Long-wave transverse instability of interfacial gravity–capillary solitary waves in a two-layer potential flow in deep water

Abstract

Interfacial gravity-capillary plane solitary waves, driven by the gravitational force in the presence of interfacial tension in a two-layer deep-water potential flow, bifurcate in the form of wavepackets with a non-zero carrier wavenumber at which the phase speed is minimized. A stability property for the interfacial gravity-capillary plane solitary waves is presented within the framework of the full Euler equations: according to a linear stability analysis based on the perturbation method, such waves are unstable under weak and long-wave disturbances in the transverse direction to the dominant wave propagation. An instability criterion is verified that the total mechanical energy of the solitary waves is a decreasing function of the solitary wavespeed, owing to the fact that the speed of the bifurcating solitary wavepackets is less than the minimum of the phase speed. This result is consistent with an earlier study on the transverse instability of the longitudinally stable interfacial gravity-capillary solitary waves from the Benjamin model equation for weakly nonlinear long interfacial elevations (Kim and Akylas, J Fluid Mech 557:237-256, 2006). The analysis is also applicable to other interfacial gravity-capillary solitary waves that may bifurcate below the minimum of the phase speed, regardless of any restrictions on fluid depths in two-layer potential flows.

Long-wave transverse instability of weakly nonlinear gravity–capillary solitary waves

“…As has also been shown in other physical settings where gravity-capillary solitary waves bifurcate below the minimum of the phase speed, there exists a positive initial instability growth rate in the long-wave limit of transversely perturbed waves to the dominant solitary-wave propagation if the total mechanical energy of solitary waves is a decreasing function of the solitary-wave speed [2][3][4][5][6][7][8]. The long-wave transverse instability of gravity-capillary solitary waves is important because it gives a dynamic generation mechanism of so-called gravity-capillary lumps [8][9][10], which denote three-dimensional (3D) fully localized solitary waves arising on the surface of a fluid flow or on the interface between two immiscible fluid-flow layers [8,[11][12][13][14].…”

“…This explains why some qualitative behaviors of the dynamic properties, such as the long-wave transverse instability of plane gravity-capillary solitary waves, between the Euler equations and other associated weakly nonlinear model equations may be similar even though there are some apparent quantitative discrepancies between them (see[9,10]). As a remark, comparing the expression of R for weakly nonlinear solitary waves, of which amplitudes are O( ), with the one from the full Euler equations[1,7], the horizontal momentum of weakly nonlinear solitary waves, not necessarily of the NLS solitary wavepackets, is estimated by +Ḡ 0 {ηḠ 0 {ζ }} ηḠ 0 {ζ }dξ + O( 3 ). (3.1)…”

On weakly nonlinear gravity–capillary solitary waves

“…In the same physical setting, it was shown that plane gravity-capillary solitary waves are unstable under long-wave disturbances in the transverse direction to the dominant solitary-wave propagation (Kim & Akylas 2007), using a perturbation method that is similar to the one used in the earlier studies by Kataoka & Tsutahara (2004a, 2004b and by Kataoka (2006Kataoka ( , 2008. A similar result in the weakly nonlinear long-wave model equation for interfacial gravity-capillary solitary waves was provided by Kim & Akylas (2006), and the generalized result for interfacial gravity-capillary solitary waves by Kim (2009). The commonly verified transverse instability criterion is consistent with an earlier result by Bridges (2001) for general water-wave problems.…”