Energy cascade rate in isothermal compressible magnetohydrodynamic turbulence

Three-dimensional direct numerical simulations are used to study the energy cascade rate in isothermal compressible magnetohydrodynamic turbulence. Our analysis is guided by a two-point exact law derived recently for this problem in which flux, source, hybrid and mixed terms are present. The relative importance of each term is studied for different initial subsonic Mach numbers M_S and different magnetic guide fields B₀. The dominant contribution to the energy cascade rate comes from the compressible flux, which depends weakly on the magnetic guide field B₀, unlike the other terms whose moduli increase significantly with M_S and B₀. In particular, for strong B₀ the source and hybrid terms are dominant at small scales with almost the same amplitude but with a different sign. A statistical analysis undertaken with an isotropic decomposition based on the SO(3) rotation group is shown to generate spurious results in the presence of B₀, when compared with an axisymmetric decomposition better suited to the geometry of the problem. Our numerical results are compared with previous analyses made with in situ measurements in the solar wind and the terrestrial magnetosheath.