Fundamentals of Ocean Climate Models

by Griffies, Stephen M.

• ISBN13: 9780691118925
• ISBN10: 0691118922
• Format: Hardcover
• Copyright: 2004-08-16
• Publisher: Princeton Univ Pr

Summary

This book sets forth the physical, mathematical, and numerical foundations of computer models used to understand and predict the global ocean climate system. Aimed at students and researchers of ocean and climate science who seek to understand the physical content of ocean model equations and numerical methods for their solution, it is largely general in formulation and employs modern mathematical techniques. It also highlights certain areas of cutting-edge research. Stephen Griffies presents material that spans a broad spectrum of issues critical for modern ocean climate models. Topics are organized into parts consisting of related chapters, with each part largely self-contained. Early chapters focus on the basic equations arising from classical mechanics and thermodynamics used to rationalize ocean fluid dynamics. These equations are then cast into a form appropriate for numerical models of finite grid resolution. Basic discretization methods are described for commonly used classes of ocean climate models. The book proceeds to focus on the parameterization of phenomena occurring at scales unresolved by the ocean model, which represents a large part of modern oceanographic research. The final part provides a tutorial on the tensor methods that are used throughout the book, in a general and elegant fashion, to formulate the equations.

Table of Contents

FOREWORD	xiii
PREFACE	xv
ACKNOWLEDGMENTS	xxv
ABOUT THE COVER	xxvii
LIST OF SYMBOLS	xxix
Chapter 1. OCEAN CLIMATE MODELS	1	(4)
1.1 Ocean models as tools for ocean science	1	(1)
1.2 Ocean climate models	2	(1)
1.3 Challenges of climate change	3	(2)
PART 1. FUNDAMENTAL OCEAN EQUATIONS	5	(148)
Chapter 2. BASICS OF OCEAN FLUID MECHANICS	7	(17)
2.1 Some fundamental ocean processes	7	(2)
2.2 The continuum hypothesis	9	(1)
2.3 Kinematics of fluid motion	10	(6)
2.4 Kinematical and dynamical approximations	16	(4)
2.5 Averaging over scales and realizations	20	(1)
2.6 Numerical discretization	21	(1)
2.7 Chapter summary	22	(2)
Chapter 3. KINEMATICS	24	(18)
3.1 Introduction	24	(1)
3.2 Mathematical preliminaries	24	(5)
3.3 The divergence theorem and budget analyses	29	(2)
3.4 Volume and mass conserving kinematics	31	(9)
3.5 Chapter summary	40	(2)
Chapter 4. DYNAMICS	42	(45)
4.1 Introduction	42	(1)
4.2 Motion on a rotating sphere	43	(4)
4.3 Principles of continuum dynamics	47	(4)
4.4 Dynamics of fluid parcels	51	(5)
4.5 Hydrostatic pressure	56	(2)
4.6 Dynamics of hydrostatic fluid columns	58	(4)
4.7 Fluid motion in a rapidly rotating system	62	(6)
4.8 Vertical stratification	68	(2)
4.9 Vorticity and potential vorticity	70	(5)
4.10 Particle dynamics on a rotating sphere	75	(5)
4.11 Symmetry and conservation laws	80	(3)
4.12 Chapter summary	83	(4)
Chapter 5. THERMO-HYDRODYNAMICS	87	(34)
5.1 General types of ocean tracers	87	(4)
5.2 Basic equilibrium thermodynamics	91	(4)
5.3 Energy of a fluid parcel	95	(10)
5.4 Global mechanical energy balance	105	(5)
5.5 Basic non-equilibrium thermodynamics	110	(1)
5.6 Thermodynamical tracers	111	(3)
5.7 Ocean density	114	(4)
5.8 Chapter summary	118	(3)
Chapter 6. GENERALIZED VERTICAL COORDINATES	121	(32)
6.1 Introduction	121	(1)
6.2 Concerning the choice of vertical coordinate	122	(6)
6.3 Generalized surfaces	128	(2)
6.4 Local orthonormal coordinates	130	(1)
6.5 Mathematics of generalized vertical coordinates	131	(5)
6.6 Metric tensors	136	(2)
6.7 The dia-surface velocity component	138	(3)
6.8 Conservation of mass and volume for parcels	141	(2)
6.9 Kinematic boundary conditions	143	(2)
6.10 Primitive equations	145	(2)
6.11 Transformation of SGS tracer flux components	147	(2)
6.12 Chapter summary	149	(4)
PART 2. AVERAGED DESCRIPTIONS	153	(62)
Chapter 7. CONCERNING UNRESOLVED PHYSICS	155	(14)
7.1 Represented dynamics and parameterized physics	155	(2)
7.2 Lateral (neutral) and vertical processes	157	(2)
7.3 Basic mechanisms for dianeutral transport	159	(2)
7.4 Dianeutral transport in models	161	(5)
7.5 Numerically induced spurious dianeutral transport	166	(1)
7.6 Chapter summary	167	(2)
Chapter 8. EULERIAN AVERAGED EQUATIONS	169	(20)
8.1 Introduction	169	(2)
8.2 The nonhydrostatic shallow ocean equations	171	(2)
8.3 Averaged kinematics	173	(1)
8.4 Averaged kinematics over finite domains	174	(5)
8.5 Averaged tracer	179	(3)
8.6 Averaged momentum budget	182	(1)
8.7 Summary of the Eulerian averaged equations	183	(2)
8.8 Mapping to ocean model variables	185	(2)
8.9 Chapter summary	187	(2)
Chapter 9. KINEMATICS OF AN ISENTROPIC ENSEMBLE	189	(26)
9.1 Parameterizing mesoscale eddies	189	(2)
9.2 Advection and skewsion	191	(3)
9.3 Volume conservation	194	(9)
9.4 Ensemble mean tracer equation	203	(3)
9.5 Quasi-Stokes transport in z-models	206	(6)
9.6 Chapter summary	212	(3)
PART 3. SEMI-DISCRETE EQUATIONS AND ALGORITHMS	215	(66)
Chapter 10. DISCRETIZATION BASICS	217	(5)
10.1 Discretization methods	217	(1)
10.2 An introduction to Arakawa grids	218	(1)
10.3 Time stepping	219	(2)
10.4 Chapter summary	221	(1)
Chapter 11. MASS AND TRACER BUDGETS	222	(15)
11.1 Summary of the continuous model equations	222	(1)
11.2 Tracer and mass/volume compatibility	223	(1)
11.3 Mass budget for a grid cell	223	(4)
11.4 Mass budget for a discrete fluid column	227	(1)
11.5 Tracer budget for a grid cell	228	(4)
11.6 Fluxes for turbulence mixed layer schemes	232	(1)
11.7 Flux plus restore boundary conditions	233	(1)
11.8 Z-like vertical coordinate models	234	(1)
11.9 Chapter summary	235	(2)
Chapter 12. ALGORITHMS FOR HYDROSTATIC OCEAN MODELS	237	(44)
12.1 Summary of the continuous model equations	237	(1)
12.2 Budget of linear momentum for a grid cell	238	(6)
12.3 Strategies for time stepping momentum	244	(4)
12.4 A leap-frog algorithm	248	(3)
12.5 Discretization of time tendencies	251	(7)
12.6 A time staggered algorithm	258	(4)
12.7 Barotropic updates with a predictor-corrector	262	(3)
12.8 Stability considerations	265	(12)
12.9 Smoothing the surface height in B-grid models	277	(1)
12.10 Rigid lid streamfunction method	278	(2)
12.11 Chapter summary	280	(1)
PART 4. NEUTRAL PHYSICS	281	(96)
Chapter 13. BASICS OF NEUTRAL PHYSICS	283	(13)
13.1 Concerning the utility of neutral physics	283	(3)
13.2 Notation and summary of scalar budgets	286	(1)
13.3 Compatibility in the mean field budgets	287	(1)
13.4 The SGS tracer transport tensor	288	(2)
13.5 Advection and skewsion	290	(1)
13.6 Neutral tracer fluxes	291	(3)
13.7 Chapter summary and a caveat on the conjecture	294	(2)
Chapter 14. NEUTRAL TRANSPORT OPERATORS	296	(32)
14.1 Neutral diffusion	296	(8)
14.2 Gent-McWilliams stirring	304	(4)
14.3 Summarizing the neutral physics fluxes	308	(1)
14.4 Flow-dependent diffusivities	309	(8)
14.5 Biharmonic operators	317	(9)
14.6 Chapter summary and some challenges	326	(2)
Chapter 15. NEUTRAL PHYSICS NEAR THE SURFACE BOUNDARY	328	(17)
15.1 Linear stability for neutral diffusion	328	(4)
15.2 Linear stability for GM stirring	332	(1)
15.3 Neutral physics near boundaries	333	(10)
15.4 Chapter summary and caveats	343	(2)
Chapter 16. FUNCTIONAL DISCRETIZATION OF NEUTRAL PHYSICS	345	(32)
16.1 Foundations for discrete neutral physics	345	(5)
16.2 Introduction to the discretization	350	(2)
16.3 A one-dimensional warm-up	352	(2)
16.4 Elements of the discrete dissipation functional	354	(7)
16.5 Triad stencils and some more notation	361	(2)
16.6 The discrete diffusion operator	363	(4)
16.7 Diffusive flux components	367	(4)
16.8 Further issues of numerical implementation	371	(3)
16.9 Chapter summary	374	(3)
PART 5. HORIZONTAL FRICTION	377	(64)
Chapter 17. HORIZONTAL FRICTION IN MODELS	379	(30)
17.1 Boussinesq and non-Boussinesq friction	379	(1)
17.2 Introduction and general framework	379	(1)
17.3 Properties of the stress tensor	380	(7)
17.4 Properties of the viscosity tensor	387	(2)
17.5 Transverse isotropy	389	(4)
17.6 Transverse anisotropy	393	(3)
17.7 Generalized orthogonal coordinates	396	(2)
17.8 Dissipation functional	398	(4)
17.9 Biharmonic friction	402	(2)
17.10 Some mathematical details	404	(3)
17.11 Chapter summary	407	(2)
Chapter 18. CHOOSING THE HORIZONTAL VISCOSITY	409	(15)
18.1 Stability and resolution considerations	409	(6)
18.2 Comparing Laplacian and biharmonic mixing	415	(1)
18.3 Smagorinsky viscosity	416	(4)
18.4 Background viscosity	420	(1)
18.5 Viscosities for anisotropic friction	421	(1)
18.6 Chapter summary	422	(2)
Chapter 19. FUNCTIONAL DISCRETIZATION OF FRICTION	424	(17)
19.1 Comments on notation	424	(1)
19.2 Summary of the various formulations	425	(1)
19.3 Horizontal friction discretization	426	(10)
19.4 Laplacian plus metric form of isotropic friction	436	(3)
19.5 Chapter summary	439	(2)
PART 6. TENSOR ANALYSIS	441	(46)
Chapter 20. ELEMENTARY TENSOR ANALYSIS	443	(23)
20.1 Introduction	443	(1)
20.2 Some practical motivation	444	(2)
20.3 Coordinates and vectors	446	(2)
20.4 The metric and coordinate transformations	448	(3)
20.5 Transformations of a vector	451	(1)
20.6 One-forms	452	(2)
20.7 Mapping between vectors and one-forms	454	(1)
20.8 Transformation of a one-form	454	(1)
20.9 Arbitrary tensors and their transformations	455	(1)
20.10 Tensorial properties of the gradient operator	456	(1)
20.11 The invariant volume element	457	(2)
20.12 Determinants and the Levi-Civita symbol	459	(2)
20.13 Surfaces embedded in Euclidean space	461	(3)
20.14 Chapter summary	464	(2)
Chapter 21. CALCULUS ON CURVED MANIFOLDS	466	(21)
21.1 Fundamental character of tensor equations	466	(2)
21.2 Covariant differentiation	468	(2)
21.3 Covariant derivative of a second order tensor	470	(1)
21.4 Christoffel symbols in terms of the metric	471	(1)
21.5 Covariant divergence of a vector	471	(1)
21.6 Covariant divergence of a second order tensor	472	(1)
21.7 Covariant Laplacian of a scalar	473	(1)
21.8 Covariant curl of a vector	473	(1)
21.9 Covariant Laplacian of a vector	473	(1)
21.10 Integral theorems	474	(1)
21.11 Orthogonal curvilinear coordinates	474	(7)
21.12 Summary of curvilinear tensor analysis	481	(6)
PART 7. EPILOGUE	487	(6)
Chapter 22. SOME CLOSING COMMENTS AND CHALLENGES	489	(4)
BIBLIOGRAPHY	493	(18)
Index	511