Seminar: A consistent model of the interactions between surface gravity waves and currents

Abstract

The interactions between surface gravity waves and currents dominate sea-surface dynamics. These are two-way interactions: currents affect wave propagation through advection and refraction, and waves affect currents including through the Stokes-Coriolis and vortex forces. Yet much of our understanding of surface wave–current interactions relies on simplified models that prescribe either the currents or the wave field. Fully coupled models are used for operational purposes, but they are heuristic and do not conserve energy or momentum. As a result, basic questions, e.g. about the energetics of the interactions, remain unanswered. 

In this talk, I will formulate a new model of the two-way interactions between surface waves and currents that conserves energy, momentum and circulation. The model couples the transport equation for wave action in position–wavevector phase space with the Craik–Leibovich system for the currents. The key to its conservation properties is consistency between the form of the wave pseudomomentum (or Stokes velocity), which controls the effect of waves on currents, and the form of the current velocity, which enters the Doppler-shift term in the action equation. This consistency emerges naturally when the model is obtained from a variational formulation of the Euler equations with a free surface. The derivation and conservation laws highlight the primacy of the Lagrangian mean velocity over the Eulerian mean velocity as representation of the currents. I will discuss applications and extensions of the model.  

Joint work with W. R. Young (Scripps).