  The nature of solar wind turbulence in the dissipation range at scales much
smaller than the large MHD scales remains under debate. Here a two-dimensional
model based on the hybrid code abbreviated as A.I.K.E.F. is presented, which
treats massive ions as particles obeying the kinetic Vlasov equation and
massless electrons as a neutralizing fluid. Up to a certain wavenumber in the
MHD regime, the numerical system is initialized by assuming a superposition of
isotropic Alfv\'en waves with amplitudes that follow the empirically confirmed
spectral law of Kolmogorov. Then turbulence develops and energy cascades into
the dispersive spectral range, where also dissipative effects occur. Under
typical solar wind conditions, weak turbulence develops as a superposition of
normal modes in the kinetic regime. Spectral analysis in the direction parallel
to the background magnetic field reveals a cascade of left-handed
Alfv\'en/ion-cyclotron waves up to wave vectors where their resonant absorption
sets in, as well as a continuing cascade of right-handed fast-mode and whistler
waves. Perpendicular to the background field, a broad turbulent spectrum is
found to be built up of fluctuations having a strong compressive component.
Ion-Bernstein waves seem to be possible normal modes in this propagation
direction for lower driving amplitudes. Also signatures of short-scale
pressure-balanced structures (very oblique slow-mode waves) are found.
