Geophysical Fluid Dynamics I: An Introduction to Atmosphere-Ocean Dynamics: Homogeneous Fluids

Preface
Contents
1 A Brief Review of Algebra and Calculus
	1.1 Vectors and Cartesian Tensors
	1.2 Algebraic Operations
		1.2.1 Scalars
		1.2.2 Vectors
		1.2.3 Second-Order Tensors (dyads)
	1.3 Differentiation and Integration
		1.3.1 Differentiation and Integration with Respect to a Scalar
		1.3.2 The `del\' Operator
		1.3.3 Identities Associated with the `del\' Operator
		1.3.4 Integral Theorems
	1.4 The Material Derivative
	1.5 Leibnitz\' Rules
2 Fluid Properties and Kinematics
	2.1 Definition of a Fluid
	2.2 Continuum Hypothesis
	2.3 Properties of Fluids
	2.4 Volume and Surface Forces
		2.4.1 Stress in a Fluid that Is at Rest
		2.4.2 Stress in a Moving Fluid
	2.5 Specification of the Flow Field
	2.6 Conservation of Mass (Continuity Equation)
		2.6.1 Continuity Equation
		2.6.2 Stream Function–2-D, Incompressible, Steady Flow
	2.7 Analysis of Relative Motion Near a Point
		2.7.1 Pure-Straining Motion
		2.7.2 Rigid Body Rotation
		2.7.3 Modes of Motion in a Fluid
3 Equations of Motion of a Fluid
	3.1 Continuity Equation–Mass Conservation
	3.2 Equation of Motion–Momentum Equation
		3.2.1 Integral Form of the Momentum Equation
	3.3 Stress Tensor in a Newtonian Fluid
	3.4 Navier–Stokes Equation
	3.5 Pressure in a Fluid
		3.5.1 Pressure in a Fluid at Rest
		3.5.2 Pressure in a Moving Homogeneous Fluid
	3.6 Equation of Motion Relative to Moving Frame of Reference
		3.6.1 Modification of Gravity and Pressure Due to Centrifugal Acceleration
	3.7 Complete Set of Governing Equations
	3.8 Vorticity Dynamics
		3.8.1 The Vorticity Diffusion Term ν2 ωA
		3.8.2 The Tipping and Stretching Term ωA cdotu u u u
		3.8.3 The Divergence Term -ωA (cdotu u u u)
		3.8.4 The Solenoidal Term 1ρ2ρtimesp
	3.9 Kelvin\'s Circulation Theorem
	3.10 Bernoulli\'s Theorem
4 Flow of a Homogeneous Incompressible Fluid
	4.1 Governing Equations
	4.2 Steady Unidirectional Flow
		4.2.1 Poiseuille Flow
		4.2.2 Flow Between Plates
	4.3 Unsteady Unidirectional Flow
		4.3.1 Example—Smoothing of a Velocity Discontinuity
		4.3.2 Flow Due to an Oscillating Plane Boundary
	4.4 Unidirectional Flow Including Coriolis Effects
		4.4.1 The Steady Ekman Layer at the Surface
		4.4.2 The Steady Ekman Layer at the Bottom
		4.4.3 The Oscillatory Ekman Boundary Layer at the Surface
5 Rotating, Homogeneous, Incompressible Fluids
	5.1 Equations of Motion for an Incompressible Homogeneous Fluid
	5.2 Inviscid Rotating Flows
		5.2.1 Inertial Motion—Unsteady (Periodic) Uniform Flow
		5.2.2 ``Elasticity\'\' in a Rotating Fluid—Restoring Effects of Coriolis
		5.2.3 Geostrophic Motion: Steady Flow with a Pressure Gradient
6 Shallow Water Theory
	6.1 Tangent Plane Approximation
	6.2 Shallow Water Approximations
		6.2.1 Scaling of the Equations
		6.2.2 Continuity of Surface Forces (Dynamic Boundary Conditions)
		6.2.3 Hydrostatic Pressure
		6.2.4 Kinematic Boundary Conditions
		6.2.5 Shallow Water Equations
		6.2.6 Conservation Properties
		6.2.7 Vorticity Conservation
		6.2.8 Energy Conservation
	6.3 The f-Plane and the β-Plane Approximations
	6.4 Simple Applications of the Potential Vorticity Conservation
		6.4.1 Geostrophic Flow
		6.4.2 Flow over a Topographic Ridge (f-Plane)
	6.5 Topographic Effects
		6.5.1 Solutions for Uniform Shelf
		6.5.2 Nonlinear Inviscid Solutions
		6.5.3 Flow over a Depth Discontinuity (β-Plane)
	6.6 Oscillatory Motions
		6.6.1 Planetary (Rossby) Waves
		6.6.2 Planetary Waves with a Mean Current
		6.6.3 Small Amplitude Motions with a Free Surface
		6.6.4 Plane Waves for Constant Depth
		6.6.5 Poincaré and Kelvin Waves
		6.6.6 Topographic Rossby Waves
7 Quasigeostrophic Theory
	7.1 An Overview and Derivation of Quasi-geostrophic  Equations
8 Elements of Ocean General Circulation
	8.1 The Ocean Circulation at Global Scale
	8.2 Elements of the Steady, Rigid Lid Ocean Circulation (f-Plane)
		8.2.1 Governing Equations in the f-Plane
		8.2.2 The Ocean Interior (f-Plane)
		8.2.3 Ekman Boundary Layers (f-Plane)
		8.2.4 Surface Ekman Layer (f-Plane)–Wind-Driven Flow
		8.2.5 Bottom Ekman Layer (f-Plane)–Interior-Driven Flow
		8.2.6 Ocean Circulation on the f-Plane
	8.3 Ocean Circulation on the β-Plane
		8.3.1 Development of the Model
		8.3.2 The Sverdrup Interior and an Approximate Solution  for the Circulation
		8.3.3 Approximate Solution for the Circulation with a Western Boundary Layer
9 Two-Layer Models
	9.1 Introduction
	9.2 Two-Layer Shallow Water Equations
	9.3 Free Periodic Motions
	9.4 Normal Modes of Oscillation
		9.4.1 Case I: Constant Depth
		9.4.2 Case II: Variable Bottom Topography
	9.5 Equations for Normal Modes
		9.5.1 Approximate Modes
		9.5.2 Barotropic Normal Mode—Leading Order
		9.5.3 Baroclinic Normal Mode—Leading Order
		9.5.4 Basin Lateral Boundary Conditions
		9.5.5 Periodic Motions in a Basin
10 Exercises
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