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12.003 Atmosphere, Ocean and Climate Dynamics

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Contents

0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

0.1.1 Natural fluid dynamics . . . . . . . . . . . . . . . . . . 10

0.1.2 Rotating fluid dynamics . . . . . . . . . . . . . . . . . 13

0.1.3 Holicism . . . . . . . . . . . . . . . . . . . . . . . . . . 16

I The Atmosphere 19

1 Characteristics of the atmosphere 21

1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2 Chemical composition of the atmosphere . . . . . . . . . . . . 22

1.3 Physical properties of air . . . . . . . . . . . . . . . . . . . . . 23

1.3.1 Moist air . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 The global energy balance 29

2.1 Effective planetary temperature (emission temperature) . . . . 29

2.2 The atmospheric absorption spectrum . . . . . . . . . . . . . . 33

2.3 The greenhouse effect . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 A simple greenhouse model . . . . . . . . . . . . . . . 36

2.3.2 A leaky greenhouse . . . . . . . . . . . . . . . . . . . . 38

2.3.3 A more opaque greenhouse . . . . . . . . . . . . . . . . 39

3 The vertical structure of the atmosphere 43

3.1 Vertical distribution of temperature and ‘Greenhouse gases’ . . 43

3.1.1 Typical temperature profile . . . . . . . . . . . . . . . 43

3.1.2 Atmospheric layers . . . . . . . . . . . . . . . . . . . . 45

3.2 The relationship between pressure and density: hydrostatic

balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Vertical structure of pressure and density . . . . . . . . . . . . 50

3
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3.3.1 Isothermal atmosphere . . . . . . . . . . . . . . . . . . 50

3.3.2 Non-isothermal atmosphere . . . . . . . . . . . . . . . 51

3.3.3 Density . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Convection 53

4.1 The nature of convection . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Convection in a shallow fluid . . . . . . . . . . . . . . . 53

4.1.2 Instability . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Convection in water (an almost-incompressible fluid) . . . . . . . 56

4.2.1 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.3 Energetics . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.4 GFD Lab II: convection . . . . . . . . . . . . . . . . . 59

4.3 Dry convection in a compressible atmosphere . . . . . . . . . . 63

4.3.1 The adiabatic lapse rate . . . . . . . . . . . . . . . . . 63

4.3.2 Potential temperature . . . . . . . . . . . . . . . . . . 65

4.4 The atmosphere under stable conditions . . . . . . . . . . . . 68

4.4.1 Gravity waves . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.2 Temperature inversions . . . . . . . . . . . . . . . . . . 71

4.5 ‘Moist’ convection . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5.1 Humidity . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.2 Saturated adiabatic lapse rate . . . . . . . . . . . . . . 74

4.5.3 Radiative-convective equilibrium . . . . . . . . . . . . 75

4.6 Convection in the atmosphere . . . . . . . . . . . . . . . . . . 76

4.6.1 Convective clouds . . . . . . . . . . . . . . . . . . . . . 76

4.6.2 Occurrence and depth of convection . . . . . . . . . . . 77

4.6.3 Where does convection occur? . . . . . . . . . . . . . . 78

5 The Meridional Structure of the Atmosphere 81

5.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.1.1 Latitudinal dependence of incoming radiation . . . . . 81

5.1.2 Latitudinal dependence of outgoing radiation . . . . . . 82

5.1.3 Meridional structure of temperature . . . . . . . . . . . 84

5.1.4 The energy balance of the atmosphere . . . . . . . . . 87

5.2 Pressure and geopotential height . . . . . . . . . . . . . . . . . 88

5.2.1 The height of pressure surfaces . . . . . . . . . . . . . 88

5.2.2 Geopotential surfaces . . . . . . . . . . . . . . . . . . . 89

5.3 Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

CONTENTS 5

5.4 Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4.1 Distribution of winds . . . . . . . . . . . . . . . . . . . 95

6 The equations of fluid motion 103

6.1 Differentiation following the motion . . . . . . . . . . . . . . . 103

6.2 Equation of motion for a nonrotating fluid . . . . . . . . . . . 106

6.2.1 Forces on a fluid parcel . . . . . . . . . . . . . . . . . . 106

6.2.2 The equation of motion . . . . . . . . . . . . . . . . . . 110

6.2.3 Hydrostatic balance . . . . . . . . . . . . . . . . . . . . 111

6.3 The continuity equation . . . . . . . . . . . . . . . . . . . . . 111

6.3.1 Incompressible flow . . . . . . . . . . . . . . . . . . . . 112

6.3.2 Compressible flow . . . . . . . . . . . . . . . . . . . . . 112

6.4 Equation of motion for a rotating fluid . . . . . . . . . . . . . 114

6.4.1 GFD Lab III: radial inflow . . . . . . . . . . . . . . . . 114

6.4.2 Transformation into rotating coordinates . . . . . . . . 119

6.4.3 The rotating equation of motion . . . . . . . . . . . . . 121

6.4.4 Experiments with Coriolis forces on a parabolic rotat-

ing table . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.4.5 Geostrophic motion . . . . . . . . . . . . . . . . . . . . 129

6.4.6 The Taylor-Proudman Theorem . . . . . . . . . . . . . 132

6.4.7 The thermal wind equation . . . . . . . . . . . . . . . 135

6.4.8 Subgeostrophic flow: the Ekman layer . . . . . . . . . 143

6.5 Putting things on the sphere . . . . . . . . . . . . . . . . . . . 148

6.5.1 GFD Lab X: An experiment on the Earth’s rotation . . 148

6.5.2 The centrifugal force, modified hydrostatic balance and

geopotential surfaces on the sphere . . . . . . . . . . . 150

6.5.3 Components of the Coriolis force on the sphere: the

Coriolis parameter . . . . . . . . . . . . . . . . . . . . 152

6.6 Geostrophic balance on the sphere . . . . . . . . . . . . . . . . 154

6.6.1 Small Rossby number flow . . . . . . . . . . . . . . . . 154

6.7 Thermal wind in pressure coordinates . . . . . . . . . . . . . . 163

6.7.1 Thermal wind expressed in terms of potential temper-

ature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7 The general circulation of the atmosphere 169

7.1 The tropical Hadley circulation . . . . . . . . . . . . . . . . . 169

7.2 The midlatitude circulation . . . . . . . . . . . . . . . . . . . 173

7.2.1 Energy stored in the thermal wind . . . . . . . . . . . 174

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7.2.2 Available potential energy . . . . . . . . . . . . . . . . 175

7.2.3 Baroclinic instability . . . . . . . . . . . . . . . . . . . 178

7.2.4 GFD Lab XI: baroclinic instability of the thermal wind 180

7.3 The ‘big picture’ of the atmospheric heat (and momentum)

budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.3.1 Energy requirements, as deduced from observations . . 182

7.3.2 Incorporating eddy transfer in to our general circula-

tion theory . . . . . . . . . . . . . . . . . . . . . . . . 185

II The Ocean 189

8 The ocean and its circulation 191

8.1 Physical characteristics of the ocean . . . . . . . . . . . . . . . 192

8.1.1 Properties of seawater; equation of state . . . . . . . . 193

8.1.2 Temperature and salinity structure . . . . . . . . . . . 196

8.1.3 The mixed layer and thermocline . . . . . . . . . . . . 201

8.2 The observed circulation . . . . . . . . . . . . . . . . . . . . . 204

8.3 Inferences from geostrophic and hydrostatic balance . . . . . . 207

8.3.1 Ocean surface structure and geostrophic flow . . . . . . 208

8.3.2 Deep geostrophic flow . . . . . . . . . . . . . . . . . . 211

9 The wind-driven circulation 213

9.1 The wind stress and Ekman layers . . . . . . . . . . . . . . . . 213

9.1.1 Balance of forces in the Ekman layer . . . . . . . . . . 215

9.1.2 Wind-driven Ekman pumping . . . . . . . . . . . . . . 216

9.2 Response of the interior ocean to Ekman pumping . . . . . . . 220

9.2.1 Interior balances . . . . . . . . . . . . . . . . . . . . . 220

9.2.2 Taylor-Proudman on the sphere . . . . . . . . . . . . . 221

9.2.3 GFD Lab XIII: Wind-driven ocean gyres . . . . . . . . 225

9.2.4 The wind-driven gyres and western boundary currents 226

9.3 The depth-integrated circulation . . . . . . . . . . . . . . . . . 228

9.3.1 Mass transport of gyres: western boundary current

speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

9.4 Effects of inhomogeneity . . . . . . . . . . . . . . . . . . . . . 231

9.4.1 Taylor-Proudman in a layered ocean . . . . . . . . . . 233

9.5 Ocean eddies . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

9.5.1 Observations of ocean eddies . . . . . . . . . . . . . . . 236

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9.5.2 Baroclinic instability in the ocean . . . . . . . . . . . . 238

10 The thermohaline circulation of the ocean 243

10.1 Sources of deep water . . . . . . . . . . . . . . . . . . . . . . . 243

10.2 Time scales and intensity of thermohaline circulation . . . . . 249

10.3 Abyssal circulation schematic deduced from ‘Taylor-Proudman’ 250

10.4 Observations of the abyssal circulation . . . . . . . . . . . . . 252

10.5 GFD Lab XIV: The thermohaline circulation . . . . . . . . . . 257

10.6 Why western boundary currents? . . . . . . . . . . . . . . . . 258

10.6.1 GFD Lab XV: Source sink flow in a rotating basin . . . 260

11 The ocean’s role in climate 263

11.1 Ocean Heat storage . . . . . . . . . . . . . . . . . . . . . . . . 263

11.2 Ocean heat transport . . . . . . . . . . . . . . . . . . . . . . . 264

11.2.1 Mechanisms of ocean heat transport . . . . . . . . . . 266