Generation of oblique interfacial waves due to resonant interaction with surface gravity waves in shallow water

Ocean in deep waters and coastal areas is stratified due to vertical gradient of density. Due to nearly distinct interface between the layers of constant density, a two-layer system is a commonly used configuration to model ocean waters. In such models, various mechanisms can lead to generation of surface and interfacial waves. Furthermore, this system admits nonlinear interactions between surface waves and internal waves. As surface waves approach coastal areas, they become long relative to water depth and through nonlinear interactions can induce long interfacial waves over fluidized seabed. This phenomenon will be studied theoretically in the present paper. The fluid is composed of two layers of density stratified, incompressible, inviscid and immiscible fluids. The depth of the top and bottom layers are assumed to be shallow relative to the typical surface wave and interfacial wave length respectively. The waves in this system are weakly nonlinear and weakly dispersive and can be described by Boussinesq-type equations. First, Boussinesq-type equations describing the displacements of the surface and interface and the depth-integrated horizontal velocities in the two layers are derived for mildly varying bathymetry. Secondly, the nonlinear resonant interactions among surface and interfacial modes are analyzed via a second order multiple scales analysis in time. Consequently, coupled transient evolution equations of wave amplitudes are derived. The results of inviscid theory indicates generation of a pair of oblique subharmonic interfacial waves due to energy gain from surface wave. In a parametric study, the influences of the angle of propagation of interfacial waves with respect to surface wave, lower layer viscosity, surface wave frequency, density difference between fluid layers, thickness of the fluid layers, and surface wave amplitude are studied.