Adaptive mesh refinement for supersonic molecular cloud turbulence

We present the results of three-dimensional numerical simulations of supersonic isotropic Euler turbulence with adaptive mesh refinement (AMR) and effective grid resolution up to 1024(3) zones. Our experiments describe nonmagnetized, driven turbulent flows with an isothermal equation of state. Mesh refinement on shocks and shear is implemented to resolve dynamically important structures and calibrated to match the turbulence statistics obtained from the equivalent uniform-grid simulations. We demonstrate that as soon as the integral and dissipation scales are sufficiently separated, further increasing the resolution does not require mesh refinement all over the computational domain. The volume of finer subgrids in our AMR simulations scales with linear resolution approximately as N(-1). Turbulence statistics derived from our AMR simulations and simulations performed on uniform grids agree surprisingly well, even though only a fraction of the volume is covered by AMR subgrids. We show that subdimensional nested "Mach cones" and U-shaped shocklets dominate the dynamics in supersonic turbulent flows. Based on these results, we discuss the fractal dimension of dissipative structures, their signature in the statistical properties of supersonic turbulence, and their role in overall flow dynamics.