The influence of fast waves and fluctuations on the evolution of the dynamics on the slow manifold

The effect of non-slow (typically fast) components of a rotating stratified Boussinesq flow on the dynamics of the slow manifold is quantified using a decomposition that isolates the part of the flow living on the slow manifold. In this system, there are three distinct asymptotic limits with corresponding reduced equations, each defining a slow manifold. All three of these distinct limits, namely rapid rotation, strong stratification, and simultaneous strong stratification and rapid rotation (quasi-geostrophy), are considered. Numerical simulations indicate that, for the geometry considered (triply periodic) and the type of forcing applied, the fluctuations 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, most of the energy transfer is between slow potential energy and slow kinetic energy, but the energetics due to the fluctuations are less clear. It is observed that the energy off the slow manifold in each case equilibrates to a quasi-steady value.