Resonant effects in weakly nonlinear geophysical fluid dynamics

Abstract

Many observed phenomena in geophysical systems, such as quasigeostrophy, and turbulence effects in rotating fluids, can be attributed to the resonances that emerge from multiple scale analysis. In this thesis the multiple scale method of asymptotic expansion is used to study resonant wave interactions in the context of quasigeostrophic geophysical systems, including and extending triad interactions. The one and two layer rotating shallow water equations and the equations for uniformly stratified fluid under the Boussinesq assumption are studied in detail, we evaluate their asymptotic expansions, and analyse their behaviour. Of particular interest is the expansion for the two layer equations, where we investigate a resonance not previously considered in the literature. We formulate general theory concerning the behaviour of the splitting of the dynamics into the fast and slow parts of the systems. We find that all layered shallow water type equations cannot have any interaction between a set of fast waves that produces a slow wave, regardless of whether they are resonant or non-resonant. In the stratified case we find that this is not true, although these interactions are constrained to a slow timescale. Building on the resonant expansion, we then reformulate the expansions to allow the inclusion of near resonant interactions. We detail a new formulation of the near resonances as the representation of higher order interactions that are sufficiently fast acting to be included at the triad order of interaction. We then demonstrate the effectiveness of this near resonant expansion by direct numerical simulation and evaluation of the rotating shallow water equations. We derive qualitatively different behaviour, found analytically in the near resonant expansion of the stratified equations, showing that many higher order interactions not accessible in the layered equations are possible in the stratified case. Finally we consider the expansions in the wavepacket framework, with the introduction of multiple spatial scales. We find that consideration of the magnitude of the difference between the group velocities of component wavepackets in a quartet interaction is sufficient to derive the higher order behaviour previously found by other methods in the literature. It then follows from this that the near resonant expansion can contain many types of interaction that are not possible between wavepackets if only exact resonances are considered.