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Overview
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.
Product Details
| ISBN-13: | 9780691187129 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 06/05/2018 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 528 |
| File size: | 4 MB |
About the Author
Read an Excerpt
Fundamentals of Ocean Climate Models
Stephen GriffiesChapter 1
OCEAN CLIMATE MODELS
The purpose of this chapter is to introduce ocean climate models and their use in climate science.
1.1 OCEAN MODELS AS TOOLS FOR OCEAN SCIENCE
A column of ocean water only 3m thick contains as much heat capacity as the full atmospheric column above (Gill, 1982). Hence, the oceans, which cover roughly 70% of the earth's surface, provide a large reservoir for heat and other constituents of the earth's climate system, such as the increasing amounts of anthropogenic carbon dioxide. Through its buffering abilities and relatively slow time scales, the ocean represents the fly wheel of the earth's climate system. That is, as goes the ocean, so goes the climate system.
A scientific understanding of the ocean's time mean state, as well as its variability about this mean and its stability to various forms of perturbations, represents a key goal of physical oceanography and climate science. Due to our inability to perform controlled experiments on large-scale systems studied in the geosciences, such as the earth's climate and its component subsystems, computer models represent a critical tool for rationalizing climate phenomena. Indeed, computer models are becoming the primary tools used to study and predict physical, chemical, and biological characteristics of the ocean fluid, reflecting the growing power of computers, improved knowledge and observations of the ocean, and enhancements in the realism of ocean model simulations.
That ocean models are increasingly being used by all sorts of scientists, includingthose without direct experience developing models, is a sign that the models have enhanced their physical integrity over the past decades to a level deserving a general respect within the broader climate science community. Correspondingly, as model usage increases, model developers have a growing responsibility to ensure that their codes are physically based, numerically sound, and well documented. Given this mandate, one aim of this book is to establish a level of ocean model documentation that goes beyond the usual technical discussion that assumes the model user is familiar with the fundamentals and understands the physical meanings of the mathematical symbols. Instead, we develop the equations from a (mostly) first principles perspective and take some care to nurture a physical understanding of the mathematics.
1.2 OCEAN CLIMATE MODELS
Models of the ocean range in complexity from idealized theoretical models whose solutions can be summarized by a few lines of mathematics, to realistic global ocean circulation models encompassing many equations and requiring thousands or millions of lines of computer code to solve. The main focus of this monograph concerns the realistic models, and in particular the fundamentals of their formulation. It is important to note that a scientific understanding of ocean climate phenomena is often realized most profoundly by the creative and judicious use of a hierarchy of models. The most realistic models play an important part in this hierarchy, but they are not the full story.
We use the term ocean climate model as a means to distinguish models that simulate the World Ocean over climatologically relevant time scales from those that simulate, say, coastal, regional, or basin scale dynamics. Distinctions between these ocean modeling subfields is decreasing, largely due to the steady growth in computer power that allows modelers to dispense with some of the simplifications required only a few years ago. Distinctions are also becoming fewer due to realizations by practitioners that elements of the ocean strongly interact across many spatial and temporal scales. That is, ocean modeling subfields overlap in crucial manners. Nonetheless, the finite nature of both scientists and their tools introduces a difference in focus, with choices made by practitioners in one subfield often unacceptable to those in another.
Simulations of the World Ocean over time scales appropriate for climate (e.g.,decades to millennia) involve extremely rich and complex arrays of flow regimes and interactions between components of the climate system. Additionally, the ocean is largely forced at its upper and lower boundaries, with interior flow relatively ideal. In particular, high latitude oceanography involves strong interactions between the ocean with sea ice and rivers, and intense air-sea heat fluxes induce deep convection and the associated formation of deep water masses. Tropical oceanography involves intense equatorial current systems with rapid adjustments to wind forcing associated with equatorial Kelvin, Rossby, and instability waves, and a powerful interannual mode of air-sea variability known as El Niño in the Pacific. Oceanography in the subtropical and subpolar latitudes is dominated by large-scale gyres with meandering and eddying boundary currents forming their western margins. Furthermore, solid earth boundaries provide a leading order influence on the ocean circulation. For example, meridional boundaries block otherwise zonal flow except within certain parts of the Southern Ocean, variations in topography cause flows to feel the bottom throughout many crucial parts of the World Ocean, and straits and sills funnel water from marginal seas, such as the Mediterranean and Greenland, into the larger ocean basins. A primary goal of ocean climate modeling is to simulate the global ocean circulation over these various regimes, given just the boundary forcing. This is a highly nontrivial goal.
As discussed in Section 6.2, there are three general model classes that have been used for ocean modeling. These classes are distinguished by the manner used to discretize the vertical direction. Indeed, as argued in Griffies et al. (2000a), the choice of vertical coordinate represents the most fundamental choice that can be made when designing an ocean model.
Since the 1960's, z-coordinate ocean models, or simply z-models, have been the dominant class for global ocean climate simulations. This is the model class with which the author is most familiar, and which forms the focus of some of the latter parts of this book. Characteristics of z-models, as well as the other two classes (isopycnal and terrain following), are described in Section 6.2. Each model class has advantages and disadvantages when simulating various flow regimes encountered in ocean climate modeling. Only two of the three (isopycnal and z) have routinely been used for global circulation studies. One reason that z-models presently dominate ocean climate modeling is that their relative simplicity has allowed them to be used for many decades, going back to the work of Bryan (1969), Bryan and Lewis (1979), and Cox (1984). In contrast, the isopycnal model class requires more sophisticated numerical schemes, whose development did not mature until the 1980's. The third model class, the terrain following sigma models, remain the model of choice for coastal oceanographers, but they have largely remained absent from simulations of global ocean climate.
Ocean climate models continue to evolve. For example, many egregious problems identified with early representatives of the different model classes are now remedied by more mature numerical treatments. Nonetheless, as argued in Section 6.2, each class has basic limitations that warrant developing models with generalized vertical coordinates. The hope is that by generalizing the treatment of the vertical, a well-designed model with this capability can reduce or remove many of the egregious problems of the less flexible models based on a single vertical coordinate choice. This remains a topic of intense research in the ocean model development community.
Research into ocean model fundamentals and algorithm development can take many years to penetrate into the common practice of major climate modeling centers. The reason is largely related to the extreme complexity involved with building coupled global climate models. It takes teams of researchers years to build and refine a coupled model, with significant feedback and compromise necessary in order to successfully mesh the needs of various component modelers. Notably, an ocean model suitable for coupled climate simulations is far more than a dynamical kernal. In addition, it must consist of a full suite of physical parameterizations of unresolved processes, diagnostics allowing its simulations to be readily analyzed, infrastructure providing a means to talk to the computer that is running the model, a superstructure with appropriate handles for interacting with other component models (e.g., sea ice and atmosphere models) necessary for climate simulations, and computational sophistication rendering it efficient on the many computer platforms employed by research laboratories and universities.
1.3 CHALLENGES OF CLIMATE CHANGE
Since the 1990's, thousands of scientists worldwide have been contributing to the development of extensive reports on climate change science, with the latest being Houghton et al. (2001). This work is in response to the increasing scientific evidence that industrial society represents a nontrivial geophysical force. Common questions that arise are: What should we expect? How much of the observed climate change is due to humans? Providing sound scientifically based answers to these and other questions is profoundly difficult. Indeed, as lucidly described in the book by Philander (1998), unequivocal answers are not forthcoming from climate science. Instead, as with weather prediction, probabilistic statements are the best the science can provide.
As discussed in Houghton et al. (2001), we are at a stage in climate science where the wide variety of climate models yield a general consensus regarding the large-scale effects of increased greenhouse gases. Quite simply, the planet is warming and will likely continue to do so, with higher latitudes feeling the relative effects more than lower latitudes. However, when quantitative questions are posed, models provide varying projections. Part of the spread is related to the chaotic nature of the climate system. Part is due to large uncertainties in future greenhouse gas emission scenarios. Yet some is due to differing details of the model formulations and their parameterizations. It is on this latter issue that climate scientists can make further progress through research and development.
Given the critical importance of models for understanding climate and predicting its future behavior, it is incumbent on model developers to impose the highest standards on model integrity. In particular, ocean climate models should incorporate realistic parameterizations and sound numerical formulations (for reviews, see Chassignet and Verron, 1998; Griffies et al., 2000a). Yet they must do so at a level of computational expense that does not overly handicap the abilities of the earth system modeler to incorporate other components of the climate system, and to fully investigate various scenarios. Within the ocean science community, this mandate to improve the models entrains hundreds of researchers such as process oriented physicists, chemists, biologists, observational oceanographers, numerical algorithm developers, software engineers, ocean climate modelers, and others. It is anticipated that the questions of climate change will continue to strongly influence and motivate all areas of climate science for many years.
Table of Contents
FOREWORD XIII
PREFACE XV
ACKNOWLEDGMENTS XXV
ABOUT THE COVER XXVII
LIST OF SYMBOLS XXIX
Chapter 1. OCEAN CLIMATE MODELS 1
1.1 Ocean models as tools for ocean science 1
1.2 Ocean climate models 2
1.3 Challenges of climate change 3
PART 1. FUNDAMENTAL OCEAN EQUATIONS 5
Chapter 2. BASICS OF OCEAN FLUID MECHANICS 7
2.1 Some fundamental ocean processes 7
2.2 The continuum hypothesis 9
2.3 Kinematics of fluid motion 10
2.4 Kinematical and dynamical approximations 16
2.5 Averaging over scales and realizations 20
2.6 Numerical discretization 21
2.7 Chapter summary 22
Chapter 3. KINEMATICS 24
3.1 Introduction 24
3.2 Mathematical preliminaries 24
3.3 The divergence theorem and budget analyses 29
3.4 Volume and mass conserving kinematics 31
3.5 Chapter summary 40
Chapter 4. DYNAMICS 42
4.1 Introduction 42
4.2 Motion on a rotating sphere 43
4.3 Principles of continuum dynamics 47
4.4 Dynamics of fluid parcels 51
4.5 Hydrostatic pressure 56
4.6 Dynamics of hydrostatic fluid columns 58
4.7 Fluid motion in a rapidly rotating system 62
4.8 Vertical stratification 68
4.9 Vorticity and potential vorticity 70
4.10 Particle dynamics on a rotating sphere 75
4.11 Symmetry and conservation laws 80
4.12 Chapter summary 83
Chapter 5. THERMO-HYDRODYNAMICS 87
5.1 General types of ocean tracers 87
5.2 Basic equilibrium thermodynamics 91
5.3 Energy of a fluid parcel 95
5.4 Global mechanical energy balance 105
5.5 Basic non-equilibrium thermodynamics 110
5.6 Thermodynamical tracers 111
5.7 Ocean density 114
5.8 Chapter summary 118
Chapter 6. GENERALIZED VERTICAL COORDINATES 121
6.1 Introduction 121
6.2 Concerning the choice of vertical coordinate 122
6.3 Generalized surfaces 128
6.4 Local orthonormal coordinates 130
6.5 Mathematics of generalized vertical coordinates 131
6.6 Metric tensors 136
6.7 The dia-surface velocity component 138
6.8 Conservation of mass and volume for parcels 141
6.9 Kinematic boundary conditions 143
6.10 Primitive equations 145
6.11 Transformation of SGS tracer flux components 147
6.12 Chapter summary 149
PART 2. AVERAGED DESCRIPTIONS 153
Chapter 7. CONCERNI NG UNRESOLVED PHYSICS 155
7.1 Represented dynamics and parameterized physics 155
7.2 Lateral (neutral) and vertical processes 157
7.3 Basic mechanisms for dianeutral transport 159
7.4 Dianeutral transport in models 161
7.5 Numerically induced spurious dianeutral transport 166
7.6 Chapter summary 167
Chapter 8. EULERIAN AVERAGED EQUATIONS 169
8.1 Introduction 169
8.2 The nonhydrostatic shallow ocean equations 171
8.3 Averaged kinematics 173
8.4 Averaged kinematics over finite domains 174
8.5 Averaged tracer 179
8.6 Averaged momentum budget 182
8.7 Summary of the Eulerian averaged equations 183
8.8 Mapping to ocean model variables 185
8.9 Chapter summary 187
Chapter 9. KINEMATICS OF AN ISENTROPIC ENSEMBLE 189
9.1 Parameterizing mesoscale eddies 189
9.2 Advection and skewsion 191
9.3 Volume conservation 194
9.4 Ensemble mean tracer equation 203
9.5 Quasi-Stokes transport in z-models 206
9.6 Chapter summary 212
PART 3. SEMI-DISCRETE EQUATIONS AND ALGORITHMS 215
Chapter 10. DISCRETIZATION BASICS 217
10.1 Discretization methods 217
10.2 An introduction to Arakawa grids 218
10.3 Time stepping 219
10.4 Chapter summary 221
Chapter 11. MASS AND TRACER BUDGETS 222
11.1 Summary of the continuous model equations 222
11.2 Tracer and mass/volume compatibility 223
11.3 Mass budget for a grid cell 223
11.4 Mass budget for a discrete fluid column 227
11.5 Tracer budget for a grid cell 228
11.6 Fluxes for turbulence mixed layer schemes 232
11.7 Flux plus restore boundary conditions 233
11.8 Z-like vertical coordinate models 234
11.9 Chapter summary 235
Chapter 12. ALGORITHMS FOR HYDROSTATIC OCEAN MODELS 237
12.1 Summary of the continuous model equations 237
12.2 Budget of linear momentum for a grid cell 238
12.3 Strategies for time stepping momentum 244
12.4 A leap-frog algorithm 248
12.5 Discretization of time tendencies 251
12.6 A time staggered algorithm 258
12.7 Barotropic updates with a predictor-corrector 262
12.8 Stability considerations 265
12.9 Smoothing the surface height in B-grid models 277
12.10 Rigid lid streamfunction method 278
12.11 Chapter summary 280
PART 4. NEUTRAL PHYSICS 281
Chapter 13. BASICS OF NEUTRAL PHYSICS 283
13.1 Concerning the utility of neutral physics 283
13.2 Notation and summary of scalar budgets 286
13.3 Compatibility in the mean field budgets 287
13.4 The SGS tracer transport tensor 288
13.5 Advection and skewsion 290
13.6 Neutral tracer fluxes 291
13.7 Chapter summary and a caveat on the conjecture 294
Chapter 14. NEUTRAL TRANSPORT OPERATORS 296
14.1 Neutral diffusion 296
14.2 Gent-McWilliams stirring 304
14.3 Summarizing the neutral physics fluxes 308
14.4 Flow-dependent diffusivities 309
14.5 Biharmonic operators 317
14.6 Chapter summary and some challenges 326
Chapter 15. NEUTRAL PHYSICS NEAR THE SURFACE BOUNDARY 328
15.1 Linear stability for neutral diffusion 328
15.2 Linear stability for GM stirring 332
15.3 Neutral physics near boundaries 333
15.4 Chapter summary and caveats 343
Chapter 16. FUNCTIONAL DISCRETIZATION OF NEUTRAL PHYSICS 345
16.1 Foundations for discrete neutral physics 345
16.2 Introduction to the discretization 350
16.3 A one-dimensional warm-up 352
16.4 Elements of the discrete dissipation functional 354
16.5 Triad stencils and some more notation 361
16.6 The discrete diffusion operator 363
16.7 Diffusive flux components 367
16.8 Further issues of numerical implementation 371
16.9 Chapter summary 374
PART 5. HORIZONTAL FRICTION 377
Chapter 17. HORIZONTAL FRICTION IN MODELS 379
17.1 Boussinesq and non-Boussinesq friction 379
17.2 Introduction and general framework 379
17.3 Properties of the stress tensor 380
17.4 Properties of the viscosity tensor 387
17.5 Transverse isotropy 389
17.6 Transverse anisotropy 393
17.7 Generalized orthogonal coordinates 396
17.8 Dissipation functional 398
17.9 Biharmonic friction 402
17.10 Some mathematical details 404
17.11 Chapter summary 407
Chapter 18. CHOOSING THE HORIZONTAL VISCOSITY 409
18.1 Stability and resolution considerations 409
18.2 Comparing Laplacian and biharmonic mixing 415
18.3 Smagorinsky viscosity 416
18.4 Background viscosity 420
18.5 Viscosities for anisotropic friction 421
18.6 Chapter summary 422
Chapter 19. FUNCTIONAL DISCRETIZATION OF FRICTION 424
19.1 Comments on notation 424
19.2 Summary of the various formulations 425
19.3 Horizontal friction discretization 426
19.4 Laplacian plus metric form of isotropic friction 436
19.5 Chapter summary 439
PART 6. TENSOR ANALYSIS 441
Chapter 20. ELEMENTARY TENSOR ANALYSIS 443
20.1 Introduction 443
20.2 Some practical motivation 444
20.3 Coordinates and vectors 446
20.4 The metric and coordinate transformations 448
20.5 Transformations of a vector 451
20.6 One-forms 452
20.7 Mapping between vectors and one-forms 454
20.8 Transformation of a one-form 454
20.9 Arbitrary tensors and their transformations 455
20.10 Tensorial properties of the gradient operator 456
20.11 The invariant volume element 457
20.12 Determinants and the Levi-Civita symbol 459
20.13 Surfaces embedded in Euclidean space 461
20.14 Chapter summary 464
Chapter 21. CALCULUS ON CURVED MANIFOLDS 466
21.1 Fundamental character of tensor equations 466
21.2 Covariant differentiation 468
21.3 Covariant derivative of a second order tensor 470
21.4 Christoffel symbols in terms of the metric 471
21.5 Covariant divergence of a vector 471
21.6 Covariant divergence of a second order tensor 472
21.7 Covariant Laplacian of a scalar 473
21.8 Covariant curl of a vector 473
21.9 Covariant Laplacian of a vector 473
21.10 Integral theorems 474
21.11 Orthogonal curvilinear coordinates 474
21.12 Summary of curvilinear tensor analysis 481
PART 7. EPILOGUE 487
Chapter 22. SOME CLOSING COMMENTS AND CHALLENGES 489
BIBLIOGRAPHY 493
Index 511
What People are Saying About This
Stephen Griffies addresses real issues that have plagued ocean models for many years. His thorough examination of the desired properties of numerical schemes puts ocean models on a much sounder physical footing. He has been instrumental in developing these ideas. Bringing them together in a single volume will be useful to students and researchers alike.
— Kelvin Richards, University of Hawaii
This clear, well-written book comprehensively covers the most important advances in ocean models for climate over the past ten years. The author's qualifications to write it are second to none.
— Peter Gent, National Center for Atmospheric Research
"Stephen Griffies addresses real issues that have plagued ocean models for many years. His thorough examination of the desired properties of numerical schemes puts ocean models on a much sounder physical footing. He has been instrumental in developing these ideas. Bringing them together in a single volume will be useful to students and researchers alike."—Kelvin Richards, University of Hawaii"A leader in the field, Stephen Griffies is very thorough in his methods and understands the workings of ocean models, and their theoretical underpinning, as much as anyone in the world at the present time. His book has evolved as research has evolved, and consequently it is right at the forefront of our current understanding."—Richard J. Greatbatch, Dalhousie University"This clear, well-written book comprehensively covers the most important advances in ocean models for climate over the past ten years. The author's qualifications to write it are second to none."—Peter Gent, National Center for Atmospheric Research
A leader in the field, Stephen Griffies is very thorough in his methods and understands the workings of ocean models, and their theoretical underpinning, as much as anyone in the world at the present time. His book has evolved as research has evolved, and consequently it is right at the forefront of our current understanding.
— Richard J. Greatbatch, Dalhousie University
Recipe
"Stephen Griffies addresses real issues that have plagued ocean models for many years. His thorough examination of the desired properties of numerical schemes puts ocean models on a much sounder physical footing. He has been instrumental in developing these ideas. Bringing them together in a single volume will be useful to students and researchers alike."—Kelvin Richards, University of Hawaii
"A leader in the field, Stephen Griffies is very thorough in his methods and understands the workings of ocean models, and their theoretical underpinning, as much as anyone in the world at the present time. His book has evolved as research has evolved, and consequently it is right at the forefront of our current understanding."—Richard J. Greatbatch, Dalhousie University
"This clear, well-written book comprehensively covers the most important advances in ocean models for climate over the past ten years. The author's qualifications to write it are second to none."—Peter Gent, National Center for Atmospheric Research