# Fluid Dynamics of the Mid-Latitude Atmosphere

# Fluid Dynamics of the Mid-Latitude Atmosphere

ISBN: 978-1-118-52600-2 August 2014 Wiley-Blackwell 432 Pages

## Description

This book gives a coherent development of the current understanding of the fluid dynamics of the middle latitude atmosphere. It is primarily aimed at post-graduate and advanced undergraduate level students and does not assume any previous knowledge of fluid mechanics, meteorology or atmospheric science. The book will be an invaluable resource for any quantitative atmospheric scientist who wishes to increase their understanding of the subject. The importance of the rotation of the Earth and the stable stratification of its atmosphere, with their implications for the balance of larger-scale flows, is highlighted throughout.Clearly structured throughout, the first of three themes deals with the development of the basic equations for an atmosphere on a rotating, spherical planet and discusses scale analyses of these equations. The second theme explores the importance of rotation and introduces vorticity and potential vorticity, as well as turbulence. In the third theme, the concepts developed in the first two themes are used to give an understanding of balanced motion in real atmospheric phenomena. It starts with quasi-geostrophic theory and moves on to linear and nonlinear theories for mid-latitude weather systems and their fronts. The potential vorticity perspective on weather systems is highlighted with a discussion of the Rossby wave propagation and potential vorticity mixing covered in the final chapter.

Preface xi

Select bibliography xv

The authors xix

**1 Observed flow in the Earth’s midlatitudes 1**

1.1 Vertical structure 1

1.2 Horizontal structure 4

1.3 Transient activity 11

1.4 Scales of motion 14

1.5 The Norwegian frontal model of cyclones 15

Theme 1 Fluid dynamics of the midlatitude atmosphere 25

**2 Fluid dynamics in an inertial frame of reference 27**

2.1 Definition of fluid 27

2.2 Flow variables and the continuum hypothesis 29

2.3 Kinematics: characterizing fluid flow 30

2.4 Governing physical principles 35

2.5 Lagrangian and Eulerian perspectives 36

2.6 Mass conservation equation 38

2.7 First Law of Thermodynamics 40

2.8 Newton’s Second Law of Motion 41

2.9 Bernoulli’s Theorem 45

2.10 Heating and water vapour 47

**3 Rotating frames of reference 53**

3.1 Vectors in a rotating frame of reference 53

3.2 Velocity and Acceleration 55

3.3 The momentum equation in a rotating frame 56

3.4 The centrifugal pseudo-force 57

3.5 The Coriolis pseudo-force 59

3.6 The Taylor–Proudman theorem 61

**4 The spherical Earth 65**

4.1 Spherical polar coordinates 65

4.2 Scalar equations 67

4.3 The momentum equations 68

4.4 Energy and angular momentum 70

4.5 The shallow atmosphere approximation 73

4.6 The beta effect and the spherical Earth 74

**5 Scale analysis and its applications 77**

5.1 Principles of scaling methods 77

5.2 The use of a reference atmosphere 79

5.3 The horizontal momentum equations 81

5.4 Natural coordinates, geostrophic and gradient wind balance 83

5.5 Vertical motion 87

5.6 The vertical momentum equation 89

5.7 The mass continuity equation 91

5.8 The thermodynamic energy equation 92

5.9 Scalings for Rossby numbers that are not small 95

**6 Alternative vertical coordinates 97**

6.1 A general vertical coordinate 97

6.2 Isobaric coordinates 100

6.3 Other pressure-based vertical coordinates 103

6.4 Isentropic coordinates 106

**7 Variations of density and the basic equations 109**

7.1 Boussinesq approximation 109

7.2 Anelastic approximation 111

7.3 Stratification and gravity waves 113

7.4 Balance, gravity waves and Richardson number 115

7.5 Summary of the basic equation sets 121

7.6 The energy of atmospheric motions 122

Theme 2 Rotation in the atmosphere 125

**8 Rotation in the atmosphere 127**

8.1 The concept of vorticity 127

8.2 The vorticity equation 129

8.3 The vorticity equation for approximate sets of equations 131

8.4 The solenoidal term 132

8.5 The expansion/contraction term 134

8.6 The stretching and tilting terms 135

8.7 Friction and vorticity 138

8.8 The vorticity equation in alternative vertical coordinates 144

8.9 Circulation 145

**9 Vorticity and the barotropic vorticity equation 149**

9.1 The barotropic vorticity equation 149

9.2 Poisson’s equation and vortex interactions 151

9.3 Flow over a shallow hill 155

9.4 Ekman pumping 159

9.5 Rossby waves and the beta plane 160

9.6 Rossby group velocity 166

9.7 Rossby ray tracing 170

9.8 Inflexion point instability 172

**10 Potential vorticity 177**

10.1 Potential vorticity 177

10.2 Alternative derivations of Ertel’s theorem 180

10.3 The principle of invertibility 182

10.4 Shallow water equation potential vorticity 186

**11 Turbulence and atmospheric flow 189**

11.1 The Reynolds number 189

11.2 Three-dimensional flow at large Reynolds number 194

11.3 Two-dimensional flow at large Reynolds number 196

11.4 Vertical mixing in a stratified fluid 201

11.5 Reynolds stresses 203

Theme 3 Balance in atmospheric flow 209

**12 Quasi-geostrophic flows 211**

12.1 Wind and temperature in balanced flows 211

12.2 The quasi-geostrophic approximation 215

12.3 Quasi-geostrophic potential vorticity 219

12.4 Ertel and quasi-geostrophic potential vorticities 221

**13 The omega equation 225**

13.1 Vorticity and thermal advection form 225

13.2 Sutcliffe Form 231

13.3 Q-vector form 233

13.4 Ageostrophic flow and the maintenance of balance 238

13.5 Balance and initialization 240

**14 Linear theories of baroclinic instability 245**

14.1 Qualitative discussion 245

14.2 Stability analysis of a zonal flow 247

14.3 Rossby wave interpretation of the stability conditions 256

14.4 The Eady model 264

14.5 The Charney and other quasi-geostrophic models 271

14.6 More realistic basic states 275

14.7 Initial value problem 281

**15 Frontogenesis 291**

15.1 Frontal scales 291

15.2 Ageostrophic circulation 294

15.3 Description of frontal collapse 299

15.4 The semi-geostrophic Eady model 305

15.5 The confluence model 307

15.6 Upper-level frontogenesis 309

**16 The nonlinear development of baroclinic waves 311**

16.1 The nonlinear domain 311

16.2 Semi-geostrophic baroclinic waves 312

16.3 Nonlinear baroclinic waves on realistic jets on the sphere 320

16.4 Eddy transports and zonal mean flow changes 323

16.5 Energetics of baroclinic waves 332

**17 The potential vorticity perspective 337**

17.1 Setting the scene 337

17.2 Potential vorticity and vertical velocity 340

17.3 Life cycles of some baroclinic waves 342

17.4 Alternative perspectives 346

17.5 Midlatitude blocking 350

17.6 Frictional and heating effects 352

**18 Rossby wave propagation and potential vorticity mixing 361**

18.1 Rossby wave propagation 361

18.2 Propagation of Rossby waves into the stratosphere 363

18.3 Propagation through a slowly varying medium 365

18.4 The Eliassen–Palm flux and group velocity 370

18.5 Baroclinic life cycles and Rossby waves 372

18.6 Variations of amplitude 373

18.7 Rossby waves and potential vorticity steps 375

18.8 Potential vorticity steps and the Rhines scale 381

Appendices 389

Appendix A: Notation 389

Appendix B: Revision of vectors and vector calculus 393

B.1 Vectors and their algebra 393

B.2 Products of vectors 394

B.3 Scalar fields and the grad operator 396

B.4 The divergence and curl operators 397

B.5 Gauss’ and Stokes’ theorems 398

B.6 Some useful vector identities 401

Index 403