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The role of short-wave instabilities on geostrophic turbulence is studied ina simplified model consisting of three layers in the quasi-geostrophicapproximation. The linear stability analysis shows that short-waveinstabilities are created by the interplay between the shear in the upper andthe lower layers. If the stratification is non-uniform, in particular surfaceintensified, the linear growth rate is larger for short-wave instabilities thanfor long-wave instabilities and the layers are essentially decoupled, with thesmall scales growing independently. The fully developed homogeneous turbulenceis studied in a number of numerical experiments. Results show that in both thecase of equal layer depths and surface intensified stratification an inversecascade in kinetic energy is observed. The modal kinetic energy spectra for thecase with surface intensified stratification show higher energy for higherbaroclinic numbers at small scales, due to the decoupling of the layers. As aresult, while the case with equal layer depths shows large barotropicinstabilities with large scale gradients of potential temperature, the surfaceintensified stratification is characterized by a transition from surfacedynamics, characterized by a patchy distribution of vorticity, to interiordynamics, characterized by vorticity filamentation. The effect of theshort-wave instabilities can be seen in the probability distribution functionsof the potential vorticity anomaly, which reduces to a Gaussian distributionwhen the growth rate of the short-wave instabilities is larger than the growthrate for the long-wave instabilities. The surface intensified stratificationalso alters the vertical structure of the potential vorticity fluxes and showsdeviations of the fluxes from a scaling obtained assuming that the turbulenceacts as a downgradient diffusion.

On the role of non-uniform stratification and short-wave instabilities in three-layer quasi-geostrophic turbulence