Exact Pollard-like internal water waves

Abstract

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates above the thermocline. The solution is provided in Lagrangian coordinates. In the process we derive the appropriate dispersion relation for the internal water waves in a stable stratification and discuss the particles paths. An analysis of the dispersion relation for the constructed model identifies one mode of the internal water waves.

Stratified periodic water waves with singular density gradients

“…Recently also other exact and explicit solutions of stratified flows in different geophysical regimes have been found, cf. [12,16,17,[29][30][31]36].…”

Nonhydrostatic Pollard-like internal geophysical waves

“…In this new model the transition from the oscillation of the thermocline to the still deep water is realised by means of a current flowing beneath the thermocline. Therefore, the flow characteristics are quite different to those presented in [35] and result in a new additional second slow wave mode being a nonlinear phenomenon.…”

“…The important aspect of this solution is that it includes the effects of the Earth's rotation. The author presented an exact internal water wave solution in [35] for a model assuming that the water is still beneath the thermocline. Consequently, we develop a more realistic scenario, which requires the imposition of transitional layers.…”

“…Pollard modified Gerstner's solution and derived a new extended solution applicable at higher latitudes to rotating flows [38]. In this new solution the planetary vorticity affects the waves and causes a slight cross-wave tilt to the wave orbital motion [14,35,38]. The particles paths are still described as closed circles in the fixed reference framework [14,38].…”

“…We derive a dispersion relation for the nonhydrostatic model and by a suitable non-dimensional change of variables we transform it to a polynomial equation of degree six. The degree of the resulting polynomial is dependent on the complexity of the model, therefore introducing additional layers beneath the thermocline we increase the degree of the polynomial -the polynomial derived for a model with one layer beneath the thermocline is of order four, which is the case presented in [35]. An analysis of the polynomial identifies four roots, which are equivalent to two modes of the internal water wave in the dimensional terms; one is a standard internal gravity wave modified slightly by the rotation of Earth and one is a slow mode describing almost inertial circles made by a particle of water.…”