A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

Abstract

Please cite only the published version using the reference above.See http://opus.bath.ac.uk/ for usage policies.Please scroll down to view the document. A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear By Anakewit Boonkasame and Paul A. MilewskiThe Miyata-Choi-Camassa (MCC) system of equations describing long internal nonhydrostatic and nonlinear waves at the interface between two layers of inviscid fluids of different densities bounded by top and bottom walls is mathematically ill-posed despite the fact that physically stable internal waves are observed matching closely those of MCC. A regularization to the MCC equations that yields a computationally simple well-posed system for time-dependent evolution is proposed here. The regularization is accomplished by keeping the full hyperbolic part of MCC and exchanging spatial and temporal derivatives in one of the linearized dispersive terms. Solitary waves of MCC over a wide range of parameters are used as a benchmark to check the accuracy of the model. Our model includes the possibility of a background shear, and we show that, contrary to the no shear case, solitary waves can cross the midlevel between the top and the bottom walls and may have different polarity from the case with no background shear. Time-dependent solutions of the regularization stable model are presented, including interactions of its solitary waves, and classical and modified Korteweg-de Vries equations for small amplitude waves with the inclusion of background shear are derived. Throughout the paper, the Boussinesq approximation is taken, although the results can be extended to the non-Boussinesq case.

Long ring waves in a stratified fluid over a shear flow

“…Two-layer example: dispersion relation and wavefronts 3.1. Two-layer model In order to clarify the general theory developed in the previous section and to illustrate the different effect of a shear flow on the wavefronts of surface and internal ring waves, here we discuss a simple piecewise-constant setting, frequently used in theoretical and laboratory studies of long internal and surface waves (figure 3; see, for example, Long 1955;Lee & Beardsley 1974;Miyata 1985;Weidman & Zakhem 1988;Choi & Camassa 1999;Ramirez et al 2002;Choi 2006;Grue 2006;Voronovich et al 2006;Chumakova et al 2009;Alias, Grimshaw & Khusnutdinova 2014;Arkhipov, Safarova & Khabakhpashev 2014;Boonkasame & Milewski 2014 and references therein). In these theoretical and laboratory studies, the model is often chosen to yield explicit formulae, but is regarded as a reasonable abstraction for a background flow with smooth density and shear profiles across the interface, in the long-wave approximation.…”

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

“…Many attempts have been made to "regularize" the Green-Naghdi model, that is proposing new models with formally the same precision as the original model, but which are not subject to high-frequency Kelvin-Helmholtz instabilities, even without surface tension [11][12][13][14][15]. The strategies adopted in these works rely on change of unknowns and/or Benjamin-Bona-Mahony type tricks; see [16, section 5.2] for a thorough presentation of such methods in the free-surface setting.…”

“…We show in Section 2.1 that our models-both under the form (3)-(4) and (7)-also admit a Hamiltonian structure, so that our models could be derived from Hamilton's principle on an approximate Lagrangian. Such an approach has been worked out in [30][31][32] (see also [33]) in the one-layer case, in [34] in the regime of small-amplitude long waves or small steepness, in [35] in the two-layer case with free surface, and lacks for existing regularized Green-Naghdi systems in the literature [11][12][13]15]. We then enumerate the group of symmetries of the system (Section 2.2) that originates from the full Euler system (see [36]), and deduce the related conserved quantities (Section 2.3).…”

Numerical study of interfacial solitary waves propagating under an elastic sheet

“…The waves are similar to the pure gravity interfacial solitary waves under the ‘rigid lid’ approximation [6,29], because the amplitude of the interface is relatively large compared with the displacement of the free surface. Figure 7 presents bifurcation diagrams and typical profiles of the solitary waves with a set of parameters $H=\frac{1}{3}$, R =0.9, E b =0.5.…”

“…The most conspicuous feature is a broadening of the wave, namely the midsections of the interface and the free surface develop plateaus, when the Froude number approaches a limiting value. The plateaus can become infinitely long; therefore, the flow in the far field and the flow in the middle can be referred to as parallel conjugate flows [30,29]. From this point of view, we can calculate the limiting value of F by solving the following algebraic equation: $false(8+Rfalse)F^6-12false(1+Hfalse)F^4+2false(3-Rfalse){false(1+Hfalse)}^2{F}^2-false(1-Rfalse){false(1+Hfalse)}^3=0.$ This relation was first given by Dias & Il'ichev, and readers are referred to [16] for the detailed derivation (they derived (3.11) in the absence of an elastic cover, but their result remains valid for our problem, because the elastic sheet does not affect the result when the free surface and the interface are flat).…”