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

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