Free-surface effects induced by internal solitons forced by shearing currents

Free-surface effects induced by internal solitary waves (ISWs) propagating through a background shear current are theoretically investigated in a shallow water framework. Following the Hamiltonian approach, we implement a mathematical formulation valid for a free surface, two-layer stratified system in the presence of mobile layers. Associated to the undisturbed condition characterized by the background fluid at rest, solitons with both positive and negative polarities are considered. To reproduce the typical oceanic conditions, we focus on theoretical predictions for a density ratio set to 0.99. Although under Boussinesq conditions a rigid-lid at the top of the theoretical domain is considered a good approximation, our analysis shows that main ISWs features may change when the fluid system, forced by shear currents, is modeled with a free surface. Signs assumed by three dimensionless quantities, i.e., the deformation parameter, the background velocity, and the undisturbed amplitude, allowed us to uniquely predict how the undisturbed solitons modify their interfacial profiles and change their celerity, in response to the background forcing. Our results show that waves celerity predicted by the rigid-lid model can be lower than the one resulting from the Hamiltonian formulation, although this never occurs in the absence of mobile layers. Theoretical predictions reveal that shearing current can induce deviations from the standard ISWs free-surface profile. We discuss the role of nonhydrostatic pressures in inducing free-surface short disturbances, in the form of undulated or multihumped profiles. The typical phase-opposition between interfacial and surface displacements can be lost as the free-surface displacements assume multihumped configurations.