Energy and enstrophy dissipation in steady state 2d turbulence

Upper bounds on the bulk energy dissipation rate γ and enstrophy dissipation rate χ are derived for the statistical steady state of body forced two-dimensional (2d) turbulence in a periodic domain. For a broad class of externally imposed body forces it is shown that γ⩽kUReC)1/2 and χ⩽kf3U(C+CRe) where U is the root-mean-square velocity, k is a wavenumber (inverse length scale) related with the forcing function, and Re=U/νk. The positive coefficients C and C are uniform in the kinematic viscosity ν, 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 constant-energy-flux forces acting at finite wavenumbers. Implications of our results are discussed.