Ricardo Barros

• Ph.D. in Applied Mathematics
• Senior Lecturer at Loughborough University

Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustm...

In this paper, we investigate mode-2 solitary waves in a three-layer stratified flow model. Localised travelling wave solutions to both the fully nonlinear problem (Euler equations), and the three-layer Miyata–Choi–Camassa equations are found numerically and compared. Mode-2 solitary waves with speeds slower than the linear mode-1 long-wave speed a...

Oceanic internal waves often have curvilinear fronts and propagate over various currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a background linear shear current. The leading order of this theory leads to the angular adjustment equation - a nonlinear first-order differentia...

In this paper, we investigate mode-2 solitary waves in a three-layer stratified flow model. Travelling wave solutions to both the fully nonlinear problem (Euler equations), and a three-layer extension of the strongly nonlinear two-layer Miyata-Choi-Camassa equations are found numerically. Mode-2 solitary waves with speeds slower than the linear mod...

To describe large amplitude internal solitary waves in a two-layer system, we consider the high-order unidirectional (HOU) model that extends the Korteweg–de Vries equation with high-order nonlinearity and leading-order nonlinear dispersion. While the original HOU model is valid only for weakly nonlinear waves, its coefficients depending on the dep...

We perform the stability analysis for a free surface fluid current modeled as two finite layers of constant vorticity, under the action of gravity and absence of surface tension. In the same spirit as Taylor [“Effect of variation in density on the stability of superposed streams of fluid,” Proc. R. Soc. London, Ser. A 132, 499 (1931)], a geometrica...

We consider a strongly nonlinear long wave model for large amplitude internal waves in a three-layer flow between two rigid boundaries. The model extends the two-layer Miyata-Choi-Camassa (MCC) model (Miyata, is able to describe the propagation of long internal waves of both the first and second baroclinic modes. Solitary-wave solutions of the mode...

We perform the stability analysis for a free surface fluid current modeled as two finite layers of constant vorticity, under the action of gravity and absence of surface tension. In the same spirit as Taylor ["Effect of variation in density on the stability of superposed streams of fluid," Proc. R. Soc. A 132, 499 (1931)], a geometrical approach to...

It is well known since Wu and Wu (in: Proceedings of the 14th symposium on naval hydrodynamics, National Academy Press, Washington, pp 103–125, 1982) that a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, continuously and periodically, a succession of solitary waves propagating ahead of the disturban...

We investigate the stability of thin liquid curtains with respect to two-dimensional perturbations. The dynamics of perturbations with wavelengths exceeding (or comparable to) the curtain's thickness are examined using the lubrication approximation (or a kind of geometric optics). It is shown that, contrary to the previous theoretical results, but...

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with a free surface proposed by Choi and Camassa and Barros et al. . Its solitary-wave solutions were the object of the work by Barros and Gavrilyuk , who proved that such solutions are governed by a Hamiltonian system with two degr...

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)],...

h i g h l i g h t s • A model is derived for 2D strongly nonlinear internal waves in a two-layer system. • The model is regularized to remove ill-posedness due to shear instability. • The dynamics of vorticity described by the regularized model is discussed. • The model is further extended to include the effects of bottom topography. • Some asympto...

We perform the stability analysis for stratified shear flows whose density transition layer is much thinner than, and possibly, displaced with respect to, the velocity shear layer for which Holmboe instability along with the well-known Kelvin-Helmholtz (KH) instability is known to be present. Here, we provide a more complete picture of stability ch...

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...

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 in...

We consider the two-layer shallow water equations in the presence of the top free surface and find explicit conditions for which the system is hyperbolic. It is commonly believed that, analogously to the rigid-lid case, this can only happen for small relative speeds. Using both the root location criteria for a quartic equation and a geometrical app...

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...

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 fa...

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 combi-nation 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 f...

Internal waves propagate in the oceans where denser, colder and saltier deep waters meet warmer, fresher and less dense upper waters. These waves evolve due to the bottom topography, currents, and others, and they may contribute significantly to mixing and transpoting energy in the ocean. Packets of large amplitude internal waves have been observed...