A Multiscale Camassa--Holm Equation
Darryl D Holm, Maneesh Kumar Singh
TL;DR
This paper introduces a multiscale generalization of the Camassa-Holm equation to model complex fluid turbulence phenomena, supported by numerical simulations demonstrating emergent singular solutions.
Contribution
It presents a novel multiscale geodesic flow system extending the Camassa-Holm equation with new singular solutions and turbulence modeling capabilities.
Findings
Numerical simulations on S^1 show complex multiscale behavior.
Emergent solutions generalize peakons of the original CH equation.
The model captures cascade-like energy transfer in turbulence.
Abstract
A system of equations for Multiscale Geodesic Flow (MGF) is introduced whose solutions illustrate the paradigm of whorls within whorls within whorls, introduced by L. F. Richardson in 1922 to describe the cascade of energy in fluid turbulence. Numerical simulations are given for MGF on , where the MGF equation comprises a multiscale generalisation of the Camassa-Holm (CH) equation whose emergent singular solutions generalise the peakon solutions of the CH.
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Taxonomy
TopicsNonlinear Waves and Solitons · Advanced Mathematical Modeling in Engineering · Homotopy and Cohomology in Algebraic Topology