Energy and enstrophy dissipation in steady state 2-d turbulence

Abstract

Upper bounds on the bulk energy dissipation rate $\epsilon$ and enstrophy dissipation rate $\chi$ are derived for the statistical steady state of body forced two dimensional turbulence in a periodic domain. For a broad class of externally imposed body forces it is shown that $\epsilon \le k_{f} U^3 Re^{-1/2}(C_1+C_2 Re^{-1})^{1/2}$ and $\chi \le k_{f}^{3}U^3 (C_1+C_2 Re^{-1})$ where $U$ is the root-mean-square velocity, $k_f$ is a wavenumber (inverse length scale) related with the forcing function, and $Re = U /\nu k_f$. The positive coefficients $C_1$ and $C_2$ are uniform in the the kinematic viscosity $\nu$, the amplitude of the driving force, and the system size. We compare these results with previously obtained bounds for body forces involving only a single length scale, or for velocity dependent a constant-energy-flux forces acting at finite wavenumbers. Implications of our results are discussed.