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LECTURES ON DYNAMICALMETEOROLOGYRoger K. SmithVersion: June 16, 2014

Contents1 INTRODUCTION1.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .562 EQUILIBRIUM AND STABILITY93 THE EQUATIONS OF MOTION3.1 Effective gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2 The Coriolis force . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3 Euler’s equation in a rotating coordinate system . . . . . . . . . . .3.4 Centripetal acceleration . . . . . . . . . . . . . . . . . . . . . . . .3.5 The momentum equation . . . . . . . . . . . . . . . . . . . . . . . .3.6 The Coriolis force . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.7 Perturbation pressure . . . . . . . . . . . . . . . . . . . . . . . . . .3.8 Scale analysis of the equation of motion . . . . . . . . . . . . . . . .3.9 Coordinate systems and the earth’s sphericity . . . . . . . . . . . .3.10 Scale analysis of the equations for middle latitude synoptic systems4 GEOSTROPHIC FLOWS4.1 The Taylor-Proudman Theorem . . . . . .4.2 Blocking . . . . . . . . . . . . . . . . . . .4.3 Analogy between blocking and axial Taylor4.4 Stability of a rotating fluid . . . . . . . . .4.5 Vortex flows: the gradient wind equation .4.6 The effects of stratification . . . . . . . . .4.7 Thermal advection . . . . . . . . . . . . .4.8 The thermodynamic equation . . . . . . .4.9 Pressure coordinates . . . . . . . . . . . .4.10 Thickness advection . . . . . . . . . . . . .4.11 Generalized thermal wind equation . . . . . . . . . . . .columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1616161819202021222325.2830343538384145464748495 FRONTS, EKMAN BOUNDARY LAYERS AND VORTEX FLOWS 545.1 Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Margules’ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Viscous boundary layers: Ekman’s solution . . . . . . . . . . . . . . . 592

3CONTENTS5.4 Vortex boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . .6 THE VORTICITY EQUATION FOR A HOMOGENEOUS FLUID6.1 Planetary, or Rossby Waves . . . . . . . . . . . . . . . . . . . . . . .6.2 Large scale flow over a mountain barrier . . . . . . . . . . . . . . . .6.3 Wind driven ocean currents . . . . . . . . . . . . . . . . . . . . . . .6.4 Topographic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.5 Continental shelf waves . . . . . . . . . . . . . . . . . . . . . . . . . .636768747579817 THE VORTICITY EQUATION IN A ROTATING STRATIFIEDFLUID837.1 The vorticity equation for synoptic-scale atmospheric motions . . . . 858 QUASI-GEOSTROPHIC MOTION898.1 More on the approximated thermodynamic equation . . . . . . . . . . 928.2 The quasi-geostrophic equation for a compressible atmosphere . . . . 938.3 Quasi-geostrophic flow over a bell-shaped mountain . . . . . . . . . . 949 SYNOPTIC-SCALE INSTABILITY AND CYCLOGENESIS9.1 The middle latitude ‘westerlies’ . . . . . . . . . . . . . . . . . . . .9.2 Available potential energy . . . . . . . . . . . . . . . . . . . . . . .9.3 Baroclinic instability: the Eady problem . . . . . . . . . . . . . . .9.4 A two-layer model . . . . . . . . . . . . . . . . . . . . . . . . . . .9.4.1 No vertical shear, UT 0, i.e., U1 U3 . . . . . . . . . . . . .9.4.2 No beta effect (β 0), finite shear (UT 6 0). . . . . . . . . .9.4.3 The general case, UT 6 0, β 6 0. . . . . . . . . . . . . . . .9.5 The energetics of baroclinic waves . . . . . . . . . . . . . . . . . . .9.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.7 Large amplitude waves . . . . . . . . . . . . . . . . . . . . . . . . .9.8 The role of baroclinic waves in the atmosphere’s general 12112112512613210 DEVELOPMENT THEORY10.1 The isallobaric wind . . . . .10.2 Confluence and diffluence . . .10.3 Dines compensation . . . . . .10.4 Sutcliffe’s development theory10.5 The omega equation . . . . .11 MORE ON WAVE MOTIONS,11.1 The nocturnal low-level jet . .11.2 Inertia-gravity waves . . . . .11.3 Filtering . . . . . . . . . . . .FILTERING134. . . . . . . . . . . . . . . . . . . . . . 136. . . . . . . . . . . . . . . . . . . . . . 140. . . . . . . . . . . . . . . . . . . . . . 143.

4CONTENTS12 GRAVITY CURRENTS, BORES AND OROGRAPHIC FLOW12.1 Bernoulli’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .12.2 Flow force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.3 Theory of hydraulic jumps, or bores. . . . . . . . . . . . . . . . . .12.4 Theory of gravity currents . . . . . . . . . . . . . . . . . . . . . . .12.5 The deep fluid case . . . . . . . . . . . . . . . . . . . . . . . . . . .12.6 Flow over orography . . . . . . . . . . . . . . . . . . . . . . . . . .14614715015115315615713 AIR MASS MODELS OF FRONTS15913.1 The translating Margules’ model . . . . . . . . . . . . . . . . . . . . 16113.2 Davies’ (Boussinesq) model . . . . . . . . . . . . . . . . . . . . . . . 16614 FRONTS AND FRONTOGENESIS14.1 The kinematics of frontogenesis . . . .14.2 The frontogenesis function . . . . . . .14.3 Dynamics of frontogenesis . . . . . . .14.4 Quasi-geostrophic frontogenesis . . . .14.5 Semi-geostrophic frontogenesis . . . . .14.6 Special specific models for frontogenesis14.7 Frontogenesis at upper levels . . . . . .14.8 Frontogenesis in shear . . . . . . . . .16916917417818118418619119115 GENERALIZATION OF GRADIENT WIND BALANCE19615.1 The quasi-geostrophic approximation . . . . . . . . . . . . . . . . . . 19715.2 The balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . 19915.3 The Linear Balance Equations . . . . . . . . . . . . . . . . . . . . . . 200A ALGEBRAIC DETAILS OF THE EADY PROBLEM SOLUTION202B APPENDIX TO CHAPTER 10205C POISSON’S EQUATION207

Chapter 1INTRODUCTIONThere are important differences in approach between the environmental sciencessuch as meteorology, oceanography and geology, and the laboratory sciences such asphysics, chemistry and biology. Whereas the experimental physicist will endeavourto isolate a phenomenon and study it under carefully controlled conditions in thelaboratory, the atmospheric scientist and oceanographer have neither the ability tocontrol a phenomenon under study, nor to study it in isolation from other phenomena.Furthermore, meteorological and oceanographical analysis tend to be concerned withthe 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 observational method (e.g. measurement of wind by tracking balloons), the data availablefor the study of a particular atmospheric or oceanographic phenomenon is frequentlytoo sparse in both space and time. For example, most radiosonde and rawin (radarwind) stations are land based, and even then are often five hundred kilometers ormore 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 stationnetwork in both hemispheres is shown in Fig. 1.1. Even more important, some observations 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 ofa thunderstorm, it cannot be expected to provide data which is representative of theair mass in which the thunderstorm is embedded. Whilst objective analysis techniques are available to assist in the interpretation of data, meteorological analysescontinue to depend in varying degrees on the experience and theoretical knowledgeof the analyst.In the study of meteorology we can identify two extremes of approach: the descriptive approach, the first aim of which is to provide a qualitative interpretation ofa 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 somephysical processes (ensured by the use of appropriate equations) and less immediately with an accurate and detailed representation of the observations. Normally,progress in understanding comes from a blend of these approaches; descriptive study5

CHAPTER 1. INTRODUCTION6begins with the detailed data and proceeds towards dynamical consistency whereasthe theory is always dynamically consistent and proceeds towards explaining more ofthe data. In this way, the two approaches complement each other; more or less qualitative data can be used to identify important processes that theory should modeland theoretical models suggest more appropriate ways of analyzing and interpretingthe data.Since the ocean, like the atmosphere, is a rotating stratified fluid, atmosphericand oceanic motions have many features in common and although this course isprimarily about atmospheric dynamics, from time to time we shall discuss oceanicmotions as well.1.1ScalesThe atmosphere and oceans are complex fluid systems capable of supporting manydifferent 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 scalesof the order of a thousand kilometres or more and persist for many days. Smallcumulus clouds, however, have dimensions of about a kilometre and lifetimes of afew 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 inhours and that of the Pacific Ocean has a period measured in days. Other typesof 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, theso-called planetary-, or Rossby waves, whereas gravity waves, caused, for example,by the airflow over mountains or hills, have wavelengths typically on the order ofkilometres and periods of tens of minutes.In order to make headway in the theoretical study of atmospheric and oceanicmotions, 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 fromthe host of all possible motions.In this course we shall attempt to discuss a range of phenomena which combineto make the atmosphere and oceans of particular interest to the fluid dynamicist aswell as the meteorologist, oceanographer, or environmental scientist.TextbooksThe recommended reference text for the course is: J. R. Holton: An Introduction to Dynamic Meteorology 3rd Edition (1992) byAcademic Press. Note that there is now a 4th addition available, dated 2004.I shall frequently refer to this book during the course.

CHAPTER 1. INTRODUCTION7Four 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 CambridgeUniv. 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 to Chapters 1-3 and 7-9 of Houghton’s book and Chapters3, 8 and 9 of Wallace and Hobbs (1977).

CHAPTER 1. INTRODUCTION8Figure 1.1: Location of upper air stations where measurements of temperature, humidity, pressure, and wind speed and direction are made as functions of height usingballoon-borne radiosondes. At most stations, full measurements are made twice daily,at 0000 and 1200 Greenwich mean time (GMT); at many stations, wind measurements are made also at 0600 and 1800 GMT, from (Phillips, 1970).

Chapter 2EQUILIBRIUM ANDSTABILITYConsider an atmosphere in hydrostatic equilibrium at rest1 . The pressure p(z) atheight z is computed from an equation which represents the fact that p(z) differsfrom p(z δz) by the weight of air in the layer from z to z δz; i.e., in the limit asdz 0,dp gρ,(2.1)dzρ(z) being the density of air at height z. Using the perfect gas equation, p ρRT , itfollows that Z zdz ′,(2.2)p(z) ps exp ′0 H(z )where H(z) RT (z)/g is a local height scale and ps p(0) is the surface pressure.Remember, that for dry air, T is the absolute temperature; for moist air it is thevirtual temperature in deg. K. Also pressure has units of Pascals (Pa) in Eq. (2.1),although meteorologists often quote the pressure in hPa (100 Pa) or millibars 2 (mb).At this point you should try exercises (2.1)-(2.4).The potential temperature θ, is defined as the temperature a parcel of air wouldhave if brought adiabatically to a pressure of 1000 mb; i.e., κ1000θ T,(2.3)pwhere p is the pressure in mb and κ 0.2865. It is easy to calculate θ knowing pand T if one has a calculator with the provision for evaluating y x . It is important toremember to convert T to degrees K. Equation (2.3) is derived as follows. Consider1It is not essential to assume no motion; we shall see later that hydrostatic balance is satisfiedvery accurately in the motion of large-scale atmospheric systems.2The conversion factor is easy: 1 mb 1 hPa.9

CHAPTER 2. EQUILIBRIUM AND STABILITY10a parcel of air with temperature T and pressure p. Suppose that it is given a smallamount of heat dq per unit mass and as a consequence its temperature and pressurechange by amounts dT and dp, respectively. The first law of thermodynamics givesdq cp dT αdp,(2.4)where α 1/ρ is the specific volume (volume per unit mass) and cp is the specificheat at constant pressure. Using the perfect gas equation to eliminate α, Eq. (2.4)can be written,cp dTdpdq ,RTR Tp(2.5)In an adiabatic process there is no heat input, i.e., dq 0. Then Eq. (2.5) canbe integrated to giveκ ln p ln T constant,(2.6)where κ R/cp . Since θ is defined as the value of T when p 1000 mb, the constantin Eq. (2.6) is equal to κ ln 1000 ln θ, whereupon κ ln(1000/p) ln(θ/T ). Equation(2.3) follows immediately.Since a wide range of atmospheric motions are approximately adiabatic3 , thepotential temperature is an important thermodynamic variable because for suchmotions it is conserved following parcels of air. In contrast the temperature may notbe, as in the case of a parcel of air which experiences a pressure change due to verticalmotion. The potential temperature is also a fundamental quantity for characterizingthe stability of a layer of air as we now show.Suppose that a parcel of air at A is displaced adiabatically through a height dz toposition B (see Fig. 2.1). Its temperature and pressure will change, but its potentialtemperature will remain constant, equal to its original value θ(z) when at A. Sincethe pressure at level B is p(z dz), the temperature of the parcel at B will be givenby κp(z dz)TB θ(z).(2.7)1000The temperature of the parcel’s environment at level B is κp(z dz)T (z dz) θ(z dz).1000(2.8)The buoyancy force per unit mass, F , experienced by the parcel at B is, accordingto Archimedes’ principle,3Such motions are frequently referred to as isentropic. This is because specific entropy changesds are related to heat changes dq by the formula ds dq/T . Using Eq. (2.5) it follows readily thatds cp ln θ; in other words, constant entropy s implies constant potential temperature

CHAPTER 2. EQUILIBRIUM AND STABILITY11Figure 2.1: Schematic of a vertical parcel displacement.weight of air weight of air in parcelmass of air in parcelgρ(z dz)V gρB V ρB V,F where V is the volume of the parcel at level B and ρB its density at level B. Cancelling V and using the perfect gas law ρ p(z dz)/RT , the above expressiongivesF gTB T (z dz),T (z dz)and using Eqs. (2.7) and (2.8), this becomesF gθ(z) θ(z dz).θ(z dz)This expression can be written approximately asg dθF dz N 2 dz.(2.9) θ dzEquation (2.9) defines the Brunt-Väisälä frequency or buoyancy frequency, N.If the potential temperature is uniform with height, the displaced parcel experiences no buoyancy force and will remain at its new location. Such a layer of air isneutrally stable. If the potential temperature increases with height, a parcel displacedupwards (downwards) experiences a negative (positive) restoring force and will tendto return to its equilibrium level. Thus dθ/dz 0 characterizes a stable layer of air.In contrast, if the potential temperature decreases with height, a displaced parcel

CHAPTER 2. EQUILIBRIUM AND STABILITY12would experience a force in the direction of the displacement; clearly an unstablesituation. Substantial unstable layers are never observed in the atmosphere becauseeven a slight degree of instability results in convective overturning until the layerbecomes neutrally stable.During the day, when the ground is heated by solar radiation, the air layers nearthe ground are constantly being overturned by convection to give a neutrally stablelayer with a uniform potential temperature. At night, if the wind is not too strong,and especially if there is a clear sky and the air is relatively dry, a strong radiationinversion forms in the lowest layers. An inversion is one in which not only thepotential temperature, but also the temperature increases with height; such a layeris very stable.The lapse rate Γ is defined as the rate of decrease of temperature with height, dT /dz. The lapse rate in a neutrally stable layer is a constant, equal to about 10K km 1 , or 1 K per 100 m; this is called the dry adiabatic lapse rate (see exercise2.5 below). It is also the rate at which a parcel of dry air cools (warms) as it rises(subsides) adiabatically in the atmosphere. However, if a rising air parcel becomessaturated at some level, the subsequent rate at which it cools is less than the dryadiabatic lapse rate because condensation leads to latent heat release.The Brunt-Väisälä frequency N may be interpreted as follows. Suppose a parcelof air of mass m in a stable layer of air is displaced vertically through a distance ξ.According to Eq. (2.9) it will experience a restoring force equal to mN 2 ξ. Hence,assuming it retains its identity during its displacement without any mixing with itsenvironment, its equation of motion is simply md2 ξ/dt2 mN 2 ξ; in other words, itwill execute simple harmonic motion with frequency N and period 2π/N. It is notsurprising that N turns out to be a key parameter in the theory of gravity wavesin the atmosphere. Since for a fixed displacement, the restoring force increases withN, the latter quantity can be used as a measure of the degree of stability in anatmospheric layer. Note that for an unstable layer, N is imaginary and instabilityis reflected in the existence of an exponentially growing solution to the displacementequation for a parcel.An example of the variation of potential temperature with height in the atmosphere is shown in Fig. 2.2a. The radiosonde sounding on which it is based wasmade at about 0500 h local time at Burketown in northern Queensland, Australiaon a day in October. The principal features are:(i) a low-level stable layer between the surface and about 1.5 km, a Brunt-Väisälä,or buoyancy period of about 6.3 minutes. This layer is probably a result ofan influx of cooler air at low levels by the sea breeze circulation during theprevious day, the profile being modified by radiative transfer overnight.(ii) a nearly neutral layer from 1.5 km to just above 4 km. This is presumably theremnant of the “well-mixed” layer caused by convective mixing over the landon the previous day. The layer is capped by a sharp inversion between about4

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 approac