We investigated the properties of plasma turbulence at ion scales in the solar wind context.
We concentrated on the behaviour of the Hall physics and the pressure strain interaction
and their anisotropy owing to the ambient magnetic field. We studied the results of a three-dimensional hybrid simulation of decaying
plasma turbulence using the K\'arm\'an-Howarth-Monin (KHM) equation, which quantifies different turbulent processes. The isotropised KHM analysis shows that kinetic plus magnetic (kinetic+magnetic) energy
decays at large scales; this energy cascades from large to small
scales via the magneto-hydrodynamic non-linearity that is partly continued via the Hall coupling
around the ion scales. The cascading kinetic+magnetic energy is partly dissipated
at small scales via resistive dissipation.
This standard dissipation is complemented by the
pressure-strain interaction, which plays the role of an effective dissipation
mechanism and starts to act at relatively large scales.
The pressure--strain interaction has two components, compressive and incompressive.
Compressive interaction is connected with the velocity dilatation,
which mostly reversibly exchanges kinetic+magnetic and internal energies. Incompressive interaction mostly irreversibly converts
the kinetic+magnetic energy to internal energy. The compressive
effects lead to important oscillations of the turbulence properties, but
the compressibility is strongly reduced when averaged over a time
period spanning a few periods of the oscillations.
The ambient magnetic field induces a strong spectral anisotropy.
The turbulent fluctuations exhibit larger scales along the magnetic field
compared to the perpendicular directions. The KHM results show the corresponding
anisotropy of turbulent processes: their characteristic scales shift to
larger scales in the quasi-parallel direction with respect to the ambient magnetic field
compared to the quasi-perpendicular direction. This anisotropy is weak at large
scales owing to the initial isotropic spectrum, and becomes progressively stronger
at small scales.