Wave–current interaction on a free surface

Abstract

The classical water wave equations (CWWEs) comprise two boundary conditions for the two-dimensional flow on the free surface of a bulk three-dimensional (3D) incompressible potential flow in the volume bounded by the free surface, which itself moves under the restoring force of gravity. One of these two boundary conditions provides the kinematic definition of the vertical velocity of the surface elevation. The other boundary condition is the dynamic Bernoulli law that governs the evaluation of the bulk velocity potential on the free surface. The present paper applies these two boundary conditions as constraints in the action integral for Hamilton's variational principle, along with a non-hydrostatic pressure constraint that imposes incompressible flow on the free surface. The stationary variations in Hamilton's principle then yield closed dynamical equations of free surface flow whose divergence-free velocity admits nonzero vorticity and whose nonhydrostatic pressure matches the pressure of the 3D bulk flow when evaluated on the free surface. A minimal coupling approach is proposed to model the mutual interactions of the waves and currents. The dynamical effects of horizontal buoyancy gradients are also considered in this context. For any combination of these model variables, the resulting system of variational equations admits a Lie–Poisson Hamiltonian formulation. Finally, stochastic versions of these model equations are derived by assuming that the material loop for their Kelvin circulation theorem in each case follows stochastic Lagrangian histories in a Stratonovich sense.