Applied PDE Seminar: Christian Kharif, Weakly nonlinear capillary waves on a shear current: Stability  and bifurcation

Submitted by Anastassiya Semenova on

Speaker: Christian Kharif, Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE/Ecole Centrale Mediterranee)

Date: April 18, 2024

Title:  Weakly nonlinear capillary waves on a shear current: Stability  and bifurcation

Abstract: The effect of wind blowing above the air-sea interface is twofold: (i) it generates waves that take place within the first millimeters to meters, and  (ii) it generates a shear flow in the uppermost layer of the water. Consequently, these waves propagate in the presence of vorticity. These small  scales contribute to the sea surface stress and consequently participate in the air-sea momentum transfer. An accurate description of the surface stress is important in modelling and forecasting ocean wave dynamics. In addition,  the knowledge of their properties is crucial for satellite remote sensing applications.  A nonlinear Schrodinger equation for pure capillary waves propagating at the free surface of a vertically sheared current has been derived to study the stability and bifurcation of capillary Stokes waves on arbitrary depth.  A linear stability analysis of weakly nonlinear capillary Stokes waves on arbitrary depth has shown that (i) the growth rate of modulational instability  increases as the vorticity decreases whatever the dispersive parameter kh where k is the carrier wavenumber and h the depth (ii) the growth rate is  significantly amplified for shallow water depths and (iii) the instability band- width widens as the vorticity decreases. A particular attention has been paid  to damping due to viscosity and forcing effects on modulational instability. In addition, a linear stability analysis to transverse perturbations in deep water has been carried out, demonstrating that the dominant modulational instability is two-dimensional whatever the vorticity. Near the minimum of linear phase velocity in deep water, we have shown that generalized capillary solitary waves bifurcate from linear capillary Stokes waves when the vorticity is positive.

Location: Residents of Lewis Hall can attend talks in the Wan Conference Room (Lewis 208) at 11:30am. The talk will also be streamed live via Zoom.