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

Jolly, M.S.; Wirosoetisno, D.


M.S. Jolly


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.


Jolly, M., & Wirosoetisno, D. (2023). The Batchelor–Howells–Townsend spectrum: Three-dimensional case. Physica D: Nonlinear Phenomena, 445, Article 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
Public URL


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