We discuss the phenomenology of the split energy cascade in a
three-dimensional thin fluid layer by mean 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 turbu-
lence, and of 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 pure 2D phenomenon, as the
coupling with the 3D velocity field is necessary to guarantee the constancy of
fluxes
