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A unified view of topological invariants of barotropic & baroclinic fluids & their application to formal stability analysis of three-dimensional ideal gas flows

Date: 
Friday 27th July 2012 - 11:45 to 12:05
Venue: 
INI Seminar Room 1
Session Chair: 
Chas Williamson
Abstract: 
Integrals of an arbitrary function of the vorticity, two-dimensional topological invariants of an ideal barotropic fluid, take different guise from the helicity. Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. Baroclinic fluid flows admit, as the Casimir invariants, a class of integrals including an arbitrary function of the entropy and the potential vorticity. A consideration is given to them from the view point of Noether's theorem. We then develop a new energy-Casimir convexity method for a baroclinic fluid, and establish a novel linear stability criterion, to three-dimensional disturbances, for equilibria of general rotating flows of an ideal gas without appealing to the Boussinesq approximation. By exploiting a larger class of the Casimir invariants, we have succeeded in ruling out a term including the gradient of a dep endent variable from the energy-Casimir function. For zonally symmetric flows, the resulting criterion is regarded as an extended Richardson number criterion for stratified rotating shear flows with compressibility taken into account.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons