An Introduction to Dynamic Meteorology/ Holton, James R.

1. Introduction
1.1 Dynamic Meteorology 1
1.2 Conservation of Momentum 4
1.2.1 Pressure Gradient Force 5
1.2.2 Viscous Force 6
1.2.3 Gravitational Force 8
1.3 Noninertial Reference Frames and "Apparent" Forces 9
1.3.1 Centripetal Acceleration and Centrifugal Force 10
1.3.2 Gravity Revisited 11
1.3.3 The Coriolis Force and the Curvature Effect 14
1.3.4 Constant Angular Momentum Oscillations 17
1.4 Structure of the Static Atmosphere 18
1.4.1 The Hydrostatic Equation 18
1.4.2 Pressure as a Vertical Coordinate 20
1.4.3 A Generalized Vertical Coordinate 22
1.5 Kinematics 23
1.6 Scale Analysis 25
Suggested References 26
Problems 26
Matlab Exercises 28
2. Basic Conservation Laws
2.1 Total Differentiation 31
2.1.1 Total Differentiation ofa Vector in a Rotating System 33
2.2 TheVectorial Form of the Momentum Equation In Rotating
Coordinates 35
2.3 Component Equations in Spherical Coordinates 37
2.4 Scale Analysis of the Equations of Motion 41
2.4.1 Geostrophic Approximation and Geostrophic Wind 42
2.4.2 Approximate Prognostic Equations: The Rossby Number 43
2.4.3 The Hydrostatic Approximation 44
2.5 The Continuity Equation 45
2.5.1 A Eulerian Derivation 46
2.5.2 A Lagrangian Derivation 47
2.5.3 Scale Analysis of the Continuity Equation 48
2.6 The Thermodynamic Energy Equation 50
VII
(TnT)
2.7 Thermodynamics of the Dry Atmosphere 53
2.7.1 Potential Temperature 53
2.7.2
The Adiabatic Lapse Rate 54
2.7.3 Static Stability 54
2.7.4 Scale Analysis of the Thermodynamic Energy Equation 56
2.8 The Boussinesq Approximation 57
2.9 Thermodynamics of the Moist Atmosphere 58
2.9.1 Equivalent Potential Temperature 59
2.9.2 The Pseudoadiabatic Lapse Rate 61
2.9.3 Conditional Instability 52
Suggested References 54
Problems 55
Matlab Exercises 55
3. Elementary Applications of the Basic Equations
3.1 Basic Equations In Isobaric Coordinates 67
3.1.1 The Horizontal Momentum Equation 67
3.1.2 The Continuity Equation 53
3.1.3 The Thermodynamic Energy Equation 69
3.2 Balanced Flow
3.2.1 Natural Coordinates 70
3.2.2 Geostrophic
Flow 71
3.2.3 Inertial Flow 72
3.2.4 Cyclostrophic Flow 73
3.2.5 The Gradient Wind Approximation 74
3.3 Trajectories and Streamlines 78
3.4
The Thermal Wind 81
3.4.1 Barotropic and Baroclinic Atmospheres 84
3.5 Vertical Motion 84
3.5.1 The Kinematic Method 85
3.5.2 The Adiabatic Method 87
3.6 Surface Pressure Tendency 87
Problems gg
Matlab Exercises 92
4. Circulation, Vorticity, and Potential Vorticity
4.1 The Circulation Theorem 95
4.2 Vorticity 100
4.2.1 Vorticity in Natural Coordinates 102
4.3 The Vorticity Equation 104
4.3.1 Cartesian Coordinate Form 104
4.3.2 The Vorticity Equation in Isobaric Coordinates 106
4.3.3 Scale Analysis ofthe Vorticity Equation 107
4.4 Potential Vorticity 110
4.5 Shallow Water Equations 115
4.5.1 Barotropic Potential Vorticity 118
Contents
Contents
4.6 Ertel Potential Vorticity in Isentropic Coordinates
4.6.1 Equations of Motion in Isentropic Coordinates
4.6.2 The Potential Vorticity Equation
4.6.3 Integral Constraints on Isentropic Vorticity
Suggested References ^22
Problems
Matlab Exercises
5. Atmospheric Oscillations
5.1 The Perturbation Method
5.2 Properties of Waves
5.2.1 Fourier Series
5.2.2 Dispersion and Group Velocity
5.2.3 Wave Properties in Two and Three Dimensions
5.2.4 AWave Solution Strategy
5.3 Simple WaveTypes
5.3.1 Acoustic or SoundWaves
5.3.2 Shallow Water Waves
5.4 Internal Gravity (Buoyancy) Waves
5.4.1 Pure Internal Gravity Waves
5.5 Linear Waves of a Rotating Stratified Atmosphere
5.5.1 Pure Inertial Oscillations
5.5.2 Rossby and Inertia-Gravity Waves
5.6 Adjustment to Geostrophic Balance
5.7 Rossby Waves '
5.7.1 Free Barotropic Rossby Waves
5.7.2 Forced Topographic Rossby Waves
Suggested References ^
Problems
Matlab Exercises
6. Quasi-geostrophic Analysis
6.1 The Observed Structure ofExtratroplcal Circulations
6.2 Derivation ofthe Quasi-Geostrophic Equations
6.2.1 Preliminaries
6.3 Potential Vorticity Derivation ofthe QG Equations
6.4 Potential Vorticity Thinking . • ia«
6.4.1 PV Inversion, Induced Flow, and Piecewise nversion
6.4.2 PV Conservation and the QG "Height Tendency" Equation 194
6.5 Vertical Motion (w) Thinking
6.6 Idealized Model of a Baroclinic Disturbance
6.7 Isobaric Form of the QG Equations
Suggested References
Problems
Matlab Exercises
QD
120
120
121
121
122
124
127
128
130
131
133
135
136
136
139
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145
150
150
152
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187
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210
CZ) Contents
7. Baroclinic Development
7.1 Hydrodynamic Instability 213
7.2 Normal Mode Baroclinic Instability; ATwo-Layer Model 215
7.2.1 Linear Perturbation Analysis 217
7.2.2 Vertical Motion in Baroclinic Waves 223
7.3 The Energetics of Baroclinic Waves 227
7.3.1 Available Potential Energy 227
7.3.2 Energy Equations for the Two-Layer Model 229
7.4 Baroclinic Instability of a Continuously Stratified Atmosphere 234
7.4.1 Log-Pressure Coordinates 235
7.4.2 Baroclinic Instability: The Rayleigh Theorem 237
7.4.3 The Eady Stability Problem 241
7.5 Growth and Propagation of Neutral Modes 245
7.5.1 Transient Growth of Neutral Waves 247
7.5.2 Downstream Development 250
Suggested References 251
Problems 251
Matlab Exercises 253
8. The Planetary Boundary Layer
8.1 Atmospheric Turbulence 256
8.1.1 Reynolds Averaging 256
8.2 Turbulent Kinetic Energy 259
8.3 Planetary Boundary Layer Momentum Equations 261
8.3.1 Well-Mixed Boundary Layer 262
8.3.2 The Flux-Gradient Theory 264
8.3.3 The Mixing Length Hypothesis 264
8.3.4 The Ekman Layer 266
8.3.5 The Surface
Layer 268
8.3.6 The Modified Ekman Layer 269
8.4 Secondary Circulations and Spin Down 270
Suggested References 275
Problems 275
Matlab Exercises 276
9. Mesoscale Circulations
9.1 Energy Sources for Mesoscale Circulations 279
9.2 Fronts and Frontogenesis 280
9.2.1 The Kinematics of Frontogenesis 281
9.2.2 Semigeostrophic Theory 285
9.2.3 Cross-Frontal Circulation 287
9.3 Symmetric Baroclinic Instability 290
9.4 Mountain Waves 294
9.4.1 Wavesover Sinusoidal Topography 294
9.4.2 Flow over Isolated Ridges 297
Contents
9.4.3 Lee Waves
9.4.4 Downslope Windstorms 299
9.5 Cumulus Convection
9.5.1 Convective Available Potential Energy 302
9.5.2 Entrainment
9.6 Convective Storms
9.6.1 Development of Rotation in Supercell Thunderstorms 306
9.6.2 The Right-Moving Storm 310
9.7 Hurricanes
9.7.1 Dynamics of Mature Hurricanes 314
9.7.2 Hurricane Development 318
Suggested References
Problems ^22
Matlab Exercises
10. The General Circulation
10.1 The Nature of the Problem 326
10.2 The Zonally Averaged Circulation 32«
10.2.1 The Conventional Eulerian Mean 3JU
10.2.2 The Transformed Eulerian Mean 337
10^2.3 The Zonal-Mean Potential Vorticity Equation
10.3 The Angular Momentum Budget
10.3.1 Sigma Coordinates
10.3.2 The Zonal-Mean Angular Momentum
10.4 The Lorenz EnergyCycle
10.5 Longitudinally Dependent Time-Averaged Flow
10.5.1 Stationary Rossby Waves
10.5.2 Jet Stream and Storm Tracks
10.6 Low-Frequency Variability
10.6.1 Climate Regimes
10.6.2 Annular Modes
10.6.3 Sea Surface Temperature Anomalies
10.7 Numerical Simulation of the General Circulation
10.7.1 Dynamical Formulation
10.7.2 Physical Processes and Parameterizations
10.8 Climate Sensitivity, Feedbacks, and Uncertainty
11.1.5 The Walker Circulation 389
n .1.6 El Nino and the Southern Oscillation 390
11.1.7 Equatorial Intraseasonal Oscillation 392
11.2 Scale Analysis of Large-Scale Tropical Motions 392
11.3 Condensation Heating 398
11.4 Equatorial Wave Theory 401
11.4.1 Equatorial Rossby and Rossby-Gravity Modes 401
11.4.2 Equatorial Kelvin Waves 404
11.5 Steady Forced Equatorial Motions 406
Suggested References 409
Problems 409
Matlab Exercises 410
12. Middle Atmosphere Dynamics
12.1 Structure and Circulation of the Middle Atmosphere 413
12.2 The Zonal-Mean Circulation of the Middle Atmosphere 417
12.2.1 Lagrangian Motion of Air Parcels 418
12.2.2 The Transformed Eulerian Mean 420
12.2.3 Zonal-Mean Transport 424
12.3 Vertically Propagating Planetary Waves 426
12.3.1 Linear Rossby Waves 426
12.3.2 Rossby Wave Breaking 428
12.4 Sudden Stratospheric Warmings 430
12.5 Waves in the Equatorial Stratosphere 435
12.5.1 Vertically Propagating Kelvin Waves 436
12.5.2 Vertically Propagating Rossby-Gravity Waves 437
12.5.3 Observed Equatorial Waves 438
12.6 The Quasi-Biennial Oscillation 44O
12.7 Trace Constituent Transport 446
12.7.1 Dynamical Tracers 446
12.7.2 Chemical Tracers 447
12.7.3 Transport in the Stratosphere 448
Suggested References 45O
Problems 45O
Matlab Exercises 452
13. Numerical Modeling and Prediction
13.1 Historical Background 453
13.2 Numerical Approximation of the Equations of Motion 455
13.2.1 Finite Differences 455
13.2.2 Centered Differences: Explicit Time Differencing 457
13.2.3 Computational Stability 458
13.2.4 Implicit Time Differencing 45O
13.2.5 The Semi-Lagrangian Integration Method 462
13.2.6 Truncation Error 453
13.3 The Barotropic Vorticity Equation In Finite Differences 464
Contents GiD
13.4 The Spectral Method
13.4.1 The Barotropic Vorticity Equation in Spherical Coordinates 468
13.4.2 Rossby-Haurwitz Waves 470
13.4.3 The Spectral Transform Method 471
13.5 Primitive Equation Models 472
13.5.1 Spectral Models 473
13.5.2 Physical Parameterizations 474
13.6 Data Assimilation
13.6.1 Data Assimilation for a Single Variable 476
13.6.2 Data Assimilation for Many Variables 479
13.7 Predictability and Ensemble Forecasting 481
Suggested References
Problems
Matlab Exercises
Appendices
A. Useful Constants and Parameters 491
B. List of Symbols
C. Vector Analysis
C.1 Vector Identities ^
C.2 Integral Theorems ^
C.3 Vector Operations in Various Coordinate Systems
D. Moisture Variables
D.1 Equivalent Potential Temperature
D.2 Pseudo Adiabatic Lapse Rate
E. Standard Atmosphere Data
F. Symmetric Baroclinic Oscillations
C. Conditional Probability and Likelihood 511