2003: On the dual cascade in two dimensional turbulence

We study the dual cascade scenario for two-dimensional turbulence driven by a spectrally localized forcing applied over a nite wavenumber range [kmin; kmax] (with kmin> 0) such that the respective energy and enstrophy injection rates and satisfy k2min k 2 max. 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 > 5 and + 8, where ( ) is the asymptotic slope of the range of wavenumbers lower (higher) than the forcing wavenumber. The in\nuence 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. Key words: To appear in Physica D (2003)