Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity

Abstract

We derive a fourth-order nonlinear evolution equation (NLEE) for narrow-banded Stokes wave in finite depth in the presence of surface tension and a mean flow with constant vorticity. The two-dimensional capillary-gravity wave motion on the surface of finite depth is considered here. The analysis is limited to one horizontal dimension, parallel to the direction of wave propagation, in order to take advantage of a formulation using potential flow theory. This evolution equation is then employed to examine the effect of vorticity on the Benjamin-Feir instability (BFI) of the Stokes capillary-gravity wave trains. It is found that the vorticity modifies significantly the modulational instability and in the case of finite depth the combined effect of vorticity and capillarity is to enhance the instability growth rate influenced by capillarity when the vorticity is negative. The key point is that the present fourth-order analysis exhibits considerable deviations in the stability properties from the third-order analysis and gives better results consistent with the exact numerical results. Furthermore, the influence of linear shear current on Peregrine breather (PB) is studied.

Bragg scattering of nonlinear surface waves by sinusoidal sandbars

“…Considering the significant influence of term on stability of wave modulation (Thomas et al. 2012; Francius & Kharif 2017; Dhar & Kirby 2023), wave kinematic and dynamic properties (Pizzo et al. 2023), and the geometry of the fluid particle trajectories (Wang, Guan & Vanden-Broeck 2020), we included this term in the present study.…”

“…The first-order problem is composed of waves propagating both forwards and backwards, as Mei (1985) demonstrated, and an additional non-propagative mode, B, which represents the wave-induced mean flow and is generated in the process of nonlinear wave modulations in slow spatial and time scales (Thomas et al 2012). Considering the significant influence of term B on stability of wave modulation (Thomas et al 2012;Francius & Kharif 2017;Dhar & Kirby 2023), wave kinematic and dynamic properties (Pizzo et al 2023), and the geometry of the fluid particle trajectories (Wang, Guan & Vanden-Broeck 2020), we included this term in the present study.…”

Nonlinear Self-Modulation of Gravity-Capillary Waves on Shear Currents in Finite Depth

“…The new fourth-order outcome shows a remarkable modification in the instability behaviour from the third-order one in deep water. This paper is an extension of the paper by Dhar and Kirby [9] to include the effect of depth uniform current on modulational instability properties.…”

“…Hsu et al [16] then elaborated that paper to include capillarity, and studied both the effects of vorticity and capillarity on modulational instability. Later, Dhar and Kirby [9] derived a fourth-order nonlinear evolution equation (NLEE) for GCWs on finite depth with constant vorticity. From the studies on vorticity modified NLSEs of preceding authors, it is revealed that they considered only the effect of vorticity.…”

An investigation of the flow structure beneath solitary waves with constant vorticity on a conducting fluid under normal electric fields

“…Furthermore, the existence of stagnation points can generate flows where the pressure at the bottom boundary is out of phase with the free surface [11,12]. Many other authors have investigated waves with constant vorticity [11,13,14,15,16,17,18,19,20,21,22]. The readers are referred to the work by Nachbin and Ribeiro-Jr [23] for a review on numerical strategies adopted for capturing the flow beneath waves with constant vorticity.…”