Exact results on stationary turbulence in 2D: consequences of vorticity conservation

Abstract

We establish a series of exact results for a model of stationary turbulence in two dimensions: the incompressible Navier-Stokes equation in a periodic domain with stochastic force white-noise in time. Essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics. Our main results are as follows: (1) we show the blow-up of mean energy as ∼l 0 2 ε ν for ν→0 when there is no IR-dissipation at the large length-scale l 0; (2) with an additional IR-dissipation, we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for ν→0, assuming finite mean energy in the limit; (3) we rigorously establish the balance equations for the energy and vorticity invariants in the 2D steady-state and the forward cascade of the higher-order vorticity invariants assuming finite mean values; (4) we derive exact inequalities for scaling exponents in the 2D enstrophy range, as follows: if 〈| Δ l ω| p 〉 ∼ l ζ p , then ζ 2 ≤ 2 3 and ζ p ≤ 0 for p ≥ 3. If the minimum Hölder exponent of the vorticity h min < 0, then we establish a 2D analogue of the refined similarity hypothesis which improves these bounds. The most novel and interesting conclusion of this work is the connection established between “intermittency” in 2D and “negative Hölder singularities” of the vorticity: we show that the latter are necessary for deviations from the 1967 Kraichnan scaling to occur.