Fluid Dynamics of the Mid-Latitude Atmosphere

Having gained mathematics degrees from Cambridge and spent some post-doc years in the USA, Brian Hoskins has been at the University of Reading for more than 40 years, being made a professor in 1981, and also more recently has led a climate institute at Imperial College London.  His international activities have included being President of IAMAS and Vice-Chair of the JSC for WCRP. He is a member of the science academies of the UK, Europe, USA and China, he has received the top awards of both the Royal and American Meteorological Societies, the Vilhelm Bjerknes medal of the EGU and the Buys Ballot Medal, and he was knighted in 2007.From a background in physics and astronomy, Ian James worked in the geophysical fluid dynamics laboratory of the Meteorological Office before joining the University of Reading in 1979. During his 31 years in the Reading meteorology department, he has taught courses in dynamical meteorology and global atmospheric circulation. In 1998, he was awarded the Buchan Prize of the Royal Meteorological Society for his work on low frequency atmospheric variability. He has been President of the Dynamical Meteorology Commission of IAMAS, vice president of the Royal Meteorological Society, and currently edits the journal Atmospheric Science Letters. He now serves as an Anglican priest in Cumbria.

• Series foreword ix Preface xiSelect bibliography xvThe authors xix1 Observed flow in the Earth’s midlatitudes 11.1 Vertical structure 11.2 Horizontal structure 41.3 Transient activity 111.4 Scales of motion 141.5 The Norwegian frontal model of cyclones 15Theme 1 Fluid dynamics of the midlatitude atmosphere 252 Fluid dynamics in an inertial frame of reference 272.1 Definition of fluid 272.2 Flow variables and the continuum hypothesis 292.3 Kinematics: characterizing fluid flow 302.4 Governing physical principles 352.5 Lagrangian and Eulerian perspectives 362.6 Mass conservation equation 382.7 First Law of Thermodynamics 402.8 Newton’s Second Law of Motion 412.9 Bernoulli’s Theorem 452.10 Heating and water vapour 473 Rotating frames of reference 533.1 Vectors in a rotating frame of reference 533.2 Velocity and Acceleration 553.3 The momentum equation in a rotating frame 563.4 The centrifugal pseudo-force 573.5 The Coriolis pseudo-force 593.6 The Taylor–Proudman theorem 614 The spherical Earth 654.1 Spherical polar coordinates 654.2 Scalar equations 674.3 The momentum equations 684.4 Energy and angular momentum 704.5 The shallow atmosphere approximation 734.6 The beta effect and the spherical Earth 745 Scale analysis and its applications 775.1 Principles of scaling methods 775.2 The use of a reference atmosphere 795.3 The horizontal momentum equations 815.4 Natural coordinates, geostrophic and gradient wind balance 835.5 Vertical motion 875.6 The vertical momentum equation 895.7 The mass continuity equation 915.8 The thermodynamic energy equation 925.9 Scalings for Rossby numbers that are not small 956 Alternative vertical coordinates 976.1 A general vertical coordinate 976.2 Isobaric coordinates 1006.3 Other pressure-based vertical coordinates 1036.4 Isentropic coordinates 1067 Variations of density and the basic equations 1097.1 Boussinesq approximation 1097.2 Anelastic approximation 1117.3 Stratification and gravity waves 1137.4 Balance, gravity waves and Richardson number 1157.5 Summary of the basic equation sets 1217.6 The energy of atmospheric motions 122Theme 2 Rotation in the atmosphere 1258 Rotation in the atmosphere 1278.1 The concept of vorticity 1278.2 The vorticity equation 1298.3 The vorticity equation for approximate sets of equations 1318.4 The solenoidal term 1328.5 The expansion/contraction term 1348.6 The stretching and tilting terms 1358.7 Friction and vorticity 1388.8 The vorticity equation in alternative vertical coordinates 1448.9 Circulation 1459 Vorticity and the barotropic vorticity equation 1499.1 The barotropic vorticity equation 1499.2 Poisson’s equation and vortex interactions 1519.3 Flow over a shallow hill 1559.4 Ekman pumping 1599.5 Rossby waves and the beta plane 1609.6 Rossby group velocity 1669.7 Rossby ray tracing 1709.8 Inflexion point instability 17210 Potential vorticity 17710.1 Potential vorticity 17710.2 Alternative derivations of Ertel’s theorem 18010.3 The principle of invertibility 18210.4 Shallow water equation potential vorticity 18611 Turbulence and atmospheric flow 18911.1 The Reynolds number 18911.2 Three-dimensional flow at large Reynolds number 19411.3 Two-dimensional flow at large Reynolds number 19611.4 Vertical mixing in a stratified fluid 20111.5 Reynolds stresses 203Theme 3 Balance in atmospheric flow 20912 Quasi-geostrophic flows 21112.1 Wind and temperature in balanced flows 21112.2 The quasi-geostrophic approximation 21512.3 Quasi-geostrophic potential vorticity 21912.4 Ertel and quasi-geostrophic potential vorticities 22113 The omega equation 22513.1 Vorticity and thermal advection form 22513.2 Sutcliffe Form 23113.3 Q-vector form 23313.4 Ageostrophic flow and the maintenance of balance 23813.5 Balance and initialization 24014 Linear theories of baroclinic instability 24514.1 Qualitative discussion 24514.2 Stability analysis of a zonal flow 24714.3 Rossby wave interpretation of the stability conditions 25614.4 The Eady model 26414.5 The Charney and other quasi-geostrophic models 27114.6 More realistic basic states 27514.7 Initial value problem 28115 Frontogenesis 29115.1 Frontal scales 29115.2 Ageostrophic circulation 29415.3 Description of frontal collapse 29915.4 The semi-geostrophic Eady model 30515.5 The confluence model 30715.6 Upper-level frontogenesis 30916 The nonlinear development of baroclinic waves 31116.1 The nonlinear domain 31116.2 Semi-geostrophic baroclinic waves 31216.3 Nonlinear baroclinic waves on realistic jets on the sphere 32016.4 Eddy transports and zonal mean flow changes 32316.5 Energetics of baroclinic waves 33217 The potential vorticity perspective 33717.1 Setting the scene 33717.2 Potential vorticity and vertical velocity 34017.3 Life cycles of some baroclinic waves 34217.4 Alternative perspectives 34617.5 Midlatitude blocking 35017.6 Frictional and heating effects 35218 Rossby wave propagation and potential vorticity mixing 36118.1 Rossby wave propagation 36118.2 Propagation of Rossby waves into the stratosphere 36318.3 Propagation through a slowly varying medium 36518.4 The Eliassen–Palm flux and group velocity 37018.5 Baroclinic life cycles and Rossby waves 37218.6 Variations of amplitude 37318.7 Rossby waves and potential vorticity steps 37518.8 Potential vorticity steps and the Rhines scale 381Appendices 389Appendix A: Notation 389Appendix B: Revision of vectors and vector calculus 393B.1 Vectors and their algebra 393B.2 Products of vectors 394B.3 Scalar fields and the grad operator 396B.4 The divergence and curl operators 397B.5 Gauss’ and Stokes’ theorems 398B.6 Some useful vector identities 401Index 403