Equatorial water waves with underlying currents in the f-plane approximation

Wind generated equatorial Gerstner-type waves

“…Gerstner waves was achieved by Henry [20]. Kluczek studied the analogous threedimensional flow taking into account a variable meridional current [23]. We present a class of exact solutions in the f -plane approximation generalizing Gerstner waves.…”

“…For waves of the form (29), the traditionally imposed boundary condition is constant pressure on the free surface. Hence, zeroing the multiplier of cosine in expression (30) yields a dispersion wave equation [23]. It may be assumed, however, that under the action of wind, pressure distribution in the form of a harmonic traveling wave is maintained on the free surface:…”

“…The freedom in the sign of the phase speed is a consequence of the f -plane approximation. It is interesting to compare expression (33) with the dispersion relation of equatorial waves with an underlying current [23]. The quantity p 2 2ρAΩ…”

On three-dimensional Gerstner-like equatorial water waves

“…recent work initiated by Constantin & Johnson [24][25][26][27] which is surveyed in this issue in [28]. Secondly, with regard to Gerstner-like (that is, explicit and exact) solutions, we refer to [29][30][31] for a discussion of geophysical edge-wave solutions; we do not discuss the restriction of β-plane solutions to the f -plane, which follows upon setting β = 0, and essentially reduces solutions from being three-dimensional to twodimensional in nature [32] (although an interesting exception are the fully three-dimensional solutions derived in [33][34][35][36] which exist solely in the f -plane setting). Finally, we refer to [11] for a recent extension of Pollard's nonlinear geophysical wave solution [37] which exists at all latitudes, whereby the authors accommodate a depth-invariant current and in the process generate a…”

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.…”