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Parameterizing interaction of disparate scales: selective decay by Casimir dissipation in fluids

Presented by: 
D Holm Imperial College London
Thursday 23rd August 2012 - 15:30 to 16:30
INI Seminar Room 1
The problem of parameterizing the interactions of disparate scales in fluid flows is addressed by considering a property of two-dimensional incompressible turbulence. The property we consider is selective decay, in which a Casimir of the ideal formulation (enstrophy in 2D flows) decays in time, while the energy stays essentially constant. This paper introduces a mechanism that produces selective decay by enforcing Casimir dissipation in fluid dynamics. This mechanism turns out to be related in certain cases to the numerical method of anticipated vorticity discussed in \cite{SaBa1981,SaBa1985}. Several examples are given and a general theory of selective decay is developed that uses the Lie-Poisson structure of the ideal theory. A scale-selection operator allows the resulting modifications of the fluid motion equations to be interpreted in several examples as parameterizing the nonlinear, dynamical interactions between disparate scales. The type of modified fluid equation systems derived here may be useful in turbulent geophysical flows where it is computationally prohibitive to rely on the slower, indirect effects of a realistic viscosity, such as in large-scale, coherent, oceanic flows interacting with much smaller eddies.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons