Tracer turbulence: the Batchelor-Howells-Townsend spectrum revisited

Given a velocity field $u(x,t)$, we consider the evolution of a passive tracer $\theta$ governed by $\partial_t\theta + u\cdot\nabla\theta = \Delta\theta + g$withtime-independentsource$g(x)$\mathrm{.}When$|u|$ is small, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134) predicted that the tracer spectrum scales as $\mathrm{\mid }{\theta }_k{\mathrm{\mid }}^2\propto \mathrm{\mid }k{\mathrm{\mid }}^{-4}\mathrm{\mid }{u}_k{\mathrm{\mid }}^2$. In this paper, we prove that this scaling does indeed hold for large $\mathrm{\mid }k\mathrm{\mid }$, 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 $u$.