Exact and Explicit Internal Equatorial Water Waves with Underlying Currents

Geophysical Equatorial Edge Wave with Underlying Currents in the f-Plane Approximation

“…From (20) and (21), we understand that the crest (and the troughs) lines are orthogonal to the shoreline and parallel each other. We can also see that the crest and the trough lines are monotonic curves.…”

“…Recently, Gerstner-type solutions have been derived and adapted to model a number of different physical and geophysical fluid dynamics [14][15][16]. These solutions were shown to incorporate constant underlying currents [17][18][19][20][21], with an analysis of the effects of the mathematical modification on the resulting physical flow properties. Constantin [22] presented an explicit edge wave solution.…”

On three-dimensional Gerstner-like equatorial water waves

“…Note that in Gerstner's wave the amplitude of wave oscillations decreases as we descend in the fluid, which is the reverse of the present setting whereby the amplitude decreases exponentially as we ascend above the thermocline. Akin to the surface waves described in [17,39], the introduction of a depth-invariant current was successfully achieved for the internal wave model described in [48].…”

Nonhydrostatic Pollard-like internal geophysical waves

“…It results in number of papers deriving solutions for various geophysical waves, e.g. 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.…”

“…The motion in the layer L(t) is a small perturbation of the free surface caused primarily by the wind. The nonhydrostatic model was used previously in [7,33] to describe internal water waves represented by solutions symmetric about the equator in the β-plane approximation. The aim of this paper is to generalise [7] for an arbitrary latitude in the f -plane.…”