Numerical investigation of the Danilov inequality for two-layer quasi-geostrophic systems

In the statistical steady state of forced-dissipative turbulence governed by the 2D incompressible Navier-Stokes (NS) equation, the difference between the energy flux Pi(E)(k) and the enstrophy flux Pi(Z)(k), k(2)Pi(E)(k) - Pi(Z)(k), where k is the wavenumber, has been proved to be negative outside the forcing wave-number range. This is often referred to as the Danilov inequality and is important for determining the directions of the energy and enstrophy fluxes in the inertial ranges for 2D NS turbulence. We investigate numerically whether or not this inequality holds for two-layer quasi-geostrophic (QG) potential vorticity equations, which are a generalization of the vorticity equation for the 2D NS system. In two-layer QG systems forced by a horizontally homogeneous, baroclinically unstable basic flow, the corresponding difference between the total energy and total potential enstrophy fluxes is mathematically sign-indefinite due to the presence of the internal forcing term. Moreover, even if we concentrate on outside of the internal forcing range, the flux difference is still sign-indefinite when the dissipation mechanisms are asymmetric between the upper and lower layers. However, in our numerical experiments adopting both the internal forcing and asymmetric dissipations, the flux difference is negative across the whole wavenumber range. That is, the Danilov inequality continues to hold for two-layer QG systems, even though this cannot be proved mathematically.