Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence

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 νμ(−∆) μ. By “monoscale-like” we mean that the forcing is applied over a finite range of wavenumbers kmin ≤ k ≤ kmax, and that the ratio of enstrophy injection η ≥ 0 to energy injection ε ≥ 0 is bounded by k minε ≤ η ≤ k 2 maxε. Such a forcing is frequently considered in theoretical and numerical studies of two-dimensional turbulence. It is shown that for μ ≥ 0 the asymptotic behaviour satisfies ||u|| 1 ≤ k max ||u|| 2 , where ||u|| and ||u||21 are the energy and enstrophy, respectively. If the condition of monoscale-like forcing holds only in a time-mean sense, then the inequality holds in the time mean. It is also shown that for Navier-Stokes turbulence (μ = 1), the timemean enstrophy dissipation rate is bounded from above by 2ν1k 2 max. 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 Preprint submitted to Elsevier Science 8 February 2008 such systems. In particular, the classical dual cascade picture is shown to be invalid for forced two-dimensional Navier–Stokes turbulence (μ = 1) when it is forced in this manner. Inclusion of Ekman drag (μ = 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 (μ < 0). These constraints on permissible power laws apply for any spectrally localized forcing, not just for monoscale-like forcing.