Preface |
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xiii | |
Notation and Acronyms |
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xvii | |
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1 | (494) |
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3 | (11) |
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1.1 A Very Brief History of Time |
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3 | (1) |
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4 | (1) |
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5 | (1) |
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1.4 Fluid Composition of Earth's Atmosphere and Oceans |
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6 | (6) |
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1.5 Organisation of Chapters |
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12 | (2) |
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2 Governing Equations for Motion of a Dry Atmosphere: Vector Form |
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14 | (24) |
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14 | (1) |
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15 | (14) |
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2.3 Thermodynamics of an Ideal Gas |
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29 | (7) |
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2.4 The Governing Equations for Motion of an Ideal Gas |
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36 | (1) |
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37 | (1) |
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3 Governing Equations for Motion of a Cloudy Atmosphere: Vector Form |
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38 | (30) |
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38 | (1) |
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3.2 Representation of Water and Other Substances in the Atmosphere |
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39 | (10) |
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3.3 The Equation of State for Cloudy Air |
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49 | (3) |
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3.4 The Momentum Equation for Cloudy Air |
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52 | (1) |
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3.5 The Thermodynamic-Energy Equation for Cloudy Air |
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53 | (3) |
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3.6 Alternative Forms for the Thermodynamic-Energy Equation |
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56 | (6) |
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3.7 The Governing Equations for Motion of a Cloudy Atmosphere |
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62 | (1) |
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63 | (5) |
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Appendix: Derivation of the Equation of State for Cloudy Air from First Principles |
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65 | (3) |
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4 Governing Equations for Motion of Geophysical Fluids: Vector Form |
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68 | (29) |
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68 | (1) |
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4.2 More General Thermodynamics |
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69 | (17) |
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4.3 Functional Forms for a Cloudy-Air Parcel |
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86 | (6) |
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4.4 The Governing Equations for Motion of a Geophysical Fluid |
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92 | (1) |
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93 | (4) |
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Appendix: Specific Heat Capacities for an Ideal Gas |
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94 | (3) |
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5 Orthogonal-Curvilinear Coordinate Systems |
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97 | (22) |
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97 | (1) |
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5.2 Deep Orthogonal-Curvilinear Coordinates |
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98 | (13) |
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5.3 Shallow Orthogonal-Curvilinear Coordinates |
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111 | (7) |
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118 | (1) |
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6 Governing Equations for Motion of Geophysical Fluids: Curvilinear Form |
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119 | (14) |
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119 | (1) |
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6.2 The Governing Equations in Vector Form |
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120 | (1) |
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6.3 Axial-Orthogonal-Curvilinear Coordinates |
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121 | (1) |
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122 | (4) |
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6.5 The Governing Equations in Axial-Orthogonal-Curvilinear Coordinates |
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126 | (2) |
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6.6 The Governing Equations in Spherical-Polar and Cylindrical-Polar Coordinates |
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128 | (1) |
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6.7 Euler--Lagrange Forms of the Momentum Components |
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128 | (1) |
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129 | (4) |
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Appendix A The Governing Equations in Spherical-Polar Coordinates |
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130 | (1) |
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Appendix B The Governing Equations in Cylindrical-Polar Coordinates |
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131 | (2) |
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7 Representation of Gravity: Basic Theory and Spherical Planets |
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133 | (41) |
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133 | (8) |
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7.2 A Guide to This Chapter and to the Next One |
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141 | (2) |
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7.3 Equilibrium States for Unaccelerated Flow |
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143 | (4) |
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7.4 The Geopotential at and near Earth's Surface |
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147 | (2) |
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7.5 Newtonian Gravity and Potential Theory |
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149 | (1) |
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7.6 A Spherical Planet of Constant Density |
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150 | (6) |
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7.7 Avenues for Investigation |
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156 | (1) |
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7.8 A Spherical Planet of Variable Density |
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156 | (15) |
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171 | (3) |
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Appendix: Some Spherical Relations |
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172 | (2) |
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8 Representation of Gravity: Further Theory and Spheroidal Planets |
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174 | (42) |
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174 | (2) |
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8.2 Functional Forms for Spheroidal Planets |
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176 | (6) |
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8.3 An Ellipsoidal Planet of Constant Density |
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182 | (7) |
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8.4 Reformulation of the Procedure to Determine Newtonian Gravity Outside a Planet |
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189 | (2) |
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8.5 An Ellipsoidal Planet of Variable Density |
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191 | (20) |
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8.6 Spherical Geopotential Approximation as an Asymptotic Limit |
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211 | (3) |
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214 | (2) |
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9 Thermodynamic Potentials and Thermodynamical Consistency |
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216 | (22) |
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216 | (2) |
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9.2 Thermodynamic Potentials |
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218 | (5) |
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9.3 Basic Gibbs Thermodynamic Potentials |
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223 | (5) |
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9.4 Composite Gibbs Potentials |
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228 | (9) |
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237 | (1) |
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238 | (56) |
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238 | (1) |
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239 | (4) |
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10.3 Latent Internal Energy and Phase Transitions |
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243 | (4) |
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10.4 Water Substance in a Vacuum |
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247 | (12) |
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10.5 Cloudy Air, Possibly Containing Liquid and/or Frozen Water |
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259 | (16) |
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10.6 The Triple Point of Water in the Presence of Dry Air |
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275 | (11) |
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10.7 Definition of Some Thermodynamic Quantities |
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286 | (6) |
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292 | (2) |
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294 | (34) |
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294 | (2) |
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11.2 Oceanic Gibbs Potentials |
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296 | (3) |
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11.3 Derivation of Vallis (2017)'s Prototypical Gibbs Potential |
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299 | (13) |
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11.4 An Alternative Prototypical Gibbs Potential for an Ocean |
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312 | (7) |
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11.5 The TEOS-10 Gibbs Potential |
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319 | (8) |
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327 | (1) |
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12 Geopotential Coordinates for Modelling Planetary Atmospheres and Oceans |
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328 | (43) |
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328 | (2) |
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12.2 Geodesy and the World Geodetic System |
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330 | (8) |
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12.3 The Classical Spherical Geopotential Approximation Revisited |
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338 | (2) |
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12.4 Geopotential Approximation for Ellipsoidal Planets |
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340 | (5) |
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12.5 Further Geopotential Approximation above Earth's Geoid |
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345 | (4) |
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12.6 Further Geopotential Approximation below Earth's Geoid |
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349 | (4) |
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353 | (1) |
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12.8 Orthogonal Trajectories to the Geopotential Surfaces |
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354 | (3) |
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357 | (9) |
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366 | (5) |
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Appendix: The Equilibrium Depth of an Ocean Covering a Planet |
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369 | (2) |
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13 Vertical Coordinates and Boundary Conditions |
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371 | (42) |
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371 | (2) |
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13.2 The Deep-Fluid Equations and Boundary Conditions |
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373 | (15) |
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388 | (1) |
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389 | (4) |
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13.5 Axial-Angular-Momentum Conservation |
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393 | (2) |
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13.6 Boundary Conditions in the Vertical and Global Conservation |
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395 | (8) |
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13.7 Conservation with the Shallow Approximation |
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403 | (1) |
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13.8 An Energy-Like Invariant for Elastic Lids at Finite Pressure |
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403 | (2) |
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13.9 An Atmospheric State with Zero Pressure at Finite Height |
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405 | (5) |
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410 | (3) |
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Appendix: Some Useful Identities |
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411 | (2) |
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14 Variational Methods and Hamilton's Principle of Stationary Action |
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413 | (50) |
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413 | (1) |
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14.2 Eulerian versus Lagrangian Viewpoints for Fluid Dynamics |
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414 | (3) |
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417 | (2) |
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14.4 Functionals and Variational Principles |
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419 | (8) |
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14.5 Hamilton's Principle of Stationary Action |
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427 | (4) |
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14.6 Gravitational Attraction between Two Particles Revisited |
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431 | (4) |
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14.7 A System of Point Particles |
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435 | (4) |
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14.8 Governing Equations for Global Fluids: Vector Form |
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439 | (10) |
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14.9 Governing Equations for Global Fluids: Curvilinear Form |
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449 | (3) |
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14.10 Euler-Lagrange Equations for Global Fluids |
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452 | (4) |
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456 | (7) |
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Appendix: Variations in Axial-Orthogonal-Curvilinear Coordinates |
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456 | (7) |
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463 | (32) |
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463 | (1) |
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464 | (1) |
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15.3 Conservation Principles: Vector Form |
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465 | (15) |
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15.4 Conservation Principles: Curvilinear Form |
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480 | (7) |
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15.5 Noether's Theorem, Symmetries, and Conservation |
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487 | (7) |
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494 | (1) |
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II DYNAMICALLY CONSISTENT EQUATION SETS |
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495 | (158) |
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16 Deep and Shallow, Dynamically Consistent Equation Sets in 3D |
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497 | (34) |
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497 | (3) |
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16.2 A Unified Quartet of Dynamically Consistent Equation Sets |
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500 | (11) |
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16.3 Derivation Methodologies for Approximate Equation Sets |
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511 | (1) |
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16.4 Classical Eulerian Derivation |
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511 | (1) |
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16.5 Lagrangian Derivation Using Hamilton's Principle |
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512 | (4) |
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16.6 Lagrangian Derivation Using Euler-Lagrange Equations |
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516 | (1) |
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16.7 Equation Transition from Deep Fluids to Shallow Fluids |
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517 | (3) |
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520 | (11) |
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Appendix A Four Equation Sets in Spherical-Polar Coordinates |
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522 | (4) |
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Appendix B Four Equation Sets in Axial-Orthogonal-Curvilinear Coordinates |
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526 | (5) |
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17 Quasi-Shallow, Dynamically Consistent Equation Sets in 3D |
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531 | (31) |
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531 | (2) |
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17.2 Classical Eulerian Derivation |
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533 | (10) |
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17.3 Lagrangian Derivation |
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543 | (6) |
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17.4 A Unified Sextet of Equation Sets in Spheroidal Geometry |
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549 | (4) |
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17.5 A Unified Sextet of Equation Sets in Spherical Geometry |
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553 | (4) |
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557 | (5) |
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Appendix A Quasi-Shallow Equation Sets in Axial-Orthogonal-Curvilinear Coordinates |
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557 | (2) |
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Appendix B Variations for Quasi-Shallow Contributions |
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559 | (3) |
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18 Shallow-Water Equation Sets in 2D |
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562 | (69) |
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562 | (4) |
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18.2 Eulerian Derivation of the Basic Shallow-Water Equations |
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566 | (9) |
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18.3 Horizontal Coordinate Systems and Models of Gravity |
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575 | (9) |
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18.4 Lagrangian Density for the Basic Shallow-Water Equations |
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584 | (7) |
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18.5 Quasi-Shallow Enhancement of Lagrangian Density |
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591 | (4) |
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18.6 Euler-Lagrange Derivation of the Quasi-Shallow Enhanced Set |
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595 | (4) |
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18.7 `Quasi-Shallow' Shallow-Water Conservation Principles |
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599 | (3) |
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18.8 The `Quasi-Shallow' Shallow-Water Equations in Spherical Geometry |
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602 | (3) |
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18.9 Derivation of a Unified Quartet of Equation Sets |
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605 | (9) |
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18.10 The Unified Quartet in Spherical Geometry |
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614 | (4) |
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618 | (13) |
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Appendix A Derivation of 2D Quasi-Shallow Lagrangian Density by Vertically Averaging the 3D One |
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621 | (3) |
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Appendix B Conservation Principles for the `Quasi-Shallow' Shallow-Water Equations |
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624 | (7) |
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19 A Barotropic Potential Vorticity (BPV) Equation for Flow over a Spheroidal Planet |
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631 | (22) |
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631 | (1) |
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19.2 The Momentum and Mass-Continuity Equations in Curvilinear Form |
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632 | (2) |
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19.3 Inviscid, Horizontal, Shallow Flow in Spheroidal Geometry |
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634 | (5) |
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639 | (4) |
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19.5 The BPV Equation for a Spheroidal Planet |
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643 | (3) |
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19.6 An Alternative Derivation of the BPV Equation |
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646 | (1) |
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19.7 Dynamical Consistency |
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647 | (1) |
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19.8 The Poisson Problem for Pressure |
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648 | (1) |
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19.9 Variational Derivation of the Momentum Equations |
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649 | (1) |
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650 | (3) |
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III EXACT STEADY AND UNSTEADY NON-LINEAR SOLUTIONS |
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653 | (120) |
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20 Exact Steady Solutions of the Global Shallow-Water Equations |
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655 | (43) |
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655 | (2) |
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20.2 The Shallow-Water Equations in Spheroidal Geometry |
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657 | (1) |
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20.3 A Derivation Methodology |
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658 | (1) |
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20.4 A Physical Interpretation of hS1 (ξ2) |
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659 | (2) |
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20.5 Some Illustrative Solutions |
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661 | (7) |
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20.6 Rotated Solutions in Spherical Geometry |
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668 | (1) |
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669 | (2) |
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20.8 The Stability of Exact Solutions to Linear Perturbation |
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671 | (11) |
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20.9 Illustrative Examples of the Application of the Stability Analysis |
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682 | (13) |
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695 | (3) |
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Appendix: Rotated Coordinate Transformations |
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695 | (3) |
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21 Exact 3D Steady Solutions of Global Equation Sets |
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698 | (25) |
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698 | (1) |
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21.2 A Unified Quartet of Governing Equations |
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699 | (1) |
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21.3 Simplification for Steady, Axially Symmetric Flow |
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700 | (1) |
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21.4 Compatibility Constraints for Balance |
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701 | (2) |
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21.5 A Change of Dependent Variable |
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703 | (2) |
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21.6 Construction of Exact Steady Solutions |
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705 | (3) |
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21.7 A Generalised Thermal-Wind Equation |
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708 | (2) |
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21.8 Three Illustrative Examples |
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710 | (11) |
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721 | (2) |
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22 Exact Unsteady Solutions of the Barotropic Potential Vorticity Equation over an Ellipsoid |
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723 | (23) |
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723 | (2) |
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22.2 Derivation of Exact Unsteady Solutions |
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725 | (12) |
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22.3 A Complementary Derivation of Exact Unsteady Solutions |
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737 | (2) |
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22.4 Diagnosis of Pressure for a Particular Solution |
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739 | (3) |
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22.5 Diagnosis of Pressure for a Family of Solutions |
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742 | (2) |
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744 | (2) |
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23 Exact Unsteady Solutions in 3D over an Ellipsoidal Planet |
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746 | (27) |
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746 | (1) |
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23.2 A Quartet of Equation Sets for Unforced 3D Fluid Flow over a Rotating Ellipsoidal Planet |
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747 | (3) |
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750 | (6) |
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23.4 Exact Barotropic Solutions over an Ellipsoid |
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756 | (4) |
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23.5 A Family of Exact, Unsteady 3D Solutions |
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760 | (1) |
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23.6 A Particular Exact, Unsteady, 3D Solution |
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761 | (1) |
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23.7 The Top Boundary Condition |
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762 | (1) |
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23.8 Test Cases for Validating 3D Dynamical Cores |
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762 | (9) |
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771 | (2) |
Appendix: Vector Identities |
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773 | (4) |
References |
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777 | (7) |
Index |
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784 | |