The Batchelor–Howells–Townsend spectrum: large velocity case

Jolly, M S; Wirosoetisno, D

Abstract

We consider the behaviour of a passive tracer θ governed by ∂tθ+u⋅∇θ=Δθ+g in two space dimensions with prescribed smooth random incompressible velocity u(x, t) and source g(x). In 1959, Batchelor et al (J. Fluid Mech. 5 113) predicted that the tracer (power) spectrum should scale as |θk|2∝|k|−4|uk|2 for |k| above some κ¯(u) , with different behaviour for |k|≲κ¯(u) predicted earlier by Obukhov and Corrsin. In this paper, we prove that the BHT scaling does indeed hold probabilistically for sufficiently large |k| , asymptotically up to controlled remainders, using only bounds on the smaller |k| component.

Citation

Jolly, M. S., & Wirosoetisno, D. (2024). The Batchelor–Howells–Townsend spectrum: large velocity case. Nonlinearity, 37(7), Article 075025. https://doi.org/10.1088/1361-6544/ad5265

• Journal Article Type: Article
• Acceptance Date: May 30, 2024
• Online Publication Date: Jun 13, 2024
• Publication Date: Jul 1, 2024
• Deposit Date: May 30, 2024
• Publicly Available Date: Jun 13, 2024
• Journal: Nonlinearity
• Print ISSN: 0951-7715
• Electronic ISSN: 1361-6544
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 37
• Issue: 7
• Article Number: 075025
• DOI: https://doi.org/10.1088/1361-6544/ad5265
• Keywords: turbulence, 76F02, 60G99, passive tracers, Batchelor–Howells–Townsend spectrum, random velocity, 47A55, 35Q30
• Public URL: https://durham-repository.worktribe.com/output/2468299