Advanced Lectures on Dynamical Meteorology

ADVANCED LECTURES ON DYNAMICAL METEOROLOGY Roger K. Smith Fassung: WS 02/03 Contents 1 INTRODUCTION 3 1.1Somerevision........................... 4 1.1.1 Themomentumequationinheightcoordinates..... 4 1.2TheBoussinesqapproximation.................. 5 1.3Theanelasticapproximation................... 5 1.4Pressurecoordinates....................... 6 1.5Otherpressurecoordinatesystems................ 7 1.6Isentropiccoordinates....................... 7 1.7Sigmacoordinates......................... 7 2 SMALL-AMPLITUDE WAVES IN A STABLY-STRATIFIED ROTATING ATMOSPHERE 9 2.1 Boundary waves .......................... 17 2.2Theeffectsofshear........................ 19 2.3Theenergyequation....................... 20 2.4Simplifiedsolutionsandfilteredequations........... 21 2.5 ‘Sound-proofing’ the equations .................. 21 2.6Thehydrostaticapproximation................. 23 2.7 Sound-proofed hydrostatic approximation . ......... 24 2.8 Variation of mean density with height; the equivalent incom- pressibleatmosphere....................... 24 2.9Geostrophicmotion........................ 24 3 WAVES ON MOVING STRATIFIED FLOWS 29 3.1Freewaves............................. 30 3.2Forcedwaves........................... 31 3.3Mountainwaves.......................... 34 3.4LinearTheory........................... 35 3.5Flowoverisolatedtopography.................. 39 3.6Trappedleewaves......................... 45 1 4 ENERGETICS OF WAVES ON STRATIFIED SHEAR FLOWS 55 4.1Slowlyvaryingwavetrainsorwavepackets........... 60 5 SHEARING INSTABILITY 66 5.1 Helmholtz instability . .................... 66 5.2 Kelvin-Helmholtz instability ................... 68 5.3 The Richardson number criterion for Kelvin-Helmholz instability 71 5.4Anenergycriterion........................ 73 5.5 Instability of stratified shear flows ................ 74 5.6Howard’ssemi-circletheorem.................. 75 6 QUASI-GEOSTROPHIC WAVES 78 6.1AUnifiedTheory......................... 87 6.2 The baroclinic instability mechanism .............. 88 6.3 Perturbation energy equation for quasi-geostrophic waves . 92 6.4TheomegaequationintermsofPV............... 95 7 FRONTOGENESIS, SEMI-GEOSTROPHIC THEORY 97 7.1Frontogenesisinahorizontaldeformationfield.......... 97 7.2 Uniform potential vorticity model, q =constant.........100 7.3 Zero potential vorticity model, q ≡ 0...............102 7.4Thegeostrophicmomentumapproximation...........105 7.5TheEadyprobleminsemi-geostrophictheory..........111 8 SYMMETRIC BAROCLINIC INSTABILITY 116 8.1Richardsonnumbercriterion...................122 9 GEOSTROPHIC ADJUSTMENT 125 9.1 The nonrotating case (f =0)..................126 9.2 The rotating case (f =0)....................126 9.3Thetransientproblem......................128 9.4Energyconsiderations......................133 9.5Discussion.............................135 9.6 More on the Rossby radius of deformation, LR .........136 9.7Balancedadjustment.......................137 10 VERTICAL COORDINATE TRANSFORMATIONS 145 10.1 Possible choices for ε .......................146 10.2 Isentropic coordinates ε = θ ...................147 10.3Thehorizontalmomentumequation...............147 10.4Theverticalmomentumequation................148 10.5Thecontinuityequation.....................148 2 10.6Thevorticityequation......................150 A Derivation of Eq. (??) 152 B Poisson Equation 154 3 PREFACE These notes are an outgrowth of various advanced courses on dynamical meteorology given over a period of years at Monash University in Australia and the University of Munich. They are an extension of my earlier set ‘Lectures on Dynamical Meteorology’, to which frequent reference is made. The notes include in places material adapted from course notes kindly given to me by Dr. John Green, formerly of Imperial College, London and Dr. Michael McIntyre from the University of Cambridge. I gratefully acknowledge these two colleagues for providing me with their unpublished lecture material. I would like to thank Anna-Riitta Järvinen for her care and patience in preparing the typescript and to the many people who have pointed out errors in previous versions. Roger Smith, Munich, August 2002. 4 Chapter 1 INTRODUCTION Dynamical meteorology concerns itself with the theoretical study of atmospheric motion. It aims to provide an understanding of such motion as well as a rational basis for the prediction of atmospheric events, including short and medium weather prediction and the forecasting of climate. The atmosphere is an extremely complex system involving motions on a very wide range of space and time scales. As a result, the dynamic and thermodynamic equations which describe the motions are too general to be solved easily. Thus, invariably, they have solutions representing phenomena that may not be of interest in the study of a particular problem: for example, we do not believe that acoustic (sound) waves are important in cyclogenesis, but such motions are described by the general equations of motion. As in earlier courses (Smith, 1992; Lectures on Modern Synoptic Mete- orology, henceforth referred to as SM; Smith, 2000; Lectures on Dynami- cal Meteorology, henceforth referred to as DM) we attempt to reduce the complexity of the equations by scaling whilst aiming to retain a reasonably accurate description of motion within certain limits of temporal and spatial scales. In doing this we need first to ascertain the essential physical aspects of the motion we hope to study. Then we seek ways of making approximations that are consistent in all equations by using quantitative estimates based on our physical perception of the problem. Finally, we need to determine for what scales the estimates accurately reflect physical reality. It is necessary to be aware of the various types of waves motion which can occur in the atmosphere. Certain wave types play an important role in atmospheric dynamics and they may propagate significant amounts of energy from one place to another in a manner analogous to the propagation of energy by electromagnetic waves. Moreover, because waves are solutions of the equations, they will appear when the equations are integrated numerically. Some of these wave types cause difficulties in attempts to make numerical 5 weather predictions and we shall seek ways to modify the equations in order to filter them out; an example is a modification that makes the equations ‘sound-proof’ discussed in Chapter 2. Under some circumstances, waves may grow rapidly in amplitude, often as a result of instability. For example, Kelvin-Helmholtz instability may arise when a strong vertical shear occurs in the neighbourhood of large stable density gradient such as an inversion layer; this is one mechanism thought to be responsible for clear air turbu- lence (CAT). Another example is baroclinic instability, a process responsible for extra-tropical cyclogenesis; this instability has consequences for the pre- dictability of the atmosphere. Examples of waves we shall study are listed in Table 1. Table 1.1: General wave types in the atmosphere. Wave type Speed controlled by: acoustic waves temperature gravity waves static stability inertial waves Coriolis forces Rossby waves latitudinal variation of the Coriolis parameter 1.1 Some revision 1.1.1 The momentum equation in height coordinates The momentum equation for a rotating stratified fluid may be written as Du 1 + fk ∧ u = − ∇p + σk − D, (1.1) Dt ρ where f =2Ωsinφ is the Coriolis parameter, φ is the latitude, p = pT − p0(z) is the perturbation pressure, pT is the total pressure, ρ0(z) is the reference pressure; this satisfies the hydrostatic equation: dp0/dz = −gρ0(z), σ = g(ρ0 − ρ)/ρ is the buoyancy force per unit mass, and D is the frictional force per unit mass. 6 A further discussion of Eq. (1.1) is given in DM Chapter 3. Recall that p and σ are NOT uniquely defined, being dependent on the arbitrarily chosen definition of the reference density and pressure. 1.2 The Boussinesq approximation This is an approximation which enables considerable simplification to the equations for a stably stratified fluid when density variations are not too large. In it, density variations are considered only in as much as they give rise to buoyancy forces in 1.1). In particular, 1/ρ in (1.1) is set equal to 1/ρ,aconstantρ, ρ being, for example, the average density over the whole flow domain. Also σ ≈ g(ρ0(z) − ρ)/ρ, ρ0(z) being the horizontal average of r at height z, or the ambient density, for example. Density variations are neglected also in the continuity equation which takes the approximate form ∇·u =0. 1.3 The anelastic approximation In meteorology, the Boussinesq approximation is sometimes too restrictive. For example, in motions which occupy the entire depth of the troposphere, density variations in the vertical are considerable. The density near the tropopause is only about one quarter to one fifth of that at the surface. In such circumstances the anelastic approximation is more accurate. In this, the effect of the decrease in density with height on the fluid inertia is taken −1 −1 into account by setting ρ ≈ ρ0(z) , and the approximate forms of the buoyancy and of the continuity equation are σ = g(ρ0(z) − ρ)/ρ0(z), (1.2) and ∇·(ρ0(z)u)=0. (1.3) The anelastic approximation is best handled by a change of pressure variable to π =(p/p)κ, the so-called Exner function,wherep = 1000 mb. This is especially the case for an atmosphere which is close to adiabatic. For further information see: Ogura and Phillips (1962), Dutton and Fichtl (1969), Lipps and Helmer (1982, 1985), Durran (1989, 1990), Fiedler (1990), Lipps (1990),

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