Hamiltonian splitting for wave packet propagation in a mean flow

Abstract

Interactions between waves and mean flows are an important object of study in ocean dynamics, and theory on this topic is well developed. In this thesis, a simplified two-dimensional model of wave packets propagating through a slowly-varying random mean flow is studied. Using theory about ray-tracing and linear waves, this model can be described as a Hamiltonian system. This type of systems is characterised by conservation of the Hamiltonian function, symplectic structure and possible chaotic behaviour. In current research projects that study interactions between wave packets and a mean flow, rather simple and standard numerical methods are used to compute wave packet trajectories, which are not optimal for this type of systems. In this thesis a second-order numerical method is constructed by using a splitting model that preserves the symplectic structure of the studied system and it is shown that this method is consistent. Moreover, by using the splitting method similar qualitative behaviour of wave packet interactions with the mean flow is found as in previous studies. Further, also some other remarkable behaviour is reported and it is discovered that the studied model exhibits chaotic behaviour in the form of sensitive dependence on initial conditions. In contrast to previous used numerical methods, it is shown that the splitting method ensures that the relative error in Hamiltonian is small due to the symplecticness of the method. However, it is found that the time step in some cases needs to be infinitesimally small, in order to control peaks in the error of the Hamiltonian. This requires further research. Future studies may also look at the behaviour on longer time intervals and implementation of a higher-order splitting method by using theory about composition methods. Since many physical processes can be described by a Hamiltonian system, it would be useful in future research to investigate whether a splitting method is applicable instead of a non-symplectic general numerical solver.