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

Given a velocity field $u(x,t)$, we consider the evolution of a passive tracer $\theta$ governed by ${\mathrm{\partial }}_t\theta +u\cdot \mathrm{\nabla }\theta =\mathrm{\Delta }\theta +g$ with time-independent source $g(x)$. When $\mathrm{\parallel }u\mathrm{\parallel }$ is small in some sense, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134; henceforth BHT) predicted that the tracer spectrum scales as $\mathrm{\mid }{\theta }_k{\mathrm{\mid }}^2\propto \mathrm{\mid }k{\mathrm{\mid }}^{-4}\mathrm{\mid }{u}_k{\mathrm{\mid }}^2$. 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.