Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics: v. 60 (Applied Mathematical Sciences)

The vigorous stirring of a cup of tea gives rise, as we all know, to interesting fluid dynamical phenomena, some of which are very hard to explain. In this book our “cup of tea” contains the currents of the Earth’s atmosphere, oceans, mantle, and fluid core. Our goal is to under stand the basic physical processes which are most important in describing what we observe, directly or indirectly, in these complex systems. While in many respects our understanding is measured by the ability to predict, the focus here will be on relatively simple models which can aid our physical intuition by suggesting useful mathematical methods of investiga tion. These elementary models can be viewed as part of a hierarchy of models of increasing complexity, moving toward those which might be use fully predictive. The discussion in this book will deal primarily with the Earth. Interplanetary probes of Venus, Mars, Jupiter and Saturn have revealed many exciting phenomena which bear on geophysical fluid dynamics. They have also enabled us to see the effect of changing the values of certain parameters, such as gravity and rotation rate, on geophysical flows. On the other hand, satellite observations of our own planet on a daily and hourly basis have turned it into a unique laboratory for the study of fluid motions on a scale never dreamt of before: the motion of cyclones can be observed via satellite just as wing tip vortices are studied in a wind tunnel. I. Fundamentals.- 1. Effects of Rotation.- 1.1. The Rossby Number.- 1.2. Equations of Motion in an Inertial Frame.- 1.3. Equations in a Rotating Frame.- 1.4. Vorticity.- 1.5. Motion at Small Rossby Number: The Geostrophic Approximation.- 2. Effects of Shallowness.- 2.1. Derivation of the Equations for Shallow Water.- 2.2. Small Amplitude Motions in a Basin.- 2.3. Geostrophic Degeneracy and Rossby Waves.- 3. The Quase-Geostrophic Approximation.- 3.1. Scaling for Shallow Layers and Small Rossby Number.- 3.2. The Beta-Plane Approximation.- 3.3. The Inertial Boundary Layer.- 3.4. Quasi-Geostrophic Rossby Waves.- II. Large-Scale Atmospheric Dynamics.- 4. Effects of Stratification Baroclinic Instability.- 4.1. A Perspective of the Atmospheric General Circulation.- 4.2. Vertical Stratification.- 4.3. The Baroclinic Potential Vorticity Equation.- 4.4. Baroclinic Instability.- 4.5. Extensions of the Theory and Discussion.- 4.6. Bibliographic Notes.- 5. Changing Flow Patterns and Successive Bifurcations.- 5.1. Rotating Annulus Experiments.- 5.2. Simplified Dynamics, A Recipe.- 5.3. Analysis of Flow Regime Transitions.- 5.4. Deterministic Aperiodic Flow.- 5.5. Bibliographic Notes.- 6. Persistent Anomalies, Blocking and Predictability.- 6.1. Phenomenology of Blocking.- 6.2. Resonant Wave Interactions and Persistence.- 6.3. Multiple Stationary States and Blocking.- 6.4. Multiple Flow Regimes and Variations in Predictability.- 6.5. Low-Frequency Atmospheric Variability.- 6.6. Bibliographic Notes.- III. Dynamo Theory.- 7. Models of Geomagnetism: A Survey.- 7.1. General Features of the Earth’s Magnetic Field.- 7.2. Equations of the Hydromagnetic Dynamo.- 7.3. Kinematic Versus Hydromagnetic Theory.- 7.4. A Disc Model.- 7.5. Cowling’s Theorem in a Spherical Core.- 7.6. The Dynamo Effect as Line Stretching.- 7.7. A Necessary Condition for a Dynamo Effect.- 7.8. Bibliographic Notes.- 8. Kinematic Dynamo Theory.- 8.1. Introduction.- 8.2. The Smoothing Method.- 8.3. Application to the Dynamo Problem.- 8.4. The ?-Effect in a Perfect Conductor.- 8.5. Almost Symmetric Dynamos.- 8.6. Small Diffusivity Aspects of Kinematic Dynamos.- 8.7. Multi-Scaling and Breakdown of Smoothing.- 8.8. Bibliographic Notes.- 9. The Hydrodynamic Basis of Geomagnetism.- 9.1. Introduction.- 9.2. Weak-Field Models and Constant ?.- 9.3. Macrodynamic Model Equations.- 9.4. The Simplest Hydromagnetic Models.- 9.5. Coupling of Macrodynamics to Microscale.- 9.6. Thermal or Gravitational Source of Convection.- 9.7. Bibliographic Notes.- IV. Theoretical Climate Dynamics.- 10. Radiation Balance and Equilibrium Models.- 10.1. Radiation Budget of the Earth.- 10.2. Energy-Balance Models (EBMs): Multiple Equilibria.- 10.3. Nonlinear Stability and Stochastic Perturbations.- 10.4. Modified EBMs: Periodic and Chaotic Solutions.- 10.5. Bibliographic Notes.- 11. Glaciation Cycles: Phenomenology and Slow Processes.- 11.1. Quaternary Glaciations, Paleoclimatological Evidence.- 11.2. Ice-Sheet Dynamics.- 11.3. Geodynamics.- 11.4. Bibliographic Notes.- 12. Climatic Oscillators.- 12.1. Free Oscillations of the Climatic System.- 12.2. Hopf Bifurcation.- 12.3. Orbital Changes and Their Climatic Effect.- 12.4. Forced Oscillations: Nonlinear Resonance.- 12.5. Entrainment, Combination Tones and Aperiodic Behavior.- 12.6. Periodicity and Predictability of Climate Evolution.- 12.7. Bibliographic Notes.- References.