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

## TL;DR

This paper proves that the classical Batchelor-Howells-Townsend spectrum scaling for passive tracers holds for large wave numbers in a probabilistic setting with random velocity fields, and introduces an asymptotic correction factor.

## Contribution

It rigorously establishes the BHT spectrum scaling for large wave numbers in a probabilistic framework and proposes a correction factor accounting for velocity field time-dependence.

## Key findings

- BHT spectrum scaling holds for large wave numbers in probabilistic models
- An asymptotic correction factor is derived for time-dependent velocity fields
- The results validate classical predictions in a rigorous mathematical setting

## 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, Batchelor, Howells and Townsend (1959, J.\ Fluid Mech.\ 5:134) predicted that the tracer spectrum scales as $|\theta_k|^2\propto|k|^{-4}|u_k|^2$. 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 $u$.