A consistent phase-averaged model of the interactions between surface gravity waves and currents

Abstract

We formulate a model of the two-way interactions between surface gravity waves and ocean currents. The model couples the transport of wave action in the four-dimensional (horizontal) position--wavevector phase space with the Craik--Leibovich system for the currents. Coupling is via the Doppler shift in the dispersion relation governing action transport, and wave pseudomomentum in the Craik--Leibovich system. The velocity in the Doppler shift is a vertical integral of the Lagrangian mean velocity of the currents, with a weight that is consistent with the vertical structure of the pseudomomentum. This consistency ensures conservation of momentum and energy in the coupled wave--current system. The conservation properties of the wave--current model stem from an underlying variational structure. We derive this structure from that of the rotating Euler equations for an incompressible fluid with free surface by introducing a Lagrangian wave--mean decomposition, making simplifying approximations, and Whitham averaging. We apply the wave--current model to the problem of generation of inertial oscillations by surface waves originally considered by Hasselmann.

Figures (2)

• Figure 1: Schematic of the CWCM: the dynamical variables are the action density $\mathcal{N}(\boldsymbol{x},\boldsymbol{k})$ and the current Lagrangian mean velocity $\boldsymbol{u}^\mathrm{{L}}(\boldsymbol{x},z)$. They are coupled through the pseudomomentum $\boldsymbol{\mathsf{p}}(\boldsymbol{x},z)$ and the weighted horizontal velocity $\boldsymbol{\mathcal{U}}(\boldsymbol{x},\kappa)$ that appears in the surface wave dispersion relation.
• Figure 2: Solution of the Hasselmann problem with $\alpha=f$. Upper panel: the dashed sinusoids uL and vL are the scaled Lagrangian velocities $u^\mathrm{{L}}(z,t)/\mathsf{p}_{\star}(z)$ and $v^\mathrm{{L}}(z,t)/\mathsf{p}_{\star}(z)$ and LKE $= |\boldsymbol{u}^\mathrm{{L}}|^2/2 \mathsf{p}_{\star}^2$, is the corresponding scaled Lagrangian kinetic energy. Lower panel: the four kinetic energy densities defined in \ref{['KEz']}, all scaled with $\mathsf{p}_{\star}(z)^2$.