Kaper H. Mathematics & climate (Philadelphia, 2013). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаKaper H. Mathematics & climate / H.Kaper, H.Engler. - Philadelphia: SIAM, 2013. - xx, 295 p.: ill. - Bibliogr.: p.283-290. - Ind.: p.291-295. - ISBN 978-1-611972-60-3
 

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Оглавление / Contents
 
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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