Nonlinear stability of two-layer shallow water flows with a free surface

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability

A consistent reduction of the two-layer shallow-water equations to an accurate one-layer spreading model

The gravity-driven spreading of one fluid in contact with another fluid is of key importance to a range of topics. To describe these phenomena, the two-layer shallow-water equations is commonly employed. When one layer is significantly deeper than the other, it is common to approximate the system with the much simpler one-layer shallow water equations. So far, it has been assumed that this approximation is invalid near shocks, and one has applied additional front conditions for the shock speed. In this paper, we prove mathematically that an effective one-layer model can be derived from the two-layer equations that correctly captures the behaviour of shocks and contact discontinuities without any additional closure relations. The proof yields a novel formulation of an effective one-layer shallow water model. The result shows that simplification to an effective one-layer model is well justified mathematically and can be made without additional knowledge of the shock behaviour. The shock speed in the proposed model is consistent with empirical models and identical to the front conditions that have been found theoretically by e.g. von K\'arm\'an and by Benjamin. This suggests that the breakdown of the shallow-water equations in the vicinity of shocks is less severe than previously thought. We further investigate the applicability of the shallow water framework to shocks by studying shocks in one-dimensional lock-exchange/lock-release. We derive expressions for the Froude number that are in good agreement with the widely employed expression by Benjamin. We then solve the equations numerically to illustrate how quickly the proposed model converges to solutions of the full two-layer shallow-water equations. We also compare numerical results using our model with results from dam break experiments. Predictions from the one-layer model are found to be in good agreement with experiments.Comment: 23 pages, 17 figure

The multi-layer shallow water model with free surface: treatment of the open boundaries

Ce travail de thèse, mené en collaboration entre le SHOM et l'Université de Toulouse, s'inscrit dans un contexte d'amélioration du traitement des conditions limites ouvertes, pour le modèle de Saint-Venant multi-couches à surface libre. L'une des principales difficultés rencontrée dans cette démarche concerne la détermination des modes associés aux surfaces interne liquide/liquide: les modes baroclines. Les travaux de cette thèse s'articulent autour de deux axes principaux : Le premier traite l'analyse des éléments propres de l'opérateur différentiel, associé au modèle général. Cela permet d'assurer des conditions d'hyperbolicité et de caractère bien-posé du système d'équations. Cet axe est divisé en deux chapitres. L'analyse du modèle bi-couche est menée dans le premier chapitre : les calculs sont exacts et il y est prouvé que la différence avec le modèle à une couche est importante. Le modèle à n couches, avec n≥3, est étudié dans le second chapitre : la difficulté principale pour l'analyse de ces équations est le nombre de paramètres, ce qui nécessite de supposer des hypothèses. Un nouveau modèle multi-couches conservatif est introduit et analysé. Le second axe traite le traitement opérationnel des conditions limites ouvertes. Le caractère bien-posé du problème initial aux limites est démontré, sous certaines conditions. Ensuite, les conditions limites à prescrire sont clairement explicitées pour un domaine général et un domaine particulier : un rectangle. La comparaison des erreurs, des modèles à une, deux et quatre couches, est menée avec deux cas tests : la propagation d'une onde de gravité et d'un vortex barotrope.This PhD dissertation, conducted as a collaboration between the SHOM and the University of Toulouse, deals with improving the treatment of open boundary conditions, for the multi-layer shallow water model with free surface. One of the main difficulties with such an objective is the determination of the modes associated to the internal surfaces liquid/liquid: the baroclinic modes. The work of this thesis focusses on two axes: The first one concerns the eigenstructure of the differential operator, associated to the general model. This allows to insure conditions of hyperbolicity and local wellposedness of the system of equations. This axis is divided in two chapters. The analysis of the two-layer model is performed in the first chapter: the calculus are exact and it is proved the gap is important compared with the single-layer model. The model with n layers, n≥3, is studied in the second chapter: the main difficulty of these equations is the number of parameters, which obliges to concede assumptions. A new conservative multi-layer model is introduced and analyzed. The second axis deals with practical treatment of the open boundary conditions. The conditional local well-posedness of the initial-boundary value problem is proved. Afterwards, the boundary conditions are clarified for a general domain and a particular one: a rectangle. Comparison of the errors is performed between the single-layer model and the two and four-layer models, with two test case: the propagation of a gravity wave and a barotropic vortex