Fully nonlinear periodic internal waves in a two-fluid system of finite depth

Abstract

Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.

Exact closed-form solutions of a fully nonlinear asymptotic two-fluid model

“…The paper is concluded with Section 6 offering a discussion of properties of the obtained solutions, in particular, in comparison with semi-numerical solutions of similar kinds obtained in the earlier works [1,10,11], and an overview of related open problems and future research directions.…”

“…The latter was shown to admit, for specific parameter relationships, kink-and solitary wave-type traveling wave solutions, which were constructed numerically. In [10], periodic traveling wave solutions were obtained as numerical solutions of the same reduced ODE. No exact solutions of the Choi-Camassa model have been reported to date.…”

Propagation regimes of interfacial solitary waves in a three-layer fluid

“…These simple models are able to mirror the basic properties of the actual internal wave systems. The efficiency of the two-layer model -the simplest system basically representing the key properties of IWs -has been established in numerous analytical and numerical studies as well as by means of in situ observations and laboratory experiments (Funakoshi, 1985;Funakoshi and Oikawa, 1986;Mirie and Pennell, 1989;Camassa, 1996, 1999;Pullin and Grimshaw, 1998;Craig et al, 2004;Guyenne, 2006;Zahibo et al, 2007;Camassa et al, 2010).…”

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

“…As a result, stable solitary waves of elevation type are also possible when the lower fluid layer is thicker than the upper fluid layer in the presence of sufficiently strong (yet stable in the sense of long-wave KH) background shear. Background shear was included in a study of periodic internal waves of MCC in [18]. This paper is organized as follows.…”