Split energy cascade in turbulent thin fluid layers

We discuss the phenomenology of the split energy cascade in a three-dimensional thin fluid layer by means of high resolution numerical simulations of the Navier-Stokes equations. We observe the presence of both an inverse energy cascade at large scales, as predicted for two-dimensional turbulence, and a direct energy cascade at small scales, as in three-dimensional turbulence. The inverse energy cascade is associated with a direct cascade of enstrophy in the intermediate range of scales. Notably, we find that the inverse cascade of energy in this system is not a purely 2D phenomenon, as the coupling with the 3D velocity field is necessary to guarantee the constancy of fluxes.