Geophysical fluid dynamics 1

Preface......Page 8
Contents......Page 11
 1.1 Vectors and Cartesian Tensors......Page 15
  1.2.2 Vectors......Page 17
  1.2.3 Second-Order Tensors (dyads)......Page 19
  1.3.2 The `del\' Operator......Page 20
  1.3.3 Identities Associated with the `del\' Operator......Page 22
  1.3.4 Integral Theorems......Page 23
 1.4 The Material Derivative......Page 24
 1.5 Leibnitz\' Rules......Page 27
 2.2 Continuum Hypothesis......Page 32
 2.4 Volume and Surface Forces......Page 33
  2.4.1 Stress in a Fluid that Is at Rest......Page 37
 2.5 Specification of the Flow Field......Page 38
  2.6.1 Continuity Equation......Page 39
  2.6.2 Stream Function–2-D, Incompressible, Steady Flow......Page 41
 2.7 Analysis of Relative Motion Near a Point......Page 44
  2.7.1 Pure-Straining Motion......Page 45
  2.7.2 Rigid Body Rotation......Page 47
  2.7.3 Modes of Motion in a Fluid......Page 48
 3.2 Equation of Motion–Momentum Equation......Page 51
  3.2.1 Integral Form of the Momentum Equation......Page 53
 3.3 Stress Tensor in a Newtonian Fluid......Page 54
 3.4 Navier–Stokes Equation......Page 57
  3.5.1 Pressure in a Fluid at Rest......Page 59
 3.6 Equation of Motion Relative to Moving Frame of Reference......Page 60
  3.6.1 Modification of Gravity and Pressure Due to Centrifugal Acceleration......Page 63
 3.7 Complete Set of Governing Equations......Page 65
 3.8 Vorticity Dynamics......Page 66
  3.8.2 The Tipping and Stretching Term ωA cdotu u u u......Page 68
  3.8.4 The Solenoidal Term 1ρ2ρtimesp......Page 69
 3.9 Kelvin\'s Circulation Theorem......Page 70
 3.10 Bernoulli\'s Theorem......Page 72
 4.1 Governing Equations......Page 75
 4.2 Steady Unidirectional Flow......Page 76
  4.2.1 Poiseuille Flow......Page 77
  4.2.2 Flow Between Plates......Page 79
 4.3 Unsteady Unidirectional Flow......Page 80
  4.3.1 Example—Smoothing of a Velocity Discontinuity......Page 84
  4.3.2 Flow Due to an Oscillating Plane Boundary......Page 86
  4.4.1 The Steady Ekman Layer at the Surface......Page 88
  4.4.2 The Steady Ekman Layer at the Bottom......Page 91
  4.4.3 The Oscillatory Ekman Boundary Layer at the Surface......Page 94
 5.1 Equations of Motion for an Incompressible Homogeneous Fluid......Page 101
 5.2 Inviscid Rotating Flows......Page 102
  5.2.1 Inertial Motion—Unsteady (Periodic) Uniform Flow......Page 104
  5.2.2 ``Elasticity\'\' in a Rotating Fluid—Restoring Effects of Coriolis......Page 107
  5.2.3 Geostrophic Motion: Steady Flow with a Pressure Gradient......Page 108
 6.1 Tangent Plane Approximation......Page 116
  6.2.1 Scaling of the Equations......Page 118
  6.2.2 Continuity of Surface Forces (Dynamic Boundary Conditions)......Page 123
  6.2.3 Hydrostatic Pressure......Page 127
  6.2.5 Shallow Water Equations......Page 128
  6.2.6 Conservation Properties......Page 134
  6.2.7 Vorticity Conservation......Page 136
  6.2.8 Energy Conservation......Page 138
 6.3 The f-Plane and the β-Plane Approximations......Page 139
  6.4.1 Geostrophic Flow......Page 141
  6.4.2 Flow over a Topographic Ridge (f-Plane)......Page 144
 6.5 Topographic Effects......Page 146
  6.5.1 Solutions for Uniform Shelf......Page 150
  6.5.2 Nonlinear Inviscid Solutions......Page 152
  6.5.3 Flow over a Depth Discontinuity (β-Plane)......Page 153
  6.6.1 Planetary (Rossby) Waves......Page 158
  6.6.2 Planetary Waves with a Mean Current......Page 164
  6.6.3 Small Amplitude Motions with a Free Surface......Page 165
  6.6.4 Plane Waves for Constant Depth......Page 167
  6.6.5 Poincaré and Kelvin Waves......Page 169
  6.6.6 Topographic Rossby Waves......Page 176
 7.1 An Overview and Derivation of Quasi-geostrophic  Equations......Page 193
 8.1 The Ocean Circulation at Global Scale......Page 203
  8.2.1 Governing Equations in the f-Plane......Page 204
  8.2.2 The Ocean Interior (f-Plane)......Page 206
  8.2.3 Ekman Boundary Layers (f-Plane)......Page 208
  8.2.4 Surface Ekman Layer (f-Plane)–Wind-Driven Flow......Page 210
  8.2.5 Bottom Ekman Layer (f-Plane)–Interior-Driven Flow......Page 213
  8.2.6 Ocean Circulation on the f-Plane......Page 214
  8.3.1 Development of the Model......Page 217
  8.3.2 The Sverdrup Interior and an Approximate Solution  for the Circulation......Page 221
  8.3.3 Approximate Solution for the Circulation with a Western Boundary Layer......Page 224
 9.2 Two-Layer Shallow Water Equations......Page 231
 9.4 Normal Modes of Oscillation......Page 235
  9.4.1 Case I: Constant Depth......Page 237
  9.4.2 Case II: Variable Bottom Topography......Page 238
  9.5.1 Approximate Modes......Page 240
  9.5.2 Barotropic Normal Mode—Leading Order......Page 241
  9.5.3 Baroclinic Normal Mode—Leading Order......Page 242
  9.5.4 Basin Lateral Boundary Conditions......Page 243
  9.5.5 Periodic Motions in a Basin......Page 244
10 Exercises......Page 245
 Recommended Text Books......Page 294