Energy and enstrophy dissipation in steady state 2d turbulence

Upper bounds on the bulk energy dissipation rate ϵ 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 ϵ⩽kfU3Re−1/2(C1+C2Re−1)1/2ϵ⩽kfU3Re−1/2(C1+C2Re−1)1/2 and χ⩽kf3U3(C1+C2Re−1) where U   is the root-mean-square velocity, kfkf is a wavenumber (inverse length scale) related with the forcing function, and Re=U/νkfRe=U/νkf. The positive coefficients C1C1 and C2C2 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.