  We study the dual cascade scenario for two-dimensional turbulence driven by a
spectrally localized forcing applied over a finite wavenumber range
$[k_\min,k_\max]$ (with $k_\min > 0$) such that the respective energy and
enstrophy injection rates $\epsilon$ and $\eta$ satisfy
$k_\min^2\epsilon\le\eta\le k_\max^2\epsilon$. The classical
Kraichnan--Leith--Batchelor paradigm, based on the simultaneous conservation of
energy and enstrophy and the scale-selectivity of the molecular viscosity,
requires that the domain be unbounded in both directions. For two-dimensional
turbulence either in a doubly periodic domain or in an unbounded channel with a
periodic boundary condition in the across-channel direction, a direct enstrophy
cascade is not possible. In the usual case where the forcing wavenumber is no
greater than the geometric mean of the integral and dissipation wavenumbers,
constant spectral slopes must satisfy $\beta>5$ and $\alpha+\beta\ge8$, where
$-\alpha$ ($-\beta$) is the asymptotic slope of the range of wavenumbers lower
(higher) than the forcing wavenumber. The influence of a large-scale
dissipation on the realizability of a dual cascade is analyzed. We discuss the
consequences for numerical simulations attempting to mimic the classical
unbounded picture in a bounded domain.
