Title: Alternative derivation of exact law for compressible and isothermal magnetohydrodynamics turbulence,
Abstract: The exact law for fully developed homogeneous compressible
magnetohydrodynamics (CMHD) turbulence is derived. For an isothermal plasma,
without the assumption of isotropy, the exact law is expressed as a function of
the plasma velocity field, the compressible Alfvén velocity and the scalar
density, instead of the Elsässer variables used in previous works. The
theoretical results show four different types of terms that are involved in the
nonlinear cascade of the total energy in the inertial range. Each category is
examined in detail, in particular those that can be written either as source or
flux terms. Finally, the role of the background magnetic field $B_0$ is
highlighted and comparison with the incompressible MHD (IMHD) model is
discussed. This point is particularly important when testing the exact law on
numerical simulations and in situ observations in space plasmas.