This page intentionally left blankAtmospheric and Oceanic Fluid DynamicsFundamentals and Large-Scale CirculationFluid dynamics is fundamental to our understanding of the atmosphere and oceans. Although manyof the same principles of fluid dynamics apply to both the atmosphere and oceans, textbooks onthe topic have tended to concentrate on either the atmosphere or the ocean, or on the theory ofgeophysical fluid dynamics (GFD). However, there is much to be said for a unified discussion,and this major new textbook provides a comprehensive, coherent treatment of all these topics. It isbased on course notes that the author has developed over a number of years at Princeton and theUniversity of California.The first part of the book provides an introduction to the fundamentals of geophysical fluiddynamics, including discussions of rotation and stratification, the role of vorticity and potentialvorticity, and scaling and approximations. The second part of the book discusses baroclinic andbarotropic instabilities, wavemean flow interactions and turbulence. The third and fourth partsdiscuss the general circulation of the atmosphere and ocean. Student problems and exercises, aswell as bibliographic and historical notes, are included at the end of each chapter.Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation willprove to be an invaluable graduate textbook on advanced courses in GFD, meteorology, atmo-spheric science, and oceanography, and will also be an excellent review volume for researchers.Additional resources are available at www.cambridge.org/9780521849692Geoffrey K. Vallis is a senior scientist and professor in the Program in Atmospheric andOceanic Sciences and NOAAs Geophysical Fluid Dynamics Laboratory at Princeton University.He is also an associate faculty member at the Program in Applied and Computational Mathematics,and a former professor at the University of California, Santa Cruz. Until recently he was editor ofthe Journal of the Atmospheric Sciences. His research interests include the general circulation ofthe ocean and atmosphere, turbulence theory, and climate dynamics. He has taught a wide range oftopics at Princeton and the University of California, and he has published extensively in both theoceanographic and meteorological literature.Pre-publication praise for Atmospheric and Oceanic Fluid DynamicsGeoff Vallis Atmosphere and Ocean Dynamics will become the standard text on modern large-scale atmosphere and ocean dynamics. It covers the field from the equations of motion to moderndevelopments such as wavemean flow interaction theory and theories for the global-scale circu-lations of atmospheres and oceans. There is no book of comparable comprehensiveness, spanningthe needs of beginning graduate students and researchers alike.Tapio Schneider, California Institute of TechnologyThis clearly written, self-contained new book is a modern treatment of atmospheric and oceanicdynamics. The book starts from classical concepts in fluid dynamics and thermodynamics and takesthe reader to the frontier of current research. This is an accessible textbook for beginning studentsin meteorology, oceanography and climate sciences. Mature researchers will welcome this workas a stimulating resource. This is also the only textbook on geophysical fluid dynamics with acomprehensive collection of problems; these cement the material and expand it to a more advancedlevel. Highly recommended!Paola Cessi, Scripps Institution of Oceanography, University of California, San DiegoVallis provides a cohesive view of GFD that smoothly blends classic results with modern inter-pretations. The book strikes an ideal balance between mathematical rigor and physical intuition,and between atmosphere- and ocean-relevant applications. The use of a hierarchy of models isparticularly welcome. Each physical phenomenon is modeled with the right degree of complexity,and the reader is introduced to the value of the hierarchy at an early stage. Well-designed home-work problems spanning a broad range of difficulty make the book very appropriate for use inintroductory courses in GFD.Adam Sobel, Lamont-Doherty Earth Observatory, Columbia UniversityI have adopted this text for my course in AtmosphereOcean Dynamics because the ideas areclearly presented and up-to-date. The text provides the flexibility for the instructor to choose amonga variety of paths that take the student from the foundations of the subject to current research topics.For me as a researcher, the text is satisfying because it presents a unified view of the ideas thatunderlie the modern theory of large scale atmospheric and oceanic circulations.Paul J. Kushner, University of TorontoThe large-scale circulation in the atmosphereocean system is maintained by small-scale turbulentmotions that interact with large-scale radiative processes. The first half of the book introduces thebasic theories of large-scale atmosphereocean flows and of small-scale turbulent motions. In thesecond half, the two theories are brought together to explain how the interactions of motions ondifferent scales maintain the global-scale climate. The emphasis on turbulent motions and theireffect on larger scales makes this book a gem in the GFD literature. Finally, we have a textbookthat is up to date with our current understanding of the climate system.Raffaele Ferrari, Massachusetts Institute of TechnologyATMOSPHERIC AND OCEANICFLUID DYNAMICSFundamentals and Large-scale CirculationG E O F F R E Y K . V A L L I SPrinceton University, New JerseyCAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So PauloCambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UKFirst published in print formatISBN-13 978-0-521-84969-2ISBN-13 978-0-511-34879-2 G. Vallis 20062006Information on this title: www.cambridge.org/9780521849692This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.ISBN-10 0-511-34879-7ISBN-10 0-521-84969-1Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.Published in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orghardbackeBook (EBL)eBook (EBL)hardbackTo my parents, Jim and Doreen Vallis.ContentsAn asterisk indicates more advanced material that may be omitted on a rst reading. Adagger indicates material that is still a topic of research or that is not settled.Preface page xixNotation xxivPart I FUNDAMENTALS OF GEOPHYSICAL FLUID DYNAMICS 11 Equations of Motion 31.1 Time Derivatives for Fluids 31.1.1 Field and material viewpoints 31.1.2 The material derivative of a uid property 41.1.3 Material derivative of a volume 61.2 The Mass Continuity Equation 71.2.1 An Eulerian derivation 71.2.2 Mass continuity via the material derivative 91.2.3 A general continuity equation 111.3 The Momentum Equation 111.3.1 Advection 121.3.2 The pressure force 121.3.3 Viscosity and diusion 131.3.4 Hydrostatic balance 131.4 The Equation of State 141.5 Thermodynamic Relations 161.5.1 A few fundamentals 161.5.2 Various thermodynamic relations 181.6 Thermodynamic Equations for Fluids 22viiviii Contents1.6.1 Thermodynamic equation for an ideal gas 231.6.2 * Thermodynamic equation for liquids 261.7 * More Thermodynamics of Liquids 311.7.1 Potential temperature, potential density and entropy 311.7.2 * Thermodynamic properties of seawater 331.8 Sound Waves 371.9 Compressible and Incompressible Flow 381.9.1 Constant density uids 381.9.2 Incompressible ows 391.10 The Energy Budget 401.10.1 Constant density uid 401.10.2 Variable density uids 421.10.3 Viscous eects 431.11 An Introduction to Non-Dimensionalization and Scaling 431.11.1 The Reynolds number 442 Eects of Rotation and Stratication 512.1 Equations in a Rotating Frame 512.1.1 Rate of change of a vector 522.1.2 Velocity and acceleration in a rotating frame 532.1.3 Momentum equation in a rotating frame 542.1.4 Mass and tracer conservation in a rotating frame 542.2 Equations of Motion in Spherical Coordinates 552.2.1 * The centrifugal force and spherical coordinates 552.2.2 Some identities in spherical coordinates 572.2.3 Equations of motion 602.2.4 The primitive equations 612.2.5 Primitive equations in vector form 622.2.6 The vector invariant form of the momentum equation 632.2.7 Angular momentum 642.3 Cartesian Approximations: The Tangent Plane 662.3.1 The f-plane 662.3.2 The beta-plane approximation 672.4 The Boussinesq Approximation 672.4.1 Variation of density in the ocean 682.4.2 The Boussinesq equations 682.4.3 Energetics of the Boussinesq system 722.5 The Anelastic Approximation 732.5.1 Preliminaries 732.5.2 The momentum equation 742.5.3 Mass conservation 752.5.4 Thermodynamic equation 762.5.5 * Energetics of the anelastic equations 762.6 Changing Vertical Coordinate 772.6.1 General relations 772.6.2 Pressure coordinates 78Contents ix2.6.3 Log-pressure coordinates 802.7 Scaling for Hydrostatic Balance 802.7.1 Preliminaries 802.7.2 Scaling and the aspect ratio 812.7.3 * Eects of stratication on hydrostatic balance 822.7.4 Hydrostasy in the ocean and atmosphere 842.8 Geostrophic and Thermal Wind Balance 852.8.1 The Rossby number 852.8.2 Geostrophic balance 862.8.3 TaylorProudman eect 882.8.4 Thermal wind balance 892.8.5 * Eects of rotation on hydrostatic balance 912.9 Static Instability and the Parcel Method 912.9.1 A simple special case: a density-conserving uid 922.9.2 The general case: using potential density 932.9.3 Lapse rates in dry and moist atmospheres 952.10 Gravity Waves 982.10.1 Gravity waves and convection in a Boussinesq uid 982.11 * Acoustic-Gravity Waves in an Ideal Gas 1002.11.1 Interpretation 1012.12 The Ekman Layer 1042.12.1 Equations of motion and scaling 1052.12.2 Integral properties of the Ekman layer 1072.12.3 Explicit solutions. I: a bottom boundary layer 1092.12.4 Explicit solutions. II: the upper ocean 1122.12.5 Observations of the Ekman layer 1132.12.6 * Frictional parameterization of the Ekman layer 1143 Shallow Water Systems and Isentropic Coordinates 1233.1 Dynamics of a Single, Shallow Layer 1233.1.1 Momentum equations 1243.1.2 Mass continuity equation 1253.1.3 A rigid lid 1273.1.4 Stretching and the vertical velocity 1283.1.5 Analogy with compressible ow 1293.2 Reduced Gravity Equations 1293.2.1 Pressure gradient in the active layer 1303.3 Multi-Layer Shallow Water Equations 1313.3.1 Reduced-gravity multi-layer equation 1333.4 Geostrophic Balance and Thermal wind 1343.5 Form Drag 1353.6 Conservation Properties of Shallow Water Systems 1363.6.1 Potential vorticity: a material invariant 1363.6.2 Energy conservation: an integral invariant 1393.7 Shallow Water Waves 1403.7.1 Non-rotating shallow water waves 140x Contents3.7.2 Rotating shallow water (Poincar) waves 1413.7.3 Kelvin waves 1433.8 Geostrophic Adjustment 1443.8.1 Non-rotating ow 1453.8.2 Rotating ow 1463.8.3 * Energetics of adjustment 1483.8.4 * General initial conditions 1493.8.5 A variational perspective 1513.9 Isentropic Coordinates 1523.9.1 A hydrostatic Boussinesq uid 1523.9.2 A hydrostatic ideal gas 1533.9.3 * Analogy to shallow water equations 1543.10 Available Potential Energy 1553.10.1 A Boussinesq uid 1563.10.2 An ideal gas 1583.10.3 Use, interpretation, and the atmosphere and ocean 1594 Vorticity and Potential Vorticity 1634.1 Vorticity and Circulation 1634.1.1 Preliminaries 1634.1.2 Simple axisymmetric examples 1644.2 The Vorticity Equation 1654.2.1 Two-dimensional ow 1674.3 Vorticity and Circulation Theorems 1684.3.1 The frozen-in property of vorticity 1684.3.2 Kelvins circulation theorem 1714.3.3 Baroclinic ow and the solenoidal term 1734.3.4 Circulation in a rotating frame 1734.3.5 The circulation theorem for hydrostatic ow 1744.4 Vorticity Equation in a Rotating Frame 1754.4.1 The circulation theorem and the beta eect 1754.4.2 The vertical component of the vorticity equation 1764.5 Potential Vorticity Conservation 1784.5.1 PV conservation from the circulation theorem 1784.5.2 PV conservation from the frozen-in property 1804.5.3 PV conservation: an algebraic derivation 1824.5.4 Eects of salinity and moisture 1834.5.5 Eects of rotation, and summary remarks 1834.6 * Potential Vorticity in the Shallow Water System 1844.6.1 Using Kelvins theorem 1844.6.2 Using an appropriate scalar eld 1854.7 Potential Vorticity in Approximate, Stratied Models 1864.7.1 The Boussinesq equations 1864.7.2 The hydrostatic equations 1874.7.3 Potential vorticity on isentropic surfaces 1874.8 * The Impermeability of Isentropes to Potential Vorticity 188Contents xi4.8.1 Interpretation and application 1905 Simplied Equations for Ocean and Atmosphere 1975.1 Geostrophic Scaling 1985.1.1 Scaling in the shallow water equations 1985.1.2 Geostrophic scaling in the stratied equations 2005.2 The Planetary-Geostrophic Equations 2035.2.1 Using the shallow water equations 2035.2.2 Planetary-geostrophic equations for stratied ow 2055.3 The Shallow Water Quasi-Geostrophic Equations 2075.3.1 Single-layer shallow water quasi-geostrophic equations 2075.3.2 Two-layer and multi-layer quasi-geostrophic systems 2115.3.3 Non-asymptotic and intermediate models 2145.4 The Continuously Stratied Quasi-Geostrophic System 2155.4.1 Scaling and assumptions 2155.4.2 Asymptotics 2165.4.3 Buoyancy advection at the surface 2195.4.4 Quasi-geostrophy in pressure coordinates 2205.4.5 The two-level quasi-geostrophic system 2215.5 * Quasi-geostrophy and Ertel Potential Vorticity 2245.5.1 * Using height coordinates 2245.5.2 Using isentropic coordinates 2255.6 * Energetics of Quasi-Geostrophy 2265.6.1 Conversion between APE and KE 2275.6.2 Energetics of two-layer ows 2285.6.3 Enstrophy conservation 2295.7 Rossby Waves 2295.7.1 Waves in a single layer 2295.7.2 Rossby waves in two layers 2325.8 * Rossby Waves in Stratied Quasi-Geostrophic Flow 2345.8.1 Setting up the problem 2345.8.2 Wave motion 235Appendix: Wave Kinematics, Group Velocity and Phase Speed 2365.A.1 Kinematics and denitions 2365.A.2 Wave propagation 2375.A.3 Meaning of group velocity 239Part II INSTABILITIES, WAVEMEAN FLOWINTERACTION ANDTURBULENCE 2456 Barotropic and Baroclinic Instability 2476.1 KelvinHelmholtz Instability 2486.2 Instability of Parallel Shear Flow 2506.2.1 Piecewise linear ows 251xii Contents6.2.2 KelvinHelmholtz instability, revisited 2536.2.3 Edge waves 2546.2.4 Interacting edge waves producing instability 2546.3 Necessary Conditions for Instability 2586.3.1 Rayleighs criterion 2586.3.2 Fjrtofts criterion 2606.4 Baroclinic Instability 2616.4.1 A physical picture 2616.4.2 Linearized quasi-geostrophic equations 2636.4.3 Necessary conditions for baroclinic instability 2646.5 The Eady Problem 2656.5.1 The linearized problem 2666.5.2 Atmospheric and oceanic parameters 2686.6 Two-Layer Baroclinic Instability 2716.6.1 Posing the problem 2716.6.2 The solution 2726.7 An Informal View of the Mechanism of Baroclinic Instability 2776.7.1 The two-layer model 2786.7.2 Interacting edge waves in the Eady problem 2806.8 * The Energetics of Linear Baroclinic Instability 2826.9 * Beta, Shear and Stratication in a Continuous Model 2846.9.1 Scaling arguments for growth rates, scales and depth 2856.9.2 Some numerical calculations 2877 WaveMean Flow Interaction 2957.1 Quasi-geostrophic Preliminaries 2967.1.1 Potential vorticity ux in the linear equations 2977.2 The EliassenPalm Flux 2987.2.1 The EliassenPalm relation 2997.2.2 The group velocity property 3007.2.3 * The orthogonality of modes 3027.3 The Transformed Eulerian Mean 3047.3.1 Quasi-geostrophic form 3047.3.2 The TEM in isentropic coordinates 3067.3.3 Residual and thickness-weighted circulation 3077.3.4 * The TEM in the primitive equations 3097.4 The Non-acceleration Result 3147.4.1 A derivation from the potential vorticity equation 3147.4.2 Using TEM to give the non-acceleration result 3167.4.3 The EP ux and form drag 3177.5 Inuence of Eddies on the Mean Flow in the Eady Problem 3197.5.1 Formulation 3197.5.2 Solution 3217.5.3 The two-level problem 3247.6 * Necessary Conditions for Instability 3247.6.1 Stability conditions from pseudomomentum conservation 325Contents xiii7.6.2 Inclusion of boundary terms 3257.7 * Necessary Conditions for Instability: Use of Pseudoenergy 3277.7.1 Two-dimensional ow 3277.7.2 * Stratied quasi-geostrophic ow 3307.7.3 * Applications to baroclinic instability 3318 Basic Theory of Incompressible Turbulence 3378.1 The Fundamental Problem of Turbulence 3388.1.1 The closure problem 3388.1.2 Triad interactions in turbulence 3398.2 The Kolmogorov Theory 3418.2.1 The physical picture 3418.2.2 Inertial-range theory 3428.2.3 * Another expression of the inertial-range scaling argument 3488.2.4 A nal note on our assumptions 3498.3 Two-Dimensional Turbulence 3498.3.1 Energy and enstrophy transfer 3518.3.2 Inertial ranges in two-dimensional turbulence 3548.3.3 More about the phenomenology 3578.3.4 Numerical illustrations 3608.4 Predictability of Turbulence 3618.4.1 Low-dimensional chaos and unpredictability 3628.4.2 * Predictability of a turbulent ow 3638.4.3 Implications and weather predictability 3658.5 * Spectra of Passive Tracers 3668.5.1 Examples of tracer spectra 3679 Geostrophic Turbulence and Baroclinic Eddies 3779.1 Eects of Dierential Rotation 3779.1.1 The waveturbulence cross-over 3789.1.2 Generation of zonal ows and jets 3809.1.3 Joint eect of and friction 3829.2 Stratied Geostrophic Turbulence 3849.2.1 An analogue to two-dimensional ow 3849.2.2 Two-layer geostrophic turbulence 3859.2.3 Phenomenology of two-layer turbulence 3889.3 A Scaling Theory for Geostrophic Turbulence 3919.3.1 Preliminaries 3929.3.2 Scaling properties 3939.3.3 The halting scale and the -eect 3959.4 Phenomenology of Baroclinic Eddies in the Atmosphere and Ocean 3959.4.1 The magnitude and scale of baroclinic eddies 3969.4.2 Baroclinic eddies and their lifecycle in the atmosphere 3979.4.3 Baroclinic eddies and their lifecycle in the ocean 400xiv Contents10 Turbulent Diusion and Eddy Transport 40710.1 Diusive Transport 40810.1.1 An explicit example 40910.2 Turbulent Diusion 40910.2.1 Simple theory 40910.2.2 * An anisotropic generalization 41310.2.3 Discussion 41510.3 Two-Particle Diusivity 41510.3.1 Large particle separation 41610.3.2 Separation within the inertial range 41710.4 Mixing Length Theory 41910.4.1 Requirements for turbulent diusion 42110.4.2 A macroscopic perspective 42210.5 Homogenization of a Scalar that is Advected and Diused 42310.5.1 Non-existence of extrema 42310.5.2 Homogenization in two-dimensional ow 42410.6 Transport by Baroclinic Eddies 42510.6.1 Symmetric and antisymmetric diusivity tensors 42610.6.2 * Diusion with the symmetric tensor 42610.6.3 * The skew ux 42710.6.4 The story so far 42910.7 Eddy Diusion in the Atmosphere and Ocean 43010.7.1 Preliminaries 43010.7.2 Magnitude of the eddy diusivity 43010.7.3 * Structure: the symmetric transport tensor 43210.7.4 * Structure: the antisymmetric transport tensor 43510.7.5 Examples 43710.8 Thickness Diusion 44010.8.1 Equations of motion 44010.8.2 Diusive thickness transport 44210.9 Eddy Transport and the Transformed Eulerian Mean 44310.9.1 Potential vorticity diusion 443Part III LARGE-SCALE ATMOSPHERIC CIRCULATION 44911 The Overturning Circulation: Hadley and Ferrel Cells 45111.1 Basic Features of the Atmosphere 45211.1.1 The radiative equilibrium distribution 45211.1.2 Observed wind and temperature elds 45311.1.3 Meridional overturning circulation 45611.1.4 Summary 45711.2 A Steady Model of the Hadley Cell 45711.2.1 Assumptions 45711.2.2 Dynamics 458Contents xv11.2.3 Thermodynamics 46011.2.4 Zonal wind 46211.2.5 Properties of solution 46311.2.6 Strength of the circulation 46411.2.7 Eects of moisture 46511.2.8 The radiative equilibrium solution 46611.3 A Shallow Water Model of the Hadley Cell 46811.3.1 Momentum balance 46811.3.2 Thermodynamic balance 46811.4 Asymmetry Around the Equator 46911.5 Eddies, Viscosity and the Hadley Cell 47311.5.1 Qualitative considerations 47311.5.2 An idealized eddy-driven model 47411.6 The Hadley Cell: Summary and Numerical Solutions 47711.7 The Ferrel Cell 48012 Zonally Averaged Mid-Latitude Atmospheric Circulation 48512.1 Surface Westerlies and the Maintenance of a Barotropic Jet 48612.1.1 Observations and motivation 48612.1.2 The mechanism of jet production 48712.1.3 A numerical example 49512.2 Layered Models of the Mid-latitude Circulation 49612.2.1 A single-layer model 49712.2.2 A two-layer model 50312.2.3 Dynamics of the two-layer model 50712.3 Eddy Fluxes and an Example of a Closed Model 51312.3.1 Equations for a closed model 51312.3.2 * Eddy uxes and necessary conditions for instability 51412.4 A Stratied Model and the Real Atmosphere 51612.4.1 Potential vorticity and its uxes 51612.4.2 Overturning circulation 52212.5 The Tropopause and the Stratication of the Atmosphere 52212.5.1 A radiativeconvective model 52612.5.2 Radiative and dynamical constraints 52812.6 Baroclinic eddies and Potential Vorticity Transport 52912.6.1 A linear argument 53012.6.2 Mixing potential vorticity and baroclinic adjustment 53012.6.3 Diusive transport of potential vorticity 53212.7 Extratropical Convection and the Ventilated Troposphere 534Appendix: TEM for the Primitive Equations in Spherical Coordinates 53613 Planetary Waves and the Stratosphere 54113.1 Forced and Stationary Rossby Waves 54213.1.1 A simple one-layer case 54213.1.2 Application to Earths atmosphere 543xvi Contents13.1.3 * One-dimensional Rossby wave trains 54513.1.4 The adequacy of linear theory 54813.2 * Meridional Propagation and Dispersion 54913.2.1 Ray tracing 54913.2.2 Rossby waves and Rossby rays 55013.2.3 Application to an idealized atmosphere 55313.3 * Vertical Propagation of Rossby Waves in a Stratied Medium 55413.3.1 Model formulation 55413.3.2 Model solution 55513.3.3 Properties of the solution 55913.4 * Eects of Thermal Forcing 56013.4.1 Thermodynamic balances 56113.4.2 Properties of the solution 56213.4.3 Numerical solutions 56313.5 Stratospheric Dynamics 56613.5.1 A descriptive overview 56613.5.2 Dynamics of the overturning circulation 56913.5.3 The polar vortex and the quasi-horizontal circulation 575Part IV LARGE-SCALE OCEANIC CIRCULATION 58114 Wind-Driven Gyres 58314.1 The Depth Integrated Wind-Driven Circulation 58514.1.1 The Stommel model 58614.1.2 Alternative formulations 58714.1.3 Approximate solution of Stommel model 58914.2 Using Viscosity Instead of Drag 59314.3 Zonal Boundary Layers 59714.4 * The Nonlinear Problem 59914.4.1 A perturbative approach 60014.4.2 A numerical approach 60014.5 * Inertial Solutions 60114.5.1 Roles of friction and inertia 60314.5.2 Attempting an inertial western boundary solution 60414.5.3 A fully inertial approach: the Fofono model 60614.6 Topographic Eects on Western Boundary Currents 60814.6.1 Homogeneous model 60814.6.2 Advective dynamics 60914.6.3 Bottom pressure stress and form drag 61114.7 * Vertical Structure of the Wind-Driven Circulation 61314.7.1 A two-layer quasi-geostrophic Model 61314.7.2 The functional relationship between and q 61614.8 * A Model with Continuous Stratication 61914.8.1 Depth of the winds inuence 61914.8.2 The complete solution 620Contents xvii15 The Buoyancy-Driven Ocean Circulation 62715.1 Sideways Convection 62915.1.1 Two-dimensional convection 63015.1.2 Phenomenology of the overturning circulation 63315.2 The Maintenance of Sideways Convection 63415.2.1 The energy budget 63515.2.2 Conditions for maintaining a thermally-driven circulation 63515.2.3 Surface uxes and non-turbulent ow at small diusivities 63715.2.4 The importance of mechanical forcing 63915.3 Simple Box Models 64015.3.1 A two-box model 64015.3.2 * More boxes 64415.4 A Laboratory Model of the Abyssal Circulation 64615.4.1 Set-up of the laboratory model 64615.4.2 Dynamics of ow in the tank 64715.5 A Model for Oceanic Abyssal Flow 65015.5.1 Completing the solution 65215.5.2 Application to the ocean 65315.5.3 A two-hemisphere model 65515.6 * A Shallow Water Model of the Abyssal Flow 65615.6.1 Potential vorticity and poleward interior ow 65715.6.2 The solution 65815.7 Scaling for the Buoyancy-Driven Circulation 65915.7.1 Summary remarks on the StommelArons model 66116 The Wind- and Buoyancy-Driven Ocean Circulation 66716.1 The Main Thermocline: an Introduction 66716.1.1 A simple kinematic model 66816.2 Scaling and Simple Dynamics of the Main Thermocline 67016.2.1 An advective scale 67116.2.2 A diusive scale 67216.2.3 Summary of the physical picture 67316.3 The Internal Thermocline 67416.3.1 The M equation 67416.3.2 * Boundary-layer analysis 67616.4 The Ventilated Thermocline 68116.4.1 A reduced gravity, single-layer model 68216.4.2 A two-layer model 68316.4.3 The shadow zone 68616.4.4 The western pool 68816.5 A Model of Deep Wind-Driven Overturning 69116.5.1 A single-hemisphere model 69316.5.2 A cross-equatorial wind-driven deep circulation 69716.6 Flow in a Channel and the Antarctic Circumpolar Current 70016.6.1 Steady and eddying ow 701xviii Contents16.6.2 Vertically integrated momentum balance 70216.6.3 Form drag and baroclinic eddies 70316.6.4 An idealized adiabatic model 70816.6.5 Form stress and Ekman stress at the ocean bottom 70916.6.6 Dierences between gyres and channels 710Appendix: Miscellaneous Relationships in a Layered Model 71016.A.1 Hydrostatic balance 71116.A.2 Geostrophic and thermal wind balance 71116.A.3 Explicit cases 712References 717Index 738We must be ignorant of much, if we would know anything.Cardinal John Newman (18011890).PrefaceTHIS IS A BOOK on the uid dynamics of the atmosphere and ocean, with an emphasison the fundamentals and on the large-scale circulation, the latter meaning owsfrom the scale of the rst deformation radius (a few tens of kilometres in the ocean,several hundred kilometres in the atmosphere) to the global scale. The book is primarily atextbook; it is designed to be accessible to students and could be used as a text for graduatecourses. It may be also useful as an introduction to the eld for scientists in other areas andas a reference for researchers in the eld, and some aspects of the book have the avour ofa research monograph.Atmospheric and oceanic uid dynamics (AOFD) is fascinating eld, and simultaneouslyboth pure and applied. It is a pure eld because it is intimately tied to some of the mostfundamental and unsolved problems in uid dynamics problems in turbulence and wavemean ow interaction, problems in chaos and predictability, and problems in the generalcirculation itself. Yet it is applied because the climate and weather so profoundly aect thehuman condition, and so a great deal of eort goes into making predictions indeed thepractice of weather forecasting is a remarkable example of a successful applied science, inspite of the natural limitations to predictability that are now reasonably well understood.The eld is plainly important, for we live in the atmosphere and the ocean covers abouttwo-thirds of the Earth. It is also very broad, encompassing such diverse topics as the generalcirculation, gyres, boundary layers, waves, convection and turbulence. My goal in this bookis present a coherent selection of these topics, concentrating on the foundations but withoutshying away from the boundaries of active areas of research for a book that limits itself towhat is absolutely settled would, I think, be rather dry, a quality best reserved for martinisand humour.AOFD is closely related to the eld of geophysical uid dynamics (GFD). The latter canbe, depending on ones point of view, both a larger and a smaller eld than the former. Itis larger because GFD, in its broadest meaning, includes not just the uid dynamics of theEarths atmosphere and ocean, but also the uid dynamics of such things as the Earthsinterior, volcanoes, lava ows and planetary atmospheres; it is the uid mechanics of allxixxx Prefacethings geophysical. But at the same time the appellation GFD implies a certain austerity, andthe subject is often seen as the one that provides the fundamental principles and languagefor understanding geophysical ows without being suocated by the overwhelming detailof the real world. In this book we are guided by the ascetic spirit of GFD, and my hope isthat the reader will gain a solid grounding in the fundamentals, motivated by and with anappreciation for the problems of the real world.The book is an outgrowth of various courses that I have taught over the years, mainly atPrinceton University but also at the University of California and at summer schools or similarin Boulder and Kyoto. There are four parts to the book: fundamentals of geophysical uiddynamics; instabilities, wavemean ow interaction and turbulence; atmospheric circulation;and ocean circulation. Each corresponds, very roughly, to a one-term graduate course,although parts could also be used for undergraduates. Limitations enforced both by theneed to keep the book coherent and focused, and my own expertise or (especially) lackthereof, naturally limit the choice of topics. In particular the chapters on the circulationfocus on the steady and statistically steady large-scale circulation and perforce a number ofimportant topics are omitted tropical and equatorial dynamics, many of the eects ofmoisture on atmospheric circulation, the spin-up of the ocean circulation, atmospheric andoceanic tides, the quasi-biennial oscillation, and so on. I have however and at no extracharge, mind you discussed the large-scale circulation of both atmosphere and ocean.The similarities and dierences between the two systems are, I believe, so instructive thateven if ones interest is solely in one, there is much to be gained by studying the other. Thereferences at the end of the book are representative and not exhaustive, and almost certainlydisproportionately represent articles written in English and those with which I happen to befamiliar. For the benet of the reader interested in exploring the development of the subjectI have included references to a number of historical articles, even when the presentationgiven does not draw from them. If there are other references that are particularly relevant Itrust the reader will inform me.I have tried to keep the overall treatment of topics as straightforward and as clear as Iknow how. In particular, I have tried to be as explicit as possible in my explanations, even atthe risk of descending from pedagogy into pedantry. Relatedly, there is a certain amount ofrepetition between sections, and this serves both to emphasize the important things and tokeep chapters reasonably self-contained. The chapters are of course intellectually linked, forexample, heat transport in the atmosphere depends on baroclinic instability, but hopefullythe reader already familiar with the latter will be able to read about the former without toomuch cross-referencing. The treatment generally is fairly physical and phenomenological,and rigour in the mathematical sense is absent; I treat the derivatives of integrals and ofinnitesimal quantities rather informally, for example.The gures (many in colour) may all be downloaded from the CUP web site associated withthis book. An asterisk, *, next to a section heading means that the section may be omittedon rst reading; although normally uncontroversial, it may contain advanced material thatis not essential for subsequent sections. A dagger, , next to a section heading means thatthe section discusses topics of research. Very roughly speaking, one might interpret asasterisk as indicating there is advanced manipulation of the equations, whereas a daggermight indicate there is approximation of the equations, or an interpretation that is notuniversally regarded as settled; caveat emptor. There is some arbitrariness in such markings,especially where the section deals with a well understood model of a poorly understoodPreface xxireality. Sections so-marked may be regarded as providing an introduction to the literature,rather than a complete or nished treatment, and the section may also require knowledge ofmaterial that appears later in the book. If the asterisk or dagger is applied to a section itapplies to all the subsections within, and if a dagger or asterisk appears within a sectionthat is already marked, the warning is even more emphatic. Reading the endnotes at the endof each chapter is not needed in order to follow the arguments in the main text. Problemsmarked with black diamonds may, like similarly marked ski-slopes, be dicult, and I do notknow the solutions to all of them. Good answers to some of them may be publishable and Iwould appreciate hearing about any such work. I would also appreciate any comments onthe material presented in the text. Qui docet discit.Finally, I should say that this book owes its existence in part to my own hubris andselshness: hubris to think that others might wish to read what I have written, and selshnessbecause the enjoyable task of writing such a book masquerades as work.Summary of ContentsThe chapters within each part, and the sections within each chapter, form logical units, anddo not necessarily directly correspond to a single lecture or set number of lectures.Part I. Fundamentals of geophysical uid dynamicsChapter 1 is a brief introduction to uid dynamics and the basic equations of motion,assuming no prior knowledge of the eld. Readers with prior knowledge of uid dynamicsmight skim it lightly, concentrating on those aspects unique to the atmosphere or ocean.Chapter 2 introduces the eects of stratication and rotation, these being the two maineects that most dierentiate AOFD from other branches of uid dynamics. Fundamentaltopics such as the primitive equations, the Boussinesq equations, and Ekman layers areintroduced here, and these form the foundation for the rest of the book.Chapter 3 focuses on the shallow water equations. Many of the principles of geophysicaluid dynamics have their simplest expression in the shallow water equations, because theeects of stratication are either eliminated or much simplied. The equations thus providea relatively gentle introduction to the eld.Chapter 4 discusses vorticity and potential vorticity. Potential vorticity plays an especiallyimportant role in large-scale, rotating and stratied ows, and its conservation provides thebasis for the equation sets of chapter 5.Chapter 5 derives simplied equation sets for large-scale ows, in particular the quasi-geostrophic and planetary-geostrophic equation sets, and introduces a simple application,Rossby waves. Much of our theoretical understanding of the large-scale circulation hasarisen through the use of these equations.Part II. Instabilities, wavemean ow interaction and turbulenceChapter 6 covers barotropic and baroclinic instability, the latter being the instability thatgives rise to weather and therefore being, perhaps, the form of hydrodynamic instabilitythat most aects the human condition.Chapter 7 provides an introduction to the important topic, albeit one that is regarded asdicult, of wavemean ow interaction. That is, how do the waves and instabilities aectthe mean ow in which they propagate?xxii PrefaceChapter 8 and 9 are on the statistical theory of turbulence. Chapter 8 introduces thebasic concepts of two- and three-dimensional turbulence, and chapter 9 applies similar ideasto geostrophic turbulence.Chapter 10 discusses turbulent diusion a hoary subject, both used and abused, yetone that plays a central role in our thinking about the transport properties of eddies in theatmosphere and ocean.Part III. Large-scale atmospheric circulationChapter 11 is mostly concerned with the dynamics of the Hadley Cell, and, rather descrip-tively, with the Ferrel Cell.Chapter 12 addresses the mid-latitude circulation. The goal of this chapter is to providethe basis of an understanding of such topics as the surface westerly winds, the dynamics ofthe Ferrel Cell, the stratication of the atmosphere and the height of the tropopause. Theseare still active topics of research, so we cannot always be denitive, although many of theunderlying principles are now established, and our discussion emphasizes the fundamentalsthat have, I think, permanent value.Chapter 13 discusses stationary waves in the atmosphere, mainly produced by theinteraction of the zonal wind with mountains and landsea temperature contrasts. It alsodiscusses the vertical propagation of Rossby waves, and how these induce a stratosphericcirculation.Part IV. Large-scale oceanic circulationChapter 14 discusses the wind-driven circulation, in particular the ocean gyres, eitherignoring buoyancy eects or assuming that buoyancy forcing acts primarily to set up astratication that can be taken as a given.Chaper 15 discusses the buoyancy-driven circulation, largely neglecting the eects ofwind forcing.Chapter 16 addresses the combined eects of wind and buoyancy forcing in setting upthe stratication and in driving both the predominantly horizontal gyral circulation and theoverturning circulation. As with many sections of Part III, many of these topics are still beingactively researched, although, again, many of the underlying principles have, I think, nowbeen established and have permanent value.AcknowledgementsThis book would not have been possible without the input, criticism, encouragement andadvice of a large number of people, from students to senior scientists. Students at PrincetonUniversity, New York University, Columbia University, California Institute of Technology, MITand the University of Toronto have used earlier versions of the text in various courses, and Iam grateful for the feedback received from the instructors and students about what worksand what doesnt, as well as for numerous detailed comments. Parts of the rst few chaptersand many of the problems draw on notes prepared over the years for a graduate class atPrinceton University taught by Steve Garner, Isaac Held, Yoshio Kurihara, Paul Kushner andme. Steve Garner has been notably generous with his time and ideas with respect to thismaterial.I would like to thank following individuals for providing comments on the text or forPreface xxiiiconversational input: Alistair Adcroft, Brian Arbic, Roger Berlind, Pavel Berlo, ThomasBirner, Paola Cessi, Sorin Codoban, Agatha de Boer, Roland de Szoeke, Raaele Ferrari, BaylorFox-Kemper, Dargan Frierson, Steve Gries, Brian Hoskins, Huei-Ping Huang, Chris Hughes,Kosuke Ito and his fellow students in Kyoto, Laura Jackson, Martin Juckes, Allan Kaufman,Samar Khatiwhala, Andrew Kositsky, Paul Kushner, Joe LaCasce, Trevor McDougall, PaulOGorman, Tim Palmer, David Pearson, Lorenzo Polvani, Cathy Raphael, Adam Scaife, JohnScinocca, Rob Scott, Tiany Shaw, Sabrina Speich, Robbie Toggweiler, Yue-Kin Tsang, RossTulloch, Eli Tziperman, Jacques Vanneste, Chris Walker, Ric Williams, Michal Zieminskiand Pablo Zurita-Gotor. Some of the above provided detailed comments on several sectionsor chapters, and I am very grateful to them; a number of other people provided guresand these are acknowledged where they appear. Anders Persson and Roger Samelsonprovided a number of useful historical remarks and interpretations, and Michael McIntyre(see also McIntyre 1997) emphasized the importance of being as explicit as possible in myexplanations. I am particularly indebted to Tapio Schneider, Adam Sobel, Jrgen Theissand Andy White who read and gave detailed comments on many chapters of text, to IsaacHeld for many conversations over the years on all aspects of atmospheric and oceanicuid dynamics, to Shafer Smith for both comments and code, to Ed Gerber for severalcalculations and gures, and to Bill Young for very useful input on thermodynamics and theBoussinesq equations. I would also like to thank Matt Lloyd of CUP for his encouragement,my copy-editor, Louise Staples, for her light but careful touch, and Anna Valerio for ecientsecretarial assistance. Needless to say, I take full responsibility for the errors, both scienticand presentational, that undoubtedly remain.My rst exposure to the eld came as a graduate student in the fecund atmosphereof Atmospheric Physics Group at Imperial College in the late 1970s and early 1980s, andId like to thank everyone who was there at that time. You know who you are. I wouldalso like to thank everyone at the Geophysical Fluid Dynamics Laboaratory (GFDL) and inPrinceton Universitys Atmospheric and Oceanic Science program for creating the pleasantand stimulating work environment that made it possible to write this book. I am gratefulto the sta and scientists at the Plymouth Marine Laboratory and the U. K. MeteorologicalOce for their hospitality during a sabbatical visit. Finally, I would like to thank the NationalScience Foundation and the National Oceanic and Atmospheric Administration for providingnancial support over the years, and Bess for support of a much greater kind.Note on second printingI have taken advantage the opportunity aorded by a second printing of this book tocorrect a number of typographic errors and make a number of small corrections to thetext throughout. The pagination and equation numbering are mostly unaltered, althoughsome microtypographic improvements have changed some of the line breaks. I am grate-ful to many readers who have sent in comments and corrections, and I would partic-ularly like to acknowledge Roger Berlind for his exceptionally detailed and perceptivecomments on the entire book. Additional resources, including downloadable gures andsolutions to many of the end-of-chapter exercises, are available at www.vallisbook.org andwww.cambridge.org/9780521849692.xxiv NotationNOTATIONThe list below contains only the more important variables, or instances of non-obviousnotation. Distinct meanings are separated with a semi-colon. Variables are normally set initalics, constants (e.g, ) in roman (i.e., upright), dierential operators in roman, vectorsin bold, and tensors in bold sans serif. Thus, vector variables are in bold italics, vectorconstants (e.g., unit vectors) in bold roman, and tensor variables are in bold slanting sansserif. Physical units are set in roman. A subscript denotes a derivative only if the subscriptis a coordinate, such as x, y, z or t; a subscript 0 generally denotes a constant referencevalue (e.g., 0). The components of a vector are denoted by superscripts.Variable Descriptionb Buoyancy, g/0 or g/`.cg Group velocity, (cxg, cyg, czg).cp Phase speed; heat capacity at constant pressure.cv Heat capacity constant volume.cs Sound speed.f, f0 Coriolis parameter, and its reference value.g, g Vector acceleration due to gravity, magnitude of g.h Layer thickness (in shallow water equations).i, j, k Unit vectors in (x, y, z) directions.i An integer index.i Square root of 1.k Wave vector, with components (k, l, m) or (kx, ky, kz).kd Wave number corresponding to deformation radius.Ld Deformation radius.L, H Horizontal length scale, vertical (height) scale.m Angular momentum about the Earths axis of rotation.M Montgomery function, M = cpT +.N Buoyancy, or BruntVisl, frequency.p Pressure.Pr Prandtl ratio, f0/N.q Quasi-geostrophic potential vorticity.Q Potential vorticity (in particular Ertel PV).Q Rate of heating.Ra Rayleigh number.Re Real part of expression.Re Reynolds number, UL/.Ro Rossby number, U/fL.S Salinity; source term on right-hand side of an evolution equation.So, So Solenoidal term, solenoidal vector.T Temperature.t Time.u Two-dimensional (horizontal) velocity, (u, v).v Three-dimensional velocity, (u, v, z).x, y, z Cartesian coordinates, usually in zonal, meridional and vertical directions.Z Log-pressure, Hlogp/pR. Usually, H = 7.5km and pR = 105Pa.Notation xxvVariable Description Wave activity. Inverse density, or specic volume; aspect ratio. Rate of change of f with latitude, f/y.T, S Coecient of expansion with respect to temperature, salinity. Generic small parameter (epsilon). Cascade or dissipation rate of energy (varepsilon). Specic entropy; perturbation height; enstrophy cascade or dissipation rate.F Eliassen Palm ux, (y, z). Vorticity gradient, uyy; the ratio cp/cv. Lapse rate. Diusivity; the ratio R/cp.J Kolmogorov or Kolmogorov-like constant. Shear, e.g., U/z. Viscosity. Kinematic viscosity, /.v Meridional component of velocity. Pressure divided by density, p/; passive tracer. Geopotential, usually gz. Exner function, = cpT/ = cp(p/pR)R/cp. Vorticity., Rotation rate of Earth and associated vector. Streamfunction. Density. Potential density. Layer thickness, z/; Prandtl number /; measure of density, 1000. Stress vector, often wind stress.` Kinematic stress, ` / . Zonal component or magnitude of wind stress; eddy turnover time. Potential temperature., Latitude, longitude. Vertical component of vorticity._ab_cDerivative of a with respect to b at constant c.aba=cDerivative of a with respect to b evaluated at a = c.a Gradient operator at constant value of coordinate a, e.g., z = i x+j y.a Divergence operator at constant value of coordinate a, e.g., z = (i x+j y). Perpendicular gradient, k .curlz Vertical component of operator, curlzA = k A = xAyyAx.DDt Material derivative (generic).DgDt Material derivative using geostrophic velocity, for example /t +ug .D3Dt, D2Dt Material derivative in three dimensions and in two dimensions, for example/t +v and /t +u respectively.Part IFUNDAMENTALS OF GEOPHYSICALFLUID DYNAMICSAre you sitting comfortably? Then Ill begin.Julia Lang, Listen With Mother, BBC radio program, 19501982.CHAPTERONEEquations of MotionTHIS CHAPTER establishes the fundamental governing equations of motion for a uid,with particular attention to the uids of the Earths atmosphere and ocean. Ourapproach in many places is quite informal, and the interested reader may consultthe references given for more detail.1.1 TIME DERIVATIVES FOR FLUIDSThe equations of motion of uid mechanics dier from those of rigid-body mechanicsbecause uids form a continuum, and because uids ow and deform. Thus, even thoughboth classical solid and uid media are governed by the same relatively simple physicallaws (Newtons laws and the laws of thermodynamics), the expression of these laws diersbetween the two. To determine the equations of motion for uids we must clearly establishwhat the time derivative of some property of a uid actually means, and that is the subjectof this section.1.1.1 Field and material viewpointsIn solid-body mechanics one is normally concerned with the position and momentum ofidentiable objects the angular velocity of a spinning top or the motions of the planetsaround the Sun are two well-worn examples. The position and velocity of a particular objectis then computed as a function of time by formulating equations of the formdxidt = F(xi, t), (1.1)34 Chapter 1. Equations of Motionwhere xi is the set of positions and velocities of all the interacting objects and the operatorF on the right-hand side is formulated using Newtons laws of motion. For example, twomassive point objects interacting via their gravitational eld obeydridt = vi, dvidt = Gmj(rirj)2 ri,j, i = 1, 2; j = 3 i. (1.2)We thereby predict the positions, ri, and velocities, vi, of the objects given their masses, mi,and the gravitational constant G, and where ri,j is a unit vector directed from ri to rj.In uid dynamics such a procedure would lead to an analysis of uid motions in termsof the positions and momenta of dierent uid parcels, each identied by some label, whichmight simply be their position at an initial time. We call this a material point of view,because we are concerned with identiable pieces of material; it is also sometimes called aLagrangian view, after J.-L. Lagrange. The procedure is perfectly acceptable in principle, andif followed would provide a complete description of the uid dynamical system. However,from a practical point of view it is much more than we need, and it would be extremelycomplicated to implement. Instead, for most problems we would like to know what thevalues of velocity, density and so on are at xed points in space as time passes. (A weatherforecast we might care about tells us how warm it will be where we live, and if we are giventhat we do not particularly care where a uid parcel comes from, or where it subsequentlygoes.) Since the uid is a continuum, this knowledge is equivalent to knowing how the eldsof the dynamical variables evolve in space and time, and this is often known as the eld orEulerian viewpoint, after L. Euler.1Thus, whereas in the material view we consider the timeevolution of identiable uid elements, in the eld view we consider the time evolution ofthe uid eld from a particular frame of reference. That is, we seek evolution equations ofthe general formt(x, y, z, t) = G(, x, y, z, t), (1.3)where the eld (x, y, z, t) represents all the dynamical variables (velocity, density, tem-perature, etc.) and G is some operator to be determined from Newtons laws of motion andappropriate thermodynamic laws.Although the eld viewpoint will often turn out to be the most practically useful, thematerial description is invaluable both in deriving the equations and in the subsequentinsight it frequently provides. This is because the important quantities from a fundamentalpoint of view are often those which are associated with a given uid element: it is thesewhich directly enter Newtons laws of motion and the thermodynamic equations. It is thusimportant to have a relationship between the rate of change of quantities associated witha given uid element and the local rate of change of a eld. The material or advectivederivative provides this relationship.1.1.2 The material derivative of a uid propertyA uid element is an innitesimal, indivisible, piece of uid eectively a very smalluid parcel of xed mass. The material derivative is the rate of change of a property(such as temperature or momentum) of a particular uid element or nite mass; that isto say, it is the total time derivative of a property of a piece of uid. It is also knownas the substantive derivative (the derivative associated with a parcel of uid substance),1.1 Time Derivatives for Fluids 5the advective derivative (because the uid property is being advected), the convectivederivative (convection is a slightly old-fashioned name for advection, still used in someelds), or the Lagrangian derivative.Let us suppose that a uid is characterized by a (given) velocity eld v(x, t), whichdetermines its velocity throughout. Let us also suppose that it has another property , andlet us seek an expression for the rate of change of of a uid element. Since is changingin time and in space we use the chain rule, = t t + x x + y y + z z = t t +x . (1.4)This is true in general for any t, x, etc. The total time derivative is thenddt = t + dxdt . (1.5)If this equation is to represent a material derivative we must identify the time derivative inthe second term on the right-hand side with the rate of change of position of a uid element,namely its velocity. Hence, the material derivative of the property isddt = t +v . (1.6)The right-hand side expresses the material derivative in terms of the local rate of change of plus a contribution arising from the spatial variation of , experienced only as the uidparcel moves. Because the material derivative is so common, and to distinguish it from otherderivatives, we denote it by the operator D/Dt. Thus, the material derivative of the eld isDDt = t +(v ) . (1.7)The brackets in the last term of this equation are helpful in reminding us that (v ) is anoperator acting on .Material derivative of vector eldThe material derivative may act on a vector eld b, in which caseDbDt = bt +(v )b. (1.8)In Cartesian coordinates this isDbDt = bt +ubx +v by +wbz, (1.9)and for a particular component of b,DbxDt = bxt +ubxx +vbxy +wbxz , (1.10)or, in Cartesian tensor notation,DbiDt = bit +vjbixj= bit +vjjbi. (1.11)6 Chapter 1. Equations of Motionwhere the subscripts denote the Cartesian components and repeated indices are summed. Incoordinate systems other than Cartesian the advective derivative of a vector is not simply thesum of the advective derivative of its components, because the coordinate vectors changedirection with position; this will be important when we deal with spherical coordinates.Finally, we note that the advective derivative of the position of a uid element, r say, is itsvelocity, and this may easily be checked by explicitly evaluating Dr/Dt.1.1.3 Material derivative of a volumeThe volume that a given, unchanging, mass of uid occupies is deformed and advected bythe uid motion, and there is no particular reason why it should remain constant. Indeed,the volume will change as a result of the movement of each element of its bounding materialsurface, and will in general change if there is a non-zero normal component of the velocityat the uid surface. That is, if the volume of some uid is _ dV, thenDDt_VdV =_Sv dS, (1.12)where the subscript V indicates that the integral is a denite integral over some nite volumeV, although the limits of the integral will be functions of time if the volume is changing. Theintegral on the right-hand side is over the closed surface, S, bounding the volume. Althoughintuitively apparent (to some), this expression may be derived more formally using Leibnitzsformula for the rate of change of an integral whose limits are changing (problem 1.2). Usingthe divergence theorem on the right-hand side, (1.12) becomesDDt_VdV =_V v dV. (1.13)The rate of change of the volume of an innitesimal uid element of volume V is obtainedby taking the limit of this expression as the volume tends to zero, givinglimV01VDVDt = v. (1.14)We will often write such expressions informally asDVDt = V v, (1.15)with the limit implied.Consider now the material derivative of some uid property, say, multiplied by thevolume of a uid element, V. Such a derivative arises when is the amount per unitvolume of -substance it might, for example, be mass density or the amount of a dye perunit volume. Then we haveDDt(V) = DVDt +VDDt. (1.16)Using (1.15) this becomesDDt(V) = V_ v + DDt_, (1.17)1.2 The Mass Continuity Equation 7and the analogous result for a nite uid volume is justDDt_V dV =_V_ v + DDt_ dV. (1.18)This expression is to be contrasted with the Eulerian derivative for which the volume, and sothe limits of integration, are xed and we haveddt_V dV =_Vt dV. (1.19)Now consider the material derivative of a uid property multiplied by the mass ofa uid element, V, where is the uid density. Such a derivative arises when is theamount of -substance per unit mass (note, for example, that the momentum of a uidelement is vV). The material derivative of V is given byDDt(V) = VDDt + DDt(V) (1.20)But V is just the mass of the uid element, and that is constant that is how a uidelement is dened. Thus the second term on the right-hand side vanishes andDDt(V) = VDDt and DDt_V dV =_VDDt dV, (1.21a,b)where (1.21b) applies to a nite volume. That expression may also be derived more formallyusing Leibnitzs formula for the material derivative of an integral, and the result also holdswhen is a vector. The result is quite dierent from the corresponding Eulerian derivative,in which the volume is kept xed; in that case we have:ddt_V dV =_Vt() dV. (1.22)Various material and Eulerian derivatives are summarized in the shaded box on the followingpage.1.2 THE MASS CONTINUITY EQUATIONIn classical mechanics mass is absolutely conserved, and in solid-body mechanics we nor-mally do not need an explicit equation of mass conservation. However, in uid mechanicsuid ows into and away from regions, and uid density may change, and an equation thatexplicitly accounts for the ow of mass is one of the equations of motion of the uid.1.2.1 An Eulerian derivationWe will rst derive the mass conservation equation from an Eulerian point of view; that is tosay, our reference frame is xed in space and the uid ows through it.8 Chapter 1. Equations of MotionMaterial and Eulerian DerivativesThe material derivative of a scalar () and a vector (b) eld are given by:DDt = t +v , DbDt = bt +(v )b. (D.1)Various material derivatives of integrals are:DDt_VdV =_V_DDt + v_ dV =_V_t + (v)_dV, (D.2)DDt_VdV =_V v dV, (D.3)DDt_VdV =_VDDt dV. (D.4)These formulae also hold if is a vector. The Eulerian derivative of an integral is:ddt_VdV =_Vt dV, (D.5)so thatddt_VdV = 0 and ddt_VdV =_Vt dV. (D.6)Cartesian derivationConsider an innitesimal rectangular parallelepiped (i.e., a cuboid) control volume, V =xyz that is xed in space, as in Fig. 1.1. Fluid moves into or out of the volume throughits surface, including through its faces in the yz plane of area A = yz at coordinatesx and x +x. The accumulation of uid within the control volume due to motion in thex-direction is evidentlyyz[(u)(x, y, z) (u)(x +x, y, z)] = (u)xx,y,zxy z. (1.23)To this must be added the eects of motion in the y- and z-directions, namely_(v)y + (w)z_xy z. (1.24)This net accumulation of uid must be accompanied by a corresponding increase of uidmass within the control volume. This ist (density volume) = xy zt , (1.25)because the volume is constant. Thus, because mass is conserved, (1.23), (1.24) and (1.25)givexy z_t + (u)x + (v)y + (w)z_= 0. (1.26)1.2 The Mass Continuity Equation 9

u+(u)x x

y zuy zzyxx x +xFig. 1.1 Mass conservation in an Eulerian cuboid control volume.Because the control volume is arbitrary the quantity in square brackets must be zero and wehave the mass continuity equation:t + (v) = 0. (1.27)Vector derivationConsider an arbitrary control volume V bounded by a surface S, xed in space, with byconvention the direction of S being toward the outside of V, as in Fig. 1.2. The rate of uidloss due to ow through the closed surface S is then given byuid loss =_Sv dS =_V (v) dV, (1.28)using the divergence theorem. This must be balanced by a change in the mass M of theuid within the control volume, which, since its volume is xed, implies a density change.That isuid loss = dMdt = ddt_V dV = _Vt dV. (1.29)Equating (1.28) and (1.29) yields_V_t + (v)_ dV = 0. (1.30)Because the volume is arbitrary, the integrand must vanish and we recover (1.27).1.2.2 Mass continuity via the material derivativeWe now derive the mass continuity equation (1.27) from a material perspective. This is themost fundamental approach of all since the principle of mass conservation states simplythat the mass of a given element of uid is, by denition of the element, constant. Thus,consider a small mass of uid of density and volume V. Then conservation of mass maybe represented byDDt(V) = 0. (1.31)10 Chapter 1. Equations of MotionFig. 1.2 Mass conservation inan arbitrary Eulerian controlvolume V bounded by a sur-face S. The mass gain,_V(/t) dV is equal to themass owing into the vol-ume, _S(v) dS = _V(v) dV.Both the density and the volume of the parcel may change, soVDDt +DVDt = V_DDt + v_= 0, (1.32)where the second expression follows using (1.15). Since the volume element is arbitrary, theterm in brackets must vanish andDDt + v = 0. (1.33)After expansion of the rst term this becomes identical to (1.27). This result may be derivedmore formally by rewriting (1.31) as the integral expressionDDt_V dV = 0. (1.34)Expanding the derivative using (1.18) givesDDt_V dV =_V_DDt + v_ dV = 0. (1.35)Because the volume over which the integral is taken is arbitrary the integrand itself must van-ish and we recover (1.33). Summarizing, equivalent partial dierential equation representingconservation of mass are:DDt + v = 0, t + (v) = 0 . (1.36a,b)1.3 The Momentum Equation 111.2.3 A general continuity equationThe derivation of continuity equation for a general scalar property of a uid is similar tothat for density, except that there may be an external source or sink, and potentially ameans of transferring the property from one location to another than by uid motion, forexample by diusion. If is the amount of some property of the uid per unit volume(which we will call the concentration of the property), and if the net eect per unit volumeof all non-conservative processes is denoted by Qv[], then the continuity equation forconcentration may be written:DDt(V) = Qv[]V. (1.37)Expanding the left-hand side and using (1.15) we obtainDDt + v = Qv[], (1.38)or equivalentlyt + (v) = Qv[]. (1.39)If we are interested in a tracer that is normally measured per unit mass of uid (which istypical when considering thermodynamic quantities) then the conservation equation wouldbe writtenDDt(V) = Qm[]V, (1.40)where is the tracer mixing ratio that is, the amount of tracer per unit uid mass andQm[] represents non-conservative sources per unit mass. Then, since V is constant weobtainDDt = Qm[]. (1.41)The source term Qm[] is evidently equal to the rate of change of of a uid element.When this is so, we write it simply as , so thatDDt = . (1.42)A tracer obeying (1.42) with = 0 is said to be materially conserved. If a tracer is materiallyconserved except for the eects of non-conservative sources then it is sometimes said to besemi-materially conserved or adiabatically conserved.1.3 THE MOMENTUM EQUATIONThe momentum equation is a partial dierential equation that describes how the velocityor momentum of a uid responds to internal and imposed forces. We will derive it usingmaterial methods and informally deducing the terms representing the pressure, gravitationaland viscous forces.12 Chapter 1. Equations of Motion1.3.1 AdvectionLet m(x, y, z, t) be the momentum-density eld (momentum per unit volume) of the uid.Thus, m= v and the total momentum of a volume of uid is given by the volume integral_V mdV. Now, for a uid the rate of change of a momentum of an identiable uid mass isgiven by the material derivative, and by Newtons second law this is equal to the force actingon it. Thus,DDt_Vv dV =_VF dV, (1.43)where F is the force per unit volume. Now, using (1.21b) (with replaced by v) to transformthe left-hand side of (1.43), we obtain_V_DvDt F_ dV = 0. (1.44)Because the volume is arbitrary the integrand itself must vanish and we obtainDvDt = F, or vt +(v )v = F, (1.45a,b)having used (1.8) to expand the material derivative.We have thus obtained an expression for how a uid accelerates if subject to knownforces. These forces are however not all external to the uid itself; a stress arises from thedirect contact between one uid parcel and another, giving rise to pressure and viscousforces, sometimes referred to as contact forces. Because a complete treatment of thesewould be very lengthy, and is available elsewhere, we treat both of these very informally andintuitively.1.3.2 The pressure forceWithin or at the boundary of a uid the pressure is the normal force per unit area due to thecollective action of molecular motion. Thusd

Fp = pdS, (1.46)where p is the pressure, Fp is the pressure force and dS an innitesimal surface element.If we grant ourselves this intuitive notion, it is a simple matter to assess the inuence ofpressure on a uid, for the pressure force on a volume of uid is the integral of the pressureover the its boundary and so

Fp = _SpdS. (1.47)The minus sign arises because the pressure force is directed inwards, whereas S is a vectornormal to the surface and directed outwards. Applying a form of the divergence theorem tothe right-hand side gives

Fp = _VpdV, (1.48)where the volume V is bounded by the surface S. The pressure force per unit volume, Fp, istherefore just p, and inserting this into (1.45b) we obtainvt +(v )v = p +F, (1.49)where F, equal to (F Fp)/, represents viscous and body forces, per unit mass.1.3 The Momentum Equation 13 ( kg m1s1) ( m2s1)Air 1.8 1051.5 105Water 1.1 1031.1 106Mercury 1.6 1031.2 107Table 1.1 Experimental values of viscosity forair, water and mercury at room temperature andpressure.1.3.3 Viscosity and diusionViscosity, like pressure, is a force due to the internal motion of molecules. The eects ofviscosity are apparent in many situations the ow of treacle or volcanic lava are obviousexamples. In other situations, for example large-scale ow the atmosphere, viscosity is to arst approximation negligible. However, for a constant density uid viscosity is the onlyway that energy may be removed from the uid, so that if energy is being added in someway viscosity must ultimately become important if the uid is to reach an equilibrium whereenergy input equals energy dissipation. When tea is stirred in a cup, it is viscous eects thatcause the uid to eventually stop spinning after we have removed our spoon.A number of textbooks2show that, for most Newtonian uids, the viscous force perunit volume is approximately equal to 2v, where is the viscosity. Although not exact,this is an extremely good approximation for most liquids and gases. With this term, themomentum equation becomes,vt +(v )v = 1p +2v +Fb (1.50)where / is the kinematic viscosity, and Fb represents any body forces (per unit mass)such as gravity, g. For gases, dimensional arguments suggest that the magnitude of shouldbe given by mean free path mean molecular velocity, (1.51)which for a typical molecular velocity of 300ms1and a mean free path of 7 108m givesthe not unreasonable estimate of 2.1105m2s1, within a factor of two of the experimentalvalue (Table 1.1). Interestingly, the kinematic viscosity is less for water and mercury than itis for air.1.3.4 Hydrostatic balanceThe vertical component the component parallel to the gravitational force, g of themomentum equation isDwDt = 1pz g, (1.52)where w is the vertical component of the velocity and g = gk. If the uid is static thegravitational term is balanced by the pressure term and we havepz = g, (1.53)14 Chapter 1. Equations of Motionand this relation is known as hydrostatic balance, or hydrostasy. It is clear in this casethat the pressure at a point is given by the weight of the uid above it, provided p = 0at the top of the uid. It might also appear that (1.53) would be a good approximation to(1.52) provided vertical accelerations, Dw/Dt, are small compared to gravity, which is nearlyalways the case in the atmosphere and ocean. While this statement is true if we need only areasonable approximate value of the pressure at a point or in a column, the satisfaction ofthis condition is not sucient to ensure that (1.53) provides an accurate enough pressure todetermine the horizontal pressure gradients responsible for producing motion. We return tothis point in section 2.7.1.4 THE EQUATION OF STATEIn three dimensions the momentum and continuity equations provide four equations, butcontain ve unknowns three components of velocity, density and pressure. Obviouslyother equations are needed, and an equation of state is an expression that diagnosticallyrelates the various thermodynamic variables to each other. The conventional equation ofstate (also called the thermal equation of state) is an expression that relates temperature,pressure, composition (the mass fraction of the various constituents) and density, and wemay write it, rather generally, asp = p(, T, n), (1.54)where n is mass fraction of the nth constituent. An equation of this form is not the mostfundamental equation of state from a thermodynamic perspective, an issue we visit later,but it connects readily measurable quantities.For an ideal gas (and the air in the Earths atmosphere is very close to ideal) the conven-tional equation of state isp = RT, (1.55)where R is the gas constant for the gas in question and T is temperature. (R is related tothe universal gas constant Ru by R = Ru/m, where m is the mean molecular weight of theconstituents of the gas. Also, R = nk, where k is Boltzmanns constant and n is the numberof molecules per unit mass.) For dry air, R = 287J kg1K1. Air has virtually constantcomposition except for variations in water vapour content. A measure of this is the watervapour mixing ratio, w = w/d where w and d are the densities of water vapour anddry air, respectively, and in the atmosphere w varies between 0 and 0.03. This variationmakes the gas constant in the equation of state a weak function of the water vapour mixingratio; that is, p = ReT where Re = Rd(1 +wRv/Rd)/(1 +w) where Rd and Rv are thegas constants of dry air and water vapour. Since w 0.01 the variation of Re is quite smalland is often ignored, especially in theoretical studies.3For a liquid such as seawater no simple expression akin to (1.55) is easily derivable, andsemi-empirical equations are usually resorted to. For pure water in a laboratory setting areasonable approximation of the equation of state is = 0[1 T(T T0)], where T is athermal expansion coecient and 0 and T0 are constants. In the ocean the density is alsosignicantly aected by pressure and dissolved salts: seawater is a solution of many ions inwater chloride ( 1.9% by weight) sodium (1%), sulfate (0.26%), magnesium (0.13%) andso on, with a total average concentration of about 35 (ppt, or parts per thousand). Theratios of the fractions of these salts are more-or-less constant throughout the ocean, and1.4 The Equation of State 15their total concentration may be parameterized by a single measure, the salinity, S. Giventhis, the density of seawater is a function of three variables pressure, temperature, andsalinity and we may write the conventional equation of state as = (T, S, p), (1.56)where = 1/ is the specic volume, or inverse density. For small variations around areference value we haved =_T_S,pdT +_S_T,pdS +_p_T,Sdp = (T dT S dS p dp), (1.57)where the rightmost expression serves to dene the thermal expansion coecient T, thesaline contraction coecient S, and the compressibility coecient (or inverse bulk modulus)p. In general these quantities are not constants, but for small variations around a referencestate they may be treated as such and we have = 0_1 +T(T T0) S(S S0) p(p p0)_. (1.58)Typical values of these parameters, with variations typically encountered through theocean, are: T 2(:1.5) 104K1(values increase with both temperature and pressure),S 7.6(:0.2) 104ppt1, p 4.1(:0.5) 1010Pa1. Since the variations around themean density are small (1.58) implies that = 0_1 T(T T0) +S(S S0) +p(p p0)_. (1.59)A linear equation of state for seawater is emphatically not accurate enough for quantita-tive oceanography; the parameters in (1.58) themselves vary with pressure, temperatureand (more weakly) salinity so introducing nonlinearities to the equation. The most importantof these are captured by an equation of state of the form = 0_1 +T(1 +p)(T T0) + T2 (T T0)2S(S S0) p(p p0)_. (1.60)The starred constants T and capture the leading nonlinearities: is the thermobaricparameter, which determines the extent to which the thermal expansion depends on pres-sure, and T is the second thermal expansion coecient. Even this equation of state hasquantitative deciencies and more complicated semi-empirical formulae are often used ifhigh accuracy is needed.4An illustration of the variation of density of seawater with temper-ature, salinity and pressure is illustrated in Fig. 1.3, which uses an accurate, semi-empiricalequation of state. More discussion is to be found in section 1.7.2.Clearly, the equation of state introduces, in general, a sixth unknown, temperature, andwe will have to introduce another physical principle the rst law of thermodynamics orthe principle of energy conservation to obtain a complete set of equations. However, ifthe equation of state were such that it linked only density and pressure, without introducinganother variable, then the equations would be complete; the simplest case of all is a constantdensity uid for which the equation of state is just = constant. A uid for which thedensity is a function of pressure alone is called a barotropic uid; otherwise, it is a baroclinicuid. (In this context, barotropic is a shortening of the original phrase auto-barotropic.)Equations of state of the form p = C, where is a constant, are sometimes calledpolytropic.16 Chapter 1. Equations of Motion2224262830SalinityTemperature32 34 36 380102030384042444648Salinity32 34 36 380102030Fig. 1.3 A temperaturesalinity diagram for seawater, calculated using an accurateempirical equation of state. Contours are (density1000) kg m3, and the temperatureis potential temperature, which in the deep ocean may be less than in situ temperatureby a degree or so (see Fig. 1.4). Left panel: at sea-level (p = 105Pa = 1000mb). Rightpanel: at p = 4 107Pa, a depth of about 4km. Note that in both cases the contoursare slightly convex.1.5 THERMODYNAMIC RELATIONSIn this section we review a few aspects of thermodynamics. We provide neither a completenor an a priori development of the subject; rather, we focus on aspects that are particularlyrelevant to uid dynamics, and that are needed to derive a thermodynamic equation foruids.5Readers whose interest is solely in an ideal gas or a simple Boussinesq uid mayskim this section, and then refer back to it as needed.1.5.1 A few fundamentalsA fundamental postulate of thermodynamics is that the internal energy of a system inequilibrium is a function of its extensive properties volume, entropy, and the mass of itsvarious constituents. (Extensive means that the property value is proportional to the amountof material present, as opposed to an intensive property such as temperature.) For ourpurposes it is more convenient to divide all of these quantities by the mass of uid present,so expressing the internal energy per unit mass, I, as a function of the specic volume = 1, the specic entropy , and the mass fractions of its various components. Ourinterest is in two-component uids (dry air and water vapour, or water and salinity) so thatwe may parameterize the composition by a single parameter, S. Thus we haveI = I(, , S) , (1.61a)or an equivalent equation for entropy, = (I, , S) . (1.61b)Given the functional forms on the right-hand sides, either of these expressions constitutes acomplete description of the macroscopic state of a system in equilibrium, and we call them1.5 Thermodynamic Relations 17the fundamental equation of state. The conventional equation of state can be derived from(1.61), but not vice versa. The rst dierential of (1.61a) gives, formally,dI =_ I_,Sd+_I_,Sd +_IS_,dS. (1.62)We will now ascribe physical meaning to these dierentials.Conservation of energy states that the internal energy of a body may change becauseof work done by or on it, or because of a heat input, or because of a change in its chemicalcomposition. We write this asdI = dQ dW + dC , (1.63)where dW is the work done by the body, dQ is the heat input to the body, and dC accountsfor the change in internal energy caused by a change in its chemical composition (e.g., itssalinity, or water vapour content), sometimes called the chemical work. The quantitieson the right-hand side are imperfect dierentials or innitesimals: Q, W and C are notfunctions of the state of a body, and the internal energy cannot be regarded as the sum ofa heat and a work. We should think of heat and work as having meaning only as uxesof energy, or rates of energy input, and not as amounts of energy; their sum changes theinternal energy of a body, which is a function of its state. Equation (1.63) is sometimes calledthe rst law of thermodynamics. Let us consider the causes of variations of the quantitieson the right-hand side.Heat Input: Thermodynamics provides a relationship between the heat input and the changein the entropy of a body, namely that in an (innitesimal) quasi-static or reversibleprocess, with constant composition,T d = dQ, (1.64)where is the specic entropy of the body. The entropy is a function of the state of abody and is, by denition, an adiabatic invariant. As we are dealing with the amountof a quantity per unit mass, is the specic entropy, although we will often refer to itjust as the entropy. We may regard (1.63) as dening the heat input, dQ, by way of astatement of conservation of energy, and (1.64) then says that there is a function ofstate, the entropy, that changes by an amount equal to the heat input divided by thetemperature.6Work done: The work done by a body is equal to the pressure times the change in its volume.Thus, per unit mass, we havedW = pd, (1.65)where = 1/ is the specic volume of the uid and p is the pressure.Composition: The change in internal energy due to changes in composition is given bydC = dS, (1.66)where is the chemical potential of the solution. In the ocean compositional changes18 Chapter 1. Equations of Motion(i.e., changes in salinity) arise through precipitation and evaporation at the surface,and molecular diusion. When salinity does so change, the internal energy of a uidparcel changes by (1.66), but in practice this change is usually small compared toother changes in internal energy. The most important eect of salinity is that itchanges the density of seawater. In the atmosphere the composition of a parcel of airprimarily mainly varies according to the amount of water vapour and liquid water init; these variations cause corresponding changes in internal energy, but in the absenceof phase-changes the internal energy variations are slight. The most important eectof water vapour is that when condensation or evaporation occurs, heat is released (orrequired) that provides an entropy source in (1.64).Collecting equations (1.63) (1.66) together we havedI = T d pd+ dS . (1.67)We refer to this as the fundamental thermodynamic relation. The fundamental equation ofstate, (1.61), describes the properties of a particular uid, and the fundamental relation,(1.67), is a statement of the conservation of energy. Much of classical thermodynamicsfollows from these two expressions.1.5.2 Various thermodynamic relationsFrom (1.62) and (1.67) it follows thatT =_I_,S, p = _ I_,S, =_IS_,. (1.68a,b,c)These may be regarded as the dening relations for these variables; because of the connec-tion between (1.63) and (1.67) these are not just formal denitions, and the pressure andtemperature so dened are indeed related to our intuitive concepts of these variables and tothe motion of the uid molecules. Note that if we writed = 1T dI + pT d T dS, (1.69)it is also clear thatp = T__I,S, T1=_I_,S, = T_S_I,. (1.70a,b,c)In the derivations below, we will, unless noted, suppose that the composition of a uidparcel is xed, and drop the sux S on partial derivatives unless ambiguity might arise.Because the right-hand side of (1.67) is equal to an exact dierential, the second deriva-tives are independent of the order of dierentiation. That is,2I = 2I (1.71)and therefore, using (1.68)_T_= _p_. (1.72)1.5 Thermodynamic Relations 19This is one of the Maxwell relations, which are a collection of four similar relations thatfollow directly from the fundamental thermodynamic relation (1.67) and simple relationsbetween second derivatives. A couple of others will be useful.Dene the enthalpy of a uid byh I +p (1.73)then, for a parcel of constant composition, (1.67) becomesdh = T d +dp. (1.74)But h is a function only of and p so that in generaldh =_h_pd +_hp_dp. (1.75)Comparing the last two equations we haveT =_h_pand =_hp_. (1.76)Noting that2hp = 2hp (1.77)we evidently must have_Tp_=__p, (1.78)and this is our second Maxwell relation.To obtain the third, we writedI = T d pd = d(T) dT d(p) +dp, (1.79)ordG = dT +dp, (1.80)where G I T + p is the Gibbs function (or Gibbs free energy or Gibbs potential).Now, formally, we havedG =_GT_pdT +_Gp_Tdp. (1.81)Comparing the last two equations we see that = (G/T )p and = (G/p)T. Further-more, because2GpT = 2GT p (1.82)we have our third Maxwell equation,_p_T= _T_p. (1.83)20 Chapter 1. Equations of MotionMaxwells RelationsThe four Maxwell equations are:_T_= _p_,_Tp_=__p,_p_T= _T_p,__T=_pT_.(M.1)These imply:(T, )(p, ) _Tp____T__p_= 0. (M.2)The fourth Maxwell equation, whose derivation is left to the reader (make use of theHelmholtz free energy, F I T), is__T=_pT_, (1.84)and all four Maxwell equations are summarized in the box above. All of them followfrom the fundamental thermodynamic relation, (1.67), which is the real silver hammer ofthermodynamics.Fundamental equation of stateThe fundamental equation of state (1.61) gives complete information about a uid inthermodynamic equilibrium, and given this we can obtain expressions for the temperature,pressure and chemical potential using (1.68). These are also equations of state; however, eachof them, taken individually, contains less information than the fundamental equation becausea derivative has been taken. Equivalent to the fundamental equation of state are, using(1.74), an expression for the enthalpy as a function of its natural variables pressure, entropyand composition, or, using (1.80) the Gibbs function as a function of pressure, temperatureand composition. Of these, the Gibbs function is the most practically useful because thepressure, temperature and composition may all be measured in the laboratory. Given thefundamental equation of state, the thermodynamic state of a body is fully specied by aknowledge of any two of p, , T, and I, plus its composition. The conventional equation ofstate, (1.54), is obtained by using (1.61a) to eliminate entropy from (1.68a) and (1.68b).One simple fundamental equation of state is to take the internal energy to be a function ofdensity and not entropy; that is, I = I(). Bodies with such a property are called homentropic.Using (1.68), temperature and chemical potential have no role in the uid dynamics and thedensity is a function of pressure alone the dening property of a barotropic uid. Neitherwater nor air are, in general, homentropic but under some circumstances the ow may beadiabatic and p = p() (problem 1.10).In an ideal gas the molecules do not interact except by elastic collisions, and the volumeof the molecules is presumed to be negligible compared to the total volume they occupy.1.5 Thermodynamic Relations 21The internal energy of the gas then depends only on temperature, and not on density. Asimple ideal gas is an ideal gas for which the heat capacity is constant, so thatI = cT, (1.85)where c is a constant. Using this and the conventional ideal gas equation, p = RT (whereR is also constant), along with the fundamental thermodynamic relation (1.67), we can inferthe fundamental equation of state; however, we will defer that until we discuss potentialtemperature in section 1.6.1. A general ideal gas also obeys p = RT, but it has heatcapacities that may be a function of temperature (but only of temperature see problem1.12).Internal energy and specic heatsWe can obtain some useful relations between the internal energy and specic heat capacities,and some useful estimates of their values, by some simple manipulations of the fundamentalthermodynamic relation. Assuming that the composition of the uid is constant (1.67) isT d = dI +pd, (1.86)so that, taking I to be a function of and T,T d =_ IT_dT +__ I_T+p_d. (1.87)From this, we see that the heat capacity at constant volume (i.e., constant ) cv is given bycv T_T_=_ IT_. (1.88)Thus, c in (1.85) is equal to cv.Similarly, using (1.74) we haveT d = dh dp =_hT_pdT +__hp__dp. (1.89)The heat capacity at constant pressure, cp, is then given bycp T_T_p=_hT_p. (1.90)For later use, we dene the ratios cp/cv and R/cp.For an ideal gas h = I +RT = T(cv+R). But cp = (h/t)p, and hence cp = cv+R, and( 1)/ = . Statistical mechanics tells us that for a simple ideal gas the internal energyis equal to kT/2 per molecule, or RT/2 per unit mass, for each excited degree of freedom,where k is the Boltzmann constant and R the gas constant. The diatomic molecules N2and O2 that comprise most of our atmosphere have two rotational and three translationaldegrees of freedom, so that I 5RT/2, and so cv 5R/2 and cp 7R/2, both beingconstants. These are in fact very good approximations to the measured values for theEarths atmosphere, and give cp 103J kg1K1. The internal energy is simply cvT and theenthalpy is cpT. For a liquid, especially one like seawater that contains dissolved salts, nosuch simple relations are possible: the heat capacities are functions of the state of the uid,and the internal energy is a function of pressure (or density) as well as temperature.22 Chapter 1. Equations of Motion1.6 THERMODYNAMIC EQUATIONS FOR FLUIDSThe thermodynamic relations for example (1.67) apply to identiable bodies or systems;thus, the heat input aects the uid parcel to which it is applied, and we can apply thematerial derivative to the above thermodynamic relations to obtain equations of motion fora moving uid. But in doing so we make two assumptions.(i) That locally the uid is in thermodynamic equilibrium. This means that, although thethermodynamic quantities like temperature, pressure and density vary in space andtime, locally they are related by the thermodynamic relations such as the equation ofstate and Maxwells relations.(ii) That macroscopic uid motions are reversible and so not entropy producing. Thus, sucheects as the viscous dissipation of energy, radiation, and conduction may produceentropy whereas the macroscopic uid motion itself does not.The rst point requires that the temperature variation on the macroscopic scales must beslow enough that there can exist a volume that is small compared to the scale of macroscopicvariations, so that temperature is eectively constant within it, but that is also sucientlylarge to co