Benjamin-Ono model of an internal wave under a flat surface

Abstract

A two-layer fluid system separated by a pycnocline in the form of an internal wave is considered. The lower layer is infinitely deep, with a higher density than the upper layer which is bounded above by a flat surface. The fluids are incompressible and inviscid. A Hamiltonian formulation for the fluid dynamics is presented and it is shown that an appropriate scaling leads to the integrable Benjamin-Ono equation.

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

“…The plus sign is for the right-running waves and the minus sign is for the left-running waves. These speeds coincide with the speeds in the case of infinitely deep lower layer [13], indeed, the h-dependence comes only from the term T h which is of order δ.…”

“…The variable ξ := ρφ − ρ 1 φ 1 (13) introduced in [3,4] plays the role of a generalised momentum. We assume that ξ ∈ S(R) for all t. The (non-canonical) equations of motion are…”

“…The Lagrangian and the Hamiltonian approaches for surface and internal waves have been developed further in many papers, from which here we mention only [3,4,5,23,24,25,44,45,47]. Currents with linear profiles and constant vorticity could be included in the Hamiltonian formulation of the wave-current dynamics, recent studies are published in [10,11,12,13,18,19,20,31,33].…”

Hamiltonian description of internal ocean waves with Coriolis force

“…The BO equation [2,45] like KdV and ILWE is a classical example of an integrable equation solvable by the inverse scattering method [38]. For more information about these models one can refer to [24,3,10] and the references therein.…”

“…We have presented a consistent approach for the derivation of the simplified long-wave and intermediate long-wave models (7.5)-(7.8) based on the extension of the Hamiltonian approach in the irrotational case [20,19]. A similar Hamiltonian approach has been successful for treating internal waves with shear currents [9,10,11,12,13,32] and hopefully could work in situations where other effects affect the long -wave or the intermediate long-wave propagation. For surface waves the derivation is analogous, only the Hamiltonian H 0 is the corresponding Hamiltonian for the propagation of surface waves.…”