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U. S. DEPARTMENT OF COMMERCE- CHARLESAWYER,ecretary.5. WEATHER BUREAU

F. W. I~EICIIRLDERPUR,hiefI .

RESEARCH PAPER NO. 35,-The Application of the HydraulicAnalogy to Certain AtmosphericFlow Problems

bYMORRIS TEPPER

WASHINGTONOctober 1952

LIBRARYAPR 0 2007

National Oceanic &Atmos heric AdministrationU.S. bept. of Commerce .

For eale by the Superintandont of Documents, U. S.Government Printing Ofice, Washington25, D. C. - Price 35 cents81752

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National Oceanic and Atmospheric AdministrationWeather Bureau Research Papers

ERRATA NOTICEOne or more conditions of the original document may affect the quality ofthe image, such as:Discolored pagesFaded or light inkBinding intrudes into the textThis has been co-operative project between the NOAA Central Library andthe Climate Database Modernization Program, National Climate Data Center(NCDC). To view the original document contact the NOAA Central Libraryin Silver Spring, MD at (301) 713-2607 x 124 orLibrarv.Reference@noaa.gov.

HOV ServicesImaging Contractor12200 Kiln CourtBeltsville, M D 20704- 1387January 22,2008

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PREFACEIn studying a meteorological paper, the reader is prone to classify it in his

mind either as theoretical 01 practical; and, depending on his own back-ground, interest, and specific professional assignment, may peruse it in theminutest detail or leaf through it with hardly morc than the mildest of interest.This is quite understandable. Meteorology, today, is split between two groups.One is characterized by a researcher who is primarily intercsted in the physicalaspects of cause and effect and in the understanding of the mechanics of weat,herprocesses. He attacks his problem by the mathematical manipulation ofbasic physical principles which he has applied to some model considered tocontain the essence of the incteorological situation. The second group, on t,llaot81ierhand, is characterized by the practicing meteorologist whose primaryconcern is in the interpretation of the weather today in terms of a prognosis forthe weather tomorrow. His tools may not have qa thcmatical rigor nor adliereto strict physical principles, but they are time-tested and for the most part,they work. It is only natural then that the theoretical meteorologist wouldtend to have little patience with the empiricism of a practical paper and thepracticing meteorologist would consider of little value the mental exercise of atheoretical paper. Although the immediate efforts of thesetwo groups arc directed along diff rent lines, the final goal is basically the same,namely, to unravel the intricate patter11 of the cause and effect of the weather.Thus, the findings of either groui) must be a step forward toward this final goal.The importance of an active interchange of ideas and information between t11esegroups was emphasized at a symposium on this problem in a reccnt AinericanMoteorologicd Society meeting. The author has intended this paper to fallwithin the scope of both of these groups. In leafing through the paper, t h oreader will find both inathcrnatical formulae and synoptic maps. It would beunfortunate, however, if the sections on the mathematical development IVOreto be read only by the ~theoretica1meteorologist and the sections colltainingelleapplications only by the practical meteorologist. Such a dissection wouldundoubtedly defeat the purpose of this paper.

Thc author is particularly anxious to interest the forechstcr, the inotoorol-agist who Concerns himself with the analysis of weather information on a currentbasis, in the idcas suggested here. In the last analysis, the only measure of theusefulness of a theory is in the extent of its applicability in explaining weathersituations and in anticipating their occurrence. The cfforts prescntcd hcrc areadmittedly limited. Nevertheless, it is believed that a start has bcen made.It is hoped that those meteorologists who analyzc todays weathcr and forccasttomOrrO\vS will find some additional analytical and forecast ^tools here; byunderstanding thc basis for the proposals made here \vi11 cnlargc the rangeof thc applications; and as a result of thoir OWII rich background in synopticmeteorology will arrive at a morc complete set of useful forecasting criteria.Motoorologloiil socic ty, Wnshingto11,D. C., May 1052%

This situation is unfortunate.

I ~ y l n p o s l o n l 011 Coordllintliig Moteorologlcnl Rasonroh ntid Wo ntl lw Iorocnsthig, 115tll Nntfonnl Mcothig, Ainurfoml

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ACKNOWLEDGMENTSThe author considers it a privilege to make the following acknowledgments:To Profs. G. S. Benton, I?. H. Clauser, and R. R. Long of the JohnsHoplcins University, for their interest, encouragement, and assistance in thepreparation of this paper as a dissertation for the degree of doctor of philosophy.TOProf. J. T. Thompson, chairman of the Civil Engineering Department,Johns Hopkins University, and to Dr. H. Wexler, chief, Scientific ServicesDivision, U. S. Weather Bureau, for their personal interest and active support.To Miss B. Ackerman and Messrs. S. Falkowski, J. Hand, W. Hoeclrer,A. Miller, and D. T. Williams, all of the U. S. Weather Bureau, for thoir assist-ance in the operational phases of the observational progranis and in the analysisof the claln.

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CONTENTS Pam1x1I V114568888101013

131314152020202225253135353036373740424540

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TABLE-Table of analogies-the hydraulic analogyG'W TKO iquids Three-layer atmospherene liquid

Hypothetical compressible gasI- = onst x pyu=c "=2

C.

Incompressible fluid with a freesurface in a gravity field Three layers with an inversionseparating tn-o fluids, eachautobarotropic (0, 0'); onedimensional motion in lowerA n a1o g o u sDomains Interface separates two incom-pressible homogeneous liquids( p , p ' ) ; one dimensional mo-tion i n lower liquid; level freesurface. (See Chapter I) fluid, Coriogs force neglected;b H bh-Iupersonic flow A i>1 supercritical flow F> 1subcritical flow F ljubcritiwl flow F

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solution of t l i ~iinultancous sct of nonlinear partinlclinorcntial cquntiaiis n n d its grnpliical rcprescnta-tion wc shall follow Clauscr [SI. For a moreextmisivo discussion of tlic solu tioil of nonlincwpartial diffcroiitial oquatioris by the mothod ofcharactoristics the rcader is rofcrrcd to [3], [32],and [48], or aiiy standard book on aerodynamicsof suporsonic flow, The oinpliasis of this cliapterlics not in tlro actual solution of the basiccquatioirs, but rather in the extraordinary prop-ortics of the flow which aro expressed by theclraractei~isticines.In chaptor 11, we investigate tho equivalenceof columns 3 and 4 of tablo 1 and mal-c someromarlcs on tho goiioral typo of atmosplioricproblems which can bo handlod by tlio nrc hod-ology outlinod in chaptor I. Tlra remainder of thetreatise deals with specific applications of thistlicory and is illustrated by case liistories whichdomonstratc certain fcatures of tho atmosplirrewhich may bo explaincd vory siinply and verydirectly by tho theory as outlined iii the table

Thoro is an additional rcinarlc to br made onthe gcnoral nature of tho problem. Basically, thoplicnomonon discussed hero, the prossuro j u m pline, is an internal---~ T Y - ~ ~ Qave. Earlierdiscussions of gravity waves, wlicther in a com-prossiblo or incoinprossiblo incdium roliod asteii-sivrly on p~rturbat~ionhoory ( [ 2 5 ] , [2G], 3 6 ] , 3 T ] ,[ 5 0 ] , [55]) or on n symmetrical form for thosolutioii of a solitary finito wave ([IO], [37], [5G]) .I t will bo oscocdingly obvious that, t h o gravitytypc wavo discussed licro approsimat,os mostncarly an liydi*aulic urnp in ap~)oarmico--nrid assuch can b o dcscrilmd ncitlior as a pcrturbntiounor as n syinnictrioal wave. This londs coiisidor-nblo suppor t to t,liooxp~annto n of tlir plir\nbnicnotlby iiieaiis of tlic hydraulic txrinlogy.I t has t$i*ct~dycoii inen t ioi cd that for expwi-montd purposos it,has p1~oveiiiscfril to invcst gatotlir propcrtios of tho ilow of a singlo liquid in nclinriiiol (column 2) in attuclring pro1)lcnrs innorod ynniiiirs. Siniilnrly it is reronr~uondodtoiiivcst igtito t h o propcrtics of ttwo stmt i f i c d liquidsin t i olianiirl (column : ) in st utlyiiifi t h r propcrtiosof ccrtain ntmosplwric niotions. Such a lwgin-tiinghas boon bcgun u n d ( ~lic dit~~;tionf Dr. GoorgoI3onton in t,lict Ilpdi-nulics Lthortitol*y of thoCivil Engineering I l e y n r t ~ i i ~ c n l ,oliiis I-PopkinsIJniversit~y,and p n r t of tho tvork rcpot*totl 011h o ~ cs t h rsul of this p r o g i ~ ~n which thoauthor lind tho ploasurc of pdcipulil ig.

3

of annlogics.

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CHAPTER ITHE UNSTEADY TWO-LIQUID FLOW PROBLEM

In this chapter wo shall investigate with con-siderable detail the properties of the flow indi-cated in column 3 of the table of analogies, table1 of the Introduction . I n this flow we assume:

1.

2.

3.

4.

Two incomprcssiblo fluids, p > p ' , with hequal to the height of the interface, de-scribe the natu re of the flow domain.The horizontal velocity (u) of the lowerliquid is independent of tlic lateral coor-dinate (y) and of the vertical coordinate(4Both the surface (z=0) and the top ofthe upper layer (the free surface, z = N) arehorizontal surfaces.,The pressure is hydrostatic (i . e., verticalaccelerations ar e ncglco cd) .

I n tliesc assump tions antl in the matlicmaticnldevelopment to follow, tlic motion of the upperliquid is not considered. However, as we shallsee la ter , the assumptions listctl above togotherwith some of the conclusions which we shall reachimply a definite restriction on the na tu re of theflow of tho upper liquid.The equation of motion for t h o lower fluid isgiven by

and the,equation of cont inui ty bya h b-+- (hu)=O.dt a x

Since thc pressure is liydrostatic, we have(3) p = ( H - h ) p g+ (h -z ) p ga.nd therefore,4)

Substituting into (I),

Equat ions (2) and ( 5 ) represent, our system ofsimultaneous difi'erential equations, in the do-pendent variables u and h. For convcnienco, WBintroduce a new variable c, given bycz=( 1-$) gh.

Substituting into ( 5 ) ant1 (2) we get7) bu au a c- t u - + 2 c -=oat dx dx

ac dc aua x a xt--+2u --+c -=o.We have tlius transformed our original set ofparlial ctifl'crmtial equations in u nntl h into a setof partial difl'erential eqiiations in u and c .li'irst adding antl then subtracting equatiops(7) and (8), we arrive at

equations (0) antl (10) with the oqua-tion for a comprcmible isentropic gas given byLamb [37, p. 4821).From elementary calculus we may wr i t e tha tif j=j{r(t),t} then the total derivative

T h u s if we consider any lino l? in t h ,f-plane,the total cliangc with time of t lre fuict io n .f d o n gthis lino i n the z,t-plane is givon I)y the esprcasionnbovc.of tdic linc I' a t tlic point wlicro the t ota l derivcttivois being oomput c d Onh/ in t h e q i c c i n l case where

d r111 this expression . rdcrs t o t h o slopedt

4

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Prescribes the motion of a particle ma y we interpretdx% as a particle velocity. I n tlie more goneral casedX merely refers to the slope of a lino along whichthe total derivative of a function is desired.

Thus, equations (9) and (10) specify that alongthose linos in the z,t plane whose slopes aro.givonby d X- dxdI -u+c for (9) or -=u-c for ( lo), the func-dttions j (z, t )=u+2c or j ( x , t )= u-2c) respoctivolybaVe a zero total derivative with time?These lines are called characteristic lines.repoat then, that along tlie characteristiclines givon by

(11) dz (cliaracteristic lino)= u f i ,we have the rolation that(12) uf2C=constant=a, 8, rospoctivcly.If W e supposo for convoni(?ncc liat c>u, thenfigure 1 \vi11 represent schematically tho state-It should bo notedt h t since u nd c may both vary in tho field, notwill a and 0 vary, but tlio charactoristic linesby t l m o parametors need not beStraight lines. W e shall now introduce tlirae

which rolato to tho propertics of thosecharactoristic lines.&initio;: A wavo is a family of characteristiclinos. By (11) arid (12) wo lmvo thealtornatc definition that a wavo ex-presses a functioncil rolntionship in-volviiig tho dopondcnt variables in theDefinition: ~ i iclcntifiablo point oii tlio wave is

givoii by n spclcifi? vduo of a (or 0).Definition: rho spoad of n point 011 t l io wavo at agivori point in tho x,t-pluuo is given byt h o slopc of tlio chnimtcristic linopassing through that point.

Tlius, iii figure 1 , tlie a family, rsprwsiiig tIiofu1lctional relationsliip ~+2c=constniit, rrpro-W t s a wavo trc~vcllittg i i tho positivo z-diroutionand tho p fninily, c~s~~res s i i~gh c k functioml ro-lationsllip u--2c= Coiistmt, roprcwi~ts i wnvotlbbvcltiiig 11 tl1c 1w g i ~vc x-(Iircction. FUYicr-lQoro, tho progress of ally point of tlio a wave, sayQt , is givoii by llio cl~tiri~ctoristiciiio a t in tlio

z

given in ( 1 1 ) and (12).

2 t-plauC .

A poitit in tho r , tq)lniio rrhrs to n locntioti nnd a tlmo for tliu f low. A ttllls tioilit, ttio flow tins 1%vtslocity (21) imc1 n vniuo for c, ~ i v o i i y (0) niid thoftfilfncclIiclgllt. l%lpt I t ~ r ~ yoliit thoro is a wluo of u+c (or 21,-c).PI10 roc~llirod lo~los t k k c ) UlUS Chist.

x,t-plane, and tho spnod of tho point is givcn byt h o local slope of tlie charactcristic liiia at.We shall soe lator tlint theso dofiiiitions nro inperfect accord with omr physically iiitui tive con-cepts of what nro n wnvc, (I point 011 a wavo, andtho spocd of a point 011n wtivn.

SINGLE WAVESReferring to figurc I , it is nppnrrnt tlint shin-tioiis may miso (as ivlieii c>>u) in wliic.11 thn twofamilies of cliii,rnctcristic. lilws divcrgo snfiiciently

such that nftcr s o m i time tlia two fmiilios no longorintoixoct. Ench fnmiiy will tlion roprosrnt a singlcwavo tinvclling in tho tlirec ion indicntod by tlioslope cf tlic clinmctoristic linos. In the nwtcoro-logiccil probloms to which this thcory lins beenappliod c is usuiilly muc+ ltirgc~hnn u niid iiitloodtlio ovidciichc swnis to br t l m t w o tiro dcding witha singlv WILVOoiily.Consequently, IVC slitill now restArt, our ntten-tioii to thoso solu tioiis of our brwic problcni wliiclishall yiold siiiglo W ~ V O S idy, For conwtiioiicc WQ

talx\ tlic positivo x-direction i i i tlio dircctioii of thownvo propngation. .lhiis, \vn ( w i stby tlint we wedcding with tlio a fruiiily otily, while tho 8 familyis missing. Aiiotlier way of loolciiig ntd his is thattlin points 011 tho mvc\ of tlic f l ftitiiily are in-distinguishable olio from t h y otlior. This implies

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1t

p,=p2=p3= . . . = p .By (12)

(13) u- 2c= p= constantovcr tlic entire z,t-plane or from (6).(14) U - 2 [(1- ) gh]=constantovcr the entire z,t-plane.

If we have giver) the undisturbed values of theflow vclocity u and the interface height h, wc maysolve for the constant in (14). This equation prc-scribes that any subsequent value of u dcfincs thecorresponding lt,, and vicc vcrsa.We shnll also find uscfiil tlic following relationdcrivctl from (13)(15) Au= 2Acovw tlic cntircb z,t-plane. Tlirough thc definitionof c , ( G ) , equation (15) prescribes the relation be-tween the variation of u bctwccn two points andtlir convspontliiig variation in the interface lieiglit.

u 2c=cu iu--2c-p .

Along a given clini.rrctctisti(: inc, e. g., al,h(wtwo ecluntions yield IL solution for u arid c in tcrmsof ai and P. Sincr at is coristrmt along 11 chnrtic-tCJistiC i e , wc liavc t l ic csti-cmcly importunt N-

we hnvc tlwr1 lor f L SillblC wave(1 6)

sult that both u and c arc constant along any c l wactoristic linc. Again through (6) , this means thatnot only tho flow velocity u) ut the interfaceheight ( h ) s constant along any charactcristic l i i io.Furthermorc, from ( 1 1 ) the slopc of a charactel-istic line must be constant at cvcry point oftbecharacteristic lina.Tlms, for a singlc wavc, the charactoristic line9arc straight linos along which u and c (and frofl(6) h also) are constant.We shall now dcmonstrate that the dcfinitiofland propertics of a wavc as discussed abovc at0in accord with the customasy physical and intui.tivc ideas about waves. Consider a symmetric@]steady statc single wavc fig.3) and its position abvarious timcs. Tho physical picture of the oventis given in the diagmm to thc left. On the rightthe event has bocn transposcd on to an xdiagram, where instead of drawing the wavc profilewe have merely indicated tho height valucs at theappropriate time and place. Since the wave i9steady-state, we may draw parallel linos tllrougbthe same values of the wave height. Each linarepresents a specific wave height (c. f . , definitioflof point on a wave) and furthermore the slope oftho line is the speed of that wave height (c. f . ,definition of spcod of point on wave). Thc entir(family of lines passes through the wave at anytime (c. f., definition of wave) and cxpressos t l lcfunctional relationship that along cnch line thowave height is constant.

(See fig. 2.)

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t

intorfaco lioight (h) by its correspond-ing value of c .a valuc of u and c . At oach point(u+c). Tlicsc arc characteristic lines

ttep 2: Evpry point on the line t = O now hasconstruct a line with a slope oqual to

* along which the f low velocity (u) and

t

0.

I 2 3 4 5 6 7 0 9 IOII I 2 1 3 W f 5 @ 1 ?

0

000

c/

,/

x xFiauiis S.-Sym~no tricnl stendy-stnto slngla wavo.

stant.Civcn (a) an iiiitial vnluo of the flow I'lalilZR 4.--hZC lllVd 01 SOltltIo11, Case 1.velocity mid intCI'fRC0 height ( ~ 0 , 0 ) plus(b) tho tomporal wavo profile (or velocitydistribution) at a singlo point (x=O)(fig. 5) .

Step 1; Using (a), roplaco each value of thointerface height (h) by its corre- 4sponding value of C. 0 0

Step 9: Every point on the line x = O now hasa val ~o f u and c . A t each pointlconstr1lct a lino with a slope equal to(u+c), These n,rc the characteristiclines along which. the f l o ~ clocity(u) a nd t h o interface height ( h ) willbe const8ant.

Case 111.Given (a) an initial valuo of the flowplus (b) a distribution of interface

X+

velocity rmd intorfaco height ~ 0 ,h,) WOWRE.-Motllod of solution, i J ~ o1.

Car0 n

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- ---- --I1t

FIGURE-Mothod of solution, Cnso 111.

heights (or velocity) along any arbi-tra ry line G(x,t)=O in th e x,t-plane (fig.6).Step 1: Using (6) replace each value of theinterface height (h) b y its corre-

spo ndin g value of c.Step 2: Every point on tho line G(x,t)=onow has a value of u and c . A teach point construct a line with a,slope equal to ( u + c ) . These arethe characteristic lincs along whichthe flow velocity u)nd the inter-face height ( h )will be constant.(Note: The simplest procedure to follow is toconvert 11. nto c whenever i t appears an d t o solvefor u whenever necessary through the simple rela-tion (15) r a the r t han thc: more cumbersome 14).)

CLASSES OF WAVESI n the preceding section we noted th e procedureto be followed in constructing the characteristiclines of a pcirticular flow when certain initia l condi-tions arc givcn. T h e general pat tern formed bycliaracteristic lines can fall in any one of threeclasscs. In tho following tliscussion we shall con-sider the initial conditions as for Case I above.The argument for the other cases follows in a nident ica l rnanr~or .~

UNI F OR M F lAI W (NO WAVlS)Given: At time t=O a uniform interface height(ho) r a constarit flow velocity (uo).By (6 ) and/or (16) every point on the line t=Ohas the same value of u and c (i. e. , uo and e o > .Consequontly the characteristic lincs arc ti11

(Fig. 7 . )

T h o following is rssrntinlly n ararhiml discassion of tile ciilsses of wnvcsand of thrlr proportics. A mnthomntlcnl discussionyicldl~lgh o snmo rcsultsIs given in Apporidlx A .8

Tt

parallel and all hav e the id entical values of th evariables along them. Th us unijorm $ow (or nowave) is characterized by parallel characteristiclines.DEPRESSION-TYPE WAVE

Given: Initial values of u a n d h (uo, o)plus a ttime t=O a distribu tion of interface height ( h ) ucht h a t h decreases in tho negative 2-direction insome in terval ( s l

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8 IIIIt i: I

I II I2 I-- - - - - ----I II II IIII1

iII

.

I1 ;I

II

IItIII_I

II

FIGURE.-Doprosslon-typo wavo.

FIGURE o.--Qovotloii-typo wavo.

9

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Again, we suppose that the interface is undisturbedoutside of this interval; such thath=hl for s l x lh=ha for ~ 2 x 8

Consequently hi>h, for x15 xt

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Case 1.-Assumo u'=u+c, 01 t h o patllis of t h oDarticlos 01 tho uppor liquid coincide with thocharacteristic lines. By (1 7) tbis asslimption im-plies that tlic incIividiia1 particles in any vrrticalcolumn of t ~ i c pper liquid al\vtiys rcrntiin ovcrtho same point on tlie wave. T I ~ I I S , aI011g t h oChractoristic linos,

bh a?/?--+u' --0.at bc

bh ah b?L'-+u' --r+(lL--TT) =o . .bt bx b c

of tho uppor fluid on tlliatside of the wa ve mustbe givon by U=u: I t ,* ) for all t. Sinco h* is con-s t a n t in a*,77m u s t bc constant in a* and all t h opurticlo path lirios arc partill01tooho cliarartaristiclinv a*. 'l'lic argiiment, can now bo rcpoutod for anyo t h r cliaractcristic line a**, But t h o particlo pathlines cannot, bo parallel to two cliarbctoristic linesat t,hc samo tinio, and we roti~hatiotlior contra-diction. 'l'lius, u' ?L+c is also not valid,Since noitlm u'=u+c or u' u+c is possible,we ronc*lutlo lint, our originti1 prcrnisc is not, validand tho uppor flow must dopond on tho z coordi-nrtt,c. HOWQVW,c slid1 sliow that ovon herowe mustjimposo roslriotions. If t h o iippor liquidis not, indcponcicntjof t h o vcrtictil coordinato, we

may wi-ilx

Now dofino

-b- fI-h)T+- ( f ~ - - l 1 ) U ' O .bt 3.r -Introduciiig the varinnce C ~ = U ~ ' - ~ ~ ,22608rj"--62- - -3 11

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Finally, by applying the equation of continuity,we arrive at the following for our equation ofmotion

Combining the continuity equations for the upperand lower liquids, we have in addition(20) H- h) g+ hu= k (t) .If u2 and its horizontal drrivative arc negligiblerelative to the terms on tho left-hand side of cqua-tion (19), equations (19) and (20) reduce, asexpected to equatiorls (17) and (18) which repre-sent the conditions whore thc flow is independentof the vertical coordinate. This we have shown t obe unsatisfactory since we have no wave. Thus,u2 or its horizontal derivative must be significantand then (19) reproscnts tho acwlcration furnishedto the mean flow by t,he passage of the wave.In order to have some estimate of the requiredvariance, let us consider. a steady statc flow varyinglinearly with hcight, or

U =U , i-p 2 - h )where ui is the uppcr flow spc?ed at tho interfacoand p is a constant. Substituting this equationinto thc definition for the averagc?s,it follows that

where u;, is t h e wind speed at H . For flows whcrou;,>3uj, the valuo of thc fraction oxceeds 25 per-cent. Thus, we see that varianccs which nro notat all excessive will yield suificimtly large relativevalues t o the right-hand side of (19), and O U T equa-tions will not degenerate to those implying no

We have shown that our solution roquircs thatthe uppor fluid vary with h i g h 1, and, furthermore,tha t this restriction is not at all prohibitive to ourmodcl. On the otlicr hand, nowlic~rcn our solutionhave we inclutlctl tlio effect on the solution of t h eflow of the upper liquid. As a mattor of fact, wehave implied that tliori? is no such cfect. l'his,however, violatcs one's physical reasoning on t h onature of thc flow. It seems rcasonablo t o oxpcctthat some offoot does exist. A discussion of thisproblem is given in Appendix 13, whcrc tho stcady-state two-liquid problem is discusscd and thoefcct of tlic upper flow analyzctl in greatw detail.

wave.

12

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TI113 T11HEE-LAY 1JR ATMOSPllEREASSUMIIIONS thrcte-layer atrnospliciric model.

clifloroiit dnvclopment, see [14] and [17].)(For a slightly

T I 1 1 C ISQUAliON OF M011ONWe s1in11 start wit11 tlic cqiiatioi) of motion fort h ower f l u i c l ,

t he ~ i ~ c l i t ~ i o nf liytlrostat ic prossure in the lowerniid middlo fluid,

(3 )

As bcforr t ~ l l i.inicd quan i l i c s rcfcr t,o the uppor~oi i s i t~u r~ i c untitjity P, wlioro 6 p iufcrs to

of t>llC> w o bot~tom yers.

any partid variation in p .P

Prom (4)

similnrly from (5 ) ,

13

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From (2), wc have finally

Integrating through a Iayclr from z arbitrary in thelower fluid to z=h, we have

Since 7ri=al we m a y substitute 9) into (8)and obtain

( I O ) A;[a-($f ~ ~ ( H - I ~ ) - - a l i1 g ( h - z ) .Differentiating partially with respect to z, w e have

but

(whcrc the doubla prirnc rc~frrs o t h o t o p fluid) ;tlicrcforc the right-lit~ridsitla of cquntion ( I 1 )bccomcs

1($)=L)= onstantP n

Since

, bu bu(13) Z+U-=- a xSinw ($),is a Conshnt by (12), w e have trans-

formed thr oquation of motion for a t l 1 r ~ ~ayciautolmotropic atmosphrrc to th( qilrLtion of mo-tion for a two liquid flow, whcro the ratio of thoclcnsitics is givcn 1)y d)P h

a P a-+- ( p u > + - - - ( p w 1 ) = 0 .al ax dz

13ut

whr r r the qiitintity in parcntIic?sis roprcsc.ntjs thoc q i d o n of conlinuit>yof an incornprossiblo layer

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Expanding this tcrin, W P gc't

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On the other hand, the investigator may applythe rulp of the science already at his disposaland in this manner deduce the behavior of thephenomenon. In meteorology this usually meansthe application of certain physical laws suchas the equations of motion and continuity and theFirst and Second Laws of Thermodynamics tothe atmosphere. Unfortunately, the atmosphereinvolves so many complex features, that the oqua-tions completely describing the atmospheric flowhave to this date defied mathematical solution.However, the benefits to be derived from thistype of approach arc so numerous that the in-vestigator compromises his situation and resorts tomathematical models. These models are admit-tedly fictitious since they involve scvoral assump-tions about the atmosphcre which do not in factdescribe the atmosphere a t all. The investigator,however, is able to solve his model mathematicallyand arrive at specific results. In the last analysis,the realistic criterion as to whether or not thochoice of a model has been a good one, depends onwhether repeated applications of the results of themodel describe the behavior of the true phenomc-non and predict its future action.

We shall now proceed to analyzo the nature ofthe assumptions of our model and show whichassumptions appear to contain weakness. In thesucceeding chapters we shall show that daspitethese weaknesses, we have been able to explain agreat deal about certain atmospheric phenomena.From that point of view we may consider the modelas a successful one.

The assumptions which we have used follow:1. T h e atmosphere consists of three layerg, thelower two o j which are autobarotropic and separatedby an inversion of zero thickness. In all tho cases

to which this theory has been applied an inversionof temperature is invariably found. T h e maximumthickness of this inversion is under 50 mb. (about500 m.). . Considering the fact that the recordingof a temperature lapse by means of radiosondes isnot a continuous process and that in the evaluationof the radiosonde recorder trace considerablesmoothing is practiced, the actual width of thoinversions may indeed be even shallower. On theother hand, estimates of the amplitude of thejump indicate that a conservative value for itsnormal amplitude is about 1,000 meters. Thus,we might reasonably expect that the fact tha t theinversion is not a sharp discontinuity, should notpresent too serious a difficulty. Thus, if wo

choose a height (22) where the effect of the jumpis essentially negligible, we may consider theatmosphere as consisting of three layers: (1) Thebottom layer below the inversion, (2) the middlelayer extending from the inversion t o the hoiglitH, and 3) the rest of the atmosphere above T?.We have postulated that the lower two lrtyeraare autobarotropic. In synoptic studies it hitsbeen observed that the varticnl 11tpse ritte of

teinpcraturc is basically constitnt mcl t h t t h ostratification is usually conditionitlly unstiiblr,i. e., the vertical lapse r n t e of tempcrrtture laybetween the dry and moist adithrttic lrtpsc rates( o m 0 - 4 5 --T< 1 OX c. g . s.). I t can boeasily demonstrated that a horizontdly homo-genous lnycr, in which tlic prcssurc is 1iydrosttLticand which has a constant vertictd lapse rata oftemperature, is baro tropic and may be expressedin the form p = K p a , where K and a are constantsonly in the undisturbed Thus in clcvclop-ing tho equcttions for rttmospheric lnyers hitving itconstant lapse rate of temperntui-a, equation ( 1 1)would contain additional tcrins corituining pttrtid

b ?derivatives of the form O . These terms mayb Xbe expected to bo significant for largo values ofbK- i. e., largo horizontrd variations in pressureand density) or lttrge values of L) (i. e.,significant departures of the vcrtical lt~psc?ttto oftemperature from tlic tdirtbntic* lapse rutc).Conversely, for rclittivcly weak jumps ttnd lnpserates of tcmperitturc c~lose o tlic adirtbntic, themodel should be quito sittisfrtctory.

We might give the following physical intcrprcta-tion to this assurription of rcplacing R nonadiabaticlayer by one which is adiabatic. Sinco air parcelsin an adiabatic stratification arc i n thcrmodynarnicequilibrium, all energy changes which thoy utidcr-go are due to variations at, the invcrsioii lcvcl only,On the other li~iitln a noiiadia1)aticstratilication,energy changes arc duc to variations in tlie entire

bz -

a xbx a

3 1%~ roof of this thoorom followmi:bIb zot ----a.

By tlro gnu Inw nnd hytlroatntlc rrlntlon

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column. Thus, in replacing a nonadialmtic strati-fication by one which is adiabatic, wo arc in effect,concentrating tho ofl'octs of variations throughoutthe column, at one level, tho inversion height.2. Tlie horizontal velocity u)f the lowest layeri s i ndspcnden t qf the lateral and vertical coordan ates

(8 a n d z ) . We do not conairier the motion o j t h emiddle and top layers.sontonco as follows: Wo Ilio wave is partly afunction of tli(. invcmion liciglit8 bovc~a horizon talsurfnw, this vnriat ion in topography must liavo amarltod cfloct. In sctual practico it is recoin-nzrndcd that a ineaii olovution bo used for thoSlJ1.ftICY\ z= 0 tiiid that tlio prohlom be solvcid 0,swquircd by tlio motlol. It should be ospwt(1d,Iiowovor, tliat, t h csiiltjs woiild iiidicato oscossivcjump spcrds w l i t ~ o lio Lopogrc~pliy w o c d s thctniei~ii olovation nnd rrtluccd spccds ~ I i c r o h etopography is lowor thnn tho iiiettn.A similar probloin osists for t h o hoiglit of t h oiiivcrsioii. ?'ha tlioory implic*i ly tissulnos t~lint,heinvwsion liciglit is l u v c ~ l n t h e undist,urbcti state.rl'his too mrLy not always bo tlio ctuo. 'l'lic pro-

topog~~pl iyIitl~igosrolii S O~ LOVO~o 3,000-4,000

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cedure that is recommended is identical with theone for topography. A mean inversion height.may be selected for the computations. Again,discrepancies in the values of the jump speed aspredicted by the model must be expected wheneverthe inversion height is substantially diff went fromthe average.The latter part of the assumption, that a levelI2 exists such that - is ~~sual lyuite satis-factory. Physical in tiii tion and ohscrvationnl rvi-dence indicate that t>hcaniplitude of thc jumpdecrcases with height such that eventually somelevel H will be reached satisfying the criterion thatbH ah-

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CIIAPTEII JIIMI1:TEOROJ~OGTCAJ APPLICATIONS O F TI I I3 1I Y D R A U l ,I C ANA1,OGY

Thcrc is no i-c~coidof wlicn inan first contchm-of wator. It was probably among the first andmost obvious approaches that suggested them-selves to the serious investigator of meteorologicalplienomcna. A s pertaining to the three laycimodel discussed here, i t is possible that Helm-holtz [27] wns among the first (if not the first) tothink of atmospheric waves on inversion surfacesby analogy to internal water waves on densitydiscontinuities. Sliaw [ 5 2 ] in 1913 really ap-proached tlie cssence of tlie hydraulic analogy(table I ) in describing a line squall. ( H e did notdistinguish a t the time bctween a cold front anda nonfrorital squall line.) He said, these phenom-ena recall those of the advance of a bore up a tidalriver which could give in a like manner a suddenincrease of pressure upon a recording gage if weimagine it to be placed a t the bottom of theriver. The first known attempt a t a moredirect application of the hydraulic analogy asdcscribccl in table 1 was by McGurrin [:39] in1942. McGurrin tried to apply this analogy toclevclop a mechanism for tornado formation.While it may be debatable whether his conc:lusionsarc indeed realistic, the fact that h e arrived a tthese conclusions through the use of the h ytl rnulicanalogy is noteworthy. In 1945, Rossby [46]suggested that internal waves in t he atmosphereare capable of developing sharp forward boundaric~sby analogy to thc propcrties of dispersive wavesin the ocean. While he did not use the hydraulicanalogy in the strict scnsc in which wc liaveclcscribed it here, his discussion of atmosphericwaves with sharp forward boundaries (pressurejumps) by analogy to thc flow of water is indeedinteresting. The first att emp t a t the applicationto meteorological situations of the unsteady statefeatures of the hydraulic analogy was by Freeman[14] in 1948. Freeman argued that since Coriolisforce effects are negligible in the tropics and morc-ovor, sincc thc stratiiication of t h c ntmosplirro inthc tropics approximates a two laycr autobaro-

plntcd wc%thcr proccsscs by annlogy to tltr flowtsopiv atmosphcrc, we may apply tlir nnnlogy.IJsiiig t l i c mrnn wind i n th(1 oww I t i - y y c ~ i s ivpoiot c l e lon thc C,-lioiii*lypilot, ballooii ol)schi.vn ioiis, t i n e l t 1 1 0vnriation of tlic inversion as r c p o r t d by tho 12-hourly radiosonde observations nt one station (1Tol-landia),he was able to construct the characteristiclinos and show that t hc jump with thc proper in-t e n s i t y arrived at a station 350 inilcs away (Bid

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8. Tllcl n i r ninss int,o which the squall linepropagates is usually clinrnetcrized \)ytwo qnnsi-adiabatic lnycrs separated by a11inversion of tcnipcrnture.

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I I 3 4 (SYNOPTIC MAP 220

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23

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06Andrews Field

05B e l t s v i l l e , Md

04Frederick, Md.

07 08Tatuxent River

05 06Annapolis, Md

07Charlotte Hall

04 05Front Royal, Va

05 06Quantico, Va.

05Baltimore , Md05Chanti l ly , Va.

Rockville5 Md,

0505 06 Nashington, D.Garrenton, Va Waterf ord , Va .M I C R O B A R O G R A M S - M A R C H 15, 1950

and indicat cs t he progression of the prcs-sur0 jump line. In this figure, as insu1)soquent isodironc charts, the numbert o the right of tho station c-iidc refers tothe time of passagc of the pressure j i i inpwhile the ratio to the left refers to the risein pressure divided by the duration of therisc. in minutes. Figure 16 is the upper airsounding. It shows not only that thcstratification approximates that of thothree-layer atmosphere model, hut d s othat, the air was very dry. As a t:onse-yuoncc w c would cxpt ':t that cvc'n con-sidrrable lift by t h o prossure ju inp rriiglit,be inadcquatc to ~)roduccc:on(lotisrLLionand precipitation. T h e star s y m t d s i nfigure 14 refer to those stntions which I i i i (1recording rain gagcs but whicli t l i c l riotrecord m y procipitA ion associutctl w iththe prossure jump ine passage. I O t*tmbenoted that all of the corrcspoiitling ti*acos

show Iriiirlicv 1 wss i * o i-iscbs IIcv(i 11 1 PSH.J his intist i w l o out i8lio iypot licksis lliiit 011crise i n prc ssriro in squall lines is t l u c orily Iofalling piwipi tation or t h t b nssoc:iatotl c*oltlair downdraf t s ( see [7], [49],this particiilar case was tho only oiio of tlic12 c:ascs stutlirtl in wliicli no station ro-cortlctl procipit ntion, it is of furthcr intcrcstthat, in only o m o the r cusc (lid a2Z of thestations report pr rc ip i (ation. Tlic rc-maini ng cases con tjainct1 niinicrous intli-vitlual exarnplcs of prc~ssurcjumps un-at:c:ompanictl by piwipiLation.

5 . Sporadic: prc ssiiro jumps whit.11 could noi,t)c systcinatiaally alignotl were dvo found ,The propt~rtic~sf suc:l~urnp iieotl frirtllorinvcstigat ion. Siri(lc oiir objcotive l~crcsto study c~ascsof prcssu~*crmps whicli arc,nligiicd, LVP slirill pui-sric tliis poiiitl nofurthcr at thiu tiiiir.

r 7

..,

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PRESSURE JUMP LINEMARCH 15, 19500400- OSOOEST c

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400\ \

450

Time 03002500 \\\

and l lodgo City (lig. 18) wcrc all so vctry intcwst-iiig that a closcr cxnmiriritiori of thv cvrnt \vasmatlc. It will bo rioted that t he Goodland traticorcprcwn ts u suddon rise in prc ssurc-n prcssr~rcjump-followed by UTIoscillation which has all theiippearancos of a tlampctl wavc. A closw examitla-tion will r(.veill that tire period of t he first tw owaves is ahout 8 minutes arid that of the su~c(~od-ing wavelcts is about 5 minutes. Jhc traccis forCrrwdcn City a n d Dodg: City do riot p r o d u w theidantical wnvo structim, but, in gcnc1r1~1,t is c . l ( mthat thc ovcr-all pattwri is mor( 01 Irss t l io siinic.I h e answers o tho following two qwstioris wwe

(1) Whnt produc:rtl t h c initial and pronounocdrim in prcssrtrr n t tlirse thtw s tn t i ons?(2) Wlrrr t is t h xplitritiion for tlw ~ I t i i ~ p d

r ?Solrght :

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OOODLANO, KANSAS

0801 C S T 0901 GST

GAROEN CITY, KANSAS

0957 CST 1057CST 1157 CST

DODGE GITY, KANSAS

1221 C S TSHE&t CHANGED

MICROBAROGRAMS, MARCH 10, 1951

B y advrcLion of cold nir ns 11, v i i i i w for t h ws-SI^ rise, w o rofcr LO possil)lccold ron al cicbivity.Indocd on the dny in qucst,ion, I \ / l t i i d ~ 10, 1951,a cold front did trnvcrso the rwea. On figurr 20

w e have plottocl tlic synoptic foritiwcs of tlic clayon an 2, -dingrnin. Ihc x-axis w n s t tdrm iinr1111i~to the oriontation of both tlw ~)rc~ssnrruiup liiicand cold f ront . From tho rcgiilrw W R A N A i i d y -sis CcntJcr~ - h o m l y urfncc synopt t * 111idyscs, twas possiblo to lo cat,^ tho surfucc position of thocold front on tho diagrnm. llieso swccssivo&hourly positions arc jndica lod wit11 solid tri-angles. Tho succcssivc positions of tlic prcssure

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GOODLAND, KANSAS06250703072808020814082508480907093010031027

082809100908101010281106112811401210122813101328

09281005102811001128120012281306132814081428

M7E7E7E7E7E7E4w1w1w1w1w 4E250E250E 200

E180E 120E200w 5w5w 4w 4w 4

w 2w 5E200E200E200E 17E 17M10w 5w2E3

e

@60XXXX

9*5-0435- a)05-a)-81200

@

X@-63140 (D-8140 D-@a)

@@X

102 25 23 I \ 6 9766 977108 26 24 I*\ 2 977\ 5 978f__ 0806t 15 k0810 984

1515154 F4 F1 F 135 27 25 / 7 9841/4 F t / 17 9871/8 ZL-F / 12 988-36-1/8 ZL-SG-F 156 26 25 / 17 9901/8 ZL-SG-F / 15 9931/4 ZL-SG-F 176 26 25 / 12 995

3881010101088333

GARDEN CITY, KANSASF l o a 34 32

102 39 35095 52 36125 41 34

L-F 142 35 32L-FL-F 156 33 30DODGE CITY, KANSAS

t \ 1816I\ 14t \ 141 15t 16 41056I / 23+

t / 25+tJ 20t / 22I / 24+tJ 26t

3 F 1155 F10 1051015 13215I5 12975 F 1468 1463/4 L-F

35 35 1t40 36 t46 38 \I \49 38 I\,I /38 34 t 1t /37 33 t J

B and G with any smooth wirve in order to allowtho front con tiniious velocities will iritlicntc thatsomctimo during this :%-hoiirIypcviotl tho f r o n tspeed must have cxcccdod I00 in. p. 11. On that

I day, however, at no chvthon did the wind c~xccod45 m. p. h. behind tho front. Furthcrmore,figure 19 indicates that in each casc, we can finda later time (roprcscnted by a star) which r*cq)re-serits the cold frontal passage much m o re cloarly.

171728+20+4 111251101822+23+27+

979979970979979- io32985990990-*-991993

9829809887 9 , t , i i ~ ~986988990-*-991992991

W c must thus d im inn tc tho ntlvcc%ion of cold airas II c a i i s e for l,ho p w ~ s i ~ r ~umps. 1 1 1 ~ cans( of t d t c h rupitl risc in prcssura lirs in tliothird altwnativo-i, e , , in wavc motion. Asintlic~~tctln tlrc prrvious 11aragrapli tlic weathorobservations u t (>noli of tho three key stations(fig. 19) show that following tlio timo of t h o rapidrise in prcssurc t h e r e is, at some later t ime, arlcfinitc itidictition of 11 cold frontal passage.

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1700

1500

1300

; 100-I0900

d

z2 700cn-I2 0500Izw0 300

0100

2300

2160R A P L B F IML GLD N S C G C K D D C

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are oscillatory. To datc no cxplanation for thisphcnomcnon has becn pu t forth. For 3 > 2 theoscillations cease fig.21). If this relation holdsfor the three-laycr atmosphere as well, tlicri inthe case in question where - .8 there occurredtdic atmospheric analogue to the oscillatoryhytlraiilic jump. I t is quit(? possiblo th at all theosc*illatior~s ntlicat (it1 011 t h c b mic*robarogrmns(fig. 18) arc tlirc to ttic os(*ilI~,tr)~*yrossiIrc jump.

hh 1

hh 1

lhl tt - - i - - c - t t

Typicol profi le of o w r o k h y d r a u l i c jump (hdh,< 2).

Fiorrrts 21. Typical profllcs of statioiinry hyt lru~i l ici ~ i ~ i p s .

On tjhe other hand , i t may bn tha t the pcriod set upby the oscillatory prcssurc jump was resonantwith a vcry short natural pcriod of oscillation ofthc atmosphere (sec [12]). It is dificiilt toanswer the question at this tirnc as to which oftlicse alternativcs is the morc lilrcly.

There is one further bit of c?vitlcncc o supportthe conten tion that tho oscillat ions of tlic micro-1)arograrn rcflcct, tho oscillat ions of tlie inversionsrrrfnca rose, cooling and oontlrnsntion worild ~ J I I C Cplacc, and evcry time t d i r invrrsion sui-facc fallhcating and evaporation would occi~r c. I. [33]).Fortunately, the solar radiation rccortls fromDodge City were avai1al)le for this comparison.On figure 22 tlic ~-carc supwirriposcd tho solarradiation trace and tho microl)arog~.tim or DodgeCity. This relationship is striJcivi,g: A maximumin prossurc diic to t h o maximum risca of theinversion surface) corrrspolrds in time with aminimum valiic>of solar adin in lion 011 tlir ground(duo to tQic intrri~aptionof t h o solar brain bycontlc~nscclwalcr. vapor). A minimiim in prcasurc(duo t o a maximum fall of th(1 invorsion surface)corrc sponds in t ime with a maximum vahic ofsolar radiation on tlic grorind (duo to tdic drci~casoin cloud particles, accompaiiying the ovaporationof watcr droplrts). Tlicb fact that tho solarratliat ion oscillat ion pcrsisi A hcyond t h o p~*cssnrooscillation may iricaii that tlic formci. (in this case)

s11rface. wc wollltl oxp that ovc~i+ylinlc tho

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Interpretation: T h e interpretation of those cvcnt,saccording to the pressure jump hypo tjhcsis may beas follows. T h e cold front is quasi-stationary inwestern Kansas. A new push of cold air crossest hc Rockies (analyzed by the WBAN AnalysisCenter as a cold front aloft). This additionalpush produccs a wave on t he cold frontalsurfaceand th i s tlistitrbnnoc moves along as a gravita-tional wave. (See [l].) A mnrlccd tcmpcratiircinversion AO=8C.) a t about 5,000 feet m. 5. 1.exists ahead of the cold front. AS t h e wavc onthe cold front pushes against the air beneath l h einvwsion, it produces an elevation-typc wnvcwhich moves o u t ahcatl of tho cold front on tbeinversion. ltiis elevation-type wave 1)rraIcs intoa prcssore jump line in a mnnnor tlcsoribetl inChapter I. Mranwhilo tha original wnvc on l h ecold front surface breaks on the surface oausingthe front to accelerate cx~stwnrd. According ttoAbdullah [l], this action is c.ontlucivo to oyc:lo-genesis and indeed a wave does dcvelop on thesurface front.

Graphical Represen,tation on the z,t-Diagramfg.4): The successive positions of thc surface

cold front are plotted as oircles and a smoothcurve is drawn through t,hcsc points to rcprescmtthe progress of the cold front. lho cold frontmotion is far from constant hu t rather rcprc-sents first a constant, speed (up to about 1500CSI, April 24), then a tlccderation (u p to about1830 C S r ) , then a uniform speed (up to 2100CST), then an accoloration (up to 2400 CSrl\),then a uniform sped (up to 0330 CS7 April 25),and finally a deceleration. Undoubtedly, some ofthese variations may be d u e to tlio cmrscncss ofthe reporting station network by means of whichany pinpointing of a frontal position is difficult.I-Iowcvcr, in lieu of an alternative, we shall ncceptas correct thrsc positions as tleterminctl withoutang bias by thc WBAN Analysis Center.

Since the height of the inversion in t h e air massalicad of tho front was at 5,000 fect m. s. l., wehave includcd t h e position of t,hc intersection oft h o frontal slopc with t h e 850-mb. surface. Pra-sumably, it is that, part of thc slopc from tdtcsurface to tlie 850-mb. lcvcd whiclt would b epushing against the air hrneath the invrrsion.T h e progress of th6 850-ml). posi tion of t J i c coldfront was much more tliniciilt to follow on tltc2, t-diagram since tb( irppcr air mops arc drawnonly every 12 hoitrs. l h c April 25, 0300 GMI

(April 24, 2100 CSl) map yicldctl the one posi-tion at, I~yons. Ih ubscqurnt map yieldedanother point at 1600 GMI (0900 CSI) of) t h ediagrnm, and so a linc drawn connccting tbcsctwo points may be consitlerctl as representing t h omean mot ion of tho c d t l front nt, 850 mb. betwcent b c s r two timos. B,y constructing n t ime crossscation at nodgo City, wc could interpolat c foranother position of the cold front, a t 850 mb.(1930 CSI). We may draw a line botwrcn thispoint and the position of the front, at, 850 mb. at1500 GMT, April 24 (0900 C S T April 24, ofl thediagram) to rcprcscnt, t h o mean motion of thefront I L ~ 50 mb. 1~Awccn 1500 GMI April 24 1ant1 0130 CMI April 25. W r now have spec.ilicdcomplctdy t h e progross of thc front at 850 mh. rs-ccpt 1)ctwecn 19:30 and 2100 CSI. Tn ortlrr to jointh r two lines with IL smooth ciirvr, W P arc compellotlto draw t~ curve with II point of inflection. Thismci~ns d i a t the front first ac lcrat~ed and then

Thus, wc havr spwificd oomplctclv on anz,t-diagrani both tlic surface and 850-mb. positionsof tltc cold front. We havc roprcscnted thesepositions by c.ontinuous lines so that, we hava tliocontinuous motion sproifirtl.

T h e 1830 CST position (at NSC) of the waveon t t ic cold front surfarr ( t h o front aloft) wasavailable from t he analysis of t he surface synopt,irmaps. From prcvious maps we determine itsspcctl rintl rrp~~cscnthat spccd by a line throughits lcriown position n t 1830 CSI. All significantsynoptic lines huvo now heen represented on thr5, -di agram.

Thetheory as we have tlovrlopctl it implics that woarc tlealing wit 11 a horizon tal lower surface anduniform velocit ios tlirotigh t he vert ical) 1)clowt h c inversion layer. Nr~turnl ly, cold front, hav-ing 1 slop. othcr than vw i d , will not tillow 11sto m w t , those contlitions. We must replace t he n ,thc w1u t ~ 1 *old rontal slopc brlow thr invcrsion,with II vrrt i o d cold frontal slop^. This approsi-mntiori m a y lie rnntlc if we t8nlco the midpoint oft h positions of the c.oltl front at the surfacc aut1at 850 in?). as the position of thc verticul coldfront. On the z,t-diagram thc line stnrting fromA rcprcserits t h e untlistirrbrd motion of thc vcr-tirr~l d d front,. W e stop this line at tho pointwhci-c it in1 w s c v t s thcb line rcprcwn ing the ~ a v eon t he cold front. At this point the latter im-

We iieetl make one motlifiration, however.

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parts an acceleration to the vertioal cold front,.Similarly the line ending a t B rcprcscnts the finalmotion of the vertical cold front. We jointhese two lines with a smooth curve (without in-tersecting, of course, the 850-mb. arid surface posi-tions of the front). l i n e AB reprcscnts the mo-tion of the accelerating vertical cold front.

The wave on thecold frontal slope hit the vertical cold f ron t andaccelerated it. This acceleration set o f fan olova-tion-type wave on the inversion surface ahead ofthe front. The wave then broke into a prcssurcjump. There is a suggestion of a decelerationfollowing the acceleration. This deceleration pro-duccd a depression-type wave whicli followod theprcssurc jump in tirnc. (The details for t h e con-struction of the characteristic lines arc omittedhere, but may be found in ch. I.)The crosses and circles reflect all of t h c synopticdata used in the construction of t h r,t-diagrainand the characteristic lines. W e shnll now (win-pare our diagram with the mit~robarograms lortho stat ions located along t h e distancbc line. Thesetraces have been drawn in tlicir appivpria tc p1ac:e(pressure increasing to thc 1t:ft). Ihch srririll di-vision represents . O1 Ilg. or i i I ) o i i t .8 ml). Weshall connitlor the J J ~ C S S U ~ Cr u c w in f i n w tlc~triil:

Ness Cit2y NJrQ) As the wavr on tho c ~ l dr o n tpassed over t h o station (1830 U S T ) thcrc was amarlied pressure incrcase.

Larned ( L A B ) : Jhc wavr on t h c d t l frontpassed at about 1926 CSJ, tlpproxirriatcly tlrc titriothat the pressure bcgan rising.

Lyons (LYO) The elcvt~ton-t,ylw wrivo 11ri t lalready hecn formed and it pnsscd J JYOat oxr~t*tythe t ime that the marl

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(21-1AlTER IVOTlIER IlYDRAULlC MODELS

a

In tlic previous clinpter wc linvc spccificd onlyone incclianisin for tho forinnt,iori of n pressurejump line-the nccclcrntion of n oold front int]o nrcgioii which could be described ns n tbrca-layerntmosphcre. This mcclinnisni is tl cscribad sclie-innticnlly in figitre 25. T h o cold front nccolcrntmfrom its spccd givcii by linc A13 to t h pcod givenby linc CD. Tho curvctl linc 13C intlicntcs theaccclcrntion (its slope is conthunlly incronsing,implying its velocity s also inamsing). Weprescribe thnt thc air below thc inversion nndndjaccnt, to the (+old ront iiiovos witli t h o spcotl oft h o front. We hnvc tlicn n tlistri1)iit on of vdo(it,yalong n linc i n tlic .x,l-plnnc. If in addition weknow tlic iniLinl invcrsion hciglit, tlien by Caso111, 11. 7, wo may solvo for. t h o flow. we l1avcscon that this t,ype of a tlivtribution of conditions(1). 8) indirt~t,cs n rlovnl ion-t,ypc wn,vo. Figuro25 shows this clcvation-lypc wnve nnct tlia formn-tion of n prrssurc~jump ino moving ou t nhoatl oftho cold fro11L.

This pnttcrn is by no nicnns the only way inwhich a prcssni*c j u m p liiici may bc protliioed.Following, we slinll prcscnt otrlior inodcls for t h oformation of prcssuro j iinip linos ns wcll us nproprySnting tlcprcssion-type wnvc. This list, isfar froin c*ornploto and t hi autlior is ccrlnin tlintother motlcds (:an I)c nntl will bo suggcstod by anyinvestigator conccrning hin~sc lfwith this tlicory.

d xd t

Standing Hydraulic Jump

TIIE STANDING IBESSURE .JUMP LINEFreemnii, Bailoy, nrid Byors [ I O ] liavo found ovi-

dcncc of a pr(~sstir~ump forniccl in t i ninnner vcvysimilar to t l io formation of u standing liydrc~iilicjump at tlic base of n spillwny (fig. 20). Thisphcnorncnon, wcll linown to t h o hydrnulic cngi-neer , is for~rirdby t h o ncoclcrt~ton of W I J ~ L Wtlownt h o spillwny. 1 f this ncoclornlioii produces supcr-critical spccds in t h o liquid, t l i c ~ na t the bnsc oftho spillwny tlic wntcr lovd c*linngos t h r r ~ p lyfroin tho higli cncrgy levo1 (shootdng wnt cr n n c llow dcpth) to t h o low oiiorgy love1 (stronining

w n cr n n t l liigli dcpth). F r ~ ~ n i n n ,silcy, nndHycrs proposcc.l t I i n 1,in tJic ntniosplicrc nccclcrntingair do\w tlic c*oldfrontd slope could by nunlogy

1

Standing Prr i run JumpI w/ /~

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B

produce a standing pressure jump ahead of thefront. They present synoptic dat a and show tha tthis evidence is compatible with their hypothesis.This type of pressure jump line is more or lessstationary relative to the cold front. It may, ofcoiirse, move, but any such motion would be d u eto the fact that the cold front itself moves. Thus,if the cold front moved to thc right (fig. 261)) astation would experience a pressure sal1 with thepassage of the pressure jump line.

TIlE DOUSLIS IWESSURE JUMP LINEIt is of particular interest that the synoptic

case cited by Ih em an , Bailey, arid Bycrs m h t vcto t h c standing pressui-e jump line contained a nadditional fcaturc. A region of prccipitation 13(fig.27) traveled out ahcad of t h c front tznd, withtime, the distmcc AB increased. Thc authorssuggest that the prccipitaf ion was due to thethunderstorm activity nssoc,iatetlwitli t Iic prcssurcjump, A. However, this uloiic: canirot cxplninwhy the distance AB i nc~cnse t l with tiino. Amore complete explanatio~imay he givon asfollows:We start with an undistur1)cd invcmion Iwightlevel. We then let air accclorate down tdic cold

front slope. These accelerations produce a(bump on the inversion surace. The forwardhalf of this bump would be an elevation-type wave(see p. 8) which we have seen would move out)to the right and develop into a jump. This jumpcould then produce prccipitation in region B.With time, the length of the distance AB wouldnaturally incroasc. On the other hand the back-ward half, A, would be the standing pressure jumpdescribed earlier . This experirnent may easilybe duplicated in a water channel. As a matter offact , Freeman, Bailcg, and Byors present a photo-graph of just such hydraulic evidence in theirpaper.The double pressure jump lino was inferred inToppers [SO]model to cxplnin the E-W orientcdpressure jump lines which move northward intothe Midwest. I n this model, it was suggestedthat tlie fast-moving easterlies corning around theback side of the B e r m u d a I-Tigh and meeting theland surface of the Gulf Sta tes wo dd produce anE--W oriented pressure jump line which woultlmove northward. Actually, a double pressurejump line would be formed. One would he nstanding pressure jump whrw the supcrcriticnlflow would jump to a subcrit icnl flow at a higher.elevation and t h e second would l m the propagattionof this higher elevation northward and its dcvolop-mc n t into a pressure jump line. The authoi*,unfortunately, has no synoptic evidence to provotha t this pattern actually O TUI*S.FlJ15 1jACK WA I{ I -hil( )V I N G I I{ IZSSU I< I : JUM I 1,JNE

Another mcclrnnism for tile formntion of apressure jump line is proposcd in [B].n thesynoptic case citctl, the prc~ssi~reump line movcdin u direction opposito to tho air flow in thc lowerIn ycr, consoqucntly the choice of thc naine

backwnrd-nioving p r c ~ ~ i i r cu m p . J11c mecli-anisni for tho goiiclsis of the prcssare jiimp line isas follows: 7ho n i t * flow bvnoti tli t ho invcrsion \ Y ~ Sturnrtl from a gcri~rnd outhrI*lg low t o an cnst,cii4yilow and direotly noixitd to a s tn t ionnr ,y frontwhich wus oricwtcd N -S. As in the. c t i sc of waterflowing against a bnrrier and piwluoing a bacltwnrd-movirig liptlrr~iilicump (lig. 28), i h WILSsuggc~stctltlint hy nrmlog,y a similar siturrtion took pli iw intlie t~trnosphcrc. 7110 air 1)olow the invcxi-sionfiowcd against the stntiomry front produc ing apresstire jumj) line whicl i moved backward withrespcct to the air Ilow.

1

( I

86

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Tlio osplanation for tho gcncsis of the pressurejump line follows from figuro 9. Wo rocall tha t the1-direction is tlio direction of propagation of thoprossuro jump line. In this case 5 is positive totho east. At some initial Limo 1=0, wo havo adistribution of velocities such that the eastorliesdiminish toward tho barrior (i. e., tho stationaryfront). Since u is positive for a westcrly wind, wemight; interpret the previous remark as indicatingu,>u, for xtlLj for z i< r j an olevation-typo wave)or (b) ui>u, for xtBwave wliicli progrcssivolyovor(ook tlio cloud baiilr wns nssocintod wilh arapid drop of tho isentropic surfaces w1iic.h inturn p r o d u c ~ . ~ llic aulinbnlic hcnt ing nccessuryto d rs i ccn tc tlio c*loiid prrrtinlly in 1110 west, niidc~oiiiplotdy ovcr Wusliington. Coiiiirnititioii ofthis c*oiicliisionwns found in the solnr rctdint i o i iof t h o solnr rntlititioii trnrcs is t h n t tlioy coiitniiitwo significnnt points---onc? is n nini*lml d c c w t i s oin riitlinlioii i i i i t l t l i r otlicr is IL subsoquolit sliiirpinci-cnsv in nulint ion.

In (nblc :

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I

I

A

Floun~0.--RadlnHon traces for Wn~hlnyton, ). C. Clrvi~lund, 'ut-In-Hny, nnd Columbus, Ohlo, and Indlnnnpolls, Tnd., Deccmbcr 0, 3040. A rc'lntlvolyclear-day rndlrrtlon trnco Is Indlontcd by t h o dnshid Iltlr.

38

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tions, the ngrccmcnt in tlic tiinc valucs is rcmnrk-I TIIW EST) able. Wc inny snfcly nssociat>c tho dccrensc inof pressure pulse ,

d i n t t i o n with thc onse t of tlic cloud bank andtlic suddon incrcnsc in rndint iou with thc pnssngoDccronso Incrcnso

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CHAPTER VSUMMARY

In this paper itj has been demonstrated that byapplying the hytlraulic analogy to special classesof meteorological situations, a consistent synop-tically significant phcnomcnon may bc arrived at.This phenomenon is a pressure jump line.

The pressure jump line is a particular kind ofgravitational wave traveling along an inversionsurface. It s main charactcristic is tha t it has avery steep slope so that it appears on microbaro-grams as a jump in pressure. This vcry steepslope (estimated at about 1 : 5 as compared with1:80 for cold fronts and 1:200 for warm fronts)is responsible for marked forccd lifting of thcatmosphere. Under favorable moisture concli-tions this lift produccs violent convection andthundcrstorm activity.

In order to identify a pressurc jump line, ameteorologist must use acceleratcd microbaro-graphs. These instrumcnts give a finer resolu-tion of the time scale and as such can diffcrcn-tiate between prcssurc jumps and more gradualpressure rises. Channcls could be devclopetl bymeans of which information about pressurejumps, identified by accelerator1 microbarograms,could be communicated to a central analysis unit.By plotting thc timc of t h e jumps on a map,isochrones marking thc progrcss of the linc coiiltlbo indicated. Then by referencc to the thermo-dynamic charactcr of the atmosphcrc, thc metc-orologist could anticipat,c t h o typc of wcathcr toexpect with the progress of the prcssurc jump line.

This procctlurc involves thc lcnowlctlgc aboutthe existence of a prcssurc jump linc. Can any-thing bc said aboutj forecasting its tlcvclopmc~?Indccd thcrc can, but unfortunatc.ly not too mrich.First of all, the meteorologist miist bec~omc~us-picious whenever a cold front i s moving int)o nrcgion tlcscri1)ctl in this papcr as a threc-hy(:ratmosphcrc. It is rc,c~ommcnt~ccIhatJ in suchasps he cblioosc a clircvtion normal t o tlia colt1

front as his t-direction arid 1)egin plotting an ,s,t-diagram, similar to tlioso prcwntotl. 1icr.c. On tliistliagrarn hr sl~oiiltlplot

( I ) T h c suac:cusivc positions of the cold front.

(2) Tho weather a t thc various stations alonghis direc tion .3) A continuous plot of the prcssurc, ifavailablc, of tho various stations alongthe 2-dircction. I n the abscncc of thisinformation, lie may plot) any prcssureinformation availablc t o him.

If thc cold front bcgins to show rapid accelora-tions, the mctcorologist could thcn construct theresulting charactcristic lincs. The weather d o -mcnts along thc direction of thcsc lincs should beearcfully scrutinizcd for any indications of apressure jump linc.

We might go one stcp furthcr and loolr for carlyindications that a cold front might, be cxpaccclorate in tho near future. Onc such inmight be the appcarancc of a fresh surge of coldorair within the cold air mass itsclf (somcwhat likeAbdullahs typc of wave refcrrrcl to in ch. 111).We have scen (fig. 24) tha t such a surge bchind thocold front could protlucc suhsequcnt acof the cold front. Another sequcncccold front accelcrations, and noted in scvcral pres-sure jump oases, sccms lo b o as follows: A quasi-stationary E- W cold front, lies across tho CentralS t a t m to the Rocliics. Whcn n small low prcssurcsystem crosses ovcr the mountains south of thecold front, thc portion of thc front lying wes t oft he low ccntcr rushcs southward.tiori protlriocs IL prossure j uinp linc. Tlicsc: iwcbut two medianisnis by means of w1iic.h (:old frontsscom to accclcratc. Untlout)tjodly, hci~oirc innnyot hers. Once the prac+tic:ingmctcoi*ologistbcginsto loolc for prcssiirc jump Iiiws, it is incvit)ablo thata oompi~c~lic~nsivc~isting of mcc.hanislns whioli pro-duce ncwlrrat ions of cold fronts will follow.

W l i u t >a l ) o n ~I C H R I I I Y \ u m l i lincs that,nrc startcdhy o~ntlit~ioiisi o trolatrtl to a cold Ci-oiil? Walinvc s(:cri thn t m y mot ooi~ological itiitLt ion pro-(hi(-ing l i t hcr t ~ i i oc~~lri71ton of tbc nir Iwlow thoinversion, o r an cilcvjLtion of tJic inveI*sionsi1 h o ocould yicltl n prcssrire jump inr . It is left, t l ien,t,o the ingenuity of the intlivitl~~nlnrtcorologist~Orccognizc such a sitaation i ~ i i t l ~i~ndlct i n a incm-

This n

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ncr similar to the one suggested here for theaccelerating cold front.

Tlic bulk of this paper lias c*oncwnctl tsclf withthe pressure jump linc and a discussion of its~~ctcorologic~allOpcrticS resid tdng f rom a con-sideration of tlie analogy bctwcon a pressure j m n pline, a hydraulic jump, and a shock wave. How-ever, the analogy necd n o t stop here. The inlicr-cht propertics of tlie flows, as drsc*riIwd n table 1,sliggest some interesting extensions of tQe analogy.We might, for example, look for 11iglilyspccinlizetlproperties of the phenomena in tdicir respectivedoninins and by means of the dictionary con-tained in t)able 1 , translate these spceializcd pr01)-c r t k so as to apply in another domain . 0 n c suchprocotlure was suggcsted by Teppcr [80] in aspeculation conccrning the origin of tornadoes.

In gas flows, it has been demonstratd (sea [13]and [g]) that when two shock waves intorsect, theyrwotluco a vortcs sliect in the flow bcliind tliem.Across this sliocf,, tho flow is para1101 butJ of dif-f ~ c n tpccd so that n oonccntratcd z011c of ~ 0 1 % -ticity is formcd. Iranslatcd into 1110 Inngungc~ofthe threr laycr atniosp~wrcb, e might, s tn t c : I.lh?~

two pressure jump line s intersect, +hey produce in tibeow behind them a vortex slheet. Across this sheet, theow is paral le l but qf diyerent speed so that aconcentrated zone of vorticity is , formed . Tllc sug-gostioii was tlicn made that tornadoes could forin

in this zoiie of vorticity. Several instances con-sistent with this hypothesis were prcscntod in [59]and [GO], where it, was shown that tornadoes wereapparcntly associated in both time niid spnce withtho spaco-time locus of t h e iiitcrscction zone of two(or more) prcssuro junip lines.In conclusion, we may state that the analogydisciissed in this papcr betweon certain atmos-pheric flows arid liquid and/or gas flows opens upa new avenue of study in meteorology, one whichit is hopcd will lead to a 1xtte1-underst,andirig ofcertain me teorological processes.

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REFERENCES

1. A. J. Abdiillah, Cyclogenesis by a Purelyhl echanical Process, J o u r n a l o Mete-oroZogy, vol. 6, No. 2, April 1949, pp. 86-97.2. B. Bakhmetcff and A. Matzke, HydraulicJ um p i n Terms of Dynamic Similarity,Transact ions American Socie ty of CiuilEngineers , vol. 101, 1936, p. 630.3 . L. Bergeron, MEthode Graphiqnc GEnEralcde C alcul des Propag ations dOnclcs Plane s,M i m a i r e s Sociite Ingenieurs Cizrils de France,July-August 1937.4 . A. M. Binnic and S. G . Hookrr, The Plowunder Gravi ty of an Incomprcssiblc andInviscid Fluid Through a Constriction in aHorizontal Channel, Proceedings of theRoyal Society of L o n d o n , Series A, vol. 159,April 1937, pp. 592-608.5 . V. Bjcrlmcs, On Quasi-static Wave Motionin Barotropic Fluid Strata, GeophysiskePubl ikas joner , vol. 111, No. 3, GcopliysiskcKomm ission, Kristiana, 1923.6 . J. R. Bruman, Application of the WaterChannel-Compressible Gas Analogy, Re-port NA-47-87, North Americnn Aviation,Iriglewood, Calif., M arch 1947.storm, U. S . W eather B I I ~ C ~ I I ,lrtdiington,D. C., J une 1949, p. 129.8. F. H. Clauscr, Classroom Lcctuiv N o h :Th eor y of t h e Flow of a Comprcssiblc Gas,Jo hn s Hopli-ins Univcrsi ty , 1950-1 951.9 . R. Cou rant and I. 2G2-267.

21, N O . 8 , hl111.~11950 , pp. 243-249.21. PJ I I I I ~ I I I I I ~ ~ I ~ ~ I ,(Go//s UI l e lrol)(Lg(ttiotb / P , si o i i x de ique,

Piiris, A. I ~ O I - I ~ ~ L I I ,908.22. U. 1 , . I [IIrris, Soinc. 1rol)lrins ICnc~oriiitr~*ctlnGravity Waves in tlic A t r n o s p h c , U. S.

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Weathcr Bureau, Washington, D. C., 1951(unpublished). tady, N. Y., 1941.ment of Rejlection Angles in Regular andMach Rejlections of Slwck Waves, PhysicsDopt., Princeton Univcrsity, Mnrch 1947.

Report 55618 General Electric Co., Schenec-23. li. 13. Harrison and W. Blcalr~icy,Remeasure- 35. E. Jouguot, Qudques Problrincs do LHydrodynamiquo GC?n6rales, Jou,rn,al deMatliPmatiques Pures et Appliguees (Series8) , vol. 3, No. 1 , 1920.24. H. T. Harrison and W. I. C., Augusl 1944.43. E, Prcisrv~rk, Applim t,ion of t he Mothods ofGns I ~ y n t i m i c s ,o W n tcr Plo\rrs w ith ti FrcwSurfncc: 1. 1~101~sith No Energy Dissipa-tion, I I. Blows with h lon ic~n tun iDi~cont~i-nuitirs (13ydrnulic Siunps) 1 1 YoclmicaZ A I m -orantlum ,934, 936, Nnlionnl A dvisory Coin-mitt,rc for Acronuutics, Wnshington, D. C.,1940.

44. D. Hinbouc~liirisliy,Sur IAnalogio I3ytlrau-l i qw dcs hlouvcnic~iitsdun Fluidc Coin-prcssiblc, (otrcptc.,~ Rcntlus de 2.Ai*arhnc.des Scieiicos, vol. 195, 1032, 1111.995-099, vol.199, 1034, pp. 632-634.

45. , Itc crr~1113 om pnrn b o siis 1nCTO-cly~wniiqna dcs p t i c s v t des grondcsvi t (wes, Coinptcs I 2 c i d u s l e IAcackrnic (IPSS c i c m w , vol. 202, 1O:K i , pp. 1725-1728.

46. C.-G. Rossby, On t lw 1ropngtit.ion of $re-qucncics nnd Energy in Certain Typt~s f43

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Oceanic and Atmospheric Waves, JournalofMeteorology, vol. 2 , No. 4 , December 1945,pp. 187-204.4 7 . H. Rouse, B. V. Bhoota, and E. Y . Hsu,Design of Channel Expansions, Proceed-ings of the American Society of Civil Engin-eers, vol. 75, No, 9 , November 1949, pp.

48 . R. Sauer, Theory of Non-stationary GasFlow: I. Laminar Steady Gas Waves, 11.P l a n e G a s W a v e s w i t h C o m p r e s s i o nShock@, AAF Translation F-TS-763-RE1and F-TS-768-RE, Hq. Air Materiel Com-man d, Da yton , Ohio, August and September1946.4 9 . R. S. Scorer, Gravity Waves, Weather, vol.

/ 50. Z Sekcra, Helmholtz Waves in a LinearTemperature Field with Vertical WindShear, Journal of Meteorology,vol. 5 , No. 3,June 1948, pp. 93-102.51. A. H. Shapiro, An Appraisal of t h e HydraulicAnalogue to Gas Dynam ics, MeteorologicalReport $4, Dept . of M echanical Engineering,Massachusc t ts Inst i u c of T c c h o l o g y ,April 1949.52, W. N. Shaw, Porecasting Weather, Constableand Co., London, 1913.

53, L. Sherman, The Neglect of Compressibilityin the Flow of a G as a t Low Speeds,Jouhnal of Meteorology, vol. 6, No. 4,

54. A. K. Showalter and J. R . Fullrs, PreliminaryReport on Tornadoes, U. S . Wenther Bureau,Washington, D. C., 1943.55 . H. Solberg, W cllcn bewegungcn cincr Isotlicr-men Atmosphiire, Report of the 18thScanJdinuwian National Congress, Copen-1-1agen, 1929.

1369-1 385.

5 , N O .7 , J u l y 1950, pp. 262-263.

August 1949, pp. 286-287.

56. V . P. Starr , Mornentum and Energy In te-grals for Gravity Waves of Finite Height,Journal of Marine Research, vol. 6 , No. 3 ,December 1947, pp. 175-193.57. J. J. Stokcr, The Formation of Brcalrers antlCommunications on Pure a n d A p -plied Mathematics, vol. 1 , 1948, p. 1.58. M. Tcpper, A Proposed Meclianism of SquallLines: Th e Pressure Ju m p Line, Journal ofMeteorology, vol. 7, No. 2 , February 1950,

59. ~ , Radar and Synoptic Analysis of aTornado Situat ion, Monthly Weather Re-view, vol. 78, No. 9, September 1950, pp.

60. On the Origin of Tornadocs, Bul-letin, of the American Meteorological Society,vol. 31, No. 9 , November 1950, pp. 311-314.61. - On the Desiccation of a Cloud Bankb y a Propagated Pressure Wavc, Monthl;t/Meuther Review, vol. 79, No. 4 , April 1951,62 . ~ , The Tornado antl Scvciec StormProject, Weatherwise, vol. 4 , No. 3 , June63. and J. C. Freeman, Jr., Commentson an Analogy Between Equatorial Enstcr -l ies and Supersonic l~lows, Jouriml ofA40t~~oro logy~ol. 6 , No. 3 , June 1949. p. 226.64. TI],von K arm an, %ine Prt~lrt isdieAnwendung

tl PI Analogie zwisch en tfbersli allstrom ungin Gascn und 15t)crlrritisc:hoi~StrGmung inOnc~nen Gcrinnen, Zeitschrift f i i r Ange-wardte A4uthemutik wnd Mtchatiik, vol. 18,

65 . D. 7. Wil lkms, A Surfaro Micro-Studyof Sq I nll-Lin e Th11nd PIS ormq, A l o r u % h /Wc.ulhcr R C O ~ C W ,ol. 713, No. 1 1 , Novcmbcr

pp. 21-29.

170-1 76 .

pp. 61-70.

1951, pp. 51-53.

1938, 111). 4 9 - 5 6 .

1948, 111). 2 3 0 - 2 4 6 .

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APPENDIX ASOME MATllEMATICAL REMARKS ON TIlE SINGLE WAVE PROBLEM

In chapte r I we investigated graphically thothrec possiblc classcs of single waves resultingfrom our basic cqua tions. Hcrcw ith, we shallproscnt the same results but from a more matlie-matical point of view.'Let us investigate the local cliangc in tlic shapeof t ho wave profilc as we progress with a pointo n the wave, i. e., along a single characteristicline. Th us , wc aro intcrcstcd in (2) lioret h e differentiation is along a givon characteristicline such t ha t

L ? + ( U + C ) i)zd t at

Difl'crontiating equ ation ( O ) , C h p t c r I , yiclds

Since c is constant along t h o clinractcristic linc,wc havcblr. a cax axx-

along tlic sanic cliaractoristic linc.Tlius, to invcstignto the clii~ngcn shape of t h owave profilc as w o progrcss wit11 the point onthe mave, wc prcfer to s t tdy-(' ) Applying(1 ) ,we may write ttt dL *

Since c is constant along the characteristic linc,the esprcssion in the brackets vaiiislies, and w0have romaining( 5 )Tlic solution of (5) is given inimcdiatdy by6) z )c - l =3(t-f0)+(5J ac - l ,

w ~ m c a g)oprosrnt,s the local slope of tlic pro-filc nt t i n initial time f = f o .C i n ~ f o r m ~ f l o wn o uc tw ) , If (*) -0, tlien for

a111, (%)do. n y 4) liis incans u + c ) + ~o r the cliaracteristic lines arc parallcl.

0

ax 0 ad X

Ilepression-type wane. I n this cnse ( o.Thiis, with incrcwsing l inir [&]-'most increasc.This iiicniis that the \vavc Aat,tcns out. 13y (4)w o liavc u+c) dccrcnses or tlic characteristiclines mu st fan out .

~ i c u c c t i o r t - t / p e onae. ~n t$liis case ( )

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APPENDIX BTHE GENERAL STEADY STATE TWO-LIQUID PROBLEM

In chapter I, we succeeded in arriving at asolution for the unsteady state flow in the two-liquid problem. However, in so doing, we havehad to ignore completely the nature of the upperflow and consequently have becn unable to cvalu-ate the effect of the upper flow on the over-allpattern.We shall now restrict ourfa ttention o the steadystate problem, b ut here we shall include the upperflow. In this manner we shall attempt to learnsomething specific about the importance of theupper flow. I n particular, we shall seek theanswers to the following questions: (1) Undcrwhat conditions does the upper flow producenegligible effects? I n these circumstances wemight expect that our unsteady state solutionmay be applicable. (2) When the effect of theupper flow is not negligible, in what manner doesit affect the over-all flow pat tern? I n tliesocircumstances, the unsteady state solution maybe considered as a first approximation only.he Problem: We establish our coordinate sys-tem with tho jump, so that the liquids appear toenter the jump at ono level and leave it a t another.All velocities are relativc to the jump speed . Thevariables used in tho problem are given in figure 30.T h e subscripts 1 refer to the conditions on thatside of the jump whcrc the lower liquid appcmsto enter the jump, and the subscripts 2 refer to

t

the conditions on the ot,her side of tJhe jumpwhere the lower liquid appears to lcavo thc jump.We retain assumptions 1, 2, and 4 given at t h obeginning of chapter I. I n addition, we specifythat while the surface z=O is a horizontal surface,the height of the free surface, z=EI, need notremain constant across the jump. This lattercondition will later be amended in order tha t ourmodel rcmain physically the samc.There are two continuity equations

while by definition(2) hi+ ~ I =Hi

h2+ I2=aWe have assumed the pressure hydrostatic, sothat

3) p (at height> z in the lower liquid)=P @I- h)+ P II 1 ~ )

p (a t height 2 in the upper liquid)= p g ( I 2 - 2 )I n order to write the momentum condition, thatthe difference in thc pressure forco equals thedifference in motncntum flux, we note that thodifrcrcncc in t h o pressure force

p-p ) (1 (h;-h:)2while thc tlillcwncc in i r i ~m ~ i i t i i i nlux is g iv e n by

A M - p h 2 7 L : t + p r ] 2 ~ i ; - p ? i , l ? L ~ - p ~ l ~ :Equating A/=A/\f , using ( l ) , nnd ddining forconv cn i(inco

46

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tho mornontum condition reduces t o(5) r;(x-l) [---+ 1 -;I

s(y-1) p y--- d 3 =sr1(1-xry)Sinco wo have not restricted tlie free surfacc tob e a constant across tlic jump, equation ( 5 )represents tlie mo incntum equation for tlie gonuraltwo-liquid steady-sta to problem. In any givensituation, yl, and s arc considerod known sincethey rcfcr to undistur1)cd'propcrtios of tho liquid.Tlius, if we arc givon in addition, oitlier t h o flowvelocities (roprcsontod by li; and $1) or tl io wt~vc

profiles (roprcsentcd by n and 11) this oquationroduccs to a single equation in two unknowns.We ricod specify aiiotlior condition in order t h a tthe problcni be doterminod. I n o u r inodol thisother condition is t h a t I$, tlio hciglit of t ho h esurfaco, is a conslant. Bcforo wo app ly thisrestriction, wo shall inspect oqiiation ( 5 ) brieflyand d r a w two very simplo rclations from thisgeneral cquation.1. A trivial solution is givcn by a=y=1, the2 . Coiisido r tlic cnso s=O, or p '

lie initialvnluos of r1 and s and find the corrospond-ing valuc of al=- on tho ordinate.

Stop 2 : Entor t h o riglit-linnd sido with this valuoof o 1 and t h o proper valuo of G1. Find tliucorrcsponding poiii t 011 the b curves.Stop 3 : Follo\v tho b curve until it iiitorsccts tliepropor value of F1. Tliis will give tlicfirst, approsimntion for tlio solution of9) u q

Sr1

Stop 4:Conlpllte a*---------- S-r1 (1 +r1 - 1xJ

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SFIGUBE1.-Somopam solution for the equation

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Step 5 : Repoat Stop 2 ctc. usin* az iiistcad ofalStop 6 : Tlio series xl, rz, x3, . . ., if it convcrgos,Several numerical csamplrs in tlic rango of valuosclndor iiitcrcst liavo iridicatcd tliat> tho scriosxl, 4 x 3 . . . convrrgcs very rapidly so that thesolutioii for (9) can readily bo found.There nrc sevoral conclusions tliat, we inay drawtion (14) or figuro 31. Thc conclusions arc drawnonly for our casa, iiainoly that in wliicli I roinainscollstant.

1. In order thnt a jump O C C I I ~ n t,lic iptwfacc(i. o., x> ) t81ie inotlificd Proutlc ~iiiiiibcr,F1,must also exccod 1. By rquations (10)and (4) of this appcndis and (6) of cliaptcr I,this implies that IuII>IcII. If \v(? intciprotc1 as a critienl wave s])ood chni*nctorizingtlic coritlitioiis alicad of tho jump, thenthis result implies that tij (stutionary)jumpexists oridy Jor sujwtwitica81 f o w . Fiirtlicr-more, sincc \vo liavo consitlcrrd u1 ns theflow alioad of t h o jump rclritivo to t h ejump, t hon for n j u m p propngaling iiitloti liquid at rest lull= ~ ivlirra zj, is t h ospcod of tlic jump. Thus, the speed of t h ejump exceeds t h e critical wave speed aheadof the j u m p .

In gas flow ant1 in onc-liquid Iiow (col-uiiiiis 1 and 2 , tnblc 1) aildogous relationsexist. In addition, it can bo sliown thatfor tlicso flows, tho flow bchind t h o shockwavr, or liydraulic jump, is subsonic OFsubcritical, respectively. In two-liquidilow, tlio flow tliat1, Iuzl1 is, of coursc, not iioccsstiry fo r tllomatlicninticnl solution of tlio nioiiiemtuin oquntioii.Coiisistoiit solutions riro possi1)lo involving z< 1.Tlicsc solutiolis on occasion will yiold F1

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energy flux as the two liquids pass through theThe energy flux (per unit width per unit time)of the lower and upper liquids arc given respec-

tively by

jump.

We shall consider two cases.CASE I: u and v have the same sign. In thiscase, we postulate that the total energy fluxmust decrease through the jump since an exchangeof encrgy between the two liquids is possible.Thus, using (1) and (7) and (15)

Eliminating uz and q by the equations of con-tinuity and after a few simple algebraic manip-ulations, w e arrive atAE +-I) [22l 1]+2 22E = (1 --s)pguhh,

Substituting the momentum condition (9), thisreduces to

where E must be larger than zero.satisfied provided(a) If z>l, then the energy condition will b e

CASE 11: u and v have opposite signs. In thiscase we postulate that the energy flux of eachliquid separately must decrease through the jumpsince an exchangeof energy in the direction of flowis not possible. Thus,

puh

Eliminating u2 nd v z and applying the momentumcondition (9) l o Ae we have

The first of these is possible i f and only if %>I.This implies y 1whenever the uppor liquid is moving in tlie samedirection but faster than the jump.In the atmospheric situations to which this