Energy Balance Climate Models Written by renowned experts in the field, this first book to focus exclusively on energy balance climate models provides a concise overview of the topic. It covers all major aspects, from the simplest zero-dimensional models, proceeding to horizontally and vertically resolved models. The text begins with global average models, which are explored in terms of their elementary forms yielding the global average temperature, right up to the incorporation of feedback mechanisms and some analytical properties of interest. The eff ect of stochastic forcing is then used to introduce natural variability in the models before turning to the concept of stability theory. Other one dimensional or zonally averaged models are subsequently presented, along with various applications, including chapters on paleoclimatology, the inception of continental glaciations, detection of signals in the climate system, and optimal estimation of large scale quantities from point scale data. Throughout the book, the authors work on two mathematical levels: qualitative physical expositions of the subject material plus optional mathematical sections that include derivations and treatments of the equations along with some proofs of stability theorems. A must-have introduction for policy makers, environmental agencies, and NGOs, as well as climatologists, molecular physicists, and meteorologists.

Table of Contents:
Preface xiii 1 Climate and Climate Models 1 1.1 Defining Climate 3 1.2 Elementary Climate System Anatomy 7 1.3 Radiation and Climate 9 1.3.1 Solar Radiation 9 1.3.2 Albedo of the Earth–Atmosphere System 13 1.3.3 Terrestrial Infrared Radiation into Space (The IR or Longwave Radiation) 14 1.4 Hierarchy of Climate Models 15 1.4.1 General Circulation Models (GCMs) 16 1.4.2 Energy Balance Climate Models 17 1.4.3 Adjustable Parameters in Phenomenological Models 19 1.5 Greenhouse Effect and Modern Climate Change 20 1.6 Reading This Book 20 1.7 Cautionary Note and Disclaimer 22 Notes on Further Reading 23 Exercises 23 2 Global Average Models 27 2.1 Temperature and Heat Balance 27 2.1.1 Blackbody Earth 28 2.1.2 Budyko’s Empirical IR Formula 29 2.1.3 Climate Sensitivity 30 2.1.4 Climate Sensitivity and Carbon Dioxide 31 2.2 Time Dependence 31 2.2.1 Frequency Response of Global Climate 32 2.2.2 Forcing with Noise 35 2.2.3 Predictability from Initial Conditions 37 2.2.4 Probability Density of the Temperature 39 2.3 Spectral Analysis 40 2.3.1 White Noise Spectral Density 41 2.3.2 Spectral Density and Lagged Correlation 41 2.3.3 AR1 Climate Model Spectral Density 42 2.3.4 Continuous Time Case 42 2.4 Nonlinear Global Model 44 2.4.1 Ice-Albedo Feedback 44 2.4.2 Linear Stability Analysis: A Slope/Stability Theorem 46 2.4.3 Relaxation Time and Sensitivity 47 2.4.4 Finite Amplitude Stability Analysis 48 2.4.5 Potential Function and Noise Forcing 49 2.4.6 Relation to Critical Opalescence 52 2.5 Summary 52 Suggestions for Further Reading 53 Exercises 53 3 Radiation and Vertical Structure 57 3.1 Radiance and Radiation Flux Density 58 3.2 Equation of Transfer 61 3.2.1 Extinction and Emission 61 3.2.2 Terrestrial Radiation 62 3.3 Gray Atmosphere 63 3.4 Plane-Parallel Atmosphere 64 3.5 Radiative Equilibrium 65 3.6 Simplified Model for Water Vapor Absorber 68 3.7 Cooling Rates 72 3.8 Solutions for Uniform-Slab Absorbers 73 3.9 Vertical Heat Conduction 75 3.9.1 K > 0 77 3.10 Convective Adjustment Models 77 3.11 Lessons from Simple Radiation Models 79 3.12 Criticism of the Gray Spectrum 80 3.13 Aerosol Particles 82 Notes for Further Reading 83 Exercises 83 4 Greenhouse Effect and Climate Feedbacks 85 4.1 Greenhouse Effect without Feedbacks 85 4.2 Infrared Spectra of Outgoing Radiation 85 4.2.1 Greenhouse Gases and the Record 92 4.2.2 Greenhouse Gas Computer Experiments 92 4.3 Summary of Assumptions and Simplifications 99 4.4 Log Dependence of the CO 2 Forcing 101 4.5 Runaway Greenhouse Effect 102 4.6 Climate Feedbacks and Climate Sensitivity 105 4.6.1 Equilibrium Feedback Formalism 107 4.7 Water Vapor Feedback 108 4.8 Ice Feedback for the Global Model 109 4.9 Probability Density of Climate Sensitivity 110 4.10 Middle Atmosphere Temperature Profile 112 4.10.1 Middle Atmosphere Responses to Forcings 113 4.11 Conclusion 115 Notes for Further Reading 116 Exercises 116 5 Latitude Dependence 119 5.1 Spherical Coordinates 120 5.2 Incoming Solar Radiation 121 5.3 Extreme Heat Transport Cases 122 5.4 Heat Transport Across Latitude Circles 122 5.5 Diffusive Heat Transport 123 5.6 Deriving the Legendre Polynomials 125 5.6.1 Properties of Legendre Polynomials 127 5.6.2 Fourier–Legendre Series 128 5.6.3 Irregular Solutions 128 5.7 Solution of the Linear Model with Constant Coefficients 129 5.8 The Two-Mode Approximation 129 5.9 Poleward Transport of Heat 133 5.10 Budyko’s Transport Model 134 5.11 Ring Heat Source 136 5.12 Advanced Topic: Formal Solution for More General Transports 137 5.13 Ice Feedback in the Two-Mode Model 138 5.14 Polar Amplification through Ice Cap Feedback 140 5.15 Chapter Summary 141 5.15.1 Parameter Count 142 Notes for Further Reading 142 Exercises 142 6 Time Dependence in the 1-D Models 145 6.1 Differential Equation for Time Dependence 146 6.2 Decay of Anomalies 146 6.2.1 Decay of an Arbitrary Anomaly 147 6.3 Seasonal Cycle on a Homogeneous Planet 148 6.4 Spread of Diffused Heat 153 6.4.1 Evolution on a Plane 155 6.5 Random Winds and Diffusion 157 6.6 Numerical Methods 159 6.6.1 Explicit Finite Difference Method 159 6.6.2 Semi-Implicit Method 162 6.7 Spectral Methods 163 6.7.1 Galerkin or Spectral Method 163 6.7.2 Pseudospectral Method 164 6.8 Summary 166 6.8.1 Parameter Count 166 Notes for Further Reading 167 Exercises 167 6.9 Appendix to Chapter 6: Solar Heating Distribution 169 6.9.1 The Elliptical Orbit of the Earth 171 6.9.2 Relation Between Declination and Obliquity 172 6.9.3 Expansion of S(μ, t) 172 7 Nonlinear Phenomena in EBMs 175 7.1 Formulation of the Nonlinear Feedback Model 176 7.2 Stürm–Liouville Modes 178 7.2.1 Orthogonality of SL Modes 179 7.3 Linear Stability Analysis 180 7.4 Finite Perturbation Analysis and Potential Function 184 7.4.1 Neighborhood of an Extremum 185 7.4.2 Relation to Gibbs Energy or Entropy 187 7.4.3 Attractor Basins—Numerical Example 187 7.5 Small Ice Cap Instability 187 7.5.1 Perturbation of an Exact Ice-Free Solution 190 7.5.2 Frequency Dependence of the Length Scale 191 7.6 Snow Caps and the Seasonal Cycle 193 7.7 Mengel’s Land-Cap Model 193 7.8 Chapter Summary 196 Notes for Further Reading 199 Exercises 199 8 Two Horizontal Dimensions and Seasonality 203 8.1 Beach Ball Seasonal Cycle 203 8.2 Eigenfunctions in the Bounded Plane 205 8.3 Eigenfunctions on the Sphere 208 8.3.1 Laplacian Operator on the Sphere 208 8.3.2 Longitude Functions 209 8.3.3 Latitude Functions 209 8.4 Spherical Harmonics 211 8.4.1 Orthogonality 211 8.4.2 Truncation 212 8.5 Solution of the EBM with Constant Coefficients 212 8.6 Introducing Geography 214 8.7 Global Sinusoidal Forcing 216 8.8 Two-Dimensional Linear Seasonal Model 217 8.8.1 Adjustment of Free Parameters 219 8.9 Present Seasonal Cycle Comparison 220 8.9.1 Annual Cycle 220 8.9.2 Semiannual Cycle 220 8.10 Chapter Summary 220 Notes for Further Reading 224 Exercises 224 9 Perturbation by Noise 229 9.1 Time-Independent Case for a Uniform Planet 230 9.2 Time-Dependent Noise Forcing for a Uniform Planet 234 9.3 Green’s Function on the Sphere: f = 0 235 9.4 Apportionment of Variance at a Point 237 9.5 Stochastic Model with Realistic Geography 238 9.6 Thermal Decay Modes with Geography 243 9.6.1 Statistical Properties of TDMs 246 Notes for Further Reading 248 Exercises 249 10 Time-Dependent Response and the Ocean 253 10.1 Single-Slab Ocean 254 10.1.1 Examples with a Single Slab 255 10.1.2 Eventual Leveling of the Forcing 258 10.2 Penetration of a Periodic Heating at the Surface 259 10.3 Two-Slab Ocean 262 10.3.1 Decay of an Anomaly with Two Slabs 266 10.3.2 Response to Ramp Forcing with Two Slabs 268 10.4 Box-Diffusion Ocean Model 269 10.5 Steady State of Upwelling-Diffusion Ocean 271 10.5.1 All-Ocean Planetary Responses 273 10.5.2 Ramp Forcing 274 10.6 Upwelling Diffusion with (and without) Geography 274 10.7 Influence of Initial Conditions 276 10.8 Response to Periodic Forcing with Upwelling Diffusion Ocean 277 10.9 Summary and Conclusions 280 Exercises 282 11 Applications of EBMs: Optimal Estimation 287 11.1 Introduction 287 11.2 Independent Estimators 288 11.3 Estimating Global Average Temperature 290 11.3.1 Karhunen–Loève Functions and Empirical Orthogonal Functions 292 11.3.2 Relationship with EBMs 296 11.4 Deterministic Signals in the Climate System 298 11.4.1 Signal and Noise 299 11.4.2 Fingerprint Estimator of Signal Amplitude 299 11.4.3 Optimal Weighting 299 11.4.4 Interfering Signals 302 11.4.5 All Four Signals Simultaneously 303 11.4.6 EBM-Generated Signals 306 11.4.7 Characterizing Natural Variability 310 11.4.8 Detection Results 311 11.4.9 Discussion of the Detection Results 314 Notes for Further Reading 317 Exercises 317 12 Applications of EBMs: Paleoclimate 321 12.1 Paleoclimatology 321 12.1.1 Interesting Problems for EBMs 322 12.2 Precambrian Earth 325 12.3 Glaciations in the Permian 327 12.3.1 Modeling Permian Glacials 327 12.4 Glacial Inception on Antarctica 331 12.5 Glacial Inception on Greenland 333 12.6 Pleistocene Glaciations and Milankovitch 335 12.6.1 EBMs in the Pleistocene: Short’s Filter 338 12.6.2 Last Interglacial 346 12.6.3 EBMs and Ice Volume 348 12.6.4 What Can Be Done without Ice Volume 350 Notes for Further Reading 350 Exercises 351 References 353 Index 365