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

Jolly, MS; Wirosoetisno, D

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 in some sense, Batchelor, Howells and Townsend (J Fluid Mech 5:134, 1959) predicted that the tracer spectrum scales as |θk| 2 ∝ |k| −4|uk| 2. In this paper we prove that, for random synthetic two-dimensional incompressible velocity fields u(x, t) with given energy spectra, this scaling does indeed hold probabilistically, asymptotically almost surely for large |k| and small u. We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of u.

Citation

Jolly, M., & Wirosoetisno, D. (2020). Tracer turbulence: the Batchelor--Howells--Townsend spectrum revisited. Journal of Mathematical Fluid Mechanics, 22(2), Article 18. https://doi.org/10.1007/s00021-019-0478-6

• Journal Article Type: Article
• Acceptance Date: Dec 6, 2019
• Online Publication Date: Feb 24, 2020
• Publication Date: Jun 30, 2020
• Deposit Date: Dec 6, 2019
• Publicly Available Date: Feb 24, 2021
• Journal: Journal of Mathematical Fluid Mechanics
• Print ISSN: 1422-6928
• Electronic ISSN: 1422-6952
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 22
• Issue: 2
• Article Number: 18
• DOI: https://doi.org/10.1007/s00021-019-0478-6
• Public URL: https://durham-repository.worktribe.com/output/1281971
• Related Public URLs: https://arxiv.org/abs/1907.12633