Overstable Radiative Shock

This tests the cooling physics of shock-heated gas. The simulation setup is as follows: A $128\times4$ cell uniform grid is used with domain $[x,y]=[0-128,0-4]\times10^{15}\,$cm; boundaries are periodic in $y$, reflecting at $x=0$ and fixed inflowing at $x=1.28\times10^{17}\,$cm. The initial uniform state has $[n_{\mathrm{H}},T,v_x,v_y,v_z]=[10\ensuremath{\,\mbox{cm}^{-3}},10^{4}\,\mathrm{K},-v\,\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}},0,0]$ with no magnetic field and where $v$ varies from $100-150\,\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$. $\gamma=5/3$ is used with a mean mass per particle of $\mu=1.0m_p$ where $m_p=1.67\times10^{-24}\,$g is the proton mass. An ionisation fraction of 0.99 is imposed and the simulation is run without ionisations and recombinations. Cooling is treated using the non-equilibrium cooling curve of Sutherland (1993) for solar abundance gas. In this way recombination cooling is treated even though we do not calculate the recombination rates. This fitting function cuts off at $T_m=10\,000\,$K so for numerical stability we extrapolate it to lower temperature with a logarithmic slope of 4 (i.e. the cooling rate $\Lambda(T<T_m)= \Lambda(T_m)[T/T_m]^4$). This is a 1D problem when there are no perturbations imposed in the $y$ direction; it is run here in 2D only because it turned out to be easier to analyse the results.

For this type of simulation it has been shown that a radiative shock is overstable (oscillatory) for speeds greater than about $130-140\,\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ and for slower speeds it is stable (Gaetz, 1988; Stone, 1993; Innes, 1987; O'Sullivan, 1999). The very simple prescription used here is less detailed than this previous work but it is sufficient to capture the essential physics. Sutherland (2003) studied 2D radiative shocks to study instabilities in the perpendicular direction. For the 1D case with an interstellar cooling function, they found results consistent with previous work.

In this test gas reflects off the $x=0$ boundary and is immediately shock heated, and the shock propagates upstream. The pressure of this shock heated gas continues to drive the shock upstream until the gas cools. After this the shock continues until until its momentum is overwhelmed by the inflowing gas and then it collapses back to the $x=0$ wall and the process repeats. For low velocities this oscillatory behaviour is strongly damped but at higher velocities stable oscillations can be set up.

Figure 2.15 shows shock positions as a function of time for different inflow velocities. The location of the shock was found by following the inflowing gas until a cell was found where the velocity had changed by 30 per cent compared to the inflowing value. This is roughly at the centre of the shock. It is found that the $120\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ and $130\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ models are stable to oscillations. The $150\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ shock is overstable, and oscillations persist for as long as the simulation is run. The $140\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ model seems to be overstable at a lower level of oscillation, and with an indication of damping at late times. These results largely agree with the work cited in the previous paragraph. Innes (1987) found the stability limit in the range $130-150\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$, Gaetz (1988) found it to be $130-140\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$, and Stone (1993) found $120-130\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ in agreement with O'Sullivan (1999). The differences and similarities are probably more to do with the chemistry and cooling models than any difference in the numerical accuracy of the methods. For example if I include more elements than hydrogen, the mean mass per particle will be larger and so the number densities and hence pressures will be lower.

Figure: Shock Position as a function of time for different inflow velocities. The 100, 120 and 130 $\,\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ models are clearly stable, the 150 $\,\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ shock is overstable, and the 140 $\,\ensuremath{\,\mbox{km}\,\mbox{s}^{-1}}$ model is marginal.