We will present results from studies of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We work in the framework of fast singular limits that derives from the work of Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Ma- jda [1996] and others.
In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution, where u is a vector of all the unknowns, as
u = u α + u ′α where α represents the Ro → 0, F r → 0 or the simultaneous limit of both (QG for
quasi-geostrophy), with
P α u α = u α P α u ′α = 0 ,
and where Pαu represents the projection of the full solution onto the null space of the fast operator. We use this decomposition to find evolution equations for the components of the flow (and the 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.
We will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale is fastest? We will give an example for the quasi-geostrophic limit of the Boussinesq equations.
At the end we will briefly discuss how understanding the role of oscillations has allowed us to develop convergent algorithms for parallel-in-time methods.
Beth A. Wingate - University of Exeter
Jared Whitehead - Brigham Young University
Terry Haut - Los Alamos National Laboratory