Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances (eBook)
400 Seiten
Elsevier Science (Verlag)
978-0-08-048946-9 (ISBN)
? Incorporates analytical, numerical and experimental approaches in the geophysical fluid dynamics context
? Combination of essentials in GFD, of the description of analytical, numerical and experimental methods (tutorial part), and new results obtained by these methods (original part)
? Provides the link between GFD and mechanics (averaging method, the method of normal forms), GFD and nonlinear physics (shocks, solitons, modons, anomalous transport, periodic nonlinear waves)
The rotating shallow water (RSW) model is of wide use as a conceptual tool in geophysical fluid dynamics (GFD), because, in spite of its simplicity, it contains all essential ingredients of atmosphere and ocean dynamics at the synoptic scale, especially in its two- (or multi-) layer version. The book describes recent advances in understanding (in the framework of RSW and related models) of some fundamental GFD problems, such as existence of the slow manifold, dynamical splitting of fast (inertia-gravity waves) and slow (vortices, Rossby waves) motions, nonlinear geostrophic adjustment and wave emission, the role of essentially nonlinear wave phenomena. The specificity of the book is that analytical, numerical, and experimental approaches are presented together and complement each other. Special attention is paid on explaining the methodology, e.g. multiple time-scale asymptotic expansions, averaging and removal of resonances, in what concerns theory, high-resolution finite-volume schemes, in what concerns numerical simulations, and turntable experiments with stratified fluids, in what concerns laboratory simulations. A general introduction into GFD is given at the beginning to introduce the problematics for non-specialists. At the same time, recent new results on nonlinear geostrophic adjustment, nonlinear waves, and equatorial dynamics, including some exact results on the existence of the slow manifold, wave breaking, and nonlinear wave solutions are presented for the first time in a systematic manner.* Incorporates analytical, numerical and experimental approaches in the geophysical fluid dynamics context* Combination of essentials in GFD, of the description of analytical, numerical and experimental methods (tutorial part), and new results obtained by these methods (original part)* Provides the link between GFD and mechanics (averaging method, the method of normal forms); GFD and nonlinear physics (shocks, solitons, modons, anomalous transport, periodic nonlinear waves)
Cover 1
Copyright page 5
Preface 6
Contents 8
Chapter 1. Introduction: Fundamentals of Rotating Shallow Water Model in the Geophysical Fluid Dynamics Perspective 10
1. Introduction 11
2. Derivation of the model 12
3. General properties of the RSW model 17
4. An overview of the two-layer RSW model 31
5. Wave-vortex interactions 37
6. RSW model on the equatorial beta-plane 42
7. Concluding remarks 50
Acknowledgements 52
References 52
Chapter 2. Asymptotic Methods with Applications to the Fast-Slow Splitting and the Geostrophic Adjustment 56
1. Introduction 57
2. Nonlinear geostrophic adjustment in the unbounded domain. One-layer RSW 64
3. Nonlinear geostrophic adjustment in the unbounded domain. Two-layer RSW 82
4. Nonlinear geostrophic adjustment in the presence of a lateral boundary 89
5. Nonlinear geostrophic adjustment in the equatorial region 115
6. Summary and discussion 125
Acknowledgements 127
References 127
Chapter 3. The Method of Normal Forms and Fast-Slow Splitting 130
1. Introduction 131
2. Slow manifold in the spatial variables on the f-plane 133
3. Slow manifold in the spectral variables on the f-plane 141
4. Poincaré normal forms 146
5. Skew-gradient normal forms for gradient systems 165
6. Hamiltonian normal forms 172
7. Normal form of the Poisson bracket for one-dimensional fluid 184
8. Conclusion 192
References 193
Chapter 4. Efficient Numerical Finite Volume Schemes for Shallow Water Models 198
1. A few notions on hyperbolic systems 200
2. Finite volume schemes for conservative systems 206
3. Finite volumes for systems with source terms 223
4. Second-order well-balanced schemes 240
5. Two-dimensional finite volumes on a rectangular grid 249
6. Numerical tests 255
References 263
Chapter 5. Nonlinear Wave Phenomena in Rotating Shallow Water with Applications to Geostrophic Adjustment 266
1. Introduction 267
2. Nonlinear wave phenomena and geostrophic adjustment in 1dRSW 268
3. Nonlinear wave phenomena and geostrophic adjustment in 2-layer 1dRSW 290
4. Nonlinear wave phenomena and geostrophic adjustment in the equatorial waveguide 306
5. Summary and conclusions 326
Acknowledgements 328
References 328
Chapter 6. Experimental Reality of Geostrophic Adjustment 332
1. The Holy Graal of rotating shallow-water flows 332
2. Potential vorticity measurements: a new challenge 348
3. Simple case studies of geostrophic adjustment 361
4. What do we learn from laboratory experiments? 384
Acknowledgements 385
References 386
List of Contributors 390
Author Index 392
Subject Index 398
Chapter 1 Introduction: Fundamentals of Rotating Shallow Water Model in the Geophysical Fluid Dynamics Perspective
V. Zeitlin
LMD, Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France
1 Introduction
The oceans and the atmosphere are thin, with respect to the Earth radius, stratified fluid layers on the rotating sphere with variable bottom topography under the influence of gravity. The effects of sphericity (the small Earth's non-sphericity is usually neglected) depend on the horizontal scale of the motions of interest. They are essential for the planetary-scale motions, like tides, but for description of the so-called synoptic-scale motions with characteristic scales of the order of thousand kilometers in the atmosphere, and hundreds kilometers in the ocean, the tangent plane approximation is largely sufficient (Gill, 1982; Holton, 1979; Pedlosky, 1982). The effects of sphericity may be then introduced via the variable Coriolis parameter. Although the vertical motions may be of considerable intensity in the oceans and the atmosphere, like e.g. deep convection, on the synoptic scale average the typical horizontal velocities are at least two orders of magnitude larger than the vertical ones. Moreover, on average at these scales the vertical accelerations are negligible, and the motion within a good accuracy is in the hydrostatic balance (cf. e.g. Holton, 1979). These observations explain why the rotating shallow water model (RSW in what follows) is widely used in geophysical fluid dynamics (GFD). Taken literally, the model describes the motion of quasi two-dimensional incompressible thin fluid layer of constant density with a free surface in hydrostatic balance on the rotating plane with constant or variable Coriolis parameter, and with the centrifugal acceleration absorbed into (effective) gravity. The key ingredients of the geophysical fluid dynamics at synoptic scales, such as rotation, quasi-bidimensionality, and hydrostatic balance, are, thus, incorporated into the model. The stratification is rudimentary: the model describes a single isopycnal surface, the free surface of the fluid layer. Stratification may be taken into account in a more realistic manner by superimposing several shallow-water layers of different densities. Bottom topography is easily introduced. Thus, the model is plausible already at such purely heuristic level. We will demonstrate below how it may be obtained from the full, so called primitive GFD equations, and remind its basic properties.
The literature on RSW and its GFD applications is vast. Needless to say that beyond GFD the shallow water model has numerous and practically important applications in hydraulics. The goal of the present chapter is not to review all of the shallow-water literature (which is an impossible task already in the GFD context), but to introduce the concepts and notions necessary for understanding of the subsequent chapters and, thus, to make the whole volume self-contained. The accents in what follows are correspondingly made on the problems addressed in the following chapters, as for example the problem of splitting of balanced and non-balanced motions, which is one of the main topics of the book. Part of material of this chapter is standard and may be found in the cited textbooks, it is given below for completeness.
2 Derivation of the model
The standard derivation of the (non-rotating) shallow-water equations from the Euler equations for incompressible constant-density fluid with a free surface (Whitham, 1974) is based on the expansion in vertical to horizontal scale ratio. Rotation may be easily incorporated and we will not repeat this derivation here. Instead, we will show how the RSW and multi-layer RSW equations can be derived from the full equations of the stratified rotating fluid (“primitive equations”).
2.1 Primitive equations model of GFD
The hydrostatic primitive equations can be written in standard notation as follows:
(2.1)
(2.2)
(2.3)
Here ? is density, p is pressure, the subscript “h” denotes the horizontal part of the fields, or operators which depend on and t. The velocity field is three-dimensional: , as well as the operator nabla . These equations are written on the tangent plane to a sphere rotating with angular velocity ?. The coordinate in the meridional direction is y (northward in the geophysical context), the coordinate in the zonal direction (eastward) is x, and z is the vertical coordinate in the direction normal to the tangent plane, i.e. opposite to the effective gravity , which includes a (small) correction due to the centrifugal acceleration (see e.g. Holton, 1979; the asterisk will be omitted from now on). is the normal unit vector in z-direction. The Coriolis parameter f entering the horizontal momentum equation is equal to , if the f-plane approximation is used, and to , if the ?-plane approximation is used. Here ? is the contact point latitude which is considered as fixed or variable in the first and second approximations, respectively. Equations (2.1)–(2.3) are non-dissipative. They express momentum and mass conservation of the fluid. Molecular (or turbulent) viscosity and diffusivity may be easily introduced in the r.h.s. of the momentum and mass equations. It should be noted, however that the Reynolds numbers for free (i.e. far from boundary layers) atmospheric and oceanic motions of synoptic scale are extremely high (Pedlosky, 1982; Holton, 1979). This is why, being interested in what follows mostly in the intrinsic dynamics of the system, and not in the forced-dissipative one, we will almost never use the dissipative form of equations.
We will work for simplicity with a rigid lid and flat bottom boundary conditions for equations (2.1)–(2.3):
(2.4)
which are of frequent use in GFD. Other upper boundary conditions, like free surface boundary, or prescribed form of the pressure (geopotential) function at a given vertical level (cf. e.g. Holton, 1979) may be used as well. In (2.3) the continuity equation
(2.5)
is replaced by the incompressibility condition and mass conservation by each fluid parcel. Such (Boussinesq) approximation is totally justified in the oceanic context taking into account the negligible compressibility of water. Further simplification may be made in this case, using the fact that in the ocean
(2.6)
Here is a constant, is a background oceanic stratification, and represents the spatio-temporal density variations. Then, the variable part of the density may be neglected in the horizontal momentum equations, and we get
(2.7)
where the geopotential is introduced. Remarkably, the same equations, up to the change of sign in the hydrostatic equation, arise in the atmospheric context if the so-called pseudo-height Z, which was first introduced by Hoskins and Bretherton (1972), is used as the vertical coordinate:
(2.8)
Pseudo-height is a modified pressure coordinate (see Holton, 1979, for pressure coordinates in the atmosphere). The constants are given by , and . In these definitions, and are (constant) surface pressure and density, and are the specific heats of air at constant volume and constant pressure respectively. The variable ? has a different physical meaning in the atmosphere: it becomes up to a sign the so-called potential temperature ? which is directly related to the entropy density of air parcels (cf. Holton, 1979). The pseudo-height Z and the physical height z are related as follows: .
2.2 Vertical averaging of the primitive equations and multi-layer RSW models
The derivation of the RSW model by vertical averaging was proposed in the classical works of Jeffreys (1925) in the linear approximation, and Obukhov (1949) in the nonlinear case. We follow below the method of Obukhov by generalizing it to the multi-layer case. To lose no generality we will consider a general case of equations (2.1), together with the continuity equation (2.5), and the hydrostatic balance equation (2.2). Equations (2.1), (2.5) may be rewritten as evolution equations of the horizontal momentum density:
(2.9)
(2.10)
As a preliminary step we integrate these equations in the vertical direction between a pair of material surfaces . By definition
(2.11)
By using (2.11) and the formula
(2.12)
which is valid for any function F, we get from (2.9)
(2.13)
Analogously, from (2.10)...
| Erscheint lt. Verlag | 3.4.2007 |
|---|---|
| Sprache | englisch |
| Themenwelt | Sachbuch/Ratgeber |
| Naturwissenschaften ► Geowissenschaften ► Geophysik | |
| Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik | |
| Technik | |
| ISBN-10 | 0-08-048946-X / 008048946X |
| ISBN-13 | 978-0-08-048946-9 / 9780080489469 |
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