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Astron. Astrophys. 353, 108-116 (2000)

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3. Kelvin-Helmholtz instabilities in superwinds of primeval galaxies: simulation results

Whereas in the preceding section we described an analytic treatment of the linear evolution of the KH mode in partially ionized plasmas, in this section we show results of numerical simulations of the linear and non-linear evolution of KH modes in the appropriate parameter régime. These first self-consistent simulations of the full KH dynamics in 2-fluid systems are carried out with an explicit plasma-neutral gas fluid code (Birk & Otto 1996). As distinct from analytical estimates for the linear evolution of the KH modes we do not need to assume incompressibility, and we now include the balance equations for the thermal pressures of both the ionized and the neutral gas components (p and [FORMULA]). These equations are

[EQUATION]

[EQUATION]

where we assume the same adiabatic index [FORMULA] for both fluid components. The integration scheme is based on the Leapfrog algorithm that is of second order in space and time and has the advantage of very low numerical diffusion. Spurious magnetic reconnection due to uncontrolled numerical diffusion can be avoided almost completely (cf. discussion by Frank et al. 1996 and Birk et al. 1999). The balance equations are integrated in a dimensionless form. They are normalized by a typical magnetic field strength [FORMULA], a typical mass density [FORMULA], a characteristic length scale [FORMULA], the Alfvén velocity [FORMULA], the Alfvénic transit time [FORMULA] and the magnetic pressure [FORMULA]. As in Sect. 3 we choose [FORMULA] and [FORMULA]kpc. The wind velocity is, as before, [FORMULA] and the initial homogeneous magnetic field is chosen as [FORMULA]G. This order of magnitude can be expected from the compressional amplification of seed fields during the protogalactic collapse (cf. Lesch & Chiba 1995). It should be noted that alternative simulation runs with a somewhat stronger initial magnetic field do not differ significantly from the results shown and discussed below as long as Kelvin-Helmholtz modes can be excited. In particular, the magnetic field amplification is comparable and happens on almost the same timescale. This is to be expected since the energy density of the initial magnetic field is much smaller than the kinetic energy of the wind plasma.

In the present simulations we choose a small homogeneous background resistivity in order to smooth gradients that otherwise could give rise to numerical instabilities. For the effective collision frequency we choose [FORMULA]. The constant background resistivity chosen corresponds to a magnetic Reynolds number of [FORMULA]. The effects of varying resistivity have been discussed by Birk et al. (1999). A lower background resistivity results in a slightly higher maximum magnetic field strength. The calculations are performed in a mean velocity observer frame. The wind proceeds in the y-direction (perpendicular to the galactic disk) and the wind fragment has a velocity gradient that is directed along the x-axes (parallel to the disk). We assume a shear flow corresponding to [FORMULA] with an amplitude [FORMULA] for both the plasma and the neutral gas andthat both the plasma and the neutral gas are homogeneous, with an ionization rate of [FORMULA] ([FORMULA]; [FORMULA]).

For the simulations we use a 3-dimensional code with a start configuration mentioned above. However, here, we only resolve the 2-dimensional dynamics. The simulation box is given by [FORMULA], [FORMULA], and [FORMULA] in units of 1kpc. The numerical grid points are equidistant, and are chosen to be [FORMULA]. This means that we are not applying strict invariance in the third dimension but do not resolve the dynamics in the z-direction.

Fig. 3 illustrates the self-consistent evolution of the unstable KH mode after 50 (left plot), 100 (middle plot) and 200 (right plot) dynamical (Alfvénic transit) times, i.e. after 16Myr and 32Myr and 80Myr, respectively. Here, and in the following plots we show cuts at the [FORMULA]-half plane. The arrows indicate the flow in the x-y-plane. Magnetic field lines are indicated by the solid lines The magnetic field lines become bent and the flow pattern develops the characteristic KH vortex structure after 100 dynamical times. After 250 dynamical times the KH mode has developed non-linearly. The magnetic field in the x-y-plane at this time is shown in Fig. 4. The curly structure can easily be identified. In the vicinity of the KH boundary layer magnetic field gradients steepen, and the field strength is intensified by a factor of about 20 after 200 Alfvénic transit times for the example shown. One recognizes that field amplification is not restricted to the boundary layer but is associated with the KH vortices (cf. also Fig. 6). We note that for a lower background resistivity sharper gradients form and the field line twisting increases. The z-component of the magnetic field remains zero for the chosen two-dimensional shear flow. The dynamics of the mode, in fact, is compressible (Fig. 5). The kinetic energy of the shear flow is mainly converted into magnetic energy rather than compressional heating (Fig. 6). The neutral gas component also shows the typical KH flow pattern (Fig. 7) with amplitudes enhanced by some [FORMULA] as it is the case for the ionized fluid component. As for the ionized component no significant flow in the z-direction occurs. The neutral gas density, as the mass density of the ionized component, show depletion and compression according to the evolution of the KH mode. Some fraction of the kinetic energy of the KH mode is converted into compressional heating (Fig. 8).

[FIGURE] Fig. 3. Ionized gas flow (indicated by arrows) in the x-y-plane. The solid lines indicate the magnetic field. Snapshots of the evolution of the KH mode are shown at [FORMULA] (left plot), [FORMULA] (middle plot) and [FORMULA] (right plot). The lengths of the exposed arrows indicate flow velocities of 1.08, 1.20 and 1.28 times the initial amplitude.

[FIGURE] Fig. 4. The x- and y-component of the magnetic field after [FORMULA]. The amplitudes follow the KH vortex structure of the flow.

[FIGURE] Fig. 5. Density and temperature of the ionized gas component after [FORMULA]. The compressible KH dynamics result in regions of depletion and cooling and compressional heating.

[FIGURE] Fig. 6. Thermal pressure of the ionized gas and magnetic pressure after [FORMULA].

[FIGURE] Fig. 7. Neutral gas velocity (indicated by arrows) after [FORMULA] (left plot) and [FORMULA] (right plot). As in the ionized fluid component the KH mode structure can easily be identified. The lengths of the exposed arrows indicate a flow velocity magnitude of 1.20 and 1.21 times the initial amplitude.

[FIGURE] Fig. 8. Density and temperature of the neutral gas component after [FORMULA]. Regions of compression and depletion are located in the plane of the shear flow similar to the ones for the ionized component.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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