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A Diagnostic Model for Initial Winds in Primitive

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A Diagnostic Model for Initial Winds in Primitive 1 1 A DIAGNOSTIC MODEL FOR INITIAL WINDS IN PRIMITIVE EQUATIONS FORECASTS by Richard Asselin Department of Meteorology, Ph. D. ABSTRACT The set of primitive meteorologicai equations reduces to a single second order partial differential cquation when the accel- eration is replaced by the derivative of the geostrophic wind in the equation.of motion. From this equation a diagnostic three-dimensional wind field can be computed when the pressure distribution is known. Calculations for a barotropic atmosphere are performed fir st; the rotational part of the wind is found to be similar to that from the classical balance equation and the divergent part agrees well with synoptic experience. A forecast thus initialized contaills Lese; gravi- tational activity than one initialized with the non-divergent wind of the balance equation. With a five-level baroclinic atmosphere. the effects of mountains, surface drag, and release of latent heat are studi~d separately; the results arc consistent wlth classical concepts and other interesting features are revealed. T!;e diagnostic surfa:::e pres­ sure tendency is fo~nd to be in very good agreement with the observed pressure changes. A DIAGNOSTIC MODEL FOR INITIAL WINDS IN PRIMITIVE EQUATiONS FORECASTS by RI CHARD ASSE LIN . A the sis submitted to the Faculty of Graduate Studies and Re search in partial fulfillrnent of the requirernents for the degree of Doctor of Philosophy. Departrnent of Meteorology McGill Univer sity Montreal, Canada July. 1970 1971 A DL~GNOSTIC MODEL FOR INITIAL WINDS IN PRIMITIVE EQUATIONS FORECASTS by RI CHARD ASSE LIN . A thesis submitted ta the Faculty of Graduate Studies and Research in partial fulfillment of the requirements. for the degree of Doctor of Philosophy. Department of Meteorology McGill Univer sity Montreal, Canada July, 1970 @ l-J.cha l'Cl L.s :::;e1in 1971 ACKNOWLEDGEMENTS The author is indebted to his research supervisor, Professor A. J. Robert, for many useful discussions and for his encouragement throughout this work, as weU as to Professor B. W. Boville, who provided helpful advice and editorial assistance. l am grateful to Dr. M. Kwizak for generating my initial incentive in these studies and for his continuing suppurt, particularly in arranging for me to use the computer and other facili­ ties of the Central Analysis Office (CAO). The programs of the barotropic forecast model as well as a diagram were borrowed from his work. Miss S. Boville is responsible for the programming of the experiment with idealized data. Mr. W. S. Creswick did the graphical computations and helped to evaluate the re~u1ts. Mr. P. Sarrazin drafted the figures and Misses T. Savoie and J. Sutherland typed the manuscript. Finally, l wish to acknowledge the financial support of the Canadian Meteorological Servi.ce and of the National Research Council of Canada. ii TABLE OF CONTENTS Acknowledgements ii List of Plates; Tables and Figures vi List of Symbols ix Résumé xi Abstract xii 1. Introduction 1 2. Review of Diagnostic Methods 6 2. 1. Balance -W- Equations 6 2. 2. Geostrophic Diagnostic (G ) 8 2 2. 3. Miyakoda-Moyer Diagnostic 9 2.4. Nitta-Hovermale Method 9 3. Theory of Initialization 11 3. 1. The Simple st Model 11 3. 2. Numerical Compatibility 15 4. Historical R~.view of the G Geos.trophic Method 16 2 Part A: BAROTROPIC EXPERIMENTS· 19 5. Basic Equations and Theory 19 6. Numerical Solution of the Diagnostic Model 25 6. 1. Relaxation 25 6. 2. Ellipticity 26 7. Hig~er Order G Approximation 30 2 iii 8. Numerical Procedures for Initialization of Forecasts 31 8. 1. Compatibility Procedure 31 8. 2. Imposing New Boundary Conditions 34 9. Discussion of Results 37 9. 1. Divergence and Tendency 37 9. 2. Vorticity 43 9.3. Height Changes 46 9.4. Forecasts 50 Part B: BAROCLINIC EXPERIMENTS 54 10. Basic Primitive Equations 54 10.1. Precipitation Model 56 Il. Diagnostic Equations 59 12. Solution of Baroclinic System 64 12. 1. Discretization 64 12.2. Boundary Condition~, Vertical Structure. 65 12.3. Relaxation 67 13. Data Preparation 69 13.1. Interpolation to (j-SurfaceEV 69 ~----- .13. 2. Smoothing 70 14. Ellipticity 72 15. _ Discussion of Results 75 15.1. Effects of Orography and Drag 75 15.2. Effect of Pressure Systems, Ellipticity 85 iv 15.3. Effect of Latent Heat 96 15.4. Combined Effect from All Sources 102 16. Conclusions 109 References III Appendix A: Finitc Differences 115 Appendix B: Special Operations for Equatorial Latitudes 119 Appendix C: 123 v LIST OF PLATES. TABLES. AND FIGURES pagè Plate 1. The baroclinic diagnostic equations in full 60 Table 1. Input data for each baroclinic experiment 76 Table 2. Statistics from each baroclinic experiment 95 Figure 1. The 51. x 55 point grid 5. 2. Two graphs of the tangential wind speed in a station­ ary circular system 23 3. 500 mb analysis for OOz 21 Feb 69 and diagnosed tendency 38 4. Summaryof 25 day integration by Kwizak (1970, Energy. vorticity, divergence • 39 5. Time variation of total available potential energy in barotropic forecasts with bala~ce and G initia1i­ 2 zations 41 6. Time variation of squared divergence in barotropic forecasts with balance and G initializations 42 2 7. Absolute vorticity from balance equation 44 8. Absolute vorticity from G geostrophic initialization 45 2 9. Total height changes ma.de by bal1:\.nce ilütialization 47 10. Total height changes ~ade by G initialization 48 D 2 vi Figure 11. 72-hour forecast from balance initialization 51 12. 72-hour forecast from G initialization 52 2 13. 72-hour verifying analysis, OOZ 24 Feb 1969 53 14. The discrete structure of the baroclinic mode1 66 15. a) response of short wave filter b) response of 10w latitude smoother as functions of wave1ength 71 16. Idealized data used for experiments 7 to 12 78 17. Orientation error pattern in baroclinic model 79 18. Basic results of experiments with idealized mountain 81 19. Resu1ts of Cressman' s (1960) experiments with orography 84 20. Diagnosed surface tendency produced by rea1 oro­ graphy a10ne 86 21. Diagnosed surface tendency produced by orography and 'drag combined 87 22. 300 mb ana1ysis for 122 10 Feb 1970 88 23. 500 mb ana1ysis for 12Z 10 Feb 1970 89 24. 1000 mb analysis for 122 10 Feb. 1970 90 25. 300 mb height change~ resulting from preliminary ellipticization 92 26. 500 mb height changes resulting from preliminary ellipticizati?n 93 vii Figure 27. Results from basic baroc1inic experiment: vertical m.otion, vorticity and tendency for a small volume of space 94 28. Surface chart for 062 10 Feb 1970 97 29. Surface chart for 122 10 Feb 1970 98 30. Surface chart for 182 10 Feb 1970 99 31. Diagnostic vertical motion accompanying observed pressure distribution assuming saturated troposphere 100 32. Diagnostic vertical motion accompanying observed pressure distribution alone 101 33. Diagnostic vertical inotion accompanying observed pressure and moisture distributions with mountain and friction effects also considered 103 34. Vertical motion from the w-equaÜon inc1uding mountain and friction effects . 104 35. Surface pressure tendency from most complete diagnostic model 105 36. Observed 12-hour surface pressure change 107 37. Exact and modified functions of the Coriolis 122 parameter used in baroclinic experiments viii LIST OF SYMBOLS a element of matrix A A' 2 x 2 or 3 x 3 matr ix b element of matrix A B 3 x 3 matrix c element of matrix A C condensation rate - cp specifie heat at const. pre 55. CD surfa.ce drag coefficient d element of matrix A, D horizontal wind divergence distance between grid points F unspecHiedfunction e water vapour pressure f friction f orc e f = 2.Qsin (l, Coriolis parame G time averaged geopotential, g accelei-ation of gravit y, Geostrophic, Gravit y subscript: geostrophic H mean depth of nuid, h height of free surface heating rate unit imaginary numbcr, K kinetic energy, Kelvin index or unit vector for x L latent heat of vapourization j index or unit vector for y pep* ) pressure (at the grcu!'\d), k index or unit vertical vcctor polynomial m map factor Q = f + f absolute vorticity p element of matrix A R radhls of curvature, residual, q mixing ratio Rossby, Gas constant r clement of matrix. A 5 water vapvl:.r SOUl-ce or sink, t time subscript: stationary u velocity in x direction T temperature v velocity in y direction, u velocity in x directio!l subscript: virtual v velo city in y direction w weight W' vector wind Ct x, y grid cartesian coord. W vertical velocity z vertical direction,- height ·of constant pressure surface ix Q wave number, relaxation factor, arbitrary factor f3 arbitrary factor y arbitrary factor ô arbitrary factor, Kronecker delta A difference e: small number used as limit, arbitrary factor ~ relative vorticity about vertical axis 6 latitude k = R/cp 1\ eigen value V iteration inde.x, smoothing coefficient 1T product sign cr _ p Ip*, vertical coordinate é1: E dtr/dt, vertical velocity ~ summation sign 1> = gh or gz, geopotential ~ velocity potential \fi tendency, stream function (,J :.: dP Idt, vertical velocity Il.. earth' s angular velocity DIFFERENTIAL OPERA TORS A. total differential a partial differ ential V = i dlèJ x + j dIa y, gradient 2 ",2= dl Iè. xl + a Id yl, Laplacian J(Fl,Fl) = èiF1/dx· dFl/O y - dFl/ë)y· àFZldX, Jacobian x RESUME ~ , L'ensemble des lois gouvernant la meteorologie dyna- mique se condense en une seule équation differentielle du second ordre . " , .. " si on substitue la derivee du vent geostrophique a l'acceleration dans la loi du mouvement. On peut ensuite se servir de cette équation pour d~terminer le champ des trois composantes du vent à partir de la distribution de la.
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