Geometry, Mechanics and Control Seminar
Wave-current interaction on the free surface of a 3D incompressible ideal fluid flow
Speaker: Darryl D Holm (Imperial College London)
Date: Friday, 05 November 2021 – 15:30
Place: Online – us06web.zoom.us/j/7555463367 (ID: 755 546 3367)
Abstract:
The classical water wave equations (CWWE) comprise two boundary conditions for the 2D flow on the free surface of a bulk 3D incompressible potential flow in the volume bounded by the free surface moving 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 which governs the evaluation of the bulk velocity potential evaluated on the free surface. However, two equations are missing from CWWE: (1) a dynamical equation for the vertical velocity; and (2) a formula for the non-hydrostatic pressure on the free surface. The former is required for closure of the system and the latter is required to match the non-hydrostatic pressure in the bulk 3D incompressible potential flow of the fluid in the volume beneath the free surface. Many approximations have been introduced previously to close this system.
This lecture discusses the results of applying the two CWWE boundary conditions as constraints on the action integral for Hamilton’s variational principle, along with a non-hydrostatic pressure constraint which imposes incompressible flow on the free surface. The stationary variations in Hamilton’s principle yield closed dynamical equations of free surface flow whose divergence-free velocity admits non-zero vorticity and whose non-hydrostatic pressure matches the pressure of the 3D bulk flow when evaluated on the free surface.
The dynamical system resulting from this variational principle admits a Lie-Poisson Hamiltonian formulation which displays its geometrical structure. Horizontal gradients of buoyancy are shown to strongly couple the dynamics of waves (vertical motions) and currents (horizontal motions) on the free surface. Stochastic versions of these model equations are also derived, by assuming that the material loop for their Kelvin circulation theorem follows stochastic Lagrangian histories on the free surface in a Stratonovich sense.