Tracer turbulence: the Batchelor-Howells-Townsend spectrum revisited

Abstract

Given a velocity field $u(x,t)$, we consider the evolution of a passive tracer $\theta$ governed by ${\partial }_t\theta +u\cdot \nabla \theta =\Delta \theta +g$ with time-independent source $g(x)$. When $\Vert u\Vert$ is small, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134) predicted that the tracer spectrum scales as $|{\theta }_k{|}^2\propto |k{|}^{-4}|u_k{|}^2$. In this paper, we prove that this scaling does indeed hold for large $|k|$, in a probabilistic sense, for random synthetic two-dimensional incompressible velocity fields $u(x,t)$ with given energy spectra. We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of $u$.

15 pages