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The influence of fast waves and fluctuations on the evolution of three slow solutions of the Boussinesq equations

Wednesday 4th December 2013 - 14:45 to 15:30
INI Seminar Room 1
Co-authors: Jared P. Whitehead (Brigham Young University), Terry Haut (Los Alamos National Laboratory)

We present results from a study of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We examine three known slow limits of the rotating and stratified Boussinesq equations: strongly stratified flow ($Fr \rightarrow 0, Ro \approx O(1)$), strongly rotating flow ($Ro \rightarrow 0, Fr \approx O(1)$) and Quasi-Geostrophy ($Ro \rightarrow 0, Fr \rightarrow 0, Fr/Ro = f/N$ finite).

In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution into a component that is projected onto the null space of the fast operator and everything else. We use this decomposition to find evolution equations for the flow (and corresponding energy) on and off the slow manifold.

Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

At the end I will discuss how greater understanding of flow/fast dynamics could impact emerging numerical algorithms designed for future computer architectures.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons