Atmospheric and oceanic fluid dynamics. Fundamentals and large-scale circulation

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 21
Part II. Instabilities, wave–mean flow interaction and turbulence......Page 23
Acknowledgements......Page 24
Note on second printing......Page 25
NOTATION......Page 26
1.1.1 Field and material viewpoints......Page 31
1.1.2 The material derivative of a fluid property......Page 32
Material derivative of vector field......Page 33
1.1.3 Material derivative of a volume......Page 34
1.2.1 An Eulerian derivation......Page 35
Cartesian derivation......Page 36
1.2.2 Mass continuity via the material derivative......Page 37
1.3 THE MOMENTUM EQUATION......Page 39
1.3.2 The pressure force......Page 40
1.3.4 Hydrostatic balance......Page 41
1.4 THE EQUATION OF STATE......Page 42
1.5.1 A few fundamentals......Page 44
Composition: The change in internal energy due to changes in composition is given by......Page 45
1.5.2 Various thermodynamic relations......Page 46
Fundamental equation of state......Page 48
Internal energy and specific heats......Page 49
1.6 THERMODYNAMIC EQUATIONS FOR FLUIDS......Page 50
1.6.1 Thermodynamic equation for an ideal gas......Page 51
Potential temperature, potential density and entropy......Page 52
I. Entropy equation using pressure and density......Page 54
Approximations using pressure and density......Page 55
Approximations using pressure and temperature......Page 56
III. Entropy equation using density and temperature......Page 57
Liquids......Page 58
Potential temperature......Page 59
1.7.2 * Thermodynamic properties of seawater......Page 61
Potential temperature and a simplified equation of state for numerical models......Page 64
1.8 SOUND WAVES......Page 65
1.9.1 Constant density fluids......Page 66
Some conditions for incompressibility......Page 67
1.10.1 Constant density fluid......Page 68
1.10.2 Variable density fluids......Page 70
1.11 AN INTRODUCTION TO NON-DIMENSIONALIZATION AND SCALING......Page 71
1.11.1 The Reynolds number......Page 72
Notes......Page 73
Problems......Page 75
2.1 THE EQUATIONS OF MOTION IN A ROTATING FRAME OF REFERENCE......Page 79
2.1.1 Rate of change of a vector......Page 80
2.1.2 Velocity and acceleration in a rotating frame......Page 81
2.1.4 Mass and tracer conservation in a rotating frame......Page 82
2.2.1 * The centrifugal force and spherical coordinates......Page 83
2.2.2 Some identities in spherical coordinates......Page 85
Rate of change of unit vectors......Page 87
Momentum Equation......Page 88
2.2.4 The primitive equations......Page 89
2.2.5 Primitive equations in vector form......Page 90
2.2.6 The vector invariant form of the momentum equation......Page 91
2.2.7 Angular momentum......Page 92
* From angular momentum to the spherical component equations......Page 93
2.3.1 The f-plane......Page 94
2.4 EQUATIONS FOR A STRATIFIED OCEAN: THE BOUSSINESQ APPROXIMATION......Page 95
2.4.2 The Boussinesq equations......Page 96
Momentum equations......Page 97
Thermodynamic equation and equation of state......Page 98
* Mean stratification and the buoyancy frequency......Page 99
2.4.3 Energetics of the Boussinesq system......Page 100
2.5.1 Preliminaries......Page 101
2.5.3 Mass conservation......Page 103
2.5.5 * Energetics of the anelastic equations......Page 104
2.5.2 The momentum equation......Page 102
2.6.1 General relations......Page 105
2.6.2 Pressure coordinates......Page 106
2.7.1 Preliminaries......Page 108
2.7.2 Scaling and the aspect ratio......Page 109
2.7.3 * Effects of stratification on hydrostatic balance......Page 110
In the atmosphere......Page 112
2.8.1 The Rossby number......Page 113
2.8.2 Geostrophic balance......Page 114
2.8.3 Taylor–Proudman effect......Page 116
2.8.4 Thermal wind balance......Page 117
Pressure coordinates......Page 118
2.9 STATIC INSTABILITY AND THE PARCEL METHOD......Page 119
2.9.1 A simple special case: a density-conserving fluid......Page 120
2.9.2 The general case: using potential density......Page 121
A liquid ocean......Page 122
A dry ideal gas......Page 123
* Effects of water vapour on the lapse rate of an ideal gas......Page 124
* Equivalent potential temperature......Page 125
2.10.1 Gravity waves and convection in a Boussinesq fluid......Page 126
Hydrostatic gravity waves and convection......Page 127
2.11 * ACOUSTIC-GRAVITY WAVES IN AN IDEAL GAS......Page 128
Acoustic and gravity waves......Page 129
Vertical structure......Page 130
Hydrostatic approximation and Lamb waves......Page 131
2.12 THE EKMAN LAYER......Page 132
The Ekman number......Page 133
Momentum balance in the Ekman layer......Page 134
2.12.2 Integral properties of the Ekman layer......Page 135
Boundary conditions and solution......Page 137
Transport, force balance and vertical velocity......Page 138
Another bottom boundary condition......Page 139
Transport, surface flow and vertical velocity......Page 140
2.12.5 Observations of the Ekman layer......Page 141
2.12.6 * Frictional parameterization of the Ekman layer......Page 142
Notes......Page 143
Problems......Page 145
3.1 DYNAMICS OF A SINGLE, SHALLOW LAYER......Page 151
3.1.1 Momentum equations......Page 152
From first principles......Page 153
From the 3D mass conservation equation......Page 154
3.1.3 A rigid lid......Page 155
3.1.4 Stretching and the vertical velocity......Page 156
3.2 REDUCED GRAVITY EQUATIONS......Page 157
II The rigid lid approximation......Page 158
3.3 MULTI-LAYER SHALLOW WATER EQUATIONS......Page 159
3.3.1 Reduced-gravity multi-layer equation......Page 161
The two- and three-layer cases......Page 160
3.4 GEOSTROPHIC BALANCE AND THERMAL WIND......Page 162
3.5 FORM DRAG......Page 163
3.6.1 Potential vorticity: a material invariant......Page 164
Vorticity and circulation......Page 166
3.6.2 Energy conservation: an integral invariant......Page 167
3.7.1 Non-rotating shallow water waves......Page 168
3.7.2 Rotating shallow water (Poincaré) waves......Page 169
(i) The short wave limit. If......Page 170
3.7.3 Kelvin waves......Page 171
3.8 GEOSTROPHIC ADJUSTMENT......Page 172
3.8.1 Non-rotating flow......Page 173
3.8.2 Rotating flow......Page 174
3.8.3 * Energetics of adjustment......Page 176
3.8.4 * General initial conditions......Page 177
3.8.5 A variational perspective......Page 179
3.9.1 A hydrostatic Boussinesq fluid......Page 180
3.9.2 A hydrostatic ideal gas......Page 181
3.9.3 * Analogy to shallow water equations......Page 182
3.10 AVAILABLE POTENTIAL ENERGY......Page 183
3.10.1 A Boussinesq fluid......Page 184
3.10.2 An ideal gas......Page 186
Notes......Page 187
Problems......Page 188
4.1.1 Preliminaries......Page 191
The ‘vr’ vortex......Page 192
4.2 THE VORTICITY EQUATION......Page 193
An integral conservation property......Page 195
4.3.1 The ‘frozen-in’ property of vorticity......Page 196
Stretching and tilting......Page 198
4.3.2 Kelvin’s circulation theorem......Page 199
Stretching and circulation......Page 200
4.3.4 Circulation in a rotating frame......Page 201
4.3.5 The circulation theorem for hydrostatic flow......Page 202
4.4.1 The circulation theorem and the beta effect......Page 203
4.4.2 The vertical component of the vorticity equation......Page 204
Two-dimensional and shallow water vorticity equations......Page 205
Barotropic fluids......Page 206
The general case......Page 207
4.5.2 PV conservation from the frozen-in property......Page 208
Baroclinic fluids......Page 209
4.5.3 PV conservation: an algebraic derivation......Page 210
4.5.5 Effects of rotation, and summary remarks......Page 211
4.6.1 Using Kelvin’s theorem......Page 212
4.6.2 Using an appropriate scalar field......Page 213
4.7.1 The Boussinesq equations......Page 214
4.7.3 Potential vorticity on isentropic surfaces......Page 215
4.8 * THE IMPERMEABILITY OF ISENTROPES TO POTENTIAL VORTICITY......Page 216
Motion of the isentropic surface......Page 218
* Dynamical choices of PV flux and a connection to Bernoulli’s theorem......Page 219
Notes......Page 220
Problems......Page 222
CHAPTER FIVE Simplified Equations for the Ocean and Atmosphere......Page 225
5.1.1 Scaling in the shallow water equations......Page 226
Non-dimensional mass continuity (height) equation......Page 227
5.1.2 Geostrophic scaling in the stratified equations......Page 228
Non-dimensional equations......Page 229
Formal derivation......Page 231
Variation of the Coriolis parameter......Page 232
Simplifying the equations......Page 233
Potential vorticity......Page 234
5.3.1 Single-layer shallow water quasi-geostrophic equations......Page 235
Connection to shallow water potential vorticity......Page 237
Two interesting limits......Page 238
5.3.2 Two-layer and multi-layer quasi-geostrophic systems......Page 239
Two-layer model......Page 240
* Multi-layer model......Page 241
5.3.3 † Non-asymptotic and intermediate models......Page 242
5.4.1 Scaling and assumptions......Page 243
5.4.2 Asymptotics......Page 244
The potential vorticity equation......Page 245
Dimensional equations......Page 246
5.4.3 Buoyancy advection at the surface......Page 247
5.4.4 Quasi-geostrophy in pressure coordinates......Page 248
5.4.5 The two-level quasi-geostrophic system......Page 249
Connection to the layered system......Page 251
5.5.1 * Using height coordinates......Page 252
5.5.2 Using isentropic coordinates......Page 253
5.6 * ENERGETICS OF QUASI-GEOSTROPHY......Page 254
5.6.1 Conversion between APE and KE......Page 255
Energy in the baroclinic and barotropic modes......Page 256
5.7.1 Waves in a single layer......Page 257
Infinite deformation radius......Page 258
The mechanism of Rossby waves......Page 259
5.7.2 Rossby waves in two layers......Page 260
5.8.1 Setting up the problem......Page 262
A simple example......Page 263
5.A.1 Kinematics and definitions......Page 264
Phase speed......Page 265
Group velocity......Page 266
Superposition of two waves......Page 267
* Superposition of many waves......Page 268
Notes......Page 269
Problems......Page 270
CHAPTER SIX Barotropic and Baroclinic Instability......Page 275
6.1 KELVIN–HELMHOLTZ INSTABILITY......Page 276
6.2 INSTABILITY OF PARALLEL SHEAR FLOW......Page 278
Jump or matching conditions......Page 279
6.2.2 Kelvin–Helmholtz instability, revisited......Page 281
6.2.4 Interacting edge waves producing instability......Page 282
The mechanism of the instability — an informal view......Page 284
6.3.1 Rayleigh’s criterion......Page 286
An alternate, more general, derivation......Page 287
6.3.2 Fjørtoft’s criterion......Page 288
6.4.1 A physical picture......Page 289
6.4.2 Linearized quasi-geostrophic equations......Page 291
6.4.3 Necessary conditions for baroclinic instability......Page 292
6.5 THE EADY PROBLEM......Page 293
Boundary conditions......Page 294
Solutions......Page 295
For the atmosphere......Page 296
For the ocean......Page 297
6.6.1 Posing the problem......Page 299
6.6.2 The solution......Page 300
Solution in the general case: non-zero shear and non-zero beta......Page 302
Minimum shear for instability......Page 303
High-wavenumber cut-off......Page 304
6.7 AN INFORMAL VIEW OF THE MECHANISM OF BAROCLINIC INSTABILITY......Page 305
A simple dynamical model......Page 306
Further simplifications to the two-layer model......Page 307
6.7.2 Interacting edge waves in the Eady problem......Page 308
6.8 * THE ENERGETICS OF LINEAR BAROCLINIC INSTABILITY......Page 310
6.9 * BETA, SHEAR AND STRATIFICATION IN A CONTINUOUS MODEL......Page 312
6.9.1 Scaling arguments for growth rates, scales and depth......Page 313
Adding beta to the Eady model......Page 315
Effects of non-uniform shear and stratification......Page 316
Notes......Page 319
Problems......Page 321
CHAPTER SEVEN Wave–Mean Flow Interaction......Page 323
7.1 QUASI-GEOSTROPHIC PRELIMINARIES......Page 324
7.1.1 Potential vorticity flux in the linear equations......Page 325
7.2 THE ELIASSEN–PALM FLUX......Page 326
7.2.1 The Eliassen–Palm relation......Page 327
Group velocity property for Rossby waves......Page 328
* Group velocity property: a general derivation......Page 329
7.2.3 * The orthogonality of modes......Page 330
7.3.1 Quasi-geostrophic form......Page 332
7.3.2 The TEM in isentropic coordinates......Page 334
7.3.3 Connection between the residual and thickness-weighted circulation......Page 335
7.3.4 * The TEM in the primitive equations......Page 337
† More general forms......Page 341
7.4.1 A derivation from the potential vorticity equation......Page 342
The stratified case......Page 344
7.4.3 The EP flux and form drag......Page 345
7.5.1 Formulation......Page 347
7.5.2 Solution......Page 349
7.6 * NECESSARY CONDITIONS FOR INSTABILITY......Page 352
7.6.2 Inclusion of boundary terms......Page 353
7.7.1 Two-dimensional flow......Page 355
Parallel shear flow and Fjørtoft’s condition......Page 357
7.7.2 * Stratified quasi-geostrophic flow......Page 358
7.7.3 * Applications to baroclinic instability......Page 359
The high-wavenumber cut-off in two-layer baroclinic instability......Page 360
Notes......Page 361
Problems......Page 362
CHAPTER EIGHT Basic Theory of Incompressible Turbulence......Page 365
8.1.1 The closure problem......Page 366
8.1.2 Triad interactions in turbulence......Page 367
8.2.1 The physical picture......Page 369
8.2.2 Inertial-range theory......Page 370
The viscous scale and energy dissipation......Page 374
8.2.3 * Another expression of the inertial-range scaling argument......Page 376
8.3 TWO-DIMENSIONAL TURBULENCE......Page 377
II. An energy-enstrophy conservation argument......Page 379
III. A similarity argument......Page 381
Some properties of forced-dissipative flow......Page 382
The enstrophy inertial range......Page 384
8.3.3 † More about the phenomenology......Page 385
8.3.4 Numerical illustrations......Page 388
8.4 PREDICTABILITY OF TURBULENCE......Page 389
8.4.1 Low-dimensional chaos and unpredictability......Page 390
8.4.2 * Predictability of a turbulent flow......Page 391
II. Error growth via a direct interaction......Page 392
8.4.3 Implications and weather predictability......Page 393
8.5 * SPECTRA OF PASSIVE TRACERS......Page 394
Energy inertial range flow in three dimensions......Page 395
The viscous-advective range of large Prandtl number flow......Page 396
† The inertial-diffusive range of small Prandtl number flow......Page 397
Notes......Page 399
Problems......Page 402
9.1 EFFECTS OF DIFFERENTIAL ROTATION IN TWO-DIMENSIONAL TURBULENCE......Page 405
Scaling......Page 406
Turbulent phenomenology......Page 407
9.1.2 Generation of zonal flows and jets......Page 408
9.1.3 † Joint effect of beta and friction......Page 410
9.2.1 An analogue to two-dimensional flow......Page 412
9.2.2 Two-layer geostrophic turbulence......Page 413
Baroclinic and barotropic decomposition......Page 414
Conservation properties......Page 415
(i) Interactions at small scales......Page 416
(iii) Interactions at scales comparable to the deformation radius......Page 417
Summary of two-layer phenomenology......Page 418
9.3 † A SCALING THEORY FOR GEOSTROPHIC TURBULENCE......Page 419
9.3.1 Preliminaries......Page 420
9.3.2 Scaling properties......Page 421
9.4 † PHENOMENOLOGY OF BAROCLINIC EDDIES IN THE ATMOSPHERE AND OCEAN......Page 423
9.4.1 The magnitude and scale of baroclinic eddies......Page 424
Amplitude and Scale......Page 425
The baroclinic lifecycle......Page 426
Basic ideas......Page 428
Eddy amplitudes and scales......Page 430
Eddy lifecycles......Page 431
Notes......Page 432
Problems......Page 433
CHAPTER TEN Turbulent Diffusion and Eddy Transport......Page 435
10.1 DIFFUSIVE TRANSPORT......Page 436
10.2.1 Simple theory......Page 437
10.2.2 * An anisotropic generalization......Page 441
10.3 TWO-PARTICLE DIFFUSIVITY......Page 443
10.3.1 Large particle separation......Page 444
10.3.2 Separation within the inertial range......Page 445
A geophysical example......Page 446
10.4 MIXING LENGTH THEORY......Page 447
(iii) Tracer mixing and turbulent cascades......Page 449
10.4.2 A macroscopic perspective......Page 450
10.5.1 Non-existence of extrema......Page 451
10.5.2 Homogenization in two-dimensional flow......Page 452
10.6 † TRANSPORT BY BAROCLINIC EDDIES......Page 453
10.6.2 * Diffusion with the symmetric tensor......Page 454
10.6.3 * The skew flux......Page 455
10.6.4 The story so far......Page 457
10.7.2 Magnitude of the eddy diffusivity......Page 458
10.7.3 * Structure: the symmetric transport tensor......Page 460
The plane of eddy displacements......Page 461
10.7.4 * Structure: the antisymmetric transport tensor......Page 463
An adiabatic, potential-energy diminishing, eddy transport scheme......Page 464
10.7.5 Examples......Page 465
Thickness and its variance......Page 468
The eddy-induced and residual velocities......Page 469
10.8.2 Diffusive thickness transport......Page 470
Preliminaries......Page 471
Implementations and approximations......Page 472
Notes......Page 473
Problems......Page 474
CHAPTER ELEVEN The Overturning Circulation: Hadley and Ferrel Cells......Page 479
11.1.1 The radiative equilibrium distribution......Page 480
11.1.2 Observed wind and temperature fields......Page 481
11.1.3 Meridional overturning circulation......Page 484
11.2.1 Assumptions......Page 485
11.2.2 Dynamics......Page 486
11.2.3 Thermodynamics......Page 488
11.2.4 Zonal wind......Page 490
11.2.5 Properties of solution......Page 491
11.2.6 Strength of the circulation......Page 492
11.2.7 † Effects of moisture......Page 493
11.2.8 The radiative equilibrium solution......Page 494
11.3.2 Thermodynamic balance......Page 496
11.4 † ASYMMETRY AROUND THE EQUATOR......Page 497
11.5.1 Qualitative considerations......Page 501
11.5.2 An idealized eddy-driven model......Page 502
11.6 THE HADLEY CELL: SUMMARY AND NUMERICAL SOLUTIONS......Page 505
11.7 THE FERREL CELL......Page 508
Notes......Page 510
Problems......Page 511
CHAPTER TWELVE Zonally Averaged Mid-latitude Atmospheric Circulation......Page 513
12.1.1 Observations and motivation......Page 514
I. The vorticity budget......Page 515
II. Rossby waves and momentum flux......Page 517
* The radiation condition and Rayleigh friction......Page 518
III. The pseudomomentum budget......Page 520
IV. The Eliassen–Palm flux......Page 522
12.1.3 A numerical example......Page 523
12.2 LAYERED MODELS OF THE MID-LATITUDE CIRCULATION......Page 524
12.2.1 single-layer modelA......Page 525
Equations of motion......Page 526
Dynamics......Page 528
Final remarks on the one-layer model......Page 530
Equations of motion......Page 531
Manipulating the equations......Page 533
12.2.3 Dynamics of the two-layer model......Page 535
Momentum balance and the overturning circulation......Page 537
12.3.1 Equations for a closed model......Page 541
12.3.2 * Eddy fluxes and necessary conditions for instability......Page 542
12.4.1 Potential vorticity and its fluxes......Page 544
Potential vorticity and Eliassen–Palm fluxes......Page 546
12.5 † THE TROPOPAUSE AND THE STRATIFICATION OF THE ATMOSPHERE......Page 550
12.5.1 A radiative–convective model......Page 554
12.5.2 Radiative and dynamical constraints......Page 556
12.6 † BAROCLINIC EDDIES AND POTENTIAL VORTICITY TRANSPORT......Page 557
12.6.2 Mixing potential vorticity and baroclinic adjustment......Page 558
An idealized model......Page 560
12.7 † EXTRATROPICAL CONVECTION AND THE VENTILATED TROPOSPHERE......Page 562
APPENDIX: TEM AND ELIASSEN–PALM FLUX FOR THE PRIMITIVE EQUATIONS IN SPHERICAL COORDINATES......Page 564
Notes......Page 566
Further reading......Page 567
Problems......Page 568
CHAPTER THIRTEEN Planetary Waves and the Stratosphere......Page 569
13.1.1 A simple one-layer case......Page 570
13.1.2 Application to Earth’s atmosphere......Page 571
13.1.3 * One-dimensional Rossby wave trains......Page 573
13.1.4 The adequacy of linear theory......Page 576
13.2.1 Ray tracing......Page 577
13.2.2 Rossby waves and Rossby rays......Page 578
* A JWKB solution......Page 580
13.2.3 Application to an idealized atmosphere......Page 581
13.3.1 Model formulation......Page 582
13.3.2 Model solution......Page 583
Stationary waves......Page 584
13.3.3 Properties of the solution......Page 587
13.4 * EFFECTS OF THERMAL FORCING......Page 588
13.4.1 Thermodynamic balances......Page 589
13.4.2 Properties of the solution......Page 590
13.4.3 Numerical solutions......Page 591
13.5.1 A descriptive overview......Page 594
Wave breaking and residual flow......Page 597
Meridional overturning circulation and downward control......Page 599
13.5.3 † The polar vortex and the quasi-horizontal circulation......Page 603
Notes......Page 605
Problems......Page 607
CHAPTER FOURTEEN Wind-Driven Gyres......Page 611
14.1 THE DEPTH INTEGRATED WIND-DRIVEN CIRCULATION......Page 613
14.1.1 The Stommel model......Page 614
I. A Homogeneous model......Page 615
II. Quasi-geostrophic formulation......Page 616
Sverdrup balance......Page 617
Boundary-layer solution......Page 618
Asymptotic matching......Page 619
14.2 USING VISCOSITY INSTEAD OF DRAG......Page 621
14.3 ZONAL BOUNDARY LAYERS......Page 625
14.4 * THE NONLINEAR PROBLEM......Page 627
14.4.2 A numerical approach......Page 628
14.5 * INERTIAL SOLUTIONS......Page 629
Frictional and inertial scales......Page 631
14.5.2 Attempting an inertial western boundary solution......Page 632
The connection between the boundary layer and the interior......Page 633
14.5.3 A fully inertial approach: the Fofonoff model......Page 634
14.6.1 Homogeneous model......Page 636
14.6.2 Advective dynamics......Page 637
14.6.3 Bottom pressure stress and form drag......Page 639
Bottom pressure stress in a homogeneous gyre......Page 640
Scales of motion......Page 641
Constructing the model......Page 642
Lower layer......Page 643
A general argument......Page 644
A specific calculation......Page 646
14.8.1 Depth of the wind’s influence......Page 647
14.8.2 The complete solution......Page 648
Notes......Page 651
Problems......Page 653
CHAPTER FIFTEEN The Buoyancy-Driven Ocean Circulation......Page 655
15.1 SIDEWAYS CONVECTION......Page 657
15.1.1 Two-dimensional convection......Page 658
Non-dimensionalization and scaling......Page 659
15.1.2 † Phenomenology of the overturning circulation......Page 661
15.2 THE MAINTENANCE OF SIDEWAYS CONVECTION......Page 662
15.2.2 Conditions for maintaining a thermally-driven circulation......Page 663
* Maintaining a steady baroclinic circulation......Page 664
15.2.3 Surface fluxes and non-turbulent flow at small diffusivities......Page 665
15.2.4 The importance of mechanical forcing......Page 667
15.3.1 A two-box model......Page 668
Interpretation......Page 670
Solutions......Page 671
15.3.2 * More boxes......Page 672
15.4.1 Set-up of the laboratory model......Page 674
15.4.2 Dynamics of flow in the tank......Page 675
15.5 A MODEL FOR OCEANIC ABYSSAL FLOW......Page 678
15.6 * A SHALLOW WATER MODEL OF THE ABYSSAL FLOW......Page 684
15.6.1 Potential vorticity and poleward interior flow......Page 685
15.6.2 The solution......Page 686
15.5.1 Completing the solution......Page 680
15.5.2 Application to the ocean......Page 681
15.5.3 A two-hemisphere model......Page 683
15.7 SCALING FOR THE BUOYANCY-DRIVEN CIRCULATION......Page 687
15.7.1 Summary remarks on the Stommel–Arons model......Page 689
Notes......Page 691
Problems......Page 692
16.1 THE MAIN THERMOCLINE: AN INTRODUCTION......Page 695
16.1.1 A simple kinematic model......Page 696
16.2 SCALING AND SIMPLE DYNAMICS OF THE MAIN THERMOCLINE......Page 698
16.2.1 An advective scale......Page 699
A wind-influenced diffusive scaling......Page 700
16.2.3 Summary of the physical picture......Page 701
16.3.1 The M equation......Page 702
A one-dimensional model......Page 703
One-dimensional model......Page 704
* The three-dimensional equations......Page 707
16.4 THE VENTILATED THERMOCLINE......Page 709
16.4.1 A reduced gravity, single-layer model......Page 710
Use of potential vorticity conservation......Page 711
Using Sverdrup balance to find the total depth......Page 713
16.4.3 The shadow zone......Page 714
(i) Potential vorticity homogenization......Page 716
(ii) The ventilated pool......Page 717
16.5 † A MODEL OF DEEP WIND-DRIVEN OVERTURNING......Page 719
Wind and buoyancy forcing......Page 721
Solution in the gyres......Page 722
Solution in the channel......Page 723
A qualitative summary......Page 724
16.5.2 A cross-equatorial wind-driven deep circulation......Page 725
Descriptive solution......Page 726
Summary remarks......Page 727
16.6 † FLOW IN A CHANNEL AND THE ANTARCTIC CIRCUMPOLAR CURRENT......Page 728
16.6.1 Steady and eddying flow......Page 729
16.6.2 Vertically integrated momentum balance......Page 730
16.6.3 Form drag and baroclinic eddies......Page 731
Momentum dynamics of layers......Page 732
Momentum dynamics in height coordinates......Page 733
Mass fluxes and thermodynamics......Page 735
16.6.4 † An idealized adiabatic model......Page 736
16.6.5 Form stress and Ekman stress at the ocean bottom......Page 737
APPENDIX: MISCELLANEOUS RELATIONSHIPS IN A LAYERED MODEL......Page 738
16.A.2 Geostrophic and thermal wind balance......Page 739
A one-layer reduced-gravity model......Page 740
Notes......Page 741
Problems......Page 743
References......Page 745
Index......Page 766