On the dual cascade in two-dimensional turbulence

We study the dual-cascade scenario for two-dimensional (2D) 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 less than or equal to eta less than or equal to k(min)(2epsilon). 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 2D 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 greater than or equal to 8, 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 analysed. We discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain. (C) 2002 Elsevier Science B.V. All rights reserved.