Local cascade and dissipation in incompressible Hall magnetohydrodynamic turbulence: the Coarse-Graining approach

We derive the coarse-graining (CG) equations of incompressible Hall Magnetohydrodynamics (HMHD) turbulence to investigate the {\it local (in space)} energy cascade rate as a function of the filtering scale $\ell$. First, the CG equations are space averaged to obtain the analytical expression of the mean cascade rate. Its application to 3 dimensional (3D) simulations of (weakly compressible) HMHD shows a cascade rate consistent with the value of the mean dissipation rate in the simulations and with the classical estimates based on the "third-order" law. Furthermore, we developed an anisotropic version of CG that allows us to study the magnitude of the cascade rate along different directions with respect to the mean magnetic field. Its implementation on the numerical data with moderate background magnetic field shows a weaker cascade along the magnetic field than in the perpendicular plane, while an isotropic cascade is recovered in the absence of a background field. The strength of the CG approach is further revealed when considering the {\it local-in-space} energy transfer, which is shown theoretically and numerically to match at a given position $\bm{x}$, when locally averaged over a neighboring region, the (quasi-)local dissipation. Prospects of exploiting this new model to investigate local dissipation in spacecraft data are discussed.