  The role of short-wave instabilities on geostrophic turbulence is studied in
a simplified model consisting of three layers in the quasi-geostrophic
approximation. The linear stability analysis shows that short-wave
instabilities are created by the interplay between the shear in the upper and
the lower layers. If the stratification is non-uniform, in particular surface
intensified, the linear growth rate is larger for short-wave instabilities than
for long-wave instabilities and the layers are essentially decoupled, with the
small scales growing independently. The fully developed homogeneous turbulence
is studied in a number of numerical experiments. Results show that in both the
case of equal layer depths and surface intensified stratification an inverse
cascade in kinetic energy is observed. The modal kinetic energy spectra for the
case with surface intensified stratification show higher energy for higher
baroclinic numbers at small scales, due to the decoupling of the layers. As a
result, while the case with equal layer depths shows large barotropic
instabilities with large scale gradients of potential temperature, the surface
intensified stratification is characterized by a transition from surface
dynamics, characterized by a patchy distribution of vorticity, to interior
dynamics, characterized by vorticity filamentation. The effect of the
short-wave instabilities can be seen in the probability distribution functions
of the potential vorticity anomaly, which reduces to a Gaussian distribution
when the growth rate of the short-wave instabilities is larger than the growth
rate for the long-wave instabilities. The surface intensified stratification
also alters the vertical structure of the potential vorticity fluxes and shows
deviations of the fluxes from a scaling obtained assuming that the turbulence
acts as a downgradient diffusion.
