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Lectures on Dynamical Meteorology

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LECTURES ON DYNAMICAL METEOROLOGY Roger K. Smith Version: December 11, 2007 Contents 1 INTRODUCTION 5 1.1 Scales . 6 2 EQUILIBRIUM AND STABILITY 9 3 THE EQUATIONS OF MOTION 16 3.1 E®ective gravity . 16 3.2 The Coriolis force . 16 3.3 Euler's equation in a rotating coordinate system . 18 3.4 Centripetal acceleration . 19 3.5 The momentum equation . 20 3.6 The Coriolis force . 20 3.7 Perturbation pressure . 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 The Taylor-Proudman Theorem . 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 Vortex flows: the gradient wind equation . 38 4.6 The e®ects of strati¯cation . 41 4.7 Thermal advection . 45 4.8 The thermodynamic equation . 46 4.9 Pressure coordinates . 47 4.10 Thickness advection . 48 4.11 Generalized thermal wind equation . 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 Vortex boundary layers . 63 6 THE VORTICITY EQUATION FOR A HOMOGENEOUS FLUID 67 6.1 Planetary, or Rossby Waves . 68 6.2 Large scale flow over a mountain barrier . 74 6.3 Wind driven ocean currents . 75 6.4 Topographic waves . 79 6.5 Continental shelf waves . 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 Available potential energy . 100 9.3 Baroclinic instability: the Eady problem . 102 9.4 A two-layer model . 109 9.4.1 No vertical shear, UT = 0, i.e., U1 = U3. 114 9.4.2 No beta e®ect (¯ = 0), ¯nite shear (UT 6= 0). 114 9.4.3 The general case, UT 6= 0, ¯ 6= 0. 115 9.5 The energetics of baroclinic waves . 116 9.6 Interpretation . 117 9.7 Large amplitude waves . 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 Confluence and di²uence . 121 10.3 Dines compensation . 125 10.4 Sutcli®e's development theory . 126 10.5 The omega equation . 132 11 MORE ON WAVE MOTIONS, FILTERING 134 11.1 The nocturnal low-level jet . 136 11.2 Inertia-gravity waves . 140 11.3 Filtering . 143 CONTENTS 4 12 GRAVITY CURRENTS, BORES AND OROGRAPHIC FLOW 146 12.1 Bernoulli's theorem . 147 12.2 Flow force . 150 12.3 Theory of hydraulic jumps, or bores. 151 12.4 Theory of gravity currents . 153 12.5 The deep fluid case . 156 12.6 Flow over orography . 157 13 AIR MASS MODELS OF FRONTS 159 13.1 The translating Margules' model . 161 13.2 Davies' (Boussinesq) model . 166 14 FRONTS AND FRONTOGENESIS 169 14.1 The kinematics of frontogenesis . 169 14.2 The frontogenesis function . 174 14.3 Dynamics of frontogenesis . 178 14.4 Quasi-geostrophic frontogenesis . 181 14.5 Semi-geostrophic frontogenesis . 184 14.6 Special speci¯c models for frontogenesis . 186 14.7 Frontogenesis at upper levels . 191 14.8 Frontogenesis in shear . 191 15 GENERALIZATION OF GRADIENT WIND BALANCE 196 15.1 The quasi-geostrophic approximation . 197 15.2 The balance equations . 199 15.3 The Linear Balance Equations . 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 di®erences 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 speci¯c 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 ¯ve 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 ¯rst 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 strati¯ed 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 di®erent 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 Paci¯c 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: ² 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 ¯nd of some interest are: ² A. E. Gill: Atmosphere-Ocean Dynamics (1982) by Academic Press ² J. T. Houghton: The Physics of Atmospheres 2nd Edition (1986) by Cambridge Univ. Press ² J. Pedlosky: Geophysical Fluid Dynamics (1979) by Springer-Verlag ² 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 to Chapters 1-3 and 7-9 of Houghton's book and Chapters 3, 8 and 9 of Wallace and Hobbs (1977). CHAPTER 1. INTRODUCTION 8 Figure 1.1: Location of upper air stations where measurements of temperature, hu- midity, pressure, and wind speed and direction are made as functions of height using balloon-borne radiosondes. At most stations, full measurements are made twice daily, at 0000 and 1200 Greenwich mean time (GMT); at many stations, wind measure- ments are made also at 0600 and 1800 GMT, from (Phillips, 1970).
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