Mean Flow Velocities and Mass Transport for Equatorially-Trapped Water Waves with an Underlying Current

Abstract

In this paper we present an analysis of the mean flow velocities, and related mass transport, which are induced by certain equatorially-trapped water waves. In particular, we examine a recently-derived exact and explicit solution to the geophysical governing equations in the β-plane approximation at the equator which incorporates a constant underlying current.Mathematics Subject Classification. 76B15, 74G05, 86A05.

On three-dimensional Gerstner-like equatorial water waves

“…However, as the form of (2.2) is explicit in terms of Lagrangian variables, it transpires that the solution (2.2) is quite amenable to an analysis of its mean flow velocities and related mass transport [19,59]. The presence of a constant underlying current term is a significant complicating factor for the analysis of (2.2), undertaken in [46], and this is what we describe briefly.…”

“…where we denote by r = R(q − ct; s * , z 0 ) the functional relationship induced by relation (5.2) between the otherwise independent variables r and q, as follows from the implicit function theorem. In [46], it is shown that the mean Eulerian velocity is given by the relation That the mean Eulerian flow is westward for an adverse current is not surprising, because in the absence of the current, the mean Eulerian flow is westward in any case (cf. [59]).…”

“…[20,24,41,42], and further relevant field data in [43,44]). We note that while the underlying current in the exact solution (2.2) assumes an apparently simple form in the Lagrangian setting, it leads to significant complexifications, both mathematically and physically, in the resulting fluid motion [45,46], as we outline in a discussion on the mean flow properties in §5. This is perhaps not surprising because the nonlinear passage from Lagrangian to Eulerian coordinates is a delicate issue in general, cf.…”

“…where we denote by r = R(q − ct; s * , z 0 ) the functional relationship induced by relation (5.2) between the otherwise independent variables r and q, as follows from the implicit function theorem. In [46], it is shown that the mean Eulerian velocity is given by the relation…”

Geophysical internal equatorial waves of extreme form

“…A significant motivation for this interest has been the fact that such waves have an exact description in a variety of settings in the form of equatorially trapped Gerstner like solutions, see [23] for a review of three-dimensional Gerstner like solutions in the equatorial region. Equatorially trapped surface waves have been investigated in [7,13,14] while surface waves in the presence of underlying currents have been treated in [22,25]. Internal geophysical flows are investigated in [10,4,33,32,36,27], while instabilities in such flows are addressed in [12,26,18,24].…”