Tracer turbulence: the Batchelor-Howells-Townsend spectrum revisited release_5xwqtvmxbba5fp6xddxdwbyfma

Abstract

Given a velocity field u(x,t), we consider the evolution of a passive tracer θ governed by ∂_tθ + u·∇θ = Δθ + g with time-independent source g(x). When u is small, Batchelor, Howells and Townsend (1959, J. Fluid Mech. 5:134) predicted that the tracer spectrum scales as |θ_k|^2∝|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.
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