\bfseries{\Large Radiative Shocks \& the Oscillatory Instability }}

Introduction

Main references:
[1] Numerical Analysis of the Dynamic Stability of Radiative Shocks, Strickland, R. & Blondin, J.M. 1995, ApJ, 449, 727 ​link.
[2] A Numerical Study of the Stability of Radiative Shocks, Imamura, J. N.; Wolff, M. T.; Durisen, R. H. 1984, ApJ, 273, 667 ​link.

Background

noindent Gas is heated when it goes through a shock, and as the heated gases radiate, energy is removed from the gas. Radiatively-cooled shocks in stationary frames will often exhibit oscillatory instabilities as the sharp velocity gradient produces pressure imbalances that expand and contract. \\\\ A shock is a propagating disturbance through a medium. The shock will cause an extremely rapid increase in pressure and temperature of the fluid, and a decrease in the density of the fluid. The module simulates a 1D oscillating shock using AstroBEAR 1.0 to test the analytically predicted behavior of an oscillating shock in the interstellar medium. Radiative shocks are an important aspect in the study of astrophysical fluid dynamics. Such shocks occur in supernova remnants, stellar and galatic jets, stellar wind bubbles, accreting white dwarfs, and galaxy formation. [1] By simulating radiative-cooling shocks, we can study the properties of the gases that are affected by such shocks in the interstellar medium. \\\\ The test is a 1D flow that collides into a wall. It consists of several simulations using different cooling parameters and scalings. From the simulations, the system’s density, temperature and velocity changes are observed and analyzed. Gravitational and magnetic field effects are ignored in this test module.

Problem Setup

noindent The test module is a 1-dimensional Cartesian grid on the domain $0.0 \le x \le 1.0$. The choice of boundary conditions on the left and right edges of the grid has a critical effect on the behavior of the system. Since a flow into a hard wall is desired for this test, a uniform flow of gas with velocity $\emph{v}_0$ is initialized from an inflow/extrapolation boundary on the left. The gas will flow into a reflecting boundary at the right. From this, perturbations will naturally arise at the reflecting boundary, which will move toward the shock, perturbing it and causing oscillations. \\\\ The boundary conditions at the top and bottom are set up as periodic and because the flow only moves along the x-direction, these boundaries do not affect the system. A fixed course grid of 32x4 is specified with two levels of adaptive mesh refinement (AMR). \\\\ AstroBEAR solves the equations of radiative hydrodynamics in 1-D: \begin{equation}\frac{d\rho}{dt} + \rho\frac{dv}{dx} = 0, \end{equation} \begin{equation}\rho\frac{dv}{dt} + \frac{dP}{dx} = 0, \end{equation} \begin{equation}\frac{dP}{dt}-\frac{\gamma P}{\rho}\frac{d\rho}{dt} = -\frac{(\gamma-1)\rho^2}{\bar{m}^2}\Delta(T). \end{equation} The term: \begin{equation}\Delta(T)=C\rho^2T^\alpha \end{equation} represents gas cooling due to optically thin radiative losses via a power law cooling function. It has units of $ergs$ $s^{-1}$ $cm^{-3}$. $P$ is the gas pressure, $\rho$ is the mass density, $v$ is the velocity, and $T$ is the temperature. In our case, we apply a gas of Mach 40. $C$ is a physical constant that will be described in more details later, and $\alpha$ is the power law index we will explore to observe its effect on the oscillations of the system ($\alpha=\frac{1}{2}$ approximates bremsstrahlung cooling). Assuming an ideal gas equation of state, the specific heat ratio $\gamma$ takes a value of 5/3. \\\\ Our module tests $\alpha=0,\frac{1}{2},1$. We plot (not in real units) $\Delta(T) \propto T$:

noindent $\alpha=0$ is plotted in blue. It is constant. In this case, we expect the gas to oscillate for a while. $\alpha=\frac{1}{2}$ is magenta. $\alpha=1$ is plotted in yellow. It is the strongest cooling. We expect the gas to cool the quickest in this case, i.e. rapid dampening of oscillations. \\\\ \noindent Note that changing the value of $\alpha$ changes the units of $\Delta(T)$. We compensate the change with the constant C. [2] Define: \begin{equation} C = \Lambda_{br}T_{shock}^{(1-2\alpha)/2}, \end{equation} where $\Lambda_{br}$ is the coefficient for nonrelativistic bremsstrahlung. For us, $\Lambda_{br} = 1$ $ergs$ $cm^3$ $s^{-1}$ $g^{-1}$ $K^{-1/2}$, and $T_{shock}$ is the postshock temperature, \begin{equation}T_{shock}=\frac{3}{16}\frac{\mu m_0}{k_B}\emph{v}_0^2, \end{equation} where $\mu$ is the mean molecular weight per atom, $m_0$ is the mass of one atomic mass unit, $k_B$ is Boltzmann's constant, and $v_0$ is the initial velocity.

Scaling

noindent Due to the sensitivity of the cooling function to scaling, it is important to take careful note of the scaling parameters we used. \begin{center} \begin{tabular}{ | l | l | l | p{2cm} |} \hline \textbf{Parameter} & \textbf{Variable} & \textbf{Value} & \textbf{Units} \\ \hline Number Density & nScale & $2.99\times10^{14}$ & $number/cm^3$ \\ \hline Mass Density & rScale & $ 5.00\times10^{-10}$ & $g/cm^3$ \\ \hline Temperature & TempScale & $2.50\times10^6$ & K \\ \hline Pressure & pScale & $1.03\times10^5$ & $dynes/cm^2$ \\ \hline Length & lScale & $5.56\times10^{10}$ & cm \\ \hline Velocity & VelScale & $1.44\times10^7$ & cm/s \\ \hline Cooling & ScaleCool & $3.35\times10^{27}$ & $\frac{g*s}{(cm/s)^5}$ \\ \hline Time & RunTimesc & $3.87\times10^3$ & s \\ \hline \end{tabular} \end{center} \noindent Since $ScaleCool=\frac{rScale*lScale}{m_H^2*VelScale^3}$, the cooling function will be stronger the bigger rScale and lScale values are. Also, note that $T_{shock} \propto v_0^2$:

Numerical Results

noindent Using the equations and values outlined above, we have a post-shock temperature of $5\times10^8$ K. With this, we can find C for each $\alpha$ case. \\\\ \noindent Having find the C for each case, we can then adjust the cooling parameters accordingly to have the numbers equal.

Conclusion

adiative shocks show general trend toward stability with increasing values of $\alpha$. We are currently not able to show this for the case of $\alpha=0$ due to scaling problems in Astrobear 1.0. We hope to have a solution for all three cases in Astrobear 2.0.

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