ALBERT

Note: CONTENTS: PREFACE. - INTRODUCTION. - 1 DESCRIBING INVERSE PROBLEMS. - 1.1 Formulating Inverse Problems. - 1.2 The Linear Inverse Problem. - 1.3 Examples of Formulating Inverse Problems. - 1.4 Solutions to Inverse Problems. - 2 SOME COMMENTS ON PROBABILITY THEORY. - 2.1 Noise and Random Variables. - 2.2 Correlated Data. - 2.3 Functions of Random Variables. - 2.4 Gaussian Distributions. - 2.5 Testing the Assumption of Gaussian Statistics. - 2.6 Confidence Intervals. - 3 SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1: THE LENGTH METHOD. - 3.1 The Lengths of Estimates. - 3.2 Measures of Length. - 3.3 Least Squares for a Straight Line. - 3.4 The Least Squares Solution of the Linear Inverse Problem. - 3.5 Some Examples. - 3.6 The Existence of the Least Squares Solution. - 3.7 The Purely Underdetermined Problem. - 3.8 Mixed-Determined Problems. - 3.9 Weighted Measures of Length as a Type of A Priori Information. - 3.10 Other Types of A Priori Information. - 3.11 The Variance of the Model Parameter Estimates. - 3.12 Variance and Prediction Error of the Least Squares Solution. - 4 SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES. - 4.1 Solutions versus Operators. - 4.2 The Data Resolution Matrix. - 4.3 The Model Resolution Matrix. - 4.4 The Unit Covariance Matrix. - 4.5 Resolution and Covariance of Some Generalized Inverses. - 4.6 Measures of Goodness of Resolution and Covariance. - 4.7 Generalized Inverses with Good Resolution and Covariance. - 4.8 Sidelobes and the Backus-Gilbert Spread Function. - 4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem. - 4.10 Including the Covariance Size. - 4.11 The Trade-off of Resolution and Variance. - 5 SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS. - 5.1 The Mean of a Group of Measurements. - 5.2 Maximum Likelihood Solution of the Linear Inverse Problem. - 5.3 A Priori Distributions. - 5.4 Maximum Likelihood for an Exact Theory. - 5.5 Inexact Theories. - 5.6 The Simple Gaussian Case with a Linear Theory. - 5.7 The General Linear, Gaussian Case. - 5.8 Equivalence of the Three Viewpoints. - 5.9 The F Test of Error Improvement Significance. - 5.10 Derivation of the Formulas of Section 5.7. - 6 NONUNIQUENESS AND LOCALIZED AVERAGES. - 6.1 Null Vectors and Nonuniqueness. - 6.2 Null Vectors of a Simple Inverse Problem. - 6.3 Localized Averages of Model Parameters. - 6.4 Relationship to the Resolution Matrix. - 6.5 Averages versus Estimates. - 6.6 Nonunique Averaging Vectors and A Priori Information. - 7 APPLICATIONS OF VECTOR SPACES. - 7.1 Model and Data Spaces. - 7.2 Householder Transformations. - 7.3 Designing Householder Transformations. - 7.4 Transformations That Do Not Preserve Length. - 7.5 The Solution of the Mixed-Determined Problem. - 7.6 Singular-Value Decomposition and the Natural Generalized Inverse. - 7.7 Derivation of the Singular-Value Decomposition. - 7.8 Simplifying Linear Equality and Inequality Constraints. - 7.9 Inequality Constraints. - 8 LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS. - 8.1 L1 Norms and Exponential Distributions. - 8.2 Maximum Likelihood Estimate of the Mean of an Exponential Distribution. - 8.3 The General Linear Problem. - 8.4 Solving L1 Norm Problems. - 8.5 The L [Infinity symbol] Norm. - 9 NONLINEAR INVERSE PROBLEMS. - 9.1 Parameterizations. - 9.2 Linearizing Parameterizations. - 9.3 The Nonlinear Inverse Problem with Gaussian Data. - 9.4 Special Cases. - 9.5 Convergence and Nonuniqueness of Nonlinear L2 Problems. - 9.6 Non-Gaussian Distributions. - 9.7 Maximum Entropy Methods. - 10 FACTOR ANALYSIS. - 10.1 The Factor Analysis Problem. - 10.2 Normalization and Physicality Constraints. - 10.3 Q-Mode and R-Mode Factor Analysis. - 10.4 Empirical Orthogonal Function Analysis. - 11 CONTINUOUS INVERSE THEORY AND TOMOGRAPHY. - 11.1 The Backus-Gilbert Inverse Problem. - 11.2 Resolution and Variance Trade-off. - 11.3 Approximating Continuous Inverse Problems as Discrete Problems. - 11.4 Tomography and Continuous Inverse Theory. - 11.5 Tomography and the Radon Transform. - 11.6 The Fourier Slice Theorem. - 11.7 Backprojection. - 12 SAMPLE INVERSE PROBLEMS. - 12.1 An Image Enhancement Problem. - 12.2 Digital Filter Design. - 12.3 Adjustment of Crossover Errors. - 12.4 An Acoustic Tomography Problem. - 12.5 Temperature Distribution in an Igneous Intrusion. - 12.6 L1, L2, and L [infinity symbol] Fitting of a Straight Line. - 12.7 Finding the Mean of a Set of Unit Vectors. - 12.8 Gaussian Curve Fitting. - 12.9 Earthquake Location. - 12.10 Vibrational Problems. - 13 NUMERICAL ALGORITHMS. - 13.1 Solving Even-Determined Problems. - 13.2 Inverting a Square Matrix. - 13.3 Solving Underdetermined and Overdetermined Problems. - 13.4 L2 Problems with Inequality Constraints. - 13.5 Finding the Eigenvalues and Eigenvectors of a Real Symmetric Matrix. - 13.6 The Singular-Value Decomposition of a Matrix. - 13.7 The Simplex Method and the Linear Programming Problem. - 14 APPLICATIONS OF INVERSE THEORY TO GEOPHYSICS. - 14.1 Earthquake Location and the Determination of the Velocity Structure of the Earth from Travel Time Data. - 14.2 Velocity Structure from Free Oscillations and Seismic Surface Waves. - 14.3 Seismic Attenuation. - 14.4 Signal Correlation. - 14.5 Tectonic Plate Motions. - 14.6 Gravity and Geomagnetism. - 14.7 Electromagnetic Induction and the Magnetotelluric Method. - 14.8 Ocean Circulation. - APPENDIX A: Implementing Constraints with Lagrange Multipliers. - APPENDIX B: L2 Inverse Theory with Complex Quantities. - REFERENCES. - INDEX

Note: Contents Prologue Acknowledgments List of Symbols PART I Foundations 1 INTRODUCTION: The Basic Challenge 1.1 The Climate System 1.2 Some Basic Observations 1.3 External Forcing 1.3.1 Astronomical Forcing 1.3.2 Tectonic Forcing 1.4 The Ice-Age Problem 2 TECHNIQUES FOR CLIMATE RECONSTRUCTION 2.1 Historical Methods 2.1.1 Direct Quantitative Measurements 2.1.2 Descriptive Accounts of General Environmental Conditions 2.2 Surficial Biogeologic Proxy Evidence 2.2.1 Annually Layered Life Forms 2.2.2 Surface Geomorphic Evidence 2.3 Conventional Nonisotopic Stratigraphic Analyses of Sedimentary Rock and Ice 2.3.1 Physical Indicators 2.3.2 Paleobiological Indicators (Fossil Faunal Types and Abundances) 2.4 Isotopic Methods 2.4.1 Oxygen Isotopes 2.4.2 Deuterium and Beryllium in Ice Cores 2.4.3 Stable Carbon Isotopes 2.4.4 Strontium and Osmium Isotopes 2.5 Nonisotopic Geochemical Methods 2.5.1 Cadmium Analysis 2.5.2 Greenhouse Gas Analysis of Trapped Air in Ice Cores 2.5.3 Chemical and Biological Constituents and Dust Layers in Ice Cores 2.6 Dating the Proxy Evidence (Geochronometry) 3 A SURVEY OF GLOBAL PALEOCLIMATIC VARIATIONS 3.1 The Phanerozoic Eon (Past 600 My) 3.2 The Cenozoic Era (Past 65 My) 3.3 The Plio-Pleistocene (Past 5 My) 3.4 Variations during the Last Ice Age: IRD Events 3.5 The Last Glacial Maximum (20 ka) 3.6 Postglacial Changes: The Past 20 ky 3.7 The Past 100 Years 3.8 The Generalized Spectrum of Climatic Variance 3.9 A Qualitative Discussion of Causes 4 GENERAL THEORETICAL CONSIDERATIONS 4.1 The Fundamental Equations 4.2 Time Averaging and Stochastic Forcing 4.3 Response Times and Equilibrium 4.4 Spatial Averaging 4.5 Climatic-Mean Mass and Energy Balance Equations 4.5.1 The Water Mass Balance 4.5.2 Energy Balance 5 SPECIAL THEORETICAL CONSIDERATIONS FOR PALEOCLIMATE: Structuring a Dynamical Approach 5.1 A Basic Problem: Noncalculable Levels of Energy and Mass Flow 5.2 An Overall Strategy 5.3 Notational Simplifications for Resolving Total Climate Variability 5.4 A Structured Dynamical Approach 5.5 The External Forcing Function, F 5.5.1 Astronomical/Cosmic Forcing 5.5.2 Tectonic Forcing 6 BASIC CONCEPTS OF DYNAMICAL SYSTEMS ANALYSIS: Prototypical Climatic Applications 6.1 Local (or Internal) Stability 6.2 The Generic Cubic Nonlinearity 6.3 Structural (or External) Stability: Elements of Bifurcation Theory 6.4 Multivariable Systems 6.4.1 The Two-Variable Phase Plane 6.5 A Prototype Two-Variable Model 6.5.1 Sensitivity of Equilibria to Changes in Parameters: Prediction of the Second Kind 6.5.2 Structural Stability 6.6 The Prototype Two-Variable System as a Stochastic-Dynamical System: Effects of Random Forcing 6.6.1 The Stochastic Amplitude 6.6.2 Structural Stochastic Stability 6.7 More Than Two-Variable Systems: Deterministic Chaos PART II Physics of the Separate Domains 7 MODELING THE ATMOSPHERE AND SURFACE STATE AS FAST-RESPONSE COMPONENTS 7.1 The General Circulation Model 7.2 Lower Resolution Models: Statistical-Dynamical Models and the Energy Balance Model 7.2.1 A Zonal-Average SDM 7.2.2 Axially Asymmetric SDMs 7.2.3 The Complete Time-Average State 7.3 Thermodynamic Models 7.3.1 Radiative-Convective Models 7.3.2 Vertically Averaged Models (the EBM) 7.4 The Basic Energy Balance Model 7.5 Equilibria and Dynamical Properties of the Zero-Dimensional (Global Average) EBM 7.6 Stochastic Resonance 7.7 The One-Dimensional (Latitude-Dependent) EBM 7.8 Transitivity Properties of the Atmospheric and Surface Climatic State: Inferences from a GCM 7.9 Closure Relationships Based on GCM Sensitivity Experiments 7.9.1 Surface Temperature Sensitivity 7.10 Formal Feedback Analysis of the Fast-Response Equilibrium State 7.11 Paleoclimatic Simulations 8 THE SLOW-RESPONSE "CONTROL" VARIABLES: An Overview 8.1 The Ice Sheets 8.1.1 Key Variables 8.1.2 Observations 8.2 Greenhouse Gases: Carbon Dioxide 8.3 The Thermohaline Ocean State 8.4 A Three-Dimensional Phase-Space Trajectory 9 GLOBAL DYNAMICS OF THE ICE SHEETS 9.1 Basic Equations and Boundary Conditions 9.2 A Scale Analysis 9.3 The Vertically Integrated Ice-Sheet Model 9.4 The Surface Mass Balance 9.5 Basal Temperature and Melting 9.6 Deformable Basal Regolith 9.7 Ice Streams and Ice Shelves 9.8 Bedrock Depression 9.9 Sea Level Change and the Ice Sheets: The Depression-Calving Hypothesis 9.10 Paleoclimatic Applications of the Vertically Integrated Model 9.11 A Global Dynamical Equation for Ice Mass 10 DYNAMICS OF ATMOSPHERIC CO2 10.1 The Air-Sea Flux, Q↑ 10.1.1 Qualitative Analysis of the Factors Affecting Q↑ 10.1.2 Mathematical Formulation of the Ocean Carbon Balance 10.1.3 A Parameterization for Q↑ 10.2 Terrestrial Organic Carbon Exchange, W↑G 10.2.1 Sea Level Change Effects 10.2.2 Thermal Effects 10.2.3 Ice Cover Effects 10.2.4 Long-Term Terrestrial Organic Burial, W↓G 10.2.5 The Global Mass Balance of Organic Carbon 10.3 Outgassing Processes, V↑ 10.4 Rock Weathering Downdraw, W↓ 10.5 A Global Dynamical Equation for Atmospheric CO2 10.6 Modeling the Tectonically Forced CO2 Variations, µˆ : Long-Term Rock Processes 10.6.1 The Long-Term Oceanic Carbon Balance 10.6.2 The GEOCARB Model 10.7 Overview of the Full Global Carbon Cycle 11 SIMPLIFIED DYNAMICS OF THE THERMOHALINE OCEAN STATE 11.1 General Equations 11.1.1 Boundary Conditions 11.2 A Prototype Four-Box Ocean Model 11.3 The Wind-Driven, Local-Convective, and Baroclinic Eddy Circulations 11.3.1 The Wind-Driven Circulation: Gyres and Upwelling 11.3.2 Local Convective Overturnings and Baroclinic Eddy Circulations 11.4 The Two-Box Thermohaline Circulation Model: Possible Bimodality of the Ocean State 11.4.1 The Two-Box System 11.4.2 A Simple Model of the TH Circulation 11.4.3 Meridional Fluxes 11.4.4 Dynamical Analysis of the Two-Box Model 11.5 Integral Equations for the Deep Ocean State 11.5.1 The Deep Ocean Temperature 11.5.2 The Deep Ocean Salinity 11.6 Global Dynamical Equations for the Thermohaline State: θ and Sφ PART III Unified Dynamical Theory 12 THE COUPLED FAST- AND SLOW-RESPONSE VARIABLES AS A GLOBAL DYNAMICAL SYSTEM: Outline of a Theory of Paleoclimatic Variation 12.1 The Unified Model: A Paleoclimate Dynamics Model 12.2 Feedback-Loop Representation 12.3 Elimination of the Fast-Response Variables: The Center Manifold 12.4 Sources of Instability: The Dissipative Rate Constants 12.5 Formal Separation into Tectonic Equilibrium and Departure Equations 13 FORCED EVOLUTION OF THE TECTONIC-MEAN CLIMATIC STATE 13.1 Effects of Changing Solar Luminosity and Rotation Rate 13.1.1 Solar Luminosity (S) 13.1.2 Rotation Rate (Ω) 13.2 General Effects of Changing Land-Ocean Distribution and Topography (h) 13.3 Effects of Long-Term Variations of Volcanic and Cosmic Dust and Bolides 13.4 Multimillion-Year Evolution of CO2 13.4.1 The GEOCARB Solution 13.4.2 First-Order Response of Global Ice Mass and Deep Ocean Temperature to Tectonic CO2 Variations 13.5 Possible Role of Salinity-Driven Instability of the Tectonic-Mean State 13.6 Snapshot Atmospheric and Surficial Equilibrium Responses to Prescribed y-Fields Using GCMs 14 THE LATE CENOZOIC ICE-AGE DEPARTURES: An Overview of Previous Ideas and Models 14.1 General Review: Forced vs. Free Models 14.1.1 Models in Which Earth-Orbital Forcing Is Necessary 14.1.2 Instability-Driven (Auto-oscillatory) Models 14.1.3 Hierarchical Classification in Terms of Increasing Physical Complexity 14.2 Forced Ice-Line Models (Box 1, Fig. 14-1) 14.3 Ice-Sheet Inertia Models 14.3.1 The Simplest Forms (Box 2) 14.3.2 More Physically Based Ice-Sheet Models: First Applications 14.3.3 Direct Bedrock Effects (Box 3) 14.3.4 Bedrock-Calving Effects (Box 4) 14.3.5 Basal Meltwater and Sliding (Box 5) 14.3.6 Ice Streams and Ice Shelf Effects 14.3.7 Continental Ice-Sheet Movement (Box 6) 14.3.8 Three-Dimensional (λ, φ, hI) Ice-Sheet Models 14.4 The Need for Enhancement of the Coupled Ice-Sheet/Atmospheric Climate Models 14.5 Ice-Sheet Variables Coupled with Additional Slow-Response Variables 14.5.1 Regolith Mass, mr (Box 7) 14.5.2 The Deep Ocean Te

Note: Contents List of Acronyms List of Symbols Foreword Preface Prologue Chapter 1 Introduction to Ocean Dynamics 1.1 Types, Advantages, and Limitations of Ocean Models 1.2 Recent Examples 1.3 Governing Equations 1.4 Vorticity Conservation 1.5 Nondimensional Numbers and Scales of Motion 1.6 Geostrophic Flow and Thermal Wind 1.7 Inertial Motions 1.8 Ekman Layers 1.9 Sverdrup Transport 1.10 Western Boundary Intensification (Stommel Solution) 1.11 Gyre Scale Circulation (Munk Solution) 1.12 Barotropic Currents over Topography 1.13 Baroclinic Transport over Topography 1.14 Coastal Upwelling and Fronts 1.15 Mesoscale Eddies and Variability 1.16 Thermohaline Circulation and Box (Reservoir) Models 1.17 Numerical Models Chapter 2 Introduction to Numerical Solutions 2.1 Introduction 2.1.1 Architecture 2.1.2 Computational Errors 2.2 Ordinary Differential Equations 2.2.1 Runge-Kutta Method 2.3 Partial.Differential Equations 2.3.1 Consistency, Convergence, and Stability 2.3.2 Elliptic, Hyperbolic, and Parabolic Systems 2.4 Elliptic Equations and Steady-State Problems 2.4.1 Direct Solvers 2.4.2 Iterative Solvers and Relaxation Methods 2.4.3 Preconditioned Conjugate Gradient Method 2.4.4 Multigrid Methods 2.4.5 Pseudo-transient Method 2.5 Time Dependent Problems 2.5.1 Advection Equation and Hyperbolic Systems 2.5.2 Diffusion Equation and Parabolic Systems 2.6 Finite-Difference (Grid Point) Methods 2.6.1 Staggered Grids 2.6.2 Time Differencing and Filtering 2.6.3 Computational Grids 2.7 Spectral (Spectral Transform) Methods 2.8 Finite-Element Methods 2.8.1 Spectral Element Approach 2.9 Parameterization of Subgrid Scale Processes 2.10 Lateral Open Boundary Conditions 2.11 Computational Issues 2.12 Examples 2.12.1 Inertial Oscillations 2.12.2 Thermohaline Circulation 2.12.3 Normal Modes 2.12.4 Gyre Scale Circulation 2.12.5 Advection Problems 2.12.6 M.I.T. Nonhydrostatic Global Model Chapter 3 Equatorial Dynamics and Reduced Gravity Models Solutions 3.1 Oceanic Dynamical Response to Forcing 3.2 Governing Equations 3.3 Equatorial Waves 3.3.1 Kelvin Waves 3.3.2 Yanai Waves 3.3.3 Rossby Waves 3.3.4 Inertia-Gravity (Poincare) Waves 3.4 Equatorial Currents 3.5 Reduced Gravity Model of Equatorial Processes Chapter 4 Midlatitude Dynamics and Quasi-Geostrophic Models 4.1 Linear Motions 4.1.1 Inertia-Gravity (Sverdrup/Poincare) Waves 4.1.2 Kelvin Waves 298 4.1.3 Planetary Ross by Waves 4.1.4 Topographic Rossby Waves 4.2 Continuous Stratification 4.3 Geostrophic Adjustment and Instabilities 4.3.1 Geostrophic Adjustment 4.3.2 Instabilities 4.4 Spinup 4.5 Quasi-Geostrophic Models 4.5.1 Governing Equations 4.5.2 Applications Chapter 5 High-Latitude Dynamics and Sea-Ice Models 5.1 Salient Features of Ice Cover 5.2 Momentum Equations for Sea Ice 5.3 Constitutive Law for Sea Ice (Ice Rheology) 5.3.1 Viscous-Plastic Ice Rheology 5.3.2 Elastic-Viscous-Plastic Ice Rheology 5.4 Continuity Equations for Sea Ice 5.5 Response of Sea Ice to Storm Passage 5.6 Numerics 5.6.1 Governing Equations in Orthogonal Curvilinear Coordinates 5.6.2 Solution Technique Chapter 6 Tides and Tidal Modeling 6.1 Description of Tides 6.2 Formulation: Tidal Potential 6.3 Body, Load, Atmospheric, and Radiational Tides 6.3.1 Body (Solid Earth) Tides 6.3.2 Load Tides 6.3.3 Atmospheric Tides 6.3.4 Radiational Tides 6.4 Dynamical Theory of Tides: Laplace Tidal Equations 6.5 Equilibrium Theory of Tides 6.6 Tidal Analysis: Orthotides 6.7 Tidal Currents 6.8 Global Tidal Models 6.9 Regional Tidal Models 6.10 Geophysical Implications 6.10.1 Tidal Dissipation and LOD 6.10.2 Tidal Energetics 6.11 Changes in Earth's Rotation 6.12 Baroclinic (Internal) Tides 6.13 Long-Period Tides 6.14 Shallow Water Tides and Residual Currents 6.15 Summary Chapter 7 Coastal Dynamics and Barotropic Models 7.1 Wind- and Buoyancy-Driven Currents 7.2 Tidal Motions 7.3 Continental Shelf Waves 7.4 Modeling Shelf Circulation 7.5 Barotropic Models 7.5.1 Coastal Ocean Response to Wind Forcing 7.5.2 Storm Surges and Storm Surge Modeling 7.5.3 Response to Pressure Forcing Chapter 8 Data and Data Processing 8.1 In Situ Observational Data 8.1.1 XBT, CTD, CM, ADCP, and Drifter Data 8.1.2 Historical Hydrographic Data 8.1.3 Historical Marine Surface Data 8.2 Remotely Sensed Data 8.2.1 Sea Surface Temperature from IR Sensors 8.2.2 Sea Surface Winds from Microwave Sensors 8.2.3 Chlorophyll and Optical Clarity from Color Sensors 8.2.4 Sea Surface Height from Satellite Altimetry 8.3 NWP Products 8.4 Preprocessing of Observational Data and Postprocessing of Model Output 8.4.1 Graphics and Visualization of Model Output 8.4.2 Analyses Chapter 9 Sigma-Coordinate Regional and Coastal Models 9.1 Introduction 9.2 Governing Equations 9.3 Vertical Mixing 9.4 Boundary Conditions 9.5 Mode Splitting 9.6 Numerics 9.6.1 Vertical Direction 9.6.2 Horizontal Direction 9.7 Numerical Problems 9.8 Applications 9.9 Code Structure Chapter 10 Multilevel Basin Scale and Global Models 10.1 Introduction 10.2 Governing Equations 10.3 Isopycnal Diffusion 10.4 Architecture and Other Model Features 10.5 Applications 10.6 Hybrid s-Coordinate Models 10.7 Regional z-Level Models Chapter 11 Layered and Isopycnal Models 11.1 Layered Models 11.2 Isopycnal Models Chapter 12 Ice-Ocean Coupled Models 12.1 Sea-Ice Models 12.2 Coupled Ice-Ocean Models Chapter 13 Ocean-Atmosphere Coupled Models 13.1 Coupling between the Ocean and the Atmosphere 13.2 Coupled Ocean-Atmosphere General Circulation Models 13.3 Regional Coupled Ocean-Atmosphere Models Chapter 14 Data Assimilation and Nowcasts/ Forecasts 14.1 Introduction 14.2 Direct Insertion 14.3 Nudging 14.4 Statistical Assimilation Schemes 14.4.1 Kalman Filter 14.4.2 Reduced State Space Kalman Filters 14.4.3 Optimal Interpolation (OI) Scheme 14.5 Variational Methods 14.5.1 Adjoint Models 14.6 Predictability of Nonlinear Systems-Low Order Paradigms 14.7 Nowcasts/Forecasts in the Gulf of Mexico Appendix A Equations of State A.1 Equation of State for the Ocean A.2 Equation of State for the Atmosphere Appendix B Wavelet Transforms B.1 Introduction B.1.1 Theory B.1.2 Continuous Wavelet Transforms (CWT) B.1.3 Discrete Wavelet Transforms (DWT) B.2 Examples B.3 Wavelet Transforms and Stochastic Processes B.4 Two-Dimensional Wavelet Transforms B.5 Cross Wavelet Transforms (CrWT) B.6 Error Analysis Appendix C Empirical Orthogonal Functions and Empirical Normal Modes C.1 Empirical Orthogonal Functions C.1.1 Complex EOFs C.1.2 Singular Spectrum Analysis C.1.3 Extended EOFs C.1.4 Coupled Pattern Analysis C.2 Empirical Normal Modes Appendix D Units and Constants D.1 Useful Quantities D.1.1 SI (International System of Units) Units and Conventions D.1.2 Useful Conversion Factors D.1.3 Useful Universal Constants D.1.4 Useful Geodetic Constants D.1.5 Useful Physical Constants D.1.6 Useful Dynamical Quantities D.2 Important Scales and Quantities D.2.1 Length Scales D.2.2 Timescales D.2.3 Velocity Scales D.2.4 Nondimensional Quantities D.3 Useful Websites References Biographies Index

Note: Contents: Contributors. - Foreword. - Preface. - PART 1. NUMERICAL WEATHER PREDICTION. - Medium-range forecasting at the ECMWF / Lennart Bengtsson. - 1. Introduction. - 2. The physical and mathematical basis for medium-range forecasting. - 3. Numerical methods and modeling techniques. - 4. Observations, their use and importance. - 5. Operational application and results. - 6. Problems and prospects in numerical weather prediction. - 7. Concluding remarks. - References. - Extended range forecasting / K. Miyakoda and J. Sirutis. - 1. Introduction. - 2. An evolution of 10-day forecast performance. - 3. Examples of monthly forecasts. - 4. A projection of seasonal forecasts. - 5. Postscript. - References. - Predictability / J. Shukla. - 1. Introduction. - 2. Classical predictability studies. - 3. Predictability of space-time averages. - 4. Some outstanding problems. - 5. Concluding remarks. - References. - Data Assimilation / W. Bourke, R. Seaman, and K. Puri. - 1. Introduction. - 2. Evolution of assimilation and the FGGE. - 3. Components of four-dimensional assimilation systems. - 4. Characteristics of some current Assimilation schemes. - 5. Role of four-dimensional assimilation in research and operations. - 6. Conclusion. - References. - PART 2. MESOSCALE DYNAMICS. - Predictability of mesoscale atmospheric motions / Richard A. Anthes, Ying-Hwa Kuo, David P. Baumhefner, Ronald M. Errico, and Thomas W. Bettge. - 1. Introduction. - 2. Classic predictability experiments and their relationship to mesoscale predictability. - 3. Preliminary predictability study with a mesoscale model. - 4. Discussion and comparison with a predictability study using a global model. - 5. Summary and conclusions. - References. - Thermal and orographic mesoscale atmospheric systems - an essay / Roger A. Pielke. - 1. Introduction. - 2. Summary of major research accomplishments. - 3. Research areas. - 4. Eventual goals. - References. - Advances in the theory of atmospheric fronts / I. Orlanski, B. Ross, L. Polinsky, and R. Shaginaw. - 1. Introduction. - 2. Baroclinic waves and fronts. - 3. Mature front. - 4. What observed features can be explained by theory?. - 5. What other processes are important in Frontogenesis?. - References. - PART 3. TROPICAL DYNAMICS. - Numerical modeling of tropical cyclones / Yoshio Kurihara. - 1. Introduction. - 2. Numerical models of hurricanes. - 3. Numerical simulation of tropcial cyclones. - 4. Some challenging issues in the future. - Appendix. GFDL Hurricane Model. - References. - Numerical weather prediction in low latitudes / T. N. Krishnamurti. - 1. Introduction. - 2. Initialization: dynamic, normal mode, and physical. - 3. Parameterization of physical processes. - 4. Medium-range prediction of monsoon disturbances. - 5. On the prediction of the quasi-Stationary component. - 6. Scope of future research. - References. - PART 4. TURBULENCE AND CONVECTION. - Sub-grid-scale turbulence modeling / J. W. Deardorff. - 1. Introduction: the need for grid-scale Reynolds averaging. - 2. The effect of grid-volume Reynolds averaging. - 3. The sub-grid scale Eddy Coefficient. - 4. Recent developments. - 5. Future outlook. - References. - Ensemble average, turbulence closure / George L. Mellor. - 1. Introduction. - 2. The turbulence macroscale and turbulence closure. - 3.Averaging distance for measurements in the atmosphere and oceans and for numerical models. - 4. Numerical modeling applications and horizontal diffusion. - 5. Concluding remarks. - References. - The planetary boundary layer / H. A. Panofsky. - 1. General characteristics. - 2. The equations in the PBL. - 3. The surface layer. - 4. First- and second-order closures. - 5. Boundary-layer models. - 6. Boundary-layer parameterization. - References. - Modeling studies and convection / Yoshi Ogura. - 1. Introduction. - 2. Bénard-Rayleigh Convection. - 3. Complexity of convection in the atmosphere. - 4. Shallow moist convection. - 5. Deep moist convection. - 6. Feedback effects of cumulus clouds on larger-scale environments. - 7. Concluding remarks. - References. - Index.

Note: Contents: Contributors. - Foreword by Paul J. Crutzen. - Preface by David Schimel. - Introduction. - 1 Uncertainties of Global Biogeochemical Predictions / E. D. Schulze, D. S. S. Schimel. - 1.1 Introduction. - 1.2 The IGBP Transect Approach. - 1.2.1 The Patagonian Transect. - 1.2.2 The Australian Transect. - 1.2.3 The European Transect. - 1.3 Variability in Processes. - 1.4 Biome Approach and Functional Types. - 1.5 New Approaches to Functional Diversity. - 1.6 Conclusions. - References. - 2 Uncertainties of Global Climate Predictions / L. Bengtsson. - 2.1 Introduction. - 2.2 Observational Evidence. - 2.3 Physical Rationale. - 2.3.1 Stochastic Forcing. - 2.3.2 Solar irradiation Changes. - 2.3.3 Volcanic Effects. - 2.3.4 Anthropogenic Effects. - 2.4 Response to Forcing of the Climate System. - 2.5 Results from Climate Change Prediction Experiments. - 2.6 Summary and Conclusions. - References. - 3 Uncertainties in the Atmospheric Chemical System / G. P. Brasseur, E. A. H. Holland. - 3.1 Introduction. - 3.2 Synthetic View of Chemical Processes in the Troposphere. - 3.3 The IMAGES Model. - 3.4 Changes in the Chemical Composition of the Global Troposphere. - 3.5 Concluding Remarks. - References. - 4 Inferring Biogeochemical Sources and Sinks from Atmospheric Concentrations: General Consideration and Applications in Vegetation Canopies / M. Raupach. - 4.1 Introduction. - 4.2 Scalar and Isotopic Molar Balances. - 4.2.1 General Principles. - 4.2.2 Single-Point Eulerian Equations. - 4.2.3 Source Terms for CO2. - 4.2.4 Single-Point Lagrangian Equations. - 4.3 Inverse Methods for Inferring Scalar Sources and Sinks in Canopies. - 4.3.1 General Principles. - 4.3.2 Localized Near Field Theory. - 4.3.3 The Dispersion Matrix. - 4.3.4 Turbulent Velocity Field. - 4.3.5 Solutions for Forward, Inverse and Implicit Problems. - 4.3.6 Field Tests. - 4.4 Inverse Methods and Isotopes in Canopies. - 4.4.1 Path Integrals and Keeling Plots. - 4.4.2 Inverse Lagrangian Analysis of Isotopic Composition. - 4.5 Summary and Conclusions. - Appendix A. - Appendix B. - References. - 5 Biogeophysical Feedbacks and the Dynamics of Climate / M. Claussen. - 5.1 Introduction. - 5.2 Synergisms. - 5.2.1 High Northern Latitudes. - 5.2.2 Subtropics. - 5.3 Multiple Equilibria. - 5.4 Transient Interaction. - 5.5 Perspectives. - References. - 6 Land-Ocean-Atmosphere Interactions and Monsoon Climate Change: A Paleo-Perspective / J. E. Kutzbach, Michael T. Coe, S. P. Harrison and M. T. Coe. - 6.1 Introduction. - 6.2 Response of the Monsoon to Orbital Forcing. - 6.3 Ocean Feedbacks on the Monsoon. - 6.4 Land-Surface Feedbacks on the Monsoon. - 6.5 Synergies between the Land, Ocean and Atmosphere. - 6.6 The Role of Climate Variability. - 6.7 Final Remarks. - References. - 7 Paleobiogeochemistry / I. C. Prentice, D. Raynaud. - 7.1 Introduction. - 7.2 Methane. - 7.3 Carbon Dioxide. - 7.4 Mineral Dust Aerosol. - 7.5 Scientific Challenges Posed by the Ice-Core Records. - 7.5.1 Methane. - 7.5.2 Carbon Dioxide. - 7.5.3 Mineral Dust Aerosol. - 7.6 Towards an Integrated Research Strategy for Palaeobiogeochemistry. - References. - 8 Should Phosphorus Availability Be Constraining Moist Tropical Forest Responses to Increasing CO2 Concentrations / J. Lloyd, M. I. Bird, E. M. Veenendaal and B. Kruijt. - 8.1 Introduction. - 8.2 Phosphorus in the Soils of the Moist Tropics. - 8.2.1 Soil Organic Phosphorus. - 8.2.2 Soil Inorganic Phosphorus. - 8.2.3 Soil Carbon/Phosphorus Interactions. - 8.3 States and Fluxes of Phosphorus in Moist Tropical Forests. - 8.3.1 Inputs and Losses of Phosphorus Through Rainfall, Dry Deposition and Weathering: Losses Via Leaching. - 8.3.2 Internal Phosphorus Flows in Moist Tropical Forests. - 8.3.3 Mechanisms for Enhanced Phosphorus Uptake in Low P Soils. - 8.4 Linking the Phosphorus and Carbon Cycles. - 8.4.1 To What Extent Does Phosphorus Availability Really Limit Moist Tropical Forest Productivity?. - 8.4.2 Tropical Plant Responses to Increases in Atmospheric CO2 Concentrations. - 8.4.3 Using a Simple Model to Examine CO2/Phosphorus Interactions in Tropical Forests. - References. - 9 Trees in Grasslands: Biogeochemical Consequences of Woody Plant Expansion / S. Archer, T. W. Boutton and K. A. Hibbard. - 9.1 Introduction. - 9.2 Woody Plant Encroachment in Grasslands and Savannas. - 9.3 The La Copita Case Study. - 9.3.1 Biogeographical and Historal Context. - 9.3.2 Herbaceous Retrogression and Soil Carbon Losses. - 9.3.3 Woody Plant Encroachment and Ecosystem Biogeochemistry. - 9.4 Degradation: Ecological Versus Socioeconomic. - 9.5 Implications for Ecosystem and Natural Resources Management. - 9.6 Summary. - References. - 10 Biogeochemistry in the Arctic: Patterns, Processes and Controls / S. Jonasson, F.S. Chapin, III and G. R. Shaver. - 10.1 Introduction. - 10.2 Tundra Organic Matter. - 10.2.1 Distribution of Organic Matter. - 10.2.2 Patterns and Controls of Organic Matter Turnover between Ecosystem Types. - 10.3 Tundra Nutrients. - 10.3.1 Nutrient Distribution and Controls of Nutrient Cycling. - 10.3.2 Nutrient Mineralization and Plant Nutrient Uptake. - 10.3.3 Are there Unaccounted Plant Sources of Limiting Nutrients?. - 10.4 Biogeochemical Responses to Experimental Ecosystem Manipulations. - 10.4.1 Applicability of Experimental Manipulations. - 10.4.2 Responses to Water Applications. - 10.4.3 Response to Nutrient Addition and Warming. - 10.4.4 Responses in Ecosystem Carbon Balance. - 10.5 Summary. - References. - 11 Evaporation in the Boreal Zone During Summer - Physics and Vegetation / F. M. Kelliher, I. Lloyd, C. Rebmann, C. Wirth and E. D. Schulze, D. D. Baldocchi. - 11.1 Introduction. - 11.2 Climate and Soil Water. - 11.3 Evaporation Theory. - 11.4 Evaporation During Summer and Rainfall. - 11.5 Forest Evaporation, Tree Life Form and Nitrogen. - 11.6 Conclusions. - References. - 12 Past and Future Forest Response to Rapid Climate Change / M.B. Davis. - 12.1 Introduction. - 12.2 Long-Distance Dispersal. - 12.3 Estimating Jump Distances. - 12.4 Interactions with Resident Vegetation - Constraints on Establishment. - 12.5 Interactions with Resident Vegetation - Competition for Light and Resulting Constraints on Population Growth. - 12.6 Conclusions. - References. - 13 Biogeochemical Models: Implicit vs. Explicit Microbiology / J. Schimel. - 13.1 Introduction. - 13.2 Microbiology in Biogeochemical Models. - 13.3 Dealing with Microbial Diversity in Models. - 13.4 Kinetic Effects of Microbial Population Size. - 13.5 Microbial Recovery from Stress. - 13.6 Conclusions. - References. - 14 The Global Soil Organic Carbon Pool / M. I. Bird, H. Santruckova, J. Lloyd and E. M. Veenendaal. - 14.1 Introduction: the Soil Carbon Pool and Global Change. - 14.2 Factors Affecting the Distribution of Soil Organic Carbon. - 14.3 Global Variations in the SOC Pool. - 14.4 The Limitations of Available Observational SOC Data. - 14.5 A Stratified Sampling Approach. - 14.6 Conclusions: Sandworld and Clayworld. - References. - 15 Plant Compounds and Their Turnover and Stability as Soil Organic Matter / G. Gleixner, C. Czimczik, C. Kramer, B. M. Lühker and M. W. I. Schmidt. - 15.1 Introduction. - 15.2 Pathways of Soil Organic Matter Formation. - 15.2.1 Formation and Decomposition of Biomass. - 15.2.2 The Influence of Environmental Conditions on SOM Formation. - 15.2.3 For