General Exact Law of Compressible Isentropic Magnetohydrodynamic Flows: Theory and Spacecraft Observations in the Solar Wind

Various forms of exact laws governing magnetohydrodynamic (MHD) turbulence have been derived either in the incompressibility limit, or for isothermal compressible flows. Here we propose a more general method that allows us to obtain such laws for any turbulent isentropic flow (i.e., constant entropy). We demonstrate that the known MHD exact laws (incompressible and isothermal) and the new (polytropic) one can be obtained as specific cases of the general law when the corresponding closure equation is stated. We also recover all known exact laws of hydrodynamic (HD) turbulence (incompressible, isothermal, and polytropic) from this law in the limit B = 0. We furthermore show that the difference between the two forms (isothermal and polytropic) of the MHD exact laws of interest in this work resides in some of the source terms and in the explicit form of the flux term that depends on internal energy. Finally, we apply these two forms to Parker Solar Probe data taken in the inner heliosphere to highlight how the different closure equations affect the energy cascade rate estimates.