Hamiltonian approach to the modeling of internal geophysical waves with vorticity

Abstract

)>IJH=?JWe examine a simplied model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the inuence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable at bed, and an upper lid. The 2 media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, determine the equations of motion of the linearized system and calculate the dispersion relation. Finally, limiting cases are examined to recover irrotational and single medium systems as well as an innite 2 media system.

Surface waves over currents and uneven bottom

“…We mention only the important works of Benjamin and Bridges (1997a,b), Craig and Groves (2000), Craig et al (2005b). Wavecurrent interactions in a two-layer system have been investigated by Constantin and Ivanov (2015), Constantin et al (2016), Compelli andIvanov (2015, 2017), Compelli (2016) and others. Therefore a system with both surface and internal waves over uneven bottom is of particular interest for future studies.…”

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

“…for all values of x and t. The physical reasoning for this is the absence of wave motion at infinite depth y → −∞. Euler's equations (5), (6) in terms of the introduced variables are [8]…”

“…where χ and χ 1 are the interface stream functions. Furthermore, it could be shown [5] that χ = χ 1 , and so the Bernoulli equation becomes…”

“…and introducing a non-canonical Poisson bracket as [34] The canonical Hamiltonian form can be achieved under the transformation (cf. [5,6,34])…”

Hamiltonian model for coupled surface and internal waves in the presence of currents

“…For some general facts concerning the description of waves interacting with currents we refer to the following reviews and monographs [10,49,37,52] and the references therein. The present study draws from previous single medium irrotational [54], [3], [46], [47], [48] and rotational [9], [11], [10], [12], [50], [17], [53], [42] studies as well as from studies of two-media systems such as [1], [2], [22], [21], [18], [19], [16], [15], [4], [5], [6], [7], [20], [27], [28], [29], [41], [44], [45].…”