 and R6, respectively.

(4) Consider the states in regions R3 and R6 as two prewave states and repeat procedure (2) for two slow shocks in­stead of the fast shocks to obtain the states in regions R4 and R5.

(5) Apply the conditions for a contact discontinuity to the states in regions R4 and R5 to improve the earlier guess on By2 , By4 , By7 ' and 1{1, as described below. With this improved guess, go back to procedure (1).

After the first four steps, the state in region R4 is a function of By2 , By4' and 1{1, and the state in region R5 is a function of the By7 , B y4 , and 1{1. If By2 , By4 , By7' and 1{Iare in a solution of the Riemann problem, two states in regions R4 and R5 should be the same, except for their mass densi­ties and total energies, The remaining requirements for the states in regions R4 and R5 are that (p,ux,uy,u z) should be the same, i.e.,

ux4(B y2,B y4, 1{1) = U xs(By7 ,B y4' 1{1),

uy4(B y2 ,B y4, 1{1} = uys(B y7,B y4, 1{1),

uz4(B y2 ,B y4, 1{1) = uzs(B y7,B y4, 1{1),

P4(B y2 ,B y4, 1{1) = Ps(B y7 ,B y4' 1{1).

In each step listed above, there are analytical formulations, and it is not difficult to find various Jacobian coefficients for the initial guess. Thus, the set of equations may be iteratively solved for B),2' By4' By7 , and 1{Iby Newton's method.

3664 Phys. Plasmas, Vol. 1. No. 11. November 1994

The initial guess needed in the Riemann solver may come from a linear Riemann solver. The linear Riemann solver may be based on characteristic formulations of the set of Eqs. (1)-(7). For the reference needed in our scheme, we list the characteristic formulations here.

Following the general outline in the book by Courant and Friedrichs,S we may find all characteristic curves and their associated Riemann invariants. The first characteristic curve is found to be dxl d t = 0, and its associated Riemann invariant is

(19)

The logarithm of this invariant is proportional to the specific gas entropy. Along the characteristic curve, the entropy is constant, but across it, the specific volume and entropy may have a jump. The remaining six Riemann invariants for two fast, two slow, and two Alfven waves may be expressed only as inexact differentials:

dRf ;!;=( CJ- C~)(dP±' Cf duJ + pAy(dAy±'Cf du y)

+pAz(dAz±.Cfduz), (20)

dRs±==(C;-C~)(dP±'Cs duJ+pAy(dAy±'Cs duy)

(21)

dRa±==±.Ca(Bz dUy-Bv dUz)+(B z dAy-By dA z). (22)

Along any characteristic curve, which is defined as dxl dt = C, with C being a wave speed, its associated Riemann invariant remains unchanged.

The scheme starts from zone averages of a set of vari­ables (p,p,UX,uy,uz,By,Bz)' which are defined through Eq, (10). For each of these variables, a cubic polynomial is used to find its value at an interface Xi from its zone averages on zones (Xi-2,Xi-I)' (Xi-I,Xi)' (Xi,xi+l)' (Xi+\oXi+2)' After we get the values of the variables at the interfaces, the mono­tonicity constraint originally suggested by Van Leer9 is ap­plied to these values at the interfaces. As we know, interpo­lated structures are not always monotone increasing (decreasing), even though they have been constructed from monotone data. The over- and undershoots in the interpolated internal zone structures eventually give rise to over- and un­dershoots in the zone-averaged data. Van Leer realized that an advection scheme may be made to preserve the monoto­nicity of its initial data if any nonmonotone interpolated zone structures are flattened, so that they become monotone. This leads the Van Leer's monotonicity constraint: no values in-

a :~)2"'" ... .. Pl

<a>j .... , .................. I-~-....,

<a> i+l :~1::::::::::::::::::: :::x~::::::::x:c:,;-q----., X

Xi_l Xi xa Xi+1

FIG. 2. Interpolation of a zone structure. Here pi and q are two points after the interpolation of a cubic polynomial defined by four zone averages. The curve inside the zone (Xi ,xx+ I) is the result of a parabolic interpolation. The ranges (x, ,Xl)' (Xi ,x.), and (x, ,x,) are the domains for a fast, Alfven. and slow waves traveling to the left, respectively.

W. Dai and P. R. Woodward

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.9

a. Q,

.8

.7

0 .2 .4 .6 .8 0 .2 .4 .6 .8

x X

at 0

·1

o 2 .4 .6 .8 o .2 .4 .6 .8 X X

FlG. 3. The propagation of a special slow wave. The dashed, dotted, and solid lines are the profiles at t = 0, 1, 2, respectively. The periodic boundary condition and 200 numerical zones in x E (0, 1) are used. The profiles should be compared with the simulation for dissipative MHD equations in Ref. 12.

TABLE I. The initial conditions used in the simulations.

p p Ux u y Uz By Bz

Fig. 4 x<0.05 3.897 208.5 12.79 -0.1181 -0.109 15.79 7.893 Bx=4 0.05<x<0.25 1 0.1 0 0 0 4 2

x>0.25 6 0.1 0 0 0 4 2 Fig. 5 x<O.1 2.18 0.4494 0.1854 0.4463 0.1116 3.04 0.7599 Bx=2 0.1 <x<O.4 0.1 0 0 0 4

x>O.4 5 0.1 0 0 0 4 Fig. 6 x<0.125 3.9 57.99 6.719 -0.1033 -0.0516 7.868 3.934 Bx=2 0.125<x<0.5 0.1 0 0 0 2 1

x>0.5 3.579 12.27 -3.259 0.1923 0.0961 7.463 3.732 Fig. 7 x<0.025 12.66 447.5 9.676 -0.0626 -0.0313 26.44 13.22 Bx=2 0.025<x<0.2 3.579 12.27 0 0 0 7.463 3.732

x>0.2 1 0.1 -3.259 0.1923 0.0961 2 1 Fig. 8 x<0.065 2.905 1.062 0.7287 0.3707 0.1853 0.9648 0.4824 Bx=5 0.065<x<0.935 1 0.1 0 0 0 2 1

x>0.935 2.905 1.062 -0.7287 -0.3707 -0.1853 0.9648 0.4824 Fig. 9 x<O.1 2.905 1.0623 0.7287 0.3707 0.1853 0.9648 0.4824 Bx=5 O.I<x<O.4 1 0.1 0 0 0 2 1

x>O.4 1.71 0.2554 -0.2307 -0.1015 -0.0508 1.858 0.9291 Fig. 10 x<0.06 3.797 1.169 0.4985 0.2719 0.136 0.798 0.399 Bx=5 0.06<x<0.25 1.71 0.2554 0 0 0 1.859 0.9291

x>0.25 1 1 -0.2307 -0.1015 0.0508 2 Fig. 11 x<0.06 3.89 57.99 6.719 -0.1033 -0.0516 7.868 3.934 Bx=2 0.06<x<0.275 0.1 0 0 0 2 1

x>0.275 2.031 0.4117 -0.2586 -0.4319 -0.2159 0.618 0.309 Fig. 12 x<0.05 7.764 35.69 3.594 -0.0288 -0.0144 2.408 1.204 Bx=2 0.05<x<0.25 2.022 0.4085 0 0 0 0.618 0.309

x>0.25 0.1 -0.2586 -0.4319 -0.216 2

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400 20

300 15

Co Q"

200 10

100 5 CD 0

0 0 .2 .4 .6 .8 0 .2 .4 .6 .8

X X

25 I I I I

20 ss 10 I- -

al" 15 SR 51- -

10

5 0-( : ..... ·t

0 .2 .4 .6 .8 0 .2 .4 .6 .8

X X

500 1:1 I-12

400 I- -10

I- -300 8 roM

200 I- - 6

100 I- - 4

o bt :------1". 2

o .2 .4 .6 .8 0 .2 .4 .6 .8

x X

FIG. 4. The simulation for the impact of a fast shock (M= 10) on a denser region. Dashed, dotted, and solid lines are the profiles at 1=0, 0.01. and 0.07, respectively. The initial conditions are shown in Table I. Here Bx =4.

terpolated within a zone shall lie outside the range defined by the zone averages for this zone and its two neighbors. For example, referring Fig. 2, if we obtain two values a pI and a q

at two interfaces Xi and xi+ 1 after the interpolation, we beat the point pI down to the point p.

After the mono tonicity constraint is applied, a parabola defined by a zone average <a} i ' and the values at two edges of the zone is used to interpolate the structure inside the zone for the calculation of various domain averages. For a wave propagating to the left, the domain on which the disturbance may influence the state at the interface Xi through the fast wave during a time step A t cannot be beyond the range (Xi 'XI)' The distance (XI-Xi) is approximately (C/}i At, with < C f) i being the zone-averaged fast wave speed. Simi­larly, the range (Xi ,Xa ) is the domain for an Alfven wave, and (Xi ,xs ) is the domain for a slow wave. A domain average for a fast (or Alfven or slow) wave is the average of an

3666 Phys. Plasmas, Vol. 1, No. 11, November 1994

interpolated structure on a fast (or Alfven or slow) domain. The flow is considered continuous inside each zone, but

may have significant change across an interface of numerical zones. After we find the various domain averages on fast, Alfven, and slow domains, we use the Riemann invariants to find the effective left and right states for a Riemann problem arising from the interface Xi' For example, p, p, Ux ' Uy ' u z '

By. and B z in the right state are the solution of the following set of equations:

lXf(P- PI) + lXfCf(ux-ux{)+ay(Ay- Ay/) +ayC/uy

- Uyl) + az(Az- A zI) + azCf(uz- Uzl) = 0,

lXf(P- (Phd) - lXfCf{ux - (Uxhd) + ay{Ay - (Ay)fd)

-ayCf(uy - (Uyhd) + az(Az - (Az}fd)

-azCiuz - (Uz)fd) =0.

(23)

(24)

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I I 12 .6 r- -

SS 10

- SS 8

f- -0.

.4 Q.

6

4 .2 f- -

2

I ----1----o .2 .4 .6 .8 0 .2 .4 .6 .8

X X

4

.15

3.5

or ~ .1

3 .05

2.5 0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x x

.9 1-1 L

- .9

.85 t- -~ ~ .8

.7 .8 t- '---

I -·-t----· .6

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x X

FIG. 5. The simulation for the impact of a slow shock (M=2) on a denser region. Dashed, dotted, and solid lines are the profiles at 1=0, 0.6, and 3.6, respectively. The initial conditions are shown in Table I. Here Bx =2.

a,( P - (P)sd) - asCs(ux - (Ux)fd) +ay(Ay - (Ay)fd)

- ayCs(uy - (Uy)fd) + az(A z- (Az)fd)

- a~Cs( U z - (u zhd) = 0, (26)

CuB:(u y- Uy/) - CaBy(u z- uz/) + Bz(Ay- A y/)

- By(Az - A z/) = 0, (27)

- CaB :(uv - (Uy)ad) + CaBy(u z - (Uz)ad) + Bz(Ay - (Ay)ad)

-B,,(Az-(Az)ad)=O. (28)

Phys. Plasmas, Vol. 1, No. 11, November 1994

Here the coefficients af,s and ay,z are defined as af,s= (CL - C~), and ay,z= pAy,z' respectively. Here (a)fd' (a) sd' and (a) ad stand for the domain averages on the fast, slow, and Alfven domains, respectively. The subscript "Z" stands for the value at the left interface of the zone (Xi,xi+l)' The coefficients are evaluated at the zone aver­ages.

After we found the effective left and right states, we use the Riemann solver to find the set of time-averaged fluxes needed in the Godunov scheme, Eq. (9), and update con­served quantities, i.e., mass, momentum, energy, and mag­netic flux. Again, we mention that although applicability of the Riemann solver itself is limited to weak rarefaction waves, the numerical scheme described above may correctly

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FS 10 150

8

C-100 Q. 6 CD

4 50 FS 2

0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 X X

6 20

4

15

at SR :! 2

10 0

5 ·2

0 .2 .4 .6 .8 0 .2 .4 .6 .8 X X

200

10

150 8

ri: 100 ar 6

50 4

2 0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 X X

FIG. 6. The simulation for the collision of two fast shocks (M = 10 and 5). Dashed, dotted. and solid lines are the profiles at t= 0, 0.02. and 0.12. respectively. The initial conditions are shown in Table I. Here B x = 2.

deal with strong rarefaction waves, since the Riemann solver is used only for the approximate calculation of the time­averaged fluxes during one time step.

One formulation, which is widely used in Eulerian hy­drodynamics algorithms, is that of performing a hydrody­namics calculation for one time step on a Lagrangian grid, and then mapping the results onto a fixed Eulerian grid. After zone averages are updated on the Lagrangian grid, we map the conserved quantities onto the Eulerian grid. The mapping procedure is only a transformation from one grid to the other, and keeps the conservation of the mass, momentum, energy, and magnetic flux.

Finally, a numerical viscosity is used in our code for strong shocks, as in the piecewise parabolic method. 10, I I

Without the numerical viscosity added, unphysical MHD

3668 Phys. Plasmas, Vol. 1, No. 11. November 1994

waves may be emitted into the flow behind a strong shock. It should be stressed that a Godunov scheme, without any nu­merical viscosity added, already contains error terms that produce numerical diffusion of this type of scheme. There­fore the numerical viscosity we are adding do not constitute the complete numerical viscosity of the scheme.

We have to point out that as long as an internal structure of a shock is not important for the dynamics, the scheme may correctly simulate resistive MHD phenomena, although it is based on the formulations of ideal MHD equations. The dissipative terms neglected in Eqs. (1)-(7) determine the in­ternal structure of a discontinuity. The mass, momentum, en­ergy, and magnetic flux are stilI conserved across any discon­tinuity when the dissipative terms are included. For those phenomena in which the influence of the internal structure of

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400

10

300

0. Q.

200

5

100 ~~.~

:. '0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x x

I I I 25

10 I- -20

at 15 ':! 5 I-- : -

10 : -.- ---,.. 01-- ' .... -------:: -

5 '. " '. " :. I

0 .2 .4 .6 .8 0 .2 .4 .6 .8 X X

500

400 SS 10

300

a: a! 200

5 :

100 _. - - """-"'T.

:. 0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 X X

FIG. 7. The simulation for the catchup of two fast shocks (M=5 each). Dashed, dotted, and solid lines are the profiles at t=O, 0.08 and 0.56, respectively. The initial conditions are shown in Table I. Here Bx= 2.

a shock on the dynamics is negligible, it is important for the dynamics whether or not the scheme has dissipation effects, but specific formulations for the dissipative terms are not important. In order to show the correctness of the scheme for dissipative MHD phenomena, we give one example for an intermediate shock, which was studied by WU12 through the numerical simulation for dissipative MHD equations. We ini­tially set a slow wave, as in Ref. 12, which is indicated by the dashed lines in Fig. 3. The solid and dotted lines in the figure show the profiles at t = 1 and 2, respectively, which are the same as those reported in Ref. 12 from simulations for dissipative MHD equations. A numerical grid with 200 numerical zones is used in our simulation, which is much less fine than the grid used in Ref. 12.

Phys. Plasmas, Vol. 1, No. 11, November 1994

III. INTERACTIONS BETWEEN MHD DISCONTINUITIES

In this section we will implement a set of numerical simulations for the interactions between MHD discontinui­ties. The interactions include impact of a shock on a contact discontinuity, the collision of two shocks, the catchup of one shock over another shock. In all the simulations, 512 numeri­cal zones between zero and unity, i for y, continuation boundary conditions, and 0.5 for the Courant safety number are used. In all the figures to be presented in this section, FS stands for a fast shock, FR for a fast rarefaction wave, SS for a slow shock, SR for a slow rarefaction wave, IS for a inter­mediate shock, and CD for a contact discontinuity. The total pressure P t in the figures are defined as the thermal pressure

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4 6

5 3

4 a.

2 CI.

3

FS 2

FS

0 O· .2 .4 .6 .8 0 .2 .4 .6 .8

x x

3

.5

2

at M 0 ........... , :>

0 IS

-1 -.5

0 .2 .4 .6 .8 0 .2 .4 .6 .8 1 x x

4 1.5

3

0: N .5 ID

2 0

-.5

0 .2 .4 .6 .8 0 .2 .4 .6 .8

X X

FIG. 8. The simulation for the collision of two slow shocks (M= 3 each). Dashed. dotted. and solid lines are the profiles at /=0, 0.3, and 0.9, respectively. The initial conditions are shown in Table I. Here B x = 5.

plus the magnetic pressure (B; + B;)/8 71" _ The dotted lines in the figures are given to show the correctness of the numerical code, and the solid lines are the profiles at some late time, as indicated in figure captions. Initial conditions for these simu­lations are given in Table 1.

Before studying the interactions, we first give our defi­nition for a Mach number, M, of a MHD shock, which will be used to measure the strength of a shock. A Mach number of a shock is defined as the ratio between the shock speed in the prewave state, (s-uxo), and the corresponding charac­teristic speed c evaluated at the pre wave state. Here c is the fast (or slow) wave speed for a fast (or slow) shock. A unity Mach number indicates the disappearance of the shock.

The interaction between a fast sho.:k and a contact dis­continuity may be represented by the impact of a fast shock

3670 Phys. Plasmas, Vol. 1, No. 11, November 1994

with a Mach number 10 on a denser region. The initial con­dition and the simulation results are shown in Fig. 4. Two fast shocks, with Mach numbers 1.4 and 14.8, two weak slow waves (one shock and one rarefaction), and one contact discontinuity are generated in the interaction. The impact of a slow shock with a Mach number 2 on a denser region is shown in Fig. 5. Two slow shocks with Mach numbers 1.18 and 2.43, and one contact discontinuity, are generated in Fig. 5. Two weak fast shocks involved in this interaction are out of the domain.

Figure 6 is for the collision of two fast shocks with Mach numbers 10 and 5. The solid lines in the figure show two fast shocks with Mach numbers 3.23 and 1.67, two weak slow wave (one shock and one rarefaction wave), and one contact discontinuity. Figure 7 is for the catchup of a fast

W. Dai and P. R. Woodward

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1.5 4

CD SS

3 c. FS SS Q"

2 .5 FS

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x x

3 -l I I I r--1 I-

.6

- ----_ .. --.--- -. . 2 .4

r- ~ .2

- 0

I I -.2

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x x

1.5 -I I I I ,...-'-", 1-

1.5

- ,--------.----. -0: ar r-

.5 -

.5

I I . 0 .2 .4 .6 . 8 o .2 .4 .6 .8

x x

FIG. 9. The simulation for the collision of two slow shocks with M=3 and 1.5. Dashed, dotted, and solid lines are the profiles at t=O, 0.15, and 0.75, respectively. The initial conditions are shown in Table 1. Here B,=5.

shock over another fast shock, and each of shocks has a Mach number 5. One fast shock with a Mach number 22.2, one fast rarefaction wave, two weak slow waves, and one contact discontinuity are generated from the interaction. The wave speeds for the two slow waves are too small to be well separated under the resolution of the simulation, but they may be recognized in the profiles for the magnetic field.

Interactions between slow shocks are more or less dif­ferent from those between fast shocks. The collision of two slow shocks with a Mach number 3 each is shown in Fig. 8. We have to point out that the entropy wave indicated in the profile for the density near x=0.5 in the figure is due to the pure discontinuities used in the initial condition, and it may be reduced through the refinement of the grid. Actually, we run the simulation with 1024 and 2048 numerical zones, and we find that the amplitude of the entropy wave become less

Phys. Plasmas, Vol. 1, No. 11, November 1994

and less. In this interaction, the most interesting is the gen­eration of two intermediate shocks. For the given preshock state, if we keep the Mach number of the initial shock that travels to the right, and gradually reduce the Mach number of the other slow shock, which travels to the left, two interme­diate shocks should disappear when the Mach number is smaller than some critical Mach number. In this preshock state and one Mach number 3, the critical Mach number is found to be around 1.75. Figure 9 shows the collision of two slow shocks, with Mach numbers 3 and 1.5. The collision generates two slow shocks with Mach numbers 2.33 and 1.15, one contact discontinuity, and two weak fast shocks, but generates no intermediate shocks.

Figure 10 shows the catchup of a slow shock (M = 2) over another slow shock (M= 1.5). One slow shock with a Mach number 3.08, one slow rarefaction wave, two weak

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1.2 I I I I

3- ~ -.8 CD

c. .6

ct.

SS 2- -

.4 ~ ... -... ~ .. _ ............ ~;--;

.2 ", 1 t:l , ."

0 .2 .4 .6 .8 0 .2 .4 .6 .8 )( )(

2f-J I · ~ ... J I I I · . · . ............... io--J-

.4 I- -

1.5 - - .2 I- -ar :J-

01- ................ t-: -1 I- -

j I I , , '.2 1-, , -,.,

0 .2 .4 .6 .8 0 .2 .4 .6 .8 )( X

1.2 q-~ I : .... , .. • .. .. .. .. .. .. .. 1---1'

.8 I- -.8 as 0::

.6 .6 I- -

.4 ............... :-:

FR SR

y .4 J 1 ~ ..c

0 .2 .4 .6 .8 0 .2 .4 .6 .8 X X

FIG. 10. The simulation for the catchup of two slow shocks (M = 2 and 1.5). Dashed. dotted. and solid lines are the profiles at r = 0.0.2. and 0.8 respectively. The initial conditions are shown in Table l. Here B" = 5.

fast rarefaction waves (one is out of the domain), and one contact discontinuity are generated through the catchup. A collision between fast and slow shocks with Mach numbers 10 and 2, respectively, is shown in Fig. 11. Two fast shocks, with Mach numbers 12.7 and 1.17. one slow shock, one slow rarefaction, and one contact discontinuity are generated through the collision. The catchup of a fast shock with a Mach number 8 over a slow shock with a Mach number 2 is shown in Fig. 12. One fast shock with a Mach number 6.6, one fast rarefaction wave, one slow shock, one slow rarefac­tion wave, and one contact discontinuity are generated through the catchup.

Except the interaction involving intermediate shocks, all the interactions studied through our simulations may be in­vestigated through the Riemann solver. For example, the col-

3672 Phys. Plasmas, Vol. 1. No. 11, November 1994

lision of two fast shocks displayed in Fig. 6 may be consid­ered as a Riemann problem in which the left and right states are the two post-shock states of the incident shocks. The solutions of these Riemann problems give quantitatively the same results as displayed in these figures. For the collision of two slow shocks displayed in Fig. 8, the solution of the cor­responding Riemann problem is given in Table II, which contains two fast shocks, two slow shocks, and two rota­tional discontinuities. Across either rotational discontinuity, the transverse part of the magnetic field undergoes a 180° rotation around the normal of the discontinuity. Note that the solution of the Riemann problem is exact. Again, we mention that the Riemann solver itself is for the set of ideal MHD equations, but the numerical scheme actually performs the simulation for dissipative MHD equations.

W. Dai and P. R. Woodward

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8

80

60 6 CD

FS FS 0. Q..

40 4

20 2

0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x ·X

10

6

8

6 4

at .:J

4 SS 2

2 .-.-----, SR

0 --------"",

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x x

5

80 4

60 3

0: a! 40

2

20 ~---.--~:

.: 0

0 .2 .4 .6 .8 0 .2 .4 .6 .8 x x

FIG. II. The simulation for the collision between one fast shock (M = 10) and a slow shock (M = 2). Dashed, dotted, and solid lines are the profiles at t= 0, 0.018. and 0.108. respectively. The initial conditions are shown in Table I. Here 8 x =2.

IV. SUMMARY AND DISCUSSIONS

In this paper, we have studied interactions between MHD discontinuities through numerical simulations for one­dimensional MHD equations. The interactions include the impact of a shock on a contact discontinuity, the collision of two shocks, and the catchup of a shock over another shock. Each shock in an interaction may be either a fast or a slow shock. The feature of these interactions under the parameters used in the simulations is as follows.

When a fast shock impacts on a more (or less) dense region, a more (or less) strong fast shock than the original one may be transmitted into the denser region, a fast shock (or rarefaction wave) may be reflected, and two weak slow waves may be generated from the impact. The statement above is also true for the impact of a slow shock with a

Phys. Plasmas, Vol. 1, No. 11, November 1994

contact discontinuity if two words, fast and slow, are ex­changed. The collision of two fast shocks will mainly gener­ate two fast shocks. The resulting fast shocks are weaker than original shocks. If the strength of an original shock is increased, the strength of the generated fast shock moving to the opposite direction will be decreased. The catchup of a fast shock over another fast shock will mainly generate one stronger fast shock moving in the original direction and one fast rarefaction wave moving in the opposite direction. Un­der a certain range of Mach numbers of incident shocks, a collision of two slow shocks may generate two fast shocks, two slow shocks, and one contact discontinuity. Beyond the range, two intermediate shocks will be generated. A collision of a fast and a slow shock will result in a stronger fast shock in the direction of the propagation of the original fast shock,

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Co

6

4

2

30

20 0:

10

o 2

o .2

o .2

.4 x

.4 x

.4

x

.6

.6

.6

SR

SS

.8

.8

.8

6

4

2

4

3

2

o

3

2

o

o

o

~ .... -.. ---... " " " ..

. 2

.2

.2

CD '----~

.4 .6 .8

x

.4 .6 .8 x

.4 .6 .8 x

FIG. 12. The simulation for the catchup of a fast shock (M= 8) over a slow shock (M =2). Dashed, dotted, and solid lines are the profiles at t=O, 0.03, and 0.15, respectively. The initial conditions are shown in Table 1. Here Bx =2.

and a weaker slow shock in the opposite direction. A catchup of a fast shock over a slow shock will result in weaker fast and slow shocks in the original direction.

TABLE II. The solution of the Riemann problem displayed in Fig. 8 in ideal MHD equations.

Region p p u,. u,

L R2 R3 R4 R5 R6 R7 R

2.905 1.062 0.7287 0.3707 0.1853 5.504 3.432 0.0476 0.5889 0.2944 5.504 3.432 0.4763 -0.1956 -0.0978 6.001 3.968 0 0 0 6.001 3.968 0 0 0 5.504 3.432 -0.0476 0.1956 0.0978 5.504 3.432 -0.0476 -0.5889 -0.2944 2.905 1.062 -0.7287 -0.3707 -0.1853

0.9648 3.262

-3.262 -1.705 - 1.705 -3.262

3.262 0.9648

3674 Phys. Plasmas, Vol. 1, No. 11, November 1994

B,

0.4824 1.631

-1.631 -0.8523 -0.8523 -1.631

1.631 0.4824

The numerical scheme used in our simulations is based on the formulation for ideal MHD equation plus intrinsic and added numerical viscosity/resistivity. The specific forms for the viscosity/resistivity are not important for the interactions, since the internal structure of a shock is negligible for the interactions. Actually, we have checked our results through a dissipative MHD code, which is developed through adding physical viscosity and resistivity in our numerical code. To run the dissipative code, we give initial shocks very sman internal structures. The results from the dissipative code are the same as those presented in this paper as long as the viscosity and resistivity are sufficiently small.

The one-dimensional study for the interactions is very ideaL Practical interactions occurring nature are more com­plicated in their dimensionality. In this paper, we intend only

W. Dai and P. R. Woodward

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to throw a light on the understanding of the interactions. Applications of the scheme to more realistic problems are expected.

ACKNOWLEDGMENTS

We would like to acknowledge the useful discussion with Thomas W. Jones.

This work was supported by the U.S. National Science Foundation Postdoctoral Fellowship under Grant No. NSF­ASC-9309829, by the Minnesota Supercomputer Institute, and by the Army Research Office Contract No. DAAL03-89-C-0038 with the Army High Performance Computing Re­search Center at the University of Minnesota.

Phys. Plasmas, Vol. 1, No. 11, November 1994

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