  The cascade of kinetic Alfv\'en waves (KAWs) at the sub-ion scales in the
solar wind is numerically simulated using a fluid approach that retains ion and
electron Landau damping, together with ion finite Larmor radius corrections.
Assuming initially equal and isotropic ion and electron temperatures, and an
ion beta equal to unity, different simulations are performed by varying the
propagation direction and the amplitude of KAWs that are randomly driven at a
transverse scale of about one fifth of the proton gyroradius in order to
maintain a prescribed level of turbulent fluctuations. The resulting turbulent
regimes are characterized by the nonlinearity parameter, defined as the ratio
of the characteristic times of Alfv\'en wave propagation and of the transverse
nonlinear dynamics. The corresponding transverse magnetic energy spectra
display power laws with exponents spanning a range of values consistent with
spacecraft observations. The meandering of the magnetic field lines together
with the ion temperature homogenization along these lines are shown to be
related to the strength of the turbulence, measured by the nonlinearity
parameter. The results are interpreted in terms of a recently proposed
phenomenological model where the homogenization process along field lines
induced by Landau damping plays a central role.
