We have developed a massively parallelized fully three-dimensional (3D) compressible Hall-magnetohydrodynamic (MHD) code to investigate inertial range electromagnetic wave cascades and dissipative processes in the regime, where characteristic length scales associated with plasma fluctuations are smaller than ion gyroradii. Such regime is ubiquitously present in the solar wind and many other collisionless space plasmas. Particularly, in the solar wind, the high time resolution databases depict a spectral break near the end of the 5/3 spectrum that corresponds to a high-frequency regime where the electromagnetic turbulent cascades cannot be explained by the usual MHD models. This refers to a second inertial range, where turbulent cascades follow a k~(-7/3) (where k is a wavenumber) spectrum in which the characteristic electromagnetic fluctuations evolve typically on kinetic Alfvén time scales. In this paper, we describe results from our 3D compressible Hall-MHD simulations that explain the observed k~(-7/3) spectrum in the solar wind plasma, energy cascade, anisotropy, and other spectral features.
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