The Batchelor--Howells--Townsend spectrum: three-dimensional case

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 in some sense, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134; henceforth BHT) predicted that the tracer spectrum scales as $|{\theta }_k{|}^2\propto |k{|}^{-4}|u_k{|}^2$. Following our recent work for the two-dimensional case, in this paper we prove that the BHT scaling does hold probabilistically, asymptotically for large wavenumbers and for small enough random synthetic three-dimensional incompressible velocity fields $u(x,t)$. We also relaxed some assumptions on the velocity and tracer source, allowing finite variances for both and full power spectrum for the latter.

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