Energy Balance Climate Models

Gerald R. North is University Distinguished Professor of Atmospheric Sciences Emeritus at Texas A&M University, having obtained his BS degree in physics from the University of Tennessee, PhD (1966) in theoretical physics from the University of Wisconsin, Madison. Among other positions he served eight years as research scientist at Goddard Space Flight Center before joining Texas A&M in 1986, where he served as department head 1995–2003. He is a fellow of AAAS, AGU, AMS, and recipient of several awards including the Jule G. Charney Award of the American Meteorology Society. He has served as Editor in Chief of the Reviews of Geophysics and Editor in Chief of the Encyclopedia of the Atmospheric Sciences, 2nd Edition. He has coauthored books on Paleoclimatology and Atmospheric Thermodynamics.Kwang-Yul Kim is a professor in climatology and physical oceanography at Seoul National University. Upon graduation from Texas A&M with his PhD degree in physical oceanography he was inducted into the Phi Kappa Phi Honor Society. He authored two books: Fundamentals of Fluid Dynamics and Cyclostationary EOF Analysis. He programmed several new energy balance models.

• Preface xiii1 Climate and Climate Models 11.1 Defining Climate 31.2 Elementary Climate System Anatomy 71.3 Radiation and Climate 91.3.1 Solar Radiation 91.3.2 Albedo of the Earth–Atmosphere System 131.3.3 Terrestrial Infrared Radiation into Space (The IR or Longwave Radiation) 141.4 Hierarchy of Climate Models 151.4.1 General Circulation Models (GCMs) 161.4.2 Energy Balance Climate Models 171.4.3 Adjustable Parameters in Phenomenological Models 191.5 Greenhouse Effect and Modern Climate Change 201.6 Reading This Book 201.7 Cautionary Note and Disclaimer 22Notes on Further Reading 23Exercises 232 Global Average Models 272.1 Temperature and Heat Balance 272.1.1 Blackbody Earth 282.1.2 Budyko’s Empirical IR Formula 292.1.3 Climate Sensitivity 302.1.4 Climate Sensitivity and Carbon Dioxide 312.2 Time Dependence 312.2.1 Frequency Response of Global Climate 322.2.2 Forcing with Noise 352.2.3 Predictability from Initial Conditions 372.2.4 Probability Density of the Temperature 392.3 Spectral Analysis 402.3.1 White Noise Spectral Density 412.3.2 Spectral Density and Lagged Correlation 412.3.3 AR1 Climate Model Spectral Density 422.3.4 Continuous Time Case 422.4 Nonlinear Global Model 442.4.1 Ice-Albedo Feedback 442.4.2 Linear Stability Analysis: A Slope/Stability Theorem 462.4.3 Relaxation Time and Sensitivity 472.4.4 Finite Amplitude Stability Analysis 482.4.5 Potential Function and Noise Forcing 492.4.6 Relation to Critical Opalescence 522.5 Summary 52Suggestions for Further Reading 53Exercises 533 Radiation and Vertical Structure 573.1 Radiance and Radiation Flux Density 583.2 Equation of Transfer 613.2.1 Extinction and Emission 613.2.2 Terrestrial Radiation 623.3 Gray Atmosphere 633.4 Plane-Parallel Atmosphere 643.5 Radiative Equilibrium 653.6 Simplified Model for Water Vapor Absorber 683.7 Cooling Rates 723.8 Solutions for Uniform-Slab Absorbers 733.9 Vertical Heat Conduction 753.9.1 K > 0 773.10 Convective Adjustment Models 773.11 Lessons from Simple Radiation Models 793.12 Criticism of the Gray Spectrum 803.13 Aerosol Particles 82Notes for Further Reading 83Exercises 834 Greenhouse Effect and Climate Feedbacks 854.1 Greenhouse Effect without Feedbacks 854.2 Infrared Spectra of Outgoing Radiation 854.2.1 Greenhouse Gases and the Record 924.2.2 Greenhouse Gas Computer Experiments 924.3 Summary of Assumptions and Simplifications 994.4 Log Dependence of the CO 2 Forcing 1014.5 Runaway Greenhouse Effect 1024.6 Climate Feedbacks and Climate Sensitivity 1054.6.1 Equilibrium Feedback Formalism 1074.7 Water Vapor Feedback 1084.8 Ice Feedback for the Global Model 1094.9 Probability Density of Climate Sensitivity 1104.10 Middle Atmosphere Temperature Profile 1124.10.1 Middle Atmosphere Responses to Forcings 1134.11 Conclusion 115Notes for Further Reading 116Exercises 1165 Latitude Dependence 1195.1 Spherical Coordinates 1205.2 Incoming Solar Radiation 1215.3 Extreme Heat Transport Cases 1225.4 Heat Transport Across Latitude Circles 1225.5 Diffusive Heat Transport 1235.6 Deriving the Legendre Polynomials 1255.6.1 Properties of Legendre Polynomials 1275.6.2 Fourier–Legendre Series 1285.6.3 Irregular Solutions 1285.7 Solution of the Linear Model with Constant Coefficients 1295.8 The Two-Mode Approximation 1295.9 Poleward Transport of Heat 1335.10 Budyko’s Transport Model 1345.11 Ring Heat Source 1365.12 Advanced Topic: Formal Solution for More General Transports 1375.13 Ice Feedback in the Two-Mode Model 1385.14 Polar Amplification through Ice Cap Feedback 1405.15 Chapter Summary 1415.15.1 Parameter Count 142Notes for Further Reading 142Exercises 1426 Time Dependence in the 1-D Models 1456.1 Differential Equation for Time Dependence 1466.2 Decay of Anomalies 1466.2.1 Decay of an Arbitrary Anomaly 1476.3 Seasonal Cycle on a Homogeneous Planet 1486.4 Spread of Diffused Heat 1536.4.1 Evolution on a Plane 1556.5 Random Winds and Diffusion 1576.6 Numerical Methods 1596.6.1 Explicit Finite Difference Method 1596.6.2 Semi-Implicit Method 1626.7 Spectral Methods 1636.7.1 Galerkin or Spectral Method 1636.7.2 Pseudospectral Method 1646.8 Summary 1666.8.1 Parameter Count 166Notes for Further Reading 167Exercises 1676.9 Appendix to Chapter 6: Solar Heating Distribution 1696.9.1 The Elliptical Orbit of the Earth 1716.9.2 Relation Between Declination and Obliquity 1726.9.3 Expansion of S(μ, t) 1727 Nonlinear Phenomena in EBMs 1757.1 Formulation of the Nonlinear Feedback Model 1767.2 Stürm–Liouville Modes 1787.2.1 Orthogonality of SL Modes 1797.3 Linear Stability Analysis 1807.4 Finite Perturbation Analysis and Potential Function 1847.4.1 Neighborhood of an Extremum 1857.4.2 Relation to Gibbs Energy or Entropy 1877.4.3 Attractor Basins—Numerical Example 1877.5 Small Ice Cap Instability 1877.5.1 Perturbation of an Exact Ice-Free Solution 1907.5.2 Frequency Dependence of the Length Scale 1917.6 Snow Caps and the Seasonal Cycle 1937.7 Mengel’s Land-Cap Model 1937.8 Chapter Summary 196Notes for Further Reading 199Exercises 1998 Two Horizontal Dimensions and Seasonality 2038.1 Beach Ball Seasonal Cycle 2038.2 Eigenfunctions in the Bounded Plane 2058.3 Eigenfunctions on the Sphere 2088.3.1 Laplacian Operator on the Sphere 2088.3.2 Longitude Functions 2098.3.3 Latitude Functions 2098.4 Spherical Harmonics 2118.4.1 Orthogonality 2118.4.2 Truncation 2128.5 Solution of the EBM with Constant Coefficients 2128.6 Introducing Geography 2148.7 Global Sinusoidal Forcing 2168.8 Two-Dimensional Linear Seasonal Model 2178.8.1 Adjustment of Free Parameters 2198.9 Present Seasonal Cycle Comparison 2208.9.1 Annual Cycle 2208.9.2 Semiannual Cycle 2208.10 Chapter Summary 220Notes for Further Reading 224Exercises 2249 Perturbation by Noise 2299.1 Time-Independent Case for a Uniform Planet 2309.2 Time-Dependent Noise Forcing for a Uniform Planet 2349.3 Green’s Function on the Sphere: f = 0 2359.4 Apportionment of Variance at a Point 2379.5 Stochastic Model with Realistic Geography 2389.6 Thermal Decay Modes with Geography 2439.6.1 Statistical Properties of TDMs 246Notes for Further Reading 248Exercises 24910 Time-Dependent Response and the Ocean 25310.1 Single-Slab Ocean 25410.1.1 Examples with a Single Slab 25510.1.2 Eventual Leveling of the Forcing 25810.2 Penetration of a Periodic Heating at the Surface 25910.3 Two-Slab Ocean 26210.3.1 Decay of an Anomaly with Two Slabs 26610.3.2 Response to Ramp Forcing with Two Slabs 26810.4 Box-Diffusion Ocean Model 26910.5 Steady State of Upwelling-Diffusion Ocean 27110.5.1 All-Ocean Planetary Responses 27310.5.2 Ramp Forcing 27410.6 Upwelling Diffusion with (and without) Geography 27410.7 Influence of Initial Conditions 27610.8 Response to Periodic Forcing with Upwelling Diffusion Ocean 27710.9 Summary and Conclusions 280Exercises 28211 Applications of EBMs: Optimal Estimation 28711.1 Introduction 28711.2 Independent Estimators 28811.3 Estimating Global Average Temperature 29011.3.1 Karhunen–Loève Functions and Empirical Orthogonal Functions 29211.3.2 Relationship with EBMs 29611.4 Deterministic Signals in the Climate System 29811.4.1 Signal and Noise 29911.4.2 Fingerprint Estimator of Signal Amplitude 29911.4.3 Optimal Weighting 29911.4.4 Interfering Signals 30211.4.5 All Four Signals Simultaneously 30311.4.6 EBM-Generated Signals 30611.4.7 Characterizing Natural Variability 31011.4.8 Detection Results 31111.4.9 Discussion of the Detection Results 314Notes for Further Reading 317Exercises 31712 Applications of EBMs: Paleoclimate 32112.1 Paleoclimatology 32112.1.1 Interesting Problems for EBMs 32212.2 Precambrian Earth 32512.3 Glaciations in the Permian 32712.3.1 Modeling Permian Glacials 32712.4 Glacial Inception on Antarctica 33112.5 Glacial Inception on Greenland 33312.6 Pleistocene Glaciations and Milankovitch 33512.6.1 EBMs in the Pleistocene: Short’s Filter 33812.6.2 Last Interglacial 34612.6.3 EBMs and Ice Volume 34812.6.4 What Can Be Done without Ice Volume 350Notes for Further Reading 350Exercises 351References 353Index 365