On the intermediate long wave propagation for internal waves in the presence of currents

Abstract

A model for the wave motion of an internal wave in the presence of current in the case of intermediate long wave approximation is studied. The lower layer is considerably deeper, with a higher density than the upper layer. The flat surface approximation is assumed. The fluids are incompressible and inviscid. The model equations are obtained from the Hamiltonian formulation of the dynamics in the presence of a depth-varying current. It is shown that an appropriate scaling leads to the integrable Intermediate Long Wave Equation (ILWE). Two limits of the ILWE leading to the integrable Benjamin-Ono and KdV equations are presented as well.

Hamiltonian description of internal ocean waves with Coriolis force

“…For the two choices the equations (61) are of course different. The obtained equation (61) extends the integrable Intermediate Long Wave (ILW) equation [36,42,5], there are more details for example in [3,21].…”

“…Since the operator D has an eigenvalue k, thus we shall also keep in mind that hD = O(1) and h 1 D = O(δ). This regime is studied in more details in [21] when of course f = 0. The Hamiltonian, expanded over δ is derived in [21] and could be obtained again with an asymptotic expansion of the general Hamiltonian from [9].…”

“…This regime is studied in more details in [21] when of course f = 0. The Hamiltonian, expanded over δ is derived in [21] and could be obtained again with an asymptotic expansion of the general Hamiltonian from [9]. The Hamiltonian with terms up to order δ 3 is [21]…”

Azimuthal equatorial flows in spherical coordinates with discontinuous stratification

“…While the azimuthal velocity and the pressure function in the lower layer Ω bottom are given explicitly, the pressure in the top layer Ω top and the interface and surface defining functions, h and k, respectively, appear (in general) in implicit form, cf. equations ( 15) and (17). However, for some specific choice of the density function and of the azimuthal velocity, cf.…”

On the modelling of short and intermediate water waves

“…The short-wave effects usually compete with the capillarity effects and then resonances can be observed -these have been studied quite a lot, see for example [6,9,13,17,19,23,25,30]. For the intermediate long waves or for waves on deep water the so-called Benjamin-Ono (BO) [1,7,29] and the Intermediate Long Wave Equation (ILWE) [14,18,21] are derived for the internal waves below a flat surface, which leads to some simplifications and these models are in fact integrable.…”