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

Abstract

We study two-dimensional (2D) turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form ν μ (− Δ) μ . By “monoscale-like” we mean that the forcing is applied over a finite range of wavenumbers k min≤ k≤ k max, and that the ratio of enstrophy injection η≥0 to energy injection ε≥0 is bounded by k min 2 ε≤ η≤ k max 2 ε. Such a forcing is frequently considered in theoretical and numerical studies of 2D turbulence. It is shown that for μ≥0 the asymptotic behaviour satisfies ∥ u∥ 1 2≤ k max 2∥ u∥ 2, where ∥ u∥ 2 and ∥ u∥ 1 2 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 time-mean enstrophy dissipation rate is bounded from above by 2 ν 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 2D 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.