The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

In this paper we derive an approximate multi-dimensional model of dispersive waves propagating in a two-layer fluid with free surface. This model is a "two-layer" generalization of the Green-Naghdi model. Our derivation is based on Hamilton's principle. From the Lagrangian for the full-water problem we obtain an approximate Lagrangian with accuracy O(ε 2 ), where ε is the small parameter representing the ratio of a typical vertical scale to a typical horizontal scale. This approach allows us to derive governing equations in a compact and symmetric form. Important properties of the model are revealed. In particular, we introduce the notion of generalized vorticity and derive analogues of integrals of motion, such as Bernoulli integrals, which are well known in ideal Fluid Mechanics. Conservation laws for the total momentum and total energy are also obtained.

In this paper we study the dispersive model derived in Part I, for the description of long wave propagation in two-layer flows with free surface. As in the case of the full water-wave problem, this model reproduces the resonance between short waves and long waves. The resulting wave is a generalized solitary wave, characterized by ripples in the far field in addition to the solitary pulse. In this work we focus on particular members of this family resulting from vanishing ripples. These are called embedded solitary waves and they correspond to true homoclinic orbits. Two wave regimes, characterized by elevation or depression of the interface between the layers, are presented. A critical depth ratio separates these two regimes. It is shown how this relates to a change of the global properties for the potential of the Hamiltonian system derived for traveling waves. In oceanic conditions, solitary waves are presented and their broadening is observed as the wave speed increases. We have observed that, for such waves to exist, their speed cannot exceed a certain limit value depending on the density ratio and thickness of each fluid. Finally, other sets of parameters were considered for which multihumped solitons exist, showing the richness and complexity of the Hamiltonian system considered here.

No abstract

A full set of conservation laws for the two-layer shallow water equations is presented for the one-dimensional case. We prove that all the conservation laws are linear combination of the equations for the conservation of mass and velocity (in each layer), total momentum and total energy.This result generalizes that of Montgomery and Moodie that found the same conserved quantities by restricting their search to the multinomials expressions in the layer variables. Though the question of whether or not there are only a finite number of these quantities is left as an open question by the authors. Our work puts an end to this: in fact, no more conservation laws are admitted for the two-layer shallow water equations. The key mathematical ingredient of the method proposed leading to the result is the Frobenius problem. Moreover, we present a full set of conservation laws for the classical one-dimensional shallow water model with topography, by using the same techniques.

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin-Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.

We revisit the stability analysis for three classical configurations of multiple fluid layers proposed by Goldstein ["On the stability of superposed streams of fluids of different densities," Proc. R. Soc. A. 132, 524 (1931)], Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. A 132, 499 (1931)], and Holmboe ["On the behaviour of symmetric waves in stratified shear layers," Geophys. Publ. 24, 67 (1962)] as simple prototypes to understand stability characteristics of stratified shear flows with sharp density transitions. When such flows are confined in a finite domain, it is shown that a large shear across the layers that is often considered a source of instability plays a stabilizing role. Presented are simple analytical criteria for stability of these low Richardson number flows. C 2014 AIP Publishing LLC. [http://dx.

No abstract