Gerstner waves in the presence of mean currents and rotation

Abstract

We present a Lagrangian analysis of nonlinear surface waves propagating zonally on a zonal current in the presence of the Earth’s rotation that shows the existence of two modes of wave motion. The first, ‘fast’ mode, one with wavelengths commonly found for wind waves and swell in the ocean, represents the wave–current interaction counterpart of the rotationally modified Gerstner waves found first by Pollard (J. Geophys. Res., vol. 75, 1970, pp. 5895–5898) that quite closely resemble Stokes waves. The second, slower, mode has a period nearly equal to the inertial period and has a small vertical scale such that very long, e.g. $O(10^{4}~\text{km})$ wavelength, waves have velocities etc. that decay exponentially from the free surface over a scale of $O(10~\text{m})$ that is proportional to the strength of the mean current. In both cases, the particle trajectories are closed in a frame of reference moving with the mean current, with particle motions in the second mode describing inertial circles. Given that the linear analysis of the governing Eulerian equations only captures the fast mode, the slow mode is a fundamentally nonlinear phenomenon in which very small free surface deflections are manifestations of an energetic current.