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Astron. Astrophys. 343, 641-649 (1999)

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2. Properties of MHD shocks and numerical solution of the MHD equations

2.1. Properties of MHD shocks

The complex topology of bow shock flows in the switch-on regime can be understood in terms of the properties of MHD shocks. This will be explained in the present section. Contrary to the hydrodynamic equations, which allow for only one wave mode, the MHD equations allow for three distinct wave modes, the fast magneto-acoustic wave, the Alfvén wave, and the slow magneto-acoustic wave, with (positive) anisotropic wave speeds satisfying [FORMULA] in standard notation. Three types of shocks are described by the MHD equations, connecting plasma states which are traditionally labeled from 1 to 4, with state 1 a super-fast state, state 2 sub-fast but super-Alfvénic, state 3 sub-Alfvénic but super-slow, and state 4 sub-slow (Landau & Lifshitz 1984, Anderson 1963, De Sterck et al. 1998b). Fast 1-2 MHD shocks refract the magnetic field away from the shock normal. Intermediate MHD shocks (1-3, 1-4, 2-3, and 2-4) change the sign of the component of the magnetic field which is tangential to the shock front, and thus flip magnetic field lines over the shock normal. A special case of a 1-4 intermediate shock is a 1-4 hydrodynamic (intermediate) shock, for which the magnetic field is perpendicular to the shock and does not change through the shock. Slow 3-4 MHD shocks refract the magnetic field towards the shock normal.

In De Sterck et al. (1998b) the types of the discontinuities that are present in the complex bow shock flow of Fig. 2 are clearly identified. The results of this detailed analysis can be summarized as follows, using the lettering labels of Fig. 1b. Shock parts A-B and D-E are 1-2 fast shocks, E-F is a 1-4 hydrodynamic shock, and B-C-D is a 1-3 intermediate shock. E-G is a 1=2-3=4 intermediate shock. D-G-H-I is a 2-4 intermediate shock. E-H is a tangential discontinuity. Other tangential discontinuities stretch out from points G and H along the streamlines to infinity. The reader can verify in Fig. 2 that all the intermediate shocks indeed flip magnetic field lines over the shock normal.

We remark here that the presence of intermediate shocks in this flow is an important illustration in two dimensions (2D) of many of the new theoretical results on the existence of intermediate shocks (Wu 1991, Freistuehler & Szmolyan 1995, Myong & Roe 1997). We refer to De Sterck et al. (1998b) for a discussion of this subject. Analysis of this stationary flow in terms of steady state characteristic curves and elliptic and hyperbolic regions, shows that this flow contains a steady state analog of an xt MHD compound shock (Brio & Wu 1988, Myong & Roe 1997), which is a manifestation of the non-convex nature of the MHD equations (De Sterck, Low, & Poedts, submitted to Phys. Plasmas ). It is important to note that there is still discussion about the stability of intermediate shocks against non-planar perturbations (Wu 1991, Barmin et al. 1996), and it will be interesting to see how the intermediate shocks present in our 2D planar simulation results, would survive in a three-dimensional (3D) context which allows for non-planar perturbations. This remains subject of further work.

A fast switch-on shock is a limiting case of the fast shock for which the upstream magnetic field direction coincides with the direction of the shock normal, and the downstream magnetic field makes a finite angle [FORMULA] with the shock normal. The downstream normal plasma speed exactly equals the downstream normal Alfvén speed in the shock frame. The component of the magnetic field parallel to the shock surface is thus effectively `switched on' in going from the upstream to the downstream state of the shock. The shock at point B in Fig. 1b is an example of a fast switch-on shock. From the MHD Rankine-Hugoniot relations one can derive (Kennel et al. 1989) that switch-on shocks can be encountered for upstream parameters satisfying




with [FORMULA] the plasma [FORMULA], and the Alfvénic Mach number given by [FORMULA], where v is the plasma velocity and [FORMULA] the Alfvén speed along the shock normal. For [FORMULA], the parameter domain in the [FORMULA] plane for which switch-on shocks can occur, is sketched in Fig. 3.

[FIGURE] Fig. 3. Parameter domain for which switch-on shocks are possible. For [FORMULA], switch-on shocks are possible for upstream values of [FORMULA] and [FORMULA] located in the shaded region. In Sect. 3, numerically obtained bow shock flows are presented for inflow quantities with fixed [FORMULA] and [FORMULA] varying from 1.1 to 1.9 (the diamonds on the vertical line), and with fixed [FORMULA] and [FORMULA] varying from 0.1 to 0.9 (the triangles on the horizontal line).

These properties of MHD shocks allow us to understand why the classical single shock front solution of Fig. 1a is not found for MHD bow shocks in the switch-on regime. Because of symmetry, the magnetic field line which coincides with the stagnation streamline (stretching horizontally from infinity to the stagnation point ([FORMULA]) at the cylinder) has to be a straight line. In other words, on this line, the field is not deflected by the shock. Away from this line along the shock front, the shock has to be a fast MHD shock, with the field refracted away from the normal (Fig. 1a) in order to have the post-shock flowing plasma drape around the cylinder. As we move along this fast shock front closer and closer to the intersection of the front with the stagnation line, the upstream tangential component of the magnetic field goes to zero. But when the upstream parameters lie in the switch-on domain, the downstream tangential component of the magnetic field does not vanish as we approach this intersection point, resulting in a switch-on shock with a finite turning angle [FORMULA], as illustrated in Fig. 1a. Clearly, approaching the stagnation line from its two sides along the fast shock front, would lead to two switch-on shocks of opposite deflection. This means that there is a discontinuity between the two physical states on the two sides of the stagnation field line. Such a discontinuity is not physically justified, so the concave-inward shock geometry (as in Fig. 1a) needs to be modified in order to avoid this discontinuity. In the present paper it is studied how nature accomplishes this, i.e. what alternatives to the concave-inward shock geometry the flow finds to get around the object and how this alternative depends on the parameters that characterize the flow, viz. the plasma beta [FORMULA] and the Alfvénic Mach number [FORMULA].

2.2. Numerical solution of the MHD equations

In Sect. 3, we will present numerically obtained bow shock flows around a cylinder for various parameter sets inside and outside the switch-on domain, to investigate closely the correspondence between the complex bow shock topology as it was obtained in De Sterck et al. (1998b), and the parameter domain in which switch-on shocks can occur. In Sect. 4, we will investigate the axi-symmetrical bow shock flow over a sphere. In the present section we will briefly describe the numerical solution technique used.

In our simulations a uniform field-aligned flow in planar symmetry (Sect. 3; xyz system with [FORMULA], and [FORMULA] and [FORMULA]) or axial symmetry (Sect. 4; [FORMULA] system with [FORMULA], and [FORMULA] and [FORMULA]) enters from the left and encounters a perfectly conducting rigid cylinder (Sect. 3) or sphere (Sect. 4). The magnetic field is aligned with the plasma velocity in the whole domain of the resulting stationary ideal MHD flow. The stationary bow shock flow is completely determined by the inflow [FORMULA] and [FORMULA] in the direction of the flow speed. We take the x axis horizontal, and we can freely choose [FORMULA] and [FORMULA] (implying that the Alfvén speed along the field lines [FORMULA]). The pressure and velocity can then be determined from [FORMULA] and [FORMULA]. Finally, we take [FORMULA] ([FORMULA]) and [FORMULA] ([FORMULA]). As the resulting stationary ideal MHD flow is scale invariant, we can freely choose the radius of the cylinder (sphere). We take [FORMULA] and the cylinder (sphere) is placed at the origin of the coordinate system. We simulate the flow in the upper left quadrant, on a stretched elliptic polar-like structured grid. We impose the above described uniform flow as the initial condition. We use ghost cells to specify the boundary conditions. On the left, we impose the uniform superfast incoming flow. The obstructing object is perfectly conducting. We look for a stationary flow solution with top-bottom symmetry, such that the horizontal line which extends to the center of the cylinder is the stagnation line, parallel to the incoming flow (Fig. 1). This symmetry has to be imposed in the boundary condition on the lower border of the simulation domain in order to obtain a stationary symmetrical solution. The right outflow condition is superfast, so there we extrapolate all quantities to the ghost cells. The flow evolves in time until a converged steady state bow shock solution is obtained.

We solve the equations of ideal one-fluid MHD. In `conservative form' these equations are given by


This set of equations has to be supplemented with the divergence free condition [FORMULA] as an initial condition. Here [FORMULA] and p are the plasma density and pressure respectively, [FORMULA] is the plasma velocity, [FORMULA] the magnetic field, and


is the total energy density of the plasma. I is the unity matrix. The magnetic permeability [FORMULA] in our units. These equations describe the conservation of mass, momentum, magnetic field, and energy.

As proposed by Powell et al. (1995), we have put a source term proportional to [FORMULA] in the right hand side (RHS) of Eq. 3. Discretization of this form of the equations results in a numerical scheme which conserves the [FORMULA] constraint up to a discretization error. This approach is an attractive alternative to the use of an extra artificial [FORMULA] correction in every time step obtained via solution of an elliptic equation, because it consumes less computing time and because it cures the [FORMULA] problems in a way which is more in harmony with the hyperbolicity of the MHD system.

We solve Eq. 3 using a conservative finite volume high resolution Godunov shock capturing scheme which is second order in space and time, employing a slope-limiter approach (Leveque 1992, Gombosi et al. 1994, Tóth & Odstrcil 1996, Linde et al. 1998) with minmod-limiting on the slopes of the primitive variables. The time-integration is explicit with a two-step Runge-Kutta method. The code was previously used for MHD simulations of interacting hot filaments in a tokamak (De Sterck et al. 1998a). For our present simulations, we use the Lax-Friedrichs numerical flux function (Leveque 1992, Tóth & Odstrcil 1996, Barmin et al. 1996), which is simple and robust. Contact and tangential discontinuities are not perfectly well resolved due to the relatively high numerical dissipation for these waves, but shocks are well resolved in steady state calculations. We did not use Roe's scheme (Roe & Balsara 1996) although this scheme in theory could resolve shocks and, especially, tangential discontinuities much better. We have found several problems while trying to apply this scheme to our simulation. Roe's scheme suffers from various instabilities, like the carbuncle-instability (Quirk 1994), and as a result of these numerical instabilities, steady state solutions could not be obtained with this scheme. Using the Lax-Friedrichs scheme, we obtained convergence of more than eight orders of magnitude in the norm of the density residual. We can remark that the code sometimes generates small spurious oscillations in the upstream part of the flow, as can be seen in Fig. 2. Such oscillations seem to be hard to avoid with shock-capturing numerical schemes, but fortunately they are very small.

The bow shock flow of Fig. 2 constitutes an interesting new test case for ideal MHD codes, because it is a well-defined problem with a simple set-up but with a wealth of MHD shocks and discontinuities in the resulting flow.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999