A Diagnostic Model for Initial Winds in Primitive

1 1

A DIAGNOSTIC MODEL FOR INITIAL WINDS IN

PRIMITIVE EQUATIONS FORECASTS

by Richard Asselin Department of Meteorology, Ph. D.

ABSTRACT

The set of primitive meteorologicai equations reduces when the accel- to a single second order partial differential cquation wind in the eration is replaced by the derivative of the geostrophic three-dimensional equation.of motion. From this equation a diagnostic is known. wind field can be computed when the pressure distribution

Calculations for a barotropic atmosphere are performed to that from fir st; the rotational part of the wind is found to be similar agrees well with the classical balance equation and the divergent part Lese; gravi- synoptic experience. A forecast thus initialized contaills wind of the tational activity than one initialized with the non-divergent

balance equation. of With a five-level baroclinic atmosphere. the effects

are studi~d mountains, surface drag, and release of latent heat concepts and separately; the results arc consistent wlth classical surfa:::e pres other interesting features are revealed. T!;e diagnostic with the observed sure tendency is fo~nd to be in very good agreement

pressure changes. A DIAGNOSTIC MODEL FOR INITIAL WINDS IN

PRIMITIVE EQUATiONS FORECASTS

RI CHARD ASSE LIN

. A the sis submitted to the Faculty of Graduate Studies and

Re search in partial fulfillrnent of the requirernents for the

degree of Doctor of Philosophy.

Departrnent of Meteorology

McGill Univer sity

Montreal, Canada July. 1970

1971 A DL~GNOSTIC MODEL FOR INITIAL WINDS IN

PRIMITIVE EQUATIONS FORECASTS

RI CHARD ASSE LIN

. A thesis submitted ta the Faculty of Graduate Studies and

Research in partial fulfillment of the requirements. for the

degree of Doctor of Philosophy.

Department of Meteorology

McGill Univer sity

Montreal, Canada July, 1970

@ l-J.cha l'Cl L.s :::;e1in 1971 ACKNOWLEDGEMENTS

The author is indebted to his research supervisor,

Professor A. J. Robert, for many useful discussions and for his

encouragement throughout this work, as weU as to Professor B. W.

Boville, who provided helpful advice and editorial assistance.

l am grateful to Dr. M. Kwizak for generating my

initial incentive in these studies and for his continuing suppurt,

particularly in arranging for me to use the computer and other facili

ties of the Central Analysis Office (CAO). The programs of the

barotropic forecast model as well as a diagram were borrowed from

his work.

Miss S. Boville is responsible for the programming

of the experiment with idealized data. Mr. W. S. Creswick did the

graphical computations and helped to evaluate the re~u1ts. Mr. P.

Sarrazin drafted the figures and Misses T. Savoie and J. Sutherland

typed the manuscript.

Finally, l wish to acknowledge the financial support of the Canadian Meteorological Servi.ce and of the National Research

Council of Canada.

ii TABLE OF CONTENTS

Acknowledgements ii

List of Plates; Tables and Figures vi

List of Symbols ix

Résumé xi

Abstract xii

1. Introduction 1

2. Review of Diagnostic Methods 6

2. 1. Balance -W- Equations 6

2. 2. Geostrophic Diagnostic (G ) 8 2 2. 3. Miyakoda-Moyer Diagnostic 9

2.4. Nitta-Hovermale Method 9

3. Theory of Initialization 11

3. 1. The Simple st Model 11

3. 2. Numerical Compatibility 15

4. Historical R~.view of the G Geos.trophic Method 16 2

Part A: BAROTROPIC EXPERIMENTS· 19

5. Basic Equations and Theory 19

6. Numerical Solution of the Diagnostic Model 25

6. 1. Relaxation 25

6. 2. Ellipticity 26

7. Hig~er Order G Approximation 30 2

iii 8. Numerical Procedures for Initialization of Forecasts 31

8. 1. Compatibility Procedure 31

8. 2. Imposing New Boundary Conditions 34

9. Discussion of Results 37

9. 1. Divergence and Tendency 37

9. 2. Vorticity 43

9.3. Height Changes 46

9.4. Forecasts 50

Part B: BAROCLINIC EXPERIMENTS 54

10. Basic Primitive Equations 54

10.1. Precipitation Model 56

Il. Diagnostic Equations 59

12. Solution of Baroclinic System 64

12. 1. Discretization 64

12.2. Boundary Condition~, Vertical Structure. 65

12.3. Relaxation 67

13. Data Preparation 69

13.1. Interpolation to (j-SurfaceEV 69 ~----- .13. 2. Smoothing 70

14. Ellipticity 72

15. _ Discussion of Results 75

15.1. Effects of Orography and Drag 75

15.2. Effect of Pressure Systems, Ellipticity 85

iv 15.3. Effect of Latent Heat 96

15.4. Combined Effect from All Sources 102

16. Conclusions 109

References III

Appendix A: Finitc Differences 115

Appendix B: Special Operations for Equatorial Latitudes 119

Appendix C: 123

v LIST OF PLATES. TABLES. AND FIGURES

pagè

Plate 1. The baroclinic diagnostic equations in full 60

Table 1. Input data for each baroclinic experiment 76

Table 2. Statistics from each baroclinic experiment 95

Figure

1. The 51. x 55 point grid 5.

2. Two graphs of the tangential wind speed in a station

ary circular system 23

3. 500 mb analysis for OOz 21 Feb 69 and diagnosed

tendency 38

4. Summaryof 25 day integration by Kwizak (1970,

Energy. vorticity, divergence • 39

5. Time variation of total available potential energy in

barotropic forecasts with bala~ce and G initia1i 2 zations 41

6. Time variation of squared divergence in barotropic

forecasts with balance and G initializations 42 2

7. Absolute vorticity from balance equation 44

8. Absolute vorticity from G geostrophic initialization 45 2

9. Total height changes ma.de by bal1:\.nce ilütialization 47

10. Total height changes ~ade by G initialization 48 D 2 vi Figure

11. 72-hour forecast from balance initialization 51

12. 72-hour forecast from G initialization 52 2

13. 72-hour verifying analysis, OOZ 24 Feb 1969 53

14. The discrete structure of the baroclinic mode1 66

15. a) response of short wave filter b) response of 10w

latitude smoother as functions of wave1ength 71

16. Idealized data used for experiments 7 to 12 78

17. Orientation error pattern in baroclinic model 79

18. Basic results of experiments with idealized mountain 81

19. Resu1ts of Cressman' s (1960) experiments with

orography 84

20. Diagnosed surface tendency produced by rea1 oro

graphy a10ne 86

21. Diagnosed surface tendency produced by orography

and 'drag combined 87

22. 300 mb ana1ysis for 122 10 Feb 1970 88

23. 500 mb ana1ysis for 12Z 10 Feb 1970 89

24. 1000 mb analysis for 122 10 Feb. 1970 90

25. 300 mb height change~ resulting from preliminary

ellipticization 92

26. 500 mb height changes resulting from preliminary

ellipticizati?n 93 vii Figure

27. Results from basic baroc1inic experiment: vertical

m.otion, vorticity and tendency for a small volume

of space 94

28. Surface chart for 062 10 Feb 1970 97

29. Surface chart for 122 10 Feb 1970 98

30. Surface chart for 182 10 Feb 1970 99

31. Diagnostic vertical motion accompanying observed

pressure distribution assuming saturated troposphere 100

32. Diagnostic vertical motion accompanying observed

pressure distribution alone 101

33. Diagnostic vertical inotion accompanying observed

pressure and moisture distributions with mountain

and friction effects also considered 103

34. Vertical motion from the w-equaÜon inc1uding

mountain and friction effects . 104

35. Surface pressure tendency from most complete

diagnostic model 105

36. Observed 12-hour surface pressure change 107

37. Exact and modified functions of the Coriolis 122

parameter used in baroclinic experiments

viii LIST OF SYMBOLS

a element of matrix A A' 2 x 2 or 3 x 3 matr ix

b element of matrix A B 3 x 3 matrix

c element of matrix A C condensation rate - cp specifie heat at const. pre 55. CD surfa.ce drag coefficient

d element of matrix A, D horizontal wind divergence

distance between grid points F unspecHiedfunction

e water vapour pressure f friction f orc e

f = 2.Qsin (l, Coriolis parame G time averaged geopotential,

g accelei-ation of gravit y, Geostrophic, Gravit y

subscript: geostrophic H mean depth of nuid,

h height of free surface heating rate

unit imaginary numbcr, K kinetic energy, Kelvin

index or unit vector for x L latent heat of vapourization

j index or unit vector for y pep* ) pressure (at the grcu!'\d),

k index or unit vertical vcctor polynomial

m map factor Q = f + f absolute vorticity

p element of matrix A R radhls of curvature, residual,

q mixing ratio Rossby, Gas constant

r clement of matrix. A 5 water vapvl:.r SOUl-ce or sink,

t time subscript: stationary

u velocity in x direction T temperature

v velocity in y direction, u velocity in x directio!l subscript: virtual v velo city in y direction

w weight W' vector wind

Ct x, y grid cartesian coord. W vertical velocity

z vertical direction,- height ·of constant pressure surface

ix Q wave number, relaxation factor, arbitrary factor

f3 arbitrary factor

y arbitrary factor

ô arbitrary factor, Kronecker delta

A difference

e: small number used as limit, arbitrary factor

~ relative vorticity about vertical axis

6 latitude

k = R/cp 1\ eigen value

V iteration inde.x, smoothing coefficient

1T product sign

cr _ p Ip*, vertical coordinate

é1: E dtr/dt, vertical velocity

~ summation sign

1> = gh or gz, geopotential

~ velocity potential

\fi tendency, stream function

(,J :.: dP Idt, vertical velocity

Il.. earth' s angular velocity

DIFFERENTIAL OPERA TORS

A. total differential a partial differ ential

V = i dlèJ x + j dIa y, gradient 2 ",2= dl Iè. xl + a Id yl, Laplacian

J(Fl,Fl) = èiF1/dx· dFl/O y - dFl/ë)y· àFZldX, Jacobian x RESUME

~ , L'ensemble des lois gouvernant la meteorologie dyna- mique se condense en une seule équation differentielle du second ordre

. " , .. " si on substitue la derivee du vent geostrophique a l'acceleration dans la loi du mouvement. On peut ensuite se servir de cette équation pour d~terminer le champ des trois composantes du vent à partir de la distribution de la. pression atmosphérique.

Un algorithme de solution relativement simple est développé et appliqué d'abord à une atmosphère barotrop~. Le tour billon du vent qui résulte est très semblable à celui que l'on obtient de l'équation de balance classique; quant à la divergence, elle est con- , . forme aux principes synoptiques. Les previsions faites a partir de ce vent sont moins contaminées par des ondes de gravité que lorsqu'elles " sont faites .a partir des vents non divergents de l'equation de balance.

On considère ensuite une atmosphère barocline repr~ sentée par cinq niveaux et l'on ~tudie séparement l'effet des montagnes, du frottement de surface, et de la chaleur latente libérée par la condensation. On retrouve les constantes classiques de ces situations et. de plus. quelques autres particularités de la structure verticale apparaissent. Enfin, le diagnostic de la tendance barométrique s'avère

\' ,; tres pres de la realite.

xi ABSTRACT

The set of primitive meteorological equations reduce s to a s_ingle second order partial differential equation when the accel eration is replaced by the derivative of the geostrophic wind in the equation of motion. From this equation a diagnostic three-dimensional wind field can be computed when the pressure distribution is known.

Calculations for a barotropic atmosphere are perforrned first; the rotational part of the wind is found to be similar to that from the classical balance equation and the divergent part agrees weU with synoptic experience. A forecast thus initialized contains less gravi tational activity than one initialized with the non-divergent wind of the balance eq uation.

With a five-level baroclinic atmosphere, the effects of mountains, surface drag, and release of latent heat are studied , separately; the results are conf;istent with classical·concepts and other interesting features are revealed. The diagnostic surface pres sure tendency is found to be in very good agreement with the observed pressure changes.

xii

do - 1 -

1. INTRODUCTION

. Early scientific study of the atmosphere was generally associated with a single observatory 50 that periodic daily and annual cycles were readily identified and formed the basis for both research and climatological prf;':diction. When it became possible to compare simultaneous weather observations from a nUlnber of different places other types of fluctuations of intermediate frequency with no well defined periodicities were recognized. The major role of large-scale travelling storms then became apparent and. as meteorological networks and communications systems developed. their study led to the rapid growth of synoptic weather analysis and prediction. We now know that the atmosphere is active on space and time scales ranging over many orders of magnitude and the specific phenomena with which we are concerned cover only a part of the whole spectrum.

The major weather systems are associated with deep atmospheric perturbations normally extending upward through most of the atmosphere' s mass and of spatial dimensions of thousands and ten 5 of thousands of kilometers. They differ from tidal oscillations J specifically. by having large amplitudes in the lower atmosphere and much slower phase speeds. Further research has shown that these perturbations, which we caU Rossby waves or simply meteorological waves. are solutions to the same c1assical equations as the tidesj they had been overlooked as mathematical entities until they were revealed by observations. - 2 -

Very simple relations were soon found to link certain

elements of the meteorological waves with high accuracy; in particular,

the pressure is in hydrostatic balance with the density and the Coriolis

acceleration is almost equal to the horizontal pressure gradient force

per unit mass. The latter state is c1assified as geostrophic; it is the

most important relationship in synoptic and dynamic meteorology and

has been.the subject of an excellent review by Phillips (1963).

About twenty years ago it was discovered that the m.ove-

ment of meteorological waves could be explained and predicted with

reasonable accuracy by measuring only one parameter, nam.ely the

height at which the pressure is 500 millibars, and by sohrïng only one

equation, the law of conservation of vorticity for a parce!. This was

the geostfophic barotropic mode!. Naturally, the interaction between

analysis 'and prediction soon pointed to forecast errors, which led to multi-parameter non-geostrophic baroclinic models and finally to. a return to the basic meteorological equations, referred to as the priInitive equations.

During the rapid expansion of numerical weather prediction it became evident that certain parameters, the vorticity for exam.ple, could not be observed with the accuracy required by the newer tnodels, and that the vertical component of the wind was too small on this scale to be observed at all. Faced with thisproblem, the numerical-dynatnical meteorologists turned to indirect methods to' calc ulate diagnostic winds, making use of the data redundancy implied by the known physical rela- tionships. - 3 -

The balance equation, relating the vorticity to the

pressure force is a consequence of the negligible divergence

or slowness of Rossby waves and the W-equation for the vertical

motion is a consequence of geostrophy. In two steps with these

relations we can produce a wind diagnosis from only two very simple

measurements in the atmosphere, temperature and pressure, and

thus minimize the need for the difficult and costly measurements of

synoptic winds. Sorne definite improvements in these methods are

necessary, however, before such cornputed winds can be accepted,

instead of conventional observations,in operational meteorc1ogy, even

though numerical weather prediction has had much success by using

wind observations in a very indirect fashion. This provides an incen

tive for further research in diagnostic methods in addition to the

requirement of primitive equations forecast models for very specialized and high quality initial data, particularly winds, which has prompted

several researchers to investigate and review an aspects of the problem.

For example, geostrophy, which has been the backbone of most theoretical investigations of the atmosphere has not been con sidered as a very sophisticated basis for full wind diagnosis. The power .of this most fundamental aspect of the meteorological waves will be exploited further in this the sis.

We will deal with the problem ·of c.alculating the complete wind field from a known distribution of pressure and temperature on the synoptic scale. Diagnostic relations are obtained from the meteoro logical equations by substituting ~he acce.leration of the geostrophic wind for that of the horizontal wind. Two cases are considered: Part A, - 4 - e a hornogeneous incompressible atmosphere with a free surface, based

on 500 rnb data and Part B. a baroc1inic "atmosphere with forcing

functions such as rnountains. surface drag. rad.iation and release of

latent heat.

In the first case, the computed wind is the basis for

initial conditions in a primitive equations forecast, which is compared

with the corresponding forecast starting from the non-divergent wind

of the balance equation. In Part B. the effect of each of the external

forces is studied but no forecast model was available to the author to

assess the quality of the diagnosed winds.

AH the calculations are based on a 280S-point rectangular

0 grid on a polar stereographie projection circumscribing the 13 N

parallel with corners extending to 9 0 S, Fig. 1. The grid-lcngth is

equal to 381 km at600 N and as many as five levels of information are

used to represent the baroclinic atmosphere.

Although the method to b<.. presented below is based on

- classical ideas, it is the first time to the author' s kn'owledge that this

diagnostic procedure has been applied successfuHy to a baroclinic

atxnosphere of such great horizontal extent and '.vith such varied forcings. -Fig. 1. The 51 x 55 point grid. Heavy lines indicate the corners of the enc10sed 47 x 51 point grid. - - 6 -

2. REVIEW OF DIAGNOSTIC METHODS

According to Helmholtz' s' theorem, the horizontal

wind vector can. be represented uniquely by the sum of a rotational. non-divergent part and of a divergent non-rotational part. Further-

more, under the hydrostatic assumption, which we shaH use

'throughout, the vertical velocity is a diagnostic quantity which can

be computed if the horizontal wind and the state variables are

known. Thus the complete wind field contains only two unknown pieces of information. For example, the pressure vertical velocity

( W:: dP /dt) is simply the pressure-integral of the wind-divergence calculated at constant pressure.

By scale analysis', Charney (1948, 1963) has shown that the horizontal divergence is much smaUer than the vorticity about the vertical axis (referred to simply as the vorticity) and that the vertical velocity is much smaller than the horizontal velocity, which is also an observed facto The scale for which this is valid is that of synoptic cyclones and planetary meteorological disturbances, as assumed throughout in this thesis.

2.1. Balance -W- Equations

Charney (1955, 1963) concludes that the balance equation can be used to relate the rotation:al part of the wind to the pressure force at aU latitude Si that is that the divergence and its time derivative can always be ignored in the divergence equation.

The balance equation states that the rotational part of the wind is geostrophic except in regions of high curvature. In fact the - 7 -

balance wind is very similar to the gradient wind. It is generally admitted that balance winds are about the best diagnostic non divergent winds; so they will be used as a standard in this thesis.

The .balance equation is a second order non-linear differential equation in two independent variables, usually written for the stream function. Its main disadvantage is an ellipticity

criterion appearing as a second order non-linear differential inequality for the unknown variable. The only practical method of satisfying this inequality is to modify the given geopotential (Bolin, 1955); this results in unduly large changes, especially at the level of the jet stream, which are detrimental to the ensuing forecast (Elsaesser, 1968).

Another serious disadvantage of this equation is that it is .practically impossible' to solve it in the equatorial region. Houghton and Washington (1969) have avoided this difficulty by making use of the observed vorticity in the low latitudes and solving

. the equation in reverse to get the geopotential there.

Finally, because of its non,-linearity, the balance equation is about three times longer to solve than a similar but linear differential equation.

The divergent part of the wind or the vertical motion can be obtained from the 0-'-equation, which assumes that the Laplacian of the thermal tendency \72( è; TI è) t) . is proportional to the tendenc y of the vertical der ivative of 'the vorticity 0la t( d 11-3 z). With this condition and ?J D/d t = 0, the complete wind field can be obtained by solving the UJ -equation together with the divergence

~: , .... " - ••••• ·C· •• ,,,. '" .--' •• - •• •• ~ __ - ~. - 8 .;.

equation. This diagnostic procedure is considered reasonable by

Phillips (1960) for initializing primitive equations forecasts. The

simultaneous solution of these two equations would be very time-

consuming and the author does not know of any forecast initialized

in this fashion. The current procedure is to use only the balance .

equa:tion and to set Wequal to zero, which appears to be satisfactory

for medium and long range forecasts.

2. 2. Geostrophic Diagnostic

The quasi-geostrophic nature of the wind is the basis

of the method used by the author. The two diagnostic assumptions

are d W Idt d W Idt, (W i u j v). The fields selected to carry = g = + the wind information are a stationary component W S. which is a

simple function of the geopotential only and the tendency of the geo

potentiat" 'èJf/J lèJt. The wind WS' which is not purely rotational, is

quite similar to the gradient wind; its vorticity is a slight under~stima~e

of the balance vorticity but its divergence has no particular significance.

The variable èf/J Idt, which affects almost only the divergence, is the'

solution of a complicated linear second order differential equation in

three independent variables.

The solution of this differential equation. as will be

seen later, is not part~cularly difficult and its convergence is rapid.

Unfortunately, the ellipticity of the equation and the solution of the

algebraic system for WSare governed" as for the balance equation,

by a second order non-linear differential inequality.

It will be dem,onstr3.ted inthë following pages that this 1 method which is potentially the fastest method considered here,

h~s the necessary quality in both the rotational and divergent parts 1

- 9 .:.

of the wind to initialize successfuUy short term primitive equations

forecasts including aU types of forcings, ev en the releasE' of latent

heat.

2.3. Miyakoda-Moyer Diagnostic

Miyakoda and Moyer (1968) have designed a method

of diagnosis in which the balancing assumptions are that the first

and second local time derivatives of divergence vanish (~D/() t =

02D / Oot 2 = 0). The tirne derivatives are approximated by the

Euler-backward method (Matsuno, 1966a; Kurihara, 1965), which

is a highly damping operator, especially for high frequencies.

The succe ss of their method is great. The vorticity is practically

identical to that from tpe balance equation and the vertical motion

agrees c10sely with that from the W-equation; furthermore, those

fields also agree with the exact ones, as "observed" in a general

circulation experiment. Forcings of aU kinds can easily be

accounted for. The only serious disadvantage of this diagnostic

method is that it is quite time consuming. Miyakoda and Moyer

diçl 120 time steps, each step involving the solution of two Poisson

equations, in order to achieve sufficient accuracy.

2.4. Nitta-Hovermale Method

FinaHy, Nitta and Hovermale (1969) have come up

with probably the simplest type of diagnosticmethod, in which no

constraints are imposed at aH. The rnethod consists in making

forward and backward time steps about the initial time with the

complete set of meteorological e~uation~. They use the modified - 10 -

(Kurihara, 1965; Matsuno, 1966b) Euler -backward time finite

differencing which allows the motions to eventually reach a quasi

steady state. In order not to lose completely the initial information

through damping, they arrange to restore the quantity which is felt

to be the best known, generally the geopotential. In their experiment,

whiéh they started with the "observed" geopotential and vorticity,

they were unable to recover more than about half of the "actual"

divergence and furthermore sorne of the details of the vorticity

were lost (their l?tandard was a general circulation experiment).

The diagnostic value of this method must then be considered as

marginal, at this stage. The computing time is also exc.essive

(Nitta and Hovermale did 150 time steps). This method nevertheless has the v~ry interesting property, apparently also shared by the

Miyakoda-Moyer process, that it does not have to fulfill an ellip ticity criterion.

Before ending this brief review, it may be mentioned that certain methods of objective analysi·s can render the diagnostic step unnecessary. Stephens (1965) has a~alyzed simultaneously the fields of wind and pressure. using the variational method to fulfill the balance equation as a constraint. His success was moderate and much work remains to be done before this method becomes satis factory. A spectral approach to filtered analysis and forecasting is also presently under study by Dr. T. Flattery at the Weather Bureau in Washington. - Il -

3. THEOR y OF INITIALIZA TION

To start a forecast (aU references will be to primitive

equations forecasts) initial data are required for each of the prognostic

variables, that is the pressure and the horizontal wind fields.

However, the type of initialization required depends a great deal

on the kind of forecast to be made and on its length. For example,

very long term general circulation experiments are independent of

the initial conditions. For medium length forecasts made with the

GFDL' s model, Smagorinsky and Miyakoda (1969) found that the

initial specification of the divergence, the surface pressure and

the moisture content were relatively unimportant. On the other

hand, the first to realize, dramatically!, that high accuracy in the

initial data is very important for short term forecasts was

Richardson (1922) who attempted to make a six-hour forecast using only observed data. In order to obtain more specific statements . about the quality required for the various components of the initial

. data, we will study a very simple model of the atmosphere from

which much insight can be gained. We assume, as is generally

the case at middle and high latitudes, that the geopotential is the

only accurately known parameter and that it must not be modified.

3. 1. The Simple st Model

Consider a quiescent layer of .incompressible homo

geneous fluid with a free surface of height h on which small

purely x-sinusoidal perturbations are introduced. This model is

representative of a barotropic atmosphex:e when H, the mean depth - 12 -

of the fluid, is about 10 lan, or the scale height of the atmosphere.

The governing equations are

dU f Olh = 0 ~t - v + g-- QI: ~x

ov + fu 0 (3.1) ô t =

ah + H~ 0 a t ox =

This system is easily shown to admit the .following three solutions

= L: wavelength

. t(f2 /a2 + gH)1/2 gH . - 2 2 1/2 ri> e1a [x+ (f la + gH) tJ o . (3. 2) where theIf:.so1ution, with a geostrophic structure, represents the meteorological solution (Rossby wave in a more general model) and is the desired one, and the two G-solutions are gravity-inertia waves moving hi opposite directions with phase speeds weakly dependent on wavelength (shallow gravit y waves).

By setting t = 0 in these solutions an arbitrary initial perturbation is seen to decompose itself into three modes as follows: - 13 - ft u iax iax v = f/J e R + G e o = + G+ f/J o f/J 1 1 1 1 3.3)

where -. = (fl / a 2 + gH) 1 /2 _ -if UR = 0, V V - R gH G agH

The amplitudes of each of the modes are found to be

= V - RVR + U - RUR 2VG 2UG

Thus it is seen that the amplitude of the meteorological mode is

independent of the initial specification of the divergence (in a more

complete.model this would only ~e approximately true (Hinkelmann.

1951; Phillips. 1960», and that an error in the specification of the

initial divergence will simply introduce gravit y waves.

If time is then turned on. the gravit y waves start dispersing 2 away with group velocity t gH (f2/ a + gH)1 /2. Since these waves

have high frequencies, they will be dissipated faster than the

.meteorological wave, provided that the right mechanism is available.

and eventually the meteorological wave will prevail.

Thus, forecasts long compared to the time scale of the

dissipation (of the order of a few days) can be expected to be meteor

ologically realistic (but not necessarily correCt) independently of

the initial errors in the divergence. On the other hand. forecasts

of the order of one day will remain noisy' due to initial errors in

divergence unless v'ery special ~easures are taken (e. g. use of the - 14 -

modified Euler-backward tÏlne steps). Nevertheless, provided that

these errors are smalI enough (ta satisfy the linear theory), the

meteorological forecast will be unaffected and it will be possible to

extract it from the total {orecast by means of a diagnostic procedure,

as wa_s done by Shuman (1960). Finally, if highly non-linear

interactions are considered in the model, such as irreversible

precipitation processes, the {orecast will be modified forever by

even moderate errors in the divergence.

Cçnsidering this geostrophic adjustment process,

which is the mechanism involved in the Nitta-Hovermale diagnostic

method, it is important to note that total energy is destroyed by

the dissipation of the G-waves, even though either some kinetic or

some potential energy (depending on the wavelength of the pertur

bation) c.an be generated. This bas been studied in greater detail by Rossby (1938), Cahn (1945), BoHn (1953), and Blumen (1967), but is already evident in the partitioning of the initial vector into the three eigen modes of our simple mode!. Thus it is necessary for the vorticity.which contains mast of the kinetic energy to be known fairly accurately at the start of the forecast.

We conclude {rom the above discussion that high accuracy in the initial rotational part of the wind is always important for an aCCl1rate forecast and that for the divergence, it is important not to make large errors if the {orecast is quasi-linear and to be accurate if it is highly non-linear; even more so if it is short.

(Throughout this discussion, the initial geo}?otential is the given parameter; any mo~ification of it must be considered detrimental). - 15 -

3. 2. Numerical Compatibility

Miyakoda and Moyer (1968) and especially Nitta and

Hovermale (1969) have insisted, with reason, on the fact that their

methods produce diagnostic data which are more numerically com

patible with the intended forecast model than the conventional

methods. By this they mean especially that the inevitable truncation

and evenrounding errors are identical in the diagnostic and in the

forecast models. These errors are normally small however, as

can be judged by the comparison made by Miyakoda and Moyer (1968)

between the results of the balance and (,J -equations and the corrcs

ponding quantities (vorticit)' and vertical motion) "observ:ed" in the

general circulation experiment. In the experiments related in this

thesis, a.high numerical compatibility was achieved by introducing

the diagnostic winds in the exaét finite difference equations of the

forecast model and by recomputing the pressure field through ·the

solution of the Poisson equation which resu1ts from imposing that the first local time derivative of the model' s divergence be zero.

With such a procedure, originally used by Robert (Kwizak and Robert,

1970), and later modified by the author, numerical compatibility can be achieved for any initial data, since only the model' s own finite differences are being usedj new boundary conditions can also be introduced easi1y on this occasion. The numerical compatibility procedure will be described in detail in section 8. - 16 -"

( 4. HISTORICAL REV~EW OF THE G ~EOSTROPHIC METHOD Z - In 1939 Phillipps, using"the equatioll of motion, showed that the wind W can be written as a series in terms of aU derivatives

of the geostrophic wind W g

which converges provided that the frequency of the motion is less than

the orbital frequency. (The special problems of the low latitudes are

dealt with in Appéndix B). Later on, Eliassen (1949) described sorne

filtered equations, based on neglecting aIl but the fir st two terms in

~he series, equivalent to letting d'Si /dt = d W g/dt or! approximately, Z Z d W /dt = O. (This approxirnation will be referred to as G whereas Z the usual·one-term expansion W ,y g could be labelled G ). He = 1 realized "that under this as"sumption, aU the meteorological equations

.. " reduced to a single mainly elliptic second order differential equation

for the tendency of the geopotential (he had pre,,-iously justified the

use of the pressure coordinate), but decided that it wasnot possible,

at the time, to solve it.

The approximation d W /dt = d W Idt was used by g . " Arnason, Haltiner and Frawley (1962) tq calculate the rotational part

of the wind only. "They noted that this wind is not significantly

different from a balance wind but that the big advantage is that G . . Z involves almost"a purely algebraic caiculatiori whe~eas the balance wind

requires the solution of a complicated differential equation. They

recogriized that, as with the gradient wind and the balance equation,

there is D"..:> real solution tmless ~he height field satisfies approximately - 17 -

2 2 2 v r!J> _2f as compared to :'0.5f for the balance equation

(neglecting the variation of f).

At the same time, Fj~rtoft (l962a, b) was making

proposaIs to solve the complete G filtered system and to use it 2 for forecasting, as originally thought by Eliassen (1949). A method

which he developed to solve mixed elliptic-hyperbolic type differ-

ential equations (Fj~rtoft, 19b2a), raising great hopes in various

centres, was used successfully (in a slightly modified version) by

Fjl.6rtoft and S~derberg (1965) to solve the differential equation.

However, the forecast was not good and so they reverted to the

standard procedure of ellipticization. o On a grid of 17 x 17 points, aIl north of 25 N, and with ten pressure levels, they managed to carry a forecast up to

33 1/2 hours, which compared favourably with forecasts from other . models. In this experiment, Fj~rtoft and S~derberg (loc. cit.) had to solve the diagnostic equations at each (1/2 hour) time step and'it is with this procedure that they had difficulty, being unable eventually. to apply the ellipticity criterion. These authors' results were in- conclusive as far as the forecasting ability of the G filtered model 2 is concerned and the purely diagnostic value of the model had not been studied.

Recent interest in the G geostrophic method seems 2 to stem from Hollman' s work (1966) which showed theoretically that it compared favourably with the balance equation as a diagnostic tool for a barotropic atmospherc. Korb, in 1967, performed an integration of the G diagnostic model for a barotropic 2 - 18 -

channel and found, by comparison with results app~oximated analyti-

cally, that it guaranteed the adaptation of the wind field to a given

pressure field.

Finally Woodroffe (1970), also working with a barotropic

atmosphere but now on an ordinary map projection and with real data, . made a comparison of several forecasts produced by various filtered

models, ,among them a barotropic stream-function model initialized

with balance winds and a three parameter baroclinic model; no

primitive equations forecast was considered. The forecasts from the

G z geostrophic model were significantly better than all other s except the baroclinic ones. However, due mainly to an inefficient method of

ellipticizing, it seemed doubtful whether this advantage was worth

the extra.computing time required for forecasting. The sllitability of

the method for initializing ,primitive equations forecasts was not

otherwise investigated.

Thus, apart from the unfinished experiment of FjqSrtoft

and SBderberg (1965), this will be the fir'st successful integration of

the G geostrophic diagnostic model in the baroclinic case. Other Z points of originality are: first comparative evaluation of the geo-

strophic diagnostic model as a means of initializing an independent barotropic primitive equations forecast; first use of this method on a hemispheric grid including equatorial latitudes; first introduction of variable topography; first introduction of initial latent heat sources. - 19 -

PART A: BAROTROPIC EXPERIMENTS

In order to te st the feasibility and to learn the technique

of integrating the geostrophic diagnostic system of equations over a

hemispheric domain with real data, a homogeneous incompressible

atmosphere with a free surface was first chosen.

5. BASIC EQUATIONS AND THEORY

For expediency, map cartesian coordinates as well as

reduced wind images V* = W lm will be used throughout this thesis, except where notedj furthermore, the * will be omitted everywhere.

Details of the transformation of the equations and of the differential

operators into map coordinates can be found in Haltiner, appendix A

(1968). The map distortion factor is denoted by m and its expression

for the polar stereographie projection used is

1+ sin 60 0 m= (5.1) • 1 + sin e

Assuming that the. wind is independent of height, the pertinent equation of motion is

2 d* W + W· W çm + fk x W = -'V~ (5.2) dt 2 where

" = gh, h the height of the free sur~ace and

. (5.3)

(this * will be omitted also) .. - 20 -

Thus. with the G diagnostic approximation d V/dt = d W /dt, the 2 g set of diagnostic equations becomes

è>~ +LUm.'l dUg (5.4a) at àl:.

()1lI9 + /.J...Nt?· ;) I\JQ (5.4b) at. 0 ~

"P= -rn 2". /J w (5. 5)

_ ~/J where lp == dt (5.6)

The terrn involving the variation of the map factor in (5.4) is very srnall so that the use of W is of no g consequence. We now use the definition

(5.7)

in (5. 4a and b) to eliminate the gradient of the geopotential and to express the tendency of the geostrophic wind in terms of the tendency of the geopotential as

(5.8)

Alter dividing by f (see Appendix B for the equatorial problem) and rearranging the terms we ob tain for (5. 4b) and (5. 4a) respectively

(5.9a)

(5.9b)

These are two linear algebraic equations. fo~ u and v. We can write their solution in rnatrix form by considering the vectors as columns - 21 -

(5. 10)

where = A -1 (~ + 0.5 W (5.11) g g

and

:) (5.12)

-1 The inverse A of A exists provided the determinant of A is non-

zero. that is

-1 2 P -2 4 -1 ac - bd = 1 + f m , + f m J (u • v ) T 0 (5. 13) g g g

2 where m (av /~x - du y) is the geostrophic vorticity. g g ra

lt remains to determine the parameter "fi in order to

complete the solution (5.10). This is done by substitutillg (5. 10) into

(5.5). giving rise to a secorid order linear differential equation with

variable coefficients for the tendency l.V ~

2 .1. -2 -1 2 .1 (5.14) m "V.rpf A 'V"/l-'f' = m'V' 'PW s

The boundary condition is simp1y tp = O.

This equation governs the meteorological frequency of

the height perturbations and is thus mainly responsible for the diver-

gence of the wind. The rotational part of the wind is almost completely independent of the frequency; its quality· is best assessed by comparison with the balance or gradient wind in the cas.e of a stationary circular system. We then have - 22 -

+ f R(V - V ) = 0 (5.15) v v g g

whereas the latter both give the exact solution

2 V fR(V - V ) 0 (5.16) + g =

provided that the Coriolis parameter is kept constant. R is the

radius of the vortex, V the tangential speed. The comparison was

made by Fj

Fig. 2a, b. The non-linear term in the balance equation is sometimes

evaluated geostrophically; in this case (5.l6) reduces to

V 2 fR (V - V ) 0 (5.17) g + g =

This form (dashed lines in Fig. 2) provides an interesting point of

comparison between (5.15) and (5.16). The three curves in Fig. 2a

are tangent and coincident at V If R 0 (straight flow). We note g = that (5.15) (dot da shed lines), is a good approximation to (5. 16) •

(full lines) even for sufficiently curved flow and also .that the limit

on anticyclonic curvature (related to the ellipticity of the full differ-

ential equations) is much less restrictive for (S. 15) than for (5.16);

this, however, is not necessarily a big advantage since the wind

speed is unreasonably high near the asymptote V IfR =-1. In Fig. 2b, . g the absolute value of the ordinate is the Rossby number, which is of

interest here since the geostrophic theory is normally restricted to

Rossby numbers smaller than 1.

A further point of comparison between the G geostrophic 2 and the balance winds is obtained by taking the divergence of (S. 4) which gives, whezi m = 1, e e

5 V/V • g Ca) 1 2 v/a · Cb) ·1 1· 1· 4 1,· 1 ·,. _.-...... -._.--.-.- ,. 3 \. -1 via 1 \. \ 3 \. 7 \ ,. \ 2 li. \ , \ li -1 \ ,\ li \ ,,~ 1 ! \ 1 ! 1 \ 1 . -2 .~ 1 \ ,,-...... , "'" ...... - .-.-.-.-._._._._.- ., ., -1 o " Vint , l' 2 3 Fig. Z. Two graphs of the tangential wind speed V as a function of the geostrophic wind speed V g. the radius of curvature R and the Coriolis parameter f. in a stationary circular system. Fulllines: gradient and balancegeostrophic winds. wind. Dashed lines: balance-equation with geostrophic non-linear term. Dot-dashed line: GZ - 24 -

dDg + DD = -V. (fk x \V) + J(I,l, v ) + J(u , v) = _\/2" _ (5.18) dt . g g . g 2 where D m (ou g g= g fox + av fày) is the geostrophic divergence.

Neglecting the terms on the left hand side, as is done ~or the balance equation, the only difference is that the latter has 2 J (u, v) instead of J (u, . g g v ) + J (u , v). Because of this, equation (5. 18) is linear and more often elliptic (subject to the above comments) than the balance equation (Hollmann, 1966); Korb (1967) has integrated it and found it acceptab1:e for non-divergent balancing. The details of the experiment and the results obtained by the author will now be related. - 25 -

6. NUMERICAL SOLUTION OF THE DIAGNOSTIC MODEL

In this section, only those details specific to the baro-

tropic model will be relatedi more general considerations can be

found in Appendices A, B, and C.

For efficient use of the given grid resolution, certain

of the intermediate variables were calculated in the middle of grid

squares (denoted by • ) instead of at regular grid points (+); the

functions f and m were calculated at either set, as needed. All

the finite differences were of the form (Al. 5)

OF -y dF -X -rvF , - l''VF Y àx X à y

thus transforming one set of points into the other. Definitions of the

difference operators are given in Appendix A. Cubic interpolation from the 16 surrounding + points was used to generate r/J.. Schema- tically, the calculations proceeded as follows

6. 1. Relaxation

The solution of the difference equation (analogue of

(5.14)) for If' was done by the Liebman z:nethod as follows (omitting the· ) - 26 -

2 .,.,\1+ 1 ='l'V + :-{m [(Tui~ +(~~)~ ] _ \JI") (6. 1)

where the latest values of 'fi. \Vere always used in evaluating u and

v according to (5. 10). The coefficient a is a relaxation factor

starting at a value of 1.3 for V = 0 and decreasingby' O. 1 after eaçh

ten iterations to a minimum value of 1. O. The weight given to the

central value of lJJ· by the terms within the braces is w. The relaxation was terminated when the change in "P. from one iteration -5 2 -3 to the next was less than 10 m sec , equivalent to better than

one tenth of one percent of the amplitude of 'V. itself. Approximately

fort y iterations were needed.

6. z. Ellipticity

It is known that the Liebman iterative process will

converge provided the differential equation is elliptic. NumericaUy,

this means that the matrix of the coefficients of the second order

differences of the dependent variable must be positive definite, that

is have all positive eigen values (aU negatives are also acceptable).

The conditions under which this is so are derived in Appendix C

for a three-dimensional system. A corollary is that in the two-

dim.ensional barotropic case, the sum of the diagonal elements and the determinant of the symmetric part of the matrix must be positive.

The m.atrix we are concerned with here is A -1 and the two conditions are easily found to be

(6. Za) and - 27 -

2 = l + m f-l > + m 4f-2 J (u v ~.-25f-2m 4 2 g . g, g D g >0

2 2 = 0.25 r(Z + m f-I r )2~· f- m 4Def il,) 0 r g g-J

(6. lb) where

m 4nef 2 = m 4(ovg + oug )2 + m 4(~ug _ è)Vg)2 g ox <3y ox oy is the geostrophic deformation squared. Thus, cOllsidering the relative smallness of D 2, we find that g the geostrophic vorticity must at least be greater than minus 2 f , and greater still when the deformation is large. In the light of the stationary circular flow model, Fig. 2, we see that (6. la) serves to choose th~ meteorologically - interesting branch of the solution whereas (6. 2b) establishes a

minimum departure from the asymptote (V IfR -1) g = r·equireà for (5. 13) anyway. 50, in view of the unrealistic behaviour of the solution. near the asymptote. it seems futile to try to approach it r.tlore by . solving the mixed elliptic-hyperbolic problem.

The ellipticity conditions were applied in two different

steps as follows: First, (6.2a) was enforced by· solving

as a Poisson equation by the Liebman method. The boundary condition was that the normal gradient of the correction vanishesi this in tUl'n imposes the integral constraint that the a,rea averaged change in the right hand side be zero. Thus the original normal gradient of the geopotential is pr~served near the boundàry and the changes made in - 28 -

the anticycionic regions are compensated by very small opposite changes over the whole area. (An example of the kind of change produced is presented in connection with the case of the baroc1inic

model, Fig. 26). The value of a was 1. 5 in this experiment.

The condition (6. 2b) is a non··linear second order differential inequality for the geopotential. Woodroffe (1970) has chosen to solve this inequality carefully, which is about equivalent to solving the complete balance equation. Arnason, Haltiner and Frawley (1962) simp1y changed the values of a, b, c, d without making the implicit changes in other quantities. Fj~rtoft and Sl1derberg (1965) remarked that the actual geopotential is rather insensitive to changes in its second derivatives and that the incon sistency thus induced should not be too large. In this experiment tJ).e following was done:

i) a > El

c > El

ii) a' =, ~a, c' = ~c, b' = b/~f d' = d/~

where ~ is s uch that

a'c' (6.2b)

The values selected were fI = 0.01 and C ,= 0.4. From 2 300 to 500 points, mainly at very low latitudes, were found to violate either (6.2a) or (6. 2b). It turned out in the results that the final absolute vorticity was sometimes negative, a condition which is considered unrealistic at 500 rob by experienced synopticians. Changing the - 29 - limits , and had very littl~ effect on this ph~nomenon so it t.l Ez seemed that either the coefficient a must be decreased in (6.3) or condition (6. 2b) must be solved more exactlYi the former procedure was retained for its expediency in the baroclin~c experiments. 30 -

7. HIGHER ORDER G APPROXIMATION 2

It is evident in Fig. 2a that the present G approximation 2 gives an underestimate of the rotational wind. This could be corrected by using o'Y lèJ t = ê),v lb t instead of total derivative·s. - g One can see from (5.18) that this willlead exactly to the usual non-linear balance eq uation. This modification can be achieved iteratively by replacing W. 9 O \V by W Y+ 1 • \1 W'J with W = W • It is eas y to g g show that at least in the stationary circular case, the iteration converges for V If R -0.25. g > In fact, with V = 2 the solution is already within five percent of the true (gradient wind) for -0.22 < V IfR 2. This g < higher order approximation was easily tested by replacing '.y • by \V -J g interpolated c ubically to the • points after each iteration. This line of research was not pursued very long since it seemed that the extra accuracy in the vorticity did not warrant the threefold increase in computiug time required. The \ idea would become much more intere sting if this technical disadvantage could be avoided since it would permit the introduction of all types of non-linear or implicit forcing functions which must be approximated otherwise. However, this has been left for future research. - 31 -

8. NUMERICAL PR OCEDURES FOR INITIALIZA TION OF FORECASTS

A forecast model of the barotropic atmosphere using the primitive equations for a homogeneous fluid with a free surface was available to the author for testing the initialization. This modél has been integrated by Kwizak and Robert (1970) and Kwizak (1970) who described it in full details and also presented a method of ensuring initial numerical compatibility of the data produced by the

CAO operational balance equation program (Asselin, 1967). This method will be refined here and extended to divergent initial data.

8. l. Compatibility Procedure

The exact finite difference equations of the forecast model, aëcording to Kwizak and Robert (1970) aloe

U.t + .1. 2t = Q. V XY _ K~ t Y/X (8. 1)

(8.2)

(8.3) where

2t F = 0.5 [F (t +6 t) + F (t - L.Hfl (8.4)

F = F· XY (8.5)

(8.6)

(8.7) - 32 -

~ = çp +~', çp average of ~ (8.8) o 0 '

The implicit formulation of (8. 1), (8. 2) and (8.3) is due to Robert

(1969) and is detailed in the papers of Kwizak, and Kwizak and Robert

cited earlier. Compatibility between the various elerrients of the

initi~l data is achieved by requiring that the initial tendency of the

wind divergence D, as computed by the forecast model, vanishes.

Namely, we want

aD -_ 0 .at t = 0 (8.9)

This constraint is fulfilled analytically (but usually not numerically) by balance winds; for the geostrophic G winds it is only approximately Z fulfilled, even analytically, as can be seen from (5. 18). Other filter ing constraints could be impose,d also (e. g. dD/dt = 0 was tried), but the important aspect of the' numerical compatibility is that a large amount of the gravitational character of the initial data, as the fo.recast model sees it numerically, must be elim,inated by the constraint. The latter is imposed on the initial data by a kind of reverse balance equation, at the cost of minor modifications to the geopotential. The full method to be described now is new and of general applicabilitYi for example, it could serve to calculate compatible fields of geopotential for wind fields obtained by any process whatsoever. The details of the description are naturally specifie to the actual forecast model available.

The compatibility condition (8.9) can be written

(8.10) - 33 -

since the first time step of the forecast model is a forward-implicit

one. Thus, applying the finite difference divergence operator to

(8.1) and (8. 2) we find, in view of (8. 10) -_._----_.. y . X XX GYY+G (Q'VXY -KY) - (Q'UXY +Ï0) (8.11) XX YY = X X . Y Y

The boundary condition is (8.25) given in the next sub-section. This

Poisson equation is solved by relaxation for the average geopotential

which is precisely the quantity needed for (8.1) and (8.2). However, t/J 0 is required for the time extrapolation of (8.3). We must then t = consider this equation, which is .precisely in terms of tP = 0 and t "t = Cl t but is unfortunately implicit. The author has not found a good m.ethod of solving (8.3) and (8.12) exactly for 0' even tPt = under (8.10) and has instead assumed that

(8.13)

Upon substitution of the finite difference equiva1ent of this into (8.3) ". ~ . and considering (8. 12) we find

This completes the process. If the time step were very small we might be tempted to use the time averaged geopotential G instead of tP 0 from (8.14). However,. a rapid analysis of the effect of t = this procedure shows that it effectively introduces a smoother ~n the divergence as follows: - 34 -

D = [1 + O. 25 ~o ~ t 2\l 2J D (8. 15) t=6t L1-0.25~0(jt2\il t=Q

The effect of this is very notice able with 6t = 1 hour in that the sign

of the small scale divergence is reversed from t = 0 to t = ~t, and

this oscillation naturally continues in the fOl'ecast •.

~orecasts made directly from the balance wind and

the corresponding ellipticized geopotential, or from the geostrophic

winds and the partly ellipticized one (i. e., according to (6.3) only), were so severly contaminated by short gravit y waves that the meteorological forecast had to be obtained by smoothing. With the initial compatibility procedure described above, however., despite the minor approximation (8. 13), gravit y perturbations were never of significant enough amplitude to show in the geopotential. (Forecasts of up to 72 hour s were made with the geostrophic initialization and as long as 25 days with balance winds by Kwizak (1970).)

8. 2. Imposing New Boundary Conditions

The given initial wind field does not necessarily satisfy the model' s boundary conditions. The forecast mode1 used a 47 x 51 point grid centered on the 51 x 55 point one (see Fig. 1). It had solid wall boundary conditions

U· = 0 (8.16) -x-x f V (8.17) ~ X = and computational conditions

u:X = V =·0 (8.18) X - 35 -

for walls along Y and symmetric,ally for walls along X. Condition (8.16) is satisfied if the stream function \fi and the velocity potential ' ex obey the following

'1-'. 0 = (8.19) .

~X = 0 (8.20)

on the wall, which is halfway between the outer two rows of the grid, where

u = -'{'.~ + ?~ (8.21a)

V = ~. i + 'X. ~ (8. 22a)

Thus

Y li? xYx + \11· XyX = "v y X T T y X - Y U' (8.23)

This Poisson equation is solved for lV· by. relaxation with 0/. = 0 as the boundary condition. However, the forecast model requires the wind components on the • points for the time extrapolation of (8.1) and (8.2). Consequently, we use cubic interpolation to pass from 'P. to using -X 'fi' ~ = 0 as a computational boundary condition in order to generate a complete field of 47 x 51 values. Fol' the velocity potentia1, we proceed in reverse order. We compute the divergence

of the given wind field from (8. 21a and 8. 22a)

'V. yy + ~. XX = U y + V X "XX ",yy X Y (8. 24a) and interpolate it cubically to all internaI grid-points of the 47 x 51 - 36 -

net. Next we solve the Poisson equation for

-YY -xx fnY -X) ex XX + "'X YY = cubic interpolation of \U X + Y Y (8. 24b)

with the boundary c.ondition of the second kind 'XX = o. This requires

that the average divergence be set to zero, a minor change. Thus a

complete field of X is also obtained. The desired initial wind is

simply

(8.21 b)

lJJY +yX y' = ,.,X y (8. 22b) which is made to satisfy (8. 18) upon transformation to the regular

grid-points by (8.5). The advantage of this elaborate procedure over that used °by Kwizak (1970) or K,wizak and Robert (1970) is that the wind field retains identically its original divergence and vorticity

(except for errors due to the cubic interpolation which is a fourtn. order approximation of the Taylor series expansion) while being sub- jected to new boundary values. The other boundary condition (8.17) is used for the relaxation of equation (8. Ü) of the compatibility procedure as

-X-X = f Y (8.25)

After reading this section 8, one is certainly in a position to appreciate the advantages of the Miyakoda-Moyer and

Nitta-Hovermale methods at least with respect to the very important problem of numerical compatibility. - 37 -

9. DISCUSSION OF RESULTS

9.1. Divergence and Tendency

,Calculations were actually made for four different 500 mb height analyses but only the analysis of 002, 21 February 1969 was used with the final version of the model, as described. This analysis is shown in Fig. 3 in full lines and the diagnosed tendency has been superposed in dashed Iines, scaled 50 that it can be read as a 24-hour height change, in meters. The line of zero . tendency is seen to pass through aIl troughs and ridges, with faIls ahead of troughs and rises behind as expectedi the magnitudes are aiso in agreement with synoptic experience. In the barotropic model, the divergence is almost exactly linearly related to the tendency, so that the same comments wouid apply to this quantity.

A good method of assessing the quality of a diagnosed wind field is to compare it with "observations" taken from a gen~ral circulation model at a time when it is certain that the "observations" do not contain any gravitational activity. Even the long integrations of Kwizak (1970) did n'ot possess this quality, as ·evidenced by highly periodic exchanges between the potential and kinetic energies, and the author had no other convenient source for such data. Figure 4 is taken from Kwizak (loc. cit.), page 109; ·it shows clearly two types of energy exchanges of about 13 hours and 13 days. However, an indirect measure of the quality of the initial data, at least with respect to content of gravitational energy, is precisely the amplitude of this exchange during a forecast.

The semi-implicit model of Kwizak (loc. cit.) was integrated ~o 72 hour s py 1 hour steps using either balance initial data - 38 -

:o·~=~-_

...

"1.. . : ~.: ..... -.:...•. :' ...... o.".' "...... ~.. . ",. . . .. :: ... ;...... : .. ( . 0' ...... ";'" -"... :.",/ . ---:.--. .:,.,' ..J. ~'

.. .. ~' 0° • "...... 1" .... + .-' . -0,>" .'. ./" " .. , . ;,' ...... , ./ . , . './ . -

Fig. 3. Fulllines: 500 mb ana1ysis for 002 21 Feb 1969 (interva1 5 dkm). The dashed lines are the diagnosed geopotentia1 tendency (interva1 100 m (24 hl-Il; the Une of zero tendency is the thick full line. 1

10 ;..3

.20

X xr~

for

energy

A4X

i~l

2.45)(10 0.0

2.45)(10 5560.9 5560.5

1 i

,-15560.1 ]3.

scale

600

squared

potential

___

right

and

top

the (left)

available

and

500

pertains

vorticity

hOtJrs~

energy left

top

1 -

sec-

--L_--.-:...._---:L-_~-~.LJ

400

absolute

kinetic (m

for

Scale

height

squared

,----

for

scale

(1970).

...J

300 are

divided

___

Kwizak

and

bott.om

Themiddle

area

200

scales

sec-

times

integration -...

_-1.-_:'_",

The

(rr2

day

(right)

TE.

~__

100

volume

energy

~ times total Summary divergence

--~---r~~~- -~.~

___

,..-::------ï~----~~----..:._:; [ t- ~ ~"" 1 1

Fig. L

1· r~' L li Il ,

l~~ ~

t9! 21"

"0 2

xie

IXI

6 7.

-'.6xIO 6.6x10r

1.85x10 lSOxlO 1.75x10 1.70x!O

3.S8xlcf'

3.64xI0

·3.S0XI6~

. - 40 -

introduced exactly as he describes, or G geostrophic Z winds computed by the author and introduced as shown in the previous section. Fig. S is a plot of the variation of the available potential energy with time for these two cases, while Fig. 6 is a plot of the variation of the squared divergence in the same runs. (The total energy of the model is con served accurately so that the variations of the kinetic energy give no further information). The curves for the balance initialization are blow-ups of the first 7 Z hours of Fig. 4.

ln. Fig. 5, the scale is the same for both curves, but. not the origine Even though the same height field is involved, the available potential energy (the variance) is different because of the effects of the ellipticization, compatibility procedl1re, etc., to be discussed later. The important point about Fig. 5 is that the use of G initial Z data has reduced significantly the undesirable gravitational activity in the forecast. Both forecasts were initially compatible to the same degree since both satisfied the forecast model' s own reverse balance equation; consequently, the improvement must be due chiefly to the reasonable quality of the initial div:ergence of the G geo Z strophic winds.

The squared divergence in Fig. 6 indicates that relative1y high amplitude perturbations are generated initially by the non-divergent balance winds. This activity eventually settles down to oscillations of about ZOO units around a mean of 1063. With the divergent G Z winds, the initial surge is hardly noticeable and the squared divergence has a lower average of 885 units and an amplitude of 150 units. -- -

19 .6.84xl0 ~AE '\ \ 6.80 - 6.76 r 1 6.72 HOURS f

....~ 6.68 1 (6.12) r

6.64 (6.08)

6.04

6.00 Fig. 5. Time variation of total available potential energy (AE in Fig. 4) in barotropic forecasts with balance initialization (top) and GZ geostrophic initialization (bottom). 1

and

(top)

initialization

balance

with

Fig.

(bottom).

divérgence

squared

initialization

variation

geostrophic

Time GZ

1 1

1 1 1

Fig.

1 , 1 1 k,2 1 \

1 \ 1 r- I r . 1 1 t-

.1 1

500

885

2500rl0-6 500

2000

1500

1063 1000

1000 - 43 -

9. z. Vorticity

Of·course, a barotropic forecast is dominatcd by

vorticity, not by divergence. As shown in the section on theory, the

meteorological par.t of the forecast is almost independent of the initial

divergence. After the application of the boundary c·onditions and in

ter~s of the forecast model' s finite differences, the initial absolute

vorticity from the balance and G winds were as shown in Figs. 7 Z and 8. The high similarity between the two fields is evident. In

view of the theor.y"for stationary circular flow, it would be expccted

from Fig. Za that the G absolute vorticity would be everywhere Z weaker than the balance one but might reach negative values in

'regions found hyperbolic by the balance eql1ation. This is indeed the

case if tl}e balance vorticity is computed directIy from the stream Z . . function as f + m .( ~)Xx + 'f y~-) and the G Zone (at the ; points) as (+ m Z (v i -Ü~). However,· the vorti-dties in Fige.7 and 8 have been calculated after the initial data had been introdl1ced in the forc-

cast model, and in this process more. tr·uncation errors (specifically

a ()-XY operator) have been imposed 0D: the balance data than on the C z data. This difference in truncation error is in fél.ct sufficient to . render the highly cyclonic centers of the balance wind actually weaker than the correspondin~ G ones. Z on the other hand in the regions where the balance equation would .normally be hyperbolic, .the ·G vorticity i~ slightly Z too anticyc1onic; it is believed that this could ~e avoided by using a smaller a in (6.3). At very low latitud.es. the balance equaUon program used a fudged Corfolis parameter, so that it is not surprising - 44 - J ---" "',/ /' '.

Fig. 7. Absolute vorticity from balance eguation as interpreted by forecast model (interval 2 x 10-5 sec-1).

r-'- - 45 -

Fig. 8. Absolute vorticity from GZ diagnostic as interpreted by forecast model (interval Z x 10-5 ~ec-l). - 46 -

that the vorticity should be smoother than the G one. Since a close Z scrutiny of these charts is not possible because of their format, let

us mention that the maximum difference between the two at any grid

point is about one unit (10-5 sec -1) in the cyclonic centers, four

units. (in the opposite direction) in the anticyclonic areas at very low latitudes and two units north of 300 N.

9. 3. Height Change s

It remains to discuss the modifications which had to be made to the original geopotential in order to ob tain these results, that is Figs. 9 and 10. Considering that the geopotential was the only given parameter, these modifications must be regarded as error s. It would have been very difficult to present the error s from each of the sources separately. The figures show the differences between the original analysis and the height field recovered from the incompatibility procedure; they include aU errors.

In both charts, the dominant feature is a lowering of the heights in the top left and bottom right of the grid, reaching or exceeding 5 dekameters. These changes, as well as smaller scale but as intense ones of either sign around the boundary are due in large part to the imposition of solid wall boundary conditions by the forecast model. This artificial condition requires the vanishing of the height gradient along the wall in order to maintain geostrophic balance. The slightly larger changes in. the geostrophic case arose from the low-latitude smoother.

The balance case, Fig. 9, is then seen to contain several other areas of decreased heights, not reaching 5 dkm. These - 47 -

Fig. 9. Height difference between original analysis and height field compatible with balance winds in forecast model (interval 5 dkIn). Isolated number s indicate the large st change in the vicinity.

-.1 - 48 -

Fig. 10. Height difference between original analysis and height field compatible with G Z geostrophic winds in forecast model (interval 5 dkm). Isolated numbers indicate the largest change in the vicinity. - 49 -

changes are the result of the ellipticity criterion of. the balance

equation. It is interesting to note that such negative centers prac-

tically do not exist in the geostrophic case. Fig. 10. The reason for

this is the following: only a weak ellipticization was d~me during the

G diagnostic (a 1. 5 in (6.3». so that slightly too strong anti 2 =

cy~lonic winds and vorticities resulted (refer to Fig. 2a). When the

reverse balance equation of the compatibility procedure was solved.

these were translated into highly antic}'-clonic G geostrophic vor l 1 2 ticities (f- '1 q, large and negative). that is higher heights. This

compensation process is in part also the cause of the positive changes

slightly exceeding 5 dkm 011 the anticyc10nic shear side of the wind

maximum in the mid Pacifie. A more severe ellipticizatioll by (6.3)

could possibly be more appropriate in order to obtain the best

possible compensation.

Many other errors are present in these maps. but are

of too small amplitude to be shown by the coarse contouring interval

of 5 dkrn. They are due mainly to the different finite differences of the diagnostic and forecast models (numerical incompatibility) and to relaxation errors. An example of the relaxation error alone due to the balance equation is shown in Asselin (1967).

It is the author' s opinion that the accuracy of the compatible initial height fields from either method is sufficient as long as the forecast model uses solid wall boundary èbnditions. If this is eventually changed to a more realistic condition. the error s due to ellipticization of the balance equation and to· the underestimate of the geostrophic method will also have to be reduced; the preceding theory and results offer sorne c1ues as to how this could be done. We will - 50 -

close this discussion by placing the initial errors in the perspective . . of the forecast errors, which are after aH much larger than, although admittedly partly caused by, the former.

9.4. Forecasts

In Figs. Il and 12, the 72 hour forecasts from the balance and G geostrophic 2 winds are presented; the verifying analysis is shown'in Fig. 13. The forecasts display the typical barotropic skill; the effect of the initial errors is not outstanding even near the boundaries and no ver y substantial difference is seen between the two. The results of Woodroffe (1970) may be recalled at this point, which demonstrated a statistical superiority of the geostrophic filtering over the balance one for barotropic forecasting. The results of th,ese barotropic experiments lead to the following conclusion. The geostrophic method is fast and reliable; it is at least as good and may prove superior to the classical non- divergent balance approach. Further developments seem desirable. - 51 -

Fig. Il. 72-hour height forecast from balance initialization (interval 5 dkm). - 52 -

Fig. 12. 72-hour height forecast from G initialization (interval 5 dkm). 2 - 53 -

Fig. 13. 72-hour verifying 500 mb height analysis, OOZ 24 Feb 1969 (interval 5 dkm). - 54 -

PART B: BAROCLINIC EXPERIMENTS

After the encouraging results obtained with the baro-

tropic model an extension to the baroclinic atmosphere, using

essentially the same techniques, looked promising. In view of the importance of external influences on vertical motion, orography would be included through the Phillips' (1957) vertical coordinate and provision would be made for considering all types of sinks or sources, among them the release of latent heat. The calculations would require enormous amounts of computer storage so that five levels seemed about the maximum vertical resolution which could be handled effi- ciently by the machine available, an IBM 360-65 with approximately

50,000 words for internaI storage and programs and 500,000 words of fast direct access storage on drum. Preliminary experiments were performed with two and three levels but only the final five-Ievel ver sion will be referred to in this report.

10. BASIC PRIMITIVE EQUATIONS

The equation of horizontal motion in Ci -coordinates

(Phillips, 1957) and on map projection can be written

d \V + fk x W = -V'QS - RT'VlnP* + m -I/F (10.1) dt 2 where d = 0 + m W'V' + ër d (10. 2) dt ot dcr

W = i u* + j v*, u* = ulm, v* =v lm: reduced wind images (these * will be omitted)

,...---v P/P*, P*·. sur f ace.pressure (10.3) - 55 -

m: the map scale factor

à::: cier = _l_(dP _ 0"' dP*\ (10.4) - dt p* dt dt - )

\l is calculated at constant cY on the projection and

IF is the friction force (per unit mass)

(It is understood that P or p'* are scaled by 1000 mb when they occur

in logarithms and exponentials). The equation for the vertical Dlotion

is reduced as usual to the hydrostatic equation

-RT (10.5) 0-

The equation of mass continuity can be written immediately in the form

in which it will be used

V' ~ p* 0/ + ~ p*w = 0 (10.6) Ocr'

where

(10.1)

Finally, we need an equation of thermodynamics

dT RT dP = H cp dt - P dt (10.8) where H is the external heating rate per unit masse Since in our diagnostic model aIl state variables including moisture content are

1!.nown and only the wind field is unknown, it is convenient to divide the various types of atmospheric hcating into those which are dh"ectly dependent on the wind field, and those which are note The latter Dlay - 56 -

he calculated immediately but in the first group precipitation pro- . . cesses de serve special attention.

10. 1. Precipitation Model

We will use the mixing ratio q for which the conserva-

tion .within an air parcel is governed by the equation

dq = -C + 5 (10.9) dt

where C is the condensation rate and 5 the rate of change due to other

sources or sinks.

The mixing ratio itself is defined hy

0.622e q = (l0.10) P-e

where e 1s the water vapour pl:"cssurc. At saturation, the vapour

pressure is known experimentally to he a function of temperature only,

so that the saturation mixing ratio is a function of pressure and temp-

erature. !ts total derivative is

dqs = ~qs dT + àqs dP (l0.11) dt bT dt èp dt

When precipitation is occurring, (10.9) and (10. Il) can he equated

giving

C = 5 _ Oqs dT è) qs dP (10.12) 'DT dt - 0 P dt

Heat ic; then released at the rate LC, w~ere L is the latent heat of vapourization here (freezing is not considered). Thus the total heating rate may be written as

H = LC' + Hn (10.13) - 57 -

where Hn is the heating rate due to sources other than precipitation

and C'is the actual condensation rate given by

C' = oqqs max (C, O.) (10.14)

It is customar y in large scale dynamics, and there are good physical reasons for it, to consider the phase change from vapour to liquid water as an irreversible process whence the maximum function in (ID. 14) which makes the condensation rate non-negative. The Kronecker oqqs ~ndicates whether saturation exists or note

When condensation is actually taking place, C' = C, which means that it can be expressed in terms of the "pr.essure vertical motion" W=:; dP/dt by solving together (10.8), (10.12) and (10.13), giving

c = C dP CI - 2 dt (10.15)

where s- ..&. è)qs ~p i3T CI = (10.16) , T J:... âqS 1?r oT ~,. 'Oq~ C op· ..coP èJo" 2 = (10.17) 1 .,.~.):.p ~

Thus C'is a discontinuous function of W through (10.14) and (10.15).

In a forecast model the decision (10. 14) can be made correctly by evaluating the local rate of temperature change from (10. 8) both ways and selecting the higher rate. This type of decision . is not practical in our diagn~stic mode!. - 58 -

On the other hand, . it is an observed fact that the moisture

distribution in the atmosphere is highly correlated with vertical motion.

In particular, it would be unlikely to observe "saturation" (this term

will be qualified later) in an area of large scale desce~t. Thus the

decision involved in (10. 14) is in a sense implicitly made by the Kron-

ecker delta. 50, under the assumption that the analyzed moisture

pattern is representative and that the geostrophic diagnostic model

will give reasonable estimates (at least in sign) of the atmosphere' s

large scale vertical motion, the precipitation process may be treated

as reversible and equation (10. 14) replaced by

C' = 6qqs C (10.18)

To make .our model more realistic, we further assume,

following Danard (1964) that the generally small scale moi sture pattern can be parameterized in terms of the grid-scale field by letting the. original Kronecker 6qqs vary linearly from zero to unit y for relative humidity between a lower critical value of 80 per cent and full

saturation. 50 the final temperature equation is

dT = _1 [R T _ 6 LC ] dP + ~ r H + 6 LCl (10.19) dt cp P 2 dt cp L D !J

The information required to evaluate q, qs and its total derivative in (10.10) and (10. 11) is contained in a table of the saturation vapour pressure over water (~nd over ice for T less than o -40 C) taken for each 5 degrces of temperature from the 5mithsonian

Meteorological Tables (1951). - 59

11. DIAGNOSTIC EQUATIONS

Equations (10.1), (10.5), (10.6), and (10.19), together

with the definition of cr (10.3) contain four prognostic quantities u, v, T, and P * • By introducing the diagnostic assumption d V Idt = d W /dt these are brought down to two which in fact, through the hydro g static assumption and the boundary conditions reduce to one quantity

01n =df/> + R T P* = Cd f/>), (11.1) \.fi - ot ct è) t P = const

which will be referred to simply as "the tendency".

In O"-coordinates, the geostrophic wind is

(Il. 2)

from which we easily obtain, using (11.1) that

1 aW g = C k x rV \jJ-0\7lnp* Ntt' _ dlnP* (\7RT -O"'V'lnP* èRT ~ (11.3) o t L Ocr ."0 t OCT :J

and

-1 ~ = -f k x (\7RT _ôVlnp*uRT ) (Il. 4) 0- 00"'

so that the diagnostic equation of motion may take the form (Il. 5) and 2 (11. 6) shown on Plate 1. The term 0.5 \V • \V\jm had been found

to be relatively unimportant in the barotropic experiments so it was not considered. As for the friction force, the linear a1gebraic nature

of equations (Il. 5) to (11.7) would be lost if it was other than linearly related to the wind. The dominant part of this force is generally considered to be in the boundary layer where 0' o

11.5

(11.6)

(11.7)

()

for

êJ6

equations

'X.

é)

aigebraic

_!Z-

è>:x..

linear

~I-!!I1

·d

..u~_r~~_ctd~~P*~.1.)

,v~-tt~-o-)#t~~)

three

\AI"

set

75-;:

form

CJ6G

+)-~'2.d(\/~l

--t~/t;

,.y-

equations

d'jï

-IlIrm~l)1\Ï

diagnostic

+Cf>1)Ûle!)N-~yd<#-&i(/t1)W=-

t-Imt"~i)

GI~P"

j)-'r~t'2.dN-~

-v (/-

baroclinic

-r

The

JJ..-"J(tP~!K

kt.-t iÂ

)

ô'X

û'X

Plate

+C~~J.t)·

•

o7..p

((+-8-';w."~ \

-t(n.'-?;;t~I~'{,J)

-11t~ - 61 -

2 IF = - gP fIL m 1w 1 W (11.8) RT P

in which CD is a surface drag coefficient. 1\V 1 is the magnitude of

the reduced wind image \V in the boundary layer. which is of thick- ness A P. In (11. 5) and (11. 6) the friction force (11. 8) has been linearized to

(11. 9) where

c.; = gP* CD RT AP in the boundary layer and ':fr= 0 elsewhere. The surface drag coefficient used was Cressman's (1960)j the orography was also taken from the same source. Both of these fields were extended to the 51 x 55 grid by Mr. W. S. Creswick of the CAO who went back to the original data of Berkofsky and Bertoni (1955) for the mountains and used •

Cressman' s technique to derive the drag.

The temperature equation could be written immediately by expanding the total derivatives of (10. 19) but it is preferable, in order to minimize the tr'.lncation errors which will arise in the finite differences, to first transform the space derivative s as follows

(Il. 10)

OT RT _ k 'l' a (lI. 11) aa - CpCT' ::::0' OC! ~ where k = R / cp. These f01."ms are equivalent to using potential - 62 -

temperature. As for the temperature tendency, it can be transformed as follows

(Il. 12)

by using (10.5) and(ll.l). Thus, in view of (11.10), (11.11), (11.12) the original equation (10. 19) transforms into (11. 7) of Plate l, where

(11.13)

HM = oLe l (Il. 14)

The three equation (lI. 5), (lI. 6), and (Il. 7) contain the three unknowns u, v, w, in terms of the tendency \fJ' which is a parameter. After sorne simple manipulations, we can write as in the barotropic case

(11.15)

where

\\'"3 = iu + jv +kW (11.16)

HQ dkP* ~ -f-2 R "'1-11" t .cp ox A -1 { -2 R Hp+ HM ~JnP* Vs = 1 N"~-f 1 \ ~p è> 'Ç" -{?R f4r;+H'.f 1cr' \ ..cp . (11.17) -1 and A is the inverse of A - 63 -

r (11.18)

with

2 b = Cl (~2~~g +:J; lm W gl) + Rf- o!n:* r

1 2 2 c 1 _ C m ~u"g + R C olnP* r = y ê§y

d = _Cl(tnZ~:g +~ltn'Wgl) + RCZO~nyp* p

2 (pk T + 01nP* p = -m -a PIë ax P CM dx ) 2 dlnP* r = _m "(pkQ.. T + .. ~y pK CM ~ y ) 2 k d T s = -m (cs -::l{ + CM) (11.19) ètr-Ci -k The value of Ter is set to 575 K at cr= 0 in order to

simulate a reasonable stratospheric stability in the top layer of the

model.

The details of these coefficients are not of much

interest except for two points: first, the matrix A has im.portant

elements of symmetry, and second, the manipulations have essentially

transformed only the first four coefficients ~rom derivatives at constant cr to derivatives at constant press ure so ·that they are practically equivalent to the corresponding coefficients of the barotropic mode!. The existence of A -1 will again be ensured by t~e ellipticity of the differential equation which will be forrned by substituting (11.15) into (10. 6). - 64 -

12. SOLUTION OF BAROCLINIC SYSTEM

12. 1. Discretization

The baroclinic atmosphere is represented by 2805 grid

points (Fig. 1) at each of five levels equaUy spaced on. the 0" -axis;

intermediate points and levels are used only for the final solution.

For aU computations except the relaxation, only eentered differences

over two.grid-Iengths (Al. 3) are used (one-sided on the boundaries).

In order to control aliasing, any quantity entering as a factor in a

produet is smoothed horizontaHy with (Al. 1) or (Al. 7), except for

already smooth functions. Smoothing of factors is avoided, however,

during the inversion of the matrix A (lI. 18). Multiple products are

involved in this operation but it was found best to treat the clements

of A as pure numbers rather than as representatives of spatial

functions; spurious damping or amplification resulted otherwise.

With aH input quantities defined at grid points and

levels, two grid-Iength centered finite differences re suIt in the

"stationary" wind field 0/ and the elements of A -1 being defined s also at aH levels and grid points. The final wind field 'Y3 is then

calculated from these on a staggered grid as foUows, according to

(11.15) ---x x P*U = p*US X + P*AU X~ + P*BU ~Y + P*CU ~o- (12. la) X Y (J" Y Y . Y p*V = P*VS + P):CAVyj~ + P*BV Y o/y' + p*CV \V; (12. lb)

---(1"" --($' cf p*W = p*WS + P*AWY-'~ + P*BWLp~ + P*CWO-'f~ (12. le)

The positions of these quantities with respect to thc - 65 -

regular grid points and levels is .illustrated in Fig .. 14b. The coeffi

cients AU, BU, etc. in (12. 1) are the elements of A -1 multiplied by·

f-2 according to (11.15) and iUS + jVS + kWS \V • . = s

12. 2. Boundary Conditions, Vertical Structur·e

The horizontal boundary condition is again '-fi = 0, and needs no further comment. We now need two vertical boundary condi-

tions because the differential equation is also of the second order in

W = 0 at 0"' = 0

-2 a lnP* W = m Olt = lV at (J = 1 (12. 2)" ~ in view of (lI. 1) and since a 1>/0 t = 0 at 0- = 1. The model then presents itself vertically as shown in Fig.• 14a.

This particular structure has the advantage that the boundary layer, represented by the cr- = 1 level is only 0.5,60"" = 0.1 thick; a non-centered difference is used for W there. On the other hand, the top of the model is poorly represented in terms of the horizontal wind field and divergence. The "vertical velocity" at

0- = 0.1 cannot be computed from the finite difference expression

(12. le) alone. Here we make use of the top boundary condition which we may write in finite differences as C1'

C1'

+ + ta

'1/

points

+ +

+ u. -1-

grid

U lL

U. u. +

w 71

the

·lI

+ +

u. to

(

u u.

.+ u.

+ +

+ u

respect

with

continuity

showing

model

equation

components

-0.4

-0.8

the

0--0.0

0--0.2

0-.0.6 0- 0- 0--1.0

wind

for

baroclinic

the

final

L 0

.L-_

• .1- •

.2- •

the

O o o

1.0.

W W

•

-.!9.L

arrangement

structure

staggering

vertical

(

the the ____

discrete

\V a) b) \V \V \V

The ( (

\J. \1. \J. '\J.

\J.\V

14.

Fig.

'i'

'" '" 0/ - 67 -

Wcr=o = 0 = (ws+AW'i'i+ BW\V~)cr=O + CWcY=O ("t'cr=O.z-\fO=O)/LlCf

(1Z. 3) and we assume as a computational condition that

li} _III = 0 (lZ.4) T 0= O. Z "f" Cf= 0

Substitution of these conditions in (1Z. le) at cr = O. 1 yields the required forrn

_ -X-y Wo- = 0.1 - 0.5 (WS + AWY'X + BWo/y)cr= o. Z (lZ.5)

The slight inaccuracy invo1ved in the expression of the top boundary condition does not appear to be detrirnental to the results. There is certainly no serious inconsistency involved since the nurnerical solution would be unreasonable or the relaxation would not converge. The illteresting aspect about it, however, is that the tendency lf' is not directly involved. a fact which Fj~rtoft (l96Za, b) did not recognize, and which prompted him to develop a completely diffcrent approach. lZ.3. Relaxation

The differential equation for \fi is forrned (implicitly) by substituting (lZ. 1) into the finite difference equivalent of (10. 6) as follows:

(P*U) + (p*v) + (P*W) = R (IZ. 6) X Y (J where the residual R is useà to improve the guess value of \jJ iteratively as follows

(1Z.7) - 68 -

Evidently, R and If are defined ~t grid points. The latest values of y; are alway" Itsed in evaluating the residual R, in the Liebman sequential manner, the initial value being (,y'0 = O. The coefficient a. decreases from 1. 5 to 1. 0 by steps of O. 1 each 15 iterations. The weight given to the central value of lf' by (12.6) is w. The relax- ' ation proceeds from top to bottom and then horizontally over al! grid points at each iteration.

The number of iterations was generally limited to 20, after which only from 400 to 600 pojnts (out of a possible number of 12985) remained where the change 1 \.f'V+ 1 -0/ vi was greater than 4 2 10- m sec -3; it took from 35 to 40 iterations to bring this limit -5 2 to 10 -3 m sec at aU points but the change in 'f was insignificant. In practice, even fewer iterations might well be sufficient. In the experiments related here, how'Èwer, onlya limited amount of attention was devoted to optimization; this applies also to the selection of the actual numerical parameters used throughout. - 69 -

13. DATA PREPARATION

13.1. Interpolation to Q""-Surfaces

Objective analyses over the 2805 point grid are available at the levels of 100, 200, 300, 500, 700, 850, and 10QO mb for heights and temperatures from the CAO history tapes. Dew point depression an,alyses are available at 500, 700 and 850 mb. The data required for the integration of the diagnostic model is the height and pressure at the ground (cr = 1) and the temperature on the five rj'-surfaces (0"= O. 2, 0.4, 0.6, o. 8, and 1. 0), at all grid points. When moisture is being considered, the mixing ratio is also needed at aU grid points of the bottom three levels (it is assumed to be zero abov'e, for expediency); furthermore, the temperature T should be replaced by the virtual temperature T everywhere v except for the calculation of the vapour pressure. The correction is

T - T v = 0.6 q T (13. l)

and is reaUy very smaU.

The interpolation procedure is described in detail by Shuman and Hovermale (1968). It is bascd on the belief that the operational height analyses are of higher quality than those of temperature; thus the temperature is used only to give the lapse rate between the levels, and the interpolation is made for the height of the various Œ"-surfaces or the pressure of the surface 0-= l, knowing the height of the ground. The mean temperature between the (}'-Ievels is then obtained hydrostatically from these·heights. For the purpose of the pre.sent diagnostic model,. the temperature is needed on each - 70 -

surface; it is obtained e by minimizing the difference between it and an intermediate temperature value which is generated during the interpolation proce ss.

The analyses of dew-point depression are of poor quality. Consequently, after the calculation of the mixing ratio on . the three pressure levels, the latter were simply interpolated or extrapolated linearly with respect to pressure to the lowest three 0"- surfaces.

13.2 Smoothing

Prior to this interpolation. aU height and temperature objective analyses had been submitted to two types of smoothing; the low latitude smoother described in Appendix B. 1 and a short-wave filter defined as follows:

where 1T is the product sign, indicating repeated application and the coefficients V. are 1

2 VI = 0.5 2 Y3 = -O. 5 + v-z

2V4 = -0.5 - v:-z

This filter has the response shown in Fig. ISa for a one-dimensional function F. The preliminary ellipticization of the height fields (14.4) was also done at this time. - 71 - 1.0 r------:------~~==----==_t

0.0 ~ ______~~~ ______~ ______~ __~~~=--+.~~ 2 3 4 5 6 8· 10 14 00 WAVELENGTH Fig. lSa._ Response of short wave filter as a function of wavelength (in grid ______~ ____ .. :-_.... , 1engths) • 1 0 . ~ - ~ '360

.75

0.0 __...t-.,-,""", _J.--L.-LJ_L__ L_Ll_J__ I-' __ L_.J.-L 2 6 14 18 22 26. 30 34 38 42 WAVELENGTH Fig. lSb. Response of low-latitude smoother as a function of wave1ength (in grid 1engths) at the latitudes of 360, 22°, 100, and 0°. - 72 -

14. ELLIPTICITY

Ellipticity of the differentia1 equation (12.6) is required

for the convergence of the Liebman iterative process. 'From (10.6)

and (11.15) it is readily seen that this implies that the,matrix

(b + d) (b + d) pr - s r - cp 2 --z- B":' P* as _ p2 (b + d) - 2 2 P - ar f Det..A: (b + d) (b + d) r - cp p - ar ac - bd (14. 1) 2 2

must be positive definite, where a to s are the elements of A (lI. 18),

(11. 19), and Det A is its determinant. If the last element was

ac - (b + d) 2/4 instead of ac - bd then

_ P*(-:-l-1 B--i\I (14. 2) . fl

where A is the symmetric part of A. Thus, in view of the smallness

of b - d compared to a + c (geostrophic divergence to absolute vorticity), and since we require more than minimum ellipticity anyway, we use positive definiteness of A for the ellipticity criterioll.

According to the theorem of Appendix C, the ellipticity reduces to

a+c+s>O (14.3a)

2 2 2 ac - (b + d) /4 + as - p + cs - r > 0 (14. 3b)

2 Det A = s Cac - (b + d)2/4J - ar '- cp2 + pr (b + d) > 0 (14.3c)

We note that this third condit,ion automaÜcaÏly ensures that Det A #- 0, -1 ' thus that A exists. - 73 -

A simple extensio~ of the method of .ellipticizing the barotropic diagnostic system would involve solving (14. 3a) by three- . dirnensional relaxation. This would not be difficult especially if s (which is basically the static stability) was expres sed in terms of the first and second vertical derivatives of the geopotential and the' relaXation was made on the constant pressure surfaces, before interpolation to the Q'"-surfaces. Nevertheless, for simplicity, the following was solved

2 - 2 -2 V " > -Qf III (14.4)

on the constant pressure surfaces. Assuming that s> 0 ·and that the term involving the variation of the Coriolis parameter in a + c is small co~pared to the Laplacian term, (14. 4) is much more restrictive than (14. 3a).

At any rate, the exact coefficients a to s were later re-adjusted as follows:

(14. Sa) then

al = lia, c' = J3c, b' = b/(3, d' = d/l', p' = yp, r t = ôr where 13, y, and li are such that - 74 -

2 2 >c - (14. Sb) cs - (, r c22s

where 5 is the average value of s at that level, and finally

p" = é p' , rIt = Er'

where E. is such that

s[ac - (b+ d) 2 /4J - E2 Carl 2 + cp' 2 - p'r' (b+ d)] >€3-s (14. Sc)

The actual constants used in (14.4) and (14. 5) are

Q = 1.0 0.25 é."1 = E = 30, 3, 2, 2, 2. degrees K for 0-= O. 2, 0.4, o. 6, 0.8. 1. 0 33 0.4 E. 2 = 0.2 E22= E. 0.4 (14.6) 3 =

Except for Q, the exact values for these limits did not seem to be véry important. - 75 -

15. DI SCUSSI ON OF RESULTS

AIl of the results descrihed below were obtained from

the same program, with five

Only the input data differed between runs. These specifications are

given in Table 1.

15. 1. Effects of Orography and Drag

In view of the G diagnostic assumption (Section 4), the Z calculated wind field must be almost an equilibrium one. The forcing

effect of orography on large scale motion is particularly interesting in

this respect.

1 t is well known that large mountain ranges maintain a

ridge of pressure upstream and a trough downstream and that the air

experiences changes of vorticity in these regions. Cressman (1960)

found that large errors occurring in filtered barotropic forecasts

could be accounted for partially by the mountain effect. H.e designed

an experiment where zonal flow was allowed to be di~turbed by orography " . in this model and found that indeed upstream ridging and downstream

troughing occurred, a reasonably normal flow pattern being achieved

after 48 hour s. He did not state whether this was an equilibrium state,

however, but we can presume that two days may be considered as a

reasonable time scale.

A slightly more idealized case was treated with primitive

barotropic equations by Kasahara ( 1966) where zonal flow on a 13-plane,

13 = df/de, was made to cross a circular "mountain. He indicated that a true steady state does not occur since the downstream trough initiates travelling wavesi ·however, the upstream ridge is quite permanent. His - 76 -

Table 1. Input data for each experiment.

Experiment Temperature Moisture Orogra~hy Drag Coeff and number pressure

-3 1 10 Feb 70 0 0 1. 3 x 10 -3 2 " Saturated 0 1. 3 x 10 -3 3 " 10 Feb 70 0 1. 3 x 10 -3 4 " 0 real 1. 3 x 10 5 " 0 real Cressman' s* 6 " 10 Feb 70 real Cressman' s* -3 7 zonal 0 0 1. 3 x 10 8 " 0 artificial 0 -3 9 " 0 artificial 1. 3 x 10 10 " 0 real '0 -3 11 " 0 real 1. 3 x 10 12 " 0 real Cressman' s*

,_ ... - ~ ..

*Cressman' s drag coefficient is positively correlatcd with mountain

height.

J t - 77 -

results were in good agreement with sorne important features of

laboratory dish-pan experiments of flow·over an obstacle.

I t is obvious from these experiments that zonal flow is

not an equilibrium state in the presence of blocking mountain ranges

and the results of Cressman inàicate that the adjustment can be achieved

bya filtered model in a time scale of about two days. Thus our geo

strophic diagnostic model should be applicable to this problem. We

now describe the results of a few diagnostic experiments with zonal

flow and orography, numbered from 7 to 12 (see Table 1).

Purel}' zonal fields of pressure and temperature were

constructed such that the implied geostrophic wind had at 450 N the

average mid-winter vertical profile used by many authors and originating

in Crutcher ( 1959), and the same north- south profile at 500 mb as

used by Cressman(l960). The statie stability was specified at 45 0 N

according to the mean January profile from Gates (1961) and a condition

of smoothness of the temperature in the vertical was used to generate

. a complete set of synthetic analy.ses of height and teniperature at the

seven pressure levels, simulating the. usual set of analyses. This is

shown in Fig. 16.

Experiment 7, with no mountains and constant drag,

served as an error check for the method and programs. The errors,

Fig. 17, are discussed in Appendix A and are small enough not to

influence the results of the next experiments.

In experiments 8 and 9, a bi-Gaussian north-south mountain

range, with maximum height of 3 km centered at 450 N was used as the

bottom boundary. This mountain.can be seen in north-south cross- ,

,1.

ex>

-.l

range.

mountain

-.":-

>:r.:::-.

idealized

"':,"

...

the

:::,

....

:;-

latitude.

~>.

middle

......

'.:':

y:"

the

12.

function

7 through

)

section

sec-

experiments 0

for

north-south

(max

used

profile

data

speed

area

wind

zonal

275

;

hatched

zonal

temperature

(a) (b) Idealized The

16.

•.

235 255

Fig

215

200

400 600

800

P(mb)

1000 - 79 -.

Fig. 17. Orientation error pattern in baroclinic modelinumbers are extreme values (units 10 -4m2 sec-3). - 80 -

sections in Figs. 16 and 18b and in the east-west section in Fig. 18a.

It may be considered as a model of the Rocky Mountains in aH its

elements except that it is only half as wide.

There was only a quantitative difference of at most

25 per cent between experirnent 8 and 9, showing that when the wind

is w'eak the effect of friction is not important. Figure 18 summarizes

the results of experirnent 9.

In order to appreciate these results, it must be stressed

that we are not cÇ>mputing the equilibrium state of the wind field, but. an initial state consistent with the given pressure distribution and devoid of high-frequency tendencies. Thus the main part of the cal culated wind is naturally zonal and equal to the geostrophic one.

However, its future state rnay be inferred safely by extrapolating the initial te;ndency for an appreciélble length of time.

Figure 18a is an east-west cross-section through the peak of the mountain, showing the tendency (it is convenient to think of it sirnply as the tendency of the geopotential of a constant pressure surface according to (Il. l}) in full lines ~nd the vertical velocity in dashed. The north-south section through the vertical line A - A'

(B - B') is shown in Fig. 18b, with fuHlines for the tendency (minus the tendency) and dashed for the isotachsj the section at B - B' is not shown separately since it is identical to Fig. 18b except that the sign of the tendency is reversed. The ordinate is·o- in both graphs, which

Ulay be interpreted as pressure in Fig. i8b by multiplying by 1000 mb.

It is appropriate to remember that the nort~-south velocity tendency is proportional to tb,e gradient of the geopotential tendenc y in Fig. 18a and similarly for the zonal velocity tendency in Fig. 18b. ()O -

/1'

sec-

(1 "

1480

a)èr

5 "

r II

. 1

,,'

\ /

.~_~':'

, lines:

725

•

1 . 1

"_0: ":.' ,

,'\ 1

1 \

shed 10

~g~

da _'"

<._;

(B-:B')

~""

---

1 ' 1?1

= -

short

~_',-.

A-A

l~~"

_"_.-;

,.'

~.-

3 ...

sec-

SECTION

----

--"

690

------'

...

(10-

.....

.0'

,.8

1 ,

1 , ,

tendency

Fulllines:

sec-

experimi.nt

isotachs

b) Results

710

18.

Fig.

1420

l ,

a - 82 -

The most importa~t feature of moun.tain flow is exhibited

by the rising tendencies upstream and faUing downstream in Fig. 18a:.

The vertical motion, being the rate at which particles cross the (j - surfaces,

causing the latter to rise or faU, is simply another way of looking at . . -4 2· -3 . the tendency itself. For extrapolatlOn, a tendency of 45 (10 m sec ;

the unit of tendency used for an baroclinic results) is equivalent to a

24 hour :peight change of 40 meters or to a surface pressure change of

approximately 4.5 mb in 24 hours.

Cressman (1960) had observed faUs of about 200 feet in

48 hours at 500 mb downstream of the Rockies (see Fig. 19); thus our

tendency figures are of the right magnitude. We also note that the tendency centers occur at two or three half-widths from the peak of the mountai~, as is the case in Cressman's (1960) experiments and aiso to someextent in Kasahara' s (i 966). A feature common to both these experiments is not indicated here, however. As shown by sections

A - A' and B - B' the ri se and faU centers are both north of the mountain, whereas the faU center is normally to the south. This may be a secondary effect.

Having verified that the verticaUy averaged state of our diagnosis is in agreement with the recognized barotropic behaviour, we can proceed to describe the baroclinic structure, which the author believes has not been done before. A two-ievei study of non-hydrostatic airflow over small mountains has been described by Magata (1969), however.

First, in Fig. 18 we notice that the tendency is about half as great in the mid-atmosphere as at top or bottom. This is - 83 - .

contrary to the tendency produced by pressure systems without forcing

since the tendency is generally largest at the top where the height

gradients are the large st. This structure implies cooling upstream

and warming downstream at low levels with compensating high level

warming and cooling. In Fig. 18a, there is a peculiar rever saI of·

sign of the tendency at·low levels, indicated by the 5-shaped zero-Hne.

This is in fact the beginning of two small cells of troughing upstream

and ridging downstream situated to the south of the peak and very near

the divide. These cells have a maximum amplitude of 12 units at

0'= 0.8. With respect to the slope of the tendency field, we note that

both cells slope eastward or westward towards the peak of the mountain

and that the line of maximum (minimum) tendency is farther north at

the mid-levels than at the top or bottom. The author does not pretend

to explain the reasons for this particular vertical structure, which

would require a careful study. High truncation errors caused by the

very sharp gradients of the O'"-surfaces could possibly explain sorne

. of the features. It was verified· by experiment 8, however, that the

role of friction is minore

Experiments 10, lI, and 12 wer·e made specifically

for comparison with the results of Cressman (1960) .. Figure 19 is

taken from his paper (Figure 6); it shows the 48 hour height changes

(in decafeet) produced at 500 mb by the barotropic model when zonal

flow (the same as we use~ for 500 mb) is disturbed by the influence

of mountains and surface drag. In his experiment, the influence of

mountains is dominant over friction and the introduction of a variable

drag coefficient contributes about 25 per cent less .troughing and more

ridging. - 84 -

Fig. 19. Deviations (in decafeet) from zonal flow in 48-hour filtered barotropic forecast introduced by combined mountain and friction effect; from Cressman' s fig 6 (1960). - 85 -

In our baroclinic ~iagnostic experim.ents with the rea1

mountains, we found again generally the mid-1eve1 tendency to be about

one-half that at the top or bottom. This may be kept in mind as we

show the tendency at the surface cr = 1 for the case of zero drag,

Fig. 20, and variable drag, Fig. 21. In these figures, the zero-line

is absent and the first contour is at 25 units in order to avoid the

confusion due to the error pattern, which may reach 5 units.

We note first a good agreement between our results and

those of Cressman as far as the position and relative magnitude of the

faU and rise centers are concerned. A very conspicuous difference

exists, however, around Greenland, where Cressman (see Fig. 19)

found centers of about two-thirds the amplitude of the Rockies' system,

whereas in our resu1ts the tendency doe s not exceed Il units at any

grid point in this area. Curiou·sly, however, at (5= 0.2 the tendency

over Greenland does reach an appreciable value, as indi~ated by the

windows in Figs. 20 and 21. Comparing the latter figures together,

we notice that the effect of the drag is to·produce an increase of ridging

and a decrease of troughing exactly as f01,1nd by Cressman. The resu1ts

of experiment Il with constant drag were about mid-way between those

of experiments 10 and 12.

15. 2. Effect of Pressure Systems, Ellipticity

In experiments 1 to 6 the state of the atmosphere was as observed on 122 10 Feb 70. This is .depicted by Figs. 22, 23, and

24 which are the objective height analyses at 300, 500, and 1000 mb.

(The small area enc10sed within dashed lines will be the subject of a more detailed presentation 1ater ·in Figs. 27 to 30). It is evident that - 86 -

Fig. 20. Diagnosed tendency produced at the surface hy the mountain effect only in experiment 10 (interval 25 x 10-4 m 2 sec-3 , zero-line omitted). The mountains exceed 125 dkm inside the shaded area. The inset is a section of the top level. - 87 -

Fig. 21. Diagnosed tendency produced at the suzface by the combined effect of mountains and variable surface drag in experiment 12 (interval 25 x 10-4 m 2 sec-3 , zero-line omitted). - 88 -

Fig. 22. 300 mb analysis for 12Z 10 Feb 1970 (interval 10 dkm). - 89 ..,

Fig. 23. 500 mb ana1ysis for 12Z 10 Feb 1970 (interva1 5 dkm).

, , .. - 90 -

Fig. 24. 1000 mb analysis for 122 10 Feb 1970 (interval 5 d~m). - 91 -

there is an unusually intense anticyc10nic region indicated in the

western Pacifie, especially at 300 mb (Fig. 22), which is probably

due to an erroneous observation. The ellipticity criterion is particu-

larly helpful in such situations, as can be judged from the changes

produced by the preliminary ellipticization (14.4) at 300 and 500 mb,

Figs. 25 and 26. Except for the error region, the changes are normal

or even ~maller than thos~ required by the balance equation. The

very large spreading of the changes is the result of the use of the

boundary condition of the second kind which preserves the height

gradient of the original field normal to the boundary; it is evident that excessive gradients would be created near the boundary if the heights themselves were kept constant there.

Figure 27 gives a three dimensional view of the diagnostic results from experiment 1 for the small volume indicated in Figs. 22,

23, and 24 which contains 9 x 9 x 5 grid points. The left column shows the vertical motions ci- at each of the five intermediate le~els.

In view of the absence of mountains, these can be interpreted as 1 ordinary pressure vertical motions and ~0-7 sec- is equivalent to -1 -1 10 microbar sec . The center and right columns contain the -5 -1 -4 2 -3 diagnosed vorticity (10 sec ) and tendency (10 m sec ) at each of the (}-surfaces.

At the high leve1s, the tendency field is very simple, indicating straight eastward movements of the trough, possibly with sorne intensification. However, at the ground, there are practically no rises but there is indication of sorne appreciable deepening to the north-east of the existing surface low and of the southward progression - 92 -

Fig. 25. 300 mb height changes resulted from preliminary elJjpticization (interval 5 m). Fig. 26. 500 mb height changes resulting from preliminary ellipticization (interval 5 m). . Left: t:r(10-7 1 Center. Q sec- ). . . (10-5 ' R1ght: UJ (10- r 4 m Zsec-11;sec-3').

Results f rOln ex . Fig. 27. per1ment 1 f or a small. vol ume of .space. . U1

VI> sec-

where

0 0

387 4 658 711 467 658

733 160

\f>V+~ e

10-

prints

20 20 20 20 20 10 20 10 19 20

tions

itera-

No.

33 33 33 1.0

760

984 772 772

1694

1017

points

6 0 0 0 6

292 222 0.8 222 413 366

1768

0 0 6

0.6' 153 176 196 196 220

2141

hyperbolic

1 0 0 0 1 0

0.4

998 998

1704 1029 1029 1029

Other

0.2 284 284 284 308 308

308

1.0 248

284 284

431 466

1305

3 .

0 0 0 0 0 0 1 0

78 63

0.8

1450

points

0 0 3 0

o '

1868

0 0 0 0 0

0.4 199 212 199 212 212

Unstable

1135

experiment.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.2

each

46 36 43 33 58 58 55 55 55 1.0

from

49 47 48 48 43 53 53 45 51 51 51

0.8

stability

(degrees)

Statistics

48 47 47 47 46 40 40 39 39 39 0.6

static

58 36 60 58 60 60 57 57 59 59 59

0.4

-T/ed9/êlO"

Mean

Table

401 401 401 405 405 405 408 408 411 411 411

'0.2

------

1 2

3 5 6 4 8

11 10 12

EXP

---- - 96 - ,

of the trough situated west of the Great Lakes. Indeed, this is very

close to what actually happened as indicated by the official surface

analyses of the CAO for 06Z, l2Z, and l8Z 10 Feb 70 shown in

Figs. 28, 29, and 30, and also by the isallobaric analysis for l2Z,

Fig. 36.

15.3. Effect of Latent Heat

Next, the effect of moisture on the vertical motion field

was investigated by using the following mixing ratios: 0, 5, 10, 15,

and 20 grams per kilogram at 0" = O. 2, 0.4, 0.6, o. 8, and 1. G.

These values were sufficient to saturate the tropospheric levels com-

pletely but the surface only partially. This had the effect of lowering

the static stability everywhere so much that about haH of the grid

points failed the limits é «(5"') given in (14.6) (refer to the statistics 33 of Table 2); no test was made to see if these limits could be lowered

by any significant amount. The results of this experiment can be • eva1uated by comparing the vertical motion C) at the ,mid-level

0""= 0.5 with or without saturating moisture, Figs. 31 and 32.

Fj:gure 32 is simply the result of experiment l, of which a portion

has already been shown in Fig. 27; Fig. 31 has the effect of full

saturation added. Since our precipitation model, Section 10. l, uses

a reversible process, downward motions cause the extraction of latent heat by instantaneous evaporation, which is an unrealistic event pre-

sumed not to happen if a real moisture field is used.

The effect of saturating the atmosphere in this manner is seen to double the intensity of the large scale vertical motion (a stilliarger increase might have been obtained by allowing a lower - 97 -

Fig. 28. Surface chart for 06z 10 Feb 1970 (isobars at 4 mb). - 98 -

Fig. 29. Surface chart for 122 10 Feb 1970 (isobars at 4 mb). - 99 -

Fig. 30. Surface chart for 182 10 Feb 1970 (isobars at 4 mb). - 100 -

Fig. 31. Mid-atmospheric diagnostic vertical mnotion 0- accompanying observed pressure distribution but w:iitth saturated troposphere, in experiment 2 (interval 10 x 10-7 sec-li first contours at t S·units). • 101 •

1 •

Fig. 32. Mid-atmospheric diagnostic vertical motion & accompanying observed pressure distribution without moisture or mountain effect, in experiment 1 (interval 10 x 10-7 sec-li first contours at ± 5 units).

.1 - 102 -

limit for the static stability), without basicaIly changing its shape, as expected.

In experiment 3, the actual moisture field for the 10 Feb 70 was usedi however, saturation was achieved' at only a few grid points and not at aIl levels in the vertical so that the effect was not as dramatic. A separate figure will not be shown for this case but sorne indications ean be obtained from Fig. 33, which has the combined effects of moisture, mountains and variable drag.

15.4. Combined Effect from all Sources

Experiments 4 and 5 were to test the effects of moun tains and of variable drag with real pressure systems. Generally these effects were as expected from the results of Section 15.1 and will not be displayed by separate figures. Experiment 6 incluàed the effect of the actual moisture distribution, of the actual (smoothed) earth ' s orographyand of a variable boundary layer drag beHeved.to be representative for the seale of motions considered. Provision had been made in the programs to include also the effects of sensible heat exchange at the surface of the water and' of long wave radiation from cloud tops, however, it was doubtful whether it would be appropriate at this time to consider such small terms.

In view of the relative smallness of the external forcings eonsidered, none of the fields from experiment 6 were sej:"iously different from those of experiment 1. Figure 33 shows the mid-level vertical motion and Fig. 35 the s'urface tendency from experiment 6.

Comparing vertical motions first, we can identify easily the mountain effect from Figs. 32 and 33 and also the effect of - 103 '-

Fig. 33. Mid-atmospheric diagnostic vertical motion 0- accompanying observed pressure and moisture distributions and inc1uding the effects of mountains and. friction, in experiment 6 (interval 10 x 10-7 sec-Ij first contour at t 5 units).

, y_o. - 104 -

Fig~ 34. 500 mb vertical motion from the w-equation inc1uding mountain and (riction effects .(interval ,40 x 10-~ mb sec-!, zero·Une omitted; these units compare' to 10-7 sec-1 for cr). - 105 -

Fig. 35. Diagnostic tendency at the ground (surface pressure tendency) resulting from combined interaction of pressure systems with moisture distribution, orographl and surface friction, in experiment 6linterval 50 x 10- m 2 sec-3 or approximately 5 mb (24 hr)- ). .

- - 106 -

moisture in the western Pacific where the upward vertical motion is

actually increased from minus 17 to minus 32 at the center situated at

0 0 35 N, 165 E. Pressure vertical motions were also computed fr.om

a four level W-equation inc1uding the mountain effect. The 500 mb

W -vertical motion is shown in Fig. 34. Although there is a strong

similarity with Figs. 32 and 33 we note two main differences. First,

the up and down centers are slightly c10ser together for CV than for 6-

and secondly, the amplitude of the maximum vertical motion is

appreciably larger for W than for 6-. The reasons for these

differences have not been investigated in detail and truncation errors

are certainly not the only reason. As is well known, the vertical

motion in the W-equation is forced for about 50 per cent by vertical

variations of vorticity advection. In the program used, this term is

evaluated by the simple G geostrophic approximation whereas the 1 G approximation is naturally used is our diagnostic model to compute 2 . èr; in regions of high curvature differences by a factor of two can

. easily result. The term o-dP*/dt is generally not an important source • of difference between p* v and CI,.) except in mountain areas.

Turning now to the diagnosed surface pressure tendency,

Fig. 35, we can examine it first with respect to the 1000 mb analysis,

Fig. 24. The isallobaric centers are seen to be in natural positions

with respect to the pressure systems, and of reasonable magnitude.

Next, we can compare Fig. 35 with an isallobaric hand analysis made

by subtracting the 06z surface pressure analysis from the l8Z one

and labelling the contours as if tjaey were 3 hour isallobars, Fig. 36.

Here, the position~ of all-well defined centers (over

land) verify surprisingly well, although the maximum values of the - 107 -

Fig. 36. Graphical computation of the surface pressure changes from 062 to 182 10 Feb 1970 (isallobars at 1 mb (3 hr)-I). - 108 - diagnosed tendencies again appea.r to be too weak by maybe 30 per cent.

With a reduction of the truncation errors by using finite differences over one rather than two grid-lengths, as in the barotropic experiments, there is no doubt that this discrepancy could be reduced appreciably.

The substantial differences over sea are due to disagreement between the machine objective analyses used for the computations and the subjectiv.e hand analyses used for comparison. - 109

16. CONCLUSIONS

In these experiments geostrophic approximations have

been used successfully to compute the three-dimensional wind field.

representing the large scale meteorological flow for a given pressure

distribution. The quality of the results clearly indicates that this

G geostrophic procedure provides a serious alternative to the balance 2 equation. now in general numerical weather prediction usage. and to

other methods proposed recently.

Apart from minor discrepancies attributable to the

insufficient finite difference accuracy in the baroclinic experiments.

the results are comparable to those· of the balance equation for the

vorticity. to those of the W-equation for the vertical motion and to the

actual observations for the surface pressure tendency. Although the

low latitude problems have been solved partially. the computed fields

do not contain much diagnostic ~alue in those areas qecause of the

severe smoothing performed on the initial data and of the large relative

errors. The main external forces of the atmosphere and the release of

latent heat can be taken into account and they produce a noticeable effect

on the initial winds. The problem of numerical incompatibility between

a diagnostic model for initial winds and an independent forecast model

is easily overcome by solving a rever se balance equation formulated

in terms of the finite differences of the forecast model. With this

procedure the computed geostrophic winds used as initial conditions for a barotropic forecast produce less g.ravitational noise than non divergent winds computed from the classical balance equation. - 110 -

The calculations were carried out in the operational context of the Central Analysis Office' s ·numerical analysis and prediction facility. Also. the solution a1gorithms are relatively simple and potentially very fast. Hence. most of the problems involved in the operational use of this method for the initialization of a baroclinic primitive equations forecast have been or can easily he solved. The first attémpts at initializing haroclinic forecasts will he made shortly, as soon as a working model becomes available at the Central Analysis Office.

In the context of GARP (Global Atmospheric Research Programme). two types of experiments could be done with this filtered diagnostic model in conjunction with a multi-level primitive equations forecast mode!. First, the experiment of Smagorinsky and M'iyakoda (1969) . . can he repeated in order to verify independently the importance of the initial specification of the vertical motion for short and medium • length forecasts. For this we simply make a comparison of two forecasts with the G geostrophic 2 winds as initial data, having removed the divergent part pf the wind from one set of data. Next. we could investigate the long debated question as to whether the success of the present-day operational multi-Ievel primitive equations models over .. ' the previous filtered haroc1inic models (e, g., the N. M·.·C .. ·. Model, Shuman and Hovermale J 1968) is due to the finer vertical resolution or to the use of primitive equations. This would involve an aspect which has not heen considered in this thesis, namely the time integration of the G diagnostic 2 system as a filtered forecast model according to the original idea of Eliassen (1949). These projects merit further attention. - III -

REFERENCES

Arnason, G., G. J. Haltiner, and M. J. Frawley, 1962: Higher-Order Geostrophic Wind Approximations. Mon. Wea. Rev., Vol. 90, No. 5. ------Asselin, R., 1967: The Operationa1 Solution of the Balance Equation, Tellus, Vol. 19, No. 1.

Berkofsky, L., and E. A. Bertoni, 1955: Mean Topographie Charts for the Entire Earth. Bulletin of the A. M. S., Vol. 36, No. 7. Blumen, W., 1967: On Nonlinear Geostrophic Adjustment, J.A.S., Vol. 24, No. 4.

Bolin, B., 1953: The Adjustment of a Non-Balanced Velocity Field Towards Geostrophic Equilibrium in a Stratified Fluid. Tellus, Vol. 5, No. 3.

Bolin, B., 1955: Numerical Forecasting with the Barotropic Model." Tellus, Vol. 7, No. 1. Cahn, A., 1945: An Investigation of the Free Oscillations in a Simple Current System, J. Me~., Vol. 2, No. 2. Charney, J. G., 1948: On "the Scale of Atmospheric Motions, Geof. Pub!., Vol. 17, No. 2.

Charney, J. G., 1955: The Use of the Primitive Equations of Motion in Numerical Prediction. Tellus, Vol. 7, No. 1. Charney, J. G., 1962: Integration of the Primitive and Balance Equations. Proc. of the Int. Symp. on NWP in Tokyo (1960). Met. Soc. of Japan.---- ~ ---

Charney, J. G., 1963: A Note on Large Sca1e Motions in the Tropics. J. A. S., Vol. 20, No. 6. Cressman, G. P., 1960: Improved Terrain Effects in Barotropic Fore casts. --Mon. --Wea. --Rev., Vol. 88, No. 9-12. Crutcher, H. L., 1959: Upper Winds Statistics Charts of the Northern Hemisphere. Office of the Chief of Naval Operations NAVAER 50-1C-535.

Danard, M. B., 1964: On the Influence of Re1eased Latent Heat on Cyclone Development, J.A.M., Vol. 3, No •. 1. Eliassen, A., 1949:" The Quasi-static Equations of Motion with Pressure as Independent Variable, Geof.Publ., Vol. 17, No. 3. 112 .-

Ellsaesser. H. W .• 1968: Comparative Test of Wind Laws for Numerical Weather Prediction. -_.Mon. --Wea. --Rev •• Vol. 96. No. 5. Fj~rtoft, R .• 1962a: A Numerical Method of Solving Certain Partial DifferentiaI Equations of Second Order. GeoL Publ., Vol. 24, No. 7.· --

Fj~rtoft, R., 1962b: On the Integration of a System of Geostrophically . Balanced Prognostic Equations. Proc. of the Int. Symp. on NWP in Tokyo (1960). Met. Soc. Japan:------

Fj~rtoft, R., and B. Sl:\derberg, 1965: A Prediction Experiment with Filtered Equat"iolls (Preliminary report). NCAR Manuscript No. 59, unpublished.

Gates. V1.~ L •• 1961: Static Stability Measures in the Atmosphere. J. Met., Vol. 19, No. 4.

Grammeltvedt, A., 1969: A Survey of Finit~-difference Schemes for the Primitive Equations for a Barotropic Fluid. Mon. Wea. Rev., Vol. 97, No. 5. - -.- - . Haltiner, G. J., 1968: Numerical Weather Prediction. Navy Weather Research Facility, Norfolk, Va.

Hinkelmann, K •• 1951: Der Mechanismus des Meteorologischen L~rmes, :rellus, Vol. 3, No. 4.

Hollmann, G., 1966: Zur Frager neuer diagnostischer Beziehungen zwischen Wind- und Druckfeld (Balancegleichungen) in ein"er barotropen Atmosph~re mit divergenter StrBmung. Beit. zur Phys. der Atm., Vol. 39, No. 2-4. - --

Houghton, D., and W. Washington, 1969: On Global Initialization of the Primitive E;quations: Part 1.. J. A. M •• Vol. 8, No •. 5.

,Kasahara. A •• 1966: The DynamicalInfluence oC Orography on Large Scale Motion of the Atmosphere. J ~ A. S •• Vol. 23, No. 3.

Korb. G., 1967: Anwendungen diagnost.ischer Beziehungen zwischen Wind- und Druckfeld am divergent:-barotropen Modell. Beit. zur Phys. der·Atm., Vol. 40, No. 1/2. - --

K.urihara, Y., 1965: On the Use of Implicit and Iterative Methods for the Time Integration of "the Wave ·Equation. Mon. Wea. Rev., Vol. 93, No. 1. ---

Kwizak, M., 1970: Semi-implicit Integration of a Grid-point Model of the Primitive Equations. McGill Ph. p. thesis. To be published.

Kwizak, M"., and A. Rob~rt, 1970: A Semi-implicit Scheme for Grid Point Atmospheric Models of the Primitive Equations. To be published in Mo~. Wea .. Rev. - 113 -

Magata, M., 1969: On the Study of the Airflow over Mountains by the Numerical Experiment. Proceedings of the WMO/IUGG Symp. on NWP in Tokyo, part II.

Matsuno, T., 1966a: Numerical Integration of the Primitive Equations by a Si~nulated Backward Difference Method. J ..~ the Met. Soc. of Japan, Ser. 2., Vol. 44, No. 1.

Matsuno, T., 1966b: A Finite Difference Scheme for Time Integrations of Oscillatory Equations with Second Order Accuracy and Sharp Cut-off for High Frequencies. J. of the Met. Soc. of Japan, Ser. 2, Vol. 44, No. 1.

Miyakoda, K., 1962: Contribution to the Numerical Weather Prediction Computation with Finite Differences, Jap. Journ. of Geoph. , Vol. 3, No. 1. ----

Miyakoda, K., and R. W. Moyer, 1968: A Method of Initialization for Dynamical Weather Forecasting, Tellus, Vol. 20, No. 1.

Nitta, T., and J. B. Hovermale, 1969: A Technique of Objective Analysis and Initialization for the Primitive Forecast Equations. Mon Wea. Rev., Vol. 97, No. 9. ----

Phillipps, H., 1939: Die Abweichung vom geostrophischen Wind, Meteor. Zeit., Vol. 56.

Phillips, N.A., 1957: A Coordinate System Having Sorne Special Advan tages for Numerica1 Forecasting, J. Met., Vol.· 14, No. 2.

Phillips, N. A., 1960: On the Prob1em of Initial Data for the Primitive Equations, Tellus, Vol. 12, No. 2.

Phillips, N. A., 1963: Geostrophic Motion, Reviews of Geophysics, Vol. 1, No. 2.

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Robert, A. J., 1969: rhe Integration of a Spectral Model of the Atmosphere by the Implicit Method. Proc. of the WMO/IUGG Symp. on NWP in Tokyo.

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Shuman. F. G .• 1960: Numerical Experiments with the Primitive Equations. JNWPU. u. S~ Weather ·Bureau; November.

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APPENDIX A: FINITE DIFFERENCES

Sorne general consideratïons about finite differences are collected in this appendix.

The choice of finite differences is always difficultj it

, . was desired to make the most efficient use of the given grid resolution, to minimize truncation errors, to prevent aliasing and to minimize computing time.

In the foUowing, F stands for any variable. AU functions are represented at grid points separated by a constant distance d in both directions on the projection, and indexed by i, j (and k for the vertical direction cr). Half integer values of the indices are sometimes used for which a set of intermediate (. points) are introduced diagonally between the regular grid points.

A.l. Basic F,orms

From the two elementary operator s of average and difference

-x (Al. 1) F = O.5(F1+ lll + Fi_Ill)

-1 F x = d (F i + III - Fi - Ill) (Al. l) where X (i) stands also for Y..{j) and (J(k) and where d is replaced by

L:lo- for o-(k) , aU common operators can be formed by combination for example:

(ld)-1 (F 1 - Fi _ 1) (Al. 3) F~ = i +

. -l F XX = d (Fi + 1 - l Fi + Fi _ 1) (Al. 4) - 116 -

Fi,= (2d)-1 (Fi + 1/1., j+ 1/2 + Fi+ 1/2, j _ 1'/2 - Fi'_1/2, j+ 1/2

- Fi _ 1/1., j - 1/2) (Al. 5)

-KY -1 F = (4) (Fi + 1/1., j + 1/2 + Fi + 1 /2, j - 1/2 + Fi .. 1",2, j + 1/2

(Al. 6) + F i _ I/Z, j-1/2)

XXYY F == (16)-1 (F. . + 2F + F + 2F 1 + l, J + 1 i, j + 1 i - l, j + 1 i + l, j

+ 4F. . + .ZF. 1 . + F. + 1 . 1 + 2F. . 1 + F. 1 . 1) (A 1. 7) 1, J 1 - ,J l ,J - 1, J - 1 - ,J-

A.2. Truncation Error

AU of the operators used in this work, except specifie 2 smoothers, have a truncation error proportional to d , that is they are second order approximations of the corresponding Taylor series expansion. (At the boundaries, however, non-centered approximations are used, in which ca.se the error is proportional to d). Thus the actual error is reduced by a factor of 4 'in (Al. 2) as compared to (Al., 3) since the effective distance d has been hé~.lved through the use of half integer values of the index. In view of this effect, (Al.2) or (Al. 5) were used instead of (AI. 3) when pos sible; at the same time a better use is made of the given grid resolution.

A.3. Aliasing

The product of two sinusoidal functions gives rise to two new functions, according to a very simple r.ule. It can be shown as a result that if the two functions have a wavelength shorter than four grid increments, one of the resulting functions may be shorter than two grid - 117 -

increments. As shown by Robert, Shurpan and Gerrity (1970), two grid increments is the· minimum resolvable. wavelength and fluctuations shortcr than that will appear to be actually longer. For exampIe, a wave exactly one grid-Iength long has the same amplitude at an grid points and thus appears to be a constant. This phenomenon, called' aliasing, must be avoided since it constitutes a fictitious transfer from short to long waves.

One way to prevent aliasing consists in eliminating from both factors all fluctuations shorter than four grid increments. To do this exactly would be impractical unless spectral techniques are used. Approximately the same effect can be achieved by applying averaging operators -x --xx like ( ) or () to the factors, since these have the property to reduce conside~ably the amplit':ldes of short waves without affecting too seriol;lsly the very long ones. When one of the factors is a second order finite difference, the truncatioll error plays the role of the averaging, but if a higher order nnite difference were ·used, a higher or der averaging operator would also become necessary to prevent aliasing. More light on this dilemma is thro\vn by the papers of Robert. Shuman and Gerrity (1970) and of Grammeltvedt (1969).

A.4. Total Finite Difference Errors

In the barotropic experiments. the emphasis was on reducing truncat.ion errors at tàe expens~ of aliasing. and vice versa in thebà:roèlinlc experiments. No great effort was made to calculate the errors exactly but the results of a "debugging run" with the 5-level baroclinic mode! will provide an ?,-ssessment. - 118 -

Zonal fields of the input variables were fed into the diagnostic model, such that they implied approximately mean observed January zonal winds and static stability as shown in Fig. 16. In these circumstances, it would be expected that the tendency '+' = af/,Ià t + RTOlnP*lèH would be exactly zero; yet the pattern shown·in Fig. 17 emerged after 20 iterations, approximately the same at aIl levels. Since there were no short fluctuations in any of the fields, this is not due so much to truncation errors as generally illustrated for one-dimensional functions, but rather to or~entation errors of the type discussed by Miyakoda (1962), associated with two-dimensionai functions; variations in the vertical do not seem to have caused any substantial errors in this case. Higher a.IJlplitude and change of zonal wave number at low latitudes may be due partly aiso to the asymmetry of the domain.

Comparing this error field to fields of tendency calculated for real data, for example Fig. 35, and considering that Fig. 17 results from only some . of aH possible errors we may conclude that the tendencies calculated at low latitudes are subject to almost lOO per cent errors but that the accuracy is sufficient in middle and high 1atitudes.

It is interesting in this connection to raise the question of numerical compatibility between a diagnostic modei and an independent forecast model as the iatter may well have the same type of errors, but of different pattern and amplitude from th~ former. And what about the forecasts themselves in these circumstances ! - 119 -

APPENDIX B: SPECIAL OPERATIONS FOR EQUATORIAL LATITUDES

B.l. Smoothing

It was noted in the geostrophic expansion 'that the latter converges for motions whose frequencies are less than f. Thus, in equatoriallatitudes, only very large scale perturbations should be considered. Consequently, a low latitude smoother was used to exc1ude short waves from the initial fields of geopotential (on constant pressure surfaces). It was also found necessary to apply it to the fields of W in the barotropic s mode!. The response of this sm,ooth~r for one-dimensional waves is given at a few latitudes in Fig. 15b. The numerical formulation is

F 2 2 = [~+ 0.25 d cos 4°9 ( >XX)· ( 1 +0.25 d cos 40e (

(, 2 40 .\1 + d cos 2 9 ( >xx)' 0+ d cos 4°6 (

where the products indicate repeated application. Even this drastic smoothing did not work perfectly because of the influence of the boundary which isolates the very low latitudes into the rather small corners; a simpler and faster process would be desirable.

B.2. Division by the Coriolis Parameter Aiso in connection with equatorial latitudes, special care had to be exercised whenever division by the Coriolis parameter was involved.

Mathematical Approach

The problem can be approached mathematically quite simply. Consider for ~xample the geostrophic relation - 120 -

fW =kxV/J. g f = 2fising

Taking the derivative with respect to latitude we get

df V + fd1Tg = k x \7 d<,6 da g dO dg

Now, if we assume tbat d \V Ida is g always bounded, we obtain the result that

= liIn k x V/J = !s.x"Vd<,6/d9 6.,0 f df/d9

It was postulated that this formulation also applied in aU other cases where division by f occurred. In practice, the rule was used at any grid point within 1. 5 degrees (less than half a grid length) of the equator. The total derivatives with respect to latitude were approxi

mated by centered 0 differences calculated along lines at 45 from the grid directions, which in the corners of the grid are almost coincident with the meridians. Each value cale ulated in this fashion was thèn -:xxyy averaged by applying a () operator to it. Tbis method was used for the barotropic experiments; the results were satisfactory but the procedure was judged too complicated for the value of the results.

Numerical Approach

Division by the vanishing Coriolis parameter can aIso be considered numerically from the view-point of aliasing. The function l f- is represented by high amplitude short wavelength terms when the equatorial discontinuity is considered. çonsequently, careless multi- plication by this function will result in very high aliasing errors. The ·latter may be eliII?-inated by representing' the inverse function only by - 121 -

waves longer than four grid increments. This principle was used to

justify the following simple procedure .. First the function

2 2 -10 . 2 -2 fI = f. + 4xl0 exp (-5 sm 9) (sec ). was used whenever division by f2 was required. For division by . f. the square root of fl2 with the appropriate sign (thus a discontinuous function) \yas used. This procedure was applied in the baroclinic model and it was surprisingly weIl behaved. The curves of f. fI. f2. fl2 are in Fig. 37. N - N

-2

seo

-8

2.0 10 1.5

1.0 0.5 0.0

36 ;j

and

Pole

North

='>$

functions

the

from

modified

distance

grid

corresponding

with 1

function

and

parameter)

experiments,

(C~riolis

of baroclinic

------

used

Curves

37.

4 8

Fig.

-1

1.6 1.4 0.8 1.2 0.6 0.2 0.4 1.0 0.0

seo

-0.2

10-4

...... - 123 -

APPENDIX C

C. 1. Lemma: The real numbcrs a,· b. e arc positive if (and only if)

i) a + b + e > 0 ii) ab + ae + be > 0 iii) abc) 0

Pr.oof: No genel'ality is lost by letting a = 1. Thus. to satisfy iii) let

b <- 0, e < 0 (Cl. 1)

Then, from i)

-1 < b + e <0 (Cl. 2)

whieh implies, with (Cl. 1) that

-1 < b < 0 (Cl. 3)

-1 < e < 0 (Cl. 4)

so that, in view of (Cl. 3) and (Cl. 4)

b (-be < 0 (Cl. 5)

Now, from ii) and a = 1

-be

. b <-be

which is impossible unIes sc> 0 .. which further ill?-plie s from iii) that

b > o. Th us, a, b, c > o.

C.2. Theorem: The real symmetric mat!"ix A has three positive eigen

values (is positive definite), if its determinant (P ) .. ifs trace (Pl) . 3 and 'P are allpositive 2

Pr.oof: Let

The eigen values /). (i = 1,3) of A are obtained from the characteristic 1 equation

lÀ - À 1'1 = 0 or, in full

)\.3 _ PI ';\2 + P 2 À - P 3 = 0

where

P a+b+c 1 = 2 2 2 . P .;: ab - d + ac - e + bc - f 2 2 2 2 P abc - af - be + 2efd - de 3 =

It is well known from the theory of polynomials that

Plis the sum of the roots

P 2 is the sum of th", p:-oducts of the roots two at a time

P is the product of the roots. 3

Thus, according te" the Lemma, the three roots of A are positive if

Pl' P 2' and P 3 are positive.