The Batchelor-Howells-Townsend spectrum: large velocity case

Abstract

We consider the behaviour of a passive tracer $\theta$ governed by ${\partial }_t\theta +u\cdot \nabla \theta =\Delta \theta +g$ in two space dimensions with prescribed smooth random incompressible velocity $u(x,t)$ and source $g(x)$. In 1959, Batchelor, Howells and Townsend (J.\ Fluid Mech.\ 5:113) predicted that the tracer (power) spectrum should then scale as $|{\theta }_k{|}^2\propto |k{|}^{-4}|u_k{|}^2$ for $|k|$ large depending on the velocity $u$. For smaller $|k|$, Obukhov and Corrsin earlier predicted a different spectral scaling. In this paper, we prove that the BHT scaling does indeed hold probabilistically for sufficiently large $|k|$, asymptotically up to controlled remainders, using only bounds on the smaller $|k|$ component.

23 pages, 1 figure