Energy Spectrum of Quasi-Geostrophic Turbulence

We consider the energy spectrum of a quasi-geostrophic model of forced, rotating turbulent flow. We provide a rigorous a priori bound E(k) <= Ck^{-2} valid for wave numbers that are smaller than a wave number associated to the forcing injection scale. This upper bound separates this spectrum from the Kolmogorov-Kraichnan k^{-{5/3}} energy spectrum that is expected in a two-dimensional Navier-Stokes inverse cascade. Our bound provides theoretical support for the k^{-2} spectrum observed in recent experiments

Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.Comment: 7 page

An Eulerian-Lagrangian approach to the Navier-Stokes equations

This work presents an approach to the Navier-Stokes equations that is phrased in unbiased Eulerian coordinates, yet describes objects that have Lagrangian significance: particle paths, their dispersion and diffusion. The commutator between Lagrangian and Eulerian derivatives plays an important role in the Navier-Stokes equations: it contributes a singular perturbation to the Euler equations, in addition to the Laplacian. Bounds for the Lagrangian displacements, their first and second derivatives are obtained without assumptions. Some of these rigorous bounds can be interpreted in terms of the heuristic Richardson law of pair dispersion in turbulent flows

A stochastic perturbation of inviscid flows

We prove existence and regularity of the stochastic flows used in the stochastic Lagrangian formulation of the incompressible Navier-Stokes equations (with periodic boundary conditions), and consequently obtain a \holderspace{k}{\alpha} local existence result for the Navier-Stokes equations. Our estimates are independent of viscosity, allowing us to consider the inviscid limit. We show that as $\nu \to 0$, solutions of the stochastic Lagrangian formulation (with periodic boundary conditions) converge to solutions of the Euler equations at the rate of $O(\sqrt{\nu t})$.Comment: 13 pages, no figures