Ian Grooms
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SPECIAL MATHEMATICS COLLOQUIUM
Speaker: Ian Grooms Abstract. The accuracy and efficiency of computational atmosphere-ocean models determines our ability to successfully predict the future climate. Equations that describe the dynamics at all scales are known (to a good approximation), but it is impossible to simulate all the active scales of motion, from millimeters to the global scale. Global, coupled atmosphere-ocean models must therefore `parameterize' unresolved scales. But parameterization is particularly difficult when energy flows from the unresolved scales into the resolved ones, or when the unresolved scales are intermittent in space and time. This is the case, for example, in all the ocean models used in the upcoming fifth report of the intergovernmental panel on climate change (IPCC AR5). Multiple-scales asymptotics provides a formalism for investigating the interactions of disparate scales which has been little used in the ocean-dynamics community. The results of two asymptotic analyses of equations appropriate to oceanic fluid dynamics are here summarized: the first applies to low-resolution ocean models in mid latitudes, and the second to medium-resolution models at high latitudes. Our analyses are novel in their use of necessary but not sufficient solvability conditions on the leading-order nonlinear dynamics. As a result, we are able to predict that mixing is primarily along surfaces of constant density, consistent with observations. We also predict that the energy of the unresolved eddies depends nonlocally on the resolved scales, in contrast with the assumptions of most parameterizations. Our multiple-scale asymptotic analyses produce prognostic partial differential equations for the dynamics of the resolved and unresolved scales, which connects them to a nascent multiscale numerical method called `superparameterization' (SP). SP couples the dynamics of a large-scale model to the dynamics on small-scale subdomains embedded in the computational grid of the large-scale model. It has been successfully applied to modeling unresolved cloud processes in atmospheric models and to thermal convection in ocean models, but remains computationally prohibitive. A stochastic formulation of SP is developed here that is far more computationally efficient than conventional SP, and that an be applied to a wide range of problems. Successful results are presented from tests in two difficult settings: a one-dimensional model of wave turbulence with unstable solitons, and the classic, two-layer model of quasigeostrophic (QG) turbulence. Both tests include an inverse cascade of energy from the unresolved scales into the resolved ones. Questions regarding future implementation of stochastic SP into operational ocean models are briefly discussed. |