Solitary wave fission of a large disturbance in a viscous fluid conduit

Abstract

This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large, localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications including tsunami and internal ocean wave modelling. This study is performed in the context of the viscous fluid conduit system-the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Due to buoyancy induced nonlinear self-steepening balanced by stress induced interfacial dispersion, the disturbance evolves into a slowly modulated wavetrain and further, into a sequence of solitary waves. An extension of Whitham modulation theory, termed the solitary wave resolution method, is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbance's profile, the number of emergent solitary waves, and their amplitude distribution, quantifying an extension of the well-known soliton resolution conjecture from integrable systems to non-integrable systems that often provide a more accurate modelling of physical systems. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally. The number of observed solitary waves is consistently within 1-2 waves of the prediction, and the amplitude distribution shows remarkable agreement. Universal properties of solitary wave fission in other fluid dynamics problems are identified.

The long-wave potential-vorticity dynamics of coastal fronts

“…This result may be of interest to researchers who wish to obtain dispersion relations for meanders in a western boundary current, as was done for the Gulf Stream in Lee & Cornillon (1996) and for the Kuroshio Extension in Tracey et al (2012). We also note that Pratt & Stern (1986) model the evolution of Gulf Stream meanders starting from a 'top-hat' frontal profile, which in the present model could be treated analytically using the adaptation of El's technique described in Maiden et al (2020). Another application of the ideas discussed here is the topographic arresting of coastal-trapped waves.…”

Soliton gas in integrable dispersive hydrodynamics

“…This problem is important from both the theoretical and experimental points of view. One possibility is to use the spontaneous 'soliton fission' mechanism [125], [81] inspired by the original Zabusky-Kruskal work [134] and employed in the experiments on the creation of a shallow water bidirectional anisotropic soliton gas in [105]. In modulationally unstable media a semiclassical scenario of the transition to integrable soliton turbulence via a chain of topological bifurcations of the finite-gap spectra was theoretically proposed in [38] and to some extent realised in an optics experiment reported in [83].…”

“…This can be seen in Fig. 1 displaying a laboratory realisation of a dense soliton gas in a viscous fluid conduit [79], a versatile fluid dynamics platform enabling high precision experiments on the generation and interaction of solitary waves that exhibit nearly elastic collisions [80], [81]. One can appreciate that, for a dense gas the particle interpretation of individual solitons becomes less transparent and the wave aspect of the collective soliton dynamics comes to the foreground.…”

Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

“…24) is based on a fundamental, generic property: the Whitham modulation equations admit exact reductions to a set of common, much simpler, analytically tractable equations in the limits of vanishing amplitude and vanishing wavenumber, which correspond to the harmonic and soliton DSW edges, respectively. DSW fitting has also been successfully applied to the propagation of a broad localized pulse into a constant state 33–35 …”