Mid-Latitude Atmospheric Dynamics A First

This exciting text provides a mathematically rigorous yet accessible textbook that is primarily aimed at atmospheric science majors. Its accessibility is due to the texts emphasis on conceptual understanding. The first five chapters constitute a companion text to introductory courses covering the dynamics of the mid-latitude atmosphere. The final four chapters constitute a more advanced course, and provide insights into the diagnostic power of the quasi-geostrophic approximation of the equations outlined in the previous chapters, the meso-scale dynamics of thefrontal zone, the alternative PV perspective for cyclone interpretation, and the dynamics of the life-cycle of mid-latitude cyclones. Written in a clear and accessible style Features real weather examples and global case studies Each chapter sets out clear learning objectives and tests students' knowledge with concluding questions and answers

Jonathan E. Martin is a Professor in the Department of Atmospheric and Oceanic Sciences at the University of Wisconsin-Madison where he has taught since 1994. He has received numerous accolades for his teaching including the Underkofler Excellence in Teaching Award and is a Fellow in the Teaching Academy of the University of Wisconsin. His teaching excellence is allied with research expertise in the study of mid-latitude weather systems. Professor Martin has published extensively in scholarly journals and was awarded the distinction of being named a Mark H. Ingraham Distinguished Faculty Member by the College of Letters and Science at UW-Madison.

Preface

ix

Acknowledgments

xi

1 Introduction and Review of Mathematical Tools

1

(24)

Objectives

1

(1)

1.1 Fluids and the nature of fluid dynamics

2

(1)

1.2 Review of useful mathematical tools

2

(12)

1.2.1 Elements of vector calculus

3

(6)

1.2.2 The Taylor series expansion

9

(1)

1.2.3 Centered difference approximations to derivatives

10

(2)

1.2.4 Temporal changes of a continuous variable

12

(2)

1.3 Estimating with scale analysis

14

(1)

1.4 Basic kinematics of fluids

15

(5)

1.4.1 Pure vorticity

17

(1)

1.4.2 Pure divergence

17

(1)

1.4.3 Pure stretching deformation

17

(2)

1.4.4 Pure shearing deformation

19

(1)

1.5 Mensuration

20

(1)

Selected references

21

(1)

Problems

21

(2)

Solutions

23

(2)

2 Fundamental and Apparent Forces

25

(18)

Objectives

25

(1)

2.1 The fundamental forces

26

(6)

2.1.1 The pressure gradient force

26

(1)

2.1.2 The gravitational force

27

(1)

2.1.3 The frictional force

28

(4)

2.2 Apparent forces

32

(8)

2.2.1 The centrifugal force

33

(2)

2.2.2 The Coriolis force

35

(5)

Selected references

40

(1)

Problems

40

(1)

Solutions

41

(2)

3 Mass, Momentum, and Energy: The Fundamental Quantities of the Physical World

43

(34)

Objectives

43

(1)

3.1 Mass in the Atmosphere

43

(6)

3.1.1 The hypsometric equation

45

(4)

3.2 Conservation of momentum: The equations of motion

49

(18)

3.2.1 The equations of motion in spherical coordinates

53

(12)

3.2.2 Conservation of mass

65

(2)

3.3 Conservation of energy: The energy equation

67

(6)

Selected references

73

(1)

Problems

73

(3)

Solutions

76

(1)

4 Applications of the Equations of Motion

77

(38)

Objectives

77

(1)

4.1 Pressure as a vertical coordinate

77

(6)

4.2 Potential temperature as a vertical coordinate

83

(6)

4.3 The thermal wind balance

89

(4)

4.4 Natural coordinates and balanced flows

93

(15)

4.4.1 Geostrophic flow

97

(1)

4.4.2 Inertial flow

98

(1)

4.4.3 Cyclostrophic flow

99

(3)

4.4.4 Gradient flow

102

(6)

4.5 The relationship between trajectories and streamlines

108

(3)

Selected references

111

(3)

Problems '

'11

Solutions

114

(1)

5 Circulation, Vorticity, and Divergence

115

(32)

Objectives

5.1 The Circulation theorem and its physical interpretation

117

(55)

5.2 Vorticity and potential vorticity

172

5.3 The relationship between vorticity and divergence

130

(8)

5.4 The quasi-geostrophic system of equations

138

(4)

Selected references

142

(1)

Problems

142

(2)

Solutions

144

(3)

6 The Diagnosis of Mid-Latitude Synoptic-Scale Vertical Motions

147

(40)

Objectives

147

(1)

6.1 The nature of the ageostrophic wind: Isolating the acceleration vector

148

(9)

6.1.1 Sutcliffe's expression for net ageostrophic divergence in a column

150

(4)

6.1.2 Another perspective on the ageostrophic wind

154

(3)

6.2 The Sutcliffe development theorem

157

(3)

6.3 The quasi-geostrophic omega equation

160

(6)

6.4 The Q-vector

166

(15)

6.4.1 The geostrophic paradox and its resolution

167

(4)

6.4.2 A natural coordinate version of the ->Q -vector

171

(7)

6.4.3 The along- and across-isentrope components of ->Q

178

(3)

Selected references

181

(1)

Problems

181

(5)

Solutions

186

(1)

7 The Vertical Circulation at Fronts

187

(50)

Objectives

187

(2)

7.1 The structural and dynamical characteristics of mid-latitude fronts

189

(4)

7.2 Frontogenesis and vertical motions

193

(11)

7.3 The semi-geostrophic equations

204

(7)

7.4 Upper-level frontogenesis

211

(9)

7.5 Precipitation processes at fronts

220

(9)

Selected references

229

(1)

Problems

229

(5)

Solutions

234

(3)

8 Dynamical Aspects of the Life Cycle of the Mid-Latitude Cyclone

237

(38)

Objectives

237

(1)

8.1 Introduction: The polar front theory of cyclones

237

(5)

8.2 Basic structural and energetic characteristics of the cyclone

242

(4)

8.3 The cyclogenesis stage: The QG tendency equation perspective

246

(4)

8.4 The cyclogenesis stage: The QG omega equation perspective

250

(2)

8.5 The cyclogenetic influence of diabatic processes: Explosive cyclogenesis

252

(6)

8.6 The post-mature stage: Characteristic thermal structure

258

(6)

8.7 The post-mature stage: The QG dynamics of the occluded quadrant

264

(1)

8.8 The Decay Stage

265

(4)

Selected references

269

(1)

Problems

269

(4)

Solutions

273

(2)

9 Potential Vorticity and Applications to Mid-Latitude Weather Systems

275

(36)

Objectives

275

(1)

9.1 Potential vorticity and isentropic divergence

276

(4)

9.2 Characteristics of a positive PV anomaly

280

(6)

9.3 Cyclogenesis from the PV perspective

286

(4)

9.4 The influence of diabatic heating on PV

290

(5)

9.5 Additional applications of the PV perspective

295

(12)

9.5.1 Piecewise PV inversion and some applications

295

(2)

9.5.2 A PV perspective on occlusion

297

(5)

9.5.3 A PV perspective on leeside cyclogenesis

302

(1)

9.5.4 The effects of PV superposition and attenuation

302

(5)

Selected references

307

(1)

Problems

307

(3)

Solutions

310

(1)

Appendix A: Virtual Temperature

311

(2)

Bibliography

313

(4)

Index

317

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Mid-Latitude Atmospheric Dynamics A First Course

Summary

Author Biography

Table of Contents

Supplemental Materials