Wave–current interaction on a free surface

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