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

TL;DR

This paper proves that the Batchelor-Howells-Townsend spectrum scaling law holds probabilistically for large wavenumbers in three-dimensional passive tracer advection with small random velocity fields, relaxing previous assumptions.

Contribution

It extends the BHT spectrum prediction to three dimensions and probabilistic settings, with relaxed assumptions on velocity and source variances.

Findings

BHT spectrum scaling holds probabilistically in 3D.

Asymptotic validity for large wavenumbers.

Applicable to small random velocity fields with relaxed assumptions.

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 $\|u\|$ 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.