Mathematical Approach to Climate Change and its Impacts, MAC2I

Preface......Page 7
Acknowledgements......Page 9
Contents......Page 10
 About the Editors......Page 12
 Contributors......Page 13
Part I Theme: Geophysical Fluids and Climate......Page 15
  1 Effects of Rotation......Page 16
   1.1 The Rossby Number......Page 17
   1.2 Equations of Motion in an Inertial Frame......Page 18
    1.2.1 Thermodynamics......Page 20
    1.3.1 Absolute and Relative Derivative......Page 22
    1.3.2 Relative Acceleration and Coriolis Acceleration......Page 24
   1.4 Vorticity......Page 25
    1.4.1 Kelvin's Theorem......Page 27
    1.4.2 Ertel's Theorem......Page 29
    1.5.1 The Taylor-Proudman Theorem......Page 33
    1.5.2 Geostrophic Motion......Page 34
  2 Effects of Shallowness......Page 35
   2.1 Derivation of the Equations for Shallow Water......Page 36
   2.2 Reduced-Gravity Shallow-Water Models......Page 41
   2.3 Small-Amplitude Motions in a Basin......Page 46
    2.3.1 Kelvin Waves......Page 49
    2.4.2 Sloping Bottom Topography......Page 51
   3.1 Introduction and Motivation......Page 54
    3.2.1 Observations......Page 57
    3.2.2 A Simple Model of the Double-Gyre Circulation......Page 60
    3.2.3 Bifurcations in the Double-Gyre Problem......Page 61
   3.3 Non-autonomous and Random Dynamical Systems......Page 64
    3.3.1 Background and Motivation......Page 65
    3.3.2 Pullback and Random Attractors......Page 67
    3.4.1 Time-Dependent Forcing......Page 75
   3.5 Concluding Remarks......Page 79
    3.5.1 A Generalized Definition of Climate Sensitivity......Page 80
    3.5.2 PBAs, Invariant Measures and Wasserstein Distance......Page 81
  Appendix 1: RDS Theory and Random Attractors......Page 83
  Appendix 2: Mixing in Random Dynamical Systems......Page 85
  Appendix 3: Low-Frequency Variability (LFV) and Mixing......Page 86
  References......Page 87
   1.1 Energy Balance Models and the Problems We Consider......Page 95
   1.2 Relation to Literature and Presentation of Our Main Results......Page 97
   2.1 Budyko Type Models with Memory......Page 99
   2.3 Plan of the Paper......Page 101
   3.1 Functional Framework......Page 102
   3.2 The Concept of Mild Solution for the Sellers Type Model (14)......Page 103
   3.4 Inverse Problem Results: Determination of the Insolation Function......Page 105
    3.4.1 Pointwise Observation and Uniqueness Result......Page 106
    3.4.2 Localized Observation and Stability Result......Page 107
  4 Main Result for the Budyko Type Model......Page 108
   4.1 The Notion of Mild Solutions for the Budyko Model (32)......Page 109
  5 Open Questions......Page 110
   6.2 Proof of Theorem 2: Local Existence of Mild Solutions......Page 111
    6.2.1 The Functional Setting and Main Tools......Page 112
    6.2.2 Step 1: Γ Maps XR into Itself if t* >0 Is Sufficiently Small......Page 114
    6.2.3 Step 2: Γ Is a Contraction if t* (0,1) Is Sufficiently Small......Page 116
    6.2.4 Step 3: Additional Regularity of the Solution and Conclusion of the Proof of Proposition 2......Page 117
   6.3 Proof of Theorem 2: Uniqueness......Page 118
   6.4 Proof of Theorem 2: The Maximal Solution Is Global in Time......Page 120
    6.4.1 L∞ Bound of the Solution......Page 121
    6.4.2 Proof of Proposition 4......Page 122
  7 Proof of Theorem 3......Page 124
   7.1 Step 1: The Linear Problem Satisfied by v......Page 125
   7.2 Step 2: q=q̃ on (-1,x0)......Page 126
   8.1 Step 1: The Problem Solved by the Difference w:=u-ũ......Page 128
   8.2 Step 2: A Useful Property of the Source Term K*......Page 129
   8.3 Step 3: A Carleman Estimate on the Problem Solved by z:=wt......Page 130
   8.4 Step 4: An Estimate from Above......Page 131
   8.6 Step 6: Conclusion......Page 132
   9.1 The Strategy to Prove Theorem 5......Page 133
   9.2 Proof of Lemma 5......Page 135
   9.4 Proof of Lemma 7......Page 136
   9.5 Proof of Lemma 8......Page 137
  References......Page 140
Part II Theme: Hydrology......Page 143
 Bedload Transport Processes in a Coastal Sand-Bed River: The Study Case of Fiumi Uniti River in the Northern Adriatic......Page 144
  2 Study Area......Page 145
  3 Methods and Instrumentation......Page 147
  4 Mathematical and Statistical Approach......Page 149
  5 Results and Discussion......Page 150
  6 Conclusions......Page 154
  References......Page 155
Part III Theme: Glaciology......Page 157
 Mathematical Modeling of Rock Glacier Flow with Temperature Effects......Page 158
  1 Introduction......Page 159
  2 Preliminaries......Page 163
  3 Constitutive Assumptions......Page 164
   3.1 Internal Energy for the Mixture......Page 165
   3.4 Cauchy Stress for the Mixture......Page 166
  Appendix......Page 168
  References......Page 170
  1 Introduction......Page 171
  2 The Basic Equations of Mixture Theory......Page 174
  3 Balance of Mass......Page 176
  4 Balance of Linear Momentum......Page 177
  6 Balance of Energy......Page 178
  8 Volume Additivity Constraint......Page 179
  9 Issues Concerning Boundary Conditions for Mixtures......Page 180
  10 Simplified Equations......Page 181
  11 Constitutive Relations: Model for Water and Ice......Page 182
  12 Constitutive Assumption for the Interaction Terms......Page 183
  13 Governing Equations......Page 184
  References......Page 185
 Modelling Sea Ice and Melt Ponds Evolution: Sensitivity to Microscale Heat Transfer Mechanisms......Page 187
  1 Introduction......Page 188
   2.1 The Sea-Ice-Thickness/Melt-Pond-Depth System......Page 189
    2.1.1 Melting Rate......Page 190
    2.1.2 Seepage Rate......Page 193
    2.1.3 Meltwater Flux......Page 194
   2.2 Numerical Implementation......Page 195
  3 Results......Page 196
   3.1 Melt Pond Areas Evolution: Role of the Turbulent Heat Flux Scaling Inside the Pond......Page 198
   3.2 Morphology of Melt Ponds: Role of Lateral Melting......Page 200
  4 Conclusions and Perspectives......Page 202
  References......Page 203
Part IV Theme: Ecology......Page 207
  1 Introduction......Page 208
  2 Methods......Page 209
  3 Results and Discussion......Page 211
  4 Conclusions......Page 214
  References......Page 215
 Mathematical Tools for Controlling Invasive Species in Protected Areas......Page 217
  1 Introduction......Page 218
   2.1 Wild Boars (Sus scrofa)......Page 220
   2.3 Brook Trout (Salvelinus fontinalis)......Page 221
  3 Wild-Boar Control Through Dynamical System Theory......Page 222
  4 Wild-Boar Control Through Z-type Dynamic Approach......Page 225
  5 Wild-Boar Control Through Optimal Control Theory......Page 229
  6 A Future Perspective: Optimal Spatio-Temporal Resources Allocation......Page 233
  7 Conclusions......Page 239
  References......Page 240