Atmospheric Dynamics

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