On surface water waves and tsunami propagation

In dieser Arbeit werden die reibungslosen Bewegungsgleichungen für wasser Wellen mit physikalischer Motivation eingeführt. Es folgt ein Studium der Eigenschaften dieser Gleichungen, die durch anwendung asymptotischer Näherungen zur Korteweg-de Vries Gleichung führen. Schließlich wird die Korteweg-de Vries Gleichung hinsichtlich ihrer Anwendung im Bereich der Tsunami Modellierung untersucht.This work introduces the inviscid governing equations for water waves from a physically motivated standpoint, in as accessible a manner as possible. From there, certain asymptotic regimes are explored, leading to the Korteweg-de Vries equation. Elaborations are made on applications to tsunami modeling, while taking care to point out shortcomings in the analytical approach as well as unresolved difficulties in reconciling the intriguing nature of water with mathematics

Deterministic wave forecasting with the Zakharov equation

Six months embargo applies.This study investigates deterministic wave forecasting from the perspective of the Zakharov equation. Forecasts based on linear dispersion, weakly nonlinear amplitude dispersion, and the Zakharov equation are compared for reference ocean surfaces generated from JONSWAP and Pierson-Moskowitz spectra. This approach allows for the role of nonlinearity to be isolated, and demonstrates the success of simple frequency corrections in forecasting wave-fields up to moderate steepness. The role of second-order bound waves is investigated by means of analytical formulae, and their impact on forecast accuracy is illustrated for a range of forecast times

Instability of waves in deep water — A discrete Hamiltonian approach

The stability of waves in deep water has classically been approached via linear stability analysis, with various model equations, such as the nonlinear Schrödinger equation, serving as points of departure. Some of the most well-studied instabilities involve the interaction of four waves – so called Type I instabilities – or five waves – Type II instabilities. A unified description of four and five wave interaction can be provided by the reduced Hamiltonian derived by Krasitskii (1994). Exploiting additional conservation laws, the discretised Hamiltonian may be used to shed light on these four and five wave instabilities without restrictions on spectral bandwidth. We derive equivalent autonomous, planar dynamical systems which allow for straightforward insight into the emergence of instability and the long time dynamics. They also yield new steady-state solutions, as well as discrete breathers associated with heteroclinic orbits in the phase space