Friday, July 26
Ocean dynamics is a fundamental component of the Earth's climate, accounting for half the heat transfer rate from the equator to the poles and containing most of the long-time effects. The ocean is a complex system whose time scales range from hours to millenia and whose space scales range from meters to thousands of kilometers. The finest horizontal scale presently available in global ocean modeling is about 20 km. Thus, numerical models do not resolve the scales upon which many important processes occur, e.g., convective overturning. When important physical processes are not resolved, the model solution may not achieve the appropriate physical balances. An alternative is to build the balances into an approximate equation set, e.g., by using asymptotic expansions in the small parameters that characterize the ocean such as Rossby number, Froude number, aspect ratio, and stratification ratio. However, small terms in PDE do not necessarily have negligible effects on their solutions, especially if the integration time is long enough. Recent exchanges of ideas between mathematicians and ocean modelers have brought a new global geometric viewpoint to this problem, and a new appreciation of how fluid-dynamical conservation laws, e.g., potential vorticity, arise from the Hamiltonian structure of the underlying inviscid equations of motion, at each level of approximation. The speaker will survey some of these issues and discuss both analytical and computational studies.
Mathematical Challenges in Ocean Modeling
Darryl D. Holm
Los Alamos National Laboratory
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