Using 3D direct numerical simulations (DNSs), we investigate the turbulent energy cascade rate under the compressible magnetohydrodynamic (CMHD) approach with an isothermal closure relation. Based on exact laws derived from the von Kármán-Howarth equation for incompressible and compressible fully developed turbulence (see, Politano & Pouquet 1998; Andrés & Sahraoui 2017), we compute the flux, source, hybrid, and mixed terms from the exact relations and study their relative contributions. More specifically, we conducted a parametric study, varying both the sonic and Alfvénic Mach numbers to investigate different levels of compressibility (subsonic and supersonic regimes) and different levels of anisotropy.  In particular, we developed a code for the numerical analysis, which enabled the computation of the energy cascade rate in arbitrary directions by interpolating outside the grid points to obtain scale-dependent cascade statistics. Our numerical results showed that the flux terms are more strongly dependent on compressibility, with modest increases driven by the level of anisotropy. Additionally, we observe that the ratio between the purely compressible term and the incompressible cascade reaches its peak at intermediate levels of compressibility. We expect that these numerical results enhance the understanding of incompressible and compressible turbulence and their essential role in the so-called solar wind heating problem.

Andrés, N., & Sahraoui, F. 2017, Phys. Rev. E, 96, 053205.

Politano, H., & Pouquet, A. 1998a, Phys. Rev. E, 57, R21.