DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION BY NORMAN A. PHILLIPS SIXTH IMO LECTURE I WMO - No. 700 I WORLD METEOROLOGICAL ORGANIZATION 1990 © 1990 World Meteorological Organization ISBN 92-63-10700-9 NOTE The designations employed and the presentation of material in this publication do not imply the expression of any opinion whatsoever on the part of the Secretariat of the World Meteorological Organization concerning the legal status of any country, territory, city or area, or of its authorities, or concerning the delimitation of its frontiers or boundaries. CONTENTS Page Foreword .................... V Summary (English, French, Russian, Spanish) VII Chapter 1 - Introduction ........... 1.1 The atmospheric equations of motion . 2 1.2 Uses of the linearized equations of motion 3 Chapter 2 - Simple plane waves on a resting atmosphere 5 2.1 Gravity and acoustic waves ......... 7 2.2 Rossby waves . 12 2.3 Preliminary consideration of group velocity 16 2.4 The Kelvin wave ............... 23 2.5 Vertical response to flow over ridges .... 24 Chapter 3 - Global solutions for a resting atmosphere 30 3.1 Separation of variables . 30 3.2 The role of the equivalent depth h 33 3.3 The vertical modes . 36 3.3.1 Eckart's thermosphere .. 38 3.3.2 Practical considerations . 41 3.4 Horizontal eigenfunctions 44 3.5 Group velocities from the Laplace Tidal Equation 52 3.6 Observed large-scale oscillations . 56 Chapter 4 - Dispersion in the presence of zonal currents 58 4.1 The channel dispersion model of Charney and Eliassen 58 4.2 Observed patterns of zonal dispersion . 64 4.3 The Charney-Drazin (Queney) criterion 69 4.4 Critical points and turning points 71 4.5 Dominance of the continuous spectrum 75 4.6 Importance of the external mode .... 76 4. 7 Latitudinal propagation of stationary Rossby waves with zonal shear 80 4.8 Latitudinal propagation of transient Rossby waves with zonal shear 83 4.8.1 Observational studies ........................ 83 4.8.2 Latitudinal influence function for individual zonal wave numbers 86 4.8.3 Two-dimensional barotropic influence function 90 4.9 Vertical propagation with zonal wind shear ............ 94 Acknowledgements 102 Appendices . 103 A. Wave definitions 103 B. Lamb mode for a non-isothermal atmosphere 106 C. WKB derivation of the Rossby frequency 108 D. The hydrostatic approximation 110 E. Geostrophic approximations 114 References 118 FOREWORD Starting with the Fifth World Meteorological Congress, in 1967, anIMO Lecture has been delivered at each Congress to commemorate the International Meteorological Organization (IMO), the predecessor of WMO. Each ofthese lectures reviewed progress in a fundamental aspect of meteorology. All ofthem were prepared by acknowledged experts in the chosen fields. This is the sixth in the series ofiMO Lectures. It deals with basic theory underlying numerical forecasting as currently practised. Keeping with the traditional high standard of these lectures, the present one was delivered by an eminent scientist, Professor N. A. Phillips, who has played a leading part, during a period extending over more than thirty years, in developing atmospheric circulation models and in studies of the characteristics of atmospheric motions upon which model formulation depends. The present publication contains the full text of his monograph. The problem of transmission of influences from one place to another is fundamental to numerical weather prediction and will remain so for the foreseeable future, since analysis errors that are unavoidable in areas where observations are insufficient propagate to other areas of the forecast field. Professor Phillips traces the development of the concepts of dispersion of energy and group velocity in the atmosphere, beginning with the propagation properties of the simplest plane waves in an atmosphere at rest. The influences of the global geometry of the Earth and the vertical stratification of the atmosphere are then described and it is finally shown that realistic atmospheric dispersion properties, both horizontally and vertically, can be achieved by including the effects of a zonal basic current. An important application of these theoretical results is to show that attempts to increase the useful length of forecasts will require that more attention be given to the middle and upper stratosphere. There are many other practical applications in explaining a number of atmospheric phenomena such as the forcing ofthe quasi-biennial oscillation in the zonal wind in the equatorial stratosphere, propagation of effects from a low-latitude source into middle latitudes ("tele-connections") and upward transmission of wave energy into the stratosphere, which may cause sudden stratospheric warmings in certain conditions. The present monograph constitutes a thorough and comprehensive review of the theoretical basis of dispersion processes in the atmosphere and their role in large-scale weather prediction, bringing together the vital steps made in this field over the past forty years in parallel with the developments in dynamical forecasting of the atmosphere. Like its predecessors in the series of IMO Lectures, the monograph will be welcomed by meteorologists both as a stimulus to further thought and as a valuable reference source. I wish to take this opportunity, on behalf of the World Meteorological Organization, to express deep appreciation to Professor Phillips for the high standard of his work and contribution to the IMO Lecture series. G. 0. P. 0BASI Secretary-General ~a.""""'.. -- SUMMARY My invitation from the World Meteorological Organization was to write a monograph on the subject "The scientific basis of weather prediction: its past, present, and future". I soon recognized that weather .prediction had been developed to the point where its scientific basis was too broad to be confined in a single monograph. Furthermore, many ofthe scientific aspects of weather prediction are well covered in existing textbooks, and several specialized books and review articles that have appeared in recent years present their subjects in far greater depth than I could bring to bear. With the tacit agreement of the World Meteorological Organization, I have therefore confined this monograph to a narrower focus. An effort has been made, however, to retain some of the universality and historical flavour implicit in the original title. The important role in forecasting that would be played by the transmission of influences from one place to another by large-scale motions was recognized by Ertel in 1941. In 1945 Rossby introduced into meteor­ ology the important concepts of group velocity and dispersion, and in 1949 Charney made quantitative applications of these concepts to the design of the first modern attempt at numerical weather prediction. This transmission ofinfluences remains an important consideration today, even though many operational weather models treat the complete range oflongitude and latitude. It will remain important for the foreseeable future for several reasons: analysis errors that are unavoidable in regions of few observations can propagate to other regions of the globe, and attempts to increase the useful length offorecasts will inevitably demand that more attention be given to the middle and upper stratosphere. Charney's application of group-velocity arguments was to the case of a 24-hour forecast, and he assumed that only transient waves of moderate wave length need be considered. Nowadays forecasts are made for periods sufficiently long to require also the forecasting of quasi-stationary features, together with their occasional changes. This monograph must therefore pay some attention to the many studies of recent years that have addressed the structure and behaviour of quasi-steady patterns of circulation. Chapter 1 introduces the dispersion problem as one of the many elements determining the quality of weather forecasts. After a simple presentation of the equations of motion, the chapter closes with a recapi­ tulation of the successes that have attended the application oflinearized forms of the equations of motion in meteorology. The plan of the remaining three chapters is to progress from the propagation properties of the simplest plane waves superimposed on a resting atmosphere, through the effects introduced by modal structures that recognize the global geometry of the Earth and the major stratification changes that exist in the vertical, and to conclude with the more realistic dispersion properties that are achieved by including the effect ofa zonal basic current. Many readers will find this progression needlessly slow; but I am convinced that the result is a clearer picture of the development of meteor-ological thought (with contributions from meteorologists of many nationalities), a healthier awareness of the complexity of the atmosphere, and an appreciation ofthe fortunate hydrodynamical circumstances in our atmosphere that allow useful weather predictions to be made. In Chapter 2, the properties of plane waves are described, using the convenient notation introduced by Eckart. The assumption of constant coefficients in the linear perturbation equations yields trigonometric solutions with a fifth-order polynomial to determine five frequencies for a given three-dimensional wave number. The largeness of the buoyancy frequency compared with the angular velocity of the Earth justifies the neglect in the equations of motion of the Coriolis terms proportional to the cosine of the latitude, a step of great convenience. It can then be seen that