Internal waves and the Poincaré equation

Abstract

Internal waves are waves propagating through a body of fluid. They are of importance in the field of geophysical fluid mechanics since their enabling mechanisms (density stratification and rotation) are predominant in Earth's oceans and atmosphere. This thesis consists of several chapters. The first chapter is a general introduction to fluid mechanics and provides a derivation of the Poincaré equation. The two dimensional version of this equation is the spatial analogue of the well know wave equation. Nonetheless it features some extraordinary properties. Some noteworthy aspects of solutions to the equation are reviewed, most notably the occurrence of 'wave attractors', exhibiting a fractal structure. The second chapter discusses the ill-posedness of the problem of solving the equation in closed domains. An efficient discretisation of the Poincaré equation is presented and a regularisation scheme is proposed which deals with the ill-posed nature of the problem. The third chapter considers the addition of a viscous term to the Poincaré equation, thereby transforming the partial differential equation from a hyperbolic equation to an elliptic one. It is believed that addition of such a viscous term has a regularising effect, a hypothesis which we test by numerical computation using a finite element discretisation. Numerically, and using perturbation techniques, we obtain relationships between the viscosity, the damping and the frequency of the waves. Also, we establish bounds on the stability of the solutions, measuring in some sense the degree of ill-posedness. Using the error bound as a criterion we were able, for modest values of the viscosity, to obtain meaningful solutions to the Poincaré problem. For more realistic, lower values of the viscosity we recommend an addition regulatisation. The final chapter describes laboratory experiments, performed at the 'Coriolis turntable' at LEGI, Grenoble. The aim of the experiments was to study the enhanced mixing of density stratification, resulting from reflections of internal waves at boundaries. The practical importance of results on this mixing mechanism lies in the fact that there is some (as yet) unexplained mixing in the Earth's oceans. By analysis of the data obtained during the experiments we establish that internal waves are forced by Ekman pumping at a corner of the domain. This mechanism is also analytically described. Furthermore we establish the formation of wave attractors by considering visualizations of the harmonic components of the flow in vertical planes. From probes injected into the fluid we deduce a dramatically enhanced mixing of the density stratification. Other sources of mixing being too weak to explain this, we offer the hypothesis that internal waves are the main source of this mixing.