This textbook is intended for both undergraduate and graduate courses in meteorology and atmospheric sciences, as well as for researchers working on theoretical and numerical aspects of weather and climate or on geophysical fluid dynamics. The treatment is concise, thorough, and self-contained. All necessary concepts are introduced, and the reader is given explicit guidance on all mathematical steps.
The book begins with a derivation of the equations of motion. These are then used to discuss fundamental aspects of weather and climate. The mechanisms behind vortical motions, that are known from the daily weather map, are discussed. Shallow-water theory is introduced as a tool for an efficient analysis of key concepts, such as atmospheric waves and synoptic-scale vortices. Quasigeostrophic theory is described and then used to explain the occurrence and mechanisms of extratropical weather by means of baroclinic instability. The specific properties of the atmospheric boundary layer are discussed, with a focus on the interaction between turbulence and mean flows. This is followed by a detailed look at the global atmospheric circulation, highlighting its control by Rossby waves and gravity waves.
At the same time, the reader is introduced to essential concepts that find applications in the field, such as balance by geostrophic and hydrostatic equilibrium, the role of entropy and potential temperature, potential vorticity, the Kelvin theorem, instability theory, the Reynolds equations, Eliassen-Palm and pseudo-momentum flux, multi-scale asymptotics, WKB theory, wave action, the transformed Eulerian mean, critical layers, and wave refraction.
The text is supplemented by appendices on important mathematical concepts and further elaborations of the main text. Chapter summaries and reading recommendations help the reader not merely to keep focus on the essentials, but just as well to broaden the horizon.
Author(s): Ulrich Achatz
Edition: 1
Publisher: Springer-Verlag GmbH Deutschland
Year: 2022
Language: English
Pages: 558
City: Berlin, Germany
Tags: Atmospheric Motion, Thermodynamics of Air, Vortex Dynamics, Shallow-Water Equation, Gravity Waves
Introduction
Contents
1 The Basic Equations of Atmospheric Motion
1.1 Time Derivatives in Fluids
1.1.1 The Fluid Description According to Euler and Lagrange
1.1.2 The Material Derivative of a Fluid Element
1.1.3 The Material Derivative of Volume Integrals
1.1.4 Summary
1.2 The Equation of Continuity
1.2.1 A Eulerian Derivation
1.2.2 A Lagrangian Derivation Using the Material Derivative
1.2.3 Summary
1.3 The Momentum Equation
1.3.1 The Volume Forces
1.3.2 Surface Forces (1): The Pressure Gradient Force
1.3.3 Surface Forces (2): Friction
1.3.4 The Total Momentum Equation
1.3.5 Summary
1.4 The Equations of Motion in a Rotating Frame of Reference
1.4.1 The Time Derivative in a Rotating Frame of Reference
1.4.2 The Momentum Equation in the Rotating Frame of Reference
1.4.3 The Equation of Continuity in the Rotating Frame of Reference
1.4.4 Summary
1.5 The Equations of Motion on the Sphere
1.5.1 Velocity and Material Derivative in Spherical Coordinates
1.5.2 The Transformed Equations of Motion
1.5.3 Summary
1.6 Synoptic Scale Analysis
1.6.1 The Geostrophic Equilibrium
1.6.2 The Hydrostatic Equilibrium
1.6.3 Summary
1.7 Recommendations for Further Reading
2 Elementary Thermodynamics and Energetics of Dry Air
2.1 Fundamentals
2.1.1 Thermodynamic Systems
2.1.2 Thermodynamic State and Thermodynamic Equilibrium
2.1.3 Temperature
2.1.4 Equations of State
2.1.5 Energy Change of a Thermodynamic System
2.1.6 Summary
2.2 The Fundamental Laws of Thermodynamics
2.2.1 The First Law of Thermodynamics and Internal Energy
2.2.2 The Heat Capacities of an Ideal Gas
2.2.3 Adiabatic and Isothermal Changes of State of an Ideal Gas
2.2.4 The Second Law of Thermodynamics
2.2.5 The Carnot Process
2.2.6 Entropy as State Variable
2.2.7 Entropy and Potential Temperature of Dry Air
2.2.8 Summary
2.3 The Prognostic Equations for Temperature and Entropy in Dry Air
2.3.1 Prediction of Temperature
2.3.2 The Prediction of Entropy and Potential Temperature
2.3.3 The Equations in a Rotating Frame of Reference
2.3.4 Spherical Coordinates
2.3.5 Summary
2.4 Potential Temperature and Static Stability
2.4.1 Stable and Unstable Stratification
2.4.2 Buoyancy Oscillations
2.4.3 Summary
2.5 Recommendations for Further Reading
3 Elementary Properties and Applications of the Basic Equations
3.1 Summary of the Basic Equations
3.2 The Importance of the Basic Equations for Weather Prediction
3.3 Conservation Laws
3.3.1 Conservation of Energy
3.3.2 Conservation of Angular Momentum
3.3.3 Summary
3.4 The Primitive Equations
3.5 The Primitive Equations in Pressure Coordinates
3.5.1 Arbitrary Vertical Coordinates
3.5.2 Pressure Coordinates
3.5.3 Summary
3.6 Balanced Flows
3.6.1 The Natural Coordinates
3.6.2 Geostrophic Flow
3.6.3 Inertial Flow
3.6.4 Cyclostrophic Flow
3.6.5 The Gradient Wind
3.6.6 Summary
3.7 Thermal Wind
3.8 Recommendations for Further Reading
4 Vortex Dynamics
4.1 Vorticity
4.1.1 Relative, Absolute, and Planetary Vorticity
4.1.2 Vortex Lines, Vortex Tubes, and Vortex Flux
4.1.3 Summary
4.2 Circulation
4.2.1 Relative and Absolute Circulation
4.2.2 The General Circulation Theorem
4.2.3 Summary
4.3 The Kelvin Theorem
4.4 The Vorticity Equation
4.4.1 The Derivation
4.4.2 Vortex-Tube Stretching and Vortex Tilting
4.4.3 Summary of the Impacts on Relative Vorticity
4.4.4 The Frozen-In Property of Absolute Vorticity
4.4.5 Summary
4.5 Potential Vorticity
4.5.1 An Algebraic Derivation of the Prognostic Equation for Potential Vorticity
4.5.2 A Derivation of Potential-Vorticity Conservation from the General Circulation Theorem
4.5.3 Summary
4.6 Vortex Dynamics and the Primitive Equations
4.6.1 The Primitive Equations in Isentropic Coordinates
4.6.2 The Primitive Vorticity Equation in Isentropic Coordinates
4.6.3 The Potential Vorticity of the Primitive Equations
4.6.4 Flow over a Mountain Ridge
4.6.5 Summary
4.7 Recommendations for Further Reading
5 The Dynamics of the Shallow-Water Equations
5.1 Derivation of the Equations
5.1.1 The Momentum Equation
5.1.2 The Continuity Equation
5.1.3 Summary
5.2 Conservation Properties
5.2.1 Energy Conservation
5.2.2 Potential Vorticity
5.2.3 Summary
5.3 Quasigeostrophic Dynamics
5.3.1 The Tangential β-Plane
5.3.2 Scaling the Shallow-Water Equations on the β-Plane
5.3.3 The Quasigeostrophic Approximation: Derivation by Scale Asymptotics
5.3.4 The Quasigeostrophic Approximation: Derivation from the Conservation of Shallow-Water Potential Vorticity
5.3.5 Summary
5.4 Wave Solutions of the Shallow-Water Equations
5.4.1 Perturbation Approach
5.4.2 Waves on the f-Plane
5.4.3 Waves on the β Plane: Quasigeostrophic Rossby Waves
5.4.4 Summary
5.5 Geostrophic Adjustment
5.5.1 The General Solution of the Linear Shallow-Water Equations on an f Plane
5.5.2 The Adjustment Process
5.5.3 Summary
5.6 Recommendations for Further Reading
6 Quasigeostrophic Dynamics of the Stratified Atmosphere
6.1 Quasigeostrophic Theory and Its Potential Vorticity
6.1.1 Analysis of Momentum and Continuity Equation
6.1.2 Analysis of the Entropy Equation
6.1.3 Quasigeostrophic Potential Vorticity in the Stratified Atmosphere
6.1.4 The Relationship with General Potential Vorticity
6.1.5 Quasigeostrophic Theory in Pressure Coordinates
6.1.6 A Quasigeostrophic Two-Layer Model
6.1.7 Summary
6.2 Quasigeostrophic Energetics
6.2.1 The Continuously Stratified Atmosphere
6.2.2 The Two-Layer Model
6.2.3 Summary
6.3 Rossby Waves in the Stratified Atmosphere
6.3.1 Rossby Waves in the Two-Layer Model
6.3.2 Rossby Waves in an Isothermal Continuously Stratified Atmosphere
6.3.3 Summary
6.4 Baroclinic Instability
6.4.1 Baroclinic Instability in the Two-Layer Model
6.4.2 Baroclinic Instability in a Continuously Stratified Atmosphere
6.4.3 Summary
6.5 Recommendations for Further Reading
7 The Planetary Boundary Layer
7.1 Anelastics and the Boussinesq Theory
7.1.1 The Anelastic Equations
7.1.2 The Boussinesq Equations
7.1.3 Summary
7.2 Instabilities in the Boundary Layer
7.2.1 The Taylor–Goldstein Equation
7.2.2 Neutral Stratification (N2 = 0)
7.2.3 No Shear (doverlineu/dz = 0) and Constant Stratification N2
7.2.4 The General Case: The Richardson Criterion of Howard and Miles
7.2.5 Summary
7.3 The Averaged Equations of Motion
7.3.1 Turbulence and Mean Flow
7.3.2 The Reynolds Equations
7.3.3 Summary
7.4 Gradient Ansatz and Mixing Length
7.5 The Turbulent Kinetic Energy
7.5.1 The Prognostic Equation
7.5.2 Sources and Sinks
7.5.3 Summary
7.6 The Prandtl Layer
7.6.1 The Momentum Flux
7.6.2 The Wind Profile
7.6.3 The Influence of Stratification
7.6.4 Summary
7.7 The Ekman Layer
7.7.1 The Ekman-Spiral
7.7.2 Ekman Pumping
7.7.3 Summary
7.8 Recommendations for Further Reading
8 The Interaction Between Rossby Waves and the Mean Flow
8.1 Basics of Quasigeostrophic Theory
8.1.1 The Governing Equations
8.1.2 Conservation Properties
8.1.3 The Quasigeostrophic Enstrophy Equation Within Linear Dynamics
8.1.4 Summary
8.2 Rossby-Wave Propagation
8.2.1 Wave Propagation Within WKB Theory
8.2.2 Rossby-Wave Propagation into the Stratosphere
8.2.3 Summary
8.3 The Eliassen–Palm Flux
8.3.1 Definition
8.3.2 The Eliassen–Palm Relationship
8.3.3 Wave Action and Eliassen–Palm Flux Within WKB Theory
8.3.4 Summary
8.4 The Transformed Eulerian Mean (TEM)
8.4.1 The TEM in the Context of Quasigeostrophy
8.4.2 The Mass-Weighted Circulation in Isentropic Coordinates
8.4.3 The Relation Between the Residual Circulation and the Mass-Weighted Circulation
8.4.4 Summary
8.5 The Non-acceleration Theorem
8.6 Recommendations for Further Reading
9 The Meridional Circulation
9.1 Some Essentials of the Empirical Basis
9.2 The Hadley Circulation
9.2.1 The Basic Equations of a Model Without Wave Driving
9.2.2 A Solution Without Meridional Circulation
9.2.3 Hide's Theorem
9.2.4 A Simplified Description of the Hadley Cell
9.2.5 The Summer–Winter Asymmetry
9.2.6 The Wave-Driven Hadley Circulation
9.2.7 Summary
9.3 The Circulation in the Midlatitudes
9.3.1 The Phenomenology of the Ferrel Cell
9.3.2 Eddy Fluxes and Barotropic Jet Stream
9.3.3 A Two-Layer Model
9.3.4 The Continuously Stratified Atmosphere
9.3.5 Summary
9.4 Recommendations for Further Reading
10 Gravity Waves and Their Impact on the Atmospheric Flow
10.1 Some Empirical Facts
10.2 The Fundamental Wave Modes of an Atmosphere at Rest
10.2.1 Equations of Motion and Energetics
10.2.2 Free Waves on the f-Plane in an Isothermal Atmosphere
10.2.3 Summary
10.3 The Interaction Between Mesoscale Gravity Waves and a Synoptic-Scale Flow
10.3.1 A Reformulation of the Dynamical Equations
10.3.2 Scaling for Synoptic-Scale Flow and for Inertia-Gravity Waves
10.3.3 Non-dimensional Equations and WKB Ansatz
10.3.4 Leading-Order Results: Equilibria, Dispersion and Polarization Relations, Eikonal Equations
10.3.5 The Next Order of the Equations
10.3.6 Wave Action
10.3.7 Wave Impact on the Synoptic-Scale Flow
10.3.8 Generalization to Gravity-Wave Spectra: Phase-Space Wave-Action Density
10.3.9 Conservation Properties
10.3.10 Summary
10.4 Critical Levels and Reflecting Levels
10.4.1 Critical Levels
10.4.2 Reflecting Levels
10.4.3 Summary
10.5 The Middle-Atmosphere Gravity-Wave Impact
10.5.1 Extension of the TEM by Gravity-Wave Effects
10.5.2 The Gravity-Wave Effect on the Residual Circulation and on the Zonal-Mean Flow
10.5.3 Summary
10.6 References and Recommendations for Further Reading
11 Appendices
11.1 Appendix A: Useful Elements of Vector Analysis
11.1.1 The Gradient
11.1.2 The Divergence and the Integral Theorem from Gauss
11.1.3 The Curl and the Integral Theorem from Stokes
11.1.4 Some Identities
11.1.5 Recommendations for Further Reading
11.2 Appendix B: Rotations
11.2.1 Recommendations for Further Reading
11.3 Appendix C: Isotropic Tensors
11.3.1 Isotropic Tensors of Rank One
11.3.2 Isotropic Tensors of Rank Two
11.3.3 Isotropic Tensors of Rank Three
11.3.4 Isotropic Tensors of Rank Four
11.3.5 Recommendations for Further Reading
11.4 Appendix D: Spherical Coordinates
11.4.1 The Local Basis Vectors
11.4.2 The Gradient in Spherical Coordinates
11.4.3 The Divergence in Spherical Coordinates
11.4.4 The Curl in Spherical Coordinates
11.4.5 Recommendations for Further Reading
11.5 Appendix E: Fourier Integrals and Fourier Series
11.5.1 Fourier Integrals
11.5.2 Fourier Series
11.5.3 Recommendations for Further Reading
11.6 Appendix F: Zonally Symmetric Rossby Waves in the Quasigeostrophic Two-Layer Model
11.7 Appendix G: Explicit Solution of the Initial-Value Problem of Baroclinic Instability in a Quasigeostrophic Two-Layer Model
11.8 Appendix H: Polarization Relations of the Geostrophic Mode and all f-Plane Modes Without Buoyancy Oscillations
11.9 Appendix I: The Higher Harmonics of a Gravity-Wave Field in WKB Theory
11.9.1 Leading-Order Results
11.9.2 Next-Order Results
11.9.3 Recommendations for Further Reading
Literature
Index
