Nonlinear transfer and spectral distribution of energy in α turbulence

Two-dimensional turbulence governed by the so-called $\alpha$ turbulence equations, which include the surface quasi-geostrophic equation ($\alpha=1$), the Navier--Stokes system ($\alpha=2$), and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating ($\alpha=3$), 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.