  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.
