List of Figures ................................................ xi
Preface ...................................................... xvii
1 Climate and Mathematics ...................................... 1
1.1 Earth's Climate System .................................. 1
1.2 Modeling Earth's Climate ................................ 2
1.3 Conceptual Models ....................................... 3
1.4 Climate and Statistics .................................. 5
1.5 Climate Variability and Climate Change .................. 6
1.6 Models from Data ........................................ 8
1.7 Exercises .............................................. 10
2 Earth's Energy Budget ....................................... 13
2.1 Solar Radiation ........................................ 13
2.2 Energy Balance Models .................................. 14
2.3 Basic Model ............................................ 16
2.4 Greenhouse Effect ...................................... 17
2.5 Multiple Equilibria .................................... 18
2.6 Budyko's Model ......................................... 19
2.7 Snowball Earth ......................................... 20
2.8 Bifurcation ............................................ 21
2.9 Exercises .............................................. 23
3 Oceans and Climate .......................................... 29
3.1 Ocean Circulation ...................................... 29
3.2 Temperature ............................................ 31
3.3 Salinity ............................................... 32
3.4 Box Models ............................................. 33
3.5 One-Dimensional Model .................................. 36
3.6 Exercises .............................................. 39
4 Dynamical Systems ........................................... 41
4.1 Autonomous Differential Equations ...................... 41
4.2 Geometrical Objects .................................... 44
4.3 Critical Points ........................................ 48
4.4 Periodic Orbits ........................................ 49
4.5 Dynamics near Critical Points .......................... 51
4.6 Planar Case ............................................ 52
4.7 Nonlinear Systems ...................................... 56
4.8 Exercises .............................................. 58
5 Bifurcation Theory .......................................... 63
5.1 Bifurcation ............................................ 63
5.2 Examples ............................................... 64
5.3 From Examples to the General Case ...................... 73
5.4 Bifurcation Points ..................................... 74
5.5 Hopf Bifurcation Theorem ............................... 74
5.6 Exercises .............................................. 75
6 Stommel's Box Model ......................................... 77
6.1 Stommel's Two-Box Model ................................ 77
6.2 Dynamical System ....................................... 78
6.3 Bifurcation ............................................ 82
6.4 Comments ............................................... 82
6.5 Exercises .............................................. 83
7 Lorenz Equations ............................................ 87
7.1 Lorenz Model ........................................... 87
7.2 Preliminary Observations ............................... 88
7.3 Equilibrium Solutions .................................. 89
7.4 Numerical Experiments .................................. 91
7.5 Exercises .............................................. 92
8 Climate and Statistics ...................................... 95
8.1 Challenges for Statistics .............................. 95
8.2 Proxy Data ............................................. 98
8.3 Reanalysis ............................................ 101
8.4 Model Skill ........................................... 102
8.5 Exercises ............................................. 103
9 Regression Analysis ........................................ 105
9.1 Statistical Modeling .................................. 105
9.2 Linear Regression ..................................... 108
9.3 Simple Linear Regression .............................. 110
9.4 Regression Diagnostics ................................ 111
9.5 Exercises ............................................. 113
10 Mauna Loa Co2 Data ......................................... 117
10.1 Keeling's Observational Study ......................... 117
10.2 Assembling the Data ................................... 118
10.3 Analyzing the Data .................................... 118
10.4 Exercises ............................................. 122
11 Fourier Transforms ......................................... 123
11.1 Fourier Analysis ...................................... 123
11.2 Trigonometric Interpolation ........................... 124
11.3 Discrete Fourier Transform ............................ 125
11.4 Fast Fourier Transform ................................ 126
11.5 Power Spectrum ........................................ 127
11.6 Correlation and Autocorrelation ....................... 128
11.7 Fourier Series and Fourier Integrals .................. 130
11.8 Milankovitch's Theory of Glacial Cycles ............... 131
11.9 Exercises ............................................. 137
12 Zonal Energy Budget ........................................ 141
12.1 Zonal Energy Balance Model ............................ 141
12.2 Legendre Polynomials .................................. 146
12.3 Spectral Method ....................................... 148
12.4 Equilibrium Solutions ................................. 150
12.5 Temperature Profile ................................... 151
12.6 Assessment ............................................ 152
12.7 Exercises ............................................. 154
13 Atmosphere and Climate ..................................... 159
13.1 Earth's Atmosphere .................................... 159
13.2 Pressure .............................................. 160
13.3 Temperature ........................................... 160
13.4 Atmospheric Circulation ............................... 162
13.5 Exercises ............................................. 164
14 Hydrodynamics .............................................. 165
14.1 Coriolis Effect ....................................... 165
14.2 State Variables ....................................... 169
14.3 Continuity Equation ................................... 170
14.4 Equation of Motion .................................... 171
14.5 Other Prognostic Variables ............................ 173
14.6 Equation of State ..................................... 173
14.7 Coupling Ocean and Atmosphere ......................... 174
14.8 Need for Approximations ............................... 174
14.9 Shallow Water Equations ............................... 174
14.10 Further Approximations ............................... 177
14.11 Boussinesq Equations ................................. 178
14.12 Exercises ............................................ 179
15 Climate Models ............................................. 183
15.1 Climate Models as Dynamical Systems ................... 183
15.2 Dimension Reduction: Lorenz Model ..................... 185
15.3 Abstract Climate Models ............................... 189
15.4 Exercises ............................................. 191
16 El Niño-Southern Oscillation ............................... 193
16.1 El Niño ............................................... 193
16.2 Recharge-Oscillator Model ............................. 195
16.3 Delayed-Oscillator Model .............................. 198
16.4 Delay Differential Equations .......................... 204
16.5 Numerical Investigations .............................. 206
16.6 Exercises ............................................. 208
17 Cryosphere and Climate ..................................... 213
17.1 Cryosphere ............................................ 213
17.2 Glaciers, Ice Sheets, and Ice Shelves ................. 214
17.3 Sea Ice ............................................... 215
17.4 Exercises ............................................. 220
18 Biogeochemistry ............................................ 223
18.1 Biosphere and Climate ................................. 223
18.2 Carbon Cycle .......................................... 224
18.3 Carbon Transport into the Deep Ocean .................. 227
18.4 Ocean Plankton ........................................ 228
18.5 Algal Blooms .......................................... 230
18.6 Exercises ............................................. 235
19 Extreme Events ............................................. 237
19.1 Climate and Weather Extremes .......................... 237
19.2 Exceedance ............................................ 240
19.3 Tail Probabilities and Return Periods ................. 242
19.4 Order Statistics, Extreme Value Distribution .......... 244
19.5 Exercises ............................................. 248
20 Data Assimilation .......................................... 251
20.1 Data Assimilation and Climate ......................... 251
20.2 Example ............................................... 252
20.3 Bayesian Approach ..................................... 254
20.4 Sequential Data Assimilation .......................... 255
20.5 Kaiman Filtering ...................................... 257
20.6 Numerical Example ..................................... 259
20.7 Extensions ............................................ 261
20.8 Data Assimilation for the Lorenz System ............... 263
20.9 Concluding Remarks .................................... 265
20.10 Exercises ............................................ 265
A Units and Symbols .......................................... 269
В Glossary ................................................... 273
С MATLAB Codes ............................................... 279
C.l MATLAB Code for Lorenz Equations ...................... 279
C.2 MATLAB Code for Regression Analysis ................... 280
C.3 MATLAB Code for Delay Differential Equations .......... 280
Bibliography .................................................. 283
Index ......................................................... 291
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