The Batchelor–Howells–Townsend spectrum: Three-dimensional case

Jolly, M.S.; Wirosoetisno, D.

Abstract

Given a velocity field u(x, t), we consider the evolution of a passive tracer governed by @t + u · ∇ = + 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 BHT59) predicted that the tracer spectrum scales as |k|2 ∝ |k|−4|uk|2. Following our recent work for the two-dimensional case, in this paper we prove that the BHT59 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.

Citation

Jolly, M., & Wirosoetisno, D. (2023). The Batchelor–Howells–Townsend spectrum: Three-dimensional case. Physica D: Nonlinear Phenomena, 445, Article 133615. https://doi.org/10.1016/j.physd.2022.133615

• Journal Article Type: Article
• Acceptance Date: Dec 7, 2022
• Online Publication Date: Dec 13, 2022
• Publication Date: 2023-03
• Deposit Date: Jan 4, 2023
• Publicly Available Date: Dec 14, 2023
• Journal: Physica D: Nonlinear Phenomena
• Print ISSN: 0167-2789
• Electronic ISSN: 1872-8022
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 445
• Article Number: 133615
• DOI: https://doi.org/10.1016/j.physd.2022.133615
• Public URL: https://durham-repository.worktribe.com/output/1184028