  We study two-dimensional turbulence in a doubly periodic domain driven by a
monoscale-like forcing and damped by various dissipation mechanisms of the form
$\nu_{\mu}(-\Delta)^{\mu}$. By ``monoscale-like'' we mean that the forcing is
applied over a finite range of wavenumbers $k_\min \leq k \leq k_\max$, and
that the ratio of enstrophy injection $\eta \geq 0$ to energy injection
$\epsilon \geq 0$ is bounded by $k_\min^2 \epsilon \leq \eta \leq k_\max^2
\epsilon$. It is shown that for $\mu\geq 0$ the asymptotic behaviour satisfies
(eqnarray) \norm u_1^2&\leq&k_\max^2\norm u^2,(eqnarray) where $\norm u^2$ and
$\norm u_1^2$ are the energy and enstrophy, respectively. It is also shown that
for Navier-Stokes turbulence ($\mu = 1$), the time-mean enstrophy dissipation
rate is bounded from above by $2\nu_1 k_\max^2$. These results place strong
constraints on the spectral distribution of energy and enstrophy and of their
dissipation, and thereby on the existence of energy and enstrophy cascades, in
such systems. In particular, the classical dual cascade picture is shown to be
invalid for forced two-dimensional Navier--Stokes turbulence ($\mu=1$) when it
is forced in this manner. Inclusion of Ekman drag ($\mu=0$) along with
molecular viscosity permits a dual cascade, but is incompatible with the
log-modified -3 power law for the energy spectrum in the enstrophy-cascading
inertial range. In order to achieve the latter, it is necessary to invoke an
inverse viscosity ($\mu<0$).
