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zoom link : https://stockholmuniversity.zoom.us/j/68666214865
The sea-state variability at scales of order 1-100 km is mostly due to the current field U(x,t). It is well-known that variable currents induce a refraction of surface gravity waves. Assuming slowing varying currents, this phenomenon is described in a Hamiltonian framework by the geometric ray theory. The wave-mean flow interaction breaks the energy conservation however the wave action, defined as the ratio of the wave energy and the intrinsic angular frequency, is a conserved quantity. It can be regarded as a phase space distribution A(x,k,t), where k is the wavenumber, then its conservation law is nothing but the Liouville theorem.
In their paper, Villas-Bôas and Young modeled the surface current by a random process with stationary and spatially homogeneous statistics. As the ratio of U and the group velocity is a small parameter, they performed a multiple scale analysis and averaged the action balance equation over an ensemble of currents. They obtained for the mean action an advection-diffusion equation in the phase space (x,k). Precisely, there is advection in x at the group speed and diffusion in k with a diffusion tensor which depends on the spectrum of the current. The key point is that there is no radial diffusion. Further assuming that the statistics of U is isotropic, the authors showed that only its rotational part makes a contribution to the rotational diffusion.
Note: in eqs. 4.1 and 4.2 read \bar{A} instead of A.