The dynamics of rotating fluids:
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
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Oxford
Oxford University Press
[2024]
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvi, 510 Seiten Illustrationen, Diagramme |
| ISBN: | 9780198886310 9780198886303 |
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Contents PART I AN INTRODUCTION TO FLUID DYNAMICS AND WAVES 1. A Qualitative Introduction to Rotating Fluids 1.1 Some Naturally Occurring Rotating Flows 1.2 Two Classic Laboratory Experiments: Taylor Columns and Inertial Waves 1.3 Two More Experiments: Ekman Boundary Layers and Stirred Cups of Tea 2. A Crash Course on Incompressible Fluid Dynamics 2.1 An Eulerian Description of Motion and the Convective Derivative 2.2 Mass Conservation and the Streamfunction 2.3 More Kinematics: Characterizing the Spin and Deformation of a Fluid Element 2.3.1 Viscous Shear Stresses and the Need to Distinguish Between Spin and Deformation 2.3.2 The Rate-of-Deformation Tensor 2.3.3 Vorticity and the Intrinsic Spin of Fluid Elements 2.4 Newton’s Law of Viscosity and the Navier-Stokes Equation 2.4.1 The Stress Tensor and Cauchy’s Equation of Motion 2.4.2 Newton’s Law of Viscosity and the No-Slip Condition 2.4.3 The Navier-Stokes Equation and the Reynolds Number 2.5 Boundary Layers, Boundary-Layer Separation, and Turbulence 2.6 The Viscous Dissipation of Mechanical Energy 2.7 The Navier-Stokes Equation in Cylindrical Polar Coordinates 2.8 Viscous Vortex Dynamics 2.8.1 A Transport Equation for Vorticity 2.8.2 The Advection, Stretching, and Diffusion of Vorticity 2.8.3 Where Does VorticityCome From? 2.8.4 The Biot-Savart Law 2.8.5 Flows Without Vorticity: The Dangers of Potential Flow Theory 2.9 The Classical Theory of Inviscid Vortex Dynamics 2.9.1 Kelvin’s Theorem 2.9.2 Helmholtz’s Laws 2.9.3 Helicity and Helicity Conservation 3 3 5 11 15 15 18 19 19 22 22 24 24 26 28 31 34 36 38 39 40 42 43 46
48 48 49 52
X CONTENTS 2.10 Axisymmetric Flow with Swirl 2.10.1 The Poloidal-Azimuthal Decomposition of the Viscous Vorticity Equation 2.10.2 Inviscid Flow: The Squire-Long Equation and a Glimpse at Inertial Waves 2.11 Buoyancy-Driven Flow and the Boussinesq Approximation 2.12 The Analogy Between Buoyancy and Swirl 2.12.1 A Formal Analogy 2.12.2 An Illustrative Example: The Bursting Vortex and the Buoyant Thermal 3. Waves and Waves in Fluids 3.1 The Wave Equation and d’Alembert s Solution 3.2 Some Wave-Bearing Systems Not Governed by the Wave Equation 3.2.1 Flexural Vibrations of a Beam 3.2.2 Surface Gravity Waves on Water of Arbitrary Depth 3.3 Dispersive Versus Non-dispersive Waves: d’Alembert Revisited 3.4 The Concept of Group Velocity for Dispersive Systems 3.5 An Example of Dispersion in Three Dimensions: Internal Gravity Waves 79 54 54 56 59 61 61 63 67 67 69 69 70 73 74 PART II THE THEORY OF ROTATING FLUIDS 4. Moving into a Rotating Frame of Reference, the Taylor-Proudman Theorem, and the Formation of Taylor Columns 4.1 Moving into a Rotating Frame of Reference and the Coriolis Force 4.2 Governing Equations and the Rossby and Ekman Numbers 4.3 Rapid Rotation, the Taylor-Proudman Theorem, and Taylor Columns 4.3.1 The Geostrophic Force Balance, Taylor’s Experiment, and Taylor Columns 4.3.2 A Flaw in the Usual Rationalization of Taylor Columns: The Role of Waves 4.4 Taylor Columns Associated with the Axial Movement of an Object 5. Ekman Boundary Layers 5.1 Three Types of EkmanLayers 5.1.1 The Nonlinear Solutions of Karman and Bödewadt 5.1.2 Generalizing the Solutions of Karman and
Bödewadt 5.1.3 Combined Karman and Bödewadt Layers: The Rotor-Stator Problem 5.1.4 The Linear Solution of Ekman 87 87 89 90 91 92 95 99 99 100 103 104 105
CONTENTS XI 5.2 Secondary Flows in Teacups, Rivers, and Ducts 5.3 Spin-Down and Spin-Up in a Cylindrical Container 5.3.1 Spin-Down 5.3.2 Spin-Up 5.4 Vertical Shear Layers and Spin-Down Revisited 5.5 Boundary Layers Generated by Differentially Rotating Spheres 107 110 110 114 117 122 6. Inertial Waves I: Inviscid Progressive Waves 6.1 The Physical Origin of Inertial Waves 6.1.1 Rayleigh’s Analogy to Buoyancy (Revisited) 6.1.2 The Coriolis Force as a Restoring Force 6.2 The Fundamental Properties of Inertial Waves 6.2.1 The Group Velocity and Spatial Structure of Inertial Waves 6.2.2 The Formation of Taylor Columns by Low-Frequency Waves 6.2.3 The Spontaneous Focussing of Inertial Waves to Form Columnar Wave Packets 6.2.4 The Generation and Segregation of Helicity 6.3 The Physical Nature of Inertial Waves Revisited 6.3.1 An Elastic Response to the Axial Compression of Fluid Columns 6.3.2 The Bending of Absolute Vortex Lines and the Generation of Helical Flow 6.4 A More Detailed Look at the Dispersion Pattern Generated by a Gaussian Vortex 6.5 The Dispersion Patterns Generated by Buoyant Blobs in a Boussinesq Fluid 6.5.1 A Single Buoyant Blob Drifting Normal or Parallel to the Rotation Axis 6.5.2 A Layer of Drifting Buoyant Blobs 6.6 Waves in a Rotating, Stratified Fluid and Near-Inertial Waves in the Oceans 6.7 Evanescent Inertial Waves 6.8 The Reflection of Inertial Waves from Plane Surfaces 6.9 Finite Amplitude Inertial Waves 7. Inertial Waves II: Inviscid, Linear Modes in Bounded, Axisymmetric Domains 7.1 General Properties of Inertial Modes in Bounded Domains 7.1.1 From
Progressive Waves to Modes 7.1.2 The Orthogonality of Distinct Modes and the Equipartition of Kinetic Energy 7.1.3 Angular Momentum Conservation and the Mean Circulation Theorem 127 127 127 129 131 131 135 136 142 143 144 146 148 153 153 156 159 163 165 169 173 173 173 174 176
xii CONTENTS 7.2 Inertial Modes in a Cylinder: An Illustrative Example 7.2.1 Axisymmetric Modes as Standing Waves 7.2.2 Non-Axisymmetric Modes as Azimuthally Drifting Waves 7.3 Some Comments on Modes in a Sphere 7.3.1 Four Classes of Solutions 7.3.2 The Spin-Over Mode 7.3.3 A Footnote: The Completeness of Inertial Modes in a Sphere 8. Rossby Waves 8.1 8.2 8.3 8.4 Rossby Waves Between Plane, Non-Parallel Boundaries Rossby Waves in a Sliced Cylinder Potential Vorticity and the Rossby Wave Equation The Physical Mechanism of Rossby Waves 9. Rotating, Shallow-Water Flow 9.1 The Shallow-Water Equations for an Inviscid Fluid 9.1.1 The Non-Rotating Shallow-Water Equations 9.1.2 Adding Rotation: Potential Vorticity Revisited 9.1.3 An Energy Equation 9.2 Small-Amplitude Perturbations in a Rotating, Shallow-Water System 9.2.1 Linearized Dynamics, Geostrophic Flow, and the Rossby Deformation Radius 9.2.2 Inertia-Gravity Waves and Geostrophic Adjustment 9.2.3 Kelvin Waves at a Boundary 9.3 The Shallow-Water Equations in the Quasi-Geostrophic Limit 9.3.1 The Quasi-Geostrophic Shallow-Water (QGSW) Equations 9.3.2 Potential Vorticity Inversion 9.3.3 An Energy Equation for QGSW Flow 9.4 Adding Topography and the ß-Effect to the QGSW Equations: Rossby Waves II 9.4.1 Generalized Potential Vorticity Conservation 9.4.2 Rossby Waves on the B-plane 10. Precession 10.1 An Example of Precessing Flow: Motion in Planetary Cores 10.2 A Crash Course on Rigid-Body Precession 10.2.1 Eulers Equations 10.2.2 Torque-Free Precession of an Axisymmetric Body 10.2.3 Euler Angles 10.2.4 Torque-Free Precession of
an Axisymmetric Body Revisited 10.2.5 Forced-Precession: The Heavy Top 179 179 182 186 187 191 192 195 195 199 202 203 205 205 205 207 208 210 210 212 214 217 217 220 222 223 223 225 229 229 230 231 234 238 241 242
CONTENTS 10.3 Free and Forced Precession of the Earth 10.3.1 The Chandler Wobble 10.3.2 Equinox Precession 10.4 Forced Precession of the Earths Fluid Core 10.4.1 A Third Frame of Reference: The Precession Frame 10.4.2 Poincares Inviscid Analysis 10.4.3 Viscous Boundary Layers 10.4.4 The Breakdown of Ekman Layers at Critical Latitudes 246 246 248 251 252 254 259 262 11. Instability I: Taylor-Couette Flow 11.1 The Centrifugal Instability of Rayleigh and Taylor 11.1.1 Rayleighs Criterion for Inviscid, Axisymmetric Disturbances 11.1.2 A Counter-Example: Two-Dimensional, Inviscid Disturbances 11.1.3 Viscous Instability: Taylors Analysis of Flow in an Annulus 11.1.4 The Experimental Evidence for Taylor-Couette Flow 11.1.5 The Stability of the Boundary Layer on a Rotating Cylinder 11.2 The Influence of Axial Flow onStability 11.2.1 A Heuristic Energy Argument 11,2.2 Rayleigh’s Analogy Between Swirl and Buoyancy Revisited 12. Instability II: Rotating Convection xiii 267 267 267 271 272 276 278 280 281 282 285 12.1Convection Without Rotation: The Rayleigh-Bénard Problem 285 12.1.1 The Experiments of James Thomson and Henri Benard 285 12.1.2 Rayleigh’s Stability Analysis I: Framing the Eigenvalue Problem 287 12.1.3 Rayleigh’s Stability Analysis II: Slip Boundaries Top and Bottom 289 12.1.4 More Realistic Cases: No-slip Boundaries 291 12.1.5 An Approximate Energy Analysis for No-slip Boundaries Top and Bottom 292 12.1.6 Nonlinear Saturation 294 12.2 Rotating Rayleigh-Bénard Convection I: Non-Oscillatory Instability 296 12.2.1 A Linear Stability Analysis 296 12.2.2 The Origin of the
Criterion (Ra)cril ~ Ta2/3 300 12.2.3 An Interpretation of the Stability Criterion in Terms of Energy 302 12.2.4 The Spatial Structure of the Unstable Mode 305
XÎV CONTENTS 12.3 Rotating Rayleigh-Bénard Convection II: Oscillatory Instability 309 12.3.1 A Linear Stability Analysis for Slip Boundaries 309 12.3.2 Marginal Stability: The Importance of Prandtl Number and Role of Inertial Waves 311 12.3.3 When is a Marginal Oscillatory Mode Preferred Over Stationary Convection? 312 12.4 The Busse Annulus and Thermal Rossby Waves 315 12.4.1 The Busse Annulus 315 12.4.2 A Linear Stability Analysis and Thermal Rossby Waves 316 321 13. Vortex Breakdown 13.1 Observations of Vortex Breakdown in Pipes and on Delta Wings 13.2 A Suggestive Model Problem of Flow in a Diverging Pipe 13.3 A Toy Model of Vortex Breakdown on the Surface of a Delta Wing 13.4 Theories of Vortex Breakdown 321 322 328 333 14. A Glimpse at Rapidly Rotating Turbulence 337 337 14.1 Some Observations of Rapidly Rotating Turbulence 14.2 Linear Structure Formation in Inhomogeneous, Freely Decaying Turbulence 14.3 Linear Structure Formation in Homogeneous, Freely Decaying Turbulence 14.3.1 What Can Two-Point Velocity Correlations Capture? 14.3.2 The Experimental Evidence 14.4 Weakly Nonlinear Interactions and Resonant Triad Theory 14.5 The Cyclone, Anticyclone Asymmetry 338 341 342 345 346 350 PART III ILLUSTRATIVE EXAMPLES OF ROTATING FLOWS IN NATURE 355 15. Tornadoes, Dust Devils, and Tidal Vortices 15.1 Tornadoes and Waterspouts 15.1.1 Observational Evidence as to the Nature of Tornadoes 15.1.2 The Ekman-Like Boundary Layer Beneath a Rankine Vortex 15.1.3 Implications for Tornadoes 15.1.4 Waterspouts 15.1.5 The Evidence for Vortex Breakdown in Tornadoes and Waterspouts 355
355 358 362 363 365
CONTENTS XV 15.2 Dust Devils 15.3 Tidal Vortices 367 370 16. Tropical Cyclones 377 16.1 The Observed Properties of Tropical Cyclones 16.2 Moist Convection and the Energetics of Tropical Cyclones 16.3 A Toy Model of a‘Dry’Tropical Cyclone 16.3.1 The Model: Its Motivation, Governing Equations, and Weaknesses 382 16.3.2 The Global Structure of the Flow 16.3.3 The Mechanism of Eye Formation 16.3.4 Scaling Laws for Wind Speed and the Criterion for Eye Formation 389 16.3.5 Points of Connection to, and Departure from, Real Tropical Cyclones 392 16.4 Estimating the Maximum Wind Speed in Real Tropical Cyclones 377 380 382 385 386 393 17. Convective Motion in the Earths Core and the Geodynamo 397 17.1 The Structure of the Planets 17.1.1 The Structure of the Earth, Sources of Motion, and the Geomagnetic Field 17.1.2 The Structure of the Other Solar System Planets and Their Magnetic Fields 17.2 Flow in the Liquid Core of the Earth 17.2.1 Dimensionless Groups and Dominant Forces in the Earth’s Core 17.2.2 Numerical Simulations of the Core Flow and the Role of Columnar Vortex Pairs 17.2.3 The Tendency for the Buoyancy Flux to Concentrate Around the Equatorial Plane 17.2.4 The Hidden Role of Inertial Waves in the North-South Segregation of Helicity 17.3 The Simplified Version of Maxwell’s Equations Required for Dynamo Theory 17.3.1 Charge Conservation and the Laws of Ohm, Ampère, and Faraday 17.3.2 The Reduced Form of Maxwell’s Equations Required for Dynamo Theory 17.3.3 An Evolution Equation for the Magnetic Field 17.3.4 An Energy Equation 17.3.5 Maxwells Stresses, Faraday’s Tension, and
Alfvén's Waves 17.4 An Introduction to Geodynamo Theory 17.4.1 Some Kinematics: The Need for a Large Rm, the Ω-effect, and Cowlings Theorem 17.4.2 Dynamics at Last: The Taylor Constraint 397 397 400 401 401 404 406 407 412 413 415 417 419 420 423 423 430
XVÎ CONTENTS 17.4.3 More Kinematics: Integral Equations for the Axial and Azimuthal Magnetic Fields 432 17.4.4 Parkers Lift-and-Twist Mechanism and the Alpha Effect 17.4.5 Two Classes of Planetary Dynamos 17.4.6 The Numerical Simulations of the Geodynamo and the Equatorial Jet 17.5 Scaling Laws for the Geodynamo 17.5.1 Force and Energy Balances in an a2 Dynamo 17.5.2 Scaling Laws for the Rossby and Lehnert Numbers in the Numerical Dynamos 17.6 The Collapse of the Dipole Field in the Numerical Dynamos for »/Ω5 0.4 17.7 Some Comments on Reversals of the Geodynamo 435 440 442 444 444 447 449 450 18. Zonal (East-West) Winds and Rossby-Wave Turbulence 18.1 A Return to the ß-plane: Turbulence Versus Rossby Waves 18.2 Zonal Flows and Jets on the ß-plane 18.2.1 A Cartoon of the Early Stages of Development of Zonal Flow: The Vallis Dumbbell 18.2.2 Strengthening the Zonal Jets: The Jet-Sharpening Model of Dritschel and McIntyre 18.2.3 Some Observations from the Numerical Experiments 18.3 Energy Spectra in ß-plane Turbulence 18.4 Jupiter’s Zonal Flows and the Need for Deeper Models 455 455 458 19. Accretion Discs in Astrophysics 19.1 Accretion Discs and Why They Form 19.2 The Angular Momentum Budget and Energy Dissipation in a Steady Disc 19.3 A Glimpse at Protoplanetary Discs and Discs in Binary Star Systems 19.4 A Non-Magnetic Description of Discs and the a Disc Model 19.5 Turbulence I: The Chandrasekhar-Velikhov Instability in Hot Discs 19.6 Turbulence II: The Origins of Turbulence in Protoplanetary Discs 473 473 Appendix Appendix Appendix Appendix 1 2 3 4 Subject Index Vector
Identities and Theorems Navier-Stokes Equation in Cylindrical Polar Coordinates Geophysical Data The Physical Properties of Common Fluids 458 460 463 465 468 476 480 482 488 493 497 501 503 505 507 |
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| id | DE-604.BV049695166 |
| illustrated | Illustrated |
| indexdate | 2024-12-02T07:01:38Z |
| institution | BVB |
| isbn | 9780198886310 9780198886303 |
| language | English |
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| physical | xvi, 510 Seiten Illustrationen, Diagramme |
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| spelling | Davidson, P. A. 1957- Verfasser (DE-588)173480845 aut The dynamics of rotating fluids P.A. Davidson Rotating fluids Oxford Oxford University Press [2024] xvi, 510 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Rotierende Flüssigkeit (DE-588)4124081-9 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 s Rotierende Flüssigkeit (DE-588)4124081-9 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=035037684&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Davidson, P. A. 1957- The dynamics of rotating fluids Rotierende Flüssigkeit (DE-588)4124081-9 gnd Strömungsmechanik (DE-588)4077970-1 gnd |
| subject_GND | (DE-588)4124081-9 (DE-588)4077970-1 |
| title | The dynamics of rotating fluids |
| title_alt | Rotating fluids |
| title_auth | The dynamics of rotating fluids |
| title_exact_search | The dynamics of rotating fluids |
| title_full | The dynamics of rotating fluids P.A. Davidson |
| title_fullStr | The dynamics of rotating fluids P.A. Davidson |
| title_full_unstemmed | The dynamics of rotating fluids P.A. Davidson |
| title_short | The dynamics of rotating fluids |
| title_sort | the dynamics of rotating fluids |
| topic | Rotierende Flüssigkeit (DE-588)4124081-9 gnd Strömungsmechanik (DE-588)4077970-1 gnd |
| topic_facet | Rotierende Flüssigkeit Strömungsmechanik |
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