  To explain energy dissipation via turbulence in collisionless, magnetized
plasmas, the existence of a dual real- and velocity-space cascade of
ion-entropy fluctuations below the ion gyroradius has been proposed. Such a
dual cascade, predicted by the gyrokinetic theory, has previously been observed
in gyrokinetic simulations of two-dimensional, electrostatic turbulence. For
the first time we show evidence for a dual phase-space cascade of ion-entropy
fluctuations in a three-dimensional simulation of hybrid-kinetic,
electromagnetic turbulence. Some of the scalings observed in the energy spectra
are consistent with a generalized theory for the cascade that accounts for the
spectral anisotropy of critically balanced, intermittent, sub-ion-Larmor-scale
fluctuations. The observed velocity-space cascade is also anisotropic with
respect to the magnetic-field direction, with linear phase mixing along
magnetic-field lines proceeding mainly at spatial scales above the ion
gyroradius and nonlinear phase mixing across magnetic-field lines proceeding at
perpendicular scales below the ion gyroradius. Such phase-space anisotropy
could be sought in heliospheric and magnetospheric data of solar-wind
turbulence and has far-reaching implications for the dissipation of turbulence
in weakly collisional astrophysical plasmas.
