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Shallow-Water Equations and Related Topics

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CHAPTER 1 Shallow-Water Equations and Related Topics Didier Bresch UMR 5127 CNRS, LAMA, Universite´ de Savoie, 73376 Le Bourget-du-Lac, France Contents 1. Preface .................................................... 3 2. Introduction ................................................. 4 3. A friction shallow-water system ...................................... 5 3.1. Conservation of potential vorticity .................................. 5 3.2. The inviscid shallow-water equations ................................. 7 3.3. LERAY solutions ........................................... 11 3.3.1. A new mathematical entropy: The BD entropy ........................ 12 3.3.2. Weak solutions with drag terms ................................ 16 3.3.3. Forgetting drag terms – Stability ............................... 19 3.3.4. Bounded domains ....................................... 21 3.4. Strong solutions ............................................ 23 3.5. Other viscous terms in the literature ................................. 23 3.6. Low Froude number limits ...................................... 25 3.6.1. The quasi-geostrophic model ................................. 25 3.6.2. The lake equations ...................................... 34 3.7. An interesting open problem: Open sea boundary conditions .................... 41 3.8. Multi-level and multi-layers models ................................. 43 3.9. Friction shallow-water equations derivation ............................. 44 3.9.1. Formal derivation ....................................... 44 3.10. Oscillating topography ........................................ 58 3.10.1. Oscillating topography inside PDEs ............................. 58 3.10.2. High-oscillations through the domain ............................. 63 3.10.3. Wall laws and oceanography ................................. 76 4. No-slip shallow-water equations ...................................... 78 4.1. Scaling the incompressible Navier–Stokes system .......................... 81 4.2. Asymptotic expansion of solutions .................................. 83 4.3. The shallow-water model ....................................... 85 HANDBOOK OF DIFFERENTIAL EQUATIONS Evolutionary Equations, Volume 5 Edited by C.M. Dafermos and M. Pokorny´ c 2009 Elsevier B.V. All rights reserved DOI: 10.1016/S1874-5717(08)00208-9 1 2 D. Bresch 5. Coupling models — Some comments .................................... 90 5.1. Multi-fluid models .......................................... 90 5.2. Pollutant propagation models ..................................... 94 5.3. Sedimentation ............................................. 97 Acknowledgments ............................................... 98 References ................................................... 98 Abstract These notes are devoted to the study of some problems related to shallow-water equations. In particular, we will review results from derivations of shallow-water systems which depend on a- priori conditions, the compressible-incompressible limit around a constant or inconstant height profile (quasi-geostrophic and lake equations), oscillating topography and boundary effects, variable depth vanishing on the shore, open boundary conditions, coupling models such as pollutants, sedimentation, multiphasic models. Keywords: Shallow water, quasi-geostrophic equations, lake equations, oscillating topography, viscous parametrization, boundary effects, degenerate systems, singular perturbations, coupling models Shallow-water equations and related topics 3 1. Preface These notes originate from lectures given during a Summer-School organized by Professor Eric´ LOMBARDI through the Groupement de Recherche: Equations´ d’Amplitudes et Propriet´ es´ Qualitatives in Roscoff 2004 (FRANCE) and during a post-graduate course given at the Institute of Mathematical Sciences (HONG-KONG) in November 2006 at the invitation of Professor ZHOUPING XIN. Their main objective is to present, at the level of beginners, an introduction to some problems related to shallow-water type equations coming from Navier–Stokes equations with free surface. In the author’s opinion, shallow- water equations are not well understood from the modeling, numerical and mathematical points of view. The reason for this is not because shallow-water approximations are hard to find but rather because it is pretty easy to find a great many different ones. The various mathematical results and properties differ from one another, and the choice of one approximation in preference to another may depend on the problems that are to be treated. As an illustration, we will mention in the introduction two-different kinds of shallow- water system which depend on the boundary conditions assumed on the bottom with regard to Navier–Stokes equations with free surface. As a sequel to this, we will focus on these two approximations. We will present some structural properties, well-posed results, asymptotic analysis and open problems. We will also discuss miscellaneous possible extensions/questions such as: Time dependency of bottom, propagation of pollutants, multiphasic flows. We remark that it is impossible to be exhaustive when talking about shallow-water systems and, for instance, will not discuss water-waves and non-rotational flows even if really important results have been recently obtained on such topics. We also refer readers to [25], [27] and [28] and the book [165] with references cited-therein for non-viscous shallow-water type models. It is a pleasure for me to thank Professors Constantine DAFERMOS and Milan POKORNY´ for their suggestions to write up these notes. I also warmly thank close collaborators and friends namely: L. CHUPIN, T. COLIN, B. DESJARDINS, E. FERNANDEZ´ -NIETO, D. GERARD´ -VARET, J.-M. GHIDAGLIA, M. GISCLON, E. GRENIER, F. GUILLEN-GONZALEZ, R. KLEIN, J. LEMOINE, G. METIVIER´ , V. MILISIC, P. NOBLE, J. SIMON, M.A. RODRIGUEZ-BELLIDO. I have not forgotten the PDEs team in Chambery´ (UMR CNRS 5127/Universite´ de Savoie) namely C. BOURDARIAS, D. BUCUR, D. DUTYKH, S. GERBI, E. OUDET, C. ROBERT and P. VIGNEAUX with whom several mathematical studies are still in progress. Research by the author on the subjects which are presented here was supported by the region´ Rhone-ˆ Alpes in 2004 and by the French Ministry through the ACI: “Analyse mathematique´ des parametrisations´ en oceanographie”´ from 2004 to 2007 (thanks to the LJK and LAMA Secretaries: JUANA,CATHY,CLAUDINE,HEL´ ENE` ,NADINE). I also wish to thank the MOISE INRIA Project in Grenoble managed by Eric´ BLAYO. Dedication. This review paper is dedicated to the memory of Professor Alexander V. KAZHIKHOV, one of the most inventive applied mathematician in compressible fluid mechanics. I also dedicate this chapter to my father Georges BRESCH and to my wife Catherine BRESCH-BAUTHIER. 4 D. Bresch 2. Introduction The shallow-water equations, generally, model free surface flow for a fluid under the influence of gravity in the case where the vertical scale is assumed to be much smaller than the horizontal scale. They can be derived from the depth-averaged incompressible Navier–Stokes equations and express conservation of mass and momentum. The shallow- water equations have applications to a wide range of phenomena other than water waves, e.g. avalanches and atmosphere flow. In the case of free surface flow when the shallow- water approximation is not valid, it is common to model the surface waves using several layers of shallow-water equations coupled via the pressure, see for instance [120], [121] and [159]. Viscous shallow-water type equations have attracted quite a lot of attention in recent years. The aim of this paper is to discuss several problems related to general forms of various shallow-water type equations coming from Navier–Stokes equations with free surface. Note that many other regimes can be investigated including for instance strong dispersive effects, see [7], however these will not be considered here. For the reader’s convenience, let us mention in this introduction two different shallow- water type systems mainly derived from two different bottom boundary condition choices for Navier–Stokes equations with free surface: friction boundary conditions and no-slip boundary conditions. Friction boundary conditions. Let be a two-dimensional space domain that will be defined later. A friction shallow-water system, with flat bottom topography, is written in (see for instance J.-F. GERBEAU and B. PERTHAME [96], F. MARCHE [127]) as follows 8 @t h C div.hv/ D 0; > ? > rh f v <@t .hv/ C div.hv ⊗ v/ D −h − h Fr2 Ro (1) > 2 2 > C div.hD.v// C r.hdiv v/ − We hr1h C D : Re Re where h denotes the height of the free surface, v the vertical average of the horizontal velocity component of the fluid and f a function which depends on the latitude y in order to describe the variability of the Coriolis force. In the second equation, ? denotes π ? D − D a rotation of 2 , namely G . G2; G1/ when G .G1; G2/. The space variable is denoted x D .x; y/ 2 , the gradient operator r D .@x ;@y/ and the Laplacian operator 2 2 1 D @x C @y . The dimensionless numbers Ro, Re, Fr, We denote the Rossby number, the Reynolds number, the Froude number and the Weber number, respectively. Damping terms D coming from friction may also be included. We will discuss this point in Section 3 and also the Coriolis force assumption. In 1871, Saint-Venant wrote in a note, see [76], about a system which describes the flow of a river and corresponds to the inviscid shallow-water model written in System (7) neglecting the Coriolis force.
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