The Batchelor–Howells–Townsend spectrum: Three-dimensional case
Jolly, M.S.; Wirosoetisno, D.
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. https://doi.org/10.1016/j.physd.2022.133615
|Journal Article Type||Article|
|Acceptance Date||Dec 7, 2022|
|Online Publication Date||Dec 13, 2022|
|Deposit Date||Jan 4, 2023|
|Publicly Available Date||Dec 14, 2023|
|Journal||Physica D: Nonlinear Phenomena|
|Peer Reviewed||Peer Reviewed|
This file is under embargo until Dec 14, 2023 due to copyright restrictions.
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