..AFCRC-TR-58-228(I)ASTIA Document No. AD 152509

GEOPHYSICAL RESEARCH PAPERS

No. 58

THEORY OF LARGE-SCALE ATMOSPHERIC DIFFUSI'ON

AND ITS APPLICATION TO AIR TRAJECTORIES

VOLUME I

SAMUEL B. SOLOT

and

EUGENE M. DARLING, JR.

June 1958

PROJECT 7616

Atmospheric Analysis LaboratoryGEOPHYSICS RESEARCH DIRECTORATE

AIR FORCE CAMBRIDGE RESEARCH CENTERAIR RESEARCH AND DEVELOPMENT COMMAND

,. UNITED STATES AIR FORCEBEDFORD, MASS.

................. .................. .............................-.... ,

FOREWORD

This Geophysics Research Paper is published in

three separate volumes.

In Volume I, the theory of large-scale atmospheric

diffusion is developed. The application of this theory to air

trajectories is presented in Volumes II and III.

Volume II contains probability tables for various

constant values of mean zonal wind.

Volume III consists of probability tables for North

America and Eurasia.

ACKNOWLEDGEMENT

The authors wish to express their sincere appre-

ciation to Messrs. Robert E. Chabot and Joseph Hess who

performed the laborious hand caiculations required to pre-

pare the basic data for niachine computation.

L

!.I"

,q

ABSTRACT

G. 1. Taylor's theory of diffusion by continuous movement is

adapted to motions on the scale of the general circulation. The re-

* sultant theory pertains to diffusion, by large-scale eddies, of air

particles constrained to move on a constant-level surface. This

theory provides a means for determining the probability field as a

function of time for a particle originating from a given point on the

surface of a sphere.

The bivariate normal density function describes the prob-

ability of a displacement with components x (West -East) and y

(North - South). This function reducei to the circular normal form

- wheii suitable empirical and theoretical simplifications are intro-

duced, concerninig the mean zonal wind, the standard deviation of

7% . . ~ dis plac vnexit comiponents, and the correlation between displacemelnt

coinponent deviations.

This denisity function is integrated in polar formi and mapped

on a lprojvction of the earth's surface by means of a suitable spher-

ical transformnation. The resulting formi- of the distribution function

i;s applicabl e to problemns of atomice fallout.

fin applying diffusioti theory to balloon operationis, the concemL

oi dowinstreati probabil ity detisity functioni is introduc ed. 1'hiMa fule -

* tioit detxi s i tt-h~ probability dtitnuty w ith respovt to aidtucie 0 of i 1an !-.a

wt~st dia&3)la i Ceet at leadt A s large as a spotcified value, occurrinig

P1q-1'tvd inl Vo iuniv It alid Ill for (it) Various constalit Values of

keanw and (hi) North Anirieca anid 1 :.ia for January, April,

Julyr, md Octobor att 40, 000( to 80, 1)00 fet't.

CONTENTS

Page

Foreword. ii

Abstract iii

Illustrations vii

Symbols ix

1.Introduction I

2, Probability Theory of Displacements 3

3. Analogy Between the Wind Problem and the Dis-placement Problem- 8

4. Polar Integration of the Probability Density Function 9

5. Simplified Solution of the Displacement Deviations 136. Complete Solution of the Displacement Deviations 15

7. Solution of the Mean Displacement 18

8. Mapping of the~ Probability Field 199. The Downstreamw Probability Function 28

APPFNL)IX 1. Wind Analysis 31

APPENDIX It. Upper Win-ds Over Europe and NorthAinerica 37

A PP E*NIX III. Variaitiun of the Standard Deviation With

A PPE.N LX IV. Sphe~rical Traiisformation 49

E iI( N~V ENE S 51

*. I I U S T R A T 1 N'! S

Figure Page

1. Displacemenit of an Air Particle 42. Mean Wind and Standard Deviation, North America,

January, 50, 000 Feet 11* "'3. Circular Normal Distribution of Winds, North America,

January, 50, 000 Feet, 40 0 N 124. Solution to E]cs. (4) and (5), Appendix IV, for 0 = 40°N 14

5. Solutions for the Deviation Displacement 0. 50 ProbabilityCurve 16

6. The Solution for the Mean Displacement

7. Mapping of th, Final Solution for the 0. 50 ProbabilityCurvu -,

8. Probability Field, January, 50, 000 Feet; Initial Point,40°N, 120)W; Time, 1 Day Z2

Q 9. Probability Field, January, 50, 000 Feet; Initial Point,40 0 N, 120 0 W; Time, ? Days 23

10. Probability Yield, January, 50,000 Feet; Initial Point,-0 0 N. 120 0 W; Time, 3 Days 24

11. Probability Field, July, S0,000 t'eut; Initial Point,40 0 N, 120 0 W; Time, 1 ay 25

"'"12. Probability Field, July, 50,000 Feet; Initial Point,'0 0 N, 10 0 W; Time, 2 [')A y4 26

13. Probaility V'iold, July, SO, 000 F' et; Initial Point40"N,... ; Time. 3 0a y, s

14. EX 111''- ol.- ic h owntis trva I-ruhzability Comiputa tion for:x V l0 1alitude; y 10" to 110N; U 30 Kiuti 0

"..15. l- oti ' uf Wind 'Spekd at any l,cv.l t th, Sp.ed at ii Itn_f~~i~i (-0, 0060 Vcot). Avvi.at- for' 5 United Staltes t Sgtati w:." @ ~( Alprox iately 37nN) )

I6. RvL intun c 'We'en :1ta Ia r Witi aud Vvctur Win d in

1 7.'lL I i t. B v't.'f co V vtr W ind a t d Stan da rd V vc t rI jev hit t o

*" l 1 8. Se',u~ S tt-0 Vari,ti tsn 4f Wi nds' ," N4 r thb A in r"41)'. . . 1 9. : i-a (, I i'. r iat -i \V W ind ( 'v ti ' ... -;u ro!p .,

X*. "

94a

................

ILLUSTRATIONS (CONT.)

Figure Page

20. Latitudinal Variation of Winds Over North America 40

21. Latitudinal Variation of Winds Over Europe 41

2. Vertical Profile of Winds Over North America 42

23. Vertical Profile of Winds Over Europe 44

TA B LES

Table Page

1. Mean Wind Speed (Knots), 50, 000 ft, January 5

2. Coefficient of Correlation Between u and v 5

3. Computatioil of Displacemnent Deviation (Complete Solution) 18

4. Correlation of Tost Data (96-mb Winds [ kta ) Over Europe) 35

5. Mean Z.onal Wind. i and Standard L0uviationi, 6 -North

Arnerica aiid Europe, 38

S Y MB OLS

x, y displacement components it the West - East and North -

South directions, respectively (Note: eastward andnorthward displacenents are taken as positive)

X, Y climatological mean displacements in the x and ydirections, respectively

x = x-X displacement component deviation in the West - Eastdirection

y' = y-Y displacement component deviation in the North - Southdirection

x , C- standard deviation ol x and y, respectively

S- standard deviation of displacements

u, v - wind components in the x and y directions, respectively

U, V .- climatological mean winds in the u and v directions,respectively

Ut u-U wind component deviation il t WVest- East directiol

v v-V wind component deviatiun in the North - South .diroctiuin

p(m, 11) - bivariate tnormal probability density function for ' -

,( m c r~~t i on c o e ffi ittit h tw e e Lit v a rlia bil . -

nut 11\ d' dviation of u ad v. reslct ively

: - ' tdnd;i d deviation of tof N ,*4d

*,i U ('l'. , GV(1)" furmctions ,vI, i ch dr't, rmin, the time dp,.ndocv e

Y V

( ., (y I magn td, if L -' the cw0i' di pla c v~l, also -Ia tude

,o - n.irtnl~ur *, o dq-" ti, . r,.spc. tixe

an -i .*v

IL

SYMBOLS (CONT.I

lU P(m,n) - bivariate normal distribution function for the vari-

ables m, n

mean longitude displacement

- longitude, measured eastward from a given initiallongitude

= ) -,, -deviation longitude

- latitude

a - longitude in a rotated system of spherical coordinates

U =- a value of U averaged over latitude between theinitial latitude, and q, an arbitrary latitude

-"ds, _ - a value of 6.1 appropriate for the population of devia-* tion dispiacements fron- 0 to S

DPDF - dow,nstrvam probability density function

,- -auto-correlation function

a snlall timue iiwrement

t- time

S T - timo in days

a - a counstant, depending un the ui e of an oddy

*:::

.I"4

THEORY OF LARGE-SCALE ATMOSPHERIC DIFFUSION

AND ITS APPLICATION TO AIR TRAJECTORIES

1. INTRODUCTION

In recent years the constant-level balloon has developed into a vehicle

of major scientific importance. Its payload may consist of a variety 6f meas-

uring devices which utilize appropriate communications systems to telemeter

information back to earth. In many cases this kind of data can be obtained in

no other way. Indeed, even the patl's of these balloons are of great scientific

, interest, since they represent soluticns to the equations of large-scale atmos-

pheric motions in a Lagravgean system of coordinates.

One of the nost attractive features of the constant -level balloon is a

direct consequence ot" its sire. It is large enou.gh to be insensitive to small-

scale eddies, while responding freely to the large-scale motions of the atmos-

phere. Paradoxically, it is this very freedom of movetmellt which imposes thie

most seve et* oprational litnitatim on he baliln .4ystem, namely uncontrol-

*0 lability. Th, horizontal path o a cntrolled ,to : mail extent it) two Ways;

(a) by varying th, time of lamic hiug, and thus pretPi,.ri-niig the initial hori-

1zontal vtoc itv at altitude, ,J !h) by V..krying thw atituth during flight. V Kcpt

for thesi' measures, knowl,.dgv oi the futur, positions of tallooils must largely

i d,"pend upon metrorlo(gic;'l ir -dictionl.

LXtrietlrI' htagt Mtiwv thii1,i , ot4ir peric'ds gral:r than a fe.w &LIvs coll--.vntional prediction mithoris.t ap tiitrttiable. For ths e longer time uttervals

4tatisti,-a! nthtl.ds must he UsHd. To b,' valid, $ueh methods timnt b, ba S od

* on a Net of fundam4'ntal ;4talt ti.As of large- scale atimospheric motion.s. This

,* i but one xiamjld o1 the nt-d for such statistics.

.. Author's in,%tiuncript approvecd 19 NIarch 1958.

B '' ". :'

;com: 1.p:ki for cVumbine(d nc('tOi-ks of balloons,

1i,. il fromi ncerl itku; .!t I)rut'dltL'rlilk't timeI inlterVValk SUCII pr

rku 11COSSarily costly and niust bev thorou:,hlly ivtlu~ttked. 'To plrforrn

~ wt l i~ntjob of eva lua tioni, onke must ag,,ain have a c sto. hb e

ul-iV Lic s onl alt rlbophorc Ilotionls. Inl tact, th. Lick of such info rrni.-tion has

I' 1011o y detterred recal iza ti on of the full potential value kI f al loon s yst urns.

mus pa pe-r pr esk-unts a set of statistics un oi ttiIIospIICFIC trajuc tories ill

.1 i en 'v hc h as becu oundmostus 1ful for solving thku pr' bl ems discse

ao\ kv e idi other r elated probl ems.

t 'tie most straightformvard approach to theV priublI em of distributionl of

ra .Jct wry enld-poinits is to comp-,[utC it large nlUnbe r of trajectories, having a

c Iorigin, from a seriUs o~f upperWl-le.vel ch.arts. Here-, either the ob-'I vind orthecomute guIsL);tro phik cW ilds , or both, are used as hsi

uat i h accuracy o)f such computations will depend to Suome extent onl the

* tie i~' valbetween succoessive maps. A numb.- of iuch studies have been

[ I, n..Id the.Se tlir, useful ill givinit at leas;L t gros, estimat of thet diittribu-

.I tu.1 t io til . Io evtev, in a doit ion to the magn~itutde ot this task. t ho:r at-

vie.-'rouslimtatonsto this akpproach:

a.It is extrt emody difftwul tor Vcomlplktv ztccuvate tra -

b. WVith thyv typ(' oi diakt at wmr dspkisal , it i.. lOrIostto aclli re a s.Imple Sut fir ivitly 144,0t, to be v

* c. vn ill thoi. low. c. xhiert the tt~ttw . tlttd -

111S VAnI tic uvovetol, Ol a pph%1 icaiitty vi thit t'sult s is 1.4-r cuto a pa rtrcul r tegion in pal a pa dir ular h right,

;t t v'i'ard to tie, first U matin it is mily fAir to stato that a ufficie ly

I .~ vttdc ould oviorconwi thi6 rebtriction if thore were to sy~iuuvatic Croan-

* i.e NMoby Dick flights are an% imortaot sour. it! of datta. Although alny

r 1balloons have beo'n flown~, only .. smtal percentage of these. flights

* i ienlylong~ duiratitn t,,) be tisel. Thus~ the l.s t two littitins

Alain illa forcc~. Vurtetivivo the'ie is .4 deli ut e hla4u ill these data ba-

1 .l~ ie a e'.ats wer k1I'1h -uavCnti4)d to Sulected( SIuiktiOIS Wh101 tile West-

1 ere strong1qest.

From the for egoing Cdiscussion it is evident thazt an a d(,(uate thecory* of diffu~ion onl the scale of the general circulation is icdd hspapr

* attempts to derive such a theory. This is by no rneans a comnpletc theory,

* since certain sirriplifyinW assumptions and approxiniiations have been in.-

troduced. The criterion adop ed in introducing these approxim-ations was:

to maintain compatibility between the degree of refinemnent of' the theory

- and the quality of the input data.

Any theory of diffusion of air particles by the general circulation ed-

* deslust provide a mneans for deerrnining the proba--bility field as a function

o' time for a particle released from- a *'iven point onl a sphere's surface.

Initially. the probability of the particlL -s location is 100 percent at the ori-

gin and 0 elsewhere. As tim-e passes, the particle's location is specified

by a probability field. After a very long period Ot time, the probability

* field tends toward uniformnity over the sphere's entire surface and the par-

til~ origin cea-.ses to be important.1 1

G. I. Taylor's theory- of diffusion by continuous movemnent is a solu-

tion of, this problem "or small -scale eddies. The theory pr es ented here is

an adaptation of Taylor's theory for motions wn the scale of the general cir-

culation. In dev eloping our theory, we incorpor.Ated restilts obtaitied froml

* empirical studies of the trajectory distribution funiction.

Tbhe probability density function Canl, of course, be integrated withi re-

s pect to space and time in a no inber of ways to obtain the desi red distributioni

a ~function. in planfninl~ the p set ioiof the r etica I calculations , weu found

thalt echtl SPL'f-ifiC ciaS ssOf aIppl icatiton required a pa rticular niethod of inte-

* - gration to obtain the .1nos t us et ul formi of thec distribution function. One pt-t e

s entat ion is inc ludced wllic't, wv i lt H ot Very us ll for' conlstant-level balloiiO

operatlions, niay have iniportant a pp1 icalions to Lit' pm old et of di ffuS iotl kJ

particulate rnlatter , SuchI as fi~ IAlout from11 iat on1i C wcip

tConsic anm air 1p.11icj const ralid to Ilove Uri eott10 - lve srt.ar v*

This particle is located at 0 at time

t =0 (Figure 1). What is the proba- I

bility of the particle being located at

point P, at t = T ? (Note that in this

statement of the problem we prescribe

only the end-point of a displacement,

not the path.) Making the usual assump- y

tion that the twc displacement compo-

nents, x and y, are normally distrib-

uted about their respective means, X 0

and Y, the probability of displacement,

OP, may be expressed by the equation

.." for the bivariate normal density func- particle

. ion,,I (!x-X 2 (x-X) (y-Y) y-Y,,

p(x ) = 7 "-+ ())7.xv V I-0

-, t"P 'dx x - L x dy y

where 6 anid D are the standard deviation of x and y, rcspectiv ely, ai t, iX y

the coefficient of correlation betwoen the quantities x" (x-X) and yl (y- Y).

13y definition

x = i (t (2 ad Y V (ito 0

whore .1, and V are compotents o" the cIlhnEtologicdtl mean whid elncounte dby

the particlu in its path.

Il: the tt rato tpher e and upp or troposphere, V is timall and diftictuUt to d,-

torminl. This is especially true, in middite hatitud-s. [in Gct it may btt a.u.nmed,

with lit~tl lozs ofo gen r~ality, that the Iean wind in th 6.v rogion h is It c

wit tlOw mOrt clo iely o i\wnk:l Ot! nivtm winid v\ rtiatiu%.

l hle I is ,I typict I st fpl , nv o n uppvr-lIv, ,,v inuls, t.%en 1ziru tit

lI Undlluu of (.kew,-ysics, NottE thAt (tl' V\Ar ation's i 'snicrably greater ,th

I"" PI,ect to latit vo thal% with rtspo,ct Io longitutdi.

'4 ..

* . .,

Table I: Mean wind speed (knots), 50, 000 ft., January

LATITUDE WEST LONGITUDE LONGITUDE RANGE1

i20 100 80 60

70 34 33 33 34 1

60 36 29 26 29 10

50 38 41 44 43 6

40 71 84 83 67 17

30 61 58 48 37 24

LAT RANGE 37 55 57 38 1. .

It is evident that if we confine our discussion to specific geographical regions,

to a reasonable approximation, we need only consider the latitudinal variation

of the mean wind. Therefore, we introduce the following modeling approxima-

tions:

Y =V =0 (4) and U= U ) (5)

where i, 1, and k are height, season, and geographical region (such as North

America, Eurasia, etc. ), respectively, and 0 is latitude.

It can be shown theoretically that p(x', y t ), the coefficient of correlation

between the displacement components, is proportional to p(u' v'), the corre-

kAtion coefficient betwven the wind components, where ut = (u-U). and v' = v.

Brooks5, et i, 3,huch, and Court 6 conducted indepundent studies of statist-

ic."al prc e rties of the winds in the free atmosphre over broad regions. All""" three, authorities agr,., tht p(u', v') is smal. Table 2 is a typical example

*. extracted from Court's paper,

Table 2: t;ot'ficient of correlation between u and v

i 11'. PANAMA CANA. ZON I MT. CLEMENS, MICIH. GOOSE BAY, ,A B.

I(unl) Wintrne iteer Summor Wintor Summer

.'* . Q000 ,124 .054 10 I -. 119 . 022• 10 ,07 .04 1 .091 .0it) -. 087 .027

. .259 -. 174 .129 -. 154 .112 -. 063

. ° . .... .* ,. . . ..S. . . , . . . . . . . . .

In E q. (1), pwhich appears in both denon-inator and e:-ponent. -aust

be comnpared to unity. Since the first and last terms in the exponent within

the brackets are of the same order, the value of p within the bracket must

be compared to unity, From a complete study of the evidence it becomes ap-

:2 parent that, compared to 1, p is small and p is very small. Thus we

introduce the assumption that

0 ~ (6)

The time dependence of () and G is fully discussed in Appendix IIIhere the equations 6 G t6an are developed. It is also

I (t u yn y G(t)dv5

shown in Appendix Ill that empirical evidence warrants the assumption that

G (t) G (t). Therefore, it is evident that 6 =(j where q is the ratiox~ Y y x

(5u

Consider now the quantities 6 and 6 .With the sample sizes genier-u v

ally avcailable, 6 can be determnined with much greater precision than 6ri v

Therefore, it is reasonable to as sumne that anly error i the estimated value

of q Wvill be reflected in 6 rather than in 6 ~. Let uIS the-n inlveStigiite theyX

conscquenCes of as suiring that q -= 1. If q is actuatlly greater than 1, then

the dispersion in the y-direction is greater than we~ have assumed. On thle

*. othei' hand, if (I is actually imaller than 1, then WVe have overestimated

the disperSionl inl the Y -dir ectionl.

In applyinkg the then rv to balloon systemis, we are usually interested

inl obtaining" the great est potss jl e cone ent ratiOnl (thtat is. loast dispersion)

of displcements. As Will be shoW n lat er, the rtesult s of sev era I studies in -

dlicate that, in igenoval, (1- 1. Thus it may be seeni that ainy terrors resulting

froml the (I' I atssumlptionl Will bo In the coilservdttivv direction wiht.11 the(

thcenry is applied to balloonl syStemjs. W e will I O\V diskcuss thet antdeOf

the orors.

b 1ols et a I i ttI S i V oIy inI ve 8t i Ite and itand conludked

* ~ thtthr WAS n110 oicLtdftrni botween these. :ktaititios in the free:tner. I IIks ut-11 cnit It dtaited invekstigatitill, onl a hiolisphleric

b-ss of tho ew md statistik. 8o I'lA sillgle yvar. IHis finldings a(grut with1 those

of Brooks' in middle and high latitudes. However, he found that in low lati-tudes q is significantly less than 1.', Court's more recent studies show thatthere is a consistent variation of q with elevation at widely separated sta-tions. The ratio, q " is 1 at the Jet Stream level (about 40, 000 ft) and de-creases gradually with increasing elevation. Court also found that this effect

is very pronounced at low latitudes, but it is much smaller at middle andhigh latitudes. Let us confine our attention to middle and highlatitudes andto elevations between 40, 000 and 80, 000 feet. All authorities menticred above

.. agreed that the erroxr in assuming 6 = b is negligible at-40, 000 feet. Our

own studies show no significant difference between 6v and 6 u, even at 80, 000feet. These results are not necessarily in conflict with Court's data. Heused data from single stations, whereas our estimates were based on popu-lations of winds taken along single latitudes but from a number of adjacent

2 =2longitudes. By definition, 0 v - . It has already been shown that

the regional va'ue of ( )2 is 0 (Eq. 4 ). For a single point, however, (V)2

is not negligible., Therefore, other things being equal, the value of 6 willv

always be larger for a region than for a single point within the region.

The foregoing discussion leads to the assumption that.6 6 = ~ (7)

y

.We mnay now rewrite E q. (1) in various alternate forms:

p (x, y, t) p (x, t) p(y, t) - I{(exp . .. ... (8)

p~x~) = C ..ix)~1(9)V_ r 0 1261

,p "Y,_

1 ':2

We may write i (8) in lpolar form by introducing a Ox I + y andw.-) integrating fr.rn( 0 to 2"r d

S D

,,

. .6

The preceding equations are familiar forms of the circular normal density:' unctor.,Broos etal,3 4

function. Brooks, et al, ' used these equations to compute the probability

N o~f wVinds.

3. ANALOGY BETWEEN THE WIND PROBLEM AND THE DISPLACEMENTPROBLEM

The displacement problem can be viewed as a logical extension of the

wind problem. We will compare the solutions to these two problems at every

step to illustrate their relationship. In this analogy we will retain the sin-

plifying assumptions and approximations which have already been established.

The appropriate wind equations which correspond to Eqs. (8),'(n, (10),

and (11) are:

r-

p(u v) = p(u) p(v) = [exp (' +v 1 8a)

1 L ]"" (u) (9a)

v'r 6, 26i

p() = XP (I1a)

. i t

where 6, is the standard wind deviation and c 2 (u') + v

NW e note that the indepoendent variables for the wind equations are u and

v; but for the displac inent e(uations, in addition to x and y, the va'riable, t.

a lso appears. In gentral both the wind and displact ment probability density

.- :-,' functions art. compho-tety speciiied when two primary paranieters, U and 6,

are gien. I lowvver, the empirically determined relationship bI':twteen U and

" dv,'lopd ii Apendi II, reduces the re-quilrCd number of primary para.

meters to a single onc, namely, U. 'Ve defint, secondary paranetors as those

"'C ...'

variables which represent the time and space dependence of the mean wind it-

self. In this paper the secondary parameters are height, season, region (in

the sense defined previously), and latitude for the wind statistic and similarly,

height, season, region, and initial latitude for the displacement statistic.

In the wind problem the specified wind (u, v) can be represented as a two-

dimensional vector in plane surface. However, the vector x, y must be mapped

on the earth's surface which is spherical. Therefore, at some point in the so-

lution of the displacement problem we must introduce a transformation to spher-

ical coordinates. One additional complication arises which is confined exclus-

ively to the displacement problem. Since the component y represents a change

j ,in latitude, and since both U and bi are functions of latitude, we must somehow

incorporate in the probability function a continuously varying population of winds.

Just how we overcome these difficulties will be discussed in the next section.

4. POLAR INTEGRATION OF THE PROBABILiTY DENSITY FUNCTION

*. Any desired form of the distribution function may be obtained by an ap-

propriate integration of Eqs. (8) to (11) and (8a) to (lla). A useful orm of the

wind distribution function is derived from integration of Eq. (lLa). A rather

complete discussion of this function will be found in Brooks, et al. 3,4

Integrating Eq. (Ila) with respect to c:

P(C) exp. c d I exp ( (Za)

This function may be interpreted as follows: Any wind with probability, P. is

the vector sum of U, -th vector mean wind, and a deviation vector of nagni-

tud'e, C, where

o C 1/2 (13a)

An analogous polar integration of the displacement density function(Eq. 11)

* is useful in cortain types of problems. To demonstrate the niethod of computing

this form of distribution function, we will apply the theory to a specific example.

9

The problem we have selected is the mapping of the 0. 50 probability curve

for the following specificationis:

Region. North AnxericaHeight: 50, 000 FeetSeason: January

Initial Latitude: 40NTime: 1 Day

The profiles of U and 6. are showvn in Figure 2. As a preliminary to the

discussion of the displacement function, we solve the 0. 50 probability prob-0 0

lem for the wind occurring at 40 N. We note that the meani wind at 40 N is

75 knots and the standard deviation is 35. 8 knots (Figure 2). From Eq. (13a):

C 35. 8 Ln 1 4 1/ 35. 8 x 1. 176 =42 kts. (14a)

The solution is illustrated in Figure 3. The probability of occurrence is at

least 0. 50 for all winds which can be represenLed by vectors with a common

origin at 0, 0 and whose end-points lie within the designated circle, Thus

the circle of radius, 42 knots, and origin at the end-point of the vector, U =75

kniots, is the locus of the limiting vector end-points for a probabiliLy of 0. 50.

The solution consists of two independent parts: (a) a mean wind vector, U.

A11d (b) a de.,viation wind vector, ul, ef niagiiitude, G. Figure 3 also clearly

shows that the. component u =(U + ul).

t or the displacem-ent distribution function we seek a sizznilar solution.

)ta wVe further rcquire that the 0. 50h probability curve be. mapped onl a spher-

ik,41 earth, For this purpose i.. .s conve~nient to use one degrve of latitude on

Oe earth's8 burface ats a unit of lengthl, anld one4 (liy as it tim-e Unit. It is also

,onlvenienlt to ;1do(pt a Coordinate Systein ill which ;"is Whe lonlgitudo coordinate

tnwasuriod eastward t roki the initial lotigitude, anid Q) is the true latitude. Here

Siii the wind prohi em, te solutioti is resolved into two indepedt arts

liv coiputat ii~ of (a) at iean is Alceiet , anid (6) a deviation displace -

111,e1t With c onipullenits ., 1 0). Also, . "i'v + ? . As \V5 pr.e -

* i ously stat ed, anly dlisplac ellivia in latitudo inevulyes it chaige iin botli U and

. huLS, unhlile thet winld pruhiemi where U S indept-ident of V , in tlli et"

Is I A lne l tio 0

1'

55

50A

LU45

40

* L 5

00

10 20 40405

DEG. LAT./DAY 102304

KNOTS I I10 20 30 40 50 60 70 80 90 100

* i~~~lu - 2. \!c~in wvilli .1'A i.-' Indat d tic* 1alton, -trh A. i t ,~-iui*v

50-201-

40-

30-

00

-20

-10,--30

50-1-201 0OGL/OY 10 20 30 40 50

* KNOTS -10 20 30 40 50 60 70 80 90 100 11O 120

Ftgur r'i. Citk u la r normnal d w ibut ion of w indsi, North Ameuricai,

Ov''ry el

For reasons which will become apparent later, it is nccessary to solve

the deviation part of the problem first. Figure 4 is a solution, for 40 N lati-

tude, of the transformation equations (Eqs. [ 4] and [51) developed in Appen-

dix IV. The radiating curves are arcs of great circles, marked off in values

of s, the linear distance from the origin. Each great circle is identified by a,

a spherical angle measured from an arbitrary meridian, Note that both / and

0 depend a and s. Thus we may write the following symbolic relations.

(Note: The actual relations are Eqs. [4] and [51 of Appendix IV.)

=A (a, s)

B (a, s)

The general method of solving the deviation part of the problem consists of:

(a) Integrating -q. (11) with respect to s along each value of a.

(b) Determining S , the appropriate value of s for P = 0. 50,

(c) Finding the /and f0 values corresponding to S , using Fig-ure 4,or Eqs. (4) and (5) of Appendix !V.

Before proceeding, however, we must first decide on a method of accounting

for the variation of 6 with latitude. We shall, therefore, describe two dif-

ferent methods of solution and compare the results obtamied from each for this

particular problem. The first method, called the "Simplified solution, '' is

quasi-linea r, while the second method or "complete solution" io non-linear.

From a computational staitdpoint, the essential difference betwiueei the two meth-

ods is: in the simplified solition the 6 profile is introduced iifter integration

of the probability function, whereas ill the conipleotc Soliutiun the ( proftie enters

tho probi, m t3,tort: intgration.

'IV e now introduce the circumfilex syllhl ( ) Lo dtnotc a quantity which.

4 is av.ragnd Ubetwe, e t the initial latitude (in this ca o N 4 0N) and at itude f

It will be und.rstood that the owthd of averag I is to! i C ifid in each ,casev

O. TMiltli1) :LlU N 11% %II , .iI't.,A(F;MI.Ni bk\ IATION

4 eVit 'lfat d ,v L o uln"t'' th;t the iltInll willd IS verv tt ere the iAIn. !ite, ,

S V) Lis)-"; , . t s u - Os ! - ,0 9'20 " ,

..

.,. . ..

0(

0 PO

00

Co co00

00

N

t-I-

0 (N

030

0L t Z Lt'Ul fn~

and

S d0 (in 4)/= G (1) i (in 4)

and, since G (1) = 0.86,

S 0.86 x 1. 176 (5 = 1.01 6. (13)

where 6. is expressed in degrees latitude per day. Since S is independent

of a, the solution of the deviation displacements for constant mean wind is acircle on the earth's surface with a radius which is a function of U'. If we

compare Eqa. (13) and (13a), also Eq. (14a), we see that S is directly pro-

portional to C. In fact, S = 0. 86C. We have seen that C represents the ra-

C dius of the wind deviation circle. However, the radius of the displacement180deviation ircle is not S, but T sin S. This difference aris,a because theradius lies in the plane of the circle and not on the curved surface of the earth

where S lies. Figure 5 is an enlargement of Figure 4 on which the deviation

displacements of.the 0. 50 probability curve have been plotted for various valuesof U, Since these curves are axially symmetrical, only half of the figure is

s hown,

At this point we introduce the profile of U . We assume that the solutionat latitude, 0 , is the samie au the aolution fur a constant nmean wind of "lag-

aitude, U whCWUr

f U d0

The profil of U is shown in Figure 2, page 11. Tho napping .3* the 0. 50 prob-

ability curve for the deviation part of the solution is shown in Figure 5.

u. COMPI.t1TIE SOLUTION OF" THE DISPI-ACE.:M ENT 1JL.V IATIONS

Egquation (i) is rewritten:

p (t) = exp . (lib)

0z1

-2 > " -_ ...-: ..,, ,, ,.- , , .. -v_ , .. , . .--, -,, ,, . . " " " .". .,

1680

5070

lo

I-60

09

100345,-

As20

35-

j 2005202

X.5 (DEVIATIONY LONGTUDE

.10 15 0 2

(DVAIO OGIUE

6 is a value of 6 appropriate for the population of displacement deviations

from 0 to S. Within the integration, 6 is a constant.* Integrating Eq. (lb):

P (L, S) : s exp - ds

* - exp - S... (1Zb)

Sa (n 4)1/21 = 1 4 (13b)

= . I (14b)

We defiae &1= d ds/S.

1J

in general, 6. will vary with a. Thus, unlike the solution for a constantmean wind, the conplete solution is not circular,

" The reason for this stems from the definition of a probability distribution

-.' ction. To show this, -we will integrate Eq. (11) in a straightforward man-

ner, considering 6 6(s): SA5

. P(S)r . / exp . .. doj 6

Integrating by parts:

"(S) 1- expA- +xp ,- - ;dd

P(~41+exl-d\,'. ."A s 8 - oo,

,. . .3 t d

By definition, P I at S ; but, in general, the integral termn on the right

does not vanish. Therefore, the above equation is not a probability function,

17

'I 7

h '..,Jx ..'^" ..'!t ,.,. ,:, -: : . . ,.".,. * ."x.. '. '-",,. '-' "*.:•* ; , .' , .,"L',''

Table 3 shows the results for this example, obtained by numerical

integration.

Table 3. Computation of displacement deviation (complete solution)

DEGREES DEGREES DEGREES

LONGITUDE, LA TITU DE LONGITU DEROTAT ED

SYSTEM

Q S

0 13.33 13.46 26.5 32.5

45 14.08 14. 22 29.4 36.2

90 14,61 14.76 38.5 1 40.2

135 12.40 12.52 47.4 32.2

180 11.65 11.87 51.9 30.0

225 12.39 12.5?, 47.4 -32.2

0 270 24.61 14.76 38.5 -40.2

315 14.08 14.22 29.4 -36.2

The solution is plotted in Figure 5. The difference in results from the two

omethods for the deviation part of the solution is practically negligible.

7. SOLUTION OF TH- MEAN DISPLACEMENT

The mean longitude displacement for a particle moving along a latitude

circle with the speed of the mean wind fur that latitude is

:/ :(U Se" 0) 1

In the uom|plute Solution, where we consider displac * .ents along great circles, a,

S(U see t (15b)

whevre'.,4.Whe

"(U ec 0) (U sec 0) ds 11)S

~~~~~~~~~~~~~.,........... ... ....>.,.'..:1'.:', :..,............ ........ ,, ......... ,.,,""* , " .", ' ' "."" v "-.,-4 "" , " , '" . "" 4: "" ' " " ."' .", 4 , ' " ." ' " : ". -"

In the simplified, solution,

U (sec 0) T ( 16b)

',4" f U d¢ sec- 0-where again U -fand (sec = o sec

0 - 40 Sa

The two solutions are plotted in Figure 6. It now remains to add the two'P.S parts of the solution. The final results are shown in Figure 7. It may be seen

that the difference between the two solutions is quite small. Nearly all of the

difference is produced by errors in the simplified solution of the mean displace-

rnent, Since the mean displacement is proportional to time, there wift be atendency for the errors to increase with time. On the other hand, the fact that

S also increases with time will tend to diminish the errors. On the whole, itQ1

may be said that the simplified solution yields a good approximation.

8. MAPPING OF THE PROBABILITY FIELD

% To illustrate the nature of the probability field, we have plotted probabil-

ity maps for an initial point of 40 N and 120 0 W at 50, 000 feet for January and

July, and for time periods of 1, 2, and 3 days (Figures 8 through 13). The

computation method used to obtain this series of charts was precisely the same

as outlined for the complete solution. These maps may be interpreted or used

* in several ways. As a simple example, suppose that a constant-level balloon

is launched at 50, 000 feet in January from the initial point. If we have no Cur-

ther information, then the indicated probability gives the correct odds that the

balloon will be located within the airea enclosed by the given probability curve

i after t days. 'lhus, the probability is 0. 25 that the balloon is within the 0. 25curve versus 0. 75 that it i: outside. Similarly, the probability is 0. 99 tihat

tile ballo on is within th t). 99 curve versus 0. 01 that it lies outside. The most

l)'oVUbIlc area is enclosed by th 0.50 )robability tcurve.

a With suitable retinemeuts, this pr'esentation could possibly be used for

problems involving atomic fallout. In thits case, wt, would need distriution

funct ions at "1eVe ra I ala s pheric levels. We cannot, of course, pro vidc dtailed

,,: .. .: .. .. ,. -. ,,.. . .. .. . . . .. . , . ,. . . . . . . .... . . . . . . . . . , , . . ..

55

50 -x

45

40-

- x

30

* - COMPLETE SOLUTION

SIMPLIFIED SOLUTION

*30 35 40 45 50

*4 ..A..MEAN LONGITUDE DISPLACEMENT

i g ~ ' The 4 1 ution fo r tho e allc dis pla collielut

20

0(u

U)-

LI-

/O N

('3)/n a -

•,0. /"" N "- . ""\ " "

-?0 , " . .._.}:..-

60*~40 b .

.-**0 ' .. ,% -

* 99/0 o60 "

140' 0%

2 20

200

F 'k. 8 PROBAFIlL17Y FIELD

JAN --. 50,000 FT

-NI N IAL I'CINT, 40 N,120W

I "IJ0, --TI I 04,Y

.W

Figure 8. Probability field, January, 50,000 Feet; initial point, 40 N, IZO°W; time I day

StC.

000

#sO,

• ' " , ' ' '," . '. ' ' ' .,,. ''''-"- . , ", ,. . .",' ,.. " ,,,. " t . ' " , '' .,,.- """ * ., . : , ."

N2

)25/

400

u 0

Fgu rv 9'. Przobabil ity fit-ld, Jamiary, 10, 000 Fett in~itia l poiit -0 0 N, 12 ~'0W; tifi, 2 cd,%y6

%

% 2e

500

N~ 04

C 0 " 1 ' *

Vi r 10 Pr b blt fi ld Jan ar .50 00 Fe tinti lpo nt 4 . 12 W; time 3 da

• " ' ( "" : 20

°

""" .' L . -. -,.:.;

4 . ' " .. .' 0

5% 4 ..'

- ,

00

20

-. 5000 b

- ''A

-~ t''.. "*',"-.-" ',- "-' . -,,

,. ... ." . '. . ...'. " . ", -' , " .

VIP

20 40 -2

060

860

'40

U '20*

20'

flG IZ PROBABILITY F i&'IOO

11 .. Prodbbtity Hotld, Jiuly, S0,00 I,' t, illitlutI pkm t, 4~N ~''V i~

%*0

<4

V

20

1,

99 :>-,~

* N

. *'..

N

x- 7

I.,' . ..NC'' ~

~ // -

----- .~.--'-< .-. '-----' - - -

S

. ,.

* information on the amount of concentration since this is a small-scale phen-

omrena. However, we can provide iniformnation on the location of high concen-

tration areas.

The foregoing probability statements were made with the understanding

that our only available information is the time and place of the balloon laun'ch-

ing. Now, suppose that we are given one further bit of information, namely,

2 the initial wind velocity, Our theory permits us to construct a different set of

probability maps which take into account this information. In the first few days,

* the areas enclosing the probability curves would be smaller and the central path

would be different. With the passage of time, differences between the two sets

of maps would tend to disappear. In other words, the added information re-

duces some of the uncertainty about th-e balloon's location, but the information

- becomes less important with increasing time.

9. THE~ DOWNSTRPFAMv PROBABILITY FUNCTION

For balloon operations, the m-ost useful probability statemient is the

"Dowvnstream- Probability Denisity Function"' (DPI)F). This may be defined as

I the probability denisity with respect to latitude of an x-displacemtent at least

as large as a specified value and occurring within T days. Mathematically,this denisity functioni is expressed as follows:

D)P D1 p' I-U ) dt .(17)

Ifze w(ish to iiiterprottis fuitki t ernis of Jii'ootlop~erations, we

uvo alutg t-rodof timo. In calSe 180 \ nd y a1. ('11011t )t~jp4Lt~jflt5

froi Oe auth~i site am refers to tho tinie initerval after launcing. The

terill - 1-1 r eeea tile p roport lotial I numlber of balloonls to be founlddowilstr kilfoil x at tilm. t (1-P ers ots the rate at whiich hal -_

loolls~~ ~ ~ ~ (Io pas xttttet ) rtepiresvots tho rate at whichbalools fOWdowiilSt1*v.ktm pitit tho poillt x. Y at ticov t . V'itally, 1Vrj. (17)

N rt.It's(.1tS 0t. f~ii lnber of ba Itoonsi Which 11a.y be vXpo~ted to have

flowcd downstream past the point x, y from the time of launching to the time T

The various steps in the computations required to solve Eq. (17) for a

specific example are illustrated in Figure 14, For convenience, we substitute

for p., the L:obabilit for a small increment of y, namely, -Py, t=(P Z+/\y -

The DPDF musL be mapped on a spherical earth and the variation of C

and U with latitude must be taken into account, just as in the case of the polar

integration. It has been previousiy demonstrated that use of the simplified so-

lution introduces very little error into the results. For this reason it was de-

cided to adopt the simplified solution for computing the DPDF. Accordingly,

Eq. (17) was solved for various values of U, s, y, and t. The next step is to

eliminate x by the transformation, ? = x sec 0- For y we substitute.Q - o.

We now have a table with the parameters, U, Oo$ X, and 0. This is Table I,

Volume li. The final step is to substitute the values of U for U for each re-

gion, season, and height. Hereagain

A 0 uda0. 00$ -Q0

W,- tnow have a set of tables for each region, season, and height for various

values ut , and Q. Two such tables appear in Volume III , namely, Table i,

No nicrica, a, Ta<ible t, hiura ia.1 hl, rc *re two limitations on the downstream probability concept. In

toe fitst dace, on a spl-riical eal rth sontle air particles may take a decided

itortlhw.vard course and reappear at a lower latitude very far dowIlstk'.t.am. While.. •4 b.: ;ea ibi e to forunulat e a solution which incorporates this type of tra -

". \V( dt'Cid to o CU\ ltcs LI.it' c frtjat c.51siU[TLlti&. Thu. tho tlotn-

k' t . i V o'lumes I1 and III rtafer oiily to those m jtioi4 which aro twi en,-

Shr . Iimivtaioni o0clurs wihlen tlt laitl .1-olit willd chlla es Sign at

z I - - ttud-. 1h16 1111r4, for vxatnplte, over North Anierica at 60,0Ot00 fe-t

: t : tr and at 80, io0 fe, I in tht fall for crti i 0,tia 1,ttvUI11. (')ut 60o1tu-

t,~o ,,ablo.,, ollt" \vt+8',Vr tIALWp|'tisacolnots a.i

.. . .. . . .. .. . . . . . .. .-. . . , 4. 5 : : :., • ., ~ .. . .S. . . ...* ..... , . . . ~ -. * .... ... .. i i i ,? 7 ' "

0

0*-T IL> (0 ~

04)

I C L

L)

64cc T7-(O.)kNlfj8 I V lk0

0~ -r~ o.-30'-

APPENIDIX I

WY2ANALYSIS

To obtain a com-plete solution of the win, climatology proilem, certain

basic quantities must be plugged into the stati tical equations, These include

the variation by season, l1atitude, andt height of 0,e mean inzitantaneous wind

and its associated distri ution functions, Thlis is analogous to the solution to

a vast jigsaw nuzzte- wh.ere many imp.ortant pieces are missing,

The fi rst --tep in tche: skolutic'n was a rather compilete survey of available

literature. From tis we wet e able to deduce a somnewhat skeL-chy broad-scale

picture of tile windl distribution at uipper leveis over the earth, It became im-

Tn1ed1I a Le Ly d rd.7e.LL L th a t fu i e' - 11- u tLi I. S- 110 Vc to0 r r to inirA

or "Ibootstrap" techniques.

One of thE! d-sired qutlantities is somne measure of th: dlistribution asso-

ciated with the mean wVind, III tho wind is norrnalv distr ibutedi, a sigle para-

meter, such as the standard vector deviation or the consfancy (ratio 'between

the vector mean wind and the scalar mean windl), can be Used. For cun teu-

ience and ease of compiutation, it was decided to comlputte the Wind constanicy.

Tlo Tminimize the known bias in L'he ob~ ervod upper wmnt recoris , the differ -

enc e inethod of avera-,pini,, w. l ued throughout h1e '4tudy,

W e decided to coeeit on an imt ens ivt. a na lv sis -hr ev data s ources,

(a) a 9-year record of 1 5 Ui. W.'eathe r B'u ro, S>tations; (b) a ser. s Of 3 to

4 years Of upper aitr s vnopttA na p s o( the WNest (;tcrmnr ee ooiavie t3 evl: 00rb 2), i000 ft) , 225 rn (36, 000 Rt) and 96 tub (5-1, 000 (t);

and (c) rawv irisorde r ec ords fi-orn, Bor~ljn anid s\'t'ral other Gormaii httiws.

[ac 'A'1Of thle Ge~rman11 Syvnoidkj ic hats for the moniczhs Of lazmaecv, April,July and October was phiotographitrdtly eiari'rdk to I 5 t~e of 1511 by 20W, at

iso tar hs (line('S Of e qua IWt.1id 5 peetd) we0-k rrconstruc :tvdu i the Ctn ni r)Ut-dg cost roph ic w inrds as well Cis, ohsver vd w Id spe-eds. t"-01' 4 rIMt : nil e .hr

me an vector and srahir wVinrds worte c oniputed for ioach la! it udeo tikason, ati

height, cornbiriine, the data lint'" 4Lritde 30L W to 30t! IWOu aSinlte St, I StiC.

crj

-0 N0CLC

00

0- -U

0 C) 0

(D U)~ qs. to

CifOViiI NUA1

-,::i In all, 1000 maps were analyzed and 35 wind values o, utedi from each map.

The United States data -were summarized bv punched cards, The ratio

between the scalar w.inds at any level and son-ic standard level. was the most

stable statistic that cou ld b c' found. For this rcasen all :c dta were re-

duced to ratios between the scalar wind speeds at given evels and the cor-' n te s 31_iiniri:e -val - o rthis sta-

responding scalar -vinds at 5 ) .,-oiters .u r C; d asof t a

tistic were smootled by -- ,-ionic analysis ;... :--atched, i for Level, with

- similar data from Europe an sources. In this xav the latitudinal variation of

the scalar ratio was stablisli.ed, F-or levels above 96 mu- , German winddata were used in ti.se compa-rison, These were supplemented bv data from

nianuscript Norther,. 1-i-- is nO erc s vaun o as P5 .1c' i95 a, ; "00 mb (53,000 ft)

and 5.) nb (68, 00 t). !,g ' 15 is an cxc, C t e o one so. Lt ratio analysis.

The trough lines in the upper portion of the cliart del;sneate the boundaries of--

Hax-ing established tie s- clar wind ratiosl, it was thien possibte to con---

pet retiabe scal ar ,vi.d spoeds by n-ultioltvM,,, the "at sos with the appropriate

_i ear. ,calar wx ,ictd at r,00 mb .vhich art! kio,.t, xit't , ' ic-.. bl accuracy," This left the vector mean winds to he ft. .. i,,. [t was foun( that this

statisti was (lit ult, to deterine accurat:e :--, i lit-nited sa lnpe because

rof variations in wind direction. An nten i-v- study of the 500-rrib United States

da11ta It'd to tht discx err that a reniarkably k i it reittiotnship vxisted be-

\tw een lt sIala s c' a Nr t'- I"I-tI V I I ls . It ':x I k' t u)W- that strong winds

a l'me cnn tcctlll/s to i ,!h\b -ix'ds, SMON n - tt t' 1 the vector and

Scalar -. a:)+[ in, a incastire o! - ' l c stalwy, tmiz tatlto sittmuid rise as the wind

Silt m. i:S. [h .is ,' " (Ct.ti w, be te c e It wi; ,iLs( h fourt that the presence

of O w tht w I ,l 'tll . t ' tzlll VcUntcs ci: .1\tltitiun'i in- t-as, e in cons ticncy over and

ab, ort t etf k f" win.d speed .lono. 1hi-: I- t cc is alst, known (utlitatively." 'l'~hiete rel~ititriships art' ,i vii tin tiitilt',' Ii. li, detee<< I.) \O,,htil 'lit' Uniitedh

.itat~~~~~~~~~ th'itui ~n:U dd-drvdci'-- -2 i 'o bli tha;t this

wa," Za ii iil- 'rsst st, it" 4, iii- t, ern itt. dit x'. -- wcrnhq , -s Attenpting to v ihfy' \Vt s SU; ' slol\,tSil -41I--leD.l ta ls ,l .00 as eli ca, it tro:: ,.- Iia 'a . t-giopii, tsi vre itoy

.. e i P r t it eitl himl- V.,-ried. 'ian t*. , ih.. ',: iixx'id tt ,-i" l ) :tII l Uop

fPT iic *. .ih---., s ii,., ',, ,-ll,-.t ri's-ntis itt ts, test. I "',.--i ,.A- ,srit , '.rc triy ,'oticlude.'

:1,

.4.4

Il

40 .- r t -,.. " Ol ] , , I I-

35 A Wi, ,N 10 DEG. LAT OF MEAN JET

B MORE THAN 10 D-EG LAT. FROM JET

,; ' ,I ! -. 1

20j... - ... - - - - - - --- . .... . .,. ----....--- 4 --

, 15' '.. ..-,...I-....... ..... I .. ......... . . . .

0 5 10 I 0 30 35 40 45 50

SCALAR MEAN WIND (KNOTS)

i" i;t." l , I .tio' ),et,.% cn scalart wind and vector wind in westerlies

35

A..A'E lLRI ; ,WITHIN I) DFG LAI OF MEAN JET,

H. W3TEPLILe --MORE THAN IL DEG LAT FROM10 MEAN JICT.

, , C" HIGH LEVEL LASTERLIES. , A

C. . :i

0

N -0

.' . . ' h

-- . .' -.... .. ...

W' .40 4S 50

It 'OH %I! AN %%iNtl (K"NOTS)

~~~~~~I I ,. .. 4ta.An kndard vok-toi- doviation

...-

o , " ..... .*. .. *. .* ..... ... ......... .. ..... .-.- -#-,.,"". . -

5"-. ,. -....... .. *.*...'i.. *... .. *. .S =. i .I'5

S...--.

* S*.S S5.

that the mean wind uniquely determines its distribution function.

Table 4, Correlation of test data (96-mb winds j kts over Europe)

MONTH LATITUDE SCALAR WIND VECTOR WINDPredicted Observed

Jan 70 37.0 27.2 26.9Jan 60 46.3 37.4 36.9Jan 50 37.4 27.5 26.9Jan 40 37.6 27.6 27.0Jan 35 40.9 32.1 31.3Apr 70 27.3 19.3 17.1Apr 60 28.8 20.6 21.4Apr 50 22.9 15.3 13.4Apr 40 29.0 20.7 19.9Apr 35 35.7 27.0 28.3Jul 70 16.9 8.5 7.0Jul 60 20.0 12.4 11.6]ul 50 20, 6 13.0 1 !. 0Jul 40 24.8 17.2 16.7

* lul 35 26.8 18.9 21.0Oct 70 27.5 19.5 i9.80Oct 60 28. 6 20.4 19.8Oct 50 25.3 17.6 16.0Oct 40 27. 5 19.5 18.0- -"Oct 35 29.7 21.2 21.9

Correlation coefficient, r 0. 99

A similar relationship was derived for the easterlies from what little data

were available, Considerably less confidence can be placed in these results.

*] Nevertheless, indications are that the high-level easterlies are extremely con-

stant \vinds.

Using the data from Figure 16 we computed the relation between the vec-

tor mau wind and the standard deviation (Figure 17).

- " "-"35

-- I%

APPENDIX Il

UPPER WINDS OVER EUROPE :' ND NORTH AMERICA

The mean zonal winds and standard deviations used in this study are

presented in Table 5. Above 54, 000 feet the sparse European data. was sup-plemented by more com-plete data fromn North Amnerica, Thus the winds at

high levels in Europe, while not as reliable as th-e lower winds, may be re-

garded as a first approxim-ation to the true winds. The standard deviations

in Table 5 are taken from Figure 17, Appendix I.

Figures 18 through 23 summnarize the imrjortant variations of the winds

in Table 5, namely, seasonal, latitudinal, and vertical. The 80, 000-foot

cur'ves are typical for the region abv 0Q 000 feet Te 410, 000'-foot cre

are characteristic of the layer below,\ 50, 000 feet.

1.SEASONAL VA RIATION (Figur(,s 18 and 10)

The seasonal variation of winds at 410, 000 and 80, 000 feet over North

America at 40 N (1Pigure 18) and Euirope at 500ON (1'igure 19) was obtained by

harmonic analysis. Over boith regions ;Asml: wave wvas saffic lent to repre-

sent the 80, 000 -foot data wvhil e two waveCs \,VCI' 17r? rjir.'tl to fit the 40, 000-foot

winds.

The 80, 000 -foot s easonia va elationl for kwtll Continentsn c onisists of a

single 11-axiirni1 of w estkerli es in ii-idw id tc and a sinigl c maxim-umn of vaster-

lies ill 11lis uiler With rev or gAIS Of thl r cc~ in April and0[.ptciiiber.

A~ At 410, 00() ot tht,(lle -douk V A V S t rue ~tul 0 I t 11.: \\ tl'stt cr i CS is very pro -1unMC ed in Eu 'An4-pQ atd u iit - in i Nir h A niori ca. It'cr re twoj distincet

maxinia in [-:uropv, tho priniary 11i Nt uc-h -10 theo sccmidary Iii Novvniber; and

two rninma, tho ma10jor ill Augus,,t -loti tinr inilerur 13y c~ontrast, theNorth A'nirjcan t-tircei show%% a ~ v peiwm 'laxtilumu of 100) Ms in January

and a melr'e skigg's tlon oi a T ' naivn i nt hl' h e e fi ct of thle

SCC.ond Wave ill this cast. is to flait ctl ouit thEc ~inin ilinnunt (tilat is,' eabt-

M ry inaximni) so thalt the ea lil.tmtI, mld IS mltlly constanlt trom XMay

A . . . . . . . . . . . , .

%-

S "'6*~. , . ..

"I C.4 x ' 0 0 a,' n ' 0 C OD 'flu? tL .0 m0 w Cf 0' 0N' 0e 0 0 0 wn.0e e 0'

0 NNNN '..

0 c c 1 1 " o m nmc o Noc c r ct

0,a Dr 6C c c, t:r: c ,T

cc~ - -': -t.fin t' -0 ?rr'7> 'i O -- fif' N NfiN - -t- - -t- - -n- - - -

'o rit c?11 a, eniinneflcoo leflfl ' at

I.' -

m I". r'0t'7o.3 - ''.v N -D nre. Ni a co r- DCoDoo0'Neninc ... neenu'N 0N' j

nn . .7 .7

N ~ 7 -0Nc,0L 11NNt- ,00o0 "ID0u'fl'.'' en erN-eeee oo cca0' 0 '2 N NN-4 ' L~f)

oq --q N Nj - - - -

0A

0 , m I0 cJi''l'.N- ONr-iin 'l~ Z~00i' N" N.M

Mmmm"0

n Cd

ruin-rn - -N - e-o-'-00-'-'-L-

o -' ~ -NNNNN N ~ -NN r N N N -4 -.- "-'-'-N-NN- --- N-N-N-N-

ooI

.r 0.Z0 ""' NN-NN -".

01P

IfV

03

through September. Note that the North Aimnrican winds are more than twice

as strong as the European winds inwinter at the 40, 000-foot Level,

2. LATITUDINAL VARIATION (Figures 20 and Z1)

At 40, 000 feet both the winter and summer latitudinal profiles for North

-- - America each show a single, sharp maximum (Figure 20) while the corres-

ponding curves for Europe display double maxima (Figure 21). A comparison

of the January and July curves rev'ci1s that the European maxima at 60 N and

33 N remain fixed in position, whereas the single maximum over North Amer-0 0ica migrates northward from 40 N in January to 45 N in July. Note also the

difference in range of the mean zonal winds with respect to latitude i.n North

America as compared with Europe, The range is far greater in North Amer-

ica than in Europe, being more than three times as great in January and in

, excess of twice as large in July.

Of considerable interest is the transition zone in the vicinity of 50, 000

feet where the European winds change from a double maxima pattern below to

h.. a single maximum above, while the North American profiles change in pre-

cisely the opposite direction from a single maximum below 50, 000 feet to

double maxima above.

In January the westerlie's at 80, 000 feet over Europe reach a broad

maximum at 60 °N, white in 'North A merica the ,major ma .xmm is north of

70 N and there is a secondary peak at about .37 N. In luly the winds are

easterly above 50, 000 oot at ,I I tlat 1tMude over both i,Vtrope and North Amer-

ica. In Europe the easterlies vary otly slitght1v w itli tatitude from 25N to

75N. There is a flat maxirum in th? vicintv of N. In contrast, the

summeor highl-levol oastorlit- over North Art'.ric: have two maxirta, one

* north of 75 UN and the other sIouth of '. As it; t~he case of the winds bo-

.low 50,0)0 f,'et, the uppr.-lvvel % nts i\'Vcr \tirlt Aonorica havt a muchlarger

range tha ii t oK r ehop .t i nis the A Irt,.' t 1i , 1O t ht og more than sixt' f li 10.; LI ' V Ot'& i1 W I 11\v li,! k.lt :IV t, i' . .10 1 t' I, i I t t n t I t 01'

%

.9,"- "- *'' *I ......."' ' ".'"' " . .... ". :". :... :'.''. "":........' - .: -- -.: ., .. ... . . .. . .-. ' .'....* o '.-,....... .- ". . -'-,.* .,.... . .... . .. . . . 4.. *... , *... ... .... .. . . ' -, . .. .

40'

100 40,N 000 FT.• I " ' 40,000 FT

70 K -- 4.- - ---

65

80 - ---- . . . _ . . .. . . -- Iw 55

I-. 50 JAN

-""-" 4570...---o -- - - - - --. --... " ...- -,o-] A

Z50

* 4 -----

0o 25 _z -00 020 3040 50 60 70 80 90 100 I10

MEAN ZONAL WIND (KNOTS)

Z 75

0-NA

30~ -'-------~ -- f- " 580,000 FT j

"-""OF -"i 0 . ... ... '.I 6I

0JUL

t 450 j--- -. . . . .

3Q

J IF Olt A i -0 - 0k I I0 No 40 5o 4o 10 40 %0

MEAN ZONAL WIND (KNOYS)

iiure 18. Seasonal valriation of winds Fiure 20, Latitudinal variation of windstwe N\or-h Arorc ove.r North Armorica

0,i

4:-:.:... . .... ..... .. .. ... , -. -.. .... . . ... .., , . . . . . . . ..

500 N. LAT. __

0

2 0

1 0 -_

E E,

* -

30 M4 AEW JAS

000 00 -20 FT0 -0 0 1 0

M AN JAN.WND(NOS60 )

,w50

* 4 40-

30-i .****..

~ '.. .. . .. 0 20** 30 40 50 60 -40 * *2 -0 0 10 20 .0

I -0

- - - -- -----_ iiI I I

."D

OD >

-- 1 - -- - . .. . . .. . . c',,.,

(00(V0

(01

4-,

0 0

0

z . . . . .... c J 0o .'

- Lx

*

z%,, ,t

3. VERTICAL VARIATION (Figures 22 and 23)

The variation with height of the mean zonal wind is similar for North

America and Europe. The mean zonal wind reaches a maximum at 40, 000

feet throughout the year between 25 N and 75 N. In January the wind de-

creases with height above 40, 000 feet to a minimum, then increases again

up to some level, presently unknown. Over Europe the minimum occurs

at 70, 000 feet; over North America, at 80, 000 feet or above. The level of

minimum wind does not appear to vary significantly with latitude. In July

the wind also decreases with height above 40, 000 feet. Iri the vicinity of

60, 000 feet the direction reverses from West to East. Above this level

the easterlies increase with height. Over North America, to the north of

70 N and to the south of 30 0 N, the winds are easterly at all levels in summer.

* ,-'.* In middle latitudes in January the westerlies over North America are

* Kstronger at all levels than over Europe. The greatest difference occurs at

40, 000 feet, the level of maximum wind, where a speed of 100 kts occurs

in North America at 40 Nas compared with 45 kts in Europe at 35°N.

(Note the two latitudes selected for this comparison both lie along ox" near

the horizontal axis of the mean wind maximum for the region in question.

In summer it appears that the maximum easterlies occur above 80, 000

feet in North America and near 80, 000 feet in Europe. Of special interest is

the fact that the high-level easterlies have the greatest constancy of all the

wind regimes we have studied.

**, .4

-. . . .. 4 4 . . . . ...- 4 . 4 . 4 4 ft . . . 4 .

, a ,+ °. *t. . , . -. • 4.- . • ',\ °° ° ° .tj . * '.. 'M° . *J.. . . -4 4.* , * ,, . , . >4 . t . f, , t %. * , - . , ,

lk. °•90

..-.. E W

., 35" N . JAN. /5 N JAN

_:3. 5 N. JULY

!'"i I, X 5 ° .JUULY\4",s

-°0

50N A./ 0N A

"" 1140 n T

:. -,. , .4

wI

o w

MEAN ZONA WIN ,(KNOT/S)

.4.4.,, 4

.°.! /." /,1/

"":." E

"""5-0 -- N20 -0I0 2 0 -0 5

l4 0 - -,%

" " LA:

---4 .444. ,

4. ,4

%\.4

APPENDIX III

VARIATION OF THE STANDARD DEVIATION WITH TIME

We have adopted the classical diffusion theory developed by G. I. Taylor.

He considered a fluid containing uniformly distributed, small-scale eddies which

produce diffusion by continuous movement. We have adapted this theory to

large-scale eddies of the general circulation.If R is the correlation between u (the West - East wind component) at

time t an u at time t +p, where ie. a small time increment, then Taylor shows

that

.22 ft ftWd (x;= 2 " (U) • R, d a t (1)0 0

-*@ where 6 (x) is the standard deviation of x-displacements (that is, West - East

displac mcnts) aiid d (u) is Lh'. standard deviation oi the wind.

The solution of this equation depends npon the form of the auto-correlation

function, R , Brunt 9 derived the following exponential form for thia function by

considering a fluid where edt.ics are all the sarne size:

where aie, a constawt depending on the iize of the eddies.

Substituting Eq, (2) into Efq. (1)

,-(x) b i . d( t t.

whe l, r (Ti ... ......

00

%4.4.

' . . . . . . .. . . . . . ...... . . . . . . . . . . . . . . . . .--'- • . .

We have seen that the function G(T), which defines the ratio between 6(x)

and 6(u), is obtained by integrating the auto-correlation function, R twice

with respect to time. If the eddies are all of one size then the auto-correlation

function is exponential and the G-function has the form given in Eq. '4a).

Obviously, eddies in nature are not of a single size; therefore, R. is not

strictly exponential. However, the principal concern here is not the exact form

of R since any errors in this function are smoothed'out by the double integra-

tion required to obtain G(T). The important considcration is the degree to which

Eq. (4a) approximates the actual G-function.

To see if Eq. (4a) does in fact closely approximate the actual G-function,

we conputed G(T) directly in an Eulerian system of coordinates (where mea-

surements are made at fixed points in space as time varies).. The advantage of

this system is that computations are made from actual wind data. This affords

''4' " the opportunity of determining the G-function for as long a time period as-desired.

Actually, we are interested in the G-function for a Lagrangean system

(where measurements are taken a.long the path of a moving particle), but limited

samples of trajectories restrict computation of the function to periods of only

2 or 3 days. However, there is no reason to assume that the form of the G-func-

tion will be different in the two systems. Therefore, we may use the Eulerian

computations to determine the form of the G-function.

The Eulerian computations were made using German wind data at 50°N

for 225 millibars. Winds were resolved into u and v components and the ratios

(u w(e) e computed at 1-day intervals up to a period of 6 days. It.6 •(u) 0(v)was found that the two ratios have identical timne variations, thus G (T)=G T).

t G yFurthermore, the G-function is very closely approximated by Eq. (4a) with

-ae o.6.

While there is every reason to expect that the form of the G-function is

the same in both the Eulerian and Lagrangean systems, it should not be antici-

pated that the constant, e is the same in the two systems. In fact, we would

expect this constant to be snaller in the Lagrangean system, since space as

well as time variations are included in the -'omputation of G(T). We will now

discuss three separate estimates ,f the G-function in Lagrangean coordinates.

All three estimates show the expected smailer value of the constant, e a

~~4'6

---.-. % *.! .1h.A.- . , .

-, . 4 ." . ... ,. . • .. '...- .. q. .' . *..... .. " .... " .'. .. ' ..-. ' .. . .... .' ".. ." ' ....... '.. . .

A sample of 55 Moby Dick constant-level balloon trajectories at about

200 nb was used in conjunction with associated upper air charts to compute

values of (x) = G(T) for time intervals of 12, 24, and 36 hours. A value ofd(u)

,e 0.4 was obtained, -a

Another value of e was computed from trajectory distributions cal-

culated by Sp'reen (unpublished report) from about 3000 geostrophic trajecto-

ries, constructed from upper-level charts of the West German Weather Ser-8a

vice, Bad Kissingen, Germany. The value e = 0.4 was again obtained.10

R. R. Rapp computed an empirical distribution from a sample of

179 .rajectories at 500 mb for two fall seasons. Using Taylor's development 1

-aand assuming that e = 0. 5, he calculated the circular normal distribution

for the same sample. The test revealed no significant difference between

the empirical and theoretical distributions; therefore, Rapp concluded that.-a

tht- value e = 0.5 , which he had chosen, was reasonable.-aOn the basis of the above studies we chose e = 0.4 as being the most

,represntative value. Substituting this value into Eq. (3) we obtain:

-(xT) = / (u) 1. 0917'r - 1.1919 (1 - [0.4]T) (5)

This equation specifies the variation of O(x) with tine.

0f

.4.

, 4-. .'

* . , . .4-4 -

APPENDIX IV

SPHERICAL TRANSFORMATION

Consider a coordinate system , y , and i on a spherical earth. This

coordinate system is rntated relative to the conventional coordinates x, y,

and z in such a way that the R and i axes are both inclined by an angle 0 =

2 0 to the x and z axes, respectively, while the - axis coincides with0

the v axis. (See Figure 24.) The angle, 0, is a fixed latitude on the real

earth.

/Z4 //X f

1. 1 /!/i"'..~ / YZx

.f~~ X I / ." ..-Y

Y=.4

Figure "4. Spherical coordinates Figure 25. Axes xatd z

"'A ry ,tbitra r y p o ti P ,ott the, sph t, rt, h ws coo rdizatt i .x, y, a nd z i o

,.tie Co lventio a I system - a ld x, y, and Y, ill 010 rtaUte~d syste n. W ,ish to

f'i td th ,, r t itio s h ip b ,,tw octi th rsS , tw o c olrd illate sy st el is . silic" V Y , w o

"i already liave mit! suclh rt,,atiollship, hett s,. tho problvin rt-dtwios. to. 1t,*0 ditilvi-

- ,4

.4-

Fi~gur eo hica. coowinates ,iu. AThusn

Andx , n t tht' sin 0 + , co n.

Awd tI , ~1nsh i R edg.,) (it ttnr te yi

al rzone-us i tu te p m rdi, ttitu wdn-

, J

A(1)

. +l tott~inud , , ioegosd

o, o

4

In the rotated coordinate system:

x = R sin s cos a

= R sin s sin a

= R cos s (2)

In the conventional coordinate system:

x = R cos 0 cos 'A

y = R cos 0 sin X

z = R sin 0 (3)

Eliminate x, y, z and x, y, z from Eq. (1) by substituting Eqs. (2)

and (3) to obtain 0 and ), in terms of 00 8, and a.

The final result is:E 2 .2 2 2 :

-~v cos 0 sin a I- cos 0( + cos a) I0

+ Cos 0 + ]/2 sin Za cos 2 0 cos (4)

ctn A ctn a sin 0 + ctn s cos 0CC (5)

*'S,

150

44- .. , .4

- '. . '4.2 " i • " .. .,' , . .. " ' ", ,' , .'- ....4 4 " " " , ,.. , - ' o '

REF EREN C ES

1. Taylor, G. I., "Diffusion by Continuous Movement," Proc.., Loid.Math. Soc. 20, Series 2, pp 196 (1922)

2, Solot, S. B. and Darling, E. M., Jr., 1958, Wind, Ch. 5, pp 5-56to 5-73, Handbo9k of Geophysics for Air Force Designers, Geo-physics Research Directorate, AFCRC (in preparation)

3. Brooks, C. E. P., Durst, C. S. Carruthers, N., et al, "UpperWinds Over the World, " Met. Office, Geophysical Memoirs.No__._5, Vol 10, No. 5, M. 0. 499C, London (1950)

4. Brooks, C. E, P., and Carruthers, N., Handbook of Statistical Meth-ods in Meteorology M.__O._53_, Ch. 11, Her Majesty's StationeryOffice, London (1953)

5. Buch, Hans S., "Hemispheric Wind Conditions During the Year 1950,"Final Rpt Part 2, Gen. Circ. Proj._No. AF1.9(1Z?)-l.53, Dept. ofMeteorology, Massachusetts Institute of Technology, Cambridge (19541

6. Court, Arnold, "Vertical Correlation of Wind Components, "'Sci.Rpt_-No, 1, AF19-604-2060, Cooperative Res. Found., Calif. (1951)

7. Solot, S. B. and Darling, E. M. , Jr, , "A Note on the Difference.Method of Averaging," AMS Bulletin, Vol 35, pp 484-487 (1954)

-. Deuschen Wetterdienst in der US- Zone, 1950-1953: Taglicher Wetter-bericht, Zentralami Bad Kissingen, Januar, April, Juli, Oktober

9. Brunt, David, Physical and Dynarnical Meteorogy, Oh. xii, Sect 157,pp 263-264, Camb. Univ. Press (1941)

1 0. Rpkpp, i'. R. , "The Distribution of the End Points of Air Trajectories, "

RI. No. -496-A, The Rand Corp., Santa Monica, Calif. (1954)

*'%2 .}5 .. . .... ... .. , . .

GEOPHYSICAL RESEARCH PAPERS

No. 1. Isotropic and Non-Isotropic Turbulence in the Atmospheric Surface Layer, Heinz Lettau, Geo-physics Research Directorate, December 1949.

No. 2. Effective Radiation Temperatures of the Ozonosphere over New Mexico, Adel, Geophysics

"- . R-D, December 1949.

No. 3. Diffraction Effects in the Propagation of Compressional Waves in the Atmosphere, Norman A.

Haskell, Geophysics Research Directorate, March 1950.N&.

No. 4. Evaluation of Results of Joint Air Force-Weather Bureau Cloud Seeding Trials ConductedDuring Winter and Spring 1949, Charles E. Anderson, Geophysics Research Directorate,May 1950.

No. 5. Investigation of Stratosphere Winds and Temperatures From Acoustical Propagation Studies,Albert P. Crary, Geophysics Research Directorate, June 1950.

No. 6. Air-Coupled Flexural Waves in Floating Ice, F. Press, M. Ewing, A. P. Crary, S. Katz, and J.Oliver, Geophysics Research Directorate, November 1950.

No. 7. Proceedings of the Conference on Ionospheric Research (June 1949), edited by Bradford B.Underhill and Ralph J. Donaldson, Jr., Geophysics Research Directorate, )ecember 1950.

No. 8. Proceedings of the Colloquium on Mesospheric Physics, edited by N. C. Gerson, GeophysicsResearch Directorate, July 1951.

No. 9. The l)ispersion of Surface Waves on Multi-Layered Media, Norman A. ilaskell, Geaphysics

Research Directorate, August 1951,

No. 10. The \lesurement of Stratospheric )Dousitv Distribution with the Searchlight Technique, L.,lterman. Geophysics Research Directorate, December 1951.

No. I I. Proceedings oif the Conference on Ionospheric Physics (July 1950) Part A, edited by N. C,SGel,4ou and Ralph J. I)oahtoals, Jr., Gtiophysics lesearch i)irectcratu, April 1952.

Not. 12. P rocet'diu.," of tie Coufert, ue oil lotospherie PhysiCs (July 1050) Part It, aditead by Ludwig

katr and N. C. Gerson, Geophvsics Resevach l)ireetorate , April 1952.

No. 11. Proceedlings t the Colloquiuma on Mierilwnve Meteorology, Aerosl and Cloutl Physies, vditod

b , thlph J. l)onaldson, Jr., Ge phyics Researth I)irpetortite, May 1452.

No.t 1 Atittspheric ic% Iatterus and Their Iepresntation Lt SphericalkSurfavte lhtrtutiics, It. Ilaur-

kvit, atnd Ihelctard A. (Craig, Gooplhyscs te00arelk lirectoate, July 1952.

\o, 15. h, iL-atterig ,f W, trowtgaetie 9ave- I.roll Sphetres ,u.o Spherical Shells., A. I. Adeu,

(,etophvsies 1(iesearch Directoratt,, July l42.

\it, 10. %(Ite.. on tlt, I'thr'or Vf I rg,-Sc le tlisturban'es in Atinipli-pric Flow V% ith Applicltiotus to

\n,,rieti k\ptlher Predictin. IPhilip I)uneuil Tho1ptou, \hij r, 1'. S. Air Fo wc, Geophysic s

lter, arrh l)irctorate, July 1952.

**% '.'.'R 4,. .

'-" ", " ' " ".'.." A',. ' . " ." -r~.A •W • L":.." "z:: '"'il :' : - YY . "":-

GEOPHYSICAL RESEARCH PAPERS (Continued)

No. 17. The Observed Mean Field of Motion of the Atmosphere, Yale Mintz and Gordon Dean, GeophysicsResearch Directorate, August 1952.

No. 18. The Distribution of Radiational Temperature Change in the Northern Hemisphere During March,Julius London, Geophysics Research Directorate, December 1952.

No. 19. International Symposium on Atmospheric Turbulence in the Boundary Layer, Massachusetts Insti-tute of Technology, 4-8 June 1951, edited by E. W. Hewson, Geophysics Research Directorate,December 1952.

No. 20. On the Phenomenon of the Colored Sun, Especially the "Blue" Sun of September 1950, RudolfPenndorf, Geophysics Research Directorate, April 1953.

No. 21. Absorption Coefficients of Several Atmospheric Gases, K. Watanabe, Murray Zelikoff and EdwardC. Y. Inn, Geo, ,'sics Research Directorate, June 1953.

No. 22. Asymptotic Approximation for the Elastic Normal Modes in a Stratified Solid Medium, Norman A.Haskell, Geophysics Research Directorate, August 1953.

No. 23. Forecasting Relationships Between tipper Level Flow and Surface Meteorological Processes,J. j. George, R. 0. Roche, III B. Visseher, R. J. Shafer, P. W. Funke, W. 11. Biggers and It. Mt.Whiting, Geophysics Research Directorate, August 1953.

No. 24. Contributions to the Study of Planetary Atmospheric Circulations, edited by Robert M. White,Geophysics Research Directorate, November 1953.

No. 25. The Vertical Distribution of Nlie Particles in the Troposphere, It. Penndori, Geophysics le.search Directorate, March 1954.

No. 26. Study of Atmospheric lons in a Nonequilibrium System, C. G. Stirgis, Geophysics ResearchDirectorate, April 1954.

No. 2 7. lavestigation of %ficroharometric Oscillations in Eastern Massaehnsetts, E. A. l"lauraud, A. It.Mears, i. A. Crowley, Jr., ond A. P. Crary, Geophysics Ree urch Directorate, May 1954.

No. 28. Tho Rotation-ibration Spectra of Ammonia in the 6- and 1-Microu livioa, I. G. lIr en, Jr.,Ctipt., USAF, Geopbysico Research l)irectorute, June 1954.

No. 2. ettsouuil lrend,,- of "'eTmperture l)ensity, antd Pressuro in tile Strttotpherv Ohtoinod With tieSearchlight Ilrobing Technmiqu, Louis Elterftlun, July 1954.

No. 30. Proceekdings of the Conermice on Auroral 1hyuica, edited 6y N. C. t;ermo, GeopIys4ico Ile-

* d nch Directorate, Jul 1954.

No. 31. log Mudlifictiot by Cold-WIater Seeding, Verno G, 'lank, 600p1kyrsics Research ilirector(ate,Augumt 1954.

S%

S. . GEOPHYSICAL RESEARCH PAPERS (Continued)

No. 32. Adsorption Studies of Heterogeneous Phase Transitions, S. J. Birstein, Geophysics ResearchDirectorate, December 1954.

4

No. 33. The Latitudinal and Seasonal Variations of the Absorption of Solar Radiation by Ozone,J. Pressman, Geophysics Research Directorate, December 1954.

No. 34. Synoptic Analysis of Convection in a Rotating Cylinder, D. Fultz and J. Corn, GeophysicsResearch Directorate, January 1955.

No. 35. Balance Requirements of the General Circulation, V. P. Starr and R. M. White, GeophysicsResearch Directorate, December 1954.

No. 36. The Mean Molecular Weight of the Upper Atmosphere, Warren E. Thompson, Geophysics Re-search Directorate, May 1955.

No. 37. Proceedings on the Conference on Interfacial Phenomena and Nucleation.I. Conference on Nucleation.

II. Conference on Nucleation and Surface Tension.I1. Conference on Adsorption.Edited by 11. Reiss, Geophysics Research Directorate, July 1955.

No. 38. The Stability of a Simple Baroclinic Flow With Horizontal Shear, Leon S. Pocinki, GeophysicsResearch Directorate, July 1955.

No. 39. The Chemistry and Vertical Distribution of the Oxides of Nitrogen in the Atmosphere, L.Miller, Geophysics Research Directorate, April 1955.

- No. 40, Near Infrared Transmission Through Synthetic Atmospheres, J. N. floward,Geophysics Res-search Directorate, November 1955.

No. 41. The Shift and Shape of Spectral Lines, R, G. Breeno, Geophysics Research Directorate,October 1955.

No. 42. Proceedings on the Conference on Atmospheric Electricity, II. iolzer, W. Smith, GeophysicsReseuarch Directorate, Decmehr 1955.

No. 3. XMethods and llesults of Vppor Atmospheric lesearch, J. Kaplan, G. Schilliag, I. Kullunn,Geophysics llesoarch Directorate, November 1955.

_ No. . ltminonus and Spectral Reflectance u Well as Colts of Natural Objects, It. Penod, Geo-physics Research lirectorate, February 1056.

No. 45. New Taules of %lie Scering lFuneitins for Spherical Particles, It Peilad,, It. Goldberg,Geophyvics Research Directorate, \hrch 195(3.

Nu. 46. lesslt of Nutnerieal l"orecatin g With the llarotrapic and Thrnutropie Models, 1. Gates,L. S. Pociaki, C. F. jeakins. Geophi, ica lievearch lirectotate, April 1956.

4L.........................................

GEOPHYSICAL RESEARCH PAPERS (Continued)

No. 47. A Meteorological Analysis of Clear Air Turbulence (A Report on the U. S. Synoptic High-Altitude Gust Program), H. Lake, Geophysics Research Directorate, February 1956.

No. 48. A Review of Charge Transfer Processes in Gases, S. N. Ghosh, W. F. Sheridan, J. A. Dillon,Jr., and H. D. Edwards, Geophysics Research Directorate, July 1955.

No. 49. Theory of Motion of a Thin Metallic Cylinder Carrying a High Current, C. W. Dubs, Geo-physics Research Directorate, October 1955.

No. 50. llurricane Edna, 1954: Analysis of Radar, Aircraft, and Synoptic Data, E. Kessler, III and1). Atlas, Geophysics Research Directorate, July 1956.

No. 51. Cloud Refractive Index Studies, i. M. Cunningham, V. G. Plank, and C. F. Campen, Jr.Geophysics Research Directorate, October 1956.

No. 52. A \eteorological Study of Radar Angels, V. G. Plank, Geophysics Research Directorate,\t1ugtst 1956.

No. 53. 'he Construction and IVsw of Forecast Rejisters, 1. Gringorten, 1. Lund, M. Miller, Geo--' physics Researtch Directorate, June, 1956.

-", 5. Sotiar (eomagnetic Ionospheric Parameters as Indices of Solar Activity, F.Ward Jr.,Geao-pysics Ilesegc :h l)irectorate, November, 1956,

No. 55. Preparation of Mutually Consistent Magnetic Charts, Paul Fougere, J. McClay, GeophysicsResearch Directorate, June 1957.

No, 56. Radar Synoptic Analysis of an Intense Winter Storm, Edwin Kwislor Ill, Geophysics Research-. I)irectorate, October 1957.

N . 57, \lew) \lMonthv '1( 111 and 200 mb Contours and 500 mob. 300 nib, 200 itb 'reniperatues for theNorthern Itntisphere, E. IA. %hl. April 1958.

No. 518. Theory tf li,arite-Sctle Atmospherie l)iffusioa and its Application to Air Tr-jectaties, S. I.Solat 'Atid E. M. Darliag Jr., Jule 198.

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