A Hamiltonian approach to wave-current interactions in two-layer fluids

Abstract

We provide a Hamiltonian formulation for the governing equations describing the two-dimensional nonlinear interaction between coupled surface waves, internal waves, and an underlying current with piecewise constant vorticity, in a two-layered fluid overlying a flat bed. This Hamiltonian structure is a starting point for the derivation of simpler models, which can be obtained systematically by expanding the Hamiltonian in dimensionless parameters. These enable an in-depth study of the coupling between the surface and internal waves, and how both these wave systems interact with the background current.

Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

“…Notice that now the discretized equations (15) and (16) correspond to equations (2) and (3) of Saffman [3]. However, observe that, in contrast with equation (6) of Saffman [3], which implies reflectional symmetry of the wave profile (see the discussion after equations (15), (30), and (31) in [13] on this point), such symmetry is not assumed here. In practice, the absence of reflectional symmetry makes the Jacobian matrix J (to be defined shortly) of the linearized equation for the evolution of perturbations complex valued.…”

Stratified periodic water waves with singular density gradients

“…In ocean flows, however, the density varies strongly in thin layers called pycnoclines which exhibit sharp density gradients, cf., e.g., [21,46,47]. For this reason some of the research [3,4,14,15,21,39,40,42,49] is restricted to so-called layered models which consider the flow as consisting of a finite number of vertical layers each of them having uniform density. These layers are separated by internal waves which are mainly driven by the density difference between the layers (some models also consider surface tension effects).…”

Surface tension effects in the equatorial ocean dynamics

“…[2,6,9,15,21], which is applicable in regions close to the Equator, where the ocean dynamics present distinctive features. There is already a large body of papers investigating equatorial fluid flows, see for instance [7,17,19]. …”

“…The study of wave dynamics in the equatorial region of the Pacific Ocean has attracted the attention of many authors in the recent years, as evidentiated through studies concerning mathematical modeling [2], existence of solutions [3,5,16,18,19,23,24], investigations regarding stability/instability [4,14,17] or even variational formulations [7]. The studies conducted in the mentioned works were stimulated, perhaps, by the peculiar nature of this zone-situated within a band about 2 • latitude from the Equator-manifested through the presence of underlying non-uniform currents, the existence of a diversity of ocean flows and a pronounced stratification.…”