ATMOSPHERIC AND OCEANIC FLUID DYNAMICS

Transcription

1 ATMOSPHERIC AND OCEANIC FLUID DYNAMICS Fundamentals and Large-scale Circulation G E O F F R E Y K. V A L L I S Princeton University, New Jersey CAMBRIDGE UNIVERSITY PRESS

2 An asterisk indicates more advanced material that may be omitted on a first reading. A dagger indicates material that is still a topic of research or that is not settled. Preface Notation page xix xxiv Part I FUNDAMENTALS OF GEOPHYSICAL FLUID DYNAMICS 1 1 Equations of Motion Time Derivatives for Fluids Field and material viewpoints The material derivative of a fluid property Material derivative of a volume The Mass Continuity Equation An Eulerian derivation Mass continuity via the material derivative A general continuity equation The Momentum Equation Advection The pressure force Viscosity and diffusion Hydrostatic balance The Equation of State Thermodynamic Relations A few fundamentals Various thermodynamic relations Thermodynamic Equations for Fluids 22 VII

3 viii Contents Thermodynamic equation for an ideal gas * Thermodynamic equation for liquids * More Thermodynamics of Liquids Potential temperature, potential density and entropy * Thermodynamic properties of seawater Soundwaves Compressible and Incompressible Flow Constant density fluids Incompressible flows The Energy Budget Constant density fluid Variable density fluids Viscous effects An Introduction to Non-Dimensionalization and Scaling The Reynolds number 44 2 Effects of Rotation and Stratification Equations in a Rotating Frame Rate of change of a vector Velocity and acceleration in a rotating frame Momentum equation in a rotating frame Mass and tracer conservation in a rotating frame Equations of Motion in Spherical Coordinates * The centrifugal force and spherical coordinates Some identities in spherical coordinates Equations of motion The primitive equations Primitive equations in vector form The vector invariant form of the momentum equation Angular momentum Cartesian Approximations: The Tangent Plane Thef-plane The beta-plane approximation The Boussinesq Approximation Variation of density in the ocean The Boussinesq equations Energetics of the Boussinesq system The Anelastic Approximation Preliminaries The momentum equation Mass conservation Thermodynamic equation * Energetics of the anelastic equations Changing Vertical Coordinate General relations Pressure coordinates 78

4 ix Log-pressure coordinates Scaling for Hydrostatic Balance Preliminaries Scaling and the aspect ratio * Effects of stratification on hydrostatic balance Hydrostasy in the ocean and atmosphere Geostrophic and Thermal Wind Balance The Rossby number Geostrophic balance Taylor-Proudman effect Thermal wind balance * Effects of rotation on hydrostatic balance Static Instability and the Parcel Method A simple special case: a density-conserving fluid The general case: using potential density Lapse rates in dry and moist atmospheres Gravity Waves Gravity waves and convection in a Boussinesq fluid * Acoustic-Gravity Waves in an Ideal Gas Interpretation The Ekman Layer Equations of motion and scaling Integral properties of the Ekman layer Explicit solutions. I: a bottom boundary layer Explicit solutions. II: the upper ocean Observations of the Ekman layer * Frictional parameterization of the Ekman layer Shallow Water Systems and Isentropic Coordinates Dynamics of a Single, Shallow Layer Momentum equations Mass continuity equation A rigid lid Stretching and the vertical velocity Analogy with compressible flow Reduced Gravity Equations Pressure gradient in the active layer Multi-Layer Shallow Water Equations Reduced-gravity multi-layer equation Geostrophic Balance and Thermal wind Form Drag Conservation Properties of Shallow Water Systems Potential vorticity: a material invariant Energy conservation: an integral invariant Shallow Water Waves Non-rotating shallow water waves 140

5 3.7.2 Rotating shallow water (Poincare) waves Kelvin waves Geostrophic Adjustment Non-rotating flow Rotating flow * Energetics of adjustment * General initial conditions A variational perspective Isentropic Coordinates A hydrostatic Boussinesq fluid A hydrostatic ideal gas Analogy to shallow water equations Available Potential Energy A Boussinesq fluid An ideal gas Use, interpretation, and the atmosphere and ocean 1 59 Vorticity and Potential Vorticity Vorticity and Circulation Preliminaries Simple axisymmetric examples The Vorticity Equation Two-dimensional flow Vorticity and Circulation Theorems The'frozen-in'property of vorticity Kelvin's circulation theorem Baroclinic flow and the solenoidal term Circulation in a rotating frame The circulation theorem for hydrostatic flow, Vorticity Equation in a Rotating Frame The circulation theorem and the beta effect The vertical component of the vorticity equation Potential Vorticity Conservation PV conservation from the circulation theorem PV conservation from the frozen-in property PV conservation: an algebraic derivation Effects of salinity and moisture Effects of rotation, and summary remarks * Potential Vorticity in the Shallow Water System Using Kelvin's theorem Using an appropriate scalar field Potential Vorticity in Approximate, Stratified Models The Boussinesq equations The hydrostatic equations Potential vorticity on isentropic surfaces * The Impermeability of Isentropes to Potential Vorticity 188

6 xi Interpretation and application Simplified Equations for Ocean and Atmosphere Geostrophic Scaling Scaling in the shallow water equations Geostrophic scaling in the stratified equations The Planetary-Geostrophic Equations Using the shallow water equations Planetary-geostrophic equations for stratified flow The Shallow Water Quasi-Geostrophic Equations Single-layer shallow water quasi-geostrophic equations Two-layer and multi-layer quasi-geostrophic systems t Non-asymptotic and intermediate models The Continuously Stratified Quasi-Geostrophic System Scaling and assumptions 215 5A-.2 Asymptotics Buoyancy advection at the surface Quasi-geostrophy in pressure coordinates The two-level quasi-geostrophic system * Quasi-geostrophy and Ertel Potential Vorticity * Using height coordinates Using isentropic coordinates * Energetics of Quasi-Geostrophy Conversion between APE and KE Energetics of two-layer flows Enstrophy conservation RossbyWaves Waves in a single layer Rossby waves in two layers * Rossby Waves in Stratified Quasi-Geostrophic Flow Setting up the problem Wave motion 235 Appendix: Wave Kinematics, Group Velocity and Phase Speed A.I Kinematics and definitions A.2 Wave propagation A.3 Meaning of group velocity 239 Part II INSTABILITIES, WAVE-MEAN FLOW INTERACTION AND TURBULENCE Barotropic and Baroclinic Instability Kelvin-Helmholtz Instability Instability of Parallel Shear Flow Piecewise linear flows 251

7 xii Contents Kelvin-Helmholtz instability, revisited Edge waves Interacting edge waves producing instability Necessary Conditions for Instability Rayleigh's criterion FJ0rtoft's criterion Baroclinic Instability A physical picture Linearized quasi-geostrophic equations Necessary conditions for baroclinic instability The Eady Problem The linearized problem Atmospheric and oceanic parameters Two-Layer Baroclinic Instability Posing the problem The solution An Informal View of the Mechanism of Baroclinic Instability The two-layer model Interacting edge waves in the Eady problem * The Energetics of Linear Baroclirjic Instability * Beta, Shear and Stratification in a Continuous Model Scaling arguments for growth rates, scales and depth Some numerical calculations Wave-Mean Flow Interaction Quasi-geostrophic Preliminaries Potential vorticity flux in the linear equations The Eliassen-Palm Flux The Eliassen-Palm relation \ The group velocity property * The orthogonality of modes The Transformed Eulerian Mean Quasi-geostrophic form The TEM in isentropic coordinates Residual and thickness-weighted circulation * The TEM in the primitive equations The Non-acceleration Result A derivation from the potential vorticity equation Using TEM to give the non-acceleration result The EP flux and form drag Influence of Eddies on the Mean Flow in the Eady Problem Formulation Solution The two-level problem * Necessary Conditions for Instability Stability conditions from pseudomomentum conservation 325

8 xiii Inclusion of boundary terms * Necessary Conditions for Instability: Use of Pseudoenergy Two-dimensional flow * Stratified quasi-geostrophic flow * Applications to baroclinic instability Basic Theory of Incompressible Turbulence The Fundamental Problem of Turbulence The closure problem Triad interactions in turbulence The Kolmogorov Theory The physical picture Inertial-range theory * Another expression of the inertial-range scaling argument A final note on our assumptions Two-Dimensional Turbulence Energy and enstrophy transfer Inertial ranges in two-dimensional turbulence t More about the phenomenology Numerical illustrations Predictability of Turbulence Low-dimensional chaos and unpredictability * Predictability of a turbulent flow Implications and weather predictability * Spectra of Passive Tracers Examples of tracer spectra Geostrophic Turbulence and Baroclinic Eddies Effects of Differential Rotation The wave-turbulence cross-over Generation of zonal flows and jets t Joint effect of P and friction Stratified Geostrophic Turbulence An analogue to two-dimensional flow Two-layer geostrophic turbulence Phenomenology of two-layer turbulence t A Scaling Theory for Geostrophic Turbulence Preliminaries Scaling properties The halting scale and the ^-effect t Phenomenology of Baroclinic Eddies in the Atmosphere and Ocean The magnitude and scale of baroclinic eddies Baroclinic eddies and their lifecycle in the atmosphere Baroclinic eddies and their lifecycle in the ocean 400

9 xiv Contents 10 Turbulent Diffusion and Eddy Transport Diffusive Transport An explicit example Turbulent Diffusion Simple theory * An anisotropic generalization Discussion Two-Particle Diffusivity Large particle separation Separation within the inertial range Mixing Length Theory Requirements for turbulent diffusion A macroscopic perspective Homogenization of a Scalar that is Advected and Diffused Non-existence of extrema Homogenization in two-dimensional flow t Transport by Baroclinic Eddies Symmetric and antisymmetric diffusivity tensors * Diffusion with the symmetric tensor * Skew diffusion The story so far f Eddy Diffusion in the Atmosphere and Ocean Preliminaries Magnitude of the eddy diffusivity * Structure: the symmetric transport tensor * Structure: the antisymmetric transport tensor Examples t Thickness Diffusion Equations of motion Diffusive thickness transport t Eddy Transport and the Transformed Eulerian Mean Potential vorticity diffusion 443 Part III LARGE-SCALE ATMOSPHERIC CIRCULATION The Overturning Circulation: Hadley and Ferrel Cells Basic Features of the Atmosphere The radiative equilibrium distribution Observed wind and temperature fields Meridional overturning circulation Summary A Steady Model of the Hadley Cell Assumptions Dynamics 458

10 xv Thermodynamics Zonal wind Properties of solution Strength of the circulation t Effects of moisture The radiative equilibrium solution A Shallow Water Model of the Hadley Cell Momentum balance Thermodynamic balance f Asymmetry Around the Equator Eddies, Viscosity and the Hadley Cell Qualitative considerations An idealized eddy-driven model The Hadley Cell: Summary and Numerical Solutions The Ferrel Cell Zonally Averaged Mid-Latitude Atmospheric Circulation Surface Westerlies and the Maintenance of a Barotropic Jet Observations and motivation The mechanism of jet production A numerical example Layered Models of the Mid-latitude Circulation A single-layer model A two-layer model Dynamics of the two-layer model t Eddy Fluxes and an Example of a Closed Model Equations for a closed model * Eddy fluxes and necessary conditions for instability A Stratified Model and the Real Atmosphere Potential vorticity and its fluxes Overturning circulation t The Tropopause and the Stratification of the Atmosphere A radiative-convective model Radiative and dynamical constraints t Baroclinic eddies and Potential Vorticity Transport A linear argument Mixing potential vorticity and baroclinic adjustment Diffusive transport of potential vorticity t Extratropical Convection and the Ventilated Troposphere 534 Appendix: TEM for the Primitive Equations in Spherical Coordinates Planetary Waves and the Stratosphere Forced and Stationary Rossby Waves A simple one-layer case Application to Earth's atmosphere 543

11 xvi Contents * One-dimensional Rossby wave trains The adequacy of linear theory * Meridional Propagation and Dispersion Ray tracing Rossby waves and Rossby rays Application to an idealized atmosphere * Vertical Propagation of Rossby Waves in a Stratified Medium Model formulation Model solution Properties of the solution * Effects of Thermal Forcing Thermodynamic balances Properties of the solution Numerical solutions Stratospheric Dynamics A descriptive overview t Dynamics of the overturning circulation t The polar vortex and the quasi-horizontal circulation 575 Part IV LARGE-SCALE OCEANIC CIRCULATION Wind-Driven Gyres The Depth Integrated Wind-Driven Circulation The Stommel model Alternative formulations Approximate solution of Stommel model Using Viscosity Instead of Drag Zonal Boundary Layers * The Nonlinear Problem ' A perturbative approach A numerical approach * Inertial Solutions Roles of friction and inertia Attempting an inertial western boundary solution A fully inertial approach: the Fofonoff model Topographic Effects on Western Boundary Currents Homogeneous model Advective dynamics Bottom pressure stress and form drag * Vertical Structure of the Wind-Driven Circulation A two-layer quasi-geostrophic Model The functional relationship between ifj and q * A Model with Continuous Stratification Depth of the wind's influence The complete solution 620

12 xvii 15 The Buoyancy-Driven Ocean Circulation Sideways Convection Two-dimensional convection t Phenomenology of the overturning circulation The Maintenance of Sideways Convection The energy budget Conditions for maintaining a thermally-driven circulation Surface fluxes and non-turbulent flow at small diffusivities The importance of mechanical forcing Simple Box Models A two-box model * More boxes A Laboratory Model of the Abyssal Circulation Set-up of the laboratory model Dynamics of flow in the tank A Model for Oceanic Abyssal Flow Completing the solution Application to the ocean A two-hemisphere model * A Shallow Water Model of the Abyssal Flow Potential vorticity and poleward interior flow The solution Scaling for the Buoyancy-Driven Circulation Summary remarks on the Stommel-Arons model The Wind-and Buoyancy-Driven Ocean Circulation The Main Thermocline: an Introduction A simple kinematic model Scaling and Simple Dynamics of the Main Thermocline An advective scale A diffusive scale Summary of the physical picture The Internal Thermocline The M equation * Boundary-layer analysis The Ventilated Thermocline A reduced gravity, single-layer model A two-layer model The shadow zone t The western pool t A Model of Deep Wind-Driven Overturning A single-hemisphere model A cross-equatorial wind-driven deep circulation t Flow in a Channel and the Antarctic Circumpolar Current Steady and eddying flow 701

13 xviii Contents Vertically integrated momentum balance Form drag and baroclinic eddies t An idealized adiabatic model Form stress and Ekman stress at the ocean bottom Differences between gyres and channels 710 Appendix: Miscellaneous Relationships in a Layered Model A.1 Hydrostatic balance A.2 Geostrophic and thermal wind balance A.3 Explicit cases 712 References 717 Index 738