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

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3. Parameter study of the flow around a cylinder

In this section we present numerical simulation results for symmetrical bow shock flows around a cylinder, for the values of the parameters [FORMULA] and [FORMULA] which are indicated by the triangles and diamonds in Fig. 3.

In Fig. 4 we show global views of the bow shock solutions for a fixed [FORMULA] and [FORMULA] varying from 1.1 to 1.9. It follows from Eq. 2 that the critical Alfvénic Mach number under which switch-on shocks can exist is [FORMULA]. For inflow speeds much faster than the Alfvén speed ([FORMULA]), the bow shock has the traditional single-front topology that is also encountered in hydrodynamic bow shocks. If the inflow speed drops below 1.732 however, a concave-outward dimple forms in the leading shock front and a second shock front appears. This change in shape and topology of the bow shock flow thus happens when the inflow speed becomes lower than the critical speed under which switch-on shocks are possible.

[FIGURE] Fig. 4. Stationary bow shock solutions for fixed [FORMULA] and for varying inflow speeds ([FORMULA] grids, [FORMULA]). Density contours pile up in shocks, and streamlines come in horizontally from the left. For inflow speeds much faster than the Alfvén speed ([FORMULA]), the bow shock has the traditional single-front topology that is also encountered in hydrodynamic bow shocks. If the inflow speed drops below 1.732 however, a concave-outward dimple forms in the leading shock front and a second shock front appears.

In Fig. 5 we show a detailed representation of the flow near the stagnation streamline for the bow shock solutions with varying inflow speed of Fig. 4. For inflow velocities below the critical switch-on value for the inflow speed ([FORMULA]), the leading shock front has a dimpled shape. The dimpling becomes much more pronounced as the inflow velocity decreases. Below the critical inflow speed, a second shock front appears which trails the leading shock front, and additional discontinuities are present between the two shock fronts. All the shocks and discontinuities present in the topology sketch of Fig. 1b seem to be present in all the flows. Inspection of the way in which the field lines are refracted when they pass the shocks, reveals that the shocks in all the flows are of the same type as the shocks in the model flow of Fig. 2 which were discussed in Sect. 2.1, and this conclusion is confirmed by detailed analysis of upstream and downstream Mach numbers, along the lines of the detailed analysis in De Sterck et al. (1998b). For smaller inflow velocities, the central interaction region becomes smaller and the leading shocks become weaker while the trailing shock becomes stronger. As a consequence, the shock E-G of Fig. 1b can not be identified for the flow with [FORMULA] with the resolution of Fig. 5. More detailed simulations and plots (not shown) do, however, show that shock E-G is present also for the flow with [FORMULA]. For inflow velocity [FORMULA], close to the critical velocity of [FORMULA], the secondary shock fronts become weak and shock E-G can hardly be identified with the resolution of Fig. 5. For [FORMULA] the secondary (stationary) waves are still present, but they are not steepened into shocks any more. The secondary waves have disappeared almost completely for [FORMULA], and the simple single-front bow shock topology of Fig. 1a is recovered. We can thus conclude that for all the flows with parameters in the switch-on domain ([FORMULA]), the topology of Fig. 1b is recovered. The shapes, sizes and shock strengths of the shock parts present in the topology of Fig. 1b, vary when [FORMULA] is varied within the switch-on region. The dimple effect is more pronounced for smaller inflow Alfvénic Mach number [FORMULA].

[FIGURE] Fig. 5. Detailed representation of the flow near the stagnation streamline for the bow shock solutions with varying inflow speed and fixed [FORMULA] ([FORMULA] grids). Density contours pile up in shocks, and streamlines come in horizontally from the left. Under the critical switch-on value for the inflow speed, the leading shock front dimples and a second shock front appears. Additional discontinuities can be seen between the two shock fronts.

Above we discussed how the flow manages to go around the obstructing cylinder by adjusting the bow shock shape and topology to the inflow Alfvénic Mach number. Hereby the plasma [FORMULA] value was fixed to 0.4. Below we fix the inflow Alfvénic Mach number and verify how the flow modifies the geometrical structure of the bow shock when the value of the plasma [FORMULA] is varied.

In Fig. 6 we show global views of the bow shock solutions for a fixed [FORMULA] and [FORMULA] varying from 0.1 to 0.9. It follows from Eq. 2 that the critical plasma [FORMULA] under which switch-on shocks can exist is [FORMULA]. For plasma [FORMULA] values larger than the critical value of [FORMULA], the bow shock has the traditional single-front topology that is also encountered in hydrodynamic bow shocks. If the plasma [FORMULA] drops below 0.7 however, a concave-outward dimple forms in the leading shock front and a second shock front appears. This second shock front thus appears when the plasma [FORMULA] becomes lower than the critical plasma [FORMULA] under which switch-on shocks are possible.

[FIGURE] Fig. 6. Stationary bow shock solutions for fixed [FORMULA] and for varying plasma [FORMULA] ([FORMULA] grids, [FORMULA]). Density contours pile up in shocks, and streamlines come in horizontally from the left. For plasma [FORMULA] values larger than the critical value of [FORMULA], the bow shock has the traditional single-front topology that is also encountered in hydrodynamic bow shocks. If the plasma [FORMULA] drops below 0.7 however, a concave-outward dimple forms in the leading shock front and a second shock front appears.

In Fig. 7 we show a detailed representation of the flow near the stagnation streamline for the bow shock solutions with varying plasma [FORMULA] of Fig. 6. For plasma [FORMULA] values below the critical switch-on value for the plasma [FORMULA] ([FORMULA]), the leading shock front has a dimpled shape. The dimpling becomes more pronounced as the plasma [FORMULA] is decreased. Below the critical plasma [FORMULA], a second shock front appears which trails the leading shock front, and additional discontinuities are present between the two shock fronts. All the shocks and discontinuities present in the topology sketch of Fig. 1b seem to be present in all the flows. Inspection of the way in which the field lines are refracted when the shocks are passed, reveals that the shocks in all the flows are of the same type as the shocks in the model flow of Fig. 2 which were discussed in Sect. 2.1, and this conclusion is confirmed by detailed analysis of upstream and downstream Mach numbers, along the lines of the detailed analysis in De Sterck et al. (1998b). For smaller plasma [FORMULA] values, the central interaction region in front of the cylinder becomes smaller. As a consequence, the shock E-G of Fig. 1b can not be identified for the flow with [FORMULA] with the resolution of Fig. 7. More detailed plots (not shown) do, however, show that shock E-G is present also for the flow with [FORMULA]. For plasma [FORMULA], which is the critical value, the secondary (stationary) wave has only nearly steepened into a shock. Shock E-G can not be identified for this critical value of the parameters. For [FORMULA] the secondary waves are still present, but they have not steepened into shocks any more. The secondary waves are even weaker for [FORMULA], and the simple single-front bow shock topology of Fig. 1a is recovered. We can thus conclude that for all the flows with parameter values in the switch-on domain ([FORMULA]), the topology of Fig. 1b is recovered. The shapes, sizes and shock strengths of the shock parts present in the topology of Fig. 1b, vary when [FORMULA] is varied in the switch-on regime. The dimple effect is more pronounced for smaller [FORMULA].

[FIGURE] Fig. 7. Detailed representation of the flow near the stagnation streamline for the bow shock solutions with varying plasma [FORMULA] and for fixed [FORMULA] ([FORMULA] grids). Density contours pile up in shocks, and streamlines come in horizontally from the left. Under the critical switch-on value for the plasma [FORMULA], the leading shock front dimples and a second shock front appears. Additional discontinuities can be seen between the two shock fronts.

We can thus conclude from this parameter study, that there is a close correspondence between inflow parameters for which a complex bow shock topology is found, and parameters for which switch-on shocks are possible. This proves that the complex bow shock topology is indeed closely related to the possible occurrence of switch-on shocks. Because of the symmetry reasons discussed in Sect. 2.1, switch-on shocks do not occur where the leading shock fronts intersect the stagnation line. In stead, a complex interacting shock structure with a dimpled leading front appears near that location, for the bow shock flows of Figs. 4 and 6 that have inflow parameters in the switch-on regime. Switch-on shocks are present in all these flows, however, and can be found at the locations on the leading shock fronts corresponding to point B in the topology sketch of Fig. 1b. The topology of the bow shock solution obtained in De Sterck et al. (1998b) and sketched in Fig. 1b is encountered for all the bow shock flows with parameters in the switch-on domain, and this topology is thus more generally valid than only for the single set of parameters ([FORMULA]) considered in De Sterck et al. (1998b). The shapes, sizes and shock strengths of the shock parts present in the topology of Fig. 1b, vary when [FORMULA] and [FORMULA] are varied in the switch-on regime. The dimple effect is more pronounced for low values of [FORMULA] and [FORMULA].

As a final remark, we can say that in the parameter regime under consideration, the global stand-off distance of the bow shocks (Petrinec & Russell 1997) decreases for increasing [FORMULA] while keeping [FORMULA] constant (Fig. 4) - although the stand-off distance on the stagnation line does not seem to change much (Fig. 5). The stand-off distance increases for increasing [FORMULA] while [FORMULA] is kept constant (Figs. 6 and 7).

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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