Global diffeomorphism of the Lagrangian flow-map for Equatorially-trapped internal water waves

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.…”

“…When these conditions are met, we can say that the exact solution (29) corresponds to the stationary trochoidal waves on a fluid surface maintained by the external pressure (31). If µ and k are known, we can find wave amplitude A from the second relation of system (32) and p 0 from the first one. The elevation of the free surface is defined by Y = A cos (ka − µ t); hence, for positive values of p 2 , the pressure changes in phase with the profile, and for negative p 2 in antiphase.…”

“…The case p 2 = 0 corresponds to a Gerstner wave with constant pressure on the profile. Solving the quadratic equation (32) for µ leads us directly to the dispersion relation…”

Exact solution and instability for geophysical waves at arbitrary latitude

“…We mention that the above exact solutions fail to capture, for example, strong depth variations of the flows. A mixture of analytical and topological methods were applied in [46,47,48] to prove that the Lagrangian flow-map describing a number of three-dimensional geophysical exact solutions is a global diffeomorphism, thereby rigorously establishing that the flow description is globally dynamically possible. Very recently, [11,12,13,36] investigate nonlinear three-dimensional models for flows with sufficient freedom and find, in the Eulerian framework, exact solutions that capture the most relevant geophysical features (see also the survey [37]).…”

“…By (46), the solution to the non-autonomous system (45) is obtained by multiplying the rotation matrix P (t) with the solution to the autonomous system (47)…”