An Explicit Solution for Deep Water Waves With Coriolis Effects

Abstract

We present an explicit solution for the geophysical equatorial deep water waves in the f-plane approximation.

Exact geophysical waves in stratified fluids

“…In the case when the density is constant, it is proved in [25] that the problem where k 4 0 is fixed and c 6 ¼ 0 defines for each t ! 0 a diffeomorphism from AE 0 into an infinite strip (t), which is bounded from above by a periodic graph and unbounded from below.…”

“…The physical relevance of the f-plane approximation for geophysical edge waves near the Equator has been recently discussed in [24]. We mention that the f-plane approximation for the deepwater wave problem possesses an explicit solution: Gerstner's solution to the deep water wave problem [7,8] can be generalized, cf [25], to describe deep water waves in the geophysical context. We emphasize that any solution of the f-plane approximation which has the property that the pressure is constant along the streamlines has to be a vertical translation of the solution described in [25], cf [26].…”

“…We mention that the f-plane approximation for the deepwater wave problem possesses an explicit solution: Gerstner's solution to the deep water wave problem [7,8] can be generalized, cf [25], to describe deep water waves in the geophysical context. We emphasize that any solution of the f-plane approximation which has the property that the pressure is constant along the streamlines has to be a vertical translation of the solution described in [25], cf [26]. Moreover, the solution found in [25] may be adapted to describe geophysical waves travelling over uniform currents, cf [27], where the motion of the fluid particles in dependence of the current's strength and the direction of propagation of the wave is also analysed.…”

“…We emphasize that any solution of the f-plane approximation which has the property that the pressure is constant along the streamlines has to be a vertical translation of the solution described in [25], cf [26]. Moreover, the solution found in [25] may be adapted to describe geophysical waves travelling over uniform currents, cf [27], where the motion of the fluid particles in dependence of the current's strength and the direction of propagation of the wave is also analysed. It was recently shown in [28] that there exists also an explicit solution for the geophysical edge wave problem in the f-plane approximation if the shoreline is parallel to the Equator.…”

“…This solution describes equatorial trapped waves propagating eastward in a stratified inviscid fluid. The aim of this paper is to show that the solutions found in [25] and [28] for the geophysical deep water and edge wave problem, respectively, may be used to describe also waves propagating over a stratified fluid. In the context of deep water waves, cf Section 3, and in Section 4, for the edge wave problem, we prove that the solution found for homogeneous flows describes waves propagating over a stratified fluid if and only if the density is constant on the isobaric surfaces, that is on the surfaces of constant pressure.…”