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 two frequencies correspond to acoustic waves modified by gravity and rotation, and that two frequencies represent internal gravity waves. The fifth frequency in this case of constant coefficients corresponds to stationary geostrophic flow with zero frequency. This solution gener alizes to Rossby waves, when allowance is made for the latitudinal variation of the Coriolis parameter. Advection of potential vorticity is essential for Rossby waves in a stratified medium, whereas acoustic and gravity waves have zero potential vorticity. An important simple solution for the vertical variation of the perturbation in an isothermal atmosphere is the solution first obtained by Lamb. For this case the dependent variables decrease exponentially with VIII SUMMARY height rapidly enough for the total energy of the perturbation to be finite. For the acoustic-gravity waves, the Lamb solution gives a horizontally propagating wave, and also defines the vertical structure for an external Rossby wave in a resting atmosphere. The group velocities for internal gravity waves are such that waves whose wave number vector is inclined at 45 degrees to the horizontal have group velocities of comparable slope, but with a reversed sign for the vertical component. Gravity waves with quasi-horizontal wave numbers will have quasi-vertical group velocities, and waves with quasi-vertical wave numbers will have group velocities that are almost horizontal. The former are most likely to be associated with convective motions, while the latter structures correspond to gravity waves that might be associated with the existence (or forecast existence) of synoptic-scale motions. The characteristic horizontal group velocity for the Lamb acoustic wave is the speed of sound, which, if it controlled the dispersion of synoptic-scale influences, would transmit influences throughout the world in about one day. The maximum group velocities for plane Rossby waves are calculated; they give speeds large enough to restrict a forecast for tropospheric points located in the centre of the northern hemisphere continents to at most one or two days, given the limited distribution of data over the oceans and in the stratosphere. Evidently there must be some effect that reduces this pessimistic estimate oflateral and vertical dispersion of large-scale motions. Chapter 2 also contains a simple derivation of the horizontal structure for a Kelvin wave on a rotating plane. This solution is characterized by oscillating geostrophic flow in the direction parallel to a lateral bounding wall, with frequencies appropriate to acoustic or gravity waves in a non-rotating co-ordinate system. (A similar solution exists in global solutions as an equatorially trapped mode.) The final section in Chapter 2 describes the wave response to uniform flow over a sinusoidal ridge of surface elevation. The response in the atmosphere above the ridge is either wave-like or trapped, depending on the value of the horizontal wave number k describing the sinusoidal ridge, and on the speed of the zonal flow, u. The k, u plane is divided into three regions in which acoustic, gravity, or Rossby waves can propagate vertically, and three intervening regions where the response is trapped. This behaviour of the vertical wave number will reappear in the analysis of global solutions. Chapter 3 examines the dispersion properties of stable atmospheric waves on a resting atmosphere without the assumptions of constant coefficients and plane wave solutions. This is done primarily to see whether the plane wave solutions seriously overestimate the magnitude of the group velocity for Rossby waves. Separation of variables results in one set of equations (the "shallow water equations") that govern the horizontal behaviour of the eigenfunctions, and a second set of two equations that govern the vertical dependence of the eigenfunctions. These sets of equations are connected by a separation parameter, the "equivalent depth", introduced by Taylor. The equivalent depth is positive for free oscillations. However, negative values must be considered for forced problems with small frequencies. This circumstance is similar to the vertically trapped flow that can occur in flow over mountains when the variability of the Coriolis parameter is ignored (i.e. the classical argument that explains the existence of an anticyclone over a mountain range.) A resting basic state in which the temperature corresponds to a global climatological average has discrete vertical modes only for internal acoustic waves and for the Lamb external mode (either acoustic or Rossby). Discrete vertical modes can be introduced for internal gravity waves and Rossby waves by using stronger temperature inversions. Eckart suggested that a thermosphere at the top of the atmosphere would ensure a discrete vertical spectrum for all modes. However, this is accomplished only because the square of the buoyancy frequency goes to zero as the reciprocal of the height, and it can be shown that this is a very inefficient reflector of internal gravity waves and Rossby waves. Observations show that velocities at the 50 km level correspond to energy density levels that are smaller by a factor of four or five than the kinetic energy density levels around the winter troposphere. The explanation of this decrease in energy density with height, and a reduction of the large vertical group velocities obtained previously, must be due to omission of some factor in the mathematical models used so far. This factor is the presence of a basic current, u, and is included in Chapter 4. The horizontal eigenfunctions of the Laplace Tidal Equation change character as the equivalent depth changes. The classical case is that for large positive values, for which the eigenfunctions divide into two classes; first-class motions have properties similar to acoustic or gravity waves, and second-class motions correspond to Rossby waves. Small positive values of the equivalent depth correspond to small vertical wave SUMMARY IX lengths. The horizontal eigenfunctions are then confined to low latitudes. The non-dimensional formulation by Matsuno of the perturbation equations in low latitudes can be interpreted to show that this confinement operates so as to keep the Coriolis force from dominating the buoyancy force when the vertical wave length becomes small. Zonal and vertical group velocity components can be calculated from the frequencies associated with solutions of the Laplace Tidal Equation. They have the same large values as those computed from the plane wave solutions. This also indicates that some factor that would decrease the rate of spread ofinfluence has not yet been included in the mathematical model. Large-scale wave motions that fit the theoretical properties of atmospheric perturbations have been observed by many meteorologists. In equatorial latitudes Kelvin waves and the mixed gravity-Rossby wave have been observed, with moderate vertical wave lengths. These play a role in the forcing of the quasi-biennial oscillation in the zonal wind of the equatorial stratosphere. Rossby waves for the external mode have been observed in extratropicallatitudes, for small latitudinal and small zonal wave numbers. The mixed gravity Rossby wave for the external mode does not seem to have been observed, however, perhaps because meteorological analyses use an assumption of geostrophic balance between wind and pressure force, and the theoretical frequencies are sufficiently large for daily analyses to be too infrequent to represent these systems. Chapter 4 begins by describing the "equivalent barotropic model" devised by Charney and Eliassen and the one-dimensional influence function that they constructed for 24-hour forecasts in a narrow zonal channel containing a zonal current u. The narrowness of their channel greatly reduces the dispersion ofRossby waves. But at longer forecast times, the shape of their influence function gives a graphic verification of simple group velocity arguments. Zonal dispersion of Rossby waves on upper-air analyses can be made clear by the use of a Hovmoller diagram. Simmons and Hoskins have shown theoretically how an initial localized disturbance in a baroclinic zonal current in middle latitudes will generate a downstream wave train of unstable waves, similar to the events sometimes seen on Hovmoller diagrams. Forecast errors from operational models over North America show a systematic tendency to move in from the Pacific Ocean. This process was analysed by Phillips with a model that imitated both the analysis and forecast aspects of an operational system. In 1961 Charney a:rid Drazin made meteorologists aware of the profound effect that a zonal current u can have on the vertical propagation ofRossby waves. (A parallel effect exists also for latitudinal propagation.) In its simplest form, their criterion for vertical wave propagation is the same as that discovered in 1948 by Queney for the generation of stationary Rossby waves by westerly flow over mountains. Their criterion explains the absence of stationary eddy motion in regions ofeasterly flow, such as the low-level subtropics and the summer stratosphere, and the prominence of low horizontal wave numbers in the winter strato sphere. Critical and turning points are important in the analysis of perturbations on a zonal current when the angular velocity varies with latitude or height. The initial views of atmospheric critical points were influenced by the hydrodynamical theory that was developed in the late 1940s to explain the instability ofhomogeneous shear flow, according to which the waves lost energy by viscosity in the immediate neighbourhood of the critical point. More recent views attribute more significance to reflection and non-linearities. An important conclusion is contained in a theorem derived by Dikii and Katayev. This states that the only discrete modes present in a barotropic model with a basic current of variable angular velocity will be those modes whose angular zonal speed of translation is less than that of the minimum angular velocity of the basic current (i.e., they have no critical point.) When extended by crude analogy to the case of internal Rossby waves, this theorem shows that discrete modes will exist only for motions oflarge vertical wave lengths. This is consistent with the observational evidence for free Rossby waves, and suggests why the latitudinally compressed eigenfunctions of the Laplace Tidal Equation for small equivalent depths have not been observed. The horizontal group velocities ofRossby waves are largest for the external vertical mode. The semi empirical definition by Charney and Eliassen of the vertical structure in their equivalent barotropic model has been shown by Held, Panetta and Pierrehumbert to be similar to the external mode present in westerly flow for stationary Rossby waves. The dispersion diagram for Rossby waves in a variable zonal current is complicated, but is considerably simplified if attention is restricted to stationary waves of zero frequency. X SUMMARY
Hoskins and Karoly have computed sample ray trajectories for this special case, using zonal currents similar to climatological flows. An important group of rays (described here for the northern hemisphere) are those that leave a low-latitude source toward the north-east, reach their turning latitude, and then propagate to the south-east and south until they reach their critical latitude. Some of these paths agree with the location of the teleconnection centres in the northern hemisphere that were found by Wallace and Gutzler. Eddy kinetic energy is transported equatorward in the upper troposphere, as shown by the study by Mak and by later analyses of Eliassen-Palm cross-sections. In order to examine the latitudinal propagation of forecast errors by transient Rossby waves, a numerical approach is necessary. This can be done effectively by calculating a latitudinal influence function separately for each zonal wave number and for each forecast latitude of interest. Such computations show that a smoothed climatological distribution of u interferes significantly with the free propagation ofRossby waves from one latitude to another that can occur in a zonal flow in solid rotation. When these results for different zonal wave numbers are combined by Fourier's theorem, the resulting two-dimensional influence function contains two regions of maximum sensitivity for the initial data used to make a forecast for a point in the extratropics. One is the region surrounding the point located upstream the distance -Ut. This is the barotropic equivalent ofthe baroclinically generated down stream wave train generated in the study by Simmons and Hoskins. The second region is located on the equatorial flank of the jet, to the south-west ofthe forecast point. This region acts as a source for rays (typically of zonal wave number 5-7) that reach their poleward turning point and maximum energy density at the forecast latitude. The existence of this second region is not dependent on the existence of a latitude where u is zero. The first example of upward propagation of wave energy into the stratosphere was documented by Muench. Derome has presented data showing apparent downward propagation ofgeopotential forecast error for zonal wave number two. However, when his data are normalized by the square root of the pressure, the resulting time-height cross-section looks more like a resonant amplification of a standing pattern. Further examples of upward propagation of influence are found in the studies by Matsuno, which showed how the winter stratospheric anticyclone over the Aleutians is "forced" by the flow pattern at 500 hPa, and how sudden stratospheric warmings can be created by amplification in the troposphere ofzonal wave number two. Tung and Lindzen suggest that the occasions most favourable for resonant amplification of large-scale tropospheric waves with pronounced blocking, and stratospheric warming, occur when the stratospheric jet exists at much lower elevations than its normal location around 60 km. Numerical studies suggest that wave development in the polar stratosphere is sensitive to details present in the zonally averaged potential vorticity. Several experiments with numerical models have explored the effects of increased height and resolution in the stratosphere. The results are not conclusive, but suggest that tropospheric forecasts of a week or more may at times be sensitive to these factors. Several appendices have been included to make the monograph more self-contained. RESUME
L'Organisation meteorologique mondiale (OMM) m'a demande de rediger une monographie sur «Les bases scientifiques de la prevision du temps: le passe, le present et les perspectives d'avenir». Je me suis bien vite rendu compte que la prevision du temps etait devenue un domaine si vaste qu'une seule monographie ne pourrait suffire a cerner l'ensemble de ses bases scientifiques. Par ailleurs, bon nombre des aspects scienti fiques de la prevision du temps ont deja ete traites de fa<;on exhaustive dans divers manuels et plusieurs ouvrages ou articles de revues specialisees parus ces dernieres annees presentent ce sujet de maniere bien plus approfondie queje ne serais en mesure de le faire. Je me suis done permis, avec l'accord tacite de l'OMM, de restreindre quelque peu l'horizon de ma monographie en m'effor<;ant de retenir, toutefois, le caractere universe! et historique de ce «vent du large» qui souffle a travers le titre original. En 1941, Ertel reconnaissait deja !'importance que peut revetir pour la prevision la transmission des influences, d'un endroit a l'autre, par des mouvements de grande echelle. En 1945, Rossby introduit dans le domaine de la meteorologie les notions fondamentales de vitesse de groupe et de dispersion et, en 1949, Charney entreprend, en appliquant ces principes quantitativement, la premiere tentative moderne de pre vision numerique du temps. Alors meme que la plupart des modeles operationnels de la prevision du temps couvrent aujourd'hui le spectre entier des latitudes et des longitudes, cette notion de transmission des influences reste un facteur important. Pour diverses raisons nous sommes loin encore de pouvoir nous passer de ces informations. Ainsi, par exemple, les erreurs d'analyse inevitables dans les zones ou les donnees d'observation sont rares peuvent se repercuter sur d'autres regions du globe et il est indispensable de tenir compte avec plus de soin des couches moyennes et superieures de la stratosphere si l'on veut parvenir a prolonger la duree utile des previsions. Charney avait applique la notion de vitesse de groupe dans le cas d'une prevision a echeance de 24 heures en admettant qu'il suffisait de s'en tenir aux ondes transitoires de longueur d'onde moyenne. De nosjours, les previsions portent sur des periodes suffisamment longues pour qu'il soit necessaire de prevoir egalement les caracteristiques quasi stationnaires ainsi que leurs eventuelles modifications. La presente monographie s'attachera done a passer en revue les nombreuses etudes qui ont ete effectuees ces dernieres annees sur la structure et le comportement de configurations quasi stationnaires de la circulation atmospherique. Dans le chapitre lle probleme de la dispersion est considere comme l'un des nombreux elements dont depend la qualite des previsions du temps. Apres avoir presente, de fa<;on simplifiee, les equations du mouvement, le chapitre termine sur un recapitulatif des succes qui ont marque !'application, en meteorologie, des formes lineaires des equations du mouvement. Les trois chapitres suivants progressent selon un plan bien defini en commen<;ant par la description des proprietes de propagation des ondes planes les plus simples superposees a I' atmosphere au repos, pour passer ensuite aux effets introduits par les structures modales qui caracterisent la geometrie globale de la Terre et les principaux changements de stratification selon la verticale, et terminer sur les proprietes de dispersion plus reelles que l'on obtient en tenant compte de l'effet d'un courant zonal de base. De nombreux lecteurs trouveront que la progression est inutilement lente, maisje suis persuade qu'elle permet de donner une image plus claire du developpement de la pensee meteorologique (avec la contribution de meteorologistes de maintes nationalites), une meilleure idee de la complexite de !'atmosphere et de mieux apprecier les condi tions hydrodynamiques favorables de notre atmosphere qui permettent d'etablir des previsions du temps utiles. Dans le chapitre 2, les proprietes des ondes planes sont decrites a l'aide de la notation pratique adoptee par Eckart. L'hypothese de coefficients constants dans les equations des perturbations lineaires permet d'obtenir des solutions trigonometriques a polynomes du cinquieme degre pour determiner cinq frequences d'un nombre d'onde tridimensionnel donne. L'ordre de grandeur de la frequence hydrostatique par rapport a la vitesse angulaire de la Terre permet de negliger, dans les equations du mouvement, les termes de Coriolis proportionnels au cosinus de la latitude, ce qui est tres pratique. Il apparait des lors que deux frequences correspondent aux ondes sonores modifiees sous l'effet de la gravite et de la rotation, alors que deux autres XII RESUME frequences representent les ondes de gravite internes. Les coefficients etant constants, la cinquieme frequence correspond a un ecoulement geostrophique stationnaire de frequence nulle. En generalisant et en tenant compte de la variation latitudinale du parametre de Coriolis, on obtient les ondes de Rossby. Dans le cas des ondes de Rossby en milieu stratifie, i1 est indispensable de tenir compte de l'advection du tourbillon potentiel alors que ce facteur est nul dans le cas des ondes sonores et des ondes de gravite. La solution simple obtenue pour la premiere fois par Lamb pour la variation verticale des perturbations en atmosphere isotherme presente un interet particulier. Dans ce cas precis, la vitesse de la decroissance exponentielle des variables dependantes en fonction de I' altitude est suffisante pour que l'energie totale de la perturbation soit finie. La solution de Lamb prevoit des ondes de gravite sonores a propagation horizontale et definit la structure verticale d'une onde de Rossby externe dans !'atmosphere au repos. Les ondes de gravite internes dont le vecteur du nombre d'onde fait un angle de 45 degres avec l'horizontale ont des vitesses de groupe de meme pente mais de signe oppose pour la composante verticale. Les ondes de gravite a nombres d'onde quasi horizontaux ont des vitesses de groupe quasi verticales et les ondes a nombres d'onde quasi verticaux ont des vitesses de groupe quasi horizontales. La premiere categorie d'ondes est le plus sou vent associee a des mouvements convectifs alors que la seconde correspond a des ondes de gravite qui peuvent etre associees a !'existence (ou a !'existence prevue) de mouvement a l'echelle synoptique. La vitesse de groupe horizontale caracterisant les ondes son ores de Lamb est la vitesse du son qui, si elle regissait la dispersion des influences a l'echelle synoptique transmettrait les influences au monde en tier en l'espace d'unjour. Les vitesses de groupe maximales des ondes planes de Rossby qui ont ete calculees sont si grandes qu'elles limitent les previsions a un ou deuxjours au plus pour les points de la troposphere situes au centre des continents de !'hemisphere Nord, etant donne la rarete des donnees au-dessus des oceans et dans la stratosphere. 11 est certain qu'il doit y avoir un moyen de remedier a cette estimation pessimiste de la dispersion laterale et verticale des mouvements a grande echelle. Le chapitre 2 contient egalement une simple derivee de la structure horizontale d'une onde de Kelvin se propageant dans un plan en rotation. Cette solution se caracterise par un ecoulement geostrophique oscillant suivant une direction parallele a une limite laterale a la frequence d'une onde sonore ou d'une onde de gravite dans un systeme de coordonnees qui n'est pas en rotation. (Les solutions globales comportent une solution analogue correspondant au mode piege a l'equateur). La derniere section du chapitre 2 decrit le comporte ment de l'onde dans le cas d'un ecoulement uniforme le long de la crete sinuso'idale d'un relief en surface. La propagation le long de la crete est soit ondulatoire, soit piegee, suivant la valeur du nombre d'onde horizontal k qui correspond a la crete sinuso'idale et la vitesse de l'ecoulement zonal u. Le plan k, uest divise en trois zones dans lesquelles les ondes sonores, les ondes de gravite ou les ondes de Rossby se propagent suivant la verticale et trois zones interposees ou elles sont piegees. I1 sera egalement question du comportement du nombre d'onde verticallors de !'analyse des solutions globales. Dans le chapitre 3les proprietes de dispersion des ondes atmospheriques stables dans une atmosphere au repos sont etudiees sans tenir compte de l'hypothese de coefficients constants et des solutions relatives aux ondes planes. Le but essentiel de cette approche est de verifier s'il n'y a pas surestimation exageree de l'ampleur de la vitesse de groupe des ondes de Rossby dans les solutions relatives aux ondes planes. Si l'on separe les variables, on obtient une serie d'equations (les «shallow water equations» ou «equations des hauts-fonds») qui regissent le comportement horizontal des fonctions propres et une seconde serie d'equa tions qui regissent la relation verticale des fonctions propres. Ces series d'equations sont liees par un parametre de separation, la «profondeur equivalente», introduite par Taylor. La profondeur equivalente est positive dans le cas d'oscillations libres, mais i1 faut prevoir des valeurs negatives dans les cas de fon;age de faible frequence. Cette situation est analogue acelle de l'ecoulement piege suivant la verticale au-dessus d'une montagne lorsqu'il n'est pas tenu compte de la variabilite du parametre de Coriolis (c'est-a-dire le cas classique expliquant !'existence d'un anticyclone au-dessus d'une chaine de montagnes). A l'etat initial au repos, lorsque la temperature correspond a une moyenne climatologique mondiale, les modes verticaux discrets ne s'appliquent qu'aux ondes sonores internes et au mode externe de Lamb (d'une onde sonore ou d'une onde de Rossby). En adoptant de plus fortes inversions thermiques, on obtient des modes verticaux discrets pour les ondes de gravite internes et les ondes de Rossby. Selon l'hypothese d'Eckart, une thermosphere situee au-dessus de !'atmosphere se traduirait par un spectre vertical discret pour I' ensemble des modes. Cette situation serait due au fait que le carre de la frequence de la poussee d' Archimede, qui est une fonction inverse de !'altitude, tend vers zero et il est possible de demontrer qu'il s'agit d'un RESUME XIII retlecteur tres imparfait des on des de gravite internes et des ondes de Rossby. Les observations montrent qu'a 50 km d'altitude, les vitesses correspondent a des niveaux de densite d'energie quatre ou cinq fois moins eleves que les niveaux de densite d'energie cinetique aux alentours de la troposphere d'hiver. Cette decrois sance de la densite d'energie en fonction de I'altitude et la reduction des fortes vitesses de groupe verticales obtenues prealablement doivent s'expliquer par I' omission d'un certain facteur dans les modeles mathema tiques utilises jusqu'a present. Ce facteur est la presence d'un courant fondamental, u, dont i1 sera question au chapitre 4. Les fonctions propres horizontales de !'equation de la maree de Laplace changent de caractere avec la profondeur equivalente. Le cas classique est celui des grandes valeurs positives pour lesquelles les fonctions propres se divisent en deux categories: les mouvements de premiere classe qui presentent des proprietes analogues a celles des ondes sonores ou des ondes de gravite et les mouvements de seconde classe qui correspondent aux ondes de Rossby. Les faibles valeurs positives de la profondeur equivalente correspondent a de petites longueurs d'onde verticales. Les fonctions propres horizontales correspondent alors uniquement aux basses latitudes et l'on peut deduire de la formule non-dimensionnelle des equations des perturbations aux basses latitudes de Matsuno, que c'est cette restriction qui empeche la force de Coriolis de dominer la pression hydrostatique lorsque la longueur d'onde verticale devient petite. Les composantes zonale et verticale de la vitesse de groupe peuvent se calculer a partir des frequences associees aux solutions de I' equation de la maree de Laplace. Elles presentent les memes valeurs elevees que celles qui se calculent a partir des solutions des ondes planes ce qui indique egalement qu'un certain facteur de decroissance de la vitesse de dispersion de !'influence n'a pas encore ete integre au modele mathema tique. De nombreux meteorologistes ont observe des mouvements ondulatoires de grande echelle qui repon dent aux proprietes theoriques des perturbations atmospheriques. Sous les latitudes equatoriales, on a observe des ondes de Kelvin et des ondes mixtes de gravite et de Rossby ayant des longueurs d'onde verticales moderees. Ces ondesjouent un rOle non negligeable dans le fon;age de I' oscillation quasi bisannuelle du vent zonal de la stratosphere equatoriale. Sous les latitudes extratropicales, on a observe des ondes de Rossby pour le mode externe, et pour de faibles valeurs des nombres d'onde latitudinal et zonal. 11 semblerait toutefois que l'on n'ait pas observe d'onde mixte de gravite et de Rossby pour le mode externe, sans doute parce que I' analyse meteorologique est fondee sur l'hypothese de l'equilibre geostrophique entre la force du vent et celle de la pression et que les frequences theoriques sont assez grandes pour que les analyses journalieres ne soient pas assez frequentes pour representer ces systemes. Le chapitre 4 commence par la description du «modele barotrope equivalent» mis au point par Charney et Eliassen et de la fonction d'influence unidimensionnelle qu'ils ont etablie pour effectuer des previsions a echeance de 24 heures dans un couloir etroit ou s'ecoule un courant zonal u. Le couloir etant tres etroit, la dispersion des ondes de Rossby s'en trouvait fortement reduite. Cependant, dans le cas de previsions a plus longue echeance, le trace de leur fonction d'influence constitue une verification graphique des notions simples de la vitesse de groupe. L'utilisation du diagramme d'Hovmoller permet de clarifier la dispersion zonale des ondes de Rossby dans les analyses en altitude. Simmons et Hoskins ont montre theoriquement de quelle maniere une per turbation initiale localisee dans un courant zonal barocline aux latitudes moyennes engendre en aval, un train d'ondes instables semblable aux phenomenes que l'on observe parfois sur les diagrammes d'Hovmoller. Les erreurs de prevision des modeles operationnels au-dessus de l'Amerique du Nord ont systematiquement leur origine ... (dans le Pacifique). Phillips a analyse ce processus a l'aide d'un modele simulant un systeme operationnel du point de vue de !'analyse aussi bien que de la prevision. En 1961, Charney et Drazin attirerent !'attention des meteorologistes sur l'effet important qu'un courant zonal u peut avoir sur la propagation verticale des ondes de Rossby. (11 existe un effet similaire sur la propagation latitudinale). Dans sa forme la plus simple, le critere sur lequel ils basent la propagation verticale de l'onde est identique a celui qui a ete decouvert par Queney en 1948 pour expliquer la formation des ondes stationnaires de Rossby par un courant d' ouest au-dessus des montagnes. 11 explique l'absence de mouvement turbulent stationnaire dans les zones de circulation d'est, telles que les basses couches subtropicales et la stratosphere d'ete ainsi que la predominance des nombres d'onde horizontaux de faible valeur dans la stratosphere d'hiver. XIV RESUME
Pour !'analyse des perturbations d'un courant zonal, i1 est important de tenir compte des points critiques et des points de rotation lorsque la vitesse angulaire varie en fonction de la latitude ou de I' altitude. L'idee que l'on s'est faite, au depart, des points critiques de !'atmosphere a ete fortement influencee par la theorie hydrodynamique developpee a la fin des annees 40 pour expliquer l'instabilite du courant de cisaillement homogene. Selon cette theorie, les ondes perdent de l'energie sous l'effet de la viscosite dans le voisinage immediat des points critiques. Dans les etudes plus recentes on tend a attribuer davantage de signification a la reflexion et a la non-linearite. Le theoreme etabli par Dikii et Katayev offre une conclusion interessante. I1 pose en effet que les seuls modes discrets que l'on puisse trouver dans un modele barotrope a courant fondamental de vitesse angulaire variable sont des modes dont la vitesse angulaire zonale de translation est inferieure a la vitesse angulaire minimale du courant fondamental, c'est-a-dire des modes n'ayant pas de point critique. L'application de ce theoreme, par simple analogie, aux ondes de Rossby internes, montre qu'il n'y a de modes discrets que dans les cas de mouvements de grande longueur d'onde verticale. Cette conclusion confirme I' existence des ondes de Rossby libres qui ont ete observees et explique pourquoi on n'a pas trouve de fonctions propres comprimees suivant la latitude pour !'equation de la maree de Laplace dans le cas de faibles profondeurs equivalentes. C'est pour le mode vertical externe que les vitesses de groupe horizontales des ondes de Rossby sont les plus elevees. Held, Panetta et Pierrehumbert ont montre que la definition semi-empirique adoptee par Charney et Eliassen pour la structure verticale dans leur modele barotrope equivalent rejoignait celle du mode externe que l'on trouve dans le courant d'ouest pour les ondes de Rossby stationnaires. Le diagramme de dispersion des ondes de Rossby dans un courant zonal variable est fort complique, mais son etude peut etre considerablement simplifiee si I' on s'en tient aux ondes stationnaires de frequence nulle. Hoskins et Karoly ont calcule, specialement a cet effet, des trajectoires de rayons types, en utilisant des courants zonaux assimilables aux ecoulements climatologiques. Les rayons qui partent d'une source situee a basse latitude, se dirigent vers le nord-estjusqu'a leur latitude de rotation a partir d'ou ils se propagent vers le sud-est et le sud jusqu'a leur latitude critique, constituent un groupe de rayons particulierement interessant (ils sont decrits dans cet ouvrage pour !'hemisphere Nord). Certaines de ces trajectoires concordent avec la localisation des centres de teleconnexion de !'hemisphere Nord que Wallace et Gutzler ont decouverts. L'etude de Mak et les analyses des sections transversales effectuees ulterieurement par Eliassen-Palm montrent que l'energie de turbulence est transportee vers l'equateur dans la haute troposphere. I1 est neces saire d'avoir recours a une methode numerique pour evaluer la propagation selon la latitude des erreurs de prevision dues aux ondes de Rossby transitoires. Pour cela, i1 suffit de determiner une fonction d'influence latitudinale pour chacun des nombres d'onde zonaux et pour chaque latitude de prevision prise en consi deration. Ces calculs montrent qu'une repartition climatologique lissee de uentrave de maniere significative la libre propagation des on des de Rossby d'une latitude a I' autre qui peut se faire dans un ecoulement zonal en rotation interrompue. Si I' on combine les resultats ainsi obtenus pour differents nombres d'onde zonaux en appliquant le theoreme de Fourier, on obtient une fonction d'influence bidimensionnelle avec deux zones de sensibilite maximale pour les donnees initiales utilisees pour elaborer une prevision concernant un point precis de !'atmosphere extratropicale. L'une des zones est celle qui entoure le point situe en amont a la distance -ut. I1 s'agit d~ I' equivalent barotrope du train d'ondes engendre en aval en milieu barocline dont il est question dans l'etude de Simmons et Hoskins. La seconde zone se situe sur le versant equatorial de l'aerojet, au sud-ouest du point de prevision. Cette zone est une source de rayons (ayant en general un nombre d'onde zonal se situant entre 5 et 7) qui atteignent leur point de rotation vers le' pole et leur densite maximale d'energie a la latitude de la prevision. L'existence de cette seconde zone ne depend pas de I' existence d'une latitude ou uest nulle. Ce fut Muench qui fournit le premier exemple de propagation ascendante de l'energie ondulaire vers la stratosphere. Derome a presente des donnees selon lesquelles il y aurait propagation descendante de l'erreur de prevision geopotentielle pour le nombre d'onde zonal deux. Toutefois, lorsque ses donnees sont norma lisees par la racine carree de la pression, la coupe transversale temps-altitude a la forme de !'amplification resonnante d'une configuration stationnaire. D'autres exemples de propagation ascendante des influences sont fournies par les etudes de Matsuno qui demontre que I'anticyclone stratospherique d'hiver au-dessus des Aleoutiennes est litteralement «force» par la configuration des courants a 500 hPa et qu'il est possible de creer un rechauffement subit de la stratosphere en amplifiant le nombre d'onde zonal deux dans la troposphere. Tung et Lindzen estiment que les occasions les plus favorables a une amplification resonnante des ondes tropospheriques de grande echelle avec blocage prononce et rechauffement de la stratosphere, se presentent RESUME XV lorsque l'aerojet stratospherique se trouve a une altitude bien inferieure a son altitude normale, d'environ 60 km. Les etudes numeriques indiquent que I' evolution des ondes dans la stratosphere polaire est influencee par des particularites du tourbillon potentiel moyen en chaque zone. Plusieurs experiences effectuees a l'aide de modeles numeriques ont permis d'etudier les effets d'une augmentation de I'altitude et de la definition de la stratosphere. Les resultats, sans etre concluants, permettent toutefois de supposer que ces facteurs peuvent, de temps a autre, influencer les previsions tropospheriques a echeance d'une semaine ou au-dela. Plusieurs annexes completent la presente monographie. PE3IDME
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Kor,ll;a 3TH pe3yJihTaThi ,ll;JI5l pa3JIH'IHbiX 30HaJihHhiX BOJIHOBhiX quceJI o6be,ll;HH5IIOTC5l TeopeMOH J1epBbiH npHMep pacnpOCTpaHeHH5l B BepXHeM HanpaBJieHHH 3HeprHH BOJIHhi B CTpaTOCcj:>epe 6hiJI 3ape rHCTpHpOBaH MIOHXOM. ,[(epoMe npe,ll;CTaBHJI ,ll;aHHbie, llOKa3biBaiOID,He Bep05ITHOe pacnpOCTpaHeHHe B HH)I{HeM XX PE310ME HanpaBJieHIUI reonoTeHIJ;HaJia npomocTwiecKo:ll onm6Kn ,IJ;JI51 30HaJihHoro BOJIHOBoro •mcna ,n;Ba. O,n;HaKo B CJiyqa51X, KOr,n;a ero ,IJ;aHHhie HOpMaJIH3YIOTC51 C ITOMOITJ;hiO KBa,n;paTHOro KOpH51 ,n;aBJieHII51, ITOJiyqaeMhiH pa3pe3 ITO BpeMeHII-BhiCOTe BhlrJI51,II;IIT 6oJiee ITOXOJKHM Ha pe30HaHCHOe yBeJIIJ:qeHHe ycTaHOBJieHHOrO TIIITa. ,lJ;aJihHeHITIHe npHMephl pacnpOCTpaHeHII51 B03,IJ;eHCTBII51 B BepxHeM HanpaBJieHHH MO)KHO HaHTH B HCCJie,n;o BaHII51X MaiJ;yHo, r,n;e noKa3hiBaeTcjJ:, KaKIIM o6pa3oM 3IIMHHH CTpaToc BKJIIOqeHIIe B MOHOrpa La Organizacion Meteorologica Mundial me pidio que escribiera una monografia sobre el tema: «La base cientifica de la prediccion meteorologica: su pasado, presente y futuro». Pronto tome conciencia de que la prediccion meteorologica habia evolucionado tanto, que su base cientifica era demasiado amplia como para poder resumirla en una solo monografia. Ademas, muchos de los aspectos cientificos de la prediccion meteorologica han sido tratados exhaustivamente eh libros de texto existentes, yen diversas publicaciones y articulos especializados que datan de afios recientes se han desarrollado estos temas con mas detalle de lo que podria ofrecer yo. Con el acuerdo tacito de la Organizacion Meteorologica Mundial, he restringido, pues, el alcance de esta monografia. Sin embargo, he hecho lo posible para respetar, en parte, el caracter universal e historico implicito en el titulo original. En 1941, Ertel reconoci6 el papel importante que puede desempefiar para la prediccion la transmisi6n de influencias, de un lugar a otro, mediante movimientos de gran escala. En 1945, Rossby introdujo en la meteorologia las nociones fundamentales de velocidad de grupo y de dispersion, y en 1949, Charney, al aplicar cuantitativamente dichos principios, hizo el primer intento moderno de predicci6n meteorol6gica numerica. Hoy, esta noci6n de transmisi6n de influencias sigue siendo un factor importante aun cuando la mayoria de los modelos operativos de la prediccion meteorol6gica abarcan la gama completa de longitudes y latitudes. Por diversas razones, seguiremos necesitando dicha informaci6n. Asi pues, por ejemplo, los errores de analisis inevitables en las zonas donde escasean las observaciones, pueden propagarse a otras regiones del globo, y si se quiere prolongar el alcance util de las predicciones es indispensable prestar mayor atenci6n alas capas medias y superiores de la estratosfera. Charney aplic6 la noci6n de velocidad de grupo en el caso de una predicci6n a escala de 24 horas suponiendo que bastaba considerar unicamente las ondas transitorias de longitud media. Hoy en dia, las predicciones se hacen para periodos lo suficientemente largos para que sea necesario prever tambien las caracteristicas casi estacionarias asi como de sus posibles cambios. Por consiguiente, en esta monografia tratare de examinar los numerosos estudios realizados en los ultimos afios sobre la estructura y el compor tamiento de las configuraciones casi estacionarias de la circulaci6n atmosferica. El Capitula 1 trata del problema de la dispersion como de uno de los numerosos elementos que determinan la calidad de las predicciones meteorologicas. Despues de presentar, de forma simplificada, las ecuaciones del movimiento, el capitula termina con una recapitulacion de los exitos conseguidos en la aplicaci6n, en meteorologia, de las formas lineales de las ecuaciones de movimiento. Los tres capitulos siguientes siguen un plan bien definido empezando por la descripci6n de las propie dades de propagacion de las ondas planas mas sencillas superpuestas a la atm6sfera en reposo, y despues pasan a los efectos provocados por las estructuras modales que caracterizan a la geometria global de la Tierra y a los principales cambios de estratificaci6n en la vertical, y terminar por las propiedades de dispersion mas reales que se obtienen al tomar en cuenta el efecto de una corriente zonal basica. Muchos lectores pensaran que esta forma de exponer el tema es demasiado lenta, pero estoy convencido de que permite dar una imagen mas clara de la evoluci6n del pensamiento meteorol6gico (al que han contribuido meteorologos de muchas nacionalidades), tener una idea mas clara de la complejidad de la atm6sfera y apreciar mejor las condiciones hidrodimimicas favorables de nuestra atm6sfera que permiten hacer predicciones meteorol6gicas utiles. En el Capitula 2, se describen las propiedades de las ondas planas, utilizando la anotaci6n practica adoptada por Eckart. La hip6tesis de coeficientes constantes en las ecuaciones de las perturbaciones lineales permite obtener soluciones trigonometricas en las que intervienen polinomios del quinto grado para deter minar cinco frecuencias de un numero dado de onda tridimensional. La amplitud de la frecuencia hidro estatica con respecto a la velocidad angular de la Tierra permite que se prescinda, en las ecuaciones del movimiento, de los terminos de Coriolis proporcionales al coseno de la latitud, lo cual es muy practico. Se observa por lo tanto, que dos frecuencias corresponden a las ondas sonoras modificadas por la gravedad y la rotaci6n, mientras que dos otras frecuencias representan las ondas de gravedad internas. La quinta frecuencia, siendo los coeficientes constantes, corresponde a un flujo geostr6fico estacionario de frecuencia nula. Si se XXII RESUMEN generaliza y si se tiene en cuenta la variacion latitudinal del padtmetro de Coriolis, se obtienen las ondas de Rossby. En el caso de las ondas de Rossby en un medio estratificado, es fundamental tener en cuenta la adveccion de la vorticidad potencial mientras este factor es nulo en el caso de las ondas sonoras y de las ondas de gravedad. La solucion sencilla obtenida por primera vez por Lamb para determinar la variacion vertical de las perturbaciones en atmosfera isotermica ofrece un interes especial. En este caso, la velocidad de la disminu cion exponencial de las variables dependientes en fun cion de la altitudes suficiente para que la energia total de la perturbacion sea finita. La solucion de Lamb preve ondas de gravedad de propagacion horizontal y define la estructura vertical de una onda de Rossby externa en la atmosfera en reposo. Las ondas de gravedad internas cuyo vector de numero de onda forma un angulo de 45" con la horizontal tienen velocidades de grupo de inclinacion comparable, pero de signo opuesto para la componente vertical. Las ondas de gravedad con numeros de onda casi horizon tales tienen velocidades de grupo casi verticales y las ondas con numeros de onda casi verticales tienen velocidades de grupo casi horizontales. La primera categoria de ondas esta a menudo relacionada con movimientos convectivos mientras que la segunda corresponde a ondas de gravedad que pueden estar asociadas a la presencia (o a la presencia prevista) de movimientos a escala sinoptica. La velocidad de grupo horizontal que caracteriza a las ondas sonoras de Lamb es la velocidad del sonido, que si determinase la dispersion de las influencias a escala sinoptica transmitiria las influencias al mundo entero en el plazo de un dia. Las velocidades de grupo maximas que se han calculado son tan grandes que limitan las predicciones a un dia o dos maximo para los puntos de la troposfera situados en el centra de los continentes del hemisferio norte, dada la escasez de los datos sobre los ocean os yen la estratosfera. Esta claro que de be haber algun medio de reducir esta estimacion pesimista de la dispersion lateral y vertical de los movimientos a gran escala. El Capitula 2 contiene tambien un simple derivado de la estructura horizontal de una onda de Kelvin que se propaga en un piano de rotacion. Esta solucion se caracteriza por un flujo geostrofico oscilante en una direccion paralela a un limite lateral a la frecuencia de una onda sonora ode una onda de gravedad en un sistema de coordenadas que no esta en rotacion. Las soluciones globales ofrecen una solucion analoga al modo de interceptacion ecuatorial. En la parte final del Capitula 2 se describe el comportamiento de las ondas en el caso de una corriente uniforme sobre crestas sinusoidales de elevaciones superficiales. La propagacion a lo largo de las crestas es ondulatoria o de interceptacion, segun el valor del numero de onda horizontal k · correspondiente a la cresta sinusoidal, y de acuerdo con la velocidad del flujo zonal u. El plano k, use divide en tres zonas en las que las ondas sonoras, de gravedad ode Rossby se pueden propagar verticalmente, yen tres regiones intermedias en las que se produce la interceptacion. Ese comportamiento del numero de onda vertical reaparecera en el analisis de las soluciones globales. En el Capitula 3 se analizan las propiedades de dispersion de las ondas atmosfericas estables en una atmosfera en reposo sin tener en cuenta supuestos de coeficientes constantes ni soluciones relativas a las ondas planales. El objetivo primordial de este planteamiento es comprobar si en las soluciones relativas alas ondas planales se sobrestima en exceso la magnitud de la velocidad de grupo de las ondas de Rossby. La separacion de las variables conduce a une primera serie de ecuaciones («ecuaciones de aguas poco pro fundas») que rigen el comportamiento horizontal de las funciones propias y a una segunda serie de dos ecuaciones que rigen la dependencia vertical de las funciones propias. Estas series de ecuaciones estan vinculadas por un parametro de separacion, la «profundidad equivalente», introducido por Taylor. La profundidad equivalente es positiva en el caso de oscilaciones libres. Sin embargo, deben preverse valores negativos en el caso de oscilaciones forzadas de escasa frecuencia. Esta situacion es analoga a la del flujo interceptado en sentido vertical por encima de una montafia, cuando no se tiene en cuenta la variabilidad del parametro de Coriolis (es decir, el razonamiento tradicional que explica la presencia de un anticiclon sobre una cadena de montafias). En el estado inicial en reposo, cuando la temperatura corresponde a una media climatologica mundiallos modos verticales discretos solo se aplican a las ondas sonoras internas y al modo externa de Lamb (ya sea onda son ora ode Rossby). Al adoptar inversiones termicas mas fuertes se obtienen modos verticales discretos en el caso de ondas internas de gravedad y ondas de Rossby. Segun Eckart, con una termosfera situada por encima de la atmosfera se conseguiria un espectro vertical discreto para el conjunto de los modos. Ahora bien, ello se logra unicamente porque el cuadrado de la frecuencia del empuje vertical tiende a un valor nulo en funcion inversa de la altura, y se puede demostrar que se trata de un reflector muy ineficaz de las ondas de RESUMEN XXIII gravedad intemas y de las ondas de Rossby. De las observaciones efectuadas se desprende que a 50 km de altitud, las velocidades corresponden a niveles de densidad de energia cuatro o cinco veces inferiores a los niveles de densidad de energia cinetica alrededor de la troposfera en invierno. Esta disminucion de la densidad de la energia en funcion de la altura, asi como la reduccion de las fuertes velocidades verticales de grupo obtenida anteriormente, obedecen seguramente a la omision de alg(m factor en los modelos matema ticos utilizados hasta el momento. Este factor es la presencia de una corriente basica, u, y de el se trata en el Capitula 4. Las funciones propias horizontales de la ecuacion mareal de Laplace cambian a medida que varia la profundidad equivalente. El caso clasico se presenta cuando intervienen valores positivos elevados para los cuales las funciones propias se dividen en dos clases: los movimientos de primera clase, que poseen pro piedades similares a las de las ondas sonoras o de gravedad, y los movimientos de segunda clase, que corresponden alas ondas de Rossby. Los valores positivos reducidos de la profundidad equivalente corres ponden a longitudes pequefias de onda vertical. Las funciones propias horizontales quedan confinadas, en este caso, a las latitudes bajas y se puede deducir de la formula no dimensional de las ecuaciones de perturbacion en bajas latitudes elaborada por Mats uno que esta restriccion impide a la fuerza de Coriolis que domine la fuerza de empuje vertical cuando la longitud de onda vertical se hace mas pequefia. Las componentes zonal y vertical de la velocidad de grupo pueden calcularse a partir de las frecuencias asociadas a las soluciones de la ecuacion mareal de Laplace. Presentan los mismos valores elevados que los calculados a partir de las soluciones basadas en ondas planales. De aqui se deduce tambien que no se ha sido incluido todavia en el modelo matematico ningun factor capaz de disminuir la velocidad de dispersion de la influencia. N umerosos meteorologos han observado movimientos ondulatorios macroescalares que responden alas propiedades teoricas de las perturbaciones atmosfericas. En las latitudes ecuatoriales se han observado ondas de Kelvin y ondas mixtas de gravedad y de Rossby que tenian longitudes de onda verticales moderadas. Estas intervienen en el forzamiento de la oscilacion cuasi bianual del viento zonal de la estratosfera ecuatorial. En las latitudes extratropicales se han observado ondas Rossby para el modo externa en el caso de valores bajos de los numeros de onda latitudinal y zonal. Sin embargo, no parece haberse observado la onda mixta de gravedad y de Rossby para el modo externa, quiza porque los analisis meteorologicos se basan en el supuesto del equilibrio geostrofico entre la fuerza del viento y la de la presion, y la amplitud de las frecuencias teoricas es suficiente para que los analisis diarios no tengan la frecuencia suficiente para representar estos sis temas. El Capitula 4 comienza con una descripcion del «modelo barotropico equixalente» ideado por Charney y Eliassen y de la fun cion unidimensional de influencia que estos crearon para las predicciones de 24 horas en un canal estrecho por el que pasa una corriente zonal u. La estrechez del canal reduce considerablemente la dispersion de las ondas de Rossby. Ahora bien, cuando se trata de tiempos de prediccion mas largos, el trazado de su funcion de influencia constituye una verificacion grafica de los conceptos simples basados en la velocidad de grupo. La dispersion zonal de las ondas de Rossby en los analisis de la atmosfera superior puede aclararse mediante la utilizacion del diagrama de Hovmoller. Simmons y Hoskins han demostrado teoricamente de que manera una perturbacion iniciallocalizada en una corriente zonal baroclinica en latitudes medias genera un tren descendente de ondas inestables, similar a los fenomenos que se observan a veces en los diagramas de Hovmoller. Los errores de prediccion de los modelos operacionales aplicados a America del Norte revelan una tendencia sistematica a originarse en el oceano Pacifica. Este proceso fue analizado por Phillips con un modelo que simulaba, a la vez, los aspectos de analisis y de prediccion de un sistema operacional. En 1961, Charney y Drazin llamaron la atencion de los meteorologos respecto al profunda efecto que podia tener una corriente zonal uen la propagacion vertical de las ondas de Rossby. (Se da tambien un efecto analogo en la propagacion latitudinal.) En su forma mas simple el criteria de estos cientificos en lo referente a la propagacion vertical de las ondas es el mismo que descubrio Queney, en 1948, para explicar las ondas estacionarias de Rossby generadas por una corriente de procedencia oeste encima de las montafias. Asi se explica la ausencia de movimiento de torbellino estacionario en las zonas de circulacion de procedencia este, tales como las regiones subtropicales de bajo nivel y la estratosfera de verano, y la preponderancia de numeros de onda horizontal es de escaso valor en la estratosfera de invierno. XXIV RESUMEN Los puntos criticos y los puntos de rotacion son importantes en el analisis de las perturbaciones de una corriente zonal cuando la velocidad angular varia en funcion de la latitud o la altitud. Las opiniones iniciales respecto de los puntos atmosfericos criticos se vieron influidas por la teoria hidrodinamica, concebida a finales del decenio de 1940 para explicar la inestabilidad de la corriente transversal homogenea, segun la cual las ondas perdian energia a causa de la viscosidad que se produce en las inmediaciones de los puntos criticos. Opiniones mas recientes atribuyen mayor importancia a la reflexion y a la no linealidad. Del teorema establecido por Dikii y Katayev se deduce la siguiente conclusion importante: los unicos modos discretos presentes en un modelo barotropico en el que se aplica una corriente basica de velocidad angular variable son los modos cuya velocidad angular zonal de traslacion es menor que la velocidad angular minima de la corriente basica (es decir que carecen de punto critico). Por simple analogia con el caso de las ondas internas de Rossby, este teorema demuestra que solo hay modos discretos en los casos de movimientos de gran longitud de onda vertical. Esta conclusion confirma la existencia de ondas libres de Rossby, que han podido observarse, y al propio tiempo aclara por que no se han encontrado las funciones propias comprimidas segun la latitud de la ecuacion mareal de Laplace en el caso de profundidades equivalentes pequeiias. Las velocidades horizontales de grupo de las ondas de Rossby son las mas altas para el modo vertical externa. Held, Panetta y Pierrehumbert demostraron que la definicion semiempirica adoptada por Charney y Eliassen para la estructura vertical en su modelo barotropico equivalente era similar al modo externa presente en la corriente de procedencia oeste para las ondas de Rossby estacionarias. El diagrama de dispersion de las ondas de Rossby en una corriente zonal variable es complicado, pero se simplifica, en gran medida, si se toman en cuenta las ondas estacionarias de frecuencia nula. Hoskins y Karoly calcularon trayectorias de radiacion de muestra para este caso especial, utilizando corrientes zonales similares a los flujos climatolo gicos. Un grupo importante de trayectorias (descritos en la presente publicacion para el hemisferio norte) son las que provienen de una fuente situada en una latitud baja para dirigirse hacia el noreste hasta su latitud de rotacion, desde donde se propagan hacia el sudeste y el sur hasta su latitud critica. Algunas de estas trayectorias coinciden con el emplazamiento de los centros de teleconexion del hemisferio norte, que fueron descubiertos por Wallace y Gutzler. La energia cinetica de turbulencia se traslada hacia el ecuador en la alta troposfera, como se indica en el estudio de Mak yen los analisis de las secciones transversales efectuados posteriormente por Eliassen-Palm. Para examinar la propagacion latitudinal de los errores de prediccion debidos a las ondas transitorias de Rossby, es necesario aplicar un metodo numerico. Esto se puede conseguir efectivamente calculando por separado una funcion de influencia latitudinal para cada numero de onda zonal y para cada latitud de prevision que tenga interes. Dichos calculos muestran que una distribucion climatologica suavizada de u interfiere de manera significativa con la propagacion libre de las ondas de Rossby de una latitud a otra, lo que puede ocurrir en un flujo zonal en rotacion pura. Si se combinan los resultados asi obtenidos para diferentes numeros de ondas zonales aplicando el teorema de Fourier se obtiene una funcion de influencia bidimen sional con las zonas de sensibilidad maxima para los datos iniciales utilizados con el fin de elaborar una prediccion para un pun to preciso de la atmosfera extratropical. Una es la zona que rodea el pun to situado por encima de la distancia -ut. Este es el equivalente barotropico del tren descendente de ondas generado baroclinicamente del que se da cuenta en el estudio de Simmons y Hoskins. La segunda zona se situa en el flanco ecuatorial del chorro, al sudoeste del punto de prediccion. Esta region actua como fuente de trayec torias (el numero de onda zonal tipico es de 5 a 7) que alcanzan su punto de inflexion hacia el polo y su densidad energetica maxima en la latitud prevista. La presencia de esta segunda zona no depende de la existencia de una latitud en la que el valor de u sea nulo. El primer ejemplo de propagacion ascendente de la energia ondular en la estratosfera fue documentado por Muench. Derome ha presentado datos que muestran una aparente propagacion descendente del error de prediccion geopotencial para la onda zonal numero dos. Por lo demas, cuando se normalizan sus datos por la raiz cuadrada de la presion, la seccion transversal tiempo-altura resultante tiene la forma de la amplificacion resonante de una configuracion estacionaria. Otros ejemplos de propagacion ascendente de las influencias se dan en los estudios de Matsuno, quien demostro como el anticiclon estratosferico de invierno sabre las Islas Aleutianas se ve «forzado» por la configuracion de las corrientes a 500 hPa, y c6mo se pueden producir calentamientos estratosfericos repentinos al amplificar la onda zonal numero dos en la troposfera. Tung y Lindzen estiman que los momentos mas favorables para la amplificacion resonante de ondas troposfericas macroescalares con bloqueo pronunciado y calentamiento estratosferico, se presentan cuando el chorro estratosferico actua en elevaciones mucho mas bajas, en unos 60 km, que las que corresponden a su posicion RESUMEN XXV normal. Los estudios numericos indican que la evoluci6n de las ondas en la estratosfera polar es sensible alas peculiaridades del torbellino potencial media en cada zona. En varios experimentos en los que se han utilizado modelos numericos se han estudiado los efectos causados par el aumento de altitud y resoluci6n de la estratosfera. Los resultados no son concluyentes, pero permiten suponer que las predicciones troposfericas para una semana o mas pueden verse afectadas, de vez en cuando, par estos factores. Varios anexos completan la presente monografia. CHAPTER 1 INTRODUCTION Weather forecasting is now based on the practice of numerical weather prediction. This technique contains two main elements, an analysis element and a forecast element. Analyses are made periodically of the three-dimensional fields of velocity, temperature, pressure, and moisture in the atmosphere, usually at intervals of six hours. This is done by taking a forecast for this time made from the previous analysis, and modifying it with current observations. The standard surface observations and rawinsonde observations from land and ship stations are supplemented with data from aircraft, buoys, and satellites. Data from satellites include temperature and moisture from satellites in a polar orbit and winds that are inferred from the drift of clouds observed from geostationary satellites. This analysis procedure is performed with numer ical methods that attempt to allow for the typical errors of each type of data and for errors in the six-hour forecast. The principles of "optimal analysis" formulated some years ago by Gandin (1963) play a major role in the design of an effective analysis scheme. Recent reviews of this process are available in the book edited by Bengtsson, Ghil and Kallen (1981), and in papers by Hollingsworth (1986) and Lorenc (1986). The forecast aspect of numerical weather prediction is accomplished by a numerical integration forward in time from the analysis. This forecast might be only a short one to provide a forecast to combine with observations at the next analysis time, or it might be a forecast far enough into the future to be of practical benefit. The accuracy of this forecast will depend on several factors: 1. The accuracy of the initial analysis. 2. The fidelity with which the methods used for the numerical integration represent the evolution of those spatial scales that are represented explicitly. The resolution in space and time of the numerical representation of the evolving forecast is important. However, the numerical method itself is also critical; methods must be used that are stable in the sense that they converge to the continuous solution of the physical equations as the spatial resolution is improved. 3. The extent to which the physical processes occurring in the atmosphere are known well enough to be stated in numerical form. This is especially critical with respect to motions that occur on spatial scales that are too fine to be represented explicitly in the spatial grid used in the numerical integration. These processes must be "parameterized" to describe their effect on motions oflarger scale. Moist and dry convection, and flow over rough terrain, are important examples of such processes. 4. The instability of the atmosphere at all spatial scales. The instability at scales that are represented explicitly in the numerical process will have an effect on item 2 above, i.e. the explicit accuracy of the numerical method. The instability at smaller spatial scales is equally important because it will determine the impact of the poorly known turbulent processes referred to in item 3. 5. Theoretical studies of the predictability of fluid motion indicate that a profound effect is exerted by the spatial spectrum of the typical field of motion. The non-linearity of the fluid dynamical equations will ensure that initial disturbances in the unobserved small scales will, at later times, have some effect on the larger scales that are being forecast explicitly. Motions that have a relatively flat spectrum seem to allow this contamination of the large scale to occur more rapidly than do those motions for which the spectral density decreases rapidly with increasing wave number. 6. Much of the energy of atmospheric motions is in patterns that change very slowly from season to season. For short forecasts, it may be sufficient that the forecast system merely preserve the near stationarity of these patterns. However, a system that is to make forecasts for a week or two must do a good job of predicting the motion and development of these quasi-stationary patterns of large horizontal dimensions. In other words, the forecast system for medium-range forecasts must of necessity give a good accounting of atmospheric climatology. 2 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION A complete study of the scientific principles of weather forecasting must consider all ofthe above factors. However, there is some justification for the view that at the present time, the accuracy of initial analyses is a major limitation to the successful prediction of large-scale flow patterns. It is necessary then to pose the following question: How rapidly do influences from regions ofinadequate data spread to those regions for which an accurate forecast is desired? The importance of the question was recognized early by Ertel (1941 ). His answer was very pessimistic; he concluded that one needed data from the entire globe. A more optimistic answer was obtained by Charney (1949), who asked the question only with respect to a forecast for one day. Nowadays the technique of numerical weather prediction is being extended to experimental forecasts as long as a month, so that Charney's answer is insufficient. A more thorough answer is needed. Fortunately, a great variety of studies have been made in the past forty years that begin to provide an answer. 1.1 The atmospheric equations of motion The equations of motion for the atmosphere can be stated at a variety of levels of approximation. The most common form is based on several successive approximations of a geometric nature (Phillips, 1972), namely that (a) gravity can be approximated by normal gravity so that a geocentric co-ordinate system with longitudinal symmetry can be defined by the surface of the reference spheroid and the two families of surfaces orthogonal to it; (b) the ellipticity of the reference spheroid is small enough for the geocentric co-ordinate system to be approximated in turn by a conventional spherical co-ordinate system; (c) for motions occurring within several hundred kilometres of the Earth's surface, gravity can be taken as constant, and the radius of the Earth can be approximated by a constant value. (A simpler version of(1.4)-(1.6) can be obtained from the original spherical equations by a systematic simplification of the scale factors r and r cos e, as shown in Phillips (1968, 1973)). We will use the notation: a = radius of the Earth (6 371 km); g = gravity; Q = angular velocity of the Earth; A. = longitude; (1.1) tJ = latitude; p = pressure; p = density; T = temperature. The three velocity components are defined by: dA u =a cos tJ dt dt) V a (1.2) dt dz w dt and the substantial derivative following an individual parcel is d( ) a( ) u a( ) v a( ) a( ) --=--+ --+---+w--· (1.3) dt at a cos tJ aA. a atJ az The three components of the equation of motion are: du ap -= + (2Q+ u t)) (v sin tJ - w cos tJ); (1.4) dt p a cos tJ aA. a cos INTRODUCTION 3 dv 1 ap - (2.Q sin 1) _ vw (1.5) dt pa a?J + ac~s d)u a ' dw ap ( u ) v2 =- g- p -a + 2.Q + 1) u cos 7J + - (1.6) dt z a cos a (Friction has been omitted in these equations). The conservation of mass is expressed by the equation 1 dp aw 1 ( au av cos 1)) -- (1.7) p dt = - v3 · v = - ----az- a cos 1) · a)., + a?J · The first law of thermodynamics for a perfect gas is dT dp q --=-- (1.8) dt p dt cp ' where q is the rate ofheating per unit mass and CP is the specific heat at constant pressure. For a perfect gas we also have the following two equations of state: P = pRT, (1.9) e = T ( Po \ K. K = _!!_ ) PI ' cp , where e is the potential temperature. (The effect of moisture is not included in the above equations). 1.2 Uses of the linearized equations of motion A fruitful step in approaching the hydrodynamic aspect of the forecast problem is to consider the behaviour of small-amplitude perturbations superimposed upon a simplified basic state. The amplitudes of the perturbations are assumed to be small, so that the non-linear terms in the original equations can be ignored or simplified. The resulting differential equations often have coefficients that vary, but they are linear. This approach has a long history in atmospheric and oceanic studies, beginning with the tidal investigations by Laplace (1799), Margules (1892), Hough (1898), and Love (1913). The analysis of shearing instabilities by von Helmholtz (1888) and of convective instabilities by Rayleigh (1916) were followed some years later by the Norwegian school of theoretical meteorologists. Their monumental work Physikalische Hydrodynamik (Bjerknes et al., 1933) contains many specialized perturbation analyses. Among these is an attempt to explain the extratropical cyclone as an unstable wave on the polar front. Kochin (1932) also considered this problem. These attempts were not successful, however. The practical usefulness of linearized perturbation solutions for weather forecasting was first demon strated by Rossby in 1939. A few years earlier, J. Bjerknes (1937) had shown that the latitudinal variation of the Coriolis parameter could have an important role in the motion of the wave-like disturbances in the upper troposphere that seemed to accompany cyclones on the polar front.* Rossby postulated a very simple model of the atmosphere. The velocity in his model was constrained to be horizontal and non-divergent, so that it was governed by the conservation of absolute vorticity about the vertical axis. The wave perturbation was superimposed upon a uniform basic current from west to east of speed u. The only variable source ofvorticity then was the latitudinal variation ofthe Coriolis parameter, f = 2 .Q sin 7J. The result was the famous formula: p eR= u- ¥" (1.10) *Platzman (!968) refers to papers by V. Ekman (1923, 1932) as an additional source of inspiration for Rossby's emphasis on the important role of vorticity. 4 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION eR is the zonal phase speed and k = 2 nl Lis the west-east wave number. fJ represents the latitudinal variation of the Coriolis parameter: d(2Q sin V) fJ= (1.11) adv Additional successes of the perturbation approach came rapidly after the conclusion of World War II. Lyra (1943) and Queney (1947, 1948) initiated the theory for air flow over mountains. Charney (1947) and Eady (1949) independently used the perturbation approach to show that the large-scale waves of extratropical latitudes might be explained as unstable perturbations. Their basic state differed from that used by the Norwegian school-Charney and Eady replaced the sloping discontinuity surface of the Norwegian polar front by a uniform poleward decrease of the temperature at all levels. Their mathematical approach also differed from earlier methods in that they simplified the equations in a consistent manner by considering the spatial and time scales of the phenomenon under consideration. As mentioned earlier, the instability of fluid flow has major implications for the accuracy required for the analyses used in weather prediction; even if the equations used for prediction are completely accurate, inaccuracies in the initial analysis will act as unknown initial perturbations whose unstable growth might quickly degrade the forecast. A concept of equal importance for the theory of forecasting is that of wave dispersion. This is the process whereby an initial disturbance oflimited spatial extent can spread into different regions of the fluid. This can occur without direct advection by the fluid motion. Thus, an error in the initial analysis in one part of the globe-for example, in an oceanic area devoid of observations-can propagate to other parts of the atmosphere even if the net motion of air is in a different direction. This important effect is measured by the group velocity. Rossby (1945) introduced the concept of group velocity into meteorology and his student T. Yeh (1949) explored its consequences for the dispersion in a zonal direction of an initial concentration of vorticity. A further application to all three dimensions was made by Charney in 1949. At that time, Charney had arrived at a rationale for the atmospheric model to be used in making the first modern attempt at numerical weather prediction with an electronic computer. It was the non-divergent vorticity model ofRossby except that full attention would be given to the non-linear aspects of the horizontal advection ofvorticity. A careful analysis ( 1949) had shown Charney that the best level for applying this model-the equivalent barotropic level-was at 500 hPa. But an an important question remained: How large an area ofinitial data need be considered about a point in order to make a 24-hour forecast for that point? If the initial analysis in the central part of the Pacific Ocean is unreliable, for example, how far from that point will the uncertainty spread in 24 hours? If Charney and his collaborators had chosen too small an area in which to make their computations, the first modern attempt at numerical weather prediction would have been severely degraded by the spread of errors from outside the small forecast area. If this had happened, the attempt at numerical weather prediction with the newly developed electronic computer ofvon Neumann might have been as discouraging as was Richardson's attempt 30 years earlier. Furthermore, the significance of the newly invented quasi-geostrophic theory of atmospheric motion would have received a tremendous setback. On the other hand, if a needlessly large area had been selected, the limited capacity of the electronic computer might have been exceeded. Fortunately, Charney was able to apply group velocity arguments in a quantitative manner so that a reasonable decision could be made about the minimum forecast area. In recent years group velocity and related concepts have been applied with considerable success in attempting to explain the influence of one latitude belt (for example, the tropics) on other latitudes, and the influence of the troposphere on motions higher up in the stratosphere. Karoly and Ho skins have described the group velocity properties (e.g. "rays") of three-dimensional Rossby waves, and the article by Held (1983) describes many significant applications to steady-state flow patterns. Excellent discussions of the basic concept of group velocity and rays can be found in papers by Whitham (1960), Lighthill (1965, 1966), and Bretherton ( 1971 ). Whitham refers to a discussion by Landau and Lifshitz (1959). For convenience, a brief summary is presented in Appendix A. CHAPTER 2 SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE For the basic state, we consider a resting ideal gas atmosphere in which temperature (T) and pressure (p) vary only with elevation. This variation will be described by three parameters used by Eckart (1960). (When necessary, an overbar denotes a basic state value of the variable.) d lne (2.1) N2 = g + dz ~ ~ U, ~), Cs2 = yRT; (2.2) g 1 r= --+- d ln (jJCs) Cs2 2 dz g (2 - y) 1 d'f (2.3) 2Cs2 - 4T dz d ln (Cs j/) (2- y) ; r = = 3/7. 2 dz y N 2 is the buoyancy frequency and e is the potential temperature. Cs is the speed of sound, andg is the ratio CvfCv = 7/5. G represents the effect on specific volume of a steady upward motion in an adiabatic atmo sphere.* Table 2.11ists values of these parameters for selected altitudes, taken from the 1976 US Standard Atmosphere (National Oceanic and Atmospheric Administration, 1976). G is almost constant except in the thermosphere. N 2 has large values in the stratosphere and lower thermosphere, with smaller values in the mesosphere and troposphere. For typical tropospheric values ofT = 260 Kand a lapse rate dT ldz of -6.5 degrees per kilometre, the three parameters have the values N2 = 1.2 X 10-4 s-2, CS 323 m s- 1, (2.4) r 3.4 X 10-5 m- 1 = 11(29 km). For reference, we note that the angular velocity of the Earth (.Q) has a value such that .Q2 = 5.288 X 10-9 s-2• N 2 I .Q2 is therefore about 2 X 104• The largeness of this ratio will be important for the dynamical theory of atmospheric motion. For horizontal co-ordinates, we will use Mercator co-ordinates instead oflatitude and longitude. They are defined by x = aA., (2.5) Y =a ln 1 + sin V ) . ( cos v (2.6) * The equation (3) on page 61 of Eckart's book that defines r lacks a factor of 2 in the first term. 6 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION a is the radius of the Earth, .A, is longitude and 7J is latitude. These co-ordinates are conformal and share a common horizontal scale factor h: y h = cos 7J = sech - (2. 7) a Eckart (page 55) has defined normalized values of the perturbation variables that result in very convenient equations. The original perturbation variables are Velocity: u,I Pressure: p', (2.8) Potential temperature: ()'. ) The normalized variables introduced by Eckart are U, P, and Q: u' Velocity: U=- E(z) ' Pressure: P = p' E(z), (2.9) ()' Potential temperature: Q = de E(z) dz TABLE 2.1 Distribution of temperature (T), square of the Brunt-Vaisala frequency (N2), reciprocal of the ), Eckart compressibility factor r, sound speed ( C5 the Eckart scaling factor (£) and kinematic viscosity ( v) z T N2 liT Cs £/£(0) V (km) (K) (s-2) (km) (m s-1) (I) (m2 s-1) Thermosphere I 000 I 000.0 0.96 w-4 136 634 I 107 - 800 I 000.0 0.96 136 634 8 106 - 600 999.8 0.96 136 634 2 106 - 400 999.8 0.97 136 633 5 J05 - 200 854.6 1.47 130 586 5 104 - 100 195.1 6.11 29 280 2 103 - Mesosphere 90 186.9 5.10 25 274 700 - 80 196.6 3.98 25 281 300 0.8 70 217.5 3.28 27 296 130 1.9 J0-1 60 245.5 2.78 31 314 65 5.5 I0-2 Stratosphere 50 270.7 3.34 36 330 35 1.7 w-2 40 251.1 4.91 38 317 18 4.2 I0-3 30 226.7 4.89 33 302 9 8.2 I0-4 20 216.7 4.57 30 295 4 1.6 J0-4 Troposphere 10 224.3 2.28 26 300 2 3.5 w-5 5 255.7 1.25 29 321 I 2.2 w-5 0 288.2 1.11 w-4 30 340 I 1.5 10-5 2 Tvalues are based on the 1976 US Standard Atmosphere. At all heights, Nl, r, and C5 are computed as functions of T for a perfect gas mixture typical of the troposphere. SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 7 E(z) is a scaling factor introduced by Eckart: E(z) = (jJCs)-t (2.10) E increases greatly with height, as shown in Table 2.1. This scaling produces perturbation equations whose dependence on z is such that a uniform magnitude of U, P, and Q with height corresponds to a uniform distribution of energy density with height. It also results in more symmetric equations than result from using the original perturbation variables. U, V, and W will denote the x (eastward), y (northward), and vertical components ofU. To begin with we consider only perturbations on a resting basic state. The perturbation equations are then five in number: au cs aP . - = - - -- - J W + fV, (2.11) at h ax av cs aP - = - - -- - fU (2.12) at h ay ' aw ( aP ) 2 -at = - CS - az + TP + jU + N Q, (2.13) aP at = - CS ('V 3 . u - TW), (2.14) aQ = _ w (2.15) at · V 3 is the three-dimensional gradient operator. f and j denote the vertical and horizontal components of the Coriolis vector: f = 2.Q sin 7J, j = 2 .Q cos 7J. (2.16) We shall find that the j component plays no significant role because N 2 greatly exceeds .Q 2. Equations (2.11)-(2.15) yield an "energy" integral after multiplication with U, V, W, P, and Q: 1 a - -- (U2 + p2 + N2Q2) = - C V'3·PU (2.17) 2 at s · The Coriolis terms and the r terms have dropped out. Eckart (page 54) refers to the P2 term as the elastic energy, and the Q 2 term as the thermobaric energy.* When transposed back to the original variables, this energy equation is somewhat more complicated in appearance. 1 2 -pa _ [ u I 2 + ( ---PI )2 +-g ( ~8')2] = V'3·puI I (2.18) 2 at pCs N2 e 2.1 Gravity and acoustic waves The coefficients N 2, r, Cs, h, f, and j in (2.11 )-(2.15) are variable. This complication can be avoided to some extent if we assume that the wave lengths ofthe solutions are short in comparison to the distance over which the coefficients vary. (For a fuller analysis, eigensolutions satisfying appropriate boundary conditions * For hydrostatic motion, the sum ofthe elastic and thermobaric energy can be related to the available potential energy defined by Lorenz (1955). 8 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION must be considered. This will be considered briefly in a later section.) This approximation is associated with the acronym WKB and is related to the group velocity concept that is summarized in Appendix A. We will consider solutions having the form of plane waves, in which the variables are described by the real parts of the expressions u [J exp (ilf!) V V exp (ilf/) w W exp (ilf!) (2.19) p P exp (ilf!) Q Q exp (ilf/) This results in five homogeneous equations: - iw [J - jV +JW + ikC5P 0, j(j -iwV + i/C5P 0, - N2Q (2.20) - JU - iwW + C5(im + nf5 0, C5kU + C5lV + C5(m + ii)W - wP 0, w -iwQ 0. In order that non-zero solutions exist, the determinant must vanish. This defines a fifth-order polynomial for the frequency. It has the special form w X E(w) = 0, (2.21) where E(w) is a fourth-order polynomial: E(w) = w4 - [f2 + p + N2 + Cs2(k2 + {2 + m2 + F2)] w2 + [2JCs2kF]w + + {N2 [f2 + Cs2(k2 + f2)] + Cs2 [(fm + lj)2 + f2P]}. (2.22) This yields five solutions for w. The solution w = 0 will require special treatment, and will be discussed after the other four solutions. The quartic equation, E(w) = 0, is complicated by the presence of}, the horizontal component of the Coriolis vector, which does not appear in the expression in quite the same way as f Fortunately, it can be shown that the }-terms play no significant role in determining w. This is because the normal static stability of the atmosphere, as expressed by N 2 ~ 10-4, isconsiderablylargerthanf2 orj2, which are at most of size 10-8• In the coefficient of the w 2 term, bothf2 andP are dwarfed by the N 2 term. Within the constant term, we can follow Queney (1947), and consider the three related terms that form a quadratic in the meridional wave 2 2 2 2 number!: C5 [(N + p) ! + (2fmj) l + f2m ]. The ratio of the (2fmj) !term to the sum ofthe other two terms has a maximum value of _J __ ~j_ ~ 10-2 (2.23) (N2 + p)i N Allj-terms may therefore be ignored in the constant term of(2.22). There does not appear to be any simple 2 theoretical argument to dispose of the remaining }-term, (2}C5 kr) w*. But numerical solutions for w including this term differ by a factor ofless than 0.003 from solutions that ignore all the }-terms. We therefore conclude that the horizontal component of the Coriolis vector plays little role in atmospheric motions, because the normal atmosphere has a vertical stability stratification sufficiently large for the (2.24) frequency of buoyancy oscillations greatly to exceed the angular velocity of the Earth. * Queney's arguments (1947, pp. 11-12) for discarding this term are not convincing. More elaborate arguments are presented in Queney (1950). SIMPLE PLANE W A YES ON A RESTING ATMOSPHERE 9 Eckart dubbed this "the traditional approximation". It introduces a simplification into the atmospheric equations of motion of considerable practical significance. Queney (194 7) seems to have been the first to justify this on the basis that N 2 greatly exceeds Q 2• The simplified WKB dispersion relation is w X E 0(w) = 0, (2.25) where E0 is a quadratic in the square of the frequency. The horizontal wave numbers k and l now appear only in the combination K2 = k2 + /2, (2.26) and the Eckart parameter r appears only in combination with the vertical wave number m, as M2 = m2 + T2. (2.27) The quartic equation is w2- p Eo (w2) = K2 (w2 - N2) + M2 (w2 - f2) - (w2 - N2) 0, (2.28) C2s 2 and yields two values for w : 2 w2 = [f2 + N2 + cs2 (K2 + M2)] ± ± {[{2 + N2 + Cs2(K2 + M2)]2 - 4 [f(N2 + Cs2M2) + N2Cs2K2]}!. (2.29) The positive branch yields internal acoustic waves, modified by the stratification and rotation. Their frequencies are bounded only from below; forK and m~ 0, this minimum frequency is wA2 (min) = N 2 C 2 P = N2 (2.30) + s ~40 . The negative branch yields internal gravity waves, modified by compressibility and rotation. For very long horizontal wave lengths (K2 ~ 0), the frequencies are bounded from below by wd (min) =.f. (2.31) For very short horizontal wave lengths (K2 ~ oo ), the gravity wave frequencies are bounded from above by 2 wd (max) = N The acoustic waves are oflittle importance for weather forecasting, and, as shown above, occupy a different frequency range from that occupied by the gravity waves. There is one very important exception to this. This is the Lamb wave (Lamb, 1932; p. 548). This solution 2 (without rotation) was derived by Lamb for an isothermal atmosphere, in which N , r, and C5 are constants. It is therefore obtainable in principle from (2.29). For Lamb's solution the Eckart variables (2.9) vary in the vertical as exp (- rz ). *In our notation this is equivalent to m = ir, which in turn signifies that M is zero. By putting M = 0 in (2.29), we find two possibilities: w2 = N 2 and w2 = K 2Cs2 +.f. (2.33) The first of these cannot satisfy boundary conditions at the ground and at z = oo. The second can satisfy the boundary conditions, and is the frequency for Lamb waves in the presence of the Coriolis force. In meteorology the Lamb solution is sometimes referred to as an external gravity wave, in analogy to the surface * The perturbation velocity v' and the ratio p'!p vary as exp (gz/ CpT) ~ exp (z/ 26 km). 10 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION gravity waves that can exist in addition to internal gravity waves in a stratified ocean. However, in an isothermal basic state the Lamb wave has no vertical velocity and no entropy perturbation (Wand Q = 0)*. Buoyancy forces therefore play no role in its propagation and it seems more meaningful to consider it as a horizontally propagating acoustic wave whose vertical structure simply reflects a kinematic effect of the stratification of density and temperature with height. A solution similar to it also exists in a non-isothermal atmosphere. Bretherton (1969) has provided a recipe for estimating its frequency by the Rayleigh-Ritz procedure. (See Appendix B.) It exists as a wave solution even when the hydrostatic approximation is imposed on the atmospheric equations, as is the prevalent practice in numerical weather prediction. This approximation eliminates the internal type of acoustic waves. The Lamb mode is then the highest frequency mode, which gives it the dominant role in setting the computational stability criterion for numerical in tegration of the equations of motion. 1 000 100 10 10 000 2 000 1 000 500 200 100 Figure 2.1 - Location in w, K space of the spectrum of acoustic waves and gravity waves. The ordinate is the logarithm of the frequency. labelled in cycles per day. The abscissa is the logarithm of the horizontal wave number K, labelled as to wave length in kilometres. The continuous acoustic and gravity spectra occupy the spaces Ac and Gr. Lamb is the discrete Lamb mode. Vertical wave numbers m increase upward within Ac and downward within Gr Figure 2.1 shows the regions ofK, w space that are occupied by the acoustic waves (region Ac), the gravity waves (region Gr), and the single curve for the Lamb wave. The boundaries of regions Ac and Gr are characterized by a zero value for the vertical wave number, m. In region Ac, frequencies increase upward with increasing m, whereas in region Gr, frequencies decrease with increasing m. Equation (2.29) for the two values of w 2 can be split into two simpler forms by assuming various limits for the wave numbers. For example, at high wave numbers the positive branch yields f2K2 + N2m2 w2 (acoustic) """ (K2 m2) Cs2 (2.34) + + K2 + m2 and the negative branch produces p m2 + N2K2 2 w (gravity) ""' K2 + m2 (2.35) * Note the relations (2.39) and (2.40) for Wand Q. SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 11 2 (2.34) shows how the ordinary sound wave frequency (KZ + m ) csz is increased by rotation and stratifica tion. The approximation (2.35) for the gravity waves has suppressed all effects of compressibility except that N 2 is still measured by the vertical rate of change of potential temperature. The association ofj2 with m 2 and NZ with KZ in (2.35) recognizes the approximate incompressibility of these gravity wave motions. (Large values of m imply that the motion in the gravity wave is almost horizontal; the Coriolis force is therefore the dominant restoring force. Large K 2 implies that the vertical velocity is larger than the horizontal velocity; the stabilizing effect of N 2 now provides the major restoring force.) The acoustic waves have the opposite association ofP and N 2 with the wave numbers, reflecting the parallelism ofU and the wave number that exists in sound waves. A choice of one of the two branches of (2.29) might seem to yield eight solutions that share the same magnitudes of the frequency w, the horizontal wave number IKI, and the vertical wave number Iml: w = ±lwl, K = ± IKI, and m= ± m. However four of these are merely the complex conjugates of the other four possibilities. Usually the most convenient choice is to consider only the four solutions with positive m. It is worth while at this point to consider the kinematic relations between each of the variables and the pressure variable for these plane wave solutions. The results are p = Re [P exp (i!f!)] (2.36) CS - u = wz _ p Re [(kw + ilj) P exp (ilf!)], (2.37) CS - V = wz _ p Re [(/w - ikf) P exp (ilf!)], (2.38) w CS - W = wz _ NZ Re [(m - iF) P exp (ilf!)], (2.39) - CS - Q = wz _ NZ Re [(im + T) P exp (ilf!)]. (2.40) These are for three-dimensional plane waves, in which no attention has been paid to boundary condi tions. The boundary condition that W vanishes at the ground (we ignore mountains at this time) for all x, y, and t can be satisfied if we add two of the above solutions, having opposite but equal values of m: m2 = - mi> (2.41) and "pressure" amplitudes of equal magnitude but different phase: (m1 + iF) P2 = (m1 - iF) P 1. (2.42) The result for the vertical velocity is we (2.43) W= s ~ sin (m 1z) Re [2 i(m 1 - iF) P1 exp (i!f!')], where the modified phase If/' no longer contains the term mz. These combined solutions do not propagate vertically, but Figure 2.1 still serves as a diagnostic diagram for locating the regions of acoustic and gravitational waves of this type in an isothermal atmosphere. It is instructive to express U and V as components parallel to the horizontal wave number vector and perpendicular to it (to the left). To this end we define 12 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION kU+ IV VII K (2.44) kV-lU vl_ K The single plane wave solutions (2.37) and (2.38) can then be written as wKCs Re [P exp (iiJI)], (2.45) V11 = wz- f2 j](Cs V_1_ = w2- p Im [P exp (ilf/)]. (2.46) The hodograph of the horizontal velocity vector is the curve that is traced out in a graph having U and Vas co-ordinates, by the behaviour of U and V with time at a fixed point in space. Within the context of the assumption of small amplitudes, this will also be the trajectory ofthe individual parcels of air. The formulas for V11 and V _1_ above show that at a fixed point in space, they can be written as v11 = Aw cos (wt) (2.47) and V _1_ = - Af sin (wt). (2.48) 2 A is the constant IPI KC5( w - f2) and the origin oft has been chosen for convenience. The hodograph is an ellipse, whose major axis is in the direction of the horizontal wave-number vector. The parcels move in the ellipse in an anticyclonic manner. The ellipse degenerates to a circle if the frequency becomes as small as f (i.e. for gravity waves and the Lamb wave at very large horizontal wave lengths). The horizontal motion of individual parcels is then called "inertia-circle motion". The presence of lateral boundaries will of course modify the simple kinematic picture described by (2.47) and (2.48). 2.2 Rossby waves It is very important to acknowledge the presence of the solution w = 0 (2.49) in (2.21). Its appearance is not a result of careless algebra. In this solution, (2.37)-(2.39) show that the horizontal velocities are geostrophic, and that the vertical velocity is zero. From the viewpoint oflarge-scale weather prediction, this trivial solution of (2.21) is the most important of the five solutions. Most of the hydrodynamical theory underlying the practical success of large-scale numerical weather prediction can be viewed as an elaboration and generalization of the seemingly trivial solution w = 0. The solution w = 0 of(2.21) is presumably valid for all wave numbers except possibly those so small as to invalidate the WKB method. But this is not so. To see this we reconsider the original perturbation equations (2.11)-(2.15) and impose the requirement (suggested by w = 0) that aI at = 0. W is then zero and U and V are geostrophically related toP: a For-=at 0: C5 ( aP (U,V) !!___) (2.50) fh ay' ax ) SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 13 To complete the requirement we also set aP/at to zero in (2.14). Since W is zero, this requires that the horizontal divergence of U, V must vanish in this linear steady state. If (2. 50) is substituted in the horizontal divergence expression we get: 1 V d 0 = _1_ ( ahu + ahv) = (2.51) hz ax ay f a du (2Q sin u). Steady state is therefore possible in (2.11 )-(2.15) only for V= 0, i.e. for pure zonal flow. This effect of the latitudinal variation of the Coriolis parameter is well known (Lamb, 1932; page 333). U and V are also geostrophic in the w = 0 solution of equations (2.20). But there was no evident requirement, as established in (2.51), that V had to be zero. This contradiction cannot be resolved by an appeal to the acoustic and gravity waves, either, since their frequencies cannot be made to approach zero by reducing k to zero. We recall that the Coriolis parameter was treated as being constant in the derivation of (2.21 ). Evidently the w = 0 solution must be examined more closely. A refined WKB treatment of this is performed in Appendix C. The result of that treatment is the following formula for the Rossby frequency wR: f3k WR = - (2.52) p 1 kz + Jz + -+- [(fm + j/)2 + prz] CsZ NZ The coefficient f3 is the notation given by Rossby to the gradient of the Coriolis term: /3=-1a dud (2Q sin u) = 2Q cos u (2.53) The contribution from the} term in (2.52) is again negligible, as in the acoustic and gravity waves, because NZ dwarfs QZ. We therefore drop the j terms. f3k WR (2.54) mZ rz )· kZ + fZ + p + + (CS -2 NZ In an isothermal atmosphere it can be shown that 1 1 rz (2.55) NZ = 4HZNZ czs + where H is the scale height for the pressure: -1 dp g (2.56) H P dz -RT. This gives an alternative expression for wR: f3k WR (2.57) kZ + f2 + p (mz + _1_)· NZ 4HZ 14 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The concept of the equivalent depth h will be developed in Chapter 3, where it will appear as a surrogate for the vertical wave number m. In that notation wR assumes the simpler form fJk WR =- (2.58) k2 + 12 + f2 . gh (2.52) has been obtained by an unattractively formal derivation, whose only virtues are that it is a logical use of the WKB approximations (guided, of course, by the hindsight developed by more heuristic approaches!) and that it enables the question of the horizontal Coriolis parameter 2Q cos v to be considered in a moderately rigorous manner. A much more direct and physically appealing derivation of wR is the following: We begin by deriving the equation for the time derivative of ~' the vertical component of the vorticity: ( = h-2 ( ahv _ ahU)· ax ay (2.59) We use the original linear equations (2.11 )-(2.15), after neglecting the j-terms. The result is V df = _ fV . U2 - dy · (2.60) !i_at 2 71 The horizontal divergence in turn can be evaluated from (2.14) and (2.15). -JV2·U2=!-a ( ---+rQ.P aQ 0 (2.61) at Cs az If we now define the relative potential vorticity, PV, PV = ( -- f P + f ( --aQ FQ) , (2.62) CS az and take its time derivative, we find that PV changes only because of the advection off" aPV V df (2.63) at h dy' The significance of PV in the present plan€ wave solutions can be seen by evaluating PV for the acoustic-gravity solutions defined by (2.28) and the kinematic relations (2.36)-(2.40) suitable for those waves. We find fCs ( = i(kV - lU) = K 2 P exp (ilf/), (2.64) wz - f2 and aQ _ rQ = fCs M 2 P exp (ilf/). (2.65) az w2 -N 2 SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 15 PVis then equal to K2 1 M2 ) (2.66) PV (ac-gr) = ( w 2 _ f2 - Cs 2 + w2 _ N 2 fCs P exp (ilf/). But the bracketed expression vanishes because it is the simplified acoustic-gravity wave frequency equation (2.28). Simple acoustic-gravity waves have no relative potential vorticity. (2.67) This seems to have been first stated explicitly by Obukhov (1949), for the "barotropic" case (equivalent to the Lamb solution). Let us now insert into PV the kinematic relations for the special solution for which w = 0. (This can be obtained by first putting w = 0 in (2.37)-(2.40) or, more directly, by simply setting w = 0 in the acoustic gravity formulas above.) 2 1 M ) Cs - . PV(w = 0) = - [ K2 + f2 ( Cs2 + N2 l f p exp (llf/). (2.68) Thus, thew = 0 solution has a non-zero value for the relative potential vorticity. (2.69) Furthermore, we see that if this is inserted into the equation (2.63) for 8PV/at, and the geostrophic relation for Vis used to evaluate the advection off, the formula for wR is recovered. Potential vorticity is therefore the important dynamical factor in the propagation of Rossby waves.* In its simplest form, PV reduces to the vorticity itself. Rossby's non-divergent model, by including no dynamic effects other than conservation of vorticity, captures the essence of what otherwise is a very subtle aspect of the fluid dynamics of rotating fluids. Equation (2.39) has not been used in this derivation. When thej-term is discarded, (2.39) shows that this type of wave motion is hydrostatic with a balance between the vertical pressure gradient and the buoyancy even at first order. Hydrostatic balance is normally justified by an appeal to a small value of the aspect ratio (klm). However, this assumption has not been made in arriving at the Rossby formula. The source of this strong constraint is the control exerted by the Earth's rotation through the geostrophic balance of (2.37) and (2.38). The famous rotating experiments by G. I. Taylor (1923) demonstrated this effect for a barotropic fluid. The effect is weaker in our case of a stratified fluid but in the case of the Rossby wave motion, it is still present. The equation for wR differs from the frequency formula (2.28) for internal acoustic and gravity waves by an asymmetry in the horizontal wave numbers k and l. The trace velocity in the west-east direction, given by wR/ k, is negative. The squared wave numbers in the denominator of(2.57) are also asymmetric, in that m 2 is multiplied by the factorf2/ N 2 """" 1o- 4• This is equivalent to stretching the vertical co-ordinate z by a factor N/f"""" 100. This geometric effect is typical ofatmospheric motions that are hydrostatic and in near-geostrophic equilibrium. It seems to have been noticed first by Prandtl (1936) and by Rossby (1938). The horizontal scale defined by N horizontal scale = f times the vertical scale (2.70) was given the name "horizontal radius of deformation" by Rossby. * Rossby waves can also exist in a rotating container of homogeneous incompressible fluid, if some of the containing surfaces are not perpendicular to the rotation vector. They then appear as modifications to inertia waves of zero frequency (Phillips, 1965). 16 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION For the vertical structure of the Lamb wave, we set m2 = - F 2, with the result fJk wR(Lamb) = - (2. 71) kZ + f2 + _E_ czs This gives a significantly larger frequency than that of the internal modes. The Rossby wave has an interesting history in addition to its interesting dispersion properties. The Symons Memorial Lecture by Platzman (1968) is a highly readable account of both aspects. A detailed presentation and geometrical interpretation of the dispersion properties of non-divergent Rossby waves is to be found in the paper by Longuet-Higgins (1964). Karoly and Hoskins (1982) describe ray paths for a selection of three-dimensional Rossby waves. 2.3 Preliminary consideration of group velocity Eckart (1960; p. 110) has introduced the informative step ofrepresenting the dependence of the fre quency on the wave numbers as a graph in which K and m are the abscissa and ordinate, respectively. A schematic depiction of this for the acoustic-gravity equation (2.29) is given in Figure 2.2. For the acoustic branch, curves of constant ware slightly flattened ellipses, centred at the origin. For the gravity wave branch, curves of constant w are hyperbolas, open to the K-axis. 3.------.------., 2 --- m \ \ \ \ \ \ \ I I I 2 3 k Figure 2.2 - Propagation surfaces of constant frequency for internal acoustic waves (dashed line) and internal gravity waves (solid line). as a function of horizontal (K) and vertical (m) wave numbers. vwA and vwa indicate the direction of the group velocities This type of graph is especially useful in indicating the properties of the group velocity aw aw) Cgrp = ( aK , am . (2. 72) We see that Cgrp for the acoustic waves is almost parallel to the wave number vector (K, m). For the gravity waves, this is true only for small values of the vertical wave number; over most of the diagram, Cgrp tends to be almost perpendicular to the wave number vector. SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 17 The vertical component of the group velocity plays an important role in the theory of atmospheric motions. By using (2.29), we can write aw m aw 2 m w2- p C2 - (2.73) am w am 2 s w ±ROOT ' where the optionally signed term "ROOT" in the denominator is exactly the same as the square root term appearing in (2.29). Since w 2 exceeds p, we have the following result for plane waves in a resting atmo sphere: aw w Acoustic waves: -- and - have the same sign. am m (2.74) aw w Gravity waves: -a- and- have opposite signs. ) m m The equivalent result for wR is aw w . . Rossby waves: -- and - have opposite Signs. (2.75) am m TABLE 2.2 Frequencies (cycles per day) and group velocity components (kilometres per day) for plane gravity waves, and the Lamb wave, as a function of horizontal and vertical wave lengths (kilometres) Horizontal wave length (km) Infinity 10 000 5 000 2 000 1000 500 100 10 5 2 I n: 0 3.5 7.5 19.5 40 80 400 4 003 8 005 20 014 40 030 Lz = 1 km: Freq. 1.41 1.42 1.42 1.42 1.42 1.45 2.1 15.2 28.9 68.1 107.6 cgx 0.00 1.64 3.28 8.18 16.29 32.03 111.5 149.3 143.4 108.9 53.8 Cgz 0.00 -.00 -.00 -.00 -.02 -.06 -1.1 -14.9 -28.6 -54.4 -53.8 Lz=2 km: Freq. 1.41 1.42 1.42 1.42 1.45 1.54 3.4 29.9 56.6 107.6 136.2 Cg'x 0.00 6.55 13.09 32.57 64.05 120.36 275.9 286.7 243.6 107.6 27.2 Cgz 0.00 -.00 -.01 -.03 -.13 -.48 -5.5 -57.3 -97.4 -107.6 -54.5 Lz = 5 km: Freq. 1.41 1.42 1.42 1.46 1.61 2.08 7.7 68.1 107.6 141.3 149.3 Cgx 0.00 40.84 81.33 197.48 360.19 556.88 744.9 544.4 269.2 39.0 5.8 Cgz 0.00 -.02 -.08 -.49 -1.80 -5.56 -37.2 -271.8 -268.9 -97.5 -28.7 Lz=10km: Freq. 1.41 1.42 1.45 1.61 . 2.07 3.35 15.2 107.5 136.1 149.3 151.5 Cgx 0.00 161.94 318.48 717.53 1 110.13 1 374.80 1 488.9 539.1 136.6 11.5 1.5 Cgz 0.00 -.16 -.63 -3.57 -11.04 -27.33 -148.0 -536.7 -272.1 -57.4 -15.0 Lamb wave: Freq. 1.41 3.13 5.76 14.04 27.96 55.87 279.27 2 792.6 5 585 13 963 27 926 cgx 0.00 24 914 27 072 27 784 27 891 27 917 27 926 27 926 27 926 27 926 27 926 Cgz (None) n is the planetary wave number corresponding to the indicated horizontal wave length. Based on basic state parameters ofT= 260 K. N2 = 1.23 X J0-4 s-2, r = 3.44 X J0-5 m-1, andf = 1.028 X J0-4 s- 1. The buoyancy frequency (N) and inertia frequency (f) are 151.5 and 1.41 cycles per day. The speed of sound is 323.24 m s-1 = 27 926 kilometres/day. 18 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION Table 2.2 lists the frequencies (cycles per day), and the horizontal and vertical components of Cgrp (kilometres per day) for internal gravity waves and the Lamb wave, for representative values of the wave numpers. The planetary wave number n is also shown. This is defined by equating the value of the horizontal Laplacian operator in spherical co-ordinates with the square of the horizontal wave number K: n (n + 1) = /2 + k2 = K 2. a2 (2.76) The vertical component of the group velocity has the opposite sign to the vertical wave number, m, as is apparent from Figure 2.2 and the preceding discussion. (The entries in the table emphasize this fact by the otherwise arbitrary use of the negative sign.) Several important facts of importance to weather forecasting are shown by this table. 1. Entries for the equal values Lx = Lz = 1, 2, 5, and 10 km lie along a diagonal on the right side ofthe table. These wave number vectors are inclined at 45 degrees to the horizontal. The two components of their group velocity are also almost equal. Wave numbers to the left ofthis diagonal are quasi-vertical, but have almost horizontal group velocity vectors. Wave numbers to the right of the diagonal are quasi-horizontal and have quasi-vertical group velocities. This is in agreement with Figure 2.2. 2. The most important region for large-scale weather prediction is where the horizontal wave length exceeds the vertical wave length. In this case, both components of Cgrp increase with increasing vertical wave lengths. 3. Both components ofCgrp vanish as the horizontal wave length becomes very large. These waves have a frequency equal to the inertia frequency given by w 2 = f2. 4. The Lamb wave has a zero vertical group velocity. This is because its solution is of the form F(z) exp (iiJI') where the phase IJI' depends only on x, y, and t. 5. The Lamb wave, at all wave lengths shorter than 5 000 km, has an almost constant group velocity, equal to the speed of sound. Disturbances of this nature will propagate rapidly, but will preserve their shape even if composed of many wave lengths of that size. 6. The values at the far right of the table, for horizontal wave lengths shorter than the vertical wave length, have small values of the horizontal group velocity (except for the Lamb wave). These large horizontal wave numbers are those that might be expected to be associated with convective motions, such as thunderstorms. The smallness of the horizontal group velocity here may explain why the internal aspect of these localized motions does not experience rapid horizontal dispersion into the surrounding atmosphere. The size of the vertical group velocity associated with them, on the other hand, suggests that special consideration may be necessary to consider propagation of their energy into the stratosphere. The main consequence of these results for large-scale numerical forecasting is the largeness of the horizontal part ofCgrp for the Lamb wave and for the internal gravity waves oflarge vertical wave length. If the large-scale weather-producing motions had the character of the gravity waves shown here, weather forecasting for any but the shortest periods would be impossible, because of the rapidity with which the motions would propagate from one part of the globe to another. If this were the case, it would not have been worth while to initiate the first synoptic networks in the ninetenth century. It was the long experience with synoptic weather charts that led to investment in an upper-air network, the formulation of the polar front model of weather processes, the first formulation of objective weather prediction by V. Bjerknes ( 1904) and L. Richardson (1922), Rossby's model of the long waves, and finally to the first successful numerical forecast by J. Charney and his collaborators. This would not have happened iflarge-scale weather-producing motions had the character of gravity waves. The dispersion surfaces w = constant for the internal Rossby waves can be given a simple geometric interpretation by defining the reciprocal length* 1 p)~ 2rc (2.77) kl = f ( Cs2 + N2 ""' 14000 km ' *For an isothermal atmosphere, c,- 2 + T 2/N2 is equal to C/(4R 2T). This is a more common expression in the literature. SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 19 and then defining the non-dimensional parameters k k' = k;' l l' k;' (2.78) I f m m' - N k'I (J) w' 2kl-p· (Note the Prandtl-Rossby scaling for the vertical wave number.) This allows (2.54) to be rewritten as 1 ( k' + + f'2 + m'2 = - 1. (2.79) ~I r (J) t2 This is illustrated on Figure 2.3, where the surfaces for w' = 1, 2/3, and 113 are shown in perspective, as seen from a position in the octant of negative k', negative l', and positive m'. The maximum value attained by w' is l. Surfaces of constant frequency are spheres whose radius is equal to the square root ofthe right side of(2. 79), and the sphere for frequency w' is centred on the negative k' axis at k' = - ( 1I w'). The sphere for w' (max) = 1 is a single point at k' = -1 (i.e. at k = - k1). * m' I • ,...... ------__ .__ ,...... ,.... . - - k' Figure 2.3- Surfaces of constant frequency for internal Rossby waves, in the space of the three non-dimensional wave numbers k', /', and m'. The surfaces are viewed from the octant k' < 0, !' < 0, and m'> 0. For greater clarity, the bottom half of the surfaces (m'< 0) is omitted * The space for positive k' need not be shown. The solution at point I k'l, !', m' for which w' = - I w' I, is the complex conjugate of the solution at - I k'l, -!', -m', for which w' = I w' I. These two points have the same group velocity vector and three equal trace velocities. 20 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION In dimensional values the maximum frequency for internal Rossby waves is 1 wR(max;internal) = ~ 1 ~ 1.8 X 10-ss- ~ 1.73cyclesperweek. (2.80) It occurs at the west-east wave length corresponding to ki> i.e. 14 000 km. The direction of the group velocities is given by the gradient of w' with respect to (k', !', m') except that allowance must be made for the rescaling of the vertical co-ordinate. Figure 2.3 shows the following properties : For aw/Dk: The largest values occur at small values of k and zero values of land m. These are directed westward, opposite to the direction of the x-trace velocity. At larger values of I k I, awl ak is (2.81) directed eastward. For awl a! and aw/am: These are directed opposite to the trace velocities in those directions, reflecting the appearance of land m only in the denominator of wR. Their maximum values will evidently occur at small, (2.82) but not zero, values of k, !, and m. The Rossby wave for the Lamb vertical structure is not included in the above picturization. To show it, we simply rewrite (2.71) as fJ fJ2 p k 2w )2. (2.83) ( + + /2 = 4w2 - C 2 • s Curves of constant frequency are circles of radius equal to the square root of the right side of this equation, centred on the negative k-axis at k = - ({J/2w). This is shown in Figure 2.4. The maximum frequency given by middle-latitude values off and fJ is {JC5 2C5 WR (Lamb; max) = 21 ""'a = 10 cycles per week. (2.84) The x-wave length corresponding to this is 2nC/f""' 20 000 kilometres. 3 2 I' -4 -3 -2 -1 k' Figure 2.4 - Surfaces of constant frequency for the Lamb mode ofRossby waves, in the space of the two non-dimensional wave numbers k' and !'. The direction of the horizontal group velocity is indicated by the arrows SIMPLE PLANE W A YES ON A RESTING ATMOSPHERE 21 We turn now to the group velocity for Rossby waves. The maximum values for the horizontal com ponents can be obtained by repeated differentiation of the generic formula - Pk WR·= (2.85) k2 + /2 + J.l2 2 2 J1 equalsj2/C5 for the vertical structure ofthe Lamb mode. For the internal modes it has the larger value: f2 [Cs-2 + (m2 + P)]IN2. The two horizontal components are awR f3(k2 - /2 - f..l2) (2.86) ----ak = (k2 + /2 + J.l2)2 and awR 2f3kl --=---- (2.87) a! (k2 + f2 + f..l2)2 · As shown by Figure 2.4, awR/akwill have two maxima with respect to k and/. Both of these occur at l = 0, and one of them also at k = 0. In order to recognize the finite size of the Earth, it is necessary to bound both k 2 and l away from zero. A convenient value will be set by I k I ~ k0, ! ~ k0, where f2 )~ = 2n ko = ( 2 Cs2 28 000 km (2.88) 28 000 kilometres is almost equal to the radius of the latitude circle at 45 degrees latitude. The westward directed maximum value of awRI ak with respect to k and l is then awR f3J.l2 (-; max k, l) (2.89) ak (2ko2 + J.l2)2 at the limiting wave lengths l = ± k0 ; k = - k0. (2.90) The eastward maximum occurs at l = ± k0 ; k - [3(ko2 + f..l2)] 112 (2.91) and has the value awR p (+; max k, l) (2.92) ak 8 (ko2 + J.l2) The largest value of awR/al occurs at l = ± k0 and k = -k0 : awR + p -----a/ (max k, l) - 4j..l2. (2.93) 22 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The largest values of these horizontal group velocity components with respect to the vertical wave number occurs for the Lamb mode, for which m = iT, and J1 2 = f2 I Cs2 • The numerical values in kilometres per day and in degrees of longitude per day are as follows: awR f3Cs2 ----ak (-; max) = - ~ = -3 432 km (44 degrees) per day (2.94) awR f3Cs2 max) = f2 = 1144 km (14.5 degrees) per day (2.95) ----ak (+ ; 12 awR f3Cs2 (max) = ± f2 = ± 3 432 km (31 degrees) per day (2.96) a! 4 All of these occur at frequencies of about 1. 7 cycles per week. J1 2 for internal waves with "reasonable" values of m is much larger than .f!Ct For example, m= 2n/ 10 km results inJ1 2 = 338 (/2/ Cs2). The horizontal group velocities for most internal Rossbywaves are therefore of the magnitude 100 km per day or less. Turning now to the vertical group velocity (for the internal modes) we find awR = 2kmf2P 2 (2.97) am 2 N2[ k2 + f2 + f2 ( Cs- + m : 2 P ) 12 · Since wR and k have opposite signs, this is consistent with the statement (2.82), and is in agreement with Figure 2.3. The maximum value of(2.97) with respect to m occurs at m= m0 : N2 1 mo2 = 3f2 rk2 + /2 + f2 ( Cs2 + N2p)J ' (2.98) and has a value awR 27! Pfk am (mo) = (2.99) 8N [ k' + I' + P (d,, + ~ )]' If we restrict l to I~ k0, (2.99) reaches a maximum with respect to kat 2 2 (2.100) k = +[k 0 + f2 (ds 2 + ~: )J. This produces the following maximum value f{J awR (max) 12.75 km per day am (2.101) 2 4 N rko + f2 ( ds 2 + ~: ) ] SIMPLE PLANE W A YES ON A RESTING ATMOSPHERE 23 attained at - 2n k 17 631 km 2n (2.102) m mo = 163 km' l = ± ko, . and a frequency of about 1.1 cycles per week. For later reference, we note that the west-east wave length for the maximum vertical group velocity is about equal to the circumference of the 45 degree latitude circle, divided by 1.6, suggesting a zonal wave number one or two. These maximum values for the Rossby group velocity components imply that if a one-day forecast is to be made for a point near the ground in middle latitudes, the state of the atmosphere at the beginning of the forecast must be known within the following volume: 13 kilometres upward; 3 400 kilometres to the north and south; (2.103) 3 400 kilometres to the east; 1 100 kilometres to the west. l (The west and east boundaries ofthis region should be shifted westward by an amount equal to the average wind speed times one day to account for advection by mean zonal winds.) Charney (1949), in his planning for the first modern attempt at numerical weather prediction, arrived at a somewhat smaller horizontal area, and a vertical extent of only 4.5 kilometres.* The amount of data from the international radiosonde decreases rapidly above 20 km (7 5 hPa), south of 30' north latitude, and is very scarce in the oceans. According to the group velocity reasoning given above, accurate forecasts for only one day could be expected for the troposphere, and at most for only the central part of the continents in the polar half of the northern hemisphere. The availability since 1979 of temperatures measured by satellites in a near-polar orbit, winds from aircraft flight measurements and cloud-drift winds from geostationary satellites will somewhat temper the pessimism ofthis preliminary conclusion. We shall also see that a mean zonal wind can modify the large values of aw I aland aw I am cited above. However, there is little doubt that the insufficiency of data is still a major problem in making accurate deterministic forecasts beyond one day. 2.4 The Kelvin wave The wave solutions described until now have been based on the WKB approximation and, with the exception of the Lamb vertical mode, they have had sinusoidal variations in all three directions. The Lamb mode, with its exponential vertical dependence, required, for its acceptance as a valid solution, the satis faction of a vertical boundary condition at the ground. The Kelvin wave is another example of this nature, in which the exponential variation is horizontal and the boundary condition is on a plane of constant x (or y). This solution (See Lamb, 1932; p. 319) is of major relevance for oceanic tides, because of the continental boundaries for the oceans. It is of interest for atmospheric motions because a similar type of motion is possible as a solution to the "exact" (i.e. not WKB) solution of the equations (2.11 )-(2.15), when full consideration is given to the variation of the Coriolis parameter 2Q sin '!J. * There seems to be an error in Charney's calculations of aw I am in this early paper by him on the subject: a factor (HI h) tl2 "" 0.2 7 (in his notation) was lost in his derivation (1949, p. 382). His corrected value would be 12 km per day, in close agreement with the value deduced here. 24 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION We consider the plane-wave acoustic-gravity solutions represented by (2.28) and (2.36)-(2.40). We impose a wall at a fixed value of y = y0, and consider only the solution for y ~ y0 to be relevant. If (2.28) is solved for tz =KZ- kz, where lis the wave number in the direction normal to the wall, we obtain 1 Mz j tz = - kZ + (wz - f2) ( csz - wz - N2 j' (2.104) (Recall that MZ = mz + rz.) We replace l by the imaginary number iA, with A~ 0 so that the solution vanishes as y ~ oo. Mz 1 ) Az = kZ + (wz - f2) ( wz _ NZ - csz · (2.105) The boundary condition at the wall is that V vanishes there. This will be satisfied by making V identically zero. According to (2.38), this can be achieved by choosing kf A (2.1 06) (() Since Amust be positive, we are now limited to waves in which (for the northern hemisphere) k and w have the same sign, i.e. for waves whose horizontal motion is toward the east.* Substitution of this value of Az = -tz into the original dispersion equation results in the following simplified dispersion equation for the Kelvin frequency wK.: 2wK2 = NZ + CsZ(kZ+MZ) ± {[NZ + CsZ(kZ+M2)]2 - 4NZkZCsz}! (2.107) This is the same as (2.29) except that all reference tof2 seems to have disappeared! For a fixed value of the vertical wave number m and the east-west wave number k, we get two values of wKz, corresponding to acoustic and gravitational Kelvin wave solutions. t The horizontal velocity parallel to the wall is given by U. According to (2.37), this becomes CsA CS aP U = f exp(-hJ,y)ReP expi(hkx + mz- wt) =- fh ay (2.108) Thus, although the frequency seems to be independent off, the exponential decrease with y depends onf, and the horizontal velocity parallel to the wall is geostrophic. The "relative potential vorticity" derived in (2.62) vanishes for the Kelvin wave as it does for the other acoustic and gravitational waves. 2. 5 Vertical response to flow over ridges Before entering upon this section, it is important to emphasize that the following discussion is not intended as a serious study of the details of flow over mountains. Its main purpose is to provide a simple orientation that will make it easier to understand some of the seemingly unusual eigenvalue properties of the "Laplace tidal equation". The ideas are quite old and well established; they were derived and summarized in the late 1940s by P. Queney (1948).:f: * More generally, with the bounding wall to the right (left) of the horizontal direction of wave motion in the northern (southern) hemisphere. t The possibility of an acoustic Kelvin wave progressing anticyclonically around the oceans is suggested by the above analysis. However, this possibility may be negated by acoustical complications in shallow coastal waters and a very long attenuation distance ..1.- 1• =I= Queney's summary agrees with the material of this section, except that he has omitted some of the compressibility aspects of the atmosphere in his analysis of Rossby waves. SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 25 Suppose our simple basic state has added to it a uniform translation toward the east, at the speed u m s - 1• The perturbation equations (2.11)-(2.15) would then have a/ at replaced by (a/at+ (ulh) a/ ax), and -iw would be replaced by -i(w-ku). The perturbation is forced by making this basic current flow over an infinite series of sinusoidal ridges in the ground height zg, oriented normal to the flow: Zg = Zg cos (hkx). (2.109) The boundary condition at the bottom of the atmosphere is that air parcels there stay in contact with the ground surface: dz dz -- g dt - dt at z = zg. (2.110) In our linear perturbation framework, this becomes an equation to determine W at z = 0. u az - g - (j)Cs)-! W(z=O) = h ----;;x- = -uZgk sin (hkx). (2.111) Finally, we assume that a steady state has been reached, so that w is zero. The net result is that, in the dispersion relations for acoustic-gravity and Rossby waves that we have derived so far, the frequency w is replaced by - ku. Since k, I, and u are given, the mathematical question is no longer the determination of w, but the determination of the vertical wave number m. We consider first the acoustic-gravity wave oscillations, and rewrite (2.28) with (ku) 2 in place of w 2 : NZ k2 zP k2 Mz m2 + rz cz ["' - ( q + ,, )l (2.112) s lj2 - f2 k2 The term PI k 2 in the last bracket seems strange, since the simpler form 17 2 - Cs 2 is more intuitive as a test for supersonic effects. It can be shown that it appears only because the addition ofu to the basic state should have been accompanied by a pressure gradient in the basic state directed in they-direction that will balance the Coriolis force from u. ap ay = - pfii. (2.113) Addition of this effect would introduce the following term into the fourth equation in (2.20): v' ap . f- u (-i) ay-=1yV. (2.114) (j)Cs)! s When this is done, the third bracket in (2.112) no longer containsf2/k2.* There are two aspects to selecting the proper value of m. (a) For some values of k 2 and 17 2, the right side of (2.112) will be negative. In this case, m = ± i I m I, and the proper solution is to take the plus sign, so that the solution is proportional to exp (-I ml z). This is a trapped solution. * Equations (2.12) and (2.13) also have very small additional terms proportional to the slopefii /g of the pressure surface p = p. They are negligible for ordinary values ofu. 26 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION (b) For other values ofk2 and Yf2, m 2 will be positive. m is real, and the problem is to select the sign of m. The proper choice is that sign which gives upward propagation of energy, out ofthe atmosphere. This will be referred to as a propagating solution. In considering (2.112), we recall first that r has the dimensions of a wave number, with a value corresponding to a wave length of about 180 km. We will therefore be reasonably accurate in ignoring the P term in (2.112). The sign of m 2 is then determined by the signs of the three expressions in brackets, each of 2 which is a measure ofthe size ofu relative to k 2• On a diagram having k and u as co-ordinates, each of the three expressions divides the first quadrant into two sectors, in which the sector closest to the k-axis (i.e. YF ~ 0) has a negative sign. As shown on Figure 2.5, the three expressions together divide the quadrant into six sectors, separated by the three solid curves. The trapped solutions in which m is imaginary contain the possibility of resonance. This can be seen for the acoustic and gravity waves by first noting that the solution for W is completely determined by (2.111): W = -ku (pC5)~ Zg e-lml.:: sin (hkx). (2.115) In these simple circumstances the formula for the other variables will, according to (2.36)-(2.37), require division by m - iF. This is not possible when m= ir. (2.116) The amplitudes ofthe other variables must then be determined by adding dissipation or specifying in detail how the flow over the mountain is created. The three curves given by the vanishing of the right side of(2.112) are therefore to be interpreted more precisely as resonance conditions, which, because ofthe smallness ofF 2, also denote the approximate boundaries between real and imaginary m. Sectors P1 and P2 have positive m 2, and give rise to solutions that are wave-like in z. P1 occurs when u is supersonic, and corresponds to a wave-like form of the sonic boom created by supersonic aircraft. Sector P2 represents the phenomenon of mountain lee waves. It is characterized by current speeds such that f2 N2 k2 < ij2 < 7(2· (2.117) The three sectors T1, T2, and T3 have negative m 2 and trapped solutions that decrease with height as exp (-I m I z). What about the Rossby solutions? Replacement of w by - ku in (2. 54) yields the following expression for m 2 : 2 2 2 (2.118) mR = ;2 l ~ - k - J2 (~5 2 + ~: )l Positive m 2 is now possible only for positive u (a west wind), and when its speed is such as to satisfy the inequality.* p O * As mentioned earlier, this important inequality was first derived by Queney (1948). Its broader meaning was not recognized, however, until Charney and Drazin (1961) considered the propagation ofRossby waves into the stratosphere. Pedlosky ( 1979; Chapter 6.13) has a good description of simple orographic Rossby waves. SIMPLE PLANE WAVES ON A RESTING ATMOSPHERE 27 0.5 L...__.L..__...J...___..___.__ _,__ __,__-lo...J _ __j_ ___j 19683 6561 2187 729 243 81 27 9 3 L Figure 2.5- Vertical response to uniform flow over sinusoidal ridges. The ordinate is the basic current speed in m s-1• The abscissa is the logarithm of the horizontal wave number, labelled as to wave length in kilometres. Sectors PI, P2 and P3 have a real vertical wave number, an oscillatory wave-like response with z, and propagate energy vertically. Sectors T1, T2 and T3 have an imaginary vertical wave number, an exponentially trapped response, and do not propagate energy upwards TABLE 2.3 Critical values of the zonal velocity uR for westerly flow over sinusoidal ridges of length Lx in the west-east direction and infinite extent in the north-south direction Lx (km) s uR (m s- 1) 28 300 I 64 14150 2 40 9 433 3 25 7 075 4 16 4 717 6 8.2 3 538 8 4.8 2 830 10 3.1 s is the nl\mber of such waves that would fit in the latitude circle at 45° N. Waves can propagate vertically ifu < uR. This curve is indicated by the line in the lower left part of Figure 2.5. Values ofUR for middle latitude values of/and pare listed in Table 2.3. Equation (2.118) modifies the extreme lower left portion of the trapped sector T3 into a third propagating sector when the zonal flow is from the west. This sector is responsible for a major part of the seasonal pattern of planetary waves of large wave length. 28 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The resonance condition for Rossby waves differs somewhat from (2.116). It can be shown to be 1 rz) m2 = - N2 ( Cs 2 + N2 ' (2.120) for which the frequency formula (2.54) requires that* u-( resonance) = k2fJ > uR. (2.121) This interesting combination of real and imaginary values for the vertical wave number will be encoun tered again in considering the exact eigenvalue solutions to the perturbation equations, in which full allo wance is made for the latitudinal variation of the coefficients in (2.11 )-(2.15). A matter of some practical consequence is the change brought about in Figure 2.5 by the hydrostatic approximation. In our simplified perturbation case, the hydrostatic assumption is achieved by ignoring w 2 compared with N 2• (A more general view ofthe hydrostatic approximation is given in Appendix D.) This will be satisfactory for motions with a period that is significantly longer than 2n/ N "" 10 minutes. All operational numerical prediction models use this assumption. When the equivalent substitution of - N 2/k 2 for (i:F- N 2/k2) is made in (2.112), two sectors change character: T2 becomes propagating, and (2.122) P 1 becomes trapped. The change in P1 is of little consequence, since atmospheric velocities are subsonic. But the change in T2 might be significant if the horizontal resolution ofthe model is fine enough to include orographic features with a horizontal wave length less than about 10 kilometres. The choice of sign for m in the propagating case is of some importance. From (2.17) we see that the vertical propagation of wave energy is equal to the horizontal average of the product p'w'= PW. Using (2.39) from the acoustic-gravity solutions, we obtain WC5 • - PW = Re [P exp (itp)] Re w _ NZ (m - 1T) P exp (i!fJ) J. l 2 For real m this has an average value of wmC 2 (2.123) 2 (w2 - ~ 2) lP 1• For acoustic waves, w 2 exceeds N 2• Upward flux of energy for them requires wand m to have the same sign, i.e. an upward vertical trace speed, and, according to (2. 74), an upward component to the group velocity. For gravity waves, N 2 exceeds w 2• For upward energy propagation, they therefore require w and m to have opposite signs, i.e. a downward-directed trace velocity, and, according to (2. 74), an upward component to the group velocity. In the present example of steady response to flow over orography, the substitution of - ku for w, when m is real, results in the average value PW (real m) = ii km Cs IFI 2 (2.124) 2 2 N2) 2 k (u - T * Tung and Lindzen (1979) use a "two-time" approach in which u is switched on at t = 0, in order to specify the resonant solution unambiguously. They find that the horizontal velocities increase with time and are independent of height. (An alternative artificial approach is to simply assume that Zg and the perturbation variables vary as exp (vt) where vis a small positive number.) Tung and Lindzen also derive simplified resonance conditions for Rossby waves for the case where u varies with height. SIMPLE PLANE W A YES ON A RESTING ATMOSPHERE 29 In the propagating sector P2, YF is less than N 2/k2 • PW> 0 therefore requires positiveness of the product ukm. Thus, our tacit assumption of positive u and k requires a positive value of m for upward energy propagation. (Alternatively, positive u and k corresponds to negative w, from which (2.74) would have also led to a choice of positive m.) Lines of constant phase slope upward in the upstream direction. (The lee waves associated with an isolated ridge in nature-and in theory-are located downstream from the ridge. This comes from superposition of the many wave lengths present in an isolated ridge.) When m is imaginary, PW is equal to PW (imag. m) . w Cs (I ml - T) e -21 ml::: Im (15'2 exp (2iljl')], (2.125) I 2 (wz -NZ) where 'I'' = kx + ly - wt. In this case, PW vanishes when integrated horizontally over one wave length at a fixed value of z. CHAPTER 3 GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE Knowledge of the "exact" solutions to (2.11 )-(2.15), i.e. without the simplifying approximations of the WKB approach, are important in numerical weather prediction. These solutions are useful in the technique of "initialization". This is the process of adjusting the initial fields before proceeding forward in time with the numerical forecast, so as to control any unrealistic and unwanted oscillations in the forecast (Machenhauer, 197 6; Baer, 1977). The solutions can also be useful in investigating theoretical aspects of prediction on large scales. For example, the coefficient j2 (square of the Coriolis parameter) plays an important role in the denominator of the Rossby frequency formula by setting limits on the frequencies and group velocities. Butf2 goes to zero at the Equator. It could then be important to know what the equivalent ofj2 is for perturbations whose horizontal scale is large enough to sense the complete variation in f2. 3.1 Separation of variables In order to conform to more traditional notation we return to the use oflongitude (A.) and latitude (lJ) in place of the Mercator co-ordinates x and y. au CS aP --JV =- (3.1) at a cos lJ aT' av aP -+JU = - __S_ (3.2) at a --alJ' aw -- N2Q = - c ( aP + rP) (3.3) at s az ' aP 1 ( a u a v cos lJ - l ) + aw - rw J (3.4) at = - Cs a cos lJ aT + a lJ az ' aQ - =- w. (3.5) at (We ignore the Coriolis terms that are proportional to 2 Q cos lJ.) Inspection of these equations shows that solutions in which the z-dependence is factored out will have the following form: U = Uh (A., lJ, t) Cs(z) A(z); (3.6) V = Vh (A., lJ, t) Cs(z) A(z); (3.7) P = Ph (A., lJ, t) A(z); (3.8) Q = Ph (A., lJ, t) B(z); (3.9) -aPh (A., lJ, t) W= B(z) (3.10) at GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 31 (3.1) and (3.2) immediately yield two equations of motion with no z-dependence: 1 aPh auh - fVh (3.11) at a cos 1J ~' aPh avh + fUh (3.12) at a a1J (3.3) reduces to 2 [B] ---at2+a Ph [N2B] ph = l-CS (-----clZ dA + FA )J Ph. (3.13) (Square brackets are used here to isolate functions of z.) (3.4) reduces to a more complicated form: aPh 1 (auh avhcos1J) [ (dB )l aPh [A] ~ + [Cs2 A] a cos 1} ~ + a1J = CS dz - rB J Tt (3.14) The most direct procedure now is to introduce a separation assumption for the three variables U11, VI,, and Ph, with respect to the three independent variables A, 1J and t. The appropriate forms are Uh = Re U1(1J) exp (ilf/), (3.15) Vh = Re V1(1J) exp (ilf/), (3.16) Ph = Re P1(1J) exp (ilfl), (3.17) in which the phase If/ is now a strictly linear function of longitude and time: If/ (A., t) = SA - wt. (3.18) The zonal wave number s is an integer to assure periodicity in longitude. (3.13) immediately yields the first oftwo differential equations in z that will relate A(z) and B(z) to one another. 2 2 CS ( ~ + r A ) = (N - w ) B. (3.19) Derivation of the second equation requires more labour. Insertion of the separation assumption into (3.11) and (3.12) produces the following relations between U~> VI, and PI: 1 ws dPl) (3.20) u, = a (w2 - f2) ( cos 1} pi + f (i'i} ' . 1 ( dP1 fs \ 1 (3.21) Vj = a (w2 - f2) w (i'i} + cos 1J PV. These, together with (3.17), are introduced into (3.14). We also define a non-dimensional form for the frequency w: w a= cycles per 12 hours. (3.22) 2Q 32 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION This results in the following equation: [A] P1 + r !~2 ]L(P1) = rcs( ~~ - FB) JP 1. (3.23) L(P1) is the Lap/ace tidal operator, to which we assign an eigenvalue -e: dP1 s sin tJ ~ cos _,u -- + p 1 1 d dt) a L(P1) cos tJ { dt} I u' - sin' t} · · (3.24) s2 s sin tJ 1 --- dP } cos tJ P1 + a dt) a 2 - sin2 tJ = -e P1• e is sometimes referred to as "Lamb's parameter". Since L contains the two parameters s and a, we expect that e will depend on these, as well as some number "/" that will order the values of e that may exist for a given choice of s and a. e = e1 (s, a). (3.25) /will act as a latitudinal wave number. To continue with tradition, it is customary, following Taylor (1936), to express e as 4£?2a2 e = (3.26) gh his called the "equivalent depth", because the depth of the ocean appears here in the oceanic tidal problem, as shown in (3.28) below. (The symbol h is also used for the horizontal scale factor, but the context will always make it clear which use is intended.) We can now introduce (3.24) into the unresolved equation (3.23). After dividing out the factor P 1, we have 52 C (~ - rB) = A = Cs2 )A. s dz (1- 4.QC2a2 e) (1 - gh (3.27) This is now ready for combination with (3.19) to determine the vertical structure of the perturbations. Before investigating this problem, it is useful to retrace the last steps and use the newly defined parameter e to relate the behaviour of P11 to U11 and V11 • To do this, we return to (3.14) and replace the B term on the right side by (3.27). The result can be written as Ph 4.Q2a2 a V· (Uh, vh) = - gh ( auh av11 cos tJ) (3.28) at e acostJ aA. + atJ · This equation, when combined with the two horizontal equations (3.11) and (3.12), is equivalent to the linearized system that describes hydrostatic perturbations on a resting ocean of uniform depth h. (P 11 plays the role of gravity times the perturbation in the height of the free surface.) They are often called the "shallow water equations".* But they apply also to our linear system, for which the hydrostatic assumption need not be made. *"Shallow-water" refers to the ratio (depth/horizontal scale), in contrast to the "deep-water" equations that apply to non-hydrostatic surface waves. GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 33 It is now known that the eigenvalues 8 are not limited to positive values. Longuet-Higgins (1968) has in fact shown that solutions exist over the complete range -oo<8<+oo. (3.29) Solutions with positive values of 8 occur in association with a complete range of frequencies. But solutions with negative 8 occur only in association with values of a that are less than 1 (i.e. periods longer than 12 hours). The latter solutions are required in solving some forced problems in which the frequency is specified and the vertical wave number must be solved for. Platzman (1985) has discussed a sequence of papers that treat the existence of negative equivalent depths in solutions of the Laplace Tidal Equation. He cites Silberman (1953), Sawada and Matsushima (1964), Dikii (1963), Lindzen (1966), Kato (1966), Flattery (1967) and Longuet-Higgins (1968). 3.2 The role of the equivalent depth h The relation of the various parameters at this stage can be confusing. The two vertical equations (3.19) and (3.27) contain the two parameters a 2 (i.e. w 2) and 8 or h. When solved subject to boundary conditions at the top and bottom ofthe atmosphere, they will presumably result in a series of admissible values of h, which we can denote (temporarily) as hv (m; a). m, acting as a vertical wave number, will serve to distinguish different value of the vertical eigensolutions and eigenvalues that may be valid for a fixed value of the non-dimensional frequency a. In a parallel fashion, the Laplace Tidal Equation contains the three para meters-s, a, and 8 (or h). When solved subject to the proper boundary conditions at the North and South Pole, it will determine admissible values of h, denoted (temporarily) by hL (l; s, a), in which l denotes a latitudinal wave number. The two values of the equivalent depth must agree: hv (m; a) = hL(l; s, a). (3.30) The resolution of this symbolic equality will determine a as a function of the three wave numbers, s, !, and m, just as occurred for the plane-wave WKB solutions with the wave numbers k, !, and m.* Some feeling for the role of different ranges of 8 can be obtained by exploring its meaning in the simpler context of the WKB approximations. We replace d/dz in (3.19) and (3.27) by im. This gives the relation m2 + rz (3.31) ghv czs wz- Nz For example, for the Lamb mode, in which m = iT, we get ghv (Lamb) = Cs2 (3.32) (strictly speaking, in an isothermal atmosphere). By rewriting (3.31) in the approximate form mz + rz""' mz = (wz- Nz)( ~sz - g~J' (3.33) we can consider what relation g hv might have to the several coexistent possibilities of w 2 and m2 that have occurred in both the free and forced WKB solutions of sections 2.1, 2.2, and 2.3. In the orographic problem of section 2.5 w2 was equivalent to k2 112• At this point we can consider w more generally as either a known applied frequency in a forced problem, or as the unknown parameter in an initial value problem whose solution will consist of free waves. 1 *For large-scale meteorological processes, the hydrostatic assumption is valid. It removes the dependence of hv on w , a considerable simplification. 34 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION In the limit of very high frequency, the operator L loses all reference to the rotation of the Earth and reduces to (ala)2 times the horizontal Laplacian operator (i.e. V}) on the sphere. This limit assigns the following value to ghL: w2 a2 ghL (high frequency) = (3.34) n (n + 1)' where n is the order of the Legendre polynomial associated with the spherical Laplacian operator. For less extreme frequencies in this exercise, we will use results obtained by replacing the latitude derivative in (3.24) by the WKB approximation (i a !) and reinterpreting the zonal wave number s as ak cos tJ. 1 }(2 ghL (ac-gr) = w2 - f2 (3.35) (3.34) is clearly only a further specialization of this. For simplicity we will ignore Pin (3.33) for the high-frequency case. We can then come to the following possibilities: For w2>N2: 1 1 2 m > 0 occurs with 0 < -h < C ; g s 2 (3.36) 1 1 m2 < 0 occurs with C < -h . s 2 g (Note that these correspond to sectors P1 and T2 on Figure 2.5.) At the lower frequencies w2 m2 1 1 N 2 = gh - 4H2N 2 (3.37) The first case is as follows: For f2 < w2 1 2 0 occurs with m > 4H2N 2 < gh ; (3.38) 1 1 m2 < 0 occurs with 0 < gh < 4H2N2 (These correspond to sectors P2 and T1 on Figure 2.5.) For the quasi-steady case of w 2 < .f, (3.35) requires negative gh and therefore: m2 > 0 does not occur. 1 (3.39) m2 < 0 occurs with gh < 0. } (This corresponds to sector T3 on Figure 2.5.) GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 35 If we were to identify f = 2Q sin tJ with its polar value 2,Q, this negative value for gh would agree with the discovery by Lindzen ( 1966) that negative equivalent depths were necessary to solve for the response of the atmosphere to forcing by the diurnal tide, for which w = ,Q < 2Q. It would also be compatible with the Longuet-Higgins result that negative equivalent depths for the Laplace tidal equation exist only for periods longer than 12 hours.* To examine the role of the equivalent depth in oscillations of the nature ofRossby waves, we rewrite (2.58) as fJ 2 1 wl k + K gh f2 (3.40) and (2.57) as f2m2 2 fJ 2 f ) (3.41) N2 - ( wlk - K - 4H2N 2 . We then find the following possibilities as a function of the zonal trace speed. For - fJ w K2 + f2 < k < 0: 4H2N2 1 m2 > 0 occurs with 4H2N2 < gh < oo. (3.42) For zonal trace speeds outside the above limited range of negative values: 1 1 m2 < 0 occurs with - oo < gh < 4H2 N2 (3.43) The result of this exercise is useful. It has pointed out that, except for the acoustic motions ofareas T 1 and P1 on Figure 2.5, the maximum value of gh of concern to large-scale weather prediction, for free motions, is set by C,2, the value ofgh for the Lamb wave. This is especially important when the numerical formulation of the equations is such as to allow horizontal propagation of the Lamb wave. In these circumstances, the time step ~t used in the numerical integration must be small enough that (At/Ax)2 < (11 gh). It is therefore necessary to have some idea of how large gh can be in a non-isothermal atmosphere. Appendix B reports some values based on the Rayleigh-Ritz estimation method used by Bretherton (1969). The normal standard atmosphere produces a value of9 933 metres for h. (This is equal to that in an isothermal atmosphere with T = 242 K.) A very warm (but realistic) atmospheric column produces a value of 10 319 for h. (This is equal to that in an isothermal atmosphere with T = 251 K.) Large-scale numerical weather prediction is based on the hydrostatic assumption. In our present context, this means that w2can be neglected compared to N 2in the vertical equations. There is then a simple functional relation between the equivalent depth h (or e) and the vertical wave number m defined by a simple WKB treatment. Figure 3.1 depicts the mutual relations between the eigenvalue e, the equivalent depth h, and the vertical wave length Lz = 2nlm, that exists under this assumption. The somewhat startling aspect is the large value of the eigenvalue e that is associated with moderate values of several kilometres for the vertical wave length. *The simplified deduction of a negative h given in (3.39) is similar to Lindzen's argument for the solar diurnal tide (Chapman and Lindzen, 1970; p. 138). 36 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION 1 10 100 1 000 10 000 100 000 100 100 000 50 10 000 20 1 000 10 100 5 10 2 10 100 1 000 10 000 100 000 E Figure 3.1 - Relations between the equivalent depth h (left scale, in metres), the vertical wave length Lz (right scale, in kilometres), and Lamb's parameter e = 4 Q2a2/gh 3.3 The vertical modes The vertical parts of the eigensolutions (often referred to as "modes") are determined by the two equations · cs ( ~ + r A ) = (N2 - w2) B, (3.44) C - TB) = ( 1 - A (3.45) s (~dz .5:_)gh ' together with boundary conditions at z = 0 and z = oo. A is proportional to the pressure perturbation, and B is proportional to the entropy (or vertical velocity) perturbation. The general problem is discussed by Eckart (1960) and by Dikii (1965, 1969). An important question with respect to the solutions to (3.44) and (3.45) deals with the existence of discrete solutions (or "modes") for a particular set ofvertical wave numbers, as opposed to solutions that form part of a continuous spectrum, in which the vertical wave number can vary continuously. The discrete modes, like the Lamb mode, are trapped in that they do not propagate vertically. The continuous modes, on the other hand, can radiate energy upwards to infinity. The discrete modes are then able to propagate energy to great distances in the horizontal direction, whereas the continuous modes will eventually disappear at large horizontal distances. GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 37 The existence of these two types of modes is determined for the resting basic state, by the three basic state parameters C" rand N2. These are functions of the temperature distribution T(z) in the basic state. We will normally be interested only in T(z) that is representative of a global average. The profiles listed in Table 2.1 from the US Standard Atmosphere eliminate features such as frontal, subsidence, and nocturnal inversions that occur on a local scale. (These may be locally significant, of course, but they do not merit the global arithmetic involved in the La place tidal operator that is now under consideration.) A globally averaged T (z) does have some characteristic features, however. (See Table 2.1) There are temperature minima at tropo pauselevel (216 Kat 11-20 km) and at the mesopause (187 Kat 90 km). Maxima occur at the ground (288 Kat z = 0) and at the top of the stratosphere (217 Kat 50 km). T increases to very large values above the mesopause. It might be expected that these features ofT(z) would significantly modify the simple pattern of vertical solutions that is present in an isothermal atmosphere. Instead of the single discrete Lamb wave and the continuous spectrum that is implied by the ability of m to assume any real value and represented on Figure 2.1, we might expect to find a sequence of additional discrete modes. This appears to be true only to a limited extent when a globally averaged T(z) is used. Most solutions to (3.44)-(3.45) in the meteorological literature have been made with the hydrostatic assumption, in which w2 is neglected compared with N2. The values of w or hat which these discrete solutions occur can be calculated by adding a forcing function, such as a non-zero value of Bat the ground. (See, for example, section 3.3B in the book by Chapman and Lindzen.) Solutions are calculated numerically for a range of values of h, subject to a proper boundary condition at the top of the computational domain. If the solution for a particular value of h = h1 has an amplitude that greatly exceeds that for neighbouring values of h, h1 is associated with a trapped discrete solution that does not propagate energy to infinity. Such 104,------ 103 --,,, ,------180° 102 ,, i Q} 0.. (f) E CO <( ..c 101 a.. goo 100 10- 11/ I I I I I . I I I 0 o 5 10 15 20 25 30 35 E Figure 3.2- Magnitude (solid line) and phase (dashed line) of the response at the ground to forcing by the semi-diurnal lunar tidal potential and heaving of the Earth and land, as a function of e. At larger values of e the curve slowly decreases to a value of 1.2 ate = 1 000 (From Hollingsworth, 1971) 38 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION calculations show that a true discrete solution is obtained only for h corresponding to the Lamb wave. A minor secondary maximum in response is also obtained for an h of about 7 km, but this is not a true resonant solution. Figure 3.2, for example, illustrates the resonance curves obtained by Hollingsworth (1971) in his study of of the lunar semi-diurnal atmospheric tide. (See also Figure 8 in the article by Salby, 1979.) Computations illustrating the effect of temperatures as warm as 350 Kat 50 km are given by Jacchia and Kopal (1952). The resonance theory of the solar semi-diurnal tide is reviewed by Chapman and Lindzen, (1970, Chapter 1). This theory required an equivalent depth near 8 000 metres. 3.3.1 Eckart's thermosphere The character of the spectrum is determined not only by the equations (3.44) and (3.45), but also by the boundary conditions. Eckart (1960; Chapters XIII-XV) demonstrates that all vertical eigenfunctions will be discrete ifthe atmosphere is capped by a "thermosphere". His idealized thermosphere is a region extending from z = z1h, say, to infinity, in which T(z) increases linearly with height. The following WKB argument seems to capture the essential part ofhis reasoning. Interested readers should refer to Chapters XIII -XV in his book as well as Chapter IV in the book by Dikii (1969). According to the WKB formalism, the formula we have derived for m2 (m is the vertical wave number) is not only valid for an isothermal atmosphere, but is approximately true for a non-isothermal atmosphere if m2 is allowed to vary with z. For acoustic-gravity waves m2 is given by (3.33): 1 1 2 2 2 ] [- (3.46) m (z) = - P(z) + [N (z) - w - - 2 J · gh C5 (z) According to (2. 73), the vertical group velocity awl am is proportional to m. It therefore vanishes when m(z) becomes zero. The "rays" along which energy is propagated will be reflected at the height where m vanishes. Alternatively, we may consider that an approximate version of the second-order differential equation for A(z) that would be obtained by combining (3.44) and (3.45) is given by d2A - + m2(z)A = 0 (3.47) dz2 A transition of m2 from positive to negative values as z increases means that the two independent solutions for A(z) will change from having an oscillatory character to having an exponential character. If m2 remains negative above this height, the only way in which A can remain finite as z ~ oo is for A(z) to contain only the exponential-like solution that decreases with z.* This implies that the solutions are ultimately "trapped", and discrete (because one solution in the upper reaches of the thermosphere has been lost). The atmosphere does have something approaching Eckart's thermosphere in that the temperature does increase dramatically with height above 90 km (see Table 2.1). Eckart's model therefore deserves exploration with some quantitative considerations. These will show that a "realistic" thermosphere will be an inefficient reflector of gravity and Rossby waves of even moderate horizontal length scales. From (2.1)-(2.3) we see that, in this idealized thermosphere, 1 2 T oo 'f2 ' 1 N2 oo T , (3.48) 1 C2 oo T . s * Einaudi and Hines (1970) discuss the uncertainties of the WKB approach for determining reflection of small-scale acoustic and gravity waves, as that approach is affected by the choice of independent and dependent variables. They arrive at a quantity "q.1?" as the most suitable coefficient to use for this purpose. Ifgh is replaced by its non-rotating equivalent w 2/ K' from (3.35), the resulting expression for mz differs from q¥" by a small term proportional to f- 2 dT/dz. The difference appears to be trivial in the present context. GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 39 rz will play only a small role in determining m2 and we can therefore approximate (3.46) as 2 m (z) = F 1(z) X F2(z), (3.49) in which 2 F 1(z) N (z) wz , (3.50) 1 1 F2(z) = gh - Cs2(z) . (3.51) (gh) and w2 are constants for each individual vertical eigenfunction. We will consider them to have values such that m2(z) is non-negative for heights below the thermosphere. For the acoustic-gravity waves the horizontal WKB solution (3.35) is Kz (gh)-1 = wz- f2 (3.52) This means that F 2 (z) can be rewritten as wz - (j2 + KZ CsZ) F (z) (3.53) 2 csz (w2 - f2) By recalling that w2 exceedsf2 in these wave motions, we see that the numerator in (3.53) determines the sign of F2• This is readily recognized as the dispersion relation for the Lamb mode, which separates the acoustic 2 from the gravity modes. The sign of(w - NZ) in F 1 also separates the two types ofwave motions. Thus the behaviour of F 1 and F 2 with height is as follows: Modes that are acoustic below z1h: At z < Zth: Fl < 0, F2 < 0. (3.54) dF1 dF2 At z > Zth: < 0, dz dZ>O. ) Modes that are gravitational below z1h: Atz The criterion for the acoustic "turning point height", z1P, is that F 2 becomes zero. This occurs at Acoustic modes (F2 = 0): wz- f2 C 2 = y R (ztp) (3.56) 5 T Kz Ifwe ignoref2 here, we recover the conventional acoustic result that the turning-point height is where the local sound speed equals the horizontal trace speed wlK. (j2 will cause only a slight lowering ofthe reflection level for sound waves of high frequency.) 40 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The gravity-wave result is more relevant for large-scale meteorology. Its turning point occurs when N2 becomes small enough to be less than w2.* This occurs at Gravity modes (F1 = 0): N 2(zth) T(ztp) g ( g dT ) T(zth) (3.57) w 2 cp + dz wz According to Table 2.1, N 2 in the bottom of the thermosphere is several times larger than typical tropospheric values. Thus, even a gravity wave that has the maximum possible frequency in the troposphere - Wmax ~ 2n/10 minutes~ w-z s-1 -and reaches the thermosphere, will not be reflected until it reaches an altitude where T is about five times that at the base of the thermosphere, i.e. at T ~ 900 K. A frequency this large can be reached in the troposphere only for gravity waves of short horizontal wave length; gravity waves ofinterest to large-scale weather prediction have much smaller frequencies. For example, the frequency for a gravity wave of horizontal wave length 1 000 km, a very large vertical wave length (i.e. having the maximum frequency for this horizontal wave length), and a reasonable tropospheric N2 of 10-4s-2, is 1.43 X 10-3s- 1• This, when squared, is equal to (11300) times N 2 at the base of the thermosphere. The temperature at its reflection level will be about 58 000 K (!). 2 For internal Rossby waves that are "propagating" below z1h, w is less than NZ, and - Pk ---KZ 1 w 1 m2 + rz -= ~--+---- (3.58) gh f2 csz NZ Thus, F 1 and F 2 will also be positive for "propagating" Rossby waves. F 2 will increase and F 1 will decrease as 2 C5 increases with height in the thermosphere, and the reflection criterion for Rossby waves will therefore be the same as that for gravity waves. However, the smallness of wR 2 will make the ratio N2(zuJiw2 extremely large. The only reasonable conclusion is that any realistic thermosphere is a poor reflector of gravity waves originating in the troposphere, and an extremely inefficient reflector of Rossby waves. t Dikii (1965, 1969) has proceeded beyond Eckart's theory by (among other things) calculating numerical solutions to the vertical equations for the 1961 CIRA Standard Atmosphere, up to a height of200 km. He used the variables developed by ·earlier tidal theorists; in place of z, the vertical co-ordinate was the logarithm of pressure z = - log (j}), dZ g 1 (3.59) dz = RT = H' l where RT H=-- (3.60) g *This condition is similar to the criterion found by Lindzen and Tung (1976) for horizontal ducting of gravity waves by a stable layer (large N 2) in the lower troposphere, underneath a layer where N2 "is almost zero". They go beyond this simple criterion by also considering the effect of a critical level where the trace speed matches the speed of a background wind. tIt should also be noted that a value of N 2 large enough to exceedf2 significantly is necessary to justify the neglect of the Coriolis term 2 Q cos JJ. Thus even an "exact" analysis of (3.44) and (3.45) would be in error in the upper reaches of Eckart's thermosphere. It can also be noted that the customary use of the hydrostatic approximation in the study of tidal oscillations and large-scale meteorology 2 amounts to replacing the factor F 1(z) by N (z). This automatically ensures that the Eckart thermosphere cannot reflect gravity waves or Rossby waves. GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 41 is the scale height for pressure, and the dependent variable y was proportional to the perturbation three dimensional divergence: 112 y (p) V 3 · v'. (3.61) The resulting differential equation is d2y 2 (3.62) dZ2 + 11 Y = 0 ' in which the coefficient 112 has the value (in our notation) /1 2 1 w2 1 H2 =:= - 4H2 + Cs 2 + gh (N2 - w2). . (3.63) 2 (It is convenient to write it in this manner, since 11 /H2 corresponds, approximately, to our m2). Note that (3.63) also becomes negative at very large T. For gravitational waves of short horizontal wave length (60 km), Dikii reports gravitational resonances at seven values of the equivalent depth h between 1.5 and 4.5 km (Dikii, 1969; Figure 4.6). At longer horizontal wave lengths he reports a major resonance at h around 10 km (presumably the counterpart to the Lamb wave), but also several resonances at smaller values of h. (See his Figures 4.7 and 4.8.) This is not at variance with Eckart's overall conclusion about the effect of an infinite thermosphere in producing a discrete spectrum. But it is at variance with the WKB conclusion above, which predicts that gravity waves of moderately low frequency will not reach their turning point until exceedingly warm temperatures are reached, considerably greater than the value of 1360 K at Dikii's upper boundary at 200 km.* 3.3.2 Practical considerations .The above considerations show that, except for internal acoustic waves, the only discrete vertical mode for a realistic global average T(z) is the Lamb mode. It is trapped with zero energy density at the top of the atmosphere, and does not propagate vertically. However, the large-scale motions in the troposphere depart significantly from the vertical structure ofthe Lamb mode; for example, they have considerable horizontal variations of temperature. This implies that a considerable fraction of the large-scale energy must be repre sented by the continuous part of the vertical spectrum that is associated with the present type ofbasic state (in which there is no background basic current). These modes can propagate vertically, and have a significant vertical group velocity, as discussed in sections 2.3 and 3.3. The group velocities are large enough to suggest that a forecast for a week or longer should include data up to 100 km. One might deal with the continuous modes by solving the equations numerically to a finite height at which a radiation condition (upward group velocity) is imposed. This method has recently begun to be tested in small-scale meteorology. (See, for example, Klemp and Durran, 1983.) Dissipation terms can be added to the equations. Infra-red radiation can have a moderately strong damping effect on motions in the upper stratosphere and lower mesosphere. Dickinson (1973; see also Blake and Lindzen, 1973) showed that a reasonable representation of this could even be achieved with a simple "Newtonian" cooling law, with a time constant (o:) as small as several days at 50 km. dT - o:T (3.64) dt * Dikii apparently uses the numerical device of setting the vertical derivative ofy at 200 km (the top of his computation region) equal to -ifl times the value ofy at 200 km (see his equation (4.25)). lff.l in the expression -ifl means the positive roots off1 2, it implies that he has taken the vertical trace velocity Wlfl to be directed upward at the top of the computational region. As discussed earlier in (2. 74), this corresponds to an upward group velocity for acoustic waves, but to a downward group velocity for gravitational waves. The opposite choice of sign would seem to be appropriate for gravitational solutions when this numerical approximation is to be used. My command of Russian is limited. The remarks in this footnote may therefore reflect a gross misunderstanding of Dikii's numerical method and results. His book certainly deserves wider study. 42 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION 14.-----~.-----.------.-----.------.----. 12 10 z 1~~1~0;8----~--~~~~~--l_--~~------_l ____j 4 1x1Q-7 4 1x1Q-6 4 1x1Q-5 1 Figure 3.3 -Distribution ofNewtonian cooling rate a in s- , as a function ofZ = -log (p!p0). (The solid curve is takenfrom A/pert et al., 1983) Figure 3.3 shows a distribution of a used by Alpert et al. (1983). This effect can act to decrease the amplitude of motions in the stratosphere and higher. Another effect could be viscosity. Molecular viscosity will give rise to a frictional acceleration of magnitude v Iv'l Lz- 2, where v is the kinematic vicosity tabulated in Table 2.1. Horizontal accelerations in large-scale motions will be significant when they become as large as a reasonable fraction (such as 0.2) of the Coriolis force. Since the magnitude ofthe latter isf I v' I, molecular viscosity will be important only when Lz 2 is less than the ratio v/0.2!). At 50 km this occurs only for Lz less than 30 metres, and at 90 km it occurs only when Lz is less than 200 metres. Molecular viscosity is therefore of little or no importance below 100 km. At higher altitudes it can have an effect on the reflection and the absorption of gravity waves (Yanowitch, 1967; see also Meyers and Yanowitch, 1971). At lower elevations empirical representations such as an eddy viscosity are often used. A further simplification is common in theoretical studies, in which the vertical viscosity effect is replaced by a simple drag law or "Rayleigh friction" (J acqmin and Lindzen, 1985). Salby (1980) has used an "equivalent Newtonian cooling effect" to model friction. These artifices reduce the amplitudes of motions in the upper stratosphere, but their numerical value is subject to considerable uncertainty. (Pages 263-267 in Plumb et al. (1985) contain a brief survey of observational studies that lead to estimates of energy dissipation and eddy viscosity.) A "lid" can be arbitrarily introduced at some height. This is the current practice in large-scale numerical weather prediction. Lindzen, Batten, and Kine (1968) emphasized that this makes all vertical modes discrete. It would seem to be most serious in models with few vertical degrees of freedom (few levels). Fortunately it turns out that motions in the upper stratosphere and in the mesosphere have smaller values of the energy density than those characteristic oflower levels. The Eckart amplification E(z) tabulated in Table 2.1 predicts that a wind field consisting of the continuous modes from (2.11 )-(2.15), and with uniform energy density in the vertical, will have a speed at 50 km that is typically 17 times that at 10 km. Figure 34 in Oort (1983) shows a root mean square value ofthe transient zonal velocity u' of 14 m s - 1 at 12 km GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 43 (200 hPa) in the northern hemisphere winter. A uniform distribution of disturbance energy in the vertical would imply typical speeds of 238 m s - 1 at 50 km. Figure 3.4(a) is taken from the analysis of rocketsonde observations by Finger et al. (1975) and represents an active day at this level in winter. It shows a maximum speed of only 100 m s- 1• The vertical distribution oflarge-scale energy is affected to a considerable extent by the background fields of wind that are present and that vary with height. This process was first described for Rossby waves by Charney and Drazin ( 1961 ). (Eliassen and Palm (1961) independently analysed vertical dispersion of energy Figure 3.4 -(A) Weekly mean map at0.4 hPalevel centred on 5 January 1972. All observed rocket winds for the week are plotted at each station. (Triangular barb = 50 knots, single and half barbs = 10 and 5 knots.) Zonal wave number 2 is especially pronounced in this sample. (B) For the week centred on 16 August 1972. (From Finger et al., 1975) 44 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION for stationary gravity waves). This effect will be analysed in Chapter 4. At this time it is sufficient to point out that numerical studies, for example those by Tung and Lindzen (1979) and by Held et al. (1985), have shown that new discrete modes for Rossby waves are created by suitable background wind fields. 3.4 Horizontal eigenfunctions Solutions of the Laplace Tidal Equation (3.24) were investigated in the nineteenth century by Margules (1893) and by Hough (1898). The solutions divide into two symmetry groups. In one group, P (tJ) and U (tJ) will be even functions of tJ, and V(tJ) will be an odd function of tJ. This symmetry group will have an odd number of zero crossings for V(tJ) in the interval -nl2 P~ (tJ) exp i(sA - wt); n = Is! + l. (3.65) The total two-dimensional wave number n is related to the zonal wave numbers and a latitudinal number l as follows: n = Is! + /. (3.66) One class was called "First Class" by Hough and "Erster Art" by Margules. These have the frequency law (for e~ 0) ai2 = e- 1n (n + 1) (3.67) or ghn (n + 1) wi2 = a2 (3.68) This corresponds to the WKB solutions (2.29) for acoustic-gravity waves. The other class of solutions was called "Second Class" by Hough and "Zweiter Art" by Margules. Fore~ 0 this also had spherical harmonics as eigenfunctions, but with the frequency law s an n (n + 1) ' (3.69) 2Qs wn = - n (n + 1) This is recognizable as a limiting form of the Rossby wave formula (Haurwitz, 1940). w 11 reaches a maximum value of Q fors = -1, n = 1, i.e., it rotates to the west as fast as the Earth rotates to the east. Figure 3.5 shows the behaviour ofboth classes, in this limit, as a function of zonal wave numbers for the lowest values ofthe latitudinal number, l. Solutions for positives progress eastward and those for negatives move westward. The curves are labelled with the number of zero crossings for V This labelling is straight forward in this case because in the limit e ~ 0, the horizontal wind components have no vorticity in the Class I motions, and no divergence in the Class II motions (Love, 1913). The meridional velocity component V(tJ) is therefore proportional toP~ (tJ) for the Class II motions, and proportional to d P~/dtJ for the Class I motions. In this limit the number of zero crossings for V is invariant along each curve in Figure 3.5. At larger e this GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 45 10.------.------. 5 £'1201 t 0.5 __s;;j_ 12 hrs 0.1 I ::::ao .....-- s ____.. 0. 0 1 .__....__....L.....--'-----'---'---L.----'-----L~'--....__....L...... L.....---'---'-__..l.___J -8 -4 4 8 Figure 3.5 - Frequency in cycles per 12 hours of solutions to the Laplace Tidal Equation fore--+ 0. (Note that the top set of curves on this figure is for the product e '12 a instead of a.) Abscissa is the zonal wave numbers. Plotted values show the number of zero crossings for the meridional velocity simplicity will change, as will the simple frequency formulas given by (3.68) and (3.69). However, the general aspect of Figure 3.5 -it consists oftwo classes of solution, each class being characterized by an infinite discrete series of latitudinal wave numbers that describe successively shorter latitudinal wave lengths-will be preserved as e increases. For a fixed s, the frequency of Class I motions increases as the number of zero crossings increases, while that for the Class II motions decreases with increasing zero crossings. The frequency of both classes of oscillations decreases as e increases, i.e., as the equivalent depth decreases. These properties are also characteristic of the WKB solutions derived earlier. When e is not small, the frequencies and eigenfunctions are more complicated. Hough took the important practical step of expressing them as a series of Associated Legendre Polynomials, instead ofthe trigonometric series used by Margules and La place. Because of this, the solutions are often called "Hough functions". Swarztrauber and Kasahara (1985) have developed an alternative method of numerical computation (for positive e) by applying spherical vector harmonics directly to the undifferentiated forms (3.11), (3.12), and (3.28) ofthe shallow-water equations. This technique makes it comparatively straightforward to establish the "completeness" of the solutions to (3.24): any initial field of velocity and pressure whose future behaviour is governed by the linear homogeneous two-dimensional shallow-water equations (3.11 ), (3.12), and (3.28), can, 46 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION for fixed e >0, be completely represented by the infinite discrete set of eigenfunctions ( U, V, P) having that value of e in which each eigenfunction changes with time according to its frequency.* The existence of solutions with negative values of e has been recognized by several authors. (See the comment at the end of section 3.1.) In 1968 a complete analysis was published by Longuet-Higgins. He found that the eigenvalue e existed over the complete range from - oo to + oo, although the negative values occurred only for a less than 1 (periods greater than 12 hours). (See also Flattery, 1967, and Miles, 1977.) The solutions for negative e are needed in analysing forced problems in which the forcing has a period greater than 12 hours. Longuet-Higgins established the existence of three limiting types of behaviour as e ~ + oo, and four limiting types ofbehaviour as e ~ - oo. Figure 3.6 contains a schematic listing of these limiting types and their relation to the original Class I and Class II motions at e ~ 0. t The motions of the Second Class exist for all values of e, whereas the motions of the First Class do not continue across a= 0 into the region of negative e. Matsuno (1966) investigated the properties of solutions with large positive e that could be described by an "equatorial beta plane". His solutions agree with the three limiting forms found by Longuet-Higgins for this case, except that Matsuno's include a more faithful accounting ofthe effects of moderate values of the zonal wave numbers. (His formulas have therefore been used in part of Figure 3.6.)=1= +00 -s ,1 a = £ v, (2v + 1) + s21 _ (2V+ 1)% s s a-~+ (4V+2)£V' a= £% V = 1, 2, 3, -- I V= 0, 1, 2,-- T2 T3 I T1 V=O MixedI ~re n = s (>0) I n =IsI I I I 11 a= (n(n: 1)r 0 ~ a= n(n+1)-s t /,////// E. ::A No Solotloc• ~ ~/.la T4 T5 V= 0, 1, 2, -- T6 s-2v a=1 a=s/(- £) a= 1 + (-£)V' - 2+s +2v + 2 (1 + s+2v) V= 0, 1, 2,-- (-£)V' (-£)% -00 Westward (s Figure 3.6 - Schematic view of the limiting forms of the solutions to the La place Tidal Equation, shown separately for eastward (s > 0) and westward (s < 0) waves, as a function of e *Special arrangements must be made to represent that part of the solution fors = 0 that is steady, i.e. a11 = 0. Swarztrauber and Kasahara (1985) offer a new method; others are possible. t Longuet-Higgins uses the conventions;;;> 0, with negative and positive values for a. His formulas for large negative r, contain some misprints. His type 6 solution is a linearized solution of the planetary-scale geostrophic equations derived by Burger ( 1956) and described in Appendix E. He identifies an additional limit (Type 7), fors= 0 and large negative values of e. For simplicity, this is not shown on Figure 3.6. :J: Some partial results on an equatorial beta plane were obtained by Rosenthal (1965). GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 47 The eigenfunctions for the limiting values ofe have some special properties. For large positive e they have large amplitude in low latitudes and small amplitude in polar regions. The phrase "equatorially trapped" is sometimes used to describe this. The meridional velocity is proportional to Hermite polynomials: V oc Hv(rt) exp -( ~ ) ; rt = e114 u. (3.70) For large negative e the eigenfunctions have large amplitude in polar latitudes and small amplitude in low latitudes. They are proportional to Laguerre polynomials n L(c;); c; = (- e)'/2 (2- u)2. (3.71) Longuet-Higgins presents graphs illustrating the dependence on latitude of V, U, and P for a selection of zonal wave numbers and e. Figure 3. 7 contains a selection of these graphs for the latitudinal pressure function, for zonal wave numbers = 2, for two latitudinal wave numbers, and several positive values ofe. The tendency to concentrate near the Equator as e increases is evident. goo.-,--,--,-,--,-,-,--,--,-,--,--,-.-.--.--.--.- llw lw lE S=2 r,-;;3 90 o.------,--~~~r-r---r-,--~-r-r-,----r--r--;-,---,---,---rr-r----,------rj llw S=2 n=4 oo~----L-~---L--~--~~L--L--~--L-~--~---L--~--~~---L--~--~--L-~---L--~~~~ Figure 3. 7 - Pressure eigenfunctions for zonal wave number 2, for Second Class Motions (IIw), westward moving First Class Motions (Iw), and eastward moving First Class Motions (IE). The upper three figures are for latitudinal wave number 11 - s = 1. The lower three are for 11 - s = 2. Curves are shown for values of e. (Adaptedfi"om Longuet-Higgins, 1968) 48 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION -8 -4 0 4 8 5 . 0 1111~~-r-~-r-.--.-:--.---.-----.----.--.-_:; 1.0 I _g_ 12 hrs 0.5 r- 0 at 0.1 I _.,...... s- 0.05 1...,...... -T I I I ~ I '\ I I I I I I I I I 1 -8 -4 0 4 8 Figure 3.8 -Frequency in cycles per 12 hours ofso1utions to the La place Tidal Equation fore= 8.82. Abscissa is the zonal wave number s. Plotted values are the number of zero crossings for the meridional velocity. K and M indicate the Kelvin and mixed waves Figure 3.8 displays afore = 8.82, a value appropriate for the Lamb wave.* The curves for the Class II motions are similar to those on Figure 3.5. However, the Class I curves fore= 8.82 seem to have been "cut" along the a-axis, as if the symmetry ofFigure 3.5 were destroyed by a sinking ofthe positive s side relative to the negative s side, and the loss of several zero-crossings at s = 0. The result is a tendency to produce a minimum frequency at s = - 1 for each Class I curve. This effect continues at larger e. Matsuno's formula for the T 1 type can be used to show that the minimum frequency for each T 1 curve occurs, for large e and v ~ 1, at the zonal smin defined by -2 e 114 (2v + 1)312 (3.72) Smin - 4(2v + 1)2 - 3 Computations with the Swarztrauber-Kasahara program verify this odd feature. The single curve labelled with a "K" is referred to as a Kelvin wave (Matsuno, 1966). This is because it has properties similar to those described in section 2.4; its meridional velocity .is small (see Figures 8 and 11 in the paper by Longuet-Higgins), and its frequency is (approximately) a = Se-112. (3. 73) *Values were obtained from the small e approximation derived by Su, Schlesinger and Kim (1980). These needed correction at several points fors = 0 and -1 by the "exact" values obtained from the Swarztrauber-Kasahara program ( 1985). The number of zero crossings plotted on this figure for- 2 .:;;s.:;; +2 were determined by inspection of the graphs given by Longuet-Higgins. (The discussion of this on page 531 ofLonguet-Higgins (1968) appears in error.) GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 49 Comparison ofFigures 3.5 and 3.8 show that the Kelvin wave can be viewed as a modified form of the Class I solution for n = s>O. By rewriting (3.73) as w2 = (s/a)2 gh, and substituting the vertical WKB definition (3.31) for ghv, we arrive at the formula s2 Cs2 (w2- N2) w2 = (3.74) a2 w2 - N2 - M2 Cs 2 The factors/a is equal to the east-west WKB wave number, k. If this substitution is made, we arrive at the earlier Kelvin formula (2.107). For large 8, the Kelvin wave has zero V, but its U and P vary as 2 - 11 ) (3.75) (U, P) oc exp ( -2 -- · The top curve of the Class II frequencies is marked with an "M" on Figure 3.8. With its absence of zero crossings for V, it appears, on this 8 = 8.82 diagram, to be approaching a smooth juncture with the lowest Class I point on the s = 0 line, the point that has changed its V zero crossings from 2 on Figure 3.5 to 0 on Figure 3.8. The Class I points fors> 0 that still have two zero crossings on Figure 3.8 lose those crossings when 8 increases to that for Figure 3.9. Their frequencies are given, for all s, by the formula for v = 0 in Longuet-Higgins's Type 1: aM = 8- 114 + s (28)-112. (3.76) 8 - 0.51 ...... : : ~ • : : : : :~2 2 2 2 ·1 1 n at ___Q[_ 12 hrs 0.1' 0.011----- -12 -8 -4 0 4 8 Figure 3.9 - Frequency in cycles per 12 hours of solutions to the Laplace Tidal Equation fore = I 600. Abscissa is the zonal wave numbers. Plotted values are the number of zero crossings for the meridional velocity. K and M indicate the Kelvin and mixed waves 50 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION This partial transformation of a Class II eigenfunction into a Class I solution is indicated graphically on Figure 3.6. Matsuno (1966) called this eigenfunction the "mixed Rossby mode", because it gradually changes its properties from a Rossby wave at large negatives to a gravity wave at large positives. In summary, Figure 3.8 shows that e = 8.82 is large enough to have eigenfunctions that differ detectably from those for e = 0. We now consider Figure 3.9. This is fore= 1 600, corresponding to a vertical wave length of about 13 kilometres, and is large enough for the asymptotic large formulas of Figure 3.6 to apply. (The Matsuno formulas were used to include the effect of s2, and checked with the Swarztrauber-Kasahara values.) The zero crossings shown on the figure are equal to v, the index ofthe Hermite polynomial solution (3.70) for V. The distribution is much simpler than it was on Figure 3.8. This has been accomplished by a loss of two zero crossings by the gravity waves for positives. The Kelvin wave is now completely separate from its original counterpart ate~ 0 (i.e. the lowest Class I mode for s>O), and the mixed mode is now clearly established across the entire s range. Large values of e result even for vertical wave lengths that are not extremely short, as can be seen on Figure 3.1. However, the equatorial trapping is most extreme for short vertical wave lengths. Matsuno ( 1966) deduced this behaviour by scaling the latitude co-ordinate (mJ) with a length LM:* 11 = LatJ .' LM = a e-114 M . (3.77) LM sin 1J """ 1J = 11 a As a time scale, T M• Matsuno selected a geometric mean ofa gravity wave time scale, T0 , and a rotational time scale, TR: am a (gh)-112 To N' (3. 78) TR (2.Q)-1; l TM = (To TR)l/2 = (2.Q)-lell4. (3.79) The important consequence is that these choices allowed the linearized shallow-water equation in low latitudes to be written in a non-dimensional form with no scaling parameters. This result provides a heuristic explanation of the equatorial trapping that occurs as the vertical wave length L: becomes small and e increases according to the definition: N 2L/ N 2 (2.Qa)2 --=-"""gh=mz (3.80) The absence ofany non-dimensional number in Matsuno's scaled equations implies that the motions must be such that the Coriolis force and the buoyancy force take part "equally". In particular, it would rule out the following possibility. Suppose the effective length scale of the eigenfunctions of lowest latitudinal order continued to occupy the full range of latitude as the vertical wave length was reduced (e increased). The Coriolis force would still have a significant effect typical of(say) middle-latitude values, but the small vertical wave length would have diminished the effect of the buoyancy force. The effect on the frequency would be similar to the w2 ~ p limit of the WKB results where P would continue to have middle-latitude values. Instead, we find that the effective Coriolis force and buoyancy force both decrease as e increases. * Lindzen (1967) analysed approximations to (3.24) suitable for low and middle latitudes. An equatorial approximation was also investigated by Rattray (1964). GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 51 (3. 77) shows that Matsuno's length scale LM is related to the latitudinal variation ofthe Coria/is parameter in low latitudes, instead of the horizontal length scale of the motion. Let us consider the details ofMatsuno's solutions. When transformed back into latitude, his differential equation for the meridional velocity V becomes d2 V s 2 2 + ( ea - s - ezF) V= 0. (3.81) dtF (] This has the same solutions-acoustic-gravity and Rossby waves-that Longuet-Higgins found by a more formal process: s 2 ea2 - s2 - e11 (2v + 1); v 0, 1, (3.82) (] 2 V = Hv 11 exp ( -0.511 ); (3.83) 11 = e 114 tJ. (3.84) (3.83) has an oscillatory solution equatorward ofthe turning latitude tJt, where the coefficient of V in (3.81) goes to zero, t}t2 (2v + 1)8-112 (2v + 1)( LaM y, (3.85) MEF"~IDIONAL VELOCITY EIGENFUt\ICTIONS FOR El = 1600.0 AND E2 = 40000,0 I <---LATITUDINAL WAVE NUMBER------) ETA LATl LAT2 MIXED 2 3 4 5 6 7 9 10 o.o o.o o.o 1000 0 -874 0 802 0 -759 0 735 0 -720 0.22 2.0 0.9 975 358 -769 -462 632 532 -531 -561 451 600 -382 0.44 4. 0 1.8 907\ 667 -483 -773 197 793 14 -738 -181 687 314 0,66 6. 0 2.7 803 \886 -86 -835 -320 655 553 -414 -676 191 719 0,88 8. 0 3. 6 677 996 331 -637 -710 193 771 188 -662 -466 462 1.10 1(1, (I 4. 5 543 999 682 -250 -826 -362 552 669 -157 -742 -216 1.32 12.0 5. 4 415 917, 912 207 -638 -754 29 727 463 -419 -704 1,55 14.0 6. 3 302 780 999 615 -230 -820 -510 339 762 233 -583 1.77 16.0 7 ., 210 618 962, 889 252 -553 -801 -249 551 713 30 1. 99 18.0 8,(1 138 459 837 '999 667 -78 -721 -703 -14 677 623 2.21 20.0 8. 9 87 321 668 963, 924 426 -333 -798 -576 172 736 2. 43 22.0 9. 8 52 212 494 828 '999 811) 185 -514 -817 -449 306 2.65 24.0 10.7 29 132 340 64 7 92;:-...... 999 646 -6 -635 -804 -340 2.87 26.0 11.6 16 77 220 465 763 999 ..... 927 508 -154 -712 -773 3.09 28.0 12.5 8 43 133 310 571 866 999, 864 396 -260 -752 3.31 30.0 13.4 22 76 192 391 670 906 "'999 812 312 -328 3.53 32.0 14.3 11 40 112 248 471 722 942' '999 773 256 3. 75 34.0 15,2 5 20 61 146 304 517 769 972...... ~9 745 3. 97 36.0 16.1 2 9 31 80 181 338 559 806 993 ...... 995 4. 19 38.0 17,0 0 15 41 100 203 368 591 832 999 4.42 40.0 17.9 0 0 6 20 52 113 221 390 612 840 4.64 42.0 18.8 0 () 9 25 58 123 235 403 616 4.86 44.0 19.7 0 0 () 3 11 28 63 130 241 403 5.08 46.0 20.6 () 0 () 4 12 30 66 132 239 5.30 48.0 21.5 0 () 0 0 5 13 31 67 130 5.52 so.o 22,4 0 0 0 0 0 (I 2 5 13 31 64 5.74 52.0 23.3 0 0 0 0 (I 0 2 5 13 29 5.96 54.0 24.1 0 () 0 0 0 (I 0 (I 2 5 12 6.18 56.0 25.0 0 0 0 0 0 0 0 " Figure 3.10- Amplitude of the meridional velocity V as a function of '7 for latitudinal wave numbers 0 - 10. Columns 2 and 3 list the latitude in degrees corresponding to the listed values of'7 fore= I 600 (column LAT I) and fore= 40 000 (column LA T 2). The turning ROint is indicated by the dashed line 52 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION and a decreasing solution poleward of this. (See Figure 3.1 0.) When vis large, we can assign an average wave length L 111 to the solutions ifwe divide the latitude range ut- (-ut)= 2ut by halfofthe number of zero crossings of (3.81)-i.e. by v/2. This operation gives the length 32\ l/2 Lv = a ( -v-) e-114. (3.86) We can also define the "average" Coriolis parameter as J = 0.5 (2.Q ut) = .Q (2v)112 8 -t/4. (3.87) 1 After noting that both Lv and] vary as e- /\ we can write their product as N2)112 2 ] L = 4 (gh) 112 = 4 - = - N L7. (3.88) v ( m2 n - This now has the same form as the Prandtl-Rossby relation (2. 70). (3.88) however applies to both Rossby and acoustic-gravity solutions, whereas (2. 70) was valid only for geostrophic motions. Evidently if we keep N2 constant and the latitudinal modal number v constant, but decrease the vertical wave length L" the eigen functions accommodate by reducing both the latitudinal wave length of the solutions and the effective mean value of the Coriolis parameter, and they do this so as to satisfy the Prandtl-Rossby relationship. Figure 3.10 displays the solution (3.83) for Vas a function ofv and IJ. The shortening of the length scales as e increases is demonstrated by the latitude scale corresponding to e = 1 600 and the smaller latitude-scale appropriate toe = 40 000. (The eigenfunctions for U and Pare related to V and d V/du, together with sand a. They will have the same shrinkage of the latitudinal length scale as V.). The values listed in the figure are normalized so their maximum value is 1 000. This occurs just equatorward of the turning-point latitude defined by d2 V/du2 = 0, and shown by the dashed line. The slow increase ofthe "envelope" ofl V(u)l between the Equator and this point might suggest that at any latitude ua, the dominant frequency for the meridional velocity would be that for waves whose turning point is located slightly poleward of ua· This simple con elusion is valid for internal gravity waves in the ocean, where periodic velocity oscillations with a frequency slightly above the local value of the Coriolis parameter (the so-called "inertia oscillations") form a major contribution to the observed spectrum of oceanic waves (Munk and Phillips, 1968). There is little evidence for any behaviour of this type in the atmosphere, however, presumably because such motions are dispersed too much through advection by background wind fields of larger scale. There is considerable evidence that the atmosphere does have oscillations that resemble some of the solutions of the Laplace Tidal Equation; some of this evidence will be referred to in section 3.6. However, the only observed wave patterns have small horizontal wave numbers and large equivalent depths, i.e. small e; there is no observational evidence for the large e solutions shown on Figure 3.1 0. A theoretical explanation for this is presented in Chapter 4. 3.5 Group velocities from the Laplace Tidal Equation The calculation in section 2.3 ofWKB group velocities for Rossby waves in a resting atmosphere showed that the maximum values existed at long horizontal wave lengths. The numerical values were large enough to require data from a significant fraction of the atmosphere for a forecast of more than one or two days. Since the WKB approximation assumes "short waves", it is necessary to compare this behaviour with that ofthe more exact solutions from the Laplace Tidal Equation. For the vertical component awl am, we can use the approximate relation (3.80) between the equivalent depth h and the vertical wave number m. In doing this we will assume hydrostatic balance, i.e., w2 « N2. GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 53 aw aw ae -=- am ae am 4Q2a2 aw _a_ (-1- + m2 + P) (3.89) 2 2 ae am C8 N 16Q3a2 m(e) aa ---- N2 8 a ln e ' where m(e) corresponds to 2--2 2 8 ---::;::,1 ) m - r + N ( 4Q2a2 Cs2 ,_ 0. (3.90) a can be evaluated numerically as a function ofln (e), for fixed values of sand the latitudinal wave number l, by interpolation in the tables provided by Swarztrauber and Kasahara (1985). Table 3.1lists values of awlam, e, a, and Lz ( = 2 nlm), for the maximum value of awlam that was found in this way for each combination of l and s. The values for l = 1 and 2 are somewhat smaller than were obtained from the middle-latitude WKB computations given in (2.101); the maximum value is now about 8 km per day, again for large horizontal wave lengths. The values of awlam for the mixed mode, l = 0, are larger, however. For horizontal propagation of energy, the external modes (e = 8.82) are most important. Table 3.2lists values of the frequencies, and angular values (positive eastward) for the phase speed (wls) and group velocity awl as for the external waves oflargest horizontal scale. (Values of a were obtained from the Swarztrauber Kasahara program.) For comparison, the table also includes values for e = 0, corresponding to Rossby's original model of a non-divergent atmosphere, with frequencies given by (3.69). Values of aw/as were computed by centred differences. Hoskins, Simmons, and Andrews ( 1977) have pointed out that the original concept of group velocity as simply expressing the process of constructive interference is still meaningful for the present case of discrete zonal wave numbers s. If two waves have a similar latitudinal structure but different zonal wave numbers s1 and s2, they will add constructively at time intervals l1f = 2n (s2 - SJ) (s 1w2 - S2WJ)-l, (3.91) TABLE 3.1 Vertical group velocities maximized with respect to the vertical wave number m, for large-scale Rossby and Kelvin waves aw I am (km per day) Frequency (cycles per 12 h) I s=l s=2 s=3 s=l s=2 s=3 K 28.65 56.1 82.9 0.21 0.42 0.62 0 16.61 8.88 5.01 0.34 0.26 0.21 I 7.16 8.28 6.55 0.06 0.09 0.10 2 4.37 5.21 4.54 0.04 0.06 0.07 e Lz (km) I s=l s=2 s=3 s =I s=2 s=3 K 23.3 23.2 23.3 202 202 202 0 34.0 40.0 46.1 126 109 97 I 30.2 34.0 36.7 142 126 117 2 31.4 34.0 36.7 136 126 117 I denotes the latitudinal number associated with the Rossby wave, e.g., 0 is the mixed Rossby-gravity wave. K denotes the Kelvin wave; s is the east-west wave number. The associated frequencies, eigen values, and vertical wave lengths are also given. 54 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION at successive longitude intervals 1 LU = 2n(w2 - w 1) (s 1w2 - s2w 1)- • (3.92) This site of this periodic intensification will move zonally at the rate LU w - w 2 1 (3.93) 11t s2 - (si The concept of group velocity in the zonal direction for small discrete wave numbers is therefore valid, but subject to a periodic modulation. The effect of e = 8.82 instead ofO is small in Table 3.2 for wand for awlas at I si > 2. At smaller I -si, e decreases the frequencies, as expected. The effect of non-zero e on w/s is greater, however-with a noticeable reduction for latitudinal modes I = 1 and 2, and a shift in the value ofsat which w is a maximum. For mode I = Oands = - 1 and -2, the differences caused by non-zero eare large. Figure 3.7 shows howthesearerelated to the appearance of the "mixed mode" as e increases. TABLE 3.2 Frequencies, zonal phase speeds, and zonal group velocities for Rossby waves on a resting basic state Frequency (cycles per week) e=O s: -I -2 -3 -4 -6 -8 I= 0 7.0 4.7 3.5 2.8 2.0 1.6 I= I 2.3 2.3 2.1 1.9 1.5 1.2 I= 2 1.2 1.4 1.4 1.3 1.2 1.0 e = 8.82 I= 0 5.9 4.3 3.4 2.7 2.0 1.5 I= I 1.4 1.9 1.9 1.8 1.5 1.3 I= 2 0.8 1.2 1.3 1.3 1.1 1.0 Eastward phase speed (degrees of longitude per day) s: -1 -2 -3 -4 -6 -8 e=O l = 0 -360 -120 -60 -36 -17 -10 1 = 1 -120 -60 -36 -24 -13 -8 1 = 2 -60 -36 -24 -17 -10 -7 e = 8.82 I= 0 -303 -111 -58 -35 -17 -10 1 =I -72 -48 -32 -23 -13 -8 1 = 2 -43 -30 -22 -16 -10 -6 Eastward group velocity (degrees of longitude per day) s: -1 -2 -3 -4 -6 -8 e=O 1 = 0 -120 90 48 30 15 9 I= I -60 6 12 11.1 7.9 5.5 1 = 2 -36 -6 1.7 3.9 4.1 3.5 e = 8.82 l = 0 +115 65 41 27 14 9 1 = 1 -48 -13 3.0 7.1 7.5 5.2 I= 2 -30 -11 -1.9 1.8 3.4 3.2 Values are shown for a non-divergent model (e = 0) and for the Lamb mode (e = 8.82). The latitudinal wave number is l, with the mixed mode denoted by a value of zero. GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 55 If we ignore s = -1 and -2 for l = 0, the largest aw/as values are: Eastward: For l = 0, s = -3: 41 degrees of longitude per day; For l = 1, s = -6: 7. 5 degrees of longitude per day; Westward: For l = 1, s = -1: -48 degrees oflongitude per day. These are in rough agreement with the middle-latitude WKB values summarized in (2.94)-(2.96). Information about latitudinal propagation is less accessible from the solutions to the Laplace Tidal Equation. Fortunately, information on this point is available from a study by Hoskins, Simmons, and Andrews (1977). They considered solutions to the non-divergent vorticity equation on the sphere: .!i_ = - v · V (f + (), at (3.94) in which the velocity is determined by a stream function !f!: v = k X V!f/, ( = V2!f!. (3.95) We first consider their solutions that were linearized about a resting basic state. The motion was created in one of two ways: an initial concentration ((0) ofvorticity was assumed to exist in a circular region centred on 30'N, or a source of vorticity of fixed intensity, ( 0, was maintained in the same location at 30'N. The response to the fixed initial concentration was calculated for a hemispheric region (in which all the even l modes were absent) and for the complete globe. The two responses at day 2.5 are shown in Figure 3.11. 90 (A) oL-~--~--L-~--~--~~--~--~--~~--~ 90 (B) (------...... ,...~ ll ------t2§"c-=:;--N\r~/j) Jl 11ll1LL ~h [(1!!/// ("t-~C";j)l ~ 0 90 e.- - c-- --- (G) ~. -- .... ~ '-=:~ ))\ r0 J --1 ~ ~?'1,//)~IJ -?Vu,~, 0 "" ----· {_// I //)I I I.../ I / I t - - j -90 -- - --' __l __._l_ - - -- __J --'---- __, I Figure 3.11 - Distribution of relative vorticity calculated by Ho skins et al. (1977) in the absence of a basic current. (A) the initial concentration at 30°N; (B) distribution at 2.5 days for the hemispheric solution; (C) distribution at 2.5 days for the global solution 56 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The hemispheric solution shows the first of a series of centres appearing eastward of the original centre. Significant vorticity values exist at latitude 75'N, and a temporary value of -0.1 so (max) appeared at 75'N as soon as day 1. Later charts for this case show additional centres appearing further eastward in accordance with the values specified in (3.91)-(3.93). The extreme regularity of this is attributed by the authors to the approximate uniformity of aw/as for l = 1 in the range 3 ~ Isi ~ 9. The global solution in Figure 3.11 is different. By day 2.5, significant negative vorticity values have appeared at 60'S. (As in the hemispheric case, a temporary value of -0.1 Co (max) appeared at 75 'Nbyday 1.) At day 8 (not shown) the vorticity patterns appear equally intense in both hemispheres, with no obvious indication that the original pattern was localized in the northern hemisphere. These global results suggest large values of awlal, quite compatible with the WKB value of 31 degrees of latitude per day in (2.96). The pessimistic conclusions from the WKB group velocity calculations in Chapter 2 are not weakened by the more exact global solutions for a resting atmosphere. 3.6 Observed large-scale oscillations There is a considerable body of evidence showing that there are some motions in the atmosphere that fit the theoretical properties of large-scale solutions to the Laplace Tidal Equation. Yanai and Maruyama (1966)-see also Maruyama (1967)-analysed equatorial motions at an elevation of about 20 km in the Marshall Islands having the properties of the mixed Rossby-gravity mode. The motion was westward, with a period of five days, an estimated zonal wave number of 4, and a vertical wave length of 50 hPa or more. Mak ( 1969) found that motions of this character could be produced in the upper troposphere of a simple linearized model ofthe equatorial atmosphere if that model were subject to realistic forcing at 30 degrees latitude by observed meridional transient eddy velocities. His model had two layers in the vertical, with westerlies in the upper layer over low-latitude trade winds in the lower layer. The equatorward flux of eddy energy was much reduced in the easterly winds in the lower layer of Mak's model. Wallace and Kousky (1968) discovered periodic zonal motions in the tropics at altitudes of 15-30 km, with eastward phase progression, a period a,round 15 days, and evidence of a very long wave length (s = 1). 3~r--r--~~---r~~--~-.--..-~--,---,--,--,--,~~--,--,---r--, 80 aJ 70 60 (i) w (i) a: UJ 50 t- UJ w a: ~ UJ 40 w 8" 0 :::? UJ t- C/l < :::; J: 30 Cl. Cl. :::; <( 20 10 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 TIME (DAYS) Figure 3.12 - Amplitude (solid line) and phase (dotted line) for the observed 500 hPa heights projected onto a Rossby Hough mode with zonal wave number 2 and meridional wave number corresponding to the curve labelled 2 at the bottom of Figure 3.8. (From Lindzen et al., 1984). The projected height fields have a latitudinal profile almost equal to thee = I 0 curve in the upper-left diagram of Figure 3.7. The value of z(SOO) at 35'N (the maximum point) is equal to 0.6 times the daily values plotted in this figure GLOBAL SOLUTIONS FOR A RESTING ATMOSPHERE 57 The eastward phase progression and the absence of meridional velocity in these motions identified them as Kelvin waves. (This type of motion also appeared in Mak's theoretical calculations.) Both types of observed waves, the mixed Rossby and the Kelvin, had downward phase propagation, suggesting an upward flux of wave energy into the stratosphere from a source lower down (Lindzen and Holton, 1968). These types of motion, it should be noted, are best observed in the wind field, and a direct analysis of wind observations was necessary for their detection. There are many studies, beginning with that by Eliasen and Machenhauer (1965), that document the occurrence of discrete Rossby modes in extratropicallatitudes. The theoretical modes so identified are only for the external mode with small latitudinal modes l. A thorough survey of these has been made by Madden (1979), and more recently, by Salby (1984) and by Plumb et al. (1985, pp. 253-262). The most frequently observed periods seem to be five days and 16 days. A study by Lindzen, Straus and Katz ( 1984) was based on global 500 hPa height analyses prepared by two meteorological centres. They report an intermittent but almost pervasive existence of oscillations in 500 hPa height that fit the properties of nine of the low-order Lamb-Rossby modes: l = 1,2, and 3; and Isi = 1,2, and 3.* Each ofthe modes migrated westward at a rate close to the Doppler-modified calculations by Kasahara (1980) of the zonal phase speed of external Rossby waves. An example is shown in Figure 3.12. The amplitude of the waves generally increased with the meridional wave number. During a period of active wave energy, the combined height field from all nine modes added up to produce a 500 hPa height pattern that ranged from a minimum of -72 to a maximum of + 131 metres. This implies a value of perhaps 104 m2 for the maximum horizontal variance of height at 500 hPa that is attained by these waves. This is smaller than the average variance in the transient eddy motion, for which Oort and Rasmussen (1971) report a latitudinal maximum of3 X 104 m 2• However it is comparable with the average variance of 1.2 X 104 m 2 in the stationary eddies (which also have a large horizontal scale). An important aspect of the study by Lindzen, Straus and Katz is that they obtained similar results from global analyses prepared at two analysis centres. This provides some evidence that these "free" Rossby modes were not an artifact of a particular data-assimilation system. *Apparently the evidence for the mixed mode (! = 0) was weak. The high frequency of these waves (Table 3.2) will make detection difficult with once- or twice-daily radiosondes. Furthermore, the geostrophic constraints that are used in most large-scale analyses may filter out any mixed-mode patterns that are present in the real atmosphere. CHAPTER 4 DISPERSION IN THE PRESENCE OF ZONAL CURRENTS The frequencies and zonal phase speeds ofRossby waves in a resting atmosphere are small, except for the smallest wave numbers. The WKB formula (2.54) can be interpreted in terms of spherical harmonic wave numbers s and n by inserting the expressions 2.Q cos 1)/a for p, s/a cos 1) for k, and n(n+ l)/a2 for kZ + /2. 2Q w (4.1) s = n(n+l) + f2a2 ( cs-2 + mZ ;zrz) The angular velocity wls decreases rapidly with increasing values ofthe horizontal wave number n and the vertical wave number m. At 45', the westward translational speed will be less than 25 m s- 1 for all horizontal wave numbers n equal to 5 or more, even for the Lamb vertical mode (which has m2 = - J'l). This slowness in a resting atmosphere suggests that their behaviour will be modified significantly by the average zonal winds present in the atmosphere. If these zonal winds were simple enough to be described as a solid rotation (i.e. u = constant times cos 7J), their effect on the Rossby waves would mostly be to change slightly the value ofQ in the numerator of(4.1 ). However, the average zonal currents depart significantly from solid rotation. This departure can be large enough, in fact, to allow instabilities. Variation ofu with latitude will allow barotropic instabilities to occur in which the kinetic energy of the mean zonal flow (112) u2 is converted into perturbation energy (Kuo, 1949). Variation of u with height corresponds to a latitudinal gradient of temperature, and will permit transformation of the potential energy into energy of the pertur bation (Charney, 1947; Eady, 1949). It can therefore be expected that non-uniform currents also have a profound effect on the propagation of stable waves. This effect will enter through an eastward advection of the potential vorticity of the wave, and through advection by the wave of the potential vorticity present in the basic current. 4.1 The channel dispersion mode of Charney and Eliassen The horizontal group velocity ofRossby waves is largest for the vertical mode of the Lamb solution. A simple allowance for the effect ofu would be to add a zonal advective term -ku to the Rossby frequency equation for the vertical Lamb mode. However, the longitudinally and seasonally averaged values of the zonal velocity vary significantly with latitude and elevation, so that it is not obvious what values ofu are most appropriate to use. Charney and Eliassen (1949) formulated a model, the "equivalent barotropic model", that attempts to define the best level in the vertical at which to apply a two-dimensional version of the quasi geostrophic forecast equations. (See Appendix E.) The equivalent barotropic model is based on the quasi-geostrophic vorticity equation (E10), and the assumption that the variable part of the geopotential field, rj/, varies with height (pressure) as rp' (A., 7J, p, t) = A(p) ifj (A., 7J, t). (4.2) The over bar here denotes a vertical average of the deviation of rjJ from a standard atmosphere 1 JP,ro rp' dp, (/) = Psfc 0 (4.3) A = 1. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 59 (The surface pressure Psrc can be taken as a constant.) The geostrophic velocity v and the vorticity (satisfy the same modelling assumption. The variables in the quasi-geostrophic vorticity equation (E10) are expressed in this way, and the equation is then integrated from p = 0 (where it is assumed that 6J vanishes) to Psrc· a( + v·Vf + A"v·V(- __jj__ W sfc · (4.4) at Psrc In evaluating Wsrc with the boundary condition (D21), we will ignore the effect of orography and surface friction. (Charney and Eliassen included them, however, when they discussed large-scale steady state motions.) Thus, Wsrc (p aq/) (A ) arp (4.5) Psrc = ----;; 3t sfc = R T sfc -----a{· Charney and Eliassen now introduced the idea of the "equivalent barotropic level" by noting that if(4.4) were multiplied with A2, it could b~ rewritten as foA) !!_ _5_at + v · V if + 0 - ( R T sfc at 0, (4.6) where rp, v, and (are at the level where A(p) is equal to A2• (This vorticity equation is that satisfied by a layer of homogeneous incompressible fluid with a free surface of elevation rp/g; hence the name "equivalent baro tropic".) Typical values of wind (or variable geopotential) in middle latitudes show that A is 1 at about 600 hPa, that A2 is about 1.25, and that A is 1.25 at about 500 hPa. Asrc is smaller, about 0.2. The equivalent barotropic level is therefore to be taken at 500 hPa. * Charney and Eliassen assumed that the motion takes place in a zonal channel centred at v0 = 45 degrees, and bounded by zonal walls: v0 - L1 v ~ v ~ v0 + L1 v. (4.7) The justification for this assumption was observational, in that daily changes in 500 hPa height consist mostly of positive and negative centres that generally move eastward, more or less within fixed latitude bounds. From a theoretical viewpoint, however, this assumption at first appears to be in direct conflict with the later evidence contained in the 1977 computations by Hoskins, Simmons, and Andrews that are reproduced in Figure 3.11. These computations seem to demonstrate that it would be erroneous even to limit the horizontal forecast area to a hemisphere. But the computations in Figure 3.11 are for a resting basic state (a= 0). For a variable basic current u, the critical points and turning points that will be discussed later in section 4.4 will suggest a theoretical reason for confinement of much of the energy ofRossby waves into a zonal channel in middle latitudes. If( 4.6) is linearized about a uniform basic current u, the following equation results for the deviation z' of the height of the 500 hPa surface. a a )(a2z' a2z') az' az' (4.8) (at + u TA --ai2 + ayz - f.l2 at fJ' a;:-· * (4.6), with data taken directly from 500 hPa, was the atmospheric model used by Charney, Fj0rtoft, and von Neumann in the first experimental forecasts with an electronic computer (Charney et al., 1950). It was used operationally by several centres in the 1950s. It is still used, in a global version, at the National Meteorological Center in Washington for historical calibration of improvements in forecasting obtained from advanced models. Held, Panetta and Pierrehumbert (1985) propose a different formulation in their study, for forced stationary Rossby waves. They arrive at an equivalent barotropic level around 425 hPa for those waves. 60 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The latitude co-ordinate is centred on 1J0, and scaled such that cos 1J0 dy = dlJ. The time unit is one day, and /3' 2 2 = 4n cos 1J0 = 2n is a non-dimensional form of Rossby's jJ. The parameters f1 and U are* /12 = fo2 a2 cos21Jo ( : T ) sfc ""' 0.5, (4.9) 2nu U(rad/day) ""' 0.2 u (m s- 1). (4.10) aQ cos iJ0 (The same procedure was used by Chamey (1949) in a companion paper. In that paper he outlined the rationale for using the equivalent barotropic form of the geostrophic system to make the first modem numerical prediction with an electronic computer.) (4.8) is solved under the condition that only one latitudinal mode need be considered: z' (A, y, t) = cos ly z 1(A, t). ( 4.11) This satisfies the boundary condition that the geostrophic meridional velocity component vanishes at the northern and southern boundary walls. The final equation involves only longitude A and time t: a a ) ( a2z1 az1 az, ( at+Ua;: ~- /2 z,) - /12 at = - /3' a:r-· (4.12) The authors showed that the solution to this can be formulated in terms of an influence function I (c;, t). z 1 (A, t) = z 1 (A - Ut, 0) + f" I (c;, t) z1 (A - Ut + c;, 0) de;, ( 4.13) -n where s = + 00 -isc; ivst I (c;, t) 2:: e e - 1 . (4.14) 2n s = + 00 (The sign convention for c; adopted here is opposite to that used by Chamey and Eliassen.) The forecast at time t for the point A is equal to the value at timet = 0 of the value upstream a distance Ut, plus a correction given by an integral over the entire range oflongitude. The function vs contains the entire zonal dispersion effect present in these Rossby waves: /3' + /12 u /3' 2n Vs = S ( 4.15) s2 + 12 + 112 ""' s s2 + /2 = s s2 + /2 . Figure 4.1 shows the distribution of I (c;, t) fort = 1 day and 5 days, constructed for the following parameter values, similar to those used by Chamey and Eliassen. lJo 45 degrees (/3' = 2 n); ill) 20 degrees (1 2 = 10.125); (4.16) u 13.5 m s-1 (U = 0.225 rad day- I 13 deg day- 1); /12 0.5. 2 * (R TIA)src replaces the factor C5 = yRTin the previous Rossby-Lamb formula (4.1). The product yAsrc has a value of only 0.28. The implied value ofe = 4Q2a2/gh is only 2.1, with h "' 88 km. Divergence effects are negligible except formations of the largest horizontal scales. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 61 n I I 1(~, t) I" I I I --t=1 day I I I I 2 ___ t = 5 days ( ...... I I I I --- ,..., / I - 180 ° - 120 ° - 60 ° 120 ° 180 ° Figure 4.1 -The influence function/(~, I) for I = I day (solid line) and for I = 5 days (dashed line), for barotropic flow in a mid latitude channel (13.5 m s-1 is an average between 20'N and 60'N of the winter values tabulated by Oort and Rasmussen (1971, p. 69.) The frequency and group velocity of these Rossby waves are as follows: P' + t-t 2 u w (cycles per day) sU- s (4.17) s2 + f2 + f-l2, s2 _ (/2 + /-l2) 2 awas (radians per day) = U + The group velocity has the following extreme values. ( aa~)min - u- 26 deg day-1, (4.19) at s = 1 with w/s - U = -31 deg day- 1• U + 4.3 deg day- 1, (4.20) ( :~)max at s = 5.6 with wls - U = -9 deg day- 1• 62 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION U is shown explicitly here as the simple Doppler effect because its contribution to the second term in (4.17) and (4.18) is negligible.* The curve t = 1 day in Figure 4.1 has a very simple distribution. Its shape is similar to the function 2 2 112 exp [- I~I (/ + ,U ) ] = exp (-3.26 I~I), (4.21) which is the solution for a point source J(,A.) of the operator a2 2 2 ( ----ai2 -/ - .U ) I (.A_, t) o (A-). (4.22) This in turn is the theoretical shape of the solution for 8z/at at t = 0. Figure 4.2 shows the values off(~, t) forO- lOdays. The values shown fort= Oareequal to lOO times the value of I for t = 0.01 d~ys. It there~re shows_ what !he _solu!ion would !Je for t == 1 day if that were to be CHARNEY-ELIASSEN INFLUENCE FL:'NCTION l {------Tif1E IN DAYS------} 0 2 3 4 5 b 7 B 9 10 LONG. -180 0~00 o.oo 0.02 0.06 0.17 0.32 0.4?':.55 0 .. 44 0.14 -o.zs -170 c..oo 0.00 0.02 0.08 0-21 0 .. 38 0.53 0.55 0.37 0.01 -o.38 -160 i).i)(l o.oo 0 .. 03 0 .. 11 0.26 0.44 0.5 0.53 0.27 -0.13 -o.49 -150 0.0(• 0.01 0 .. 04 o. ~4 0-32 0.51 o. 0 0.48 0.15 -0.28 -0.57 / -140 0.(:0 0.01 0.06 0.19 0.38 0-5:/0.60 0.40 o.oo -0.41 -~.60 -130 o.oo 0.01 0.08 0.24 0.46 0.67 0.58 0.29 -0.15 -0.52 ;-0.59 -120 o.oo 0.02 0.12 0.30 0.53 0.6..... 0.53 0.15 -0.31 -0.59., -0.52 /. ~ -110 0.01 Q.03 0.16 Q.38 0.6: ~...67 0.44 -0.01 -0.45 -O.c1 -0.39 -100 0.01 0.05 0.21 0.47 0.6/ ~-65 0.31 -0-~9 -0.5'/<'-~.57 -0.2~ -90 0.02 0.07 0.28 0.56 0.~ 0.59 0,14 -O.w.7 -0~62 -0.45 O.OJ. -so 0.03 o. 11 0.37 0.66 0.75 0.48 -0.06 -0.53 ,-0.62 -0.27 0.24 -70 •).06 0.16 0.48 o.76/:...n o.;;t -0.21 -o.';•~ -o.s;; -o.o3 o.45 -60 0.11 0.23 0.61 0.84 0.66 0.10 -0.49 -0.68 -0.35 Q.23 0.60 -50 0.19 0.33 0,76 0.88 0.51 -o. 16 -o. 6~ "'-o. 62 -o. 09 o. 48 o. 66 -40 0.33 0.47 0.91 0.87 0.28 -0.44 -0.76 -0.44 0.22 o-?/0.61 -30 0.58 0.67 1. 06 0.76 -0.03 -o. 71, ~- 74 -o.16 o.s3/o. 77 o.44 -20 1.03 0.93 1.16 0.53 -0.41 -o .. 89 -o. 57 o. 2= _>7B o. 71 o. 17 -10 1. 81 1. 27 1. 18 0.15 -0.82 -0-~4 -o. 23 o. oy o. 90 o .. 49 -o. 13 0 3.19 1.69 1.03 -0.41 -\. 1. 7. -o. 77 o. 20 o. 96 o. s2 o .. 13 -o. 39 0 -3,22 -4.72 -11.79 -19.64 -26.81 -32.62 -38,20 -43.91 -50.46 -51'.56 -64.46 10 -1.82 -1.97 -3.25 -2.67 -0.12~.94 6.75 5.74 3.80 1.99 20 -1.03 -0.79 -0.60 0.67 2~9 2. 72 1. 55 -o. 10 -3~4 -4.39 -3. so 30 -0.58 -0.30 0,08 o. 76 0.98 0 .. 24 -1.16 ..... -2.23 -1.98 -0.19 2.31 /' 40 33 -0.10 0.17 - -o. 0.39 0.16 -0-:;.9,._0.99 -0.70 0.50~2.74 so -o. ts -0.'03 0.12 0.13 -0.11 .,;-0.39 -0.31 Q.\29 1.09 J.-~5 0.93 60 -·). 10 -0.00 0.07 0.02 -o.t!' -0.16 o.;.<:7o. ~-76 o.57 -0.01 / - ,. .,. 70 -o. ob o.oo 0.03 -0.01 -0.07 -0.01 0.1 0.~9 0 .. 41 ,..,.0.21 0.01 ao -o. o3 o.oo 0.01 -0.02 -0.02 o.os 0.18 0.28,..f1.27 0 .. 23 0,31 90 -0.02 0.00 o.oo -0.01 o.oo 0.07 0.16 0.23 0.28 0.37 0.52 100 -O.tll O.Oi) -0.00 0.02 0.08 0.16 ...... 0 .. 24 0.34 0.47 ~.57 110 -t),Ol ;).00 0.00 o.oo 0.03 0.09 0.17 0.28 0.41 0)3 0.53 120 -t),OO (•.00 o.oo 0.0! 0.04 0.10 0.20 0.33 0.46 0.53 0.45 130 -1).00 o.oo o.oo 0 .. 01 0.05 0.12 0.24 0.38 o~'5c/ o.51 o.35 140 -(1. 00 i) .....o o.oo •) .. 02 0.06 0.15 0,26 0.43 0.52 o. 48 o. 25 150 -0.00 o.oo o.oo 0.03 0.08 0.18 0.33 0.47 0.53 0.42 0.14 t6o -o.oo o.oo o.or o.o3 o.to o. 22 o.3B o.st o.s2 o. 34 o.o1 170 -0.00 0.00 0.01 0.05 0 .. 13 0.27 0.,43 0.54 0.49 0.25 -0.12 Figure 4.2- The Charney-Eliassen influence function for days 0 to 10 inclusive. Solid and dashed curves show the motion of maxima and minima. Values fort = 0 are explained in the text *The fraction f1 2/ (s2 + /2 + f1 2) has a maximum value of only 0.043 for waves whose latitudinal wave length is 80 degrees. The non-Doppler U-term in (4.17) and (4.18) is therefore negligible. The smallness ofthis fraction for motions of the horizontal scale assumed in the quasi-geostrophic approximation (Appendix E) has two implications: 1. The term arp/at in the hydrostatic boundary condition at the ground for w, that is stated in (D.21), can be neglected for quasi geostrophic motion of all but the very largest horizontal scales. 2. Without this term, the quasi-geostrophic system of equations is invariant to a Galilean translation, in spite of the rotation of the Earth. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 63 obtained by calculating aJ/at at t = 0 and multiplying the answer by one day. It is symmetric and surprisingly close to the correct solution for t = 1. This shows that in this narrow channel the Rossby dispersive process has had little effect at t = 1 day -a feature that might have been expected from the group velocity values listed in (4.19) and (4.20). The magnitude of I for one day exceeds 0.1 only for the longitude interval - 40' < ~ < + 80'. For the typical value of U cited above, this would mean that a one-day forecast for longitude A would require initial data from 40 + 13 = 53 degrees to the west of A, and from 80 - 13 = 67 degrees to the east of A. (These numbers are somewhat larger than the values selected by Charney (1949; Figure 6 and Table 1) because in that paper Charney used a latitudinal width of only 30 degrees for the channel, with /2 + JJ 2 = 18.) The first operational numerical predictions in the United States were made with the non-divergent model at 500 hPa, for a limited area. However, when the computational region was increased to include almost the complete northern hemisphere, severe difficulties were experienced; the s = 1 modes oflarge latitudinal scale retro graded rapidly and empirical corrections were necessary (Wolff, 1958; Cressman, 1958). Rossby proposed his linear non-divergent model for large-scale motions of the atmosphere in 1939. Shortly thereafter Ertel (1941, 1944) reasoned that even short-period forecasts using the non-divergent vorticity equation were impossible unless data from the whole globe were available. His argument was based essentially on a simplified version of (4.4), which shows that the geopotential tendency at the initial time is given by the inversion of the horizontal Laplacian operator acting on vorticity advection over the entire globe. Chamey contended in 1949 that Ertel was unduly pessimistic, because the spread of any significant influence from distant regions would be controlled by the maximum group velocity, and that this velocity was limited in magnitude. Ertel's conclusion was based on his estimate ofthe contamination ofthe interior of a limited forecast region by unknown changes at the lateral boundaries of the region. These boundary changes were assumed to be zero in Chamey's channel model. In retrospect, then, it would appear that the most effective step taken by Charney to achieve a more optimistic conclusion than Ertel was his bold assumption that the changes on the lateral boundaries could be set to zero. The years between Ertel's 1941 paper and Chamey's 1949 paper had witnessed a great expansion in the construction ofdaily hemispheric maps at 700 and 500 hPa. Ertel had to reason with less knowledge of daily flow patterns. However, the later operational experience in the United States referred to above shows that Ertel's pessimism on this matter had some justification. The curve for day 5 on Figure 4.1 has more character than that for day 1. The numbers in Figure 4.2 show how this comes about. The broad maximum located at~= 120 degrees on Figure 4.1 is seen to be moving westward in Figure 4.2 with a speed of about 30 degrees per day, in agreement with the properties associated with the wave number of minimum group velocity that are listed in (4.19). The sharp wave-like pattern located just west of~ = 0 in Figure 4.1 means that significant influences from that region will have moved eastward faster than the zonal current U, and that these influences will be important to the extent that their wave length matches the 60-degree wave length shown in I. The individual maxima and minima have a phase speed of about -10 degree per day. These properties are in good agreement with those listed in (4.20) for the wave of maximum aw/a/s. These aspects of the zonal dispersion from a point source were stressed by Rossby (1949). They unify what would otherwise seem to be two contrasting problems-what is the effect of a point source on the entire range oflongitude?-and what is the effect of the entire range oflongitude on a single forecast point? A theoretical example of the propagation of a broad distribution of forecast errors in a channel model is given in the study by Phillips (1976). The channel width was again about 40 degrees of latitude, and the geostrophic equations were linearized about a zonal current independent oflatitude. But the vertical repre sentation was given by a "two-level" model instead of the equivalent barotropic model used by Charney and Eliassen, and can thereby simulate the conversion of potential energy into kinetic energy. The value of U1 in the bottom layer of the model was 7.4 degrees per day, that in the upper layer was 20 degrees per day. The average U was therefore close to the 13 degrees per day used in constructing Figures 4.1 and 4.2. The linearized equations for the height perturbations z1 and z2 can be manipulated to yield prediction equations for the 2 statistically expected variance of the errors, z1 and zl, much as is done in the Kalman type of control modelling (Gelb, 197 4). The initial statistical errors were defined by applying an idealized version of the optimum interpolation analysis method used in numerical weather prediction (Gandin, 1963) to a simulated network of radiosondes and satellite temperatures. 64 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION NORTH AMERICA 0 I 2 -...... 3 4 H ' '\ 20 s <30 ' 6 30...,.)' ' 60 7 --- 8 - 9 Figure 4.3 - Simulated root-mean-square 500 hPa height errors as a function oflongitude and time (downward. in days.) Isolines are at 20 m intervals with the exception of the I 0 and 30 m lines. Arrows sloping down to the right indicate the phase speed for the most unstable wave. The two vertical lines mark the coasts of North America. The black dots across the top indicate the simulated radiosonde network. Those at the bottom mark off the simulated analysis points. (From Phi/lips, 1976) Figure 4.3 shows the theoretical dispersion in time and longitude of a realistic set of simulated analysis errors. The hypothetical analysis errors (shown at the top of the figure) are characterized by small values over the North American sector (where the idealized analysis model used in the study assumed the existence of many radiosondes) and large values over the Pacific Ocean (where there were few radiosondes, and satellite temperatures of little accuracy). The initial values of the perturbation field in this example constitute a well-distributed source, and the only obvious connection between the spread ofthe errors to the east with elementary wave properties is the moderate agreement between the evolving pattern and the phase speed of the most unstable waves shown by the arrows in the figure. Two considerations seem to be indicated by Figure 4.3: 1. Analysis errors from the data-sparse region of the Pacific will have considerable effect on forecasts in the European area, perhaps as early as ten days. This delay will be shorter if the mean zonal wind has a value greater than the climatological winter average. 2. The results show a diminution of the errors on the west coast of North America after day 5, presumably because at this time the (smaller) initial errors from the radiosondes over eastern Asia are beginning to arrive on the west coast ofNorth America. (Note the early invasion of the Atlantic in the upper right corner by small errors from North America.) This, however, should be viewed with some scepticism, since in the real world there is no precisely defined channel to confine the initial distri bution of errors. 4.2 Observed patterns of zonal dispersion The distribution of wave energy with longitude and time can be displayed effectively on the "trough and-ridge" diagram devised by Hovmoller (1949). His example for November 1945 is reproduced as Figure 4.4. The height of the 500 hPa surface is first averaged for each synoptic map over the latitude interval 35'N-60'N, and then displayed as a function of longitude and time. In this case, the region from 150'E eastward to about O'E displays a wave-like structure, with alternating bands of higher- and lesser-than average height. These patterns generally move eastward, especially those of short wave length. Hovmoller drew attention to two typical rates of displacement. The first is relatively slow, about ten degrees per day, and marks the progression ofindividual ridges and troughs. In this month it appears most clearly between days 10 and 20 near longitudes 90-150'W. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 65 Figure 4.4 - The trough-and-ridge diagram ofHovmi:iller (1949). Contour intervals are at 100 m. above-average values being shown by horizontal hatching and below-average values by vertical hatching Hovmoller has indicated the second rate by the three short straight lines, inclined at a rate of about 28 degrees per day. These lines are associated with east-west wave lengths of about 60 degrees, and there is a tendency for the individual centres along them to increase in amplitude with time. One possibility of explaining this is the redistribution of existing energy by the zonal group velocity. A general result ofthe WKB treatment of waves is that energy is propagated with the group velocity, but in a slightly different manner than frequency and wave number. If E denotes a locally averaged value of the wave energy (which will be proportional to the square of the wave amplitude), the relation is* aE _ at = - divergence (Cgrp E). (4.23) For the present case, this relation can be derived by first multiplying (4.12) by - zi> and manipulating the result to obtain the relation aE!at = - aF/a). where 1 az 1 E = 2 I(l a:f )2 + (/2 + !P) z,2 j ' (4.24) and 1 1 1J2 J.l2 2 P' 2 F = - - a lz (az,- + U-az,)J + -az -az + U rE + (az - -- z J- - z • (4.25) a;. 1 at a;. at a;. a;. 2 1 2 1 *See, for example, Chapter 11 in Whitham (1974). In moving media, the energy E must be replaced by the "wave action". E (w - k.u)- J (Bretherton and Garrett, 1969). 66 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION E and Fare now "averaged over a wave length" under the assumption that z1 is periodic in both longitude and time. (A more formal method is to make an expansion like that used in Appendix C.) The leading term in F averages out, and the result can be written as* aE a !5!!_ -) (4.26) at a.A ( as E · The condition that E increase along the path A = constant + (aw/as) t is that (awl as) decrease with longitude. This condition can be readily evaluated from (4.18): aw ) _ 2 P' + j..l2 U ----a ( as [3 (/2 + f.-l2) - s2] < 0 (4.27) a.A as aA (s2 + 12 + f.-l2)3 For the present model, with f2 + f.-l 2 = 10, the last bracket changes sign at s around 5. This strong dependence on /2 + J.-l 2 complicates this criterion, since this wave number is in the peak of the active spectrum ofRossby waves. The successive amplification of height centres downstream of an initial development documented so clearly by Hovmoller is nonetheless a common event. An alternative interpretation is the original suggestion by Rossby that the initial development occurs through a baroclinic process, in which potential energy is converted into kinetic energy. This would act as a "point source", following which amplification of disturb ances downstream will take place in the wake of the zonal trajectory A = constant + t X ( aa:Jmax· (4.28) This speed will exceed the advective speed of the background zonal wind u by about five degrees per day, according to the value determined in (4.20). An early description of this process is found in the 1948 article by Cressman. Simmons and Hoskins (1979) have analysed and simulated this process in a numerical experiment with a hemispheric forecast model. They inserted an initial localized disturbance, independent of height, into a zonally symmetric baroclinic current u (1J, p). This zonal current increased monotonically with height and had a jet-like structure in latitude with a maximum speed of 45 m s- 1 (40 degrees per day) at 30'. The initial disturbance amplified because of baroclinic instability and moved eastward at seven degrees per day. Successive new systems developed in advance of the original one, in the wake of a point moving about 23 degrees per day to the east. This is illustrated in Figure 4.5. The centres formed about 60 degrees apart. Twenty-three degrees per day is close to the vertically averaged speed of u in the centre of the jet. Simmons and Hoskins were able to deduce some of their numerical results by applying a complex integration technique to the initial-value problem posed by an isolated unstable baroclinic wave of the type discovered by Eady (1949). They emphasized that their principal results were not very sensitive to the details of the initial disturbance or even to the somewhat different baroclinic models they applied to this problem. They point out that if numerical integration errors can be kept small, this robustness provides "evidence for the longer-term predictability of large-scale atmospheric motion". It must be remembered, however, that their basic current was simple and "known", and that the initial disturbance was not exposed to corruption from other initial disturbances. The situation will not be so optimistic for imperfectly known initial flows. A striking example of rapid eastward progression of effects is described by Hollingsworth et al. (1985). They first made preliminary comparisons between the three analyses and the three forecasts made for part of February 1979 by the British Meteorological Office, the European Centre for Medium-range Weather Forecasts (ECMWF), and the National Meteorological Center (NMC) in Washington. These comparisons seemed to show that at 0000 GMT on 18 February 1979, differences in analyses in the north-eastern part of * Rossby's derivation of(4.23) for horizontally propagating gravity waves (1945, section 4) appears to be artificial, since the amplitude ( 0 in his equation (35) is not treated as a complex quantity. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 67 Figure 4.5 - Development of the surface pressure in the numerical experiment by Simmons and Hoskins. Isobars are drawn for every 4 hPa. (From Simmons and Hoskins, 1979) Figure 4.6- Forecast dispersion of the analysis differences for 0, 2, 4, and 6 days after 0000 GMT, 18 February 1979, in the transplant experiment by Hollingsworth et al., 1985 68 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION Figure 4.7 - Top: Mean flow pattern at 500 hPa for December 1986. Bottom: Progression of standard deviation of 500 hPa errors (metres) as a function of forecast length from the regional model at the National Meteorological Center during December 1985. The heavy line marks the average eastward progression of a hypothetical line of particles located off the west coast at zero hours the Pacific Ocean led to major differences in medium-range forecasts in the Atlantic after only four days. In order to demonstrate this hypothesis more clearly, a "transplant" analysis was made by merging the NMC analysis from the north-eastern part of the Pacific with the ECMWF analysis. This modified analysis was then forecast for six days with the ECMWFforecast model. Figure 4.6 shows the difference in 300 hPa height, at 0, 2, 4, and 6 days between this transplant forecast and the forecast made with the ECMWF model from the complete original ECMWF analysis. The differences (centres of+ 30, -43, -38, 35, 56, and-62 metres) at 0 days in the north-eastern Pacific move rapidly eastward at an average speed of about 24 degrees per day. By day 6 the leading difference centre is well into northern Europe. As in the Simmons-Hoskins experiment, many of the centres have grown in magnitude. Figure 4. 7 contains a mean 500 hPa height map for December 1986, together with four maps of 500 hPa forecast height errors from the Nested Grid Model used for North American forecasts by the National Meteorological Center in Washington, D.C. The values shown are the standard deviation of the forecast errors from the set of 12-hour forecasts, the set of24-hour forecasts, etc. Each map represents 58 of the 62 possible forecasts that were made in December. (In other months the isolines are similar in broad outline, but differ in details. Computations in this area based on forecasts from two other models at Washington, from the twice-daily forecasts from the British Meteorological Office, and from daily forecasts from the European Centre for Medium-range Weather Forecasts show very similar patterns.) The error pattern moves eastward across the continent from the Pacific Ocean. The heavy north-south lines between 30'N and 50'N show the average geostrophic zonal displacement for that longitude, calculated from the contours shown on the mean map, and beginning with the heavy dashed line located at 129'W. The error isolines move faster than this mean displacement to some extent, but not in such an organized way that the rate of displacement can be DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 69 confidently compared to an estimate of the typical zonal group velocity. Furthermore, some of the progres sion of larger values across the continent is undoubtedly due to the amplification of errors by instability processes. It is not possible to rule out completely the possibility that the forecast model makes very large errors on the West Coast. Arguments against this are the similarity to the theoretical pattern shown in Figure 4.3, the example in Figure 4.6 from Hollingsworth et al., and the fact that the same broad pattern of errors is obtained from all forecast models. The conclusion seems inescapable that a major source of forecast error is analysis errors over the Pacific Ocean. 4.3 The Charney-Drazin (Queney) criterion In 1961, Charney and Drazin showed how the distribution of u with z would affect the vertical propagation of Rossby waves. They used the quasi-geostrophic system of equations that are described in Appendix E. For the present purpose we need only linearize the equation for the conservation of geostrophic . potential vorticity in the presence of a basic current u. In Cartesian geometry this is a a ) aPV - + PV' + v' -- = 0. ( at u -ax ay (4.29) PV' and PV are the geostrophic potential vorticity of the perturbation and the basic current: 1 a ( p ar;;') PV' =- "Y2rj;' + (4.30) fo p az foS az · aPV a r au 1 a ( p a,y = a.Ylf- ay + p az foS ;~ )J (4.31) a2u p- a ( p au) ay2 P az s az · The other symbols are: p . Z= - ln --- ' (4.32) Po S = N2 = H2N2. (RT)2 (4.33) 2 2 g fo fo ' 1 ar;;' v' = (4.34) fo ax fo is a constant value of the Coriolis parameter, and rj;' is the perturbation geopotential. If rj;' is redefined as proportional to G(y,Z) in the manner · · s )112 rp' = (P Re G(y, Z) exp i(kx - wt), (4.35) the following partial differential equation for G results: 2 _ f 1 aG a2G r (1 + 0) J l aPV (u - C)\ s az2 + ay2 - l k2 + 4 s G J + ay- G 0. (4.36) 70 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION C = wlk is the trace velocity in the x-direction and ois the factor o = 4 Sl/2 f d2(S-112) - d(S-112) ] l dZ2 dZ .. (4.37) ois zero in an isothermal atmosphere, and Dickinson ( 1968, Figure 6) shows that it is very small in a standard atmosphere except where the latter has a sudden change in dT/dz. Dickinson also presents seasonal cross-sections for each of the three terms in the potential vorticity gradient aPV/ay. The f3term dominates in low latitudes in all seasons, and almost everywhere in summer. In winter, on the other hand, the presence of an 80 m s - 1 westerly jet near 60 kilometres in middle latitudes contributes u terms that exceed 2 Q cos u. (At very high latitudes and elevations, aPVlay is even negative, although Dickinson's data are probably not good enough to be sure of this.) The effect of the u terms in aPV/ay can be seen, to some extent, by ignoring the p factor, assuming that S is almost constant, replacing the product HZ by the geometric height z, and then replacing z by the Prandtl Rossby scaled height z'= (Nij)z.* Under these conditions, the two u terms give a2u a2u) (4.38) - ( ay2 + az'2 • These terms give a positive contribution to aPV/ay in a region of maximum u, just as Rossby's /3-term gives a positive contribution to aPVlay in the region of maximum rotational velocity of the Earth (= Qa cos u). The simple WKB approach to (4.36) is most similar to earlier results if s- 1 a2G/aZ2 is replaced by fo2 a2G foz a2G fo2 N2H2 aZ2 = N2 az2 = - N2 m2G. (4.39) (We assume that G is proportional to exp (im z) and ignore the small o term.) The Rossby frequency is now k aPV (4.40) ku- 1 ay WR 2 fo ( 1 ) k2 + f2 + N2 4H2 + m2 The formula for the square of the vertical wave number is aPV ay J. m2 = _ 1 N2 ( 2 4H2 - fo 2 k + f2 + c-u (4.41) These differ from earlier versions of wR by the Doppler effect ku and by the new terms in aPV/ay. In particular, the formulas (2.87) and (2.97) for the south-north and vertical components of the group velocity are changed only through the replacement of f3 by aPV/ay. Chamey and Drazin worked with these equations under the additional restriction that au/ay = 0. In their simplest case of constant u, the condition for positive m2 reduces to f3 (m2 > 0): < c < u. (4.42) u--(1o) 2 k2 + /2 + 2 HN *The omitted derivative ap/aZ = - pis quantitatively important, however. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 71 For stationary waves, this reduces further to p UR· (4.43) (m2 > 0 and C - O)·· 0 < u < k2 + / + ( foHN ) 2 2 2 This is essentially Queney's 1948 criterion for a wave-like response to uniform flow over a sinusoidal mountain. (See section 2.5.) But it is now seen to play a more pervasive role in large-scale atmospheric motions. Ifit is assumed that stationary Rossby waves are created only in the troposphere, for example, (4.43) would explain why the summer stratosphere, with u <0, has little stationary Rossby wave energy. (Note Figures 3.4A and 3.4B.) The most permissive positive value of uR occurs for the minimum value of k2 + 12• This is about 2/ a 2 and yields, for middle latitude values of p, a value of38 m s- 1 for uR. Larger values of k2 + /2, such as those typical of baroclinic instability, will have smaller values of uR, sufficiently small as not to propagate far into the stratosphere (Charney and Pedlosky, 1963). This is consistent with the impression in Figure 3.4A that only eddies of large horizontal wave length are found in the upper stratosphere in winter. A similar conclusion can be reached about the latitudinal wave number. For constant u we have p (/2 > 0): u c < u. (4.44) k2 + 2 (!~rr m + ( 2~ r J< For the special case of steady motion, p (1 2 > 0; c = 0): O 2 (For the vertical Lamb mode, the denominator would be k + (j0/C5)2.) This predicts an inability ofstationary Rossby waves to propagate into or through a zone of easterly mean zonal wind, such as exists in the troposphere of low latitudes. These results describe the ability ofRossby waves to propagate energy, in terms of the zonal wind u and the wave numbers k, l, and m. An alternative view ofenergy propagation was formulated for stationary waves by Eliassen and Palm (1961 ). Their results, which are readily extended to the case of w -,6- 0, are discussed in section 4.8. 4.4 Critical points and turning points A striking introduction to the effect of background zonal currents is given in the second part of the non-divergent calculations by Hoskins, Simmons, and Andrews (1977) that were described in sec ion 3.5. In this part of their paper they considered the effect of two basic currents. One was a solid rotation: u = 0.032 .Q a cos u, with a value of15 m s _,for uat the Equator. The second basic current was more realistic. It was a smoothed representation ofa climatological distribution at 300 hPa, with a broad "jet" of27 m s-1 at 40'N, and weak easterlies of -2.5 m s- 1 at the Equator. In both cases the perturbations were created by a steady vorticity source at 30'N. The two calculations for day 5.5 are reproduced in Figure 4.8. The solid rotation basic current gives results similar to those from the global case with no basic current (see Figure 3.11 ), i.e. significant propagation into the southern hemisphere. The more realistic zonal current allows almost no reaction in the southern hemisphere. Instead of propagating across the Equator, the waves generated by the steady vorticity source seem to be halted in their southward progression, "piling up", so to speak, with an accumulation of short latitudinal wave lengths. The theoretical interpretation of this phenomenon is that it is 72 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION 90~~--~--~-,--~~---,--~-.--~--.--. 0 I I I I I - .... 1 ( 'G -90~~--~--~~--~~--~--~~--~--~~ 90 0 -90 Figure 4.8- Distribution of relative vorticity at 5 1/2 days generated by a steady positive source at 30'N. Positive isolines are solid, negative are dashed. Top: With a zonal current equivalent to a solid rotation of 11.7 degrees per day. Bottom: With a zonal current having westerly jets of27 m s- 1 at 40'N and 40'S, and equatorial easterlies of2.5 m s- 1• (From Figures 7 and 9 in Hoskins et al., 1977) caused by a critical point (line, latitude, height, etc.) where the waves encounter a zonal current whose speed u is equal to their zonal trace velocity. w u (4.46) s a cos 7J (critical point) In this example, the steady source ofvorticity is setting up a train of stationary Rossby waves, with w = 0, so that the critical latitude is where u = 0. To explore this effect of u on the north-south propagation, we retreat temporarily to the per turbation vorticity equation for a non-divergent atmosphere (e = 0). In Mercator co-ordinates x = aA, y = a ln (1 + sin 7J/cos 7J), this equation is: a(' u a(' v' a -+----+----(AV)= 0 (4.47) at cos 7J ax cos 7J ay · The perturbation velocity v' and vorticity ('are specified by a stream function, !f!', as follows. a2!f!' a2!f!'~ ,, --+--, (4.48) cos2 7J ( ax2 ay2 1 a!f!' v' -- (4.49) cos 7J ax DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 73 A V is the absolute vorticity in the basic state: d AV= 2Q sin cos2 1) dy (u cos u). (4.50) Following the notation used by Hoskins and Karoly (1981), we define u UM = cos u. (4.51) uM is a measure of the angular velocity of the basic state current. We also define a modified version of Rossby's beta parameter, dA V 2Q d ~ 1 d fJM = ~- =- cos2 JJ -- - (uM cos2 JJ)l (4.52) dy a dy cos2 JJ dy ' and define a Mercator zonal wave number by k = s (4.53) a Solutions of the form !fl' = Re !fl exp i (kx - wt) yield the equation 2 d !jf + /2 !jf = 0, (4.54) dy2 in which /2 (y) fJM _ k2. (4.55) w UM- k I is the latitudinal wave number in Mercator co-ordinates. (The horizontal gradient of the phase has components k/cos u and //cos JJ). The normal mode approach to ( 4. 54) treats k as given and seeks to determine values of wand function !fJ that satisfy appropriate boundary conditions in y. This is the approach used in the analysis of barotropic instability (Kuo, 1949), where the main interest is in complex values of the frequency w. For real values of w, an important role is played by the singularity that occurs at the critical point, uM = w/k. Figure 4.9 is a schematic picture of how /2 would vary with latitude according to (4.55). The curve for uM contains a maximum westerly wind of20 m s- 1 at 30°N and a maximum easterly wind of -4 m s- 1 at the Equator. A phase speed of 5 m s _, has been assumed for wals = w/ k. The critical latitude at J)c is marked by a heavy line. 2 2 2 Just to the north oflJc, the quantity a / + s 2 = a fJM/ (u M- wlk) becomes infinitely large, corresponding to very short north-south wave lengths. South of this latitude, a 2l 2 + s 2 is large and negative, implying a rapid exponential variation with latitude. The curve of a 2f2 + s 2 in Figure 4.9 decreases at higher latitudes. For a fixed value ofs 2 this means that / 2 will again become negative, this time by passing smoothly through a zero value. Fors = 6, for example, this would happen at the point marked JJ on Figure 4.9. This is the turning point. In contrast to the singularity where u = C, the turning point is a well-behaved singularity, at which wave energy from the region of positive /2 will be reflected back into that region. The WKB plane-wave treatment of two-dimensional plane-wave solutions to (4.4 7) has parallel properties to those discussed above for the eigenvalue problem posed by (4.54). The coefficients in (4.47) are independent oflongitude (i.e. x) and time. Therefore (as described in Appendix A) the WKB phase derivatives wand k are constant along the ray path that traces out the horizontal propagation of wave energy associated with that value of wand that value of k. 74 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION \ a212 + s2 30~ \ --4120 \ 25~ \ --4100 \ I .,··· .. ~ ro (/) 2l \ 15 •. .• 60 E • • . ' .. • 10~ I : ·- ' ' ....._....!.,..--•• ~ -~40 • UM- w/k • • 51 ·· .. ••ro 11/. I I I 0. - - - 40° _ ~ .: vc ut 5 • • Figure 4.9 - Schematic distribution of terms in the perturbation eigenvalue problem for barotropic flow with a mean zonal velocity. The scale on the left axis describes uM and uM - wlk; the scale for a2 12+ s2 is given on the right The meridional wave number I, on the other hand, will change along the ray path so that the local WKB frequency equation-which contains functions oflatitude-is everywhere satisfied. This is exactly the rela tion (4. 55) for /2. For example, waves moving poleward so as to approach their turning latitude will experience a stretching of their meridional wave length, i.e. I~ 0, and will be reflected equatorward, with the opposite sign of I. (A graphical description ofthis will be presented in section 4. 7.) The wave amplitude will increase toward the turning point, proportional to the square root of the latitudinal wave length (until this part of the WKB approximation must be corrected by an Airy function). The critical-point singularity where u = C is a mathematical complication of major proportions. It appears for several reasons-the equations have been linearized, viscosity has been ignored, and the solutions are assumed to be of a permanent type, valid for all time (i.e., 8/fJt = - C8/8x.) These are idealizations, as is the assumption of a basic current u that is independent of longitude and time. Dickinson (1968) considered the case C = 0 and extended the Charney-Drazin treatment by including the effect oflatitudinal variations in u in addition to the vertical variation of u considered by Charney and Drazin. He arrived at a picture (see Figure 4.10) oftwo paths by which Rossby waves of zero frequency that were generated in the winter troposphere could propagate into the stratosphere. His polar wave guide is located between the turning point associated with the strong stratospheric westerlies in middle latitudes (u > u R) and that in polar latitudes, where the k 2 term in (4.40) begins to dominate. (k is proportional to the zonal wave number divided by the cosine of the latitude.) This wave guide offers an explanation for the concentration of eddy energy in high latitudes that is shown in Figure 3.4. The equatorial wave guide defined by Dickinson is located between the strong westerlies in middle latitudes (u > uR) and the critical point (u = 0) in low latitudes. Dickinson's treatment of the zero wind line follows the viewpoint developed in classical theories of shear instability, namely that the presence of even a small viscosity will remove the DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 75 ,. ... / \ E ) 1001"" SINK "-_o___/ tf (:;,/ ~ ~ ,-'\I () i? 50 \ 40 ""'-- ;: \ I -' \ ~BSORPTIONI ( -20 ~ ' ' I \ / ', EQUATORIAL ' I WAVE GUIDE RAY _o POLAR WAVE , GUIDE RAY ',eo' \ ENERGY SOUfi!OE I O' s'o tf f "' 0 60 WINTER LATITUDE SUMMER Figure 4.10 - Location of the polar and equatorial wave guides deduced by Dickinson for the winter stratosphere (dashed lines). Solid lines indicate average zonal velocities in metres per second. (From Dickinson, 1968) singularity in the differential equation. The solution away from the singularity is such as to propagate wave energy toward a sink at the singularity, and propagate zonal momentum poleward away from the singularity. (Refer to the Eliassen-Palm result (4. 70).) Dickinson concludes that wave energy moving upward along a ray in the equatorial guide will be absorbed after "at best, a few tens of kilometres".* A different view of the critical singularity appeared with the paper by Benney and Bergeron (1969), followed, inter alia, by Murakami (1974), Beland (1976 and 1978) and Tung (1979). More recent results are presented by Kill worth and Mclntyre (1985). In this view, non-linearities are important in the vicinity of the singularity and play an important role in bridging the wave-like solutions (i.e. those with /2 > 0) on one side of the singularity with the evanescent solutions (i.e. with /2 < 0) on the other side. The basic current will be modified in this process. The results of these studies cannot be summarized simply, but reflection of wave energy by the singularity can occur, as well as transmission of a significant fraction of incident energy. One important aspect, recognized already by Dickinson, is that it can take a long time for an incident wave to reach a steady state in the vicinity of the singularity. Another point is that these solutions, for which C is somewhere equal to u, are part of a continuous spectrum in which C varies continuously, so that only an infinitesimal amount of energy is associated with the waves having speeds between C and C + dC. Synoptic examples of critical-point phenomena do not exist. Vorticity patterns similar to those com puted by Beland (1978, Figure 2) are not unusual, but they seem to occur only as isolated accidental examples, with no periodicity in longitude, and with no preference in latitude. 4.5 Dominance of the continuous spectrum An analysis ofthe spectrum ofnon-divergent Rossby waves (i.e., e = 0) in the presence of a variable zonal current u(v) has been made by Dikii and Katayev (1971), building upon the earlier studies ofbarotropic instability by Kuo (1949). They proved the following property of the spectrum: * Eddy viscosity is also invoked in numerical studies that may be sensitive to critical lines. Grose and Hoskins ( 1979), for example, use a horizontal diffusive effect KV 4 (to limit amplitudes in the high wave numbers that would otherwise be generated at the singularity by their barotropic model. This represents a strong damping effect. They remark that "it is open to question whether it (well) represents the processes occurring in such a region." 76 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION When the basic angular velocity a= u/a cos V is not constant, the spectrum of non-divergent Rossby waves consists of discrete modes plus a continuous spectrum. These parts of the spec trum are separated by the minimum value amin in that only modes whose angular phase velocity (4.56) wls is less than amin belong to the discrete set. Phase velocities in the continuous spectrum are bounded from above by the maximum value of a. The values of w/s depend, of course, on the distribution of a and p. But the westward phase speed of non-divergent Rossby waves with a= 0 decreases rapidly as the planetary wave number n increases. This suggests that only the horizontal modes with small values of Is I and n will be in the discrete spectrum. Numerical calculations by Kasahara (1980) show that the Dikii-Katayev criterion is also valid for Rossby waves with the Lamb mode value for e.* Let us suppose that the Dikii-Katayev theoretical result and the Kasahara numerical result can be extended to larger values ofe. We can then estimate crudely the value ofe above which there will be essentially no discrete Rossby waves with respect to the latitudinal wave number. To do this we use the Matsuno (1966) Rossby formula for large e and add a positive increment JC to its zonal phase speed to represent the eastward Doppler effect of the extratropical westerlies. The phase speed is then w 2.Q + JC. (4.57) s sz + e112 (2v + 1) On the assumption that the minimum value of a is negative, the criterion for a mode to be part of the continuous spectrum can be written (4.58) e 112 > 1---,--=-. ' - szl (2v ~ 1) We will require that all horizontal Rossby modes (except the mixed model= 0) be part of the continuous spectrum, by setting s2 and v equal to one. (The s2 term is then negligible.) This will happen if e exceeds a critical value: 4 ( .Q e > Bcont where Bcont )2 (4.59) 9 JC + laminl A value of five degrees per day for JC and for Iamin I results in a value of 576 for Bcont· According to Figure 3.1 this e occurs for a vertical wave length of about 20 kilometres and an equivalent depth of 200 metres. The size of Bcont indicates that the discreteness of the latitudinal eigenfunctions of the Laplace Tidal Equation that were analysed by Matsuno and Longuet-Higgins for large e has little, if any, significance as representing preferred modes of motion of the atmosphere; we can expect that, ofthe infinite series of discrete Rossby modes that exist for a resting atmosphere, only those of very large horizontal and vertical wave lengths will maintain their identity as distinct Rossby waves in the presence of typical distributions ofu. The structure shown on Figure 3.9 has never been observed in the atmosphere. 4.6 Importance of the external mode An instructive demonstration of the importance of the external mode for transmitting energy ofRossby waves is to consider the response of the atmosphere at one latitude to forcing by stationary effects, inde pendent of time, at another latitude. Longitudinally variable, but stationary heating from precipitation in equatorial latitudes is an important effect of this nature. The theoretical response within the tropics to this * Geisler and Dickinson (1976) present detailed calculations for the theoretical structure and frequency of the "five-day wave", <;;_orresponding to 1.4 cycles per week, and the entry fore = 8.82, l = 1, and s = - 1, in Table 3.2. They find that realistic distributions of u have little effect on the frequency of this wave. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 77 local heat source was estimated by Webster (1972). At the same time Rowntree (1972) showed that tropical heating in a numerical model would also produce effects in the extratropics. This result was consistent with the mechanism suggested in 1966 by Bjerknes in which heating in the tropics could affect flow patterns in higher latitudes. However, a first glance at the possibility of significant poleward transmission of effects from tropical heat sources is not encouraging. In the tropics, the horizontal gradients of temperature are small, so that horizontal advection of temperature cannot balance the release ofheat. Therefore, in a steady state the release ofheat in the tropics must be balanced by an upward velocity in the middle troposphere, with opposite signs of horizontal divergence near the ground and near the tropopause. This pattern of divergence will force an internal mode of response. (Note that the vertical mode structure being referred to here is that for a resting atmosphere.) Internal modes defined for a resting basic state suffer two handicaps with respect to poleward propa gation. Firstly, the meridional group velocity of internal Rossby waves is much smaller than that of the external Lamb mode. The maximum value of awRI a/ in a resting atmosphere is equal top gh /(2/) 2, where his the equivalent depth. For the external mode, h is about 10 000 metres. Puri and Bourke (1982) find that representation of heating from tropical convection requires internal modes with equivalent depths of 580 metres or smaller.* The maximum awR/al for the fastest relevant internal mode will therefore be (580/10 000) ,_, 0.06 times that of the external mode, and the group velocity of these internal modes will be at most one or two degrees of latitude per day. The second handicap can be seen from (4.45)-the range of positive u into which equatorial waves can propagate poleward before reaching their turning point is considerably smaller for an internal wave (m2 > 0) than it is for the external mode. Nonetheless, calculations with baroclinic models and realistic profiles of mean zonal wind u show that the wave response at higher latitudes to steady heating in low latitudes can be significant (Ho skins and Karoly, 1981; Webster, 1982). But in these calculations, the extratropical response is predicted to have the same sign in the lower and upper troposphere, indicating considerable energy in the external mode. This change in wave type must be explained. The theoretical mechanism for the change from an internal mode near the Equator to the external mode at higher latitudes seems to be the one suggested by Lim and Chang (1983). This is that the vertical shear au/ az in the mean zonal wind can force an exchange of energy between different vertical modes. For example, the advective term ualax appears in the linearized equation for u' as au' arjJ' I _ au' - +- - fv = - u(z) -+etcetera. (4.60) at ax ~ Modal analysis of this equation with the mode structure of a resting atmosphere will isolate a single vertical mode component for u', rjl and v' on the left side, but will involve contributions from all vertical modes of u' on the right side. Kasahara and da Silva Dias (1986) have modelled this mode-mixing process and shown that this interaction is indeed the mechanism that puts energy from tropical heat sources into the external mode.t An alternative view is possible, however, ifthe "modes" are redefined to be those for a basic state that is not resting but has a basic current that varies with z. Held, Panetta, and Pierrehumbert (hereafter referred to as HPP) show how this works for steady-state motions on a basic state in which u varies with height, but is nowhere negative. The differential equation defining their vertical modes is equivalent to the equation obtained by neglecting the small J term in (4.36), and then setting C to zero: _1 ( azG __1 G) + ( ~ aPV ) G = k2G _ azG = _ )..G. (4.61) s aZ2 4 u ay ay2 G is a scaled value of the perturbation geopotential equal to rjl times (jj/S) 112• A boundary condition at the ground is obtained from (El4) by setting w equal to zero and ignoring the heating, q. * Da Silva Dias and Bonatti (1986) find an energy peak at internal modes with an equivalent depth of about 200 metres from data over South America. 'I Kasahara and da Silva Dias have a zero wind line at about 12' in their uprofile. They seem to have avoided problems at this latitude by solving only for a solution with limited vertical and horizontal resolution. 78 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION A in (4.61) is the eigenvalue associated with a particular eigenfunction G (or if/). Negative values of A will correspond to eigenfunctions that are wave-like in the horizontal, whereas positive values of Awill correspond to eigenfunctions that vary exponentially. * The important new feature of(4.61) is the variability with zthat is introduced by the term (1/U) on the left side; for these steady motions it has replaced the constant parameter -C = - wlk that appears here for non-steady motions in a resting atmosphere. This dependence of u on z will produce different structures of the vertical eigenfunctions; as indicated earlier in section 4.3 an increase of u with height will introduce the possibility of a turning point, which will lead in turn to the possibility of a discrete mode in which G approaches zero at large Z. HPP assign the adjective "external" to the eigenfunction with the most negative value of A. They show that such a mode exists (and normally only one) for u profiles representative of the middle-latitude tropo sphere and the lower stratosphere in winter. (Additional discrete modes can be introduced by the strong westerly jet in the lower mesosphere of winter, as shown by Tung and Lindzen (1979; Figure 4). The solid curve in Figure 4.11 shows the vertical variation of rp' that HPP computed using climatological profiles of u and Tat 45"N with a vertical numerical grid structure typical of numerical prediction models. (rjJ' for profiles from 30'N and 60'N was very similar.) Figure 4.11 also shows the geopotential structure from a computation by Kasahara and Shigehisa (1983) of the lowest-order vertical mode of a typical numerical model for a resting basic state. As shown on the figure, the Kasahara-Shigehisa result is very similar to the vertical structure ofthe Lamb wave in an isothermal atmosphere, for which rp' varies inversely withp to the power R/CP. They both differ significantly from the external mode found by HPP. The latter has its maximum value near the tropopause, where u also has its maximum. The HPP external mode offers a more realistic representation of the typical variation of wind and geopotential with height for disturbances in extratropicallatitudes than does the external mode for a resting atmosphere. HPP suggest that this shape for rp' occurs because the external mode value of Ais small enough in magnitude to contain very little vertical motion, so that u aT'!ax balances 0~--~----~----~----~ HPP 400 i 600 ;I :I•I :I • I 800 : I f IKs I I 1000 · I Figure 4.11 -Vertical eigenfunctions for geopotential and velocity. The solid curve (HPP) is the external mode found by Held et al. (1985); the dashed curve (KS) is the lowest order mode found by Kasahara and Shigehisa (1983) for a finite-difference numerical model; and the dotted curve (L) is that for the Lamb wave in an isothermal atmosphere *Tung and Lindzen (1979) solve (4.61) for specified negative values of A, in order to examine vertical propagation of Rossby waves. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 79 v'a T lay in the linearized form of the thermal equation (E 14). (They also point out that this similarity probably explains the success attained by the simple Charney-Eliassen method of deriving the "equivalent barotropic" model described in section 4.1.) A major result of the HPP analysis concerns the horizontal distribution of the response to a forcing that is localized in the horizontal. To place this in perspective, we note first from (4.61) that the square ofthe scaled three-dimensional wave number m 2 + (N2 I f2) (k2 + /2) is essentially equal to the positive term (aPV/ay)/u. With respect to the signs of m2 and K 2 = (k2 + /2) we therefore have only the three possibilities shown schematically in Figure 4.12. * K2<0 K2>0 m2>0 Ill 11 m2<0 NONE Figure 4.12 - The combinations of m2 and K 2 that are permissible in steady Rossby waves in westerly baroclinic flow 50 \ l'l\\\ /11;' I \ I 1\\'-'::/ 11 I 1"'~// ( \ \ 1,,_/ ( -\ 40 f-1 \ I \ I I I I I \ ' / I \ - I I I I I I \ 1 I I I '--"' I \ I I I \ I I \ \ I > ::t: I I I <::J I 1 \ i:Li ::t: I 1\ I I\ I I I 10 0 -60 60 120 180 240 300 LONGITUDE Figure 4.13 - Zonal cross-section of the steady-state response in geopotential to zonal flow over a delta-function mountain located at zero longitude. Positive isolines are solid; negative isolines are dashed. (Copied from Figure 6.18 in Held, 1983) *Note that areas I and II on Figure 4.12 correspond to areas T3 and P3, respectively, on Figure 2.5. HPP also find that in a numerical model with a reflecting top boundary condition, the artificial reflection caused by this artificial top is not too serious because the reflected modes are mostly in area Ill, and therefore have no effect at long distances. 80 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION HPP show that modes in the continuous spectrum of areas II and III are important only at horizontal distances that are not too far from a localized source, because those modes will propagate upward to infinity. The response at large horizontal distances is therefore due only to the single external mode shown in Figure 4.11. An illustrative example is presented by Held (1983, Figure 6.18), reproduced here as Figure 4.13. This shows the response(') in a zonal cross-section to zonal flow in middle latitudes over an infinitely sharp mountain located in the figure at 0' longitude. The basic current in this computation was an idealized profile varying linearly with z from 5 m s - 1 at the ground to 20 m s - 1 at 10 km, and remaining at 20 m s- 1 from 10 km to infinity. The response far downstream is seen to be a wave train with maximum amplitude at 10 km, where the external eigenfunction has its maximum value. The most intense response, however, is seen at the top of the figure, 60' downstream from the source. This positive centre is from the continuous modes located in areas II and Ill, that are amplified with height by the p-Ill factor in (4.35). If the point orographic source at 0' is interpreted as representing the Himalayas, this centre of maximum response can be interpreted as the equivalent of the stratospheric anticyclone that is found over the Aleutians in winter. This analysis by HPP provides an additional interpretation to the earlier study by Matsuno (1970) (see also Hayashi, 1981), which was the first to explain successfully the Aleutian anticyclone. Matsuno used a mathematical model in which the first step was a Fourier decomposition in longitude, followed by forcing in the form of a prescribed value of the perturbed geopotential in the upper troposphere; whereas HPP emphasize the insight into this problem provided by ignoring the constraint of cyclic continuity in longitude. The question of steady-state vertical modes in the summer, when ii in the extratropics changes to easterlies in the lower stratosphere, and the appearance there of a critical layer does not seem to have been addressed with the possibility of reflections (from critical surfaces) in mind. Vertical modes appropriate to winter flow in the subtropics, where there is a change from low-level easterlies to westerlies at higher levels, is also an open question. Unfortunately the HPP eigenfunction approach is not easily extendable to the time-dependent case. Barotropic calculations of horizontal Rossby-wave dispersion with time dependency will therefore depend for their credibility on the partial success attained by use of the barotropic model in the first decade of numerical weather prediction, in addition to the heuristic support offered by the steady state calculations by HPP. 4. 7 Latitudinal propagation of stationary Rossby waves with zonal shear Two-dimensional propagation for non-divergent Rossby waves in the presence of a basic current u(y) is governed by (1) = k iiM - k PM l"' I 7,., ' (4.62) where the notation is the Mercator notation used by Hoskins and Karoly: cos tJ d(f+ 0 pM(y) = (4.63) a dtJ u UM(y) = cos tJ . (4.64) We also use their notation for the stationary wave number Ks: K/(y) = (k2 + /2) = UM (4.65) s PM . (This is the value of k 2 + /2 that makes w zero. Note that Ks is imaginary for negative ii). Climatological values ofii produce Ks values that decrease from infinity at the critical latitude (ii = 0) to a value of3/a or 4/ a at polar latitudes. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 81 It is convenient to make (4.62) non-dimensional by defining (k', /') = (il) (4.66) s and (JJ k' w' = =-- = k' - (4.67) uKs k'2 + /'2 . w' is graphed as a function of k' and I' on Figure 4.14. w' vanishes on the I '-axis and on the unit circle where k'2 + /'2 = 1. The group velocity is given by aw aw' (4.68) ak u ak' · Its direction is shown by the arrows on Figure 4.14, normal to the isolines of w '. (Solutions in quadrant 3 are merely the complex conjugate ofthose in quadrant 1, and have equal phase and group velocities. A similar arrangement exists for quadrants 2 and 4, so that we may ignore quadrants 2 and 3.) 2~--~------~----~--~ 0 1 l't 0 I I I I T I I I 1- I I I I -1 k' -2 -1 0 1 2 Figure 4.14 -The non-divergent Rossby frequency as a function of non-dimensional east-west (k') and south-north(/') wave numbers. Group velocity directions are shown by the arrows 82 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION At large values of k'2 + !'2, aw'/ak' approaches a constant equal to a unit vector in the k' direction. This corresponds to the purely advective case, where the wave number is so large that the Rossby effect is negligible. Inside the unit circle there exists a rich variety of group velocity vectors, in orientation and in magnitude. On the unit circle, however, where a/at'= 0, the group velocity has a simple distribution, parallel to the wave number vector.* aw' (4.69) ak' (unit circle) = 21 k' I k'. For waves of zero frequency, the group velocity establishes the direction and speed at which a train of stationary waves is established in response to a point source that is turned on at t = 0. Hoskins and Karoly (1981) present some examples of ray paths computed from (4.62) with a climato logical distribution of u at both 300 hPa and 500 hPa. Their result for rays moving north-eastward from a point source located at 15"N Gust north of the u = 0 latitude) is shown in Figure 4.15. The behaviour of the ray is readily understood from the WKB principles summarized in Appendix A, the distribution ofK, referred to above, and Figure 4.14. We first note that rays with zero frequency can leave the source only with an eastward component. Since their frequency remains zero, the image point ofthe non-dimensional wave number vector will stay on the unit circle ofFigure 4.14. Secondly, the non-dimensional zonal wave number k'will vary inversely with K, along the ray path, since the product K,k' is equal to k, and k is constant. For northward moving rays, the image point will move clockwise on the unit circle, and the non-dimensional wave number l' will decrease until finally it becomes zero. This occurs at the turning point of the ray path, marking the most northerly point reached by the ray. Reflection occurs at this point, and the image point of the vector (k', l) on Figure 4.14 will continue its clockwise motion around the unit circle. It can be seen that of the initially poleward-moving rays, those that leave the source with the smallest value of k'will have the most northerly turning point. Rays moving in an equatorward direction (but still in the latitudes where u is positive) will experience increasing values of K, decreasing values of k', and increasing values of l ' 2 (increasingly negative values of l ). Figure 4.15 - Examples of ray paths ofbarotropic Rossby waves with zonal wave numbers 1-6 in a climatological 500 hPa mean zonal flow. (Copiedfrom Figure 17 in Hoskins and Karoly, 1981) *The relative simplicity of awlak for w' = 0 is obviously a major inducement to apply WKB ray-tracing arguments to steady state barotropic flows. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 83 Their group velocity becomes oriented more and more southward. This process may stop where the critical latitude is reached (ii = 0, K, = oo ), while some rays, especially those with small k values, may propagate through the zone of easterlies. The rays shown in Figure 4.15 are of interest because they are similar to a path connecting a series of vorticity centres that Hoskins and Karoly found to be created in response to a steady low-latitude source of vorticity in a barotropic numerical model and in a baroclinic model. The meteorological interest ofthis lies in the roughly similar locations found by Wallace and Gutzler (1981) for centres of maximum correlation of geopotential anomalies between two points. Their results are shown here in Figure 4.16. The three centres labelled PNA are thought to represent responses to increased rainfall in the equatorial Pacific. The poleward propagation of this steady-state pattern requires the presence of westerly flow in low latitudes at the source. In this regard, Webster and Holton (1982) have emphasized the importance of longitudinal variations in the zonal wind u. In the northern hemisphere winter, the zonally-averaged u is negative in the upper troposphere at low latitudes, but u is from the west in the eastern part of the equatorial Pacific. Webster and Holton used a "shallow water model" (i.e. the non-linear forms of (3.11), (3.12), and (3.28)) to demonstrate that equatorial westerlies in this belt oflongitude can form a "duct" that is favourable for the transmission of stationary Rossby-wave energy in a meridional direction. Simmons (1982) and Branstator (1982) have also modelled this effect. 4.8 Latitudinal propagation of transient Rossby waves with zonal shear In this section we focus on the latitudinal propagation of Rossby waves whose frequency is not zero. 4.8.1 Observational studies The first quantitative modelling ofthe latitudinal propagation oflarge-scale waves in the atmosphere was done by Mak (1969). He used a linearized set of equations to calculate the statistics of the velocity and tao· • ·:o• . ; :' Figure 4.16- Location of major anomaly correlation systems, as deduced by Wallace and Gutzler (1981) 84 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION temperature fields in low latitudes on the assumption that they were forced by motions at 30'N and at 30'S latitude. His model equations were truncated severely in the vertical, with a representation of the velocity fields only at 750 hPa and 250 hPa_ The lateral boundary conditions at 30'N and 30'S were specified in the form of the time spectra and cospectra of the meridional velocity at those latitudes, for each of 12 wave numbers in longitude. (At 30'N, these spectra were measured from a year of stream-function maps from the National Meteorological Center. At 30'S, the absence of data forced Mak to assume that the motion was statistically similar to that at 30'N, but uncorrelated.) Calculations were made with a realistic profile of ii, with a maximum speed at 250 hPa of28 m s- 1 at 30'N and 30'S, and a minimum of-2 m s - 1 at the Equator. A comparison set of calculations was made with a ii field that was independent of latitude at each level. Mak's results with the realistic profile of ii gave realistic latitudinal profiles and magnitudes for the variances of the meridional velocity, the zonal velocity, the temperature, and for the poleward fluxes of momentum and sensible heat. Statistics calculated using ii profiles that were independent oflatitude agreed less well with data, generally by assigning too large a size to the meridional eddy velocity in tropical latitudes. This difference in behaviour due to latitudinal shear in ii and easterly zonal winds agrees with the conclusions that can be drawn from the Charney-Drazin analysis referred to earlier. (See also Charney, 1969.) Figure 4.17 shows the contribution to the variance ofu and v in the upper layer of the model and summed over latitude, as a function of zonal wave number and period, based on the realistic distribution of ii. The diagram for v2 shows two maxima. The larger one represents motions that have little amplitude at the Equator. The second maximum in v2, located at wave number 3 and periods around four days, has maximum amplitude at the Equator. Mak recognized this as fitting the properties of the mixed Rossby-gravity waves that had been observed several years earlier by Yanai and Maruyama (1966). (The equations used by Mak could describe internal gravity waves.) Much of the large maximum in u2 on the top half of Figure 4.17 represents motions whose maximum amplitude is at the Equator, especially for periods around ten days. This is similar to the Kelvin waves that were discovered at that time by Wallace and Kousky (1968). "I o. o. 1. o. 1. o. 1. 1. 1. o. o. a. 1. 1. 1. ll o. 1. 2. 2. 1. 2, 1. 1. 1. I. 1. 1. 1. 1. I. 10 1. 2. 1. 1, ), 4. 1. 2. 2. 2. 2. 2. ), 3. ), J, 6. '· 6. '· '· '· '· '· '· '· '· '· '· '· f "I z. '· lJ. 9. 11. 9. 7, 6. 6. 6. 7. 7. 7, '· '· '· 22. 16. ll. 12. ll. 10. 10. 10. J, J. J, h u9._25. 40. 20. 14. 14. 14. 15. lJ. l). 2, 2. 2. 12. 12. u. 14. 14. 11. ·u. u. 11. s. 22. 7. 5. 9. 12. So s. J. 2. l. 24. 16. 12. 12.~· 5. 2. J. 62. 9. 21, 22. 29. 12. "· 1. 2, 1. 1. ~I 15. 8. 5. 12. 17. s. 1. o. 0, o. go 30 Period 10 (days) ro ., 3 ''I o. l. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. l. 11 o. 1. ). J. 2. ). 2. 2. 2, 2, 2. 2. l. 1. 1. 10 2. ). 1. 2. "· 6. 1.. ). ), 2. 2. 2. 2. 2. 2. 1' 6. 6. s. "· 4. 4. ), ), 2, -<1.' s. 6. 5, 4, 4. 4. 1. 1. l. 8. 9. 9, 6. 6. 6. 1. 1. 1. s. s. s. 1. 1. 1. t., ), 2. 2. 2. 2. 6. ), 2. 1. 2. 6. 2, 2, 4. 2, 1. ). 3. 5. 1. 1. L 1. o. 2. ), 2. o. o. o. o. Figure 4.17 - Contribution to the latitudinally averaged variance of u (upper diagram) and 1', a function of frequency and zonal wave number. (From Figures 11 and 10 in Mak, 1969) DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 85 It appears then that equatorward propagation of wave energy from higher latitudes plays a role in maintaining the large-scale transient eddy motion present at higher elevations in low latitudes. This forcing effect from higher latitudes must be included in tropical forecasts, at least for forecasts of some duration. A very graphic illustration of this energy propagation is provided by what has since come to be called the Eliassen-Palmjlux vector, F. This is a vector lying in the meridional plane. It was first defined by Eliassen and Palm ( 1961) for stationary gravity and Rossby waves (i.e., w = 0), but it can also be defined for transient waves. (See, for example, Boyd (1976) and Andrews and Mclntyre (1976).) The approach is to evaluate the zonally averaged eddy flux of geopotential in the meridional plane, which in the hydrostatic case replaces the eddy fluxes p'v' and p'w' for the rate of doing work across a fixed surface (see e.g. (2.18)). The energy flux results for quasi-geostrophic motion are rj/v' (C- U) u'v', (4.70) _ ( fRp ) -,--T' ifJ'w, = (C - U) H2N2 v , (4. 71) where the overbar denotes a zonal average and C is the trace velocity in the zonal direction. The importance of these expressions is that u'v' and v'T' can be readily evaluated from data, whereas the terms on the left side of (4.70) and (4.71) require a knowledge ofnon-geostrophic winds. According to (4.40), for Rossby waves aPV ay c- u (4.72) k2 + {2 + fo2 (- + N2 4H2 m2) a PV/ay is almost always positive, leading to the following results: (a) The latitudinal flux of wave energy (rjJ'v') has the opposite direction to the latitudinal flux of momentum (u'v'); (4.73) (b) The upward flux of wave energy -(rjJ'w ')has the same sign as the poleward flux of sensible heat CP v'T'. (4.74) Edmond et al. (1980) assign the following definitions to the components F lJ and FP of the Eliassen-Palm flux vector: FlJ a cos u v'u'; \'I f) I (4.75) FP a cos u dH/dp a cos fRp l' 1 T 1 u I-f2N" The signs are such that FlJ and FP indicate the direction of wave energy flux in the directions of increasing latitude and increasing pressure when a PV I ay is positive. Figure 4.18 shows the direction of these fluxes, as computed by Edmon et al. from the winter circulation statistics ofOort and Rasmussen (1971) for transient waves. (The fluxes for stationary waves are similar. Those for summer are also similar but of smaller magnitude.) The figure indicates a general upward flux, presumably from energy generated by baroclinic instability, heating, and orographic effects in the lower troposphere. But an equatorward flux of energy is also indicated in the upper troposphere in middle and low latitudes. This flux indicates that wave energy will be 86 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION 100 - ... • . 200 t- .. .. .- ~ ~ ~ ~ i- ... ~ ,., ,., 300 " "' "' r " 11; \ ~ ~ t t t t • I< .. .. 500 r- " ~ ~ t t t t t r- 700 t- ~ ~ \ t f i f t t t =5 x 1Q15m3 t- lt t t i t t ~ r- l f i 4 ~ I I t t 1 t 1000 I r l I · i EQ 20N 40N 60N Figure 4.18 - Eliassen-Palm fluxes for winter transient eddies. Vectors are scaled so that the vector enclosed in the box corresponds to a horizontal component of5 x 10 15 m 3 or a vertical component of 402 m kPa. (Abstracted from Figure 1b ofEdmon, Hoslcins, and Mclntyre, 1980) either dissipated in the tropics or transmitted, in those latitudes, to higher altitudes beyond the scope of the figure. The figure is of most interest as a diagnostic indicator of normal processes in the atmosphere. However, we can note that if we are interested in the propagation offorecast errors, the figure has some relevance as suggesting paths for the ready transmission of analysis errors. In this connection, it is important to note that wR is a function only ofthe squares of the meridional wave number I and the vertical wave number m. This means that transmission of influences from point A to point B in the meridional plane is as feasible as is transmission from point B to point A. A major use of the Eliassen-Palm flux vector is for its role as a forcing function for changes in the zonally-averaged flow u, and for forcing a meridional circulation. (A primitive example is given in Phillips (1954).) This application has been developed considerably, following the stimulating analyses by Eliassen and Palm (1961) and by Charney and Drazin (1961) of the conditions underwhichFdoes not produce changes in the zonal current. A summary is given by Andrews and Mclntyre (1978). It has been especially useful in analysing the processes involved in the sudden warmings that occur in late winter in the stratosphere. 4.8.2 Latitudinal influencefunctionfor individual zonal wave numbers For a more precise answer about these latitudinal effects we must return to the question posed originally by Ertel and by Chamey-how rapidly do influences propagate in latitude? Chamey answered this question in 1949 with his estimate of the maximum latitudinal group velocity for Rossby waves. But that early answer ignored the effect oflatitudinal shear in the basic zonal current u, an effect which, we have seen, can be of considerable significance. A more effective way to answer this question is to extend the concept of the influence function that was introduced by Chamey and Eliassen and discussed in section 4.1. In their calculations of the zonal influence function they considered a zonal channel containing motions characterized DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 87 by only one latitudinal wave number and one vertical wave number. The procedure used here will be first to calculate latitudinal influence functions for the barotropic vertical mode and individual zonal wave numbers. The results for individual zonal wave numbers can then be synthesized with Fourier's theorem into a two-dimensional influence function. We therefore consider first a single zonal wave numbers for the barotropic stream function !jl: !fls = Re [P5 (X, 1) exp (isA.)] (4.76) where X= sin u; (4.77) 1 = time in days. } The equivalent barotropic equation (4.6) reduces to the following equation for Ps: (£ - aPs = - is [a£(Ps) + QPs]. f.lo2) ar (4.78) a is the non-dimensional angular velocity of the basic current ii: 2n u a (4.79) Qa cos u · £is the Laplacian operator: a s2 a (1 X2) - £ = ax ax 1- X2' (4.80) and Q is proportional to the gradient of the absolute vorticity of the basic current: a2 Q = 2n _a_ _ (4.81) Q cos u au if+ () = 4n - ax2 a [(1 - X2)]. f.lo 2 is the small barotropic divergence parameter: fo2 a2 2 flo = R T Asfc - 5 Asrc ~ 1. (4.82) sfc Forecast values of Ps are calculated from (4.78) on a numerical grid: 2 '-J-1 2 J X= J ; j = 1, ' ---, . (4.83) J J-1 j= 1 andj=J define the South and North Poles, where Psis zero. The latitudinal influence function Is for this zonal wave number is obtained from a sequence of J-2 numerical forecasts. In each of these forecasts the initial value of Psis set equal to zero everywhere except at one latitude, j', where it is given the value 1. j' takes on the successive values 2, 3, - -, (J -1) in the J-2 forecasts. In each forecast the value of Ps at time r and j = j1 is recorded, as Ps(jfi 1; j '). If we define Ps (/r; r; j') 2 Is (jfi r; j') = ;~X= (4.84) ~X J-1 we can express the results at latitude jp of the J-2 forecasts for this wave number, as the following approximation to an integral over latitude: 88 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION J-1 Ps (X = sin 1J has been used to obtain this character of an area integral.)* Ps and Is are complex quantities; the contribution of the initial field at latitudej'to the forecast at}rhas the absolute value . I JPs Contributions tojffrom neighbouring values of}' can cancel one another if the phase of either I or P varies rapidly with j '. A consideration of only the magnitude of Is is, however, sufficient for a preliminary examination focusing on the maximum possible values of the influence. Forecasts were made with a = 0, and with a corresponding to the very smooth annual 500 hPa climatological field of ii shown in Figure 4.19. This field was made symmetric about the Equator and smoothed so as to eliminate irregularities that might produce unrepresentative results. (A cubic spline was fitted to iivalues ofO, 10.0, 20.0, 8.0, and -4.0atX = ± 1, ±0.75, ±0.50, ± 0.25, andO.) There is a zero wind line at ± 7 degrees. The associated absolute vorticity profile corresponding to Figure 4.18 increases mono tonically from the South to the North Pole, and produces the vorticity gradient Q shown in Figure 4.19. goo •v 20 60° t sinev 30° ~ I I 1 I 0 m s- I I /I I I I ,~' ;; 10 20 I 0 Figure 4.19 - Distribution ofu used to compute latitudinal influence function. Q (dashed) is proportional to the gradient of absolute vorticity *A value of J = 101 (L\.X = 0.02) was used. Tests with J = 200 showed little change for forecasts up to 12 days. A 4th order Runge Kutta time integration was used with a total time step of two hours. A time step of one hour made little difference. The elliptical operator [ -1-102 on the left side of(4.78) was inverted with the procedure described in Richtmyer and Morton (1967; page 198). DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 89 90~~~~~--~~~~ 90~~~~--~~~~~ 60 s 6 60 s = 6-. ,.-- - / / 45 0 0 UsYM \, / 45 I' .01 30 30 0 o·'~ I I 8')u2 IC::c:=J 0 1 2 4 8 0 1 2 4 6 8 DAYS -+- DAYS~ 90l~TI--~I~I--TI~c=~~~ 90~~~~~--~--~~ 60 - - ... 1 L I I I 60 45 45 30 30 0 0 ~ -:- " - I I I I 0 1 2 4 6 8 0 1 2 4 6 8 DAYS~ DAYS~ Figure 4.20 - Absolute magnitude of the latitudinal influence function I for zonal wave numbers s = 2 and 6 for a point on the Equator. Results are shown for zero zonal wind and for the symmetric u illustrated in Figure 4.19 Figure 4.20 shows IIs I for a point on the Equator for r = 0 through eight days, for zonal wave number 6 (the upper two diagrams), and 2 (the lower two diagrams). Results are shown both for zero zonal winds and for the profile illustrated in Figure 4.19. (Only one hemisphere is shown on Figure 4.20 because Is is symmetric for a point on the Equator when u is symmetric.) Two main points can be made: 1. For u = 0, influences from high latitudes move rapidly equatorward. They move more rapidly for s = 2 than fors = 6, in agreement with earlier estimates of awR/al. The regularity of IIs I at this time range indicates a major contribution from the periodic discrete modes of the Laplace Tidal Equation that were discussed in section 3. 2. The results with the climatological zonal current show a significant reduction in the influence of the extratropics on a forecast for the Equator. This reduction is most pronounced fors = 6, whereas the results for s = 2 continue to exhibit some recognition of oscillations with a period around four days. These results agree well with the general theoretical conclusions from the Charney-Drazin theory and the Dikii-Katayev theorem. Figure 4.21 shows lis I for latitude 44.4'Nforzonal wave number 2. The u = Oresults again show a strong periodic influence. The results with the climatological zonal wind have some diminution of the effect of the southern hemisphere and a diminished influence of periodicities. But there is now a greatly increased 90 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION eo 1 901 -~'1 'd 60 .-- ~A 601·.....-----·~ 45 . . 45~:> ~ 30 30 0 0 30 30 45 --l s = 2 \ \ I 45 s = 2 O=v~- ~~ 1 I I I I I I ~~ 0 1 2 4 6 8 0 1 2 4 6 8 DAYS~ DAYS ~ Figure 4.21 - Absolute magnitude of the latitudinal influence function I for zonal wave number 2 for a point at 4YN. Results are shown for zero zonal wind and for the symmetric u illustrated in Figure 4.19 sensitivity at six to eight days to initial values on the equatorial flank of the jet stream, between the zero wind line and the jet maximum. Increased sensitivity over the case of zero zonal wind is also apparent for initial conditions poleward of 44.4"N. Figure 4.22 shows IIs I for 44.4"N for zonal wave number 6. It displays the same features as Figure 4.20, except in a more extreme fashion. 4.8.3 Two-dimensional barotropic influence function The new feature shown on Figures 4.21 and 4.22 is the sensitivity to initial conditions between the zero line and the jet maximum. The meaning of this will become apparent from the construction of the two dimensional influence function. This is obtained by combining the Fourier theorem for a periodic domain, +x 2n "" F A Fs = ; e-is A, /(A)= L__, s eis ' :f f(A) (4.87) -x 0 with (4.85). For a forecast of the streamfunction at A = 0, j = }p and time T, the result is 2n If/ (0, }j; T) L AX r F(jfi r; }',A') If/ (A', j'; T = 0) dA', (4.88) r 2n J 0 DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 91 90 ~ I ~ I I I I ~ 90 60 ..,...-1 - 60 ~~4 45 I<~ ~E= - ?!I 45 1~2 ._ --3 l I 1 30-1 '\... ~ 30 \ \ \ ~ 0-..J \ 0 .0\ \ ', \ ', 30 -l \_ 30 ...... - .05 '\ ' ' 45J ~ = 6 ' ...... __ I 45 j ~ = 6 60 U = 0 -- 60 U SYM 90 I I I I I I I 90 012 4 6 8 012 4 6 8 DAYS -+- DAYS -+- Figure 4.22 - Absolute magnitude of the latitudinal influence function I for zonal wave number 6 for a point at 45"N. Results are shown for zero zonal wind and for the symmetric u illustrated in Figure 4.19 0 0 -20 -111 69 530 -927 -85 217 -26 -57 375 448 88 -152 -65 458 -3~~~ u -~~~~ 291 125 -37 254 -355 4186. -171 -68 -35 1 277 -446 -794 -69 -30 -23 32 -243 382 -92 -14 7 12 Longitude- 60°~ 48 62 -93 -61 -32 -16 -9 I I ~ I I I I -180 -120 -60 0 60 120 180 Figure 4.23 -Two-dimensional barotropic influence function Ffor an eight-day forecast for a point at 44.4'N and 0 longitude (point.n. The point u is located upstream of the distance moved in eight days by a parcel in the zonal current at 44.4'N. Details ofthe solution around point u are printed in the inset box at ten degree intervals in longitude and at each latitude point used in the calculations. The location of any value exceeding 50 in magnitude is indicated by the contours. The heavy bar centred on the Equator indicates the latitudes of negative u 92 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION where F is the barotropic horizontal influence function: +x (};; F(J;; r; j', A') - 2 Re L ! 5 r; j') e-is X. (4.89) s ~ l This function is graphed in Figure 4.23, for }1 = 86, corresponding to 44.4°N and a r value of eight days. Computations were made for zonal wave numbers 1 to 36 inclusive. The large dots "f" and "U" at 45"N on this figure locate the forecast point and the upstream point A. = - a (j= 86) X 8 days. There are two major influence regions: I. A concentrated region containing variations in F oflarge amplitude and short wave lengths is centred about the upstream point u. Details of this are shown in the inset box, plotted at 10° intervals in longitude for the }-values 83-89. This region reflects the small dispersion values that are typical of Rossby wave of short wave length;* II. A more extensive region, distinctly separate from region 1, centred around 'A= - 70° and lJ = 16°. The dominant zonal wave numbers in this pattern are 5 and 6, with a south-north wave length of about 9° of latitude. It can be shown that region 11 represents effects from rays that originate in that region, move north eastward, and reach their turning latitude at 44.4°N. (These paths are similar to those fors = 5 and 6 for the stationary waves on Figure 4.15.) To see this we consider a hypothetical ray with zonal wave numbers that has a turning point at 44.4°N, and calculate the ray path backwards. According to the WKB theory, this ray has a zero latitudinal wave number at its turning point. The frequency associated with this ray is then given by w = s Q + al12 )· (4.90) (a - s2 + f2 cos2 lJ + 112 with /2 = 0, and the values of Q, a, and cos lJ appropriate to 44.4°N. At other latitudes, this ray will be characterized by latitudinal wave lengths obtained by solving 11 Q + a 2 ) - s2 f2 cos2 lJ a - + -112 (4.91) ( for /2, using the values of Q, a, and cos lJ appropriate to that latitude, and the already determined values of s and w. The components aw/as and aw/al can be calculated at each latitude, aw/a/being zero at 44.4°N. Values of the time ~r required for the ray to pass from one latitude (i.e. J) to another are readily calculated from aw/al. The associated longitudinal displacement at the rate aw/as can then be computed. These WKB ray calculations will be meaningful to the extent that the parameter wkb is small: wkb = cos2 lJ d~ ( +) « 1. (4.92) Table 4.1 lists the ray characteristics computed in this way for zonal wave number 6 with an assumed turning point at}= 86 (44.4°N). The waves for this ray have aperiodof13.65 days and a zonal trace speed wls of 4.4° oflongitude per day. The parameter wkb is not very small, being less than one only between 32°N and 13°N, and the ray interpretation is therefore not very precise. But the results agree with Figure 4.23 in that the ray traces back to the approximate location of region 11; it does so in a time of six to ten days, and it has a short *Note that Fin (4.88) differs from the Charney-Eliassen influence function (4.13) in that the contribution 1/f (..1. =- a r, Jr: r = 0) has not been separated out. · DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 93 amplitude along the ray as it moves north-eastward to its turning point is proportional to the square root of the latitudinal wave length (see, for example, Hoskins and Karoly, 1981; page 1191.) In the present case this gives a fourfold increase between 13.9'N and 42.8'N. Similar ray calculations fors = 1, 2, and 3, with a turning point at 44.4 'N result in smaller values of wkb, but they maintain long latitudinal wave lengths throughout their path, and have larger latitudinal group velocities. This suggests that these waves oflonger zonal wave length influence conditions at 44.4 'N eight days later-this influence being apparent on the right side of Figure 4.21-by repeated reflection between the mid-latitudes of both hemispheres, i.e. by acting effectively as normal modes. This interpretation is sup ported by the fact that their zonal trace speeds are negative and large enough in magnitude to satisfy the Dikii-Katayev criterion (4.56) for normal modes. Rays for s = 4 and 5 behave similarly to s = 6. For s = 7 and higher, however, the zonal trace speed relative to the zonal current at 44.4'N is so small that the ray reaches its critical latitude (where a = wls) between 44.4 and 42.8'N; i.e., according to this WKB treatment, such waves cannot propagate southward at this latitude in this basic current. Region 11 is located somewhat north ofthe zero value ofu. It can easily be demonstrated, however, that the existence of negative u is not an essential feature for the existence of region 11. Computations were made in which the u profile of Figure 4.19 was changed by the addition of a solid rotation r5u = 4 m s -t times cos u, just enough to eliminate all negative values of u. The resulting values of I and F on Figures 4.20-4.25 were unchanged except for the expected small upstream displacement of the patterns of F. Region 11 was missing when computations were made with no basic current, in agreement with the left sides of Figures 4.20-4.22. The existence of region 11 as a sensitive area for eight-day forecasts at 44.4'N is therefore to be thought of as being due to the gradients in u rather than the existence of a zero wind line. Ray computations were also made for rays poleward of a turning point at 44.4'N. Solutions were found fors~ 7 that emanated from critical latitudes ranging from 51 'N-61 'N (j = 90-95), and longitudes more than 100 degrees oflongitude upstream, with travd times of 10-20 days. These ray solutions had only a few points for which wkb < 1, however, and therefore give a very crude explanation of the structure of Fin the top left part of Figure 4.23. Figures 4.24 and 4.25 show the field ofF at eight days for forecast points at latitudes 30.0'N G= 76) and 22.3'N G= 69). Their counterparts to region 11 are more intense than that on Figure 4.23, and are not separated as distinctly from the upstream region I. Ray calculations equatorward from 30.0'N were almost as satis factory as those from 44.4'N, but those from 22.3'N had trajectories that were too meridional to match the 60° f_ D-o=o ~a~~====;;_;:, ~~~§ -23 3[9 7 14 10 -45 112 u-=~~ 40 30 21 -45 53 . -9 40 30 21 -327 -560" .1983 404 176 98 I I \ 181 1042 -39 -307 -154 -97 -50 224 -3088 955 211 86 -520 -408 - 46 1349 220 70 -180 -120 -60 0 60 120 180 Figure 4.24 - As for Figure 4.23, except for a forecast point at 30'N 94 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION 0 0 0. :::> / ::;:::::::== __) -· -266 188 -13 -21 -8 -2 0 420 -206 -366 u -137 -58 -35 -25 -44 376 1527 -1332 -227 -105 -64 -540 311 -2323 • 6159 -259 57 70 ___._ -115 -638 2036 -7241 4729 54 58 • oL 248 -458 -316 3253 -6897 3397 161 Longitude 6o 1 301 74 -812 483 2691 -5726 2507 -180 -120 -60 120 180 Figure 4.25- As for Figure 4.23, except for a forecast point at 21.1°N latitudinal wave length in region II. (Note that it fits the wave pattern in region II on Figure 4.23.) The WKB centre of region II on Figure 4.25. Ray calculations poleward of these turning-point latitudes have either large values of wkb or immediately encounter latitudes where a - wls is negative. The following conclusions can be made from the calculations in this section and in section 4.8.2: 1. The presence of a mean zonal wind u with typical latitudinal variation serves to limit the latitudinal spread of forecast errors more than would be the case if u were zero or differed little from a solid rotation. 2. A medium-range forecast in extratropical latitudes is most sensitive to initial errors in two regions: (a) Region I, centred on the upstream point defined by the mean u at the forecast latitude; (b) Region II, on the equatorial side of the jet stream about half as far upstream as Region I. Initial errors in this region will be particularly damaging when they have a moderate zonal wave number and short latitudinal scales. The existence of Region II does not depend critically on the existence of negative u. The above effects, having been derived for a barotropic model, do not include the possibility of enhanced growth of initial errors located in regions oflarge horizontal temperature, nor, since the mean zonal current was smooth, do they include the possibility ofbarotropic instability that exists when the absolute vorticity of the basic current is not a monotonic function oflatitude. To some extent the baroclinic growth ofinitial errors might be viewed as the delayed creation of initial barotropic errors, and the two regions above define those places where baroclinic growth of small initial errors could most seriously affect the forecast point. In the case of Region I, this view would be consistent with the numerical experiments and analyses by Simmons and Hoskins (1979) that were discussed in section 4.2. 4. 9 Vertical propagation with zonal wind shear The vertical propagation of influence (and forecast errors) is beset with somewhat different problems than those encountered with respect to latitudinal and zonal propagation-there is no prescribed upper limit to the atmosphere, the density decreases upward, and routine observations exist only to moderate altitudes. Furthermore, whereas downward propagation is of primary interest for the forecast problem, most studies have considered upward propagation, in order to explain the interesting stratospheric phenomena that have been observed. DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 95 .,o ,o-:J ,# -t9 I 10 .., I~ 47 H •o 1 'JI I~ I,. /u~ ~ .."3-1 I 1 I )b 2~ -1 ~ '24 1 b l I 'U 11 I. " I I/, ':lfJ -~~ I ~ . I ,," 1f',:P 50-I ll lb q I I~:, ,. ~ :··y.. ~ ~ I ...... I I,' l I' I 100 ~ 41 ... ll 4 11J~ 44 11 ~1 I z 1.~. ~ 700 ~ t,.'/7~ I'.!~ ~~~ ~tOO --~ ...... 1000 ~1 - 2 4 & 8 10 12 14 ,, 18 20 22 24 ~ 2'1 JANUARY 1~8 Figure 4.26 -Meridional kinetic energy of zonal wave number 2 at SOON during January 1958. Units are 10- 1 ergs cm - 3• (From Muench, 1965) 20 I \ \ \ \\ \ 15 ~ \ I \ \ I \I \ I t km 10 5 \ 0 2 4 6 8 10 Days- Figure 4.27- Average forecast error in geopotential height (metres) in zonal wave number 2 at 56°N for seven winter forecasts, (Constructed.fi'om Figure 17 in Derome (1981) by changing the vertical co-ordinate to log (pressure)) 96 DISPERSION PROCESSES IN LARGF-SCALE WEATHER PREDICTION An early documentation of vertical propagation was given by Muench (1965). Figure 4.26 shows the variation with height during January 1958 of the kinetic energy of the meridional velocity component associated with zonal waves 1 and 2, at 50·N. The upward progression rate indicated by the long-dashed lines is about five kilometres per day. Muench remarks that zonal wave numbers 3 and 4 showed no evidence of vertical propagation. An example of vertical propagation of forecast errors was analysed by Derome (1981). Figure 4.27 shows the distribution with time and height (log (pressure)) of the average r.m.s. error in the amplitude of the zonal wave number 2 component ofthe isobaric height at 56.N from an ensemble of seven ten-day forecasts. The forecasts were made with a 15-level forecast model in use at the European Centre for Medium Range Weather Forecasts, with a top level at 25 hPa, with initial data taken from February 1976. Wave number 2 is of special interest because routine forecasts showed a systematic tendency to weaken the amplitude of wave number 2 at high latitudes, where the normal phase of this component corresponds to a ridge over the Gulf of Alaska and over Europe.* Figure 4.27 suggests a downward propagation of the error. A dramatically different impression results, however, if the values in Figure 4.27 are normalized through 112 multiplication by the square root of the pressure-i.e. converted to approximate values of(energy density) • The results of this are shown on Figure 4.28; the suggestion now is more that of the growth with time of a 20~--~--~~~--~--~~---. 15r-----~---r----~--+----+~ k~ 10 r----- ' -r-++__[__ sr---~------~--~--~--~~ 0 . 2 4 6 8 10 Days-- Figure 4.28 - Figure 4.27 with isolines multiplied by the square root of (pressure/ 1000 hPa) *This error persists in more recent forecasts. (See Figure 28 in Jarraud, Simmond and Kanamitsu, 1986.) DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 97 pattern that is trapped between the reflecting surface at the ground and the artificial reflecting surface at the top of the model.* An early example of a study to explain how tropospheric influences determine stratospheric behaviour is that by Mats uno ( 1970). He used a linear geostrophic model, with realistic zonal currents, to calculate the response in the stratosphere to a lower boundary condition consisting ofthe zonal wave number 1 component of the geopotential field at 500 hPa, as it was observed in January 1967. His simple model produced an excellent simulation of the anticyclone that exists over Alaska in the winter stratosphere (Figure 4.29). Beaudoin and Derome (1976) repeated Matsuno's calculations and introduced some dissipation of the type described in section 3.3. Their main purpose, however, was to explore the effect of different altitudes for the top level, the type of numerical boundary condition applied there, and the vertical resolution used in the finite-difference calculations. The extent to which their experiments duplicated those of Matsuno can be summarized as follows: For calculations using a vertical resolution of 2.5 kilometres: (a) Radiation boundary condition at 65, 42.5, or 37.5 km: Satisfactory (b) Radiation boundary condition at 22.5 km: Unsatisfactory (c) Zero vertical velocity at 65 km: Satisfactory (d) Zero vertical velocity at 47.5 km: Unsatisfactory For calculations using a radiation condition at 65 kilometres: (a) Vertical resolution of 5 km: Satisfactory (b) Vertical resolution of 10 km: Unsatisfactory (Some additional results are presented in Kirkwood and Derome, 1977.) Mechoso et al. (1982) performed an experiment based on the general-circulation numerical model developed at the University of California at Los Angeles. An integration with a 15-layer version (top at 1 hPa "" 48 km) was first made for 46 days starting with conditions typical of 1 December. The atmospheric state developed at day 31 in this integration was then selected as initial conditions for the start of a 15-day Figure 4.29 -Height perturbations at 10 hPa for January 1967, at intervals of200 m. Observed values are on the left, calculated values on the right. (From Matsuno, 1970) *Numerical forecast models typically assume that d5 = dp I dt vanishes at the top of the model, and that d5 approaches zero rapidly enough for products such as WlfJ and d5 T also to vanish. This is equivalent to a reflecting lid at the top of the model (Lindzen et al., 1968). 98 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION TABLE 4.1 Ray computations for zonal wave number 6 for a ray moving poleward and reaching a turning point at latitude 44.4°N, longitude 0, at r = 0 days awl a/ awl as ll"kb Latitud!' L (degrees per (degrees per T )c (degrees) (degrees) day) day) (days) (degrees) 44.4 - 00 0.0 22.8 0.0 0 42.8 - 275 2.3 23.8 -1.4 -33 41.3 9.73 186 3.7 24.6 -1.9 -46 39.8 4.65 146 5.0 25.1 -2.3 -55 38.3 2.94 121 6.4 25.2 -2.5 -61 36.9 2.11 104 7.6 25.1 -2.7 -66 35.5 1.64 91 8.8 24.6 -2.9 -71 34.1 1.33 81 9.8 23.7 -3.0 -74 32.7 1.12 73 10.6 22.6 -3.2 -77 31.3 0.98 66 11.2 21.1 -3.3 -80 30.0 0.76 60 11.4 19.6 -3.4 -82 28.7 0.33 58 11.4 18.7 -3.5 -84 27.4 0.29 56 11.1 17.6 -3.7 -86 26.1 0.35 54 10.6 16.3 -3.8 -88 24.8 0.39 52 10.0 14.9 -3.9 -90 23.6 0.42 49 9.3 13.4 -4.0 -92 22.3 0.45 46 8.4 11.9 -4.2 -93 21.1 0.47 43 7.4 10.4 -4.3 -95 19.9 0.49 39 6.3 9.0 -4.5 -96 18.7 0.52 35 5.2 7.8 -4.7 -98 17.5 0.55 31 4.0 6.7 -5.0 -lOO 16.3 0.59 27 2.8 5.8 -5.3 -102 15.1 0.67 22 1.7 5.1 -5.9 -104 13.9 0.80 17 0.7 4.6 -6.8 -109 12.7 1.23 7 0.1 4.4 -9.8 -122 The zonal trace speed and period of the waves associated with this ray are 4.39 degrees per day and 13.7 days. 11·kb is the criterion for accuracy of the WKB calculations. L is the latitudinal wave length. integration with a truncated nine-level version whose. top level was located at 52 hPa ("" 21 km). Allowance was made for the lack of downwelling infra-red radiation at the top of the nine-level version. Differences between the two integrations developed with time, in the zon~lly-averaged fields and in the departures from zonal averages. The differences in the long zonal waves were most interesting. The truncated model exhibited less westward slope with height of the long waves, implying less poleward heat transport, and, by the Eliassen-Palm ;relation (4.71), less upward flux of wave energy. The truncated model also underestimated notably the amplitude of zonal wave number 3 in the troposphere after eight days of integration (Figure 4.30). The upward flux of energy and momentum by non-stationary waves has provided theoretical explana tions for two major phenomena of the stratosphere, the quasi-biennial oscillation and sudden warmings. Both phenomena have some implications for the answer to the question of how high a numerical forecast model must extend (and have adequate data) in order to make an accurate forecast in the lower troposphere. The quasi-biennial oscillation (QBO) consists of an almost periodic alternation between easterlies and westerlies in the middle and lower stratosphere in low latitudes. The pattern moves downward at atypical rate of one kilometre per month, and has a period of about 26 months. The theory for this was developed by Lindzen and Holton (1968). (See also Ch~ter 4 in Holton (197 5) and the historical recollections by Lindzen (1987)). It explains the oscillation as arising from the upward transport ofeasterly and westerly momentum by internal Kelvin waves and internal "mixed Rossby-gravity" waves (simple forms of which were described in Chapter 3), and the deposition ofthis momentum at the critical levels where u = C.* The source ofthe wave * The rotation of the Earth, however, is not an essential ingredient of this interaction since similar wave-mean flow interactions have been reproduced in laboratory experiments by Plumb and McEwan (1978). DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 99 30 I z 15L 25 ~ E C') 0 20 ~ -(]) 0 15 c: ,_a:s a:s 10 ,__..__.,.. > ~ 5 ·-·-._./1-· -· ol I I I I I I Days_.. 5 10 15 30 25 Lz': - 500 hPa ,50°N ~ E C') 20 0 ~ -Q) 0 15 c: ,_«1 ro 10 > 5 0 Figure 4.30 - Variance (1000 m 2) at 500 hPa accounted for by zonal wave numbers 1-5, as a function of time in a 15-day forecast. The upper diagram is from a 15-level model, the lower diagram is from a truncated nine-level model. Contributions from wave number 3 are shaded. (From Mechoso et al., 1982) motion is thought to be a statistical effect from almost random variations in the intensity of convection in the tropics. The long time-scale of the QBO eliminates it as a forecast problem for numerical predictions of con ventional length. However, the change in the latitudinal and vertical distribution of u in low latitudes associated with the QBO could have an effect on the characteristics oflarge-scale Rossby waves, through a variation in the location of the critical line where C = u in the stratosphere. Sudden warming refers to the occurrence, most often in late winter, of a remarkable change in the usual winter stratospheric flow pattern in the northern hemisphere shown on Figure 3.4A. The strong cyclonic vortex is replaced by an anticyclonic circulation (east winds), and the cold temperatures in the vortex are replaced by much warmer temperatures. Matsuno (1971) first succeeded in modelling part ofthis phenom enon. He used a modified linear form of the quasi-geostrophic equations, forced by a time-variable geopo tential field at the bottom of his model, at 250 hPa. (Reviews ofthe theory are given in the book by Holton (1975) and in the essay by Mclntyre (1982).) The changes in the mean zonal flow from westerly to easterly winds require-for their explanation at the level oflinearized wave theory-that the waves have significant amplitude in the middle stratosphere and that they possess some form of "non-uniformity". (The non uniformity is required in order to circumvent the strictures of the Eliassen-Palm and Charney-Drazin "non-interaction" theorem, which states that steady neutral non-dissipative Rossby waves have no second order effect on u.) The first requirement eliminates shorter wave lengths from consideration, because they are trapped by the wind maximum at the tropopause. The non-uniformity in the structure oflong wave lengths 100 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION 90 BO lO BO 50 t -...... _ ----~ ...... '"30 20 10 0 10 17 February 90 25 February BO w 90 ~ '"~ ~ 10 Ok--+~~+-~-+--r-1--+--~+--+~~+--+-+--+-~-+--r--r-+-- 10 Day ---- Persistence ----50hPa -.------~--- 25 hPa ...... 10 hPa Figure 4.31 -- Anomaly correlation (per cent) as a function of forecast length for four cases. Scores were evaluated over the latitude belt 20'N-82SN, for the standard level surfaces between 1000 and 200 hPa. A persistence forecast is shown by the heavy solid line. Numerical forecasts with different model tops are indicated by thin solid (50 hPa), dashed (25 hPa), and dotted (1 0 hPa) lines. (From Sirnrnons and Striifing, 1983) can be created by wave instability, by dissipation, or from the amplitude variations accompanying the propagation of a wave upward from the troposphere. These aspects were recognized by Mats uno; he therefore prescribed the geopotentia1 at his lower boundary to vary with time. Large-scale numerical models are now capable of predicting this phenomenon to some extent in a straightforward "operational" forecast. But studies show that the details of the distribution of the potential vorticity in the stratosphere are important for success in these forecasts (Butchart et al., 1982). The amplifying large-scale wave in the troposphere and lower stratosphere is believed to be an essential part of this picture of stratospheric warming. It is also thought to be related to the formation of large-scale "blocking events" in the troposphere. This is the connection through which the stratospheric sudden warming is related to an important problem in tropospheric weather prediction. The main features of this hypothesis, as synthesized from the arguments ofMatsuno (1971) and the more detailed reasoning ofTung and Lindzen (1979a), may be summarized as follows: Zonal wave number 1 (and/or 2) is arrested in its eastward or westward progression and becomes stationary ( C = 0). Ifthe vertical structure ofthe wave, as determined by the basic current u(z ), is such as to make the wave evanescent (i.e. trapped) above some height in the middle stratosphere, the now stationary wave might become resonant because of the forcing from stationary orographic uplift and DISPERSION IN THE PRESENCE OF ZONAL CURRENTS 101 stationary heating. The increase of wave amplitude with time will provide the needed freedom from the non-interaction theorem, and lead to a pronounced change of u in the stratosphere.* The quasi-biennial oscillation may play a role in this process by affecting the location of a critical line (u = C), moving this pole ward in years with easterly winds in the lower equatorial stratosphere and moving it equatorward in years with westerly winds. The reflecting properties of this surface (as opposed to its being a sink for wave energy) will determine the latitudinal half-width of the wave (Tung, 1979), the zonal trace velocity of the wave, and the extent to which the westerly winds at higher latitudes can make the wave stationary. Tung and Lindzen (1979b) find that the most effective means to obtain vertically trapped solutions for zonal wave numbers 1 and 2 is to lower the height of the stratospheric jet from its usual location around 65 km (p "" 0.2 hPa) to 45 km (p = 2 hPa). The prediction of resonance, or, equally important, the prediction ofnon-resonance, according to this scheme, will demand that the forecast model have an adequate representation of the complete stratosphere and perhaps part of the lower mesosphere. This conclusion is in agreement with the numerical experiments ofBeaudoin and Derome (1976) and those by Mechoso et al. (1982) that were referred to above. The possibility exists that an artificial model top in the lower middle stratosphere may create conditions for resonance even when conditions are not favourable in the real atmosphere. Simmons and Strlifung (1983) describe a series of four experimental ten-day forecasts designed to test prediction of the warmings that occurred in January and February 1979, and the influence of different heights ofmodel tops. The forecasts were from 16 January, 9 February, 17 February, and 25 February. Each forecast was made with three varieties of the European Centre forecast model, in which the top level was placed at 50 hP a, 25 hPa, and 10 hPa (approximately at 20 km, 25 km, and 31 km). The forecast skill in the troposphere is reproduced here as Figure 4.31. ·The three most successful forecasts are those from 16 January, 9 February and 25 February. In these the model with a top at 10 hPa shows no added skill until after eight days, where it does slightly better in all three cases. In the least successful case (17 February) on the other hand, the model with a top at 10 hPa yields the worst forecast after five days. It should be recognized, however, that even the 10 hPa top level is below the height of 45 km at which Tung and Lindzen found that a jet of 50 m s- 1 would produce a stationary resonant wave number 2. This speed was present at 10 hPa (31 km) in two of the four forecast cases reported by Simmons and Strlifing, and their 25 February case was characterized by easterly flow in the stratosphere. This raises the possibility that forecasts during this period may have been insensitive to the top of the model because the atmosphere itself had a reflecting "lid" at that level. Mechoso et al. (1986) describe the results of forecasts from 21 January and 17 February 1979. Several of their experiments compared results from integrations with the 15- and 9-layer model structures that were used in the 1982 paper (Mechoso et al., 1982). The 15-layer model predicted correctly the upward propagation of zonal wave number 1 that was active in the late January minor warming event, whereas the nine-level version did not. In the February case, where wave number 2 was most active, neither version ofthe model did well. It must be admitted that the results of these studies of vertical propagation of forecast errors are not conclusive as to how high model layers must extend, nor as to the accuracy that is required in stratospheric analyses. But they suggest very strongly that the accuracy of present-day numerical predictions for the troposphere at medium and longer ranges may suffer, on some occasions, from inadequate attention to the stratosphere. A much more systematic experimentation on these questions is needed, in which model tops are increased to include the stratopause and a representative variety of realistic wind patterns is included in the initial data in the middle and upper stratosphere. * Lindzen (personal communication) now places less weight on this resonance hypothesis, because it ignores latitudinal variations in the height of the reflecting layer. These variations were shown earlier (Lindzen and Hong, 1974) to be significant even in the atmospheric tidal problem, where the higher frequencies might be expected to have less sensitivity to mean zonal currents. ACKNOWLEDGEMENTS It has been an honour to have been selected by the World Meteorological Organization to join the group of distinguished scientists who have written the first five monographs in this series. The support and patience ofMartha Phillips was essential to me in this work. Discussions with Akira Kasahara during and after his visit at the National Meteorological Center were very helpful. Many con structive comments were also received from G. Platzman and R. Lindzen, and some brief correspondence with A. Eliassen, L. Dikii, G. Golitsin, B. Hoskins, and P. Websterwas of considerable help. Finally, I would like to acknowledge the many allowances that were made by the late John Brown and other staff of the National Meteorological Center for the time I spent on this paper. November, 1987. * * * APPENDIX A WAVE DEFINITIONS We consider that each of the dependent variables appearing in a linearized meteorological problem is expressed in the form f = Re F exp (itj!). (Al) The amplitude F is complex, but the phase If! is real. Both of them can vary in space and time. The frequency w and the wave number k are defined as the partial derivatives of the phase: alfl w =- (A2) at k = (k, !, m) 'VIjl. (A3) The basic assumption will be that the phase If! varies more rapidly than any of the amplitude factors, as well as any coefficients that appear in the equations. When the notation (A 1) is substituted into the linearized perturbation equations, terms such as af ( aF ) at = Re Tt - iwF exp (ilf!) (A4) will appear. The assumed rapid variation of the phase means that the aF!at term can be ignored: aj ~ _ Re iwF exp (ilf/), (A5) at with a similar approximation for ajlax, etc. This will result in a frequency (or dispersion) equation of the form w = Q (k, !, m; x, y, z, t). (A6) The explicit dependence on x, y, z and t reflects the possibility that the coefficients in the original perturbation equations may depend on space and time. The approximation (A5) that enabled this formula for w to be derived is assumed to be sufficiently accurate that (A5) is valid for all x, y, z, and t, i.e. that If! and its derivatives will depend on space and time so as to satisfy (A5) everywhere.* The group velocity Cgrp is defined as the vector whose components are the derivatives of w with respect to k, !, and m: aw aw aw) Cgrp = ( fik' -----a/' am ' (A7) where this notation denotes the derivatives of Q with respect to k, l, and m, holding con~tant any variable coefficients. * Some care is necessary, however, in dealing with systems of equations with more than one variable. For example, if the approximation (A5) is introduced into the two geostrophic equations (El 0) and (E14) after they have been linearized about a basic state with a zonal current u(z), the resulting frequency equation (A6) has complex coefficients. If the two equations are combined into (El6) before applying (A5), (A6) will have real coefficients. 104 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION The motion may be such as to have arrived at a stage where iv and the wave numbers have a well-defined distribution with space and time. It is then meaningful to consider the rate of change of w observed at a point moving with some arbitrary velocity U. Dw aQ -=- + U·VQ dt at (AS) ak aQ at aQ am) ( aQ ) =e: at+al at+ am at+ ate+ U·VQ. The subscript e denotes the explicit time dependence in (A6) due to the variable coefficients. But ak/at = a (a!fl/ax)/at can be rewritten as -aw/ax = -aQ/ax, and similar transformations can be made for at/at and am/at. Thus we may write Dw (aQ) dt = (- Cgrp + U)·VQ + ate· (A9) A ray is defined as a point that moves with the group velocity appropriate to the wave number and frequency present at that point, i.e. u = cgrp• For such a point we have Dw (aQ\ dt (ray) = Tt}e' (AlO) i.e., the frequency is conserved along such a trajectory, unless the coefficients in the original perturbation equations vary with time. A similar treatment of the variation ofthe wave numbers observed along a ray produces the results Dk ( aQ) dt (ray) = - ax e' (All) Dt ( aQ) dt (ray) = - ay e' (Al2) Dm ( aQ) dt (ray) = - iJi" e· (Al3) In meteorological problems, the x-, y-, and z-axes are normally taken as directed eastward, northward, and upward, and the basic state to be one of zonal symmetry. This results in coefficients that are independent ofx and t. In these circumstances it is t and m that have the task of changing along a ray in such a way as to maintain the relation (A6). Rays are also important in describing the propagation of energy. The perturbation equations will have an expression for the energy density E of the motion which will be a positive function ofthe squared amplitude(s) F. In problems with at most a uniform basic current V, it can be shown in each case that the energy density obeys the relation aE -- = _ V. ECgrp· (A14) at or DE___ - (ray ) - _ - E V · Cgrp· (A15) dt APPENDIX A 105 In cases where the basic current V varies with space, E in these two equations must be replaced by the "wave action", A (Bretherton and Garrett, 1969): E A (A16) w-k·V The trace speed denotes the speed of motion of a point of constant phase on a line oriented in an arbitrary assigned direction. The trace speeds in the direction of the three co-ordinate axes are (J) (J) (J) k' -~-,and m (A17) These are of importance when conditions at a bounding surface (for example, reflection) must be considered. A quantity ofcomparatively little significance is the phase velocity. Its magnitude is that ofthe trace speed in the direction of the wave number, wk (A18) cph = Tkf2' and its direction is that of the wave number. Its magnitude is exceeded by the trace speeds in other directions. (Note that the trace speeds in the x-, y-, and z-directions are not the components of the phase velocity.) The term "phase speed" is common in the literature. This usage is correct where there is only one direction of wave propagation and there is no distinction between phase and trace velocities. Clear and concise descriptions ofgroup velocity and related matters as they apply to Rossby waves can be found in Chapter 3 ofPedlosky (1979), and in the paper by Hoskins and Karoly (1981). APPENDIX B LAMB MODE FOR A NON-ISOTHERMAL ATMOSPHERE Of all the vertical modes that exist in a resting stratified atmosphere, the Lamb wave is the most significant for large-scale weather prediction. It is therefore necessary that its properties in a non-isothermal atmosphere be understood. One important property is the value of gh that it possesses. An estimate of this for a non-isothermal atmosphere can be obtained from the "Rayleigh-Ritz" type of procedure followed by Bretherton ( 1969). In the isothermal atmosphere, the Lamb wave solution for the vertical functions A(z) and B(z) that appear in (3.43) and (3.44) is as follows: A(z) = A0(z) exp (- T 0 z) (B1) and B(z) = B0(z) = 0. (B2) The vertical velocity is proportional to B. The subscript 0 denotes the isothermal solution, in which 2 C0 = RT0 = constant; (B3) g (2 - y) (B4) To (2 Co2) = constant. These are consistent with (3.43) and (3.44) if T is constant, and gh has the value: 2 gho = C0 • (B5) Let 6 denote changes from this isothermal solution. If only first-order changes are retained, (3.44) can be written 2 d6B ) ( C5 ) C0 ( dZ - T0 6B = A0 6 1 - gh . (B6) 2 Since B0 = 0, and gh0 = C0 , this is equivalent to writing (B7) Co ( ~~ - T0 B) = A0 (1 - ~~). We multiply this with A 0(z), and integrate from z = 0 to infinity. 0 2 cz) (B8) Co r Ao B 1- JB ( ~ + To Ao) dz j = JAo ( 1 - g~ dz. 0 0 0 B vanishes at z = 0. The first term on the left therefore vanishes unless B increases with height more rapidly than exp ( +T 0z ). The second term vanishes because of (B 1). Rewriting of the right side then produces a formula that equates gh to a weighted average of the square of the sound speed: APPENDIX B 107 x_ , -2T z 0 2 gh ("Lamb") = 2T0 e C (z) dz. (B9) J0 Bretherton reports a value of (312 m s-1)2 from a standard atmosphere. This corresponds to C, 2 for an isothermal atmosphere with a temperature of 242 K. The parameters for the Laplace Tidal Equation are: h (equivalent depth) = 9 933 metres; (B10) e = 8.82. } Data at the National Meteorological Center in Washington suggest that the warmest temperatures to be found in the troposphere occur in July and August around the Arabian Peninsula. A calculation with a temperature profile for Medina at 1200 GMT, 29 July 1986, inserted into (B9) produced the value gh (max) = (318 m s-1)2. (Bll) 318 m s-1is the sound speed for 251 K; the equivalent depth is 10 319 metres, and e = 8.5. 318 m s- 1 can be accepted as the fastest horizontal wave speed in a hydrostatic model of the atmosphere. As such it will set a limit for the time step that is used in explicit numerical integration of the hydrostatic equations of motion. APPENDIX C WKB DERIVATION OF THE ROSSBY FREQUENCY We consider that w = 0 in the fifth-order polynomial (2.21) is only an initial approximation. The task is to find corrections to the phase derivatives and the amplitudes, and we can expect that these will be related somehow to the variation of the Coriolis parameter with latitude. We denote the zero-order solutions by the subscript 0, and the first-order corrections by the subscript 1. The zero-order solutions are obtained from (2.36)-(2.40), with w 0 = 0 and W0 = 0: ik Cs p Do = - 0 0 (Cl) f Vo = + ilo Cs Po (C2) f Wo = 0, (C3) ij Cs lo ) Po (C4) Qo = (imo + r + f N2 k0, 10, and m0 are the gradients of the approximate phase !f/0• We maintain the assumption that the time derivative is still determined by the frequency, w 1• However, we must also allow spatial variations of the zero-order amplitudes - which we have until now neglected - as part of the correction process. The same attention must also be given the spatial variation of the coefficients. The first-order terms in the five equations are then as follows: . - r 1 aP 0 . - - l - 1w 1 U0 + Cs h ax + 1(k1P 0 + k0P!)J + JW 1 - Jf/1 0; (CS) . - f 1 aP o . - - l 1w 1 V0 + Cs lT ---ay + 1(/1P 0 + 10P 1) J + !0, 0; (C6) - iw, [0] + CS r a~o + i(m,Po + moP,) + r P, J- N 2 Q, - JU, 0; (C7) 0 - r 1 ( ahUo ahVo\' . ,. -; -. T . . - l - 1w 1 P0 + Cs l h2 ----;;x + ----ay!' 1(k0 L 1 + k 1 u0 + 10 ~ 1 + /1l 0) + 1(1110 + 1T) W1J= 0; (CS) - iw 1 Q0 + W1 = 0. (C9) Further manipulation is straightforward but tedious. (C4) and (C9), by elimination of Q0, provide a relation between W 1 and P0 : i(w, CS) ( imo + r + ij lo ) - w, N2 f Po. (ClO) APPENDIX C 109 (C5) and (C 1) together with (C 10) then show how V1 is related to P0 and P1: ! - aP a . - - r la . ( . la G; v, = hax + I(k 1Pa + kaP 1)- w1 l/ + ;\ma- iF+ 71)l- Pa (Cll) A similar operation with (C2) and (C6) relates U1 to P0 and P1: f _ aPa . _ _ ka _ C U 1 = - ~ - 1(l1Pa + loP 1)- w1 - Pa. (Cl2) s y 1 These are now substituted for the first-order velocity amplitudes in the as yet unused equation (CS). The terms in P1 cancel. 2 2 2 ka + la 1 r 2 j la ( j la )l }} - w 1{ 1 + C5 { p + N2 l (ma + P) + f 2ma + f Pa (C13) 2 = C5 (l aPa _ k aPa) + _1 ( ahUa + ahVa\ fh a ax a ay h2 ax ay 7 We now invoke the geostrophic relations (Cl) and (C2) to evaluate the horizontal divergence of U0 and V0• In doing so, we must consider that P0, f, and the scale factor, h, vary in the horizontal. After recalling that the wave numbers are given by k0h = alflo/ax, and l0h = alf/0/ay, we obtain _1 ( ahOa + ahVa) = iCs (- aPa + k aPa _ kadf P \ (Cl4) h2 ax ay fh 1a ax 0 ay fdy a;- When we substitute this into (C13) we find that the horizontal gradients of P0 cancel, and we are left with P0 times the following expression: P 1 . } ka df w, { ka2 + fa2 + Cs2 + N2 [Cfma + ;laF + p T2] + h dy = 0. (C15) h dy is equal to a d1J, and the resulting expression for w1 is the Rossby wave dispersion formula. fJ ka WR (C16) f2 1 ka2 + la2 + C 2 + N2 [Cfma + JloF + P F2] s fJ is the notation introduced by Rossby for the latitudinal variation of the Coriolis parameter: [J=-1a dd 1J (2.Q sin 7J) = 2.Q cos 1J (C17) The subscript 0 on the wave number can now be dropped. APPENDIX D THE HYDROSTATIC APPROXIMATION The assumption of hydrostatic balance is a major convenience in numerical weather prediction. One reason for this is that observational data from radiosondes are normally based on the hydrostatic approxi mation. (Temperature is measured as a function of pressure, not height.) Secondly, it eliminates all sound waves in the equations except the Lamb mode. This is important because a stable numerical integration process that includes sound waves will normally require that the time step & be sufficiently small, for a fixed spatial increment Jx or Jz, that the ratios Cs &/Jx and Cs &/Jz be smaller than one. Jz is normally smaller than Jx, and direct integration of the equations of motion would require extremely short time steps; for example, Jz 1000 m &< Cs ~ 300 m s-1 = 3 seconds. (D1) The hydrostatic approximation consists of neglecting the vertical acceleration in the vertical equation of motion (1.6). In the linear equations (2.20)-(2.22) this is accomplished by simply ignoring w2 compared with N 2• The quartic frequency equation (2.28) is changed into the quadratic equation K2N2 w2 (hydrostatic) = f2 + N2 m2 + T2 + C2s (D2) = p + K2Cs2 f 1 - m2 + T2 2 J. m2 + T2 + N C2s The maximum value is that for the Lamb wave, with m 2 = - T 2, and the internal acoustic frequencies have disappeared. (Note, however, that the internal gravity wave restriction that w2 Jw 1 Jp -<- -- (D3) & p Jz. (Jz denotes a characteristic vertical wavelength of the motion.) Estimation of the size of Jp must be pursued along two different paths. Path A. The three-dimensional velocity divergence is a significant factor in the motion, with no sensible cancellation between its constituent parts. Here we can use the adiabatic process relation 2 dp/dt = C5 dp/dt as follows: dp Jp dp -=- gpw = Cs2 dt = - pCs 2 V dt & v3. (D4) Jw JuJ = C 2 p - and/or - . } s ( Jz Jx APPENDIX D 111 (Note that&, Jx, and Jz now define properties ofthe motion, and no longer refer to the time and space steps used in numerical calculations.) Combining this with (D3), we find that hydrostatic balance is satisfied approximately for motions such that Jw satisfies the following three inequalities: 0£2 ) ( &2 ) ( &2 ) Jw « ( Cs2 Jz2 Jw, Cs2 Jx2 Ju, g ~ Jw (D5) Term 3 merely requires that the vertical velocity be less than that associated with free fall in a vacuum. Terms 1 and 2 require the absence of motions whose period is as short as that typical of acoustic phenomena. Term 2 is satisfied for the Lamb mode because Jw in that mode is small compared with Ju. Path B. The three-dimensional divergence is smaller than its individual terms, so that there is cancellation. Jw Ju Jz ~ Jz · (D6) In this case we must estimate the size of Jp from the horizontal equations of motion. There are two cases: B1: Dominance of the horizontal accelerations. This corresponds to Jp Ju -,Jx-. (D7) p Jt Combination of this with (D3) and (D6) leads to the geometric condition oz) 2 ( Jx « 1. (D8) (This is the condition for hydrostatic motion in traditional hydrodynamics, where the emphasis is on incompressible motion of a homogeneous non-rotating fluid.) B2: Dominance of the Coria/is force. This corresponds to Jp - ""'fJu Jx, and f& » 1. (D9) p This results in a more generous condition on the aspect ratio: Jz) 2 ( Jx « f&. (D10) For the assumed condition that}& is large, this would allow hydrostatic motion to exist with larger aspect ratios Jz/Jx than in the case Bl.* The equations of motion can be modified to include the hydrostatic equation in an implicit manner. This was done systematically by Eliassen (1949). (An earlier exposition was given by Sutcliffe and Godart in a note published by the United Kingdom Meteorological Office (1942).) Three changes are involved: pressure replaces height as the vertical co-ordinate, the variation of the geopotential rjJ = gz on an isobaric * The rotating expenments by Taylor (1923) had Jzlox greater than one, but the motion was very hydrostatic. 112 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION surface takes over the role ofpressure on a surface of constant height, and the individual rate of change of pressure dp w (D11) dt takes over the role of the vertical velocity w. The substantial derivative dF/dt defined in (1.3) is expressed as dF aF u aF v aF aF -=-+ -+---+w- (D12) dt at acosv a;, a av ap' where the partial derivatives with respect tot, A, and v are at constant p. (However, u and v are still horizontal components of the velocity.) In the equations of motion (1.4) and (1.6), the terms w cos v and u cos v are dropped. (This is justified by the largeness of N 2 compared with w 2.) In (1.4) and (1.5), the components of the horizontal pressure force per unit mass are replaced by 1 arp arp (D13) a cos v aT and a av' the derivatives being at constant pressure. The vertical equation of motion (1.6) becomes simply arp az = RT, (D14) where p Z = - log -; Po constant. (D15) Po Finally, the continuity equation (1. 7) takes the form 1 au av cos v aw vp. v = (D16) a cos v aT+ av ap' where the A and v derivatives are again at constant pressure. The first law of thermodynamics (1.8) can be rewritten as (_a_ + v . V) _!!!_ - (RT)z NZ w = !.!._ q. (D17) at P az g p cP The kinematic boundary condition at the ground must also be restated. Given the relation ap dp - ap + v . V 2P + w ---az (D18) w dt- at in the x-, y-, z-system, we use the following relations between partial derivatives (subscripts indicating which variable is being held constant): ( :~ ) z = p~ :~ ~ P' (D19) (Vp)z = p (Vf/J)p, (D20) APPENDIX D 113 and derive the following formula for w: w- = p ( Tta~ + V • Vryp ~ - pgw. (D21) This enables a statement about the vertical velocity w at the ground in the normal co-ordinate system to be translated into a boundary condition on w. APPENDIX E GEOSTROPHIC APPROXIMATIONS The large-scale motion in the extra-tropical atmosphere is almost in geostrophic balance, and has a time scale that is shorter than that associated with acoustic and gravity waves. The geostrophic approximation was first proposed as a purposeful modification of the complete equations of motion by Chamey (1947). Similar work was done independently by several other scientists, notably Sutcliffe (1947), Obukhov (1949), and Eliassen (1949). The system of equations derived by them is now called the "quasi-geostrophic" system. Formal derivations of the equations based upon expansion of the hydrostatic equations (see Appendix D) in terms of a small parameter (now called the Rossby number) appeared somewhat later, beginning with that by Monin (1958). *This methodical approach has the advantage of consistency, and examples are to be found in Chamey (1962), Phillips (1963), and Pedlosky (1979). The presentation here is a compromise. As in Appendix D, the symbol owill denote the order of magnitude of the departure of a variable from its value in a resting standard atmosphere. The nearly geostrophic nature of the horizontal motion and the rough equality between the size of the local time changes and the advective changes mean that the acceleration of the horizontal wind is small compared with the Coriolis force. This is expressed by a small value for the Rossby number Ro; ouz ox ou (El) Ro = fou = fox « 1. This implies that the variable part of the geopotential has the magnitude &/J ""' fou ox. (E2) The horizontal divergence of the geostrophic wind is proportional to the meridional component vgea of the geostrophic wind: '7 Vgeo p · Vgeo = -a- cot tJ. (E3) The motions of primary interest in the quasi-geostrophic theory have a horizontal scale ox that is smaller than that of the radius ofthe Earth, a distinction that was first made by Burger (1958). ox « a. (E4) (E3 sets an upper limit to the horizontal divergence in motion that is almost geostrophic. For the present case where (E4) applies, the horizontal divergence must be smaller than the magnitude of ou/ox. This allows us to conclude that aw ou (E5) '7P. v = - ap « ox . (There must be strong cancellation between the longitudinal derivative ofu and the latitudinal derivative of v.) The thermodynamic equation (D17) allows us to evaluate the size of wto establish the conditions under which that equation will be consistent with (E5). * The first scientist to make formal use of the ratio (convective acceleration)/(Coriolis acceleration) as a small parameter was Kibel (1940). APPENDIX E 115 &o p 1 ( a ) aq; ~P = ~P H2N2 ----at + v. vp az (E6) 1 ~u ~f/J ~z H 2N 2 ~x ~z' where His the pressure scale height, RT/g. Use of (E2) to evaluate ~f/J and use of the equality HaZ = az then leads to &o ~u f~x (~~r ~u 1 -=- =- (E7) ~p ~X ~u N2 ~X Ro Ri' where Ri is the Richardson number: NZ (E8) Ri = ( ~;j2 Since adJ/ap = -VP · v must be small compared with ~u/~x, the product Ri Ro must be large. Smallness being measured in powers of Ro, it is logical to require that Ri be as large as Ro- 2• N2~z2 Ri Ro2 = ""' 1. (E9) if~x)2 This is recognizable as the Prandtl-Rossby relation (2. 70). A formal expansion in powers ot Ro leads to successive approximations for the velocities and geopo tential that are similar to the development in Appendix C of the Rossby-wave formula. Two basic equations result. They are stated most succinctly in the notation used for the hydrostatic system in Appendix D. The first is a non-linear vorticity equation: aw (ElO) ( :t + v · vp) if + o = fo ap Cis the relative vorticity: av au cos 1J) ( (Ell) ' = a cos 1J a..t a1J • The horizontal velocity components in (E 10) and (Ell) are evaluated geostrophically with a constant value of the Coriolis parameter fo = 2Q sin 1J0 = constant. (El2) aq; . u = afo au ' (El3) 1 aq; V = af0 cos u0 ay-· The other equation is a simplified form of the thermodynamic equation (Dl7), in which the static stability factor H 2N 2 is restricted to being a function only of pressure. 116 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION a ) arp w R - +v·V --S(p)-=-q (E14) ( at P az p cp ' in which S(p) = H 2 N 2 • (E15) If 6J = dp/dt in (E10) is replaced by its value from (E14), the result is a simple equation for the geostrophic potential vorticity, PV. When heating and friction are ignored, this equation is ( :t + v · VP) P V = 0; (E16) fo a ( P arp) (E17) PV = f + C + P az s az · (Recall that p = exp (-Z).) Boundary conditions at the ground on w can be derived from (D21). In the quasi-geostrophic system the term v · VifJ drops out in comparison with arp/ at because vis (almost) geostrophic. One contribution to w at the ground will be due to sloping terrain: W - V· VZsfc· (E18) Another contribution can arise from the effect of surface friction, as derived by Charney and Eliassen (1949). The divergence of the frictional inflow toward low pressure is proportional to C: K )112 w (friction) = sin (2a) Co = ECo. (E19) ( 21 a is the cross-isobar angle, and K is the eddy austausch coefficient. In middle latitudes, the length E is 100 to 200 metres. Burger (1958) analysed the case of very large horizontal scale, where Jx =a. (E20) It differs from the above system because (E3) no longer leads to the inequality (E5). In his analysis, Burger assumed that the time scale remained of the same size as the advective time scale*, i.e. that Ju Ju a (E21) at = Jx = a For the same size of Ju, the Rossby number is now smaller than in the quasi-geostrophic scaling treated earlier. It is no longer allowable to simplify the Coriolis parameter to fo. The vorticity equation (El 0) no longer applies, and is replaced by a balance between the (geostrophic) /3-term and the product of /with the geo strophic divergence (E3): /3Vgeo = - f Vp · Vgeo· (E22) The only prediction equation is the thermal equation (D 17), in which vis evaluated geostrophically, and the full horizontal variability of (RT!g) and N 2 must be allowed for in the coefficient of w. * The time scale (period) of free Rossby waves does not satisfy the condition (E2!); their periods decrease as their wave length increases. APPENDIX E 117 If Burger's system is linearized for a resting basic state, the result is the limiting Type 6 solutions of the Laplace Tidal Equation that were analysed by Longuet-Higgins (1968) for negative equivalent depth h. They have an eastward phase velocity (see Figure 3.6), (1)6 2Q (E23) s -e where e = 4Q 2a 2/gh is large and negative. According to (3.37), negative h corresponds to a trapped solution that varies exponentially with height. These properties are different from those of the quasi -geostrophic system (E 1O)-(E19). The implication is that theoretical calculations of the response of the atmosphere to steady state forcing on a large horizontal scale should not be based on the quasi-geostrophic system. Recent computations of this type of response have, in fact, been made with linearized versions of the hydrostatic system, with no geostrophic assumption (Salby, 1980; Jacqmin and Lindzen, 1985). REFERENCES Alpert, J., Geller, M. and A very, S., 1983: The response of stationary planetary waves to tropospheric forcing. J. Atm. Sci., 40, 2467-2483. Andrews, D. and Mclntyre, M., 1976: Generalized Eliassen-Palm and Chamey-Drazin theorems for waves on axisymmetric flows in compressible atmospheres. J. Atm. Sci., 35, 175-185. Baer, F., 1977: Adjustment of initial conditions required to suppress gravity oscillations in non-linear flow. Beitr. Phys. Atmos., 50, 350-366. Beaudoin, C. and Derome, J., 1976: On the modelling of stationary planetary waves. Atmosphere, 14, 245-253. Beland, M., 1976: Numerical study of the nonlinear Rossby wave critical level development in a barotropic zonal flow. J. Atm. Sci., 33, 2066-2078. Beland, M., 1978: The evolutionofanonlinearRossbywavecriticallevel: effects ofviscosity. J. Atm. Sci., 35, 1802-1815. Bengtsson, L., Ghil, M. and Kallen, E. (Eds.), 1981: Dynamic meteorology: Data assimilation methods. Springer-Verlag, New York 330 pp. Bennett, J. and Young, J., 1971: The influence oflatitudinal wind shear upon large-scale wave propagation into the tropics. Mon. Wea. Rev., 99, 202-214. Benney, D. and Bergeron, R., 1969: A new class of non-linear waves in parallel flows. Stud. Appl. Math., 48, 181-204. Bjerknes, J., 1937: Theorie der aussertropischen Zyklonenbildung. Meteor. Zeit., 54, 462-266. Bjerknes, J., 1966: A possible response of the atmospheric Hadley circulation to equatorial anomalies of ocean temperature. Tel!us, 18, 820-829. Bjerknes, V., 1904: Das Problem von der Wettervorhersage, betrachtet vom Standpunkt der Mechanik und der Physik. Meteor. Zeit., 21, 1-7. Bjerknes, V., Bjerknes, J., Solberg, H. and Bergeron, T., 1933: Physikalische Hydrodynamik. Springer, Berlin, 797 pp. Blake, D. and Lindzen, R., 1973: Effect of photochemical models on calculated equilibria and cooling rates in the stratosphere. Mon. Wea. Rev., 101, 783-802. Booker, J. and Bretherton, F., 1967: The critical layer for internal gravity waves in a shear flow. J. Fluid Mech., 27, 513-539. Boyd, J., 1976: Noninteraction of waves with the zonally averaged flow on a spherical Earth and the interrelationship of eddy fluxes of energy, heat and momentum. J. Atm. Sci., 33, 2285-2291. Branstator, G., 1983: Horizontal energy propagation in a barotropic atmosphere with meridional and zonal structure. J. Atmos. Sci., 40, 1689-1708. Bretherton, F., 1966: The propagation of groups ofintemal waves in a shear flow. Quart. J. R. Meteor. Soc., 92, 466-480. Bretherton, F., 1969: Lamb waves in a nearly isothermal atmosphere. Quart. J. R. Meteor. Soc., 95, 754- 757. Bretherton, F., 1971: The general linear theory of wave propagation. Lectures in Applied Mathematics, vol. 13, Amer. Math. Soc., 61-102. Bretherton, F. and Garrett, C., 1969: Wavetrains in inhomogeneous moving media. Proc. Ray. Soc. London, A, 302, 529-554. Burger, A., 1958: Scale consideration of planetary motion ofthe atmosphere. Tel!us, 10, 195-205. REFERENCES 119 Burger, A., 1962: On the non-existence of critical wavelengths in a continuous baroclinic stability problem. J. Atm. Sci., 19, 31-38. Butchart, N., Clough, S., Palmer, T. and Treveljan, P., 1982: Simulation of an observed stratospheric warming with quasi-geostrophic refractive index as a model diagnostic. Quart. J. R. Meteor. Soc., 108, 475-502. Chapman, S. and Lindzen, R., 1970: Atmospheric Tides. Gordon and Breach, 200 pp. Charney, J., 1947: The dynamics of long waves in a baroclinic westerly current. J. Meteor., 4, 135-162. Charney, J., 1948: On the scale of atmospheric motions. Geofys. Pub!. (Oslo), 17, No. 2. Charney, J., 1949: On a physical basis for numerical weather prediction of large-scale motions in the atmosphere. J. Meteor., 6, 371-385. Charney, J., 1962: Integration of the primitive and balance equations. Proc. Inter. Symp. Nu mer. Wea. Pred., Tokyo, Meteorological Society of Japan, 131-152. Charney, J., 1969: A further note on large-scale motions in the tropics. J. Atm. Sci., 26, 182-185. Charney, J., 1973: Planetary fluid dynamics. In Dynamic Meteorology, P. Morel, Ed., Reidel (Dordrecht), 98-351. Charney, J. and Drazin, P., 1961: Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res., 66, 83-109. Charney, J., and A. Eliassen, 1949: A numerical method for predicting the perturbations of the middle latitude westerlies. Tellus, 1, No. 1, 38-54. Charney, J., Fj0rtoft, R. and von Neumann, J., 1950: Numerical integration of the barotropic vorticity equation. Tellus, 2, 237-254. Charney, J. and Stern, M., 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atm. Sci., 19, 159-172. Clark, J., 1972: The vertical propagation of forced atmospheric planetary waves. J. Atm. Sci., 1430- 1451. Corby, G., 1967: Laplace's tidal equations-an application of solutions for negative equivalent depth. Quart. J. R. Meteor. Soc., 93, 368-372. Cressman, G., 1948: On the forecasting of long waves in the upper westerlies. J. Meteor., 5, 44-57. Cressman, G., 1958: Barotropic divergence and very long atmospheric waves. Mon. Wea. Rev., 86, 293- 297. Crutcher, H., 1961: Meridional cross-sections of upper winds over the Northern Hemisphere. Tech. Paper No. 41, US Weather Bureau, Washington, D.C., 307 pp. Da Silva Dias, P. and Bonatti, J., 1986: Vertical mode decomposition and model resolution. Tellus, 38A, 205-214. Daley, R., Tribbia, J. and Williamson, D., 1981: The excitation oflarge-scale free Rossby waves in numerical weather predictions. Mon. Wea. Rev., 109, 1836-1861. Derome, J., 1981: On the average errors of an ensemble offorecasts. Atmos.-Ocean, 19, 103-127. Derome, J. and Dolph, C., 1970: Three-dimensional non-geostrophic disturbances in a baroclinic zonal flow. Geophys. Fluid Dyn., 1, 91-122. Dickinson, R., 1968: On the exact and approximate linear theory of vertically propagating planetary Rossby waves forced at a spherical lower boundary. Mon. Wea. Rev., 96, 405-415. Dickinson, R., 1970: Development of a Rossby wave critical level. J. Atm. Sci., 27, 627-633. Dickinson, R., 1973: A method ofparameterization for infra-red cooling between altitudes of30 and 70 kms. J. Geophys. Res., 78, 4451-4457. 120 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION Dikii, L., 1965: The terrestrial atmosphere as an oscillating system. Izv. A cad. Sci. USSR, Atm. and Oceanic Physics. English edition, vol. 1, No. 5, 275-286. Dikii, L., 1969: Teoriya kolebanii zemnoi atmosferi. Gidromet. Izdat., Leningrad, 195 pp. Dikii, L. and Katayev, V., 1971: Calculation of the planetary wave spectrum by the Galerkin method. Izv. Atm. Oceanic Physics, 7, 1031-1038 (English trans., Amer. Geophys. U., pp. 682-686.) Eady, E., 1949: Long waves and cyclone waves. Tellus, 1, No. 3, 33-52. Eckart, C., 1960: Hydrodynamics of oceans and atmospheres. Pergamon Press, Oxford, 290 pp. Edmon, H., Hoskins B. and Mclntyre, M., 1980: Eliasscn-Palm cross sections for the troposphere. J. Atm. Sci., 37, 2600-2616. Einaudi, F. and Hines, C., 1970: WKB approximation in application to acoustic-gravity waves. Canadian J. Physics, 48, 1458-1471. Ekman, V., 1923: Uber Horizontalzirkulation bei winderzeugten Meeresstromungen. Arkiv f Mat., Astron. och Fysik (Stockholm), 17, No. 26, 74 pp. Ekman, V., 1932: Studien zur Dynamik der Meeresstromungen. Gerl. Beitr. zur Geophysik, 36, 385-438. Eliassen, E. and Machenhauer, B., 1965: A study ofthe fluctuations of atmospheric planetary flow patterns represented by spherical harmonics. Tellus, 17, 220-238. Eliassen, A., 1949: The quasi-static equations of motion with pressure as independent variable. Geofys. Pub!. (Oslo), 17, No. 3. Eliassen, A. and Palm, E., 1961: On the transfer of energy in mountain waves. Geofys. Pub!. (Oslo), 22, No. 3, 1-23. Ertel, H., 1941: Die Unmoglichkeit einer exakten Wetterprognose aufGrund synoptischer Luftdruckkarten von Teilgebieten der Erde. Meteor. Z., 58, 309-313. Ertel, H., 1944: Wettervorhersage als Randwertproblem. Meteor. Z., 61, 181-190. Finger, F., Gelman, M., Griffith, D., Laver, J., Miller, A., Nag~tani, R. and Quiroz, R., 1975: Synoptic analyses, 5-, 2-, and 0.4 millibar surfaces for January 1972 through June 1973. NASA SP-3091, Washington, 205 pp. Flattery, T., 1967: Hough functions. Ph. D. thesis, University of Chicago, 168 pp. Gandin, L., 1963: Objective analysis ofmeteorological fields. Gidrometeoizdat, Leningrad. (Trans. U. S. Dept of Commerce, Washington, 242 pp.) Geisler, J. and Dickinson, R., 1976: The five-day wave on a sphere with realistic zonal winds. J. Atm. Sci., 33, 632-641. Gelb, A. (Ed.), 1974: Applied optimal estimation. The M.I.T. Press, Cambridge. 374 pp. Gent, P. and McWilliams, J., 1983: Regimes of validity for balanced models. Dyn. ofAtmos. and Oceans, 7, 167-183. Green, J., 1960: A problem in baroclinic instability. Quart. J. R. Meteor. Soc., 86, 237-251. Grose, W. and Ho skins, B., 1979: On the influence of orography on large-scale atmospheric flow. J. Atm. Sci., 36, 223-234a. Hamilton, K., 1982: Stratospheric circulation statistics. NCAR Technical Note, 191-STR, Boulder, 174 pp. Hayashi, Y., 1981: Vertical-zonal propagation of a stationary planetary wave packet. J. Atm. Sci., 38, 1197-1205. Haurwitz, B., 1940: The motion of atmospheric disturbances on the spherical earth. J. Marine Res., 3, 254-267. Held, I., 1983: Stationary and quasi-stationary eddies in the extra-tropical atmosphere. In Large-scale dynamical processes in the atmosphere, Eds. B. Hoskins and R. Pearce, Acad. Press (London), 397 pp. REFERENCES 121 Held, 1., Panetta, R. and Pierrehumbert, R., 1985: Stationary external Rossby waves in vertical shear. J. Atm. Sci., 42, 865-883. Helmholtz, H. von, 1888: Uber atmospharische Bewegungen. I. Sitzber. Ges. Wiss. Berlin, 64 7-663. Also in The mechanics ofthe earth's atmosphere, transl. by C. Abbe. Smithsonian Mise. Collections (1891), pp. 78-93. Hollingsworth, A., 1971: The effect of ocean and earth tides on the semi-diurnallunar airtide. J. Atm. Sci., 28, 1021-1044. Hollingsworth, A., 1986: Objective analysis for numerical weather prediction. WMO/IUGG NWP Symposium, Tokyo, 11-59. Hollingsworth, A., Arpe, K., Tiedtke, M., Capaldo, M. and Savijarvi, H., 1980: The performance of a medium range forecast model in winter- impact of physical parameterizations. Mon. Wea. Rev., 108, 1736-1773. Hollingsworth, A., Lorenc, A., Tracton, M., Arpe, K., Cats, G., Uppala, S. and Kallberg, P., 1985: The response of numerical weather prediction systems to FGGE Level lib data. Part I: Analyses. Quart. J. R. Meteor. Soc., 111, 1-66. Holton, J., 1975: The dynamic meteorology of the stratosphere and mesosphere. Meteor. Monographs, vol. 15, No. 37. Amer. Meteorol. Soc., Boston, 218 pp. Hoskins, B. and Bretherton, F., 1972: Atmospheric frontogenesis models: mathematical formulation and solution. J. Atm. Sci., 29, 11-37. Hoskins, B. and Karoly, D., 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atm. Sci., 38, 1179-1196. Hoskins, B., Simmons, A. and Andrews, D., 1977: Energy dispersion in a barotropic atmosphere. Quart. J. R. Meteor. Soc., 103, 553-567. Hough, S., 1898: On the application ofharmonic analysis to the dynamical theory ofthe tides. Phi!. Trans., A, 191, 139-185. Hovmoller, E., 1949: The trough-and-ridge diagram. Tellus, 1, No. 2, 62-66. Jacchia, L. and Kopal, Z., 1952: Atmospheric oscillations and the temperature profile of the upper atmosphere. J. Meteor., 9, 13-23. J acqmin, D. and Lindzen, R., 1985: The causation and sensitivity of the Northern Winter planetary waves. J. Atm. Sci., 42, 724-745. Jarraud, M., Simmons, A. and Kanamitsu, M., 1986: Sensitivity of medium-range weather forecasts to the use of an envelope orography. Tech. Rpt. 56, European Centre for Medium-range Weather Forecasts (Reading), 83 pp. Karoly, D. andHoskins, B., 1982: Three-dimensional propagation of planetary waves. J. Meteor. Soc. Japan, 60, 109-123. Kasahara, A., 1980: Effect of zonal flows on the free oscillations of a barotropic atmosphere. J. Atm. Sci., 37, 917-929. Kasahara, A. and Shigehisa, Y., 1983: Orthogonal vertical normal modes of a vertically staggered discretized atmospheric model. Mon. Wea. Rev., 111, 1724-1735. Kasahara, A. and da Silva Dias, P., 1986: Response of planetary waves to stationary heating in a global atmosphere with meridional and vertical shear. J. Atm. Sci., 43, 1893-1911. Kato, S., 1966: Diurnal atmospheric oscillations. 1. Eigenvalues and Hough functions. J. Geophys. Res., 71, 3201-3209. Kibel, 1., 1940: Prilozhenie k meteorologii uravnenii mekhaniki baroklinnoi zhidkosti. Izv. Akad. Nauk, SSSR, Ser. Geograf Geofiz., No. 5, 627-637. Killworth, P., and Mclntyre, M., 1985: Do Rossby wave critical layers absorb, reflect, or over-reflect? J. Fluid Mech., 161, 449-491. 122 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION Kirkwood, E. and Derome, J., 1977: Some effects of the upper boundary condition and vertical resolution on modelling forced stationary planetary waves. Mon. Wea. Rev., 105, 1239-1251. Klemp, J. and Durran, D., 1983: An upper boundary condition permitting internal gravity wave radiation in numerical mesoscale models. Mon. Wea. Rev., 111, 430-444. Kochin, N., 1932: Uber die StabiliHit von Margulesschen Diskontinuitatsflachen. Beitr. Phys. Frei. Atmos., 18, 129-164. Kuo, H., 1949: Dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere. J. Meteor., 6, 105-122. Lamb, H., 1932: Hydrodynamics. 6th ed., Camb. Univ. Press, 738 pp. Landau, L. and Lifshitz, E., 1959: Fluid Mechanics. Pergamon, London. Laplace, P., 1799: Mechanique celeste. Part I, Book IV. Paris. Lighthill, M., 1965: Group velocity. J. Inst. Math. Applic., 1, 1-28. Lighthill, M., 1966: Dynamics of rotating fluids: a survey. J. Fluid Mech., 26, 411-431. Lim, H. and Chang, C., 1983: Dynamics ofteleconnections and Walker circulations forced by equatorial heating. J. Atom. Sci., 40, 1897-1915. Lindzen, R., 1966: On the theory of the diurnal tide. Mon. Wea. Rev., 94, 295-301. Lindzen, R., 1967: Planetary waves on beta-planes. Mon. Wea. Rev., 95, 441-451. Lindzen, R., 1968: Rossby waves with negative equivalent depths-comments on a note by G. A. Corby. Quart. J. R. Meteor. Soc., 94, 402-407. Lindzen, R., 1987: On the development of the theory of the QBO. Bull. Amer. Meteor. Soc., 68, 329- 337. Lindzen, R., Batten, E. and Kine, J., 1968: Oscillations in atmospheres with tops. Mon. Wea. Rev., 96, 133-140. Lindzen, R. and Holton, J., 1968: A theory of the quasi-biennial oscillation. J. Atm. Sci., 25, 1095-1107. Lindzen, R. and Hong, S., 1974: Effects of mean winds and horizontal temperature gradients on solar and lunar semi-diurnal tides in the atmosphere. J. Atm. Sci., 31, 1421-1446. Lindzen, R., Straus, D. and Katz, B., 1984: An observational study oflarge-scale atmospheric Rossby waves during FGGE. J. Atm. Sci., 41, 1320-1335. Lindzen, R. and Tung, K., 1976: Banded convective activity andductedgravitywaves.Mon. Wea. Rev., 104, 1602-1617. Longuet-Higgins, M., 1964: Planetary waves on a rotating sphere. Proc. Ray. Soc., A, 279, 446-473. Longuet-Higgins, M., 1964: On group velocity and energy flux in planetary wave motions. Deep-sea Research, II, 35-42. Longuet-Higgins, M., 1968: The eigenfunctions of Laplace's tidal equations over a sphere. Phi!. Trans. Ray. Soc. London, A, 1132, 262, 511-607. Lorenc, A., 1986: Analysis methods for numerical weather prediction. Quart. J. R. Meteor. Soc., 112, 1177-1194. Lorenz, E., 1955: Available potential energy and the maintenance of the general circulation. Tell us, 7, 157- 167. Love, A., 1913: Notes on the dynamical theory of the tides. Proc. Land. Math. Soc., 12, 309-314. Lyra, G., 1943: Theorie der stationaren Leewellenstromung in freier Atmosphiire. Zeit.fur angew. Math. u. Mech., 23, 1-28. Machenhauer, B., 1976: An initialization procedure based on the elimination of gravity oscillations in a spectral, barotropic primitive equation model. Simulation of large-scale processes, Ann. Meteorol., 11, 135-138. REFERENCES 123 Madden, R., 1979: Observations of large-scale traveling waves. Rev. Geophys. Space Phys., 17, 1935- 1949. Mak, M.-K., 1969: Laterally driven stochastic motions in the tropics. J. Atm. Sci., 26, 41-64. Margules, M., 1892/3: Luftbewegungen in einer rotierenden Spharoidschale. Ber. Akad. Wiss. Wien. Part I, 101, 597-562; Part II, 102, 11-56; Part Ill, 102, 1369-1421. (English transl. by B. Haurwitz in NCAR Technical Note 156+STR, Boulder, 1980.) Maruyama, T., 1967: Large-scale disturbances in the equatorial lower stratosphere. J. Meteor. Soc. Japan, Ser. II, 45, 391-408. Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44, 25-43. Matsuno, T., 1970: Vertical propagation of stationary planetary waves in the winter Northern Hemisphere. J. Atm. Sci., 27, 871-883. Matsuno, T., 1971: A dynamical model of the stratospheric sudden warming. J. Atm. Sci., 28, 1479- 1494. Mcintyre, M., 1982: How well do we understand the dynamics ofstratospheric warmings? J. Meteor. Soc. Japan. Ser. II, 60, 37-65. Mechoso, C., Suarez, M., Yamazaki, K., Kitoh, A. and Arakawa, A., 1986: Numerical forecasts of tropospheric and stratospheric events during the winter of 1979: Sensitivity to the model's horizontal resolution and vertical extent. In Anomalous atmospheric flows and blocking(R. Benzi, B. Saltzman and A. Wiin-Nielsen, eds.), Adv. in Geophysics, 20, Acad. Press, New York, 459 pp. Mechoso, C., Suarez, M., Yamazaki, K., Spahn, J. and Arakawa, A., 1982: A study of the sensitivity of numerical forecasts to an upper boundary in the lower stratosphere. Mon. Wea. Rev., 110, 1984- 1993. Meyers, R. and Yanowitch, M., 1971 : Small oscillations of a viscous isothermal atmosphere. J. Comput. Physics, 8, 241-257. Miles, J., 1967: Laplace's tidal equations revisited. Seventh U.S. Natl. Cong. on Applied Math., Amer. Soc. Mech. Eng., New York, 27-38. Miles, J., 1977: Asymptotic eigensolutions of La place's tidal equation. Proc. Ray. Soc. London, A, 353, 377-400. Monin, A., 1958: Izmeneniia davleniia v baroklinnoi atmosfere. Izv. Akad. Nauk, SSSR, Geofiz., 4, 497- 514. Muench, H., 1965: On the dynamics of the wintertime stratosphere circulation. J. Atm. Sci., 22, 349- 360. Munk, W. and Phillips, N., 1968: Coherence and band structure of inertial motion in the sea. Rev. of Geophys., 6, 4, 447-472. Murakami, M., 1974: Influence of mid-latitudinal planetary waves on the tropics under the existence of critical latitude. J. Meteor. Soc. Japan, 52, 261-271. National Oceanic and Atmospheric Administration, 1976: U.S. Standard Atmosphere, 1976. Govt. Print. Office, Washington, 227 pp. Obukhov, A., 1949: K voprosu geostroficheskom vetra. Izv. Akad. Nauk, SSSR, Ser. Geograf Geofiz., 13, No. 4, 1949. Oort, A., 1983: Global atmospheric circulation statistics, 1958-1973. NOAA Pro£ Paper 14, Rockville, Md., 180 pp. Oort, A. and Rasmussen, E., 1971: Atmospheric Circulation Statistics. NOAA Prof. Paper 5, Rockville, Md., 323 pp. Opsteegh, J. and van den Dool, H., 1980: Seasonal differences in the stationary response of a linearized primitive equation model: Prospects for long range weather forecasting? J. Atm. Sci., 37, 2169-2185. 124 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION Orlanski, L, 1968: Instability of frontal waves. J. Atm. Sci., 25, 178-200. Pedlosky, J ., 1979: Geophysical fluid dynamics. Springer, New York, 624 pp. Phillips, N., 1954: Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level quasi-geostrophic model. Tellus, 6, 273-286. Phillips, N., 1963: Geostrophic motion. Rev. Geophys., 1, 123-176. Phillips, N., 1965: Elementary Rossby waves, Tellus, 17, 295-301. Phillips, N., 1972: Principles of large-scale numerical weather forecasting. In Dynamic Meteorology, P. Morel, Ed., Reidel (Dordrecht), 2-96. Phillips, N., 1976: The impact of synoptic observing and analysis systems on flow pattern forecasts. Bull. Amer. Meteor. Soc., 57, 1225-1240. Platzman, G., 1968: The Rossby wave (Symons Memorial Lecture). Quart. J. R. Meteor. Soc., 94, 225- 248. Platzman, G., 1985: The S-1 Chronicle: a tribute to Bemard Haurwitz. Unpublished lecture presented at Colorado State University, Fort Collins, 4 October 1985. Plumb, R., Andrews, D., Geller, M., Grose, W., O'Neill, A., Salby, M. and Vincent, R., 1985: Dynamical processes. In vol. I of Atmospheric Ozone 1985, Ozone Report No. 16, World Meteorological Organization, Geneva, 241-347. Plumb, R., and McEvans, A., 1978: The instability of a forced standing wave in a viscous stratified fluid: a laboratory analogue of the quasi-biennial oscillation. J. Atm. Sci., 35, 1827-1839. Prandtl, L., 1939: Beitrage zurMechanikder Atmosphare. M em. Assoc. Meteor. de UGGI, Edimbourg. Paris, P. Dupont. Puri, K. and Bourke, W., 1982: A scheme to retain the Hadley circulation during nonlinear normal mode initialization. Mon. Wea. Rev., 110, 327-335. Queney, P., 1947: Theory ofperturbations in stratified currents with applications to airflow over mountain barriers. Mise. Report 23, Dept. Meteorology, Univ. Chicago Press, 81 pp. Queney, P., 1948: The problem of airflow over mountains: a summary oftheoretical studies. Bull. Amer. Meteor. Soc., 29, 16-26. Queney, P., 1950: Adiabatic perturbation equations for a zonal atmospheric current. Tellus, 2, No. 2, 35-51. Rattry, M., 1964: Time dependent motion in an ocean: a unified two-layer, beta-plane approximation. Studies on Oceanography. Geophy. Inst. Tokyo Univ., pp. 19-29. Rayleigh, Lord, 1916: On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phi!. Mag., Series 6, 32. Richardson, L., 1922: Weather prediction by numerical process. Camb. Univ. Press, (London), 236 pp. (Dover reprint, New York, 1965.) Richtmyer, R. and Morton, K., 1967: Difference methods for initial-value problems. J. Wiley, New York, 405 pp. Rosenthal, S., 1965: Some preliminary theoretical considerations of tropospheric wave motions in equatorial latitudes. Mon. Wea. Rev., 93, 605-612. Rossby, C., 1938: On temperature changes in the stratosphere resulting from shrinking and stretching. Beitr. z. Phys. d. fr. Atmos., 24, 53-60. Rossby, C., 1945: On the propagation offrequencies and energy in certain types of oceanic and atmospheric waves. J. Meteor., 2, 187-204. Rossby, C., 1949: On the dispersion of planetary waves in a barotropic atmosphere. Tellus, 1, No. 1, 54-58. REFERENCES 125 Rossby, C., and collaborators, 1939: Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Marine Res., 2, 38-55. Rowntree, P., 1972: The influence of tropical east Pacific Ocean temperatures on the atmosphere. Quart. J. Roy. Meteor. Soc., 98, 290-321. Sal by, M., 1979: On the solution of the homogeneous vertical structure problem for long-period oscillations. J. Atm. Sci., 36, 2350-2359. Salby, M., 1980: The influence of realistic dissipation on planetary normal structures. J. Atm. Sci., 2186- 2199. Salby, M., 1984: Survey of planetary-scale travelling waves: the state of theory and observation. Rev. Geoph. and Space Phy., 22, 209-236. Sawada, R. and Matsushima, A., 1964: Thermally driven annual atmospheric oscillations as a cause of dynamic heating of the winter polar upper atmosphere. J. Meteor. Soc. Japan, 42, 97-108. Silberman, I., 195 3: A matrix methodfor computing atmospheric oscillations. New York University, Dept. of Meteorology and Oceanography, 13 pp. Simmons, A., 1982: The forcing of stationary wave motion by tropical diabatic heating. Quart. J. Roy. Meteor. Soc., 108, 503-534. Simmons, A. and Hoskins, B., 1979: The downstream and upstream development of unstable baroclinic waves. J. Atm. Sci., 36, 1239-1254. Simmons, A. and Striifing, R., 1983: Numerical forecasts of stratospheric warming events using a model with a hybrid vertical coordinate. Quart. J. Roy. Meteor. Soc., 109, 81-111. Su, C., Schlesinger, M. and Kim, J., 1980: Asymptotic expansion ofthe normal modes ofthe linearized shallow water equations for small Lamb's parameter. Clim. Res. Inst., Report 13, Dept. Atmos. Sci., Corvallis, Oregon, 46 pp. Sutcliffe, R., 1947: A contribution to the problem of development. Quart. J. Roy. Meteor. Soc., 73, 370- 383. Swarztrauber, P. and Kasahara, A., 1985: The vector harmonic analysis ofLaplace's tidal equations. SIAM J. Sci. Stat. Comput., 6, No. 2, 464-491. Taylor, G., 1923: Experiments on the motion of solid bodies in rotating fluid. Proc. Roy. Soc. London, A, 104, 213-218. Taylor, G., 1926: Waves and tides in the atmosphere. Proc. Roy. Soc. London, A, 126, 169-183. Taylor, G., 1936: The oscillations ofthe atmosphere. Proc. Roy. Soc. London, A, 156, 318-326. Tung, K., 1979: A theory of stationary long waves. Part Ill: Quasi-normal modes in a singular waveguide. Mon. Wea. Rev., 107, 751-774. Tung, K. and Lindzen, R., 1979a: A theory of stationary long waves. Part I: A simple theory of blocking. Mon. Wea. Rev., 107, 714-734. Tung, K. and Lindzen, R., 1979b: A theory of stationary long waves. Part II: Resonant Rossby waves in the presence ofrealistic vertical shears. Mon. Wea. Rev., 107, 735-750. van Loon, H., Taljaard, J., Jenne, R. and Crutcher, H., 1971: Climate of the upper air. Southern Hemisphere, Vol. II: Zonal geostrophic winds. NAVAIR 50-IC-56, Natl. Clim. Center, Federal Bldg, Asheville. Wallace., J. and Gutzler, D., 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter. Mon. Wea. Rev., 109, 784-812. Wallace, J., and Kousky, V., 1968: Observational evidence of Kelvin waves in the tropical stratosphere. J. Atm. Sci., 25, 900-907. Webster, P., 1972: Response of the tropical atmosphere to local steady forcing. Mon. Wea. Rev., 100, 518-540. 126 DISPERSION PROCESSES IN LARGE-SCALE WEATHER PREDICTION Webster, P., 1982: Seasonality in the local and remote atmospheric response to sea surface temperature anomalies. J. Atm. Sci., 39, 41-52. Webster, P. and Holton, J., 1982: Cross-equatorial response to middle-latitude forcing in a zonally varying basic state. J. Atm. Sci., 39, 722-733. Whitham, G., 1960: A note on group velocity. J. Fluid Mech., 9, 347-352. Whitham, G., 1974: Linear and non-linear waves. Wiley and Sons, New York, 636 pp. W olff, P ., 19 58: The errorin numericalforecasts due to retrogression of ultra-long waves. M on. W ea. Rev., 86, 245-250. Yanai, M. and Maruyama, T., 1966: Stratospheric wave disturbances propagating over the equatorial Pacific. J. Meteor. Soc. Japan, Ser. 11, 44, 291-294. Yanowitch, M., 1967: Effect of viscosity on gravity waves and the upper boundary condition. J. Fluid Mech., 29, 2, 209-231. Yeh, T., 1949: On energy dispersion in the atmosphere. J. Meteor., 6, 1-16.