A link between turbulent cascade and gyrotropic pressure instabilities in compressible and magnetized fluids.

In recent years, the Kolmogorov's statistical formalism of exact law that describes incompressible hydrodynamic turbulence, has been extended to compressible magnetized fluid described by isothermal or polytropic closure. Such exact laws permit an evaluation of the energy cascade rate, assumed within this formalism to be equivalent to the dissipation rate. Its estimation in the solar wind can help to better understand particle heating in such collisionless media. But previous exact laws are insufficient in a system led by pressure anisotropy. We propose a general exact law of Hall-MHD turbulence based on models with a pressure tensor that allows us to study various known equations of state as particular limits, derive a new one corresponding to the CGL (i.e., gyrotropic pressure tensor), and correlate the cascade rate to instable plasma conditions. In the incompressible MHD limit we provide a generalization of the Politano & Pouquet law to pressure-anisotropic plasmas.