Exact law for homogeneous compressible Hall magnetohydrodynamics turbulence

We derive the exact law for three-dimensional (3D) homogeneous compressible isothermal Hall magnetohydrodynamics (CHMHD) turbulence, without the assumption of isotropy. The Hall current is shown to introduce new flux and sources terms that act at the small scales (comparable or smaller than the ion skin depth) to significantly impact the turbulence dynamics. The new law provides an accurate means to estimate for the first time the energy cascade rate over a broad range of scales covering the MHD inertial range and the sub-ion dispersive range in 3D numerical simulations and {\it in situ} spacecraft observations of compressible turbulence. This work is particularly relevant to astrophysical flows in which small scale density fluctuations cannot be ignored such as the solar wind, planetary magnetospheres and the interstellar medium (ISM).