Stochastic parametrization of the Richardson triple

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Title: Stochastic parametrization of the Richardson triple
Authors: Holm, DD
Item Type: Journal Article
Abstract: A Richardson triple is an ideal fluid flow map (Formula presented.) composed of three smooth maps with separated time scales: slow, intermediate and fast, corresponding to the big, little and lesser whorls in Richardson’s well-known metaphor for turbulence. Under homogenization, as (Formula presented.), the composition (Formula presented.) of the fast flow and the intermediate flow is known to be describable as a single stochastic flow (Formula presented.). The interaction of the homogenized stochastic flow (Formula presented.) with the slow flow of the big whorl is obtained by going into its non-inertial moving reference frame, via the composition of maps (Formula presented.). This procedure parameterizes the interactions of the three flow components of the Richardson triple as a single stochastic fluid flow in a moving reference frame. The Kelvin circulation theorem for the stochastic dynamics of the Richardson triple reveals the interactions among its three components. Namely, (1) the velocity in the circulation integrand is kinematically swept by the large scales and (2) the velocity of the material circulation loop acquires additional stochastic Lie transport by the small scales. The stochastic dynamics of the composite homogenized flow is derived from a stochastic Hamilton’s principle and then recast into Lie–Poisson bracket form with a stochastic Hamiltonian. Several examples are given, including fluid flow with stochastically advected quantities and rigid body motion under gravity, i.e. the stochastic heavy top in a rotating frame.
Issue Date: 20-Jun-2018
Date of Acceptance: 6-Jun-2018
ISSN: 0938-8974
Publisher: Springer
Start Page: 89
End Page: 113
Journal / Book Title: Journal of Nonlinear Science
Volume: 29
Issue: 1
Copyright Statement: © 2018 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords: physics.flu-dyn
0102 Applied Mathematics
Fluids & Plasmas
Publication Status: Published
Open Access location:
Online Publication Date: 2018-06-20
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics

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