Global diffeomorphism of the Lagrangian flow-map defining equatorially trapped water waves

Abstract

The aim of this paper is to prove that a three dimensional Lagrangian flow which defines equatorially trapped water waves is dynamically possible. This is achieved by applying a mixture of analytical and topological methods to prove that the nonlinear exact solution to the geophysical governing equations, derived by Constantin in [5], is a global diffeomorphism from the Lagrangian labelling variables to the fluid domain beneath the free surface.

Geophysical internal equatorial waves of extreme form

“…The dynamic possibility of the flow (18) has been confirmed in the context of surface waves in the work [38], and in the context of internal geophysical waves in [35]. In either regime, as time advances, the mapping (18) remains a diffeomorphism from the parameter space (q, s, r) to the fluid domain when r ≥ 0.…”

Wind generated equatorial Gerstner-type waves

“…This is an important property of the Lagrangian flow description. We note that for both, the two-dimensional Gerstners wave [7,19] and a number of three-dimensional generalizations [32,33,34], a mixture of analytical and topological methods can be applied to prove that the Lagrangian flow-map describing these exact solutions is a global diffeomorphism, with the result that the flow is globally dynamically possible. In Eq.…”

On three-dimensional Gerstner-like equatorial water waves

“…We describe briefly the approach which was used in [49] to establish the global validity of (2.2) in solving (2.12); other geophysical scenarios were addressed in [50,51]. Firstly, from examining its Jacobian matrix, and applying the inverse function theorem, it can be proved that the mapping (2.2) represents a local diffeomorphism from the Lagrangian variables to the fluid domain.…”