Rotational effects in water waves

Abstract: Abstract: This thesis consists of four papers related to various aspects of water waves with vorticity. Paper I: Symmetry of steady periodic gravity water waves with vorticity. We prove that steady periodic two-dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid. Paper II: Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity. We present existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional water-wave problem in the presence of surface tension and vorticity. The established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the time-like variable. A centre-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods. Paper III: A Hamiltonian formulation of water waves with constant vorticity. We show that the governing equations for two-dimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the surface elevation. Paper IV: Hamiltonian long-wave approximations of water waves with constant vorticity Starting with the Hamiltonian formulation in Paper III we derive several long-wave approximations. These approximate models are also Hamiltonian and the connection between the symplectic structures is described by a simple transformation theory.