Collaborative Research: Mathematical Studies of Certain Geophysical Models

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Computer predictions of phenomena on large or global scales, for exampleweather or climate forecasts, need to compromise between accuracy of the predictions and available computing resources. It is therefore a grandscientific challenge to derive global climate models which are reliable and trustworthy. Exploiting certain geophysical balances, such as geostrophic balance (due to earth rotation) or hydrostatic balance (due to the shallowness of the ocean and atmosphere) geophysicists derive reasonable, yet less complex, balanced models. It is therefore essential to justify rigorously the validity of these models, for the relevant spatialand time scales. The focus of the proposed project is on the analytical,statistical and numerical properties of solutions to nonlinear ocean dynamics models and turbulent sub-grid models. The first aspect of this project is to: show existence, uniqueness and continuous dependence on initial data, to some of these reduced geophysical models. In particular, a two-layer zonal jet model, a planetary geostrophic ``thermocline'' model, the lake equations with degenerate varying bottom topography and the two-dimensional primitive equations. This is the first and the most essential step in validating the derivation of these models. In order to justify the long-time behavior of fluid dynamical models, one has to compare the statistical properties of their attracting invariant sets, rather than compare individualsolutions. To do so, it is necessary to focus on models which include some mechanism of dissipation. This project addresses questions related to the asymptotic derivation of these models and the effect of numerical dissipation on their solutions, which include boundary layer analysis. The second aspect of thisproject is to: derive new large-eddy simulation models, the so-called alpha-models, in the context of the two-layers geostrophic zonal jet models. The alpha-models are asserted to reproduce the right energy spectrum for a wide range of large scales. It is proposed to investigate this claim using rigorous analytical tools. It is also proposed to perform computational tests on the newly derived two-layers geostrophic zonal jet alpha-model to verify the above assertion. Furthermore, it is proposed to explore the implementation of the alpha-models approach as sub-grid models. The grand challenge in climate prediction is that the mathematical equations governing the ocean and atmosphere dynamics, are too difficult to study analytically, and still prohibitively expensive computationally. Indeed, it is well established, based on physical grounds and collected experimental data, that atmospheric and oceanic turbulent flows involve a broad spectrum of spatial and time scales. This in turn makes them inaccessible to the most powerful and state-of-the-art computers. However, due to the rotation of the earth and other geophysical situations, such as the shallowness of the oceans and the atmosphere - in the sense that they are much wider than they are deep - geophysicists take advantage of certain geophysical balances to derive simplifiedbalanced models. The first theme of this project is to: establish existence and regularity of solutions to some of these nonlinear reduced models. This isa crucial step in justifying the derivation of these models and theirconsistency with the physical observations for the relevant length and time scales. Furthermore, in global climate prediction one is interested in the long-time statistical features of the climate. The second theme of this project is to develop a systematic approach for deriving and studying new averaged models, in the context of ocean and atmosphere dynamics, which are reliable in reproducing the correct long-term statistics.