Cascades and fluxes in two-dimensional turbulence

We develop simple models of forced and unforced turbulence at large but finite Reynolds number. These models yield a number of predictions, such as (i) the form of the second- and third-order structure functions in the inertial range; (ii) the rate of generation of palinstrophy in forced and unforced turbulence; (iii) the rate of destruction of enstrophy in decaying turbulence; (iv) the rate of change of Loitsyansky’s integral in decaying turbulence; and, again for decaying turbulence; and (v) the existence of an inverse energy cascade, embedded within the direct enstrophy cascade. We show that these predictions are not mere artifacts of our simple models, but rather genuine features of two-dimensional turbulence. The most surprising findings relate to (iii), (iv), and (v) above, where we show that, once the turbulence is fully developed, the enstrophy dissipation rate scales as β∼⟨ω2⟩3∕2∕ln(Re), Loitsyansky’s integral grows at a rate proportional to βℓ6∕ln(Re), and the filamentation of vorticity fuels an inverse energy cascade with a flux of order βℓ2∕ln(Re). (Here, ω is the vorticity, ℓ is the integral scale, β the enstrophy dissipation rate, and Re the Reynolds number.) This suggests that, in the limit of Re→∞, the enstrophy dissipation rate tends to zero and the inverse energy cascade progressively shuts down. Loitsyansky’s integral is also an invariant in this limit, provided that ℓ remains finite as Re→∞.