Nonlinear transfer and spectral distribution of energy in α turbulence

Two-dimensional turbulence governed by the so-called α turbulence equations, which include the surface quasi-geostrophic equation (α=1), the Navier–Stokes system (α=2), and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating (α=3), is studied here in both the unbounded and doubly periodic domains. This family of equations conserves two inviscid invariants (energy and enstrophy in the Navier–Stokes case), the dynamics of which are believed to undergo a dual cascade. It is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse-cascading quantity approaches its own injection rate. Constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived. For Navier–Stokes turbulence with moderate Reynolds numbers, the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations.