On three-dimensional Gerstner-like equatorial water waves

Abstract

This paper reviews some recent mathematical research activity in the field of nonlinear geophysical water waves. In particular, we survey a number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave-current interactions, in the equatorial region. These solutions are nonlinear, three-dimensional and explicit in terms of Lagrangian variables.This article is part of the theme issue 'Nonlinear water waves'.

Exact Pollard-like internal water waves

“…Furthermore, an instability analysis of Gerstner's solution was presented in [9]. The mathematical importance of the recently derived and analysed Gerstner-like solutions is presented in a form of a review paper in [23,27,29]. For rotating flows in the Pollard solution a wave experiences a very slight cross-wave tilt to the wave orbital motion associated with the planetary vorticity.…”

Exact solution and instability for geophysical waves at arbitrary latitude

“…(s) < 0, s > 0, the necessary condition isA (s) = c 0 − ce 2k[r0−m(s)] m (s) c = c 0 − ce 2k[r0−m(s)] βs + f f c 0 + g < 0,for s > 0 small enough, which is equivalent toc 0 − ce 2k[r0−m(s)] < 0(27)for s > 0 small enough. Therefore, for a given 0 < c 0 < ce 2kr0 ,(27) holds for some s ∈ (0, s 1 ] and accordingly (25) holds for s ∈ (0, s 0 ] with s 0 < s 1 . (b) the solution c = c − < 0 and the uniform current − ĝ f < c 0 < 0.…”

“…Exact solutions play an important role in the study of geophysical flows because many apparently intangible wave motions can be regarded as the perturbations of them, and relevant information about the dynamics of more complex flows can be extracted by controlling the perturbations. The approach pioneered by Gerstner [20] (for modern detailed descriptions see [3,22]) of finding explicit exact solutions for gravity fluid flows within the Lagrangian framework [2], was extended to geophysical flows too (see the survey [27]). Gerstner-like three-dimensional solutions were obtained in the f -plane approximation at an arbitrary latitude in [45] and very recently in [14,17], the internal wave in [38,39], in the β-plane approximation in [5,7,8,23] or in a modified β-plane approximation [25,26]; for other studies, we refer the reader to [24,30,31,32,43,44].…”

Nonhydrostatic Pollard-like internal geophysical waves

“…equatorially-trapped waves [5,6,7,22,25,26], waves in the presence of underlying depth-invariant currents [9,13,20,21,23,33,34,41], and a solution for trapped waves at an arbitrary latitude [16] with an instability analysis of Gerstner-like solutions in [8,29]. The mathematical importance of those solutions is presented in [24,29,31]; cf. [2] for a discussion of the oceanographical relevance of those solutions.…”