LECTURES ON DYNAMICAL METEOROLOGY Roger K. Smith Version: June 16, 2014 Contents 1 INTRODUCTION 5 1.1 Scales ................................... 6 2 EQUILIBRIUM AND STABILITY 9 3 THE EQUATIONS OF MOTION 16 3.1 Effectivegravity.............................. 16 3.2 TheCoriolisforce............................. 16 3.3 Euler’s equation in a rotating coordinate system . 18 3.4 Centripetalacceleration .. .. .. 19 3.5 Themomentumequation.. .. .. 20 3.6 TheCoriolisforce............................. 20 3.7 Perturbationpressure. .. .. .. 21 3.8 Scale analysis of the equation of motion . 22 3.9 Coordinate systems and the earth’s sphericity . 23 3.10 Scale analysis of the equations for middle latitude synoptic systems . 25 4 GEOSTROPHIC FLOWS 28 4.1 TheTaylor-ProudmanTheorem . 30 4.2 Blocking.................................. 34 4.3 Analogy between blocking and axial Taylor columns . 35 4.4 Stability of a rotating fluid . 38 4.5 Vortexflows: thegradientwindequation . 38 4.6 Theeffectsofstratification. 41 4.7 Thermaladvection ............................ 45 4.8 Thethermodynamicequation . 46 4.9 Pressurecoordinates ........................... 47 4.10 Thicknessadvection. .. .. .. 48 4.11 Generalizedthermalwindequation . 49 5 FRONTS, EKMAN BOUNDARY LAYERS AND VORTEX FLOWS 54 5.1 Fronts ................................... 54 5.2 Margules’model.............................. 54 5.3 Viscous boundary layers: Ekman’s solution . 59 2 CONTENTS 3 5.4 Vortexboundarylayers. .. .. .. 63 6 THE VORTICITY EQUATION FOR A HOMOGENEOUS FLUID 67 6.1 Planetary,orRossbyWaves . 68 6.2 Largescaleflowoveramountainbarrier . 74 6.3 Winddrivenoceancurrents . 75 6.4 Topographicwaves ............................ 79 6.5 Continentalshelfwaves. .. .. .. 81 7 THE VORTICITY EQUATION IN A ROTATING STRATIFIED FLUID 83 7.1 The vorticity equation for synoptic-scale atmospheric motions . ... 85 8 QUASI-GEOSTROPHIC MOTION 89 8.1 More on the approximated thermodynamic equation . 92 8.2 The quasi-geostrophic equation for a compressible atmosphere .... 93 8.3 Quasi-geostrophic flow over a bell-shaped mountain . 94 9 SYNOPTIC-SCALE INSTABILITY AND CYCLOGENESIS 99 9.1 The middle latitude ‘westerlies’ . 99 9.2 Availablepotentialenergy . 100 9.3 Baroclinic instability: the Eady problem . 102 9.4 Atwo-layermodel ............................ 109 9.4.1 No vertical shear, UT = 0, i.e., U1 = U3.............. 114 9.4.2 No beta effect (β = 0), finite shear (UT =0)........... 114 9.4.3 The general case, U = 0, β =0. ................ 115 T 9.5 The energetics of baroclinic waves . 116 9.6 Interpretation ............................... 117 9.7 Largeamplitudewaves . .. .. .. 118 9.8 The role of baroclinic waves in the atmosphere’s general circulation . 119 10 DEVELOPMENT THEORY 120 10.1 The isallobaric wind . 121 10.2 Confluenceanddiffluence. 121 10.3 Dinescompensation. 125 10.4 Sutcliffe’s development theory . 126 10.5Theomegaequation ........................... 132 11 MORE ON WAVE MOTIONS, FILTERING 134 11.1 Thenocturnallow-leveljet. 136 11.2 Inertia-gravitywaves . 140 11.3Filtering.................................. 143 CONTENTS 4 12 GRAVITY CURRENTS, BORES AND OROGRAPHIC FLOW 146 12.1 Bernoulli’stheorem ............................ 147 12.2Flowforce................................. 150 12.3 Theoryofhydraulicjumps,orbores. 151 12.4 Theoryofgravitycurrents . 153 12.5Thedeepfluidcase ............................ 156 12.6 Flowoverorography . .. .. .. 157 13 AIR MASS MODELS OF FRONTS 159 13.1 ThetranslatingMargules’model . 161 13.2 Davies’(Boussinesq)model . 166 14 FRONTS AND FRONTOGENESIS 169 14.1 Thekinematicsoffrontogenesis . 169 14.2 Thefrontogenesisfunction . 174 14.3 Dynamicsoffrontogenesis . 178 14.4 Quasi-geostrophicfrontogenesis . 181 14.5 Semi-geostrophicfrontogenesis . 184 14.6 Special specific models for frontogenesis . 186 14.7 Frontogenesisatupperlevels. 191 14.8 Frontogenesisinshear . 191 15 GENERALIZATION OF GRADIENT WIND BALANCE 196 15.1 The quasi-geostrophic approximation . 197 15.2 Thebalanceequations . 199 15.3 TheLinearBalanceEquations . 200 A ALGEBRAIC DETAILS OF THE EADY PROBLEM SOLUTION202 B APPENDIX TO CHAPTER 10 205 C POISSON’S EQUATION 207 Chapter 1 INTRODUCTION There are important differences in approach between the environmental sciences such as meteorology, oceanography and geology, and the laboratory sciences such as physics, chemistry and biology. Whereas the experimental physicist will endeavour to isolate a phenomenon and study it under carefully controlled conditions in the laboratory, the atmospheric scientist and oceanographer have neither the ability to control a phenomenon under study, nor to study it in isolation from other phenomena. Furthermore, meteorological and oceanographical analysis tend to be concerned with the assimilation of a body of data rather than with the proof of specific laws. Besides the problems of instrument error and inherent inaccuracies in the obser- vational method (e.g. measurement of wind by tracking balloons), the data available for the study of a particular atmospheric or oceanographic phenomenon is frequently too sparse in both space and time. For example, most radiosonde and rawin (radar wind) stations are land based, and even then are often five hundred kilometers or more apart and make temperature and/or wind soundings only a few times a day, some only once. To illustrate this point the regular upper air observing station network in both hemispheres is shown in Fig. 1.1. Even more important, some ob- servations may be unrepresentative of the scale of the phenomenon being analyzed. If, for example, a radiosonde is released too close to, or indeed, in the updraught of a thunderstorm, it cannot be expected to provide data which is representative of the air mass in which the thunderstorm is embedded. Whilst objective analysis tech- niques are available to assist in the interpretation of data, meteorological analyses continue to depend in varying degrees on the experience and theoretical knowledge of the analyst. In the study of meteorology we can identify two extremes of approach: the de- scriptive approach, the first aim of which is to provide a qualitative interpretation of a large fraction of the data, with less attention paid to strict dynamical consistency; and the theoretical approach which is concerned mainly with self-consistency of some physical processes (ensured by the use of appropriate equations) and less immedi- ately with an accurate and detailed representation of the observations. Normally, progress in understanding comes from a blend of these approaches; descriptive study 5 CHAPTER 1. INTRODUCTION 6 begins with the detailed data and proceeds towards dynamical consistency whereas the theory is always dynamically consistent and proceeds towards explaining more of the data. In this way, the two approaches complement each other; more or less qual- itative data can be used to identify important processes that theory should model and theoretical models suggest more appropriate ways of analyzing and interpreting the data. Since the ocean, like the atmosphere, is a rotating stratified fluid, atmospheric and oceanic motions have many features in common and although this course is primarily about atmospheric dynamics, from time to time we shall discuss oceanic motions as well. 1.1 Scales The atmosphere and oceans are complex fluid systems capable of supporting many different types of motion on a very wide range of space and time scales. For example, the huge cyclones and anticyclones of middle latitudes have horizontal length scales of the order of a thousand kilometres or more and persist for many days. Small cumulus clouds, however, have dimensions of about a kilometre and lifetimes of a few tens of minutes. Short surface waves on water have periods measured in seconds, while the slopping around (or seiching) of a large lake has a period measured in hours and that of the Pacific Ocean has a period measured in days. Other types of wave motion in the ocean have periods measured in months. In the atmosphere, there exist types of waves that have global scales and periods measured in days, the so-called planetary-, or Rossby waves, whereas gravity waves, caused, for example, by the airflow over mountains or hills, have wavelengths typically on the order of kilometres and periods of tens of minutes. In order to make headway in the theoretical study of atmospheric and oceanic motions, we must begin by identifying the scales of motion in which we are interested, in the hope of isolating the mechanisms which are important at those scales from the host of all possible motions. In this course we shall attempt to discuss a range of phenomena which combine to make the atmosphere and oceans of particular interest to the fluid dynamicist as well as the meteorologist, oceanographer, or environmental scientist. Textbooks The recommended reference text for the course is: J. R. Holton: An Introduction to Dynamic Meteorology 3rd Edition (1992) by • Academic Press. Note that there is now a 4th addition available, dated 2004. I shall frequently refer to this book during the course. CHAPTER 1. INTRODUCTION 7 Four other books that you may find of some interest are: A. E. Gill: Atmosphere-Ocean Dynamics (1982) by Academic Press • J. T. Houghton: The Physics of Atmospheres 2nd Edition (1986) by Cambridge • Univ. Press J. Pedlosky: Geophysical Fluid Dynamics (1979) by Springer-Verlag • J. M. Wallace and P. V. Hobbs: Atmospheric Science: An Introductory Survey • (1977) by Academic Press. Note that there is now a second addition available, dated 2006. I refer you especially