Introduction to Geophysical Fluid Dynamics (eBook)

Physical and Numerical Aspects

Jean-Marie Beckers, Benoit Cushman-Roisin (Autoren)

eBook Download: PDF | EPUB

2011 | 2. Auflage
875 Seiten
Elsevier Science (Verlag)
978-0-08-091678-1 (ISBN)

Infos

Introduction to Geophysical Fluid Dynamics provides an introductory-level exploration of geophysical fluid dynamics (GFD), the principles governing air and water flows on large terrestrial scales. Physical principles are illustrated with the aid of the simplest existing models, and the computer methods are shown in juxtaposition with the equations to which they apply. It explores contemporary topics of climate dynamics and equatorial dynamics, including the Greenhouse Effect, global warming, and the El Nino Southern Oscillation. - Combines both physical and numerical aspects of geophysical fluid dynamics into a single affordable volume - Explores contemporary topics such as the Greenhouse Effect, global warming and the El Nino Southern Oscillation - Biographical and historical notes at the ends of chapters trace the intellectual development of the field - Recipient of the 2010 Wernaers Prize, awarded each year by the National Fund for Scientific Research of Belgium (FNR-FNRS)

Benoit Cushman-Roisin is Professor of Engineering Sciences at Dartmouth College, where he has been on the faculty since 1990. His teaching includes a series of introductory, mid-level, and advanced courses in environmental engineering. He has also developed new courses in sustainable design and industrial ecology. Prof. Cushman-Roisin holds a B.S. in Engineering Physics, Summa Cum Laude, from the University of LiŠge, Belgium (1978) and a Ph.D. in Geophysical Fluid Dynamics from the Florida State University (1980). He is the author of two books and numerous refereed publications on various aspects of environmental fluid mechanics. In addition to his appointment at Dartmouth College, Prof. Cushman-Roisin maintains an active consultancy in environmental aspects of fluid mechanics and energy efficiency He is co-founder of e2fuel LLC, a startup dedicated to aviation and trucking fuel from clean sources. He is the founding editor of Environmental Fluid Mechanics.

Inhalt

Front Cover 1
Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects 4
Copyright 5
Table of Contents 6
Foreword 14
Preface 16
Preface of the First Edition 18
I Fundamentals 20
1 Introduction 22
1.1 Objective 22
1.2 Importance of Geophysical Fluid Dynamics 23
1.3 Distinguishing Attributes of Geophysical Flows 25
1.4 Scales of Motions 27
1.5 Importance of Rotation 29
1.6 Importance of Stratification 31
1.7 Distinction between the Atmosphere and Oceans 33
1.8 Data Acquisition 36
1.9 The Emergence of Numerical Simulations 38
1.10 Scales Analysis and Finite Differences 42
1.11 Higher-Order Methods 47
1.12 Aliasing 52
Analytical Problems 54
Numerical Exercises 54
2 The Coriolis Force 60
2.1 Rotating Framework of Reference 60
2.2 Unimportance of the Centrifugal Force 63
2.3 Free Motion on a Rotating Plane 66
2.4 Analogy and Physical Interpretation 69
2.5 Acceleration on a Three-Dimensional Rotating Planet 71
2.6 Numerical Approach to Oscillatory Motions 74
2.7 Numerical Convergence and Stability 78
2.7.1 Formal Stability Definition 80
2.7.2 Strict Stability 80
2.7.3 Choice of a Stability Criterion 80
2.8 Predictor-Corrector Methods 82
2.9 Higher-Order Schemes 84
Analytical Problems 88
Numerical Exercises 91
3 Equations of Fluid Motion 96
3.1 Mass Budget 96
3.2 Momentum Budget 97
3.3 Equation of State 98
3.4 Energy Budget 99
3.5 Salt and Moisture Budgets 101
3.6 Summary of Governing Equations 102
3.7 Boussinesq Approximation 102
3.8 Flux Formulation and Conservative Form 106
3.9 Finite-Volume Discretization 107
Analytical Problems 111
Numerical Exercises 113
4 Equations Governing Geophysical Flows 118
4.1 Reynolds-Averaged Equations 118
4.2 Eddy Coefficients 120
4.3 Scales of Motion 122
4.4 Recapitulation of Equations Governing Geophysical Flows 125
4.5 Important Dimensionless Numbers 126
4.6 Boundary Conditions 128
4.6.1 Kinematic Conditions 131
4.6.2 Dynamic Conditions 133
4.6.3 Heat, Salt, and Tracer Boundary Conditions 135
4.7 Numerical Implementation of Boundary Conditions 136
4.8 Accuracy and Errors 139
4.8.1 Discretization Error Estimates 140
Analytical Problems 144
Numerical Exercises 145
5 Diffusive Processes 150
5.1 Isotropic, Homogeneous Turbulence 150
5.1.1 Length and Velocity Scales 151
5.1.2 Energy Spectrum 154
5.2 Turbulent Diffusion 156
5.3 One-Dimensional Numerical Scheme 159
5.4 Numerical Stability Analysis 163
5.5 Other One-Dimensional Schemes 169
5.6 Multi-Dimensional Numerical Schemes 173
Analytical Problems 176
Numerical Exercises 177
6 Transport and Fate 182
6.1 Combination of Advection and Diffusion 182
6.2 Relative Importance of Advection: The Peclet Number 186
6.3 Highly Advective Situations 187
6.4 Centered and Upwind Advection Schemes 188
6.5 Advection–Diffusion with Sources and Sinks 202
6.6 Multidimensional Approach 205
Analytical Problems 215
Numerical Exercises 217
II Rotation Effects 222
7 Geostrophic Flows and Vorticity Dynamics 224
7.1 Homogeneous Geostrophic Flows 224
7.2 Homogeneous Geostrophic Flows Over an Irregular Bottom 227
7.3 Generalization to Nongeostrophic Flows 229
7.4 Vorticity Dynamics 231
7.5 Rigid-Lid Approximation 234
7.6 Numerical Solution of the Rigid-Lid Pressure Equation 236
7.7 Numerical Solution of the Streamfunction Equation 240
7.8 Laplacian Inversion 243
Analytical Problems 250
Numerical Exercises 252
8 The Ekman Layer 258
8.1 Shear Turbulence 258
8.1.1 Logarithmic Profile 259
8.1.2 Eddy Viscosity 261
8.2 Friction and Rotation 262
8.3 The Bottom Ekman Layer 264
8.4 Generalization to Nonuniform Currents 266
8.5 The Ekman Layer over Uneven Terrain 269
8.6 The Surface Ekman Layer 270
8.7 The Ekman Layer in Real Geophysical Flows 273
8.8 Numerical Simulation of Shallow Flows 276
Analytical Problems 284
Numerical Exercises 286
9 Barotropic Waves 290
9.1 Linear wave dynamics 290
9.2 The Kelvin Wave 292
9.3 Inertia-Gravity Waves (Poincér Waves) 295
9.4 Planetary Waves (Rossby Waves) 297
9.5 Topographic Waves 302
9.6 Analogy between Planetary and Topographic Waves 306
9.7 Arakawa's Grids 308
9.8 Numerical Simulation of Tides and Storm Surges 319
Analytical Problems 328
Numerical Exercises 331
10 Barotropic Instability 336
10.1 What Makes a Wave Grow Unstable? 336
10.2 Waves on Shear Flow 337
10.3 Bounds on Wave Speeds and Growth Rates 341
10.4 A Simple Example 343
10.5 Nonlinearities 347
10.6 Filtering 350
10.7 Contour Dynamics 353
Analytical Problems 359
Numerical Exercises 360
III Stratification Effects 364
11 Stratification 366
11.1 Introduction 366
11.2 Static Stability 367
11.3 A Note on Atmospheric Stratification 368
11.4 Convective Adjustment 373
11.5 The Importance of Stratification: The Froude Number 375
11.6 Combination of Rotation and Stratification 377
Analytical Problems 380
Numerical Exercises 380
12 Layered Models 384
12.1 From Depth to Density 384
12.2 Layered Models 388
12.3 Potential Vorticity 393
12.4 Two-Layer Models 393
12.5 Wind-Induced Seiches in Lakes 398
12.6 Energy Conservation 400
12.7 Numerical Layered Models 402
12.8 Lagrangian Approach 406
Analytical Problems 409
Numerical Exercises 410
13 Internal Waves 414
13.1 From Surface to Internal Waves 414
13.2 Internal-wave Theory 416
13.3 Structure of an Internal Wave 418
13.4 Vertical Modes and Eigenvalue Problems 420
13.4.1 Vertical Eigenvalue Problem 423
13.4.2 Bounds on Frequency 423
13.4.3 Simple Example of Constant N2 424
13.4.4 Numerical Decomposition into Vertical Modes 426
13.4.5 Waves Concentration at a Pycnocline 429
13.5 Lee Waves 431
13.5.1 Radiating Waves 433
13.5.2 Trapped Waves 435
13.6 Nonlinear Effects 435
Analytical Problems 438
Numerical Exercises 440
14 Turbulence in Stratified Fluids 444
14.1 Mixing of Stratified Fluids 444
14.2 Instability of a Stratified Shear Flow: The Richardson Number 448
14.3 Turbulence Closure: k-Models 454
14.4 Other Closures: k-e and k-klm 468
14.5 Mixed-layer Modeling 469
14.6 Patankar-Type Discretizations 474
14.7 Wind Mixing and Penetrative Convection 477
14.7.1 Wind Mixing 478
14.7.2 Penetrative Convection 480
Analytical Problems 485
Numerical Exercises 486
IV Combined Rotation and Stratification Effects 490
15 Dynamics of Stratified Rotating Flows 492
15.1 Thermal Wind 492
15.2 Geostrophic Adjustment 494
15.3 Energetics of Geostrophic Adjustment 499
15.4 Coastal Upwelling 501
15.4.1 The Upwelling Process 501
15.4.2 A Simple Model of Coastal Upwelling 503
15.4.3 Finite-Amplitude Upwelling 505
15.4.4 Variability of the Upwelling Front 508
15.5 Atmospheric Frontogenesis 509
15.6 Numerical Handling of Large Gradients 521
15.7 Nonlinear Advection Schemes 526
Analytical Problems 531
Numerical Exercises 535
16 Quasi-Geostrophic Dynamics 540
16.1 Simplifying Assumption 540
16.2 Governing Equation 541
16.3 Length and Timescale 546
16.4 Energetics 549
16.5 Planetary Waves in a Stratified Fluid 551
16.6 Some Nonlinear Effects 558
16.7 Quasi-Geostrophic Ocean Modeling 561
Analytical Problems 565
Numerical Exercises 566
17 Instabilities of Rotating Stratified Flows 572
17.1 Two Types of Instability 572
17.2 Inertial Instability 573
17.3 Baroclinic Instability—the Mechanism 580
17.4 Linear Theory of Baroclinic Instability 585
17.5 Heat Transport 593
17.6 Bulk Criteria 595
17.7 Finite-Amplitude Development 598
Analytical Problems 603
Numerical Exercises 604
18 Fronts, Jets and Vortices 608
18.1 Fronts and Jets 608
18.1.1 Origin and Scales 608
18.1.2 Meanders 611
18.1.3 Multiple Equilibria 616
18.1.4 Stretching and Topographic Effects 616
18.1.5 Instabilities 619
18.2 Vortices 620
18.3 Turbulence 630
18.4 Simulations of Geostrophic Turbulence 632
Analytical Problems 637
Numerical Exercises 640
V Special Topics 644
19 Atmospheric General Circulation 646
19.1 Climate Versus Weather 646
19.2 Planetary Heat Budget 646
19.3 Direct and Indirect Convective Cells 650
19.4 Atmospheric Circulation Models 656
19.5 Brief Remarks on Weather Forecasting 661
19.6 Cloud Parameterizations 661
19.7 Spectral Methods 663
19.8 Semi-Lagrangian Methods 668
Analytical Problems 671
Numerical Exercises 672
20 Oceanic General Circulation 676
20.1 What Drives the Oceanic Circulation 676
20.2 Large-Scale Ocean Dynamics (Sverdrup Dynamics) 679
20.2.1 Sverdrup Relation 681
20.2.2 Sverdrup Transport 682
20.2.3 Thermal Wind and Beta Spiral 683
20.2.4 A Bernoulli Function 685
20.2.5 Potential Vorticity 686
20.3 Western Boundary Currents 688
20.4 Thermohaline Circulation 692
20.4.1 Subduction 692
20.4.2 Ventilated Thermocline Theory 695
20.4.3 Scaling of the Main Thermocline 696
20.5 Abyssal Circulation 696
20.6 Models 700
20.6.1 Coordinate Systems 705
20.6.2 Subgrid-Scale Processes 712
Analytical Problems 714
Numerical Exercises 715
21 Equatorial Dynamics 720
21.1 Equatorial Beta Plane 720
21.2 Linear Waves Theory 722
21.3 El Niño – Southern Oscillation (ENSO) 726
21.3.1 The Ocean 730
21.3.2 The Atmosphere 734
21.3.3 The Coupled Model 734
21.4 ENSO Forecasting 735
Analytical Problems 739
Numerical Exercises 740
22 Data Assimilation 744
22.1 Need for Data Assimilation 744
22.2 Nudging 749
22.3 Optimal Interpolation 750
22.4 Kalman Filtering 758
22.5 Inverse Methods 762
22.6 Operational Models 769
Analytical Problems 773
Numerical Exercises 775
VI Web site Information 780
A. Elements of Fluid Mechanics 782
A.1 Budgets 782
A.2 Equations in Cylindrical Coordinates 787
A.3 Equations in Spherical Coordinates 788
A.4 Vorticity and Rotation 789
Analytical Problems 790
Numerical Exercise 791
B. Wave Kinematics 792
B.1 Wavenumber and Wavelength 792
B.2 Frequency, Phase Speed, and Dispersion 795
B.3 Group Velocity and Energy Propagation 797
Analytical Problems 800
Numerical Exercises 800
C. Recapitulation of Numerical Schemes 802
C.1 The Tridiagonal System Solver 802
C.2 1D Finite-Difference Schemes of Various Orders 804
C.3 Time-Stepping Algorithms 805
C.4 Partial-Derivatives Finite Differences 806
C.5 Discrete Fourier Transform and Fast Fourier Transform 806
Analytical Problems 811
Numerical Exercises 812
References 814
Index 834
A 834
B 835
C 835
D 836
E 837
F 838
G 839
H 839
I 839
J 840
K 840
L 841
M 841
N 842
O 842
P 843
Q 843
R 844
S 844
T 846
U 846
V 846
W 847
Y 847
Z 847

Textauszug

Chapter 1

Introduction

Benoit Cushman-Roisin,    Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire 03755, USA

Jean-Marie Beckers,    Département d’Astrophysique, Géophysique et Océanographie, Université de Liège, B-4000 Liège, Belgium

Abstract

This opening chapter defines the discipline known as geophysical fluid dynamics, stresses its importance, and highlights its most distinctive attributes. A brief history of numerical simulations in meteorology and oceanography is also presented. Scale analysis and its relationship with finite differences are introduced to show how discrete numerical grids depend on the scales under investigation and how finite differences permit the approximation of derivatives at those scales. The problem of unresolved scales is introduced as an aliasing problem in discretization.

Keywords

Aliasing; computational fluid dynamics (CFD); data acquisition; discretization; forecasting; hurricanes; Jupiter; Lewis Fry Richardson; meteorological office; numerical simulations; rotation; scales of motion; stratification; Walsh Cottage

1.1 Objective

The object of geophysical fluid dynamics is the study of naturally occurring, large-scale flows on Earth and elsewhere, but mostly on Earth. Although the discipline encompasses the motions of both fluid phases – liquids (waters in the ocean, molten rock in the outer core) and gases (air in our atmosphere, atmospheres of other planets, ionized gases in stars) – a restriction is placed on the scale of these motions. Only the large-scale motions fall within the scope of geophysical fluid dynamics. For example, problems related to river flow, microturbulence in the upper ocean, and convection in clouds are traditionally viewed as topics specific to hydrology, oceanography, and meteorology, respectively. Geophysical fluid dynamics deals exclusively with those motions observed in various systems and under different guises but nonetheless governed by similar dynamics. For example, large anticyclones of our weather are dynamically germane to vortices spun off by the Gulf Stream and to Jupiter’s Great Red Spot. Most of these problems, it turns out, are at the large-scale end, where either the ambient rotation (of Earth, planet, or star) or density differences (warm and cold air masses, fresh and saline waters), or both assume some importance. In this respect, geophysical fluid dynamics comprises rotating-stratified fluid dynamics.

Typical problems in geophysical fluid dynamics concern the variability of the atmosphere (weather and climate dynamics), ocean (waves, vortices, and currents), and, to a lesser extent, the motions in the earth’s interior responsible for the dynamo effect, vortices on other planets (such as Jupiter’s Great Red Spot and Neptune’s Great Dark Spot), and convection in stars (the sun, in particular).

1.2 Importance of Geophysical Fluid Dynamics

Without its atmosphere and oceans, it is certain that our planet would not sustain life. The natural fluid motions occurring in these systems are therefore of vital importance to us, and their understanding extends beyond intellectual curiosity—it is a necessity. Historically, weather vagaries have baffled scientists and laypersons alike since times immemorial. Likewise, conditions at sea have long influenced a wide range of human activities, from exploration to commerce, tourism, fisheries, and even wars.

Thanks in large part to advances in geophysical fluid dynamics, the ability to predict with some confidence the paths of hurricanes (Figs. 1.1 and 1.2) has led to the establishment of a warning system that, no doubt, has saved numerous lives at sea and in coastal areas (Abbott, 2004). However, warning systems are only useful if sufficiently dense observing systems are implemented, fast prediction capabilities are available, and efficient flow of information is ensured. A dreadful example of a situation in which a warning system was not yet adequate to save lives was the earthquake off Indonesia’s Sumatra Island on 26 December 2004. The tsunami generated by the earthquake was not detected, its consequences not assessed, and authorities not alerted within the 2 h needed for the wave to reach beaches in the region. On a larger scale, the passage every 3–5 years of an anomalously warm water mass along the tropical Pacific Ocean and the western coast of South America, known as the El-Niño event, has long been blamed for serious ecological damage and disastrous economical consequences in some countries (Glantz, 2001; O’Brien, 1978). Now, thanks to increased understanding of long oceanic waves, atmospheric convection, and natural oscillations in air–sea interactions (D’Aleo, 2002; Philander, 1990), scientists have successfully removed the veil of mystery on this complex event, and numerical models (e.g., Chen, Cane, Kaplan, Zebiak & Huang, 2004) offer reliable predictions with at least one year of lead time, that is, there is a year between the moment the prediction is made and the time to which it applies.

Figure 1.1 Hurricane Frances during her passage over Florida on 5 September 2004. The diameter of the storm was about 830 km, and its top wind speed approached 200 km per hour. Courtesy of NOAA, Department of Commerce, Washington, D.C.

Figure 1.2 Computer prediction of the path of Hurricane Frances. The calculations were performed on Friday, 3 September 2004, to predict the hurricane path and characteristics over the next 5 days (until Wednesday, 8 September). The outline surrounding the trajectory indicates the level of uncertainty. Compare the position predicted for Sunday, 5 September, with the actual position shown on Fig. 1.1. Courtesy of NOAA, Department of Commerce, Washington, D.C.

Having acknowledged that our industrial society is placing a tremendous burden on the planetary atmosphere and consequently on all of us, scientists, engineers, and the public are becoming increasingly concerned about the fate of pollutants and greenhouse gases dispersed in the environment and especially about their cumulative effect. Will the accumulation of greenhouse gases in the atmosphere lead to global climatic changes that, in turn, will affect our lives and societies? What are the various roles played by the oceans in maintaining our present climate? Is it possible to reverse the trend toward depletion of the ozone in the upper atmosphere? Is it safe to deposit hazardous wastes on the ocean floor? Such pressing questions cannot find answers without, first, an in-depth understanding of atmospheric and oceanic dynamics and, second, the development of predictive models. In this twin endeavor, geophysical fluid dynamics assumes an essential role, and the numerical aspects should not be underestimated in view of the required predictive tools.

1.3 Distinguishing Attributes of Geophysical Flows

Two main ingredients distinguish the discipline from traditional fluid mechanics: the effects of rotation and those of stratification. The controlling influence of one, the other, or both leads to peculiarities exhibited only by geophysical flows. In a nutshell, this book can be viewed as an account of these peculiarities.

The presence of an ambient rotation, such as that due to the earth’s spin about its axis, introduces in the equations of motion two acceleration terms that, in the rotating framework, can be interpreted as forces. They are the Coriolis force and the centrifugal force. Although the latter is the more palpable of the two, it plays no role in geophysical flows; however, surprising this may be.1 The former and less intuitive of the two turns out to be a crucial factor in geophysical motions. For a detailed explanation of the Coriolis force, the reader is referred to the following chapter in this book or to the book by Stommel and Moore (1989). A more intuitive explanation and laboratory illustrations can be found in Chapter 6 of Marshall and Plumb (2008).

In anticipation of the following chapters, it can be mentioned here (without explanation) that a major effect of the Coriolis force is to impart a certain vertical rigidity to the fluid. In rapidly rotating, homogeneous fluids, this effect can be so strong that the flow displays strict columnar motions; that is, all particles along the same vertical evolve in concert, thus retaining their vertical alignment over long periods of time. The discovery of this property is attributed to Geoffrey I. Taylor, a British physicist famous for his varied contributions to fluid dynamics. (See the short biography at the end of Chapter 7.) It is said that Taylor first arrived at the rigidity property with mathematical arguments alone. Not believing that this could be correct, he then performed laboratory experiments that revealed, much to his amazement, that the theoretical prediction was indeed correct. Drops of dye released in such rapidly rotating, homogeneous fluids form vertical streaks, which, within a few rotations, shear laterally to form spiral sheets of dyed fluid (Fig. 1.3). The vertical coherence of these sheets is truly fascinating!

Figure 1.3 Experimental evidence of the rigidity of a rapidly rotating, homogeneous fluid. In a spinning vessel filled with clear water, an initially amorphous cloud of aqueous dye is transformed in the course of several rotations into perfectly vertical sheets,...

Produkt-Details

• Erscheint lt. Verlag: 26.8.2011
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik
• Themenwelt: Technik
• ISBN-10: 0-08-091678-3 / 0080916783
• ISBN-13: 978-0-08-091678-1 / 9780080916781