Numerical Integra 00 s Lau

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NUMERICAL INTEGRATION EXPERIMENTS WITHA BAROTROPIC PRIMITIVE EQUATION MODEL

Jimmy Ray Slaughter


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VAL POSTGRADUATE SCHOOLonterey, California

TNumerical Integration Experiments With

aBarotropic Primitive Equation Model

by

Jimmy Ray Slaughter

Thesis Advisor G. J. HaltinerMarch 1972

Approved loh. public. fiaXiah^', dib&ubutlon unlunotzd.


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Numerical Integration Experiments WithaBarotropic Primitive Equation Model

by

Jimmy Ray SlaughterLieutenant Commander, United States NavyB.S., United States Naval Academy, 1963

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN METEOROLOGYfrom the

NAVAL POSTGRADUATE SCHOOLMarch 1972


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DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CALIFORNIA 93943-5002

ABSTRACT

Experiments are carried out with various time and spacedifferencing schemes applied to the barotropic primitiveequations using both real data and a particular stream func-tion which is an analytic solution to the nondivergent baro-tropic vorticity equation. With both types of data, therewere some significant differences in the forecasts producedby the various schemes. Replacement of the widely used leapfrog (centered) scheme by others which eliminate or lessensome of its inherent errors at the expense of more computertime or storage appears to be justified at such time whencomputer capacity no longer restricts operational use ofthese more time consuming schemes.


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TABLE OF CONTENTS

I. INTRODUCTION 9II. PROGNOSTIC EQUATIONS 10

A. SPACE DIFFERENCING 101. Second Order Scheme 102. Fourth Order Scheme 11

B. TIME DIFFERENCING 111. Leap Frog 112. Runge Kutta 113. Euler Backward 12

III. BOUNDARY CONDITIONS 13A. PARTICULAR STREAM FUNCTION 13

1. North-South Boundary Conditions 132. East-West Boundary Conditions 133. Variation for Fourth Order SpaceDifference 14

B. ACTUAL 500 MILLIBAR DATA 14IV. GRID 18

A. A PARTICULAR STREAM FUNCTION 18B. FNWC 63x63 HEMISPHERIC 18

V. INITIAL CONDITIONS 19A. A PARTICULAR STREAM FUNCTION 19B. ACTUAL 500 MILLIBAR DATA 20

VI. RESULTS 22A. A PARTICULAR STREAM FUNCTION 22


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B. ACTUAL 500 MILLIBAR DATA 22C. COMPUTER REQUIREMENTS 23

VII. CONCLUSIONS 28BIBLIOGRAPHY 35INITIAL DISTRIBUTION LIST 36DD FORM 1473 37


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LIST OF TABLESTABLE I. Table of root mean square errors inposition and central height of thethree low centers at 24, 48, and 72hours as forecast by various differ-encing schemes (on actual data) incomparison to the correspondinganalysis. 21


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LIST OF FIGURES

1. Illustrating the 63x63 square grid andboundaries as applied to a particularstream function. The dimensionlesssymbols x and y are used for notationconvenience . 162. Illustrating the 65x65 expansion of thegrid in Fig. 1 in order to apply fourthorder space differencing over the samearea. The dimensionless symbols x and

y are used for notation convenience. 173. Initial height field as produced by aparticular analytic stream function on

a 63x63 grid. Only the central contoursof the three features are illustrated.-- 21

4. Relative position and central height ofa particular stream function as forecastby various differencing schemes. 24

5. Forecast and analyzed relative positionsand central heights for an actual data500 millibar feature designated lownumber one. 256. Forecast and analyzed relative positionsand central heights for an actual data

500 millibar feature designated lownumber two. 267. Forecast and analyzed relative positions

and central heights for an actual data500 millibar feature designated lownumber three. 278. One feature in an initial height fieldproduced from a particular stream func-tion illustrating the central heightand 390 meter contour. 309. Forecast height in meters of centralvalues and positions of centers and 390

meter contours for a particular streamfunction by the leap frog scheme. 31


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10. Forecast height in meters of centralvalues and positions of centers and 390meter contours for a particular streamfunction by the Runge Kutta scheme. 32

11. Forecast height in meters of centralvalues and positions of centers and 390meter contours for a particular streamfunction by the Euler backward scheme. . 3312. Forecast height in meters of centralvalues and positions of centers and 390meter contours for a particular streamfunction by the leap frog fourth order

space differencing. 34


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LIST OF SYMBOLS

c Phase speed.d Grid distance.d' Grid distance at 60 degrees latitude.f Coriolis parameter.f Coriolis parameter at 60 degrees latitudeFNWC Fleet Numerical Weather Central.g Force of gravity.h Height of meteorological surface.L Wavelength in the x direction.xL Half a wavelength in the y direction.m Map factor .u Velocity component in the x direction.v Velocity component in the y direction.v Initial meridional wind speed.a Relaxation coefficient.C Relative vorticity. Geopo tentialty Stream function.^ Initial stream function amplitude.


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I. INTRODUCTION

In selecting a particular differencing scheme for appli-cation to numerical weather prediction, a major considera-tion is computer time and storage requirements. Therefore,the selected scheme is a compromise between verification ac-curacy and computer limitations. At present, the leap frog(centered) scheme is widely used in numerical weather predic-tion and will be used as a standard for purposes of compari-son. Present computer limitations and methods of verifica-tion tend to preclude seeking more accurate solutions basedon mathematical considerations alone. The purpose of thisstudy is to look beyond present computer capabilities andexamine several other differencing schemes which reduce oreliminate some of the inaccuracies in the leap frog scheme.This scheme has three time levels and, therefore, a spuriouscomputational mode arises from the finite differencingscheme. This problem can be eliminated by the applicationof two level time schemes such as the Euler backward orRunge Kutta methods which have on computational mode. Afourth order space differencing scheme is applied to reducetruncation error and improve phase speeds. Should any ofthe schemes produce a significant change in forecast rela-tive to the leap frog method, it must be determined if thedifference represents a sufficient increase in accuracy towarrant the added computer time and storage requirements.


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II . PROGNOSTIC EQUATIONS

The equations of motion and the continuity equation fora barotropic fluid in the momentum flux form are9(uh) 2 9 (uuh)^ 2 9 (uvh) _ , v 9hfr-r + m -r + m "^ fvh = - mgh-xdt dx m ay m dx9(vh) . 2 8 (vuh) J 2 8 (vvh) , , ,9h~ + m -r + m tt + fuh = - mgh-r9 t 9 x 9 y m 9 y9h 2 9 (uh) 9 (vh), nz + m [it + -z J -9t 9xm dy m

(1)

(2)

(3)

where h is the depth of the fluid (or height of a pressuresurface in the atmosphere), g is the force of gravity, f isthe coriolis parameter and u and v are horizontal velocitycomponents

A. SPACE DIFFERENCING1 . Second Order Scheme

With u, v, and h given at each grid point, thedifferential equations may be approximated by

uh - . uh . .(uh) . . 2 //jr/ _l w 1+1>3. 1J\it = -m /4df (u. ., .+ u..)( ' J + J )m mt 3 - -m /Ad[(u. +1> . + u..)(; (4)uhh . -- (u. . .+ u. .)( X 1,3 + 1Jl-l, j 13 m m )vh . . . n vh . .+ (u. .._+ * )t-J-*+1+ 1J )1,3+1 13 m mvh . . ' vh . .- (u. n + u..)( 1,J i+ 1J )] + f..(vh)..1,3-1 13 m m 13 '13- mgh. . (h. , . - h. . .)/2d = F. .13 1+1,3 i-l3 13

where 3 increases with x and i decreases with y. (See Fig.1) Similar expressions are used for the other two equa-

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tions. This method of space differencing, devised by A. Ara^kawa (see, for example, [1]), prevents nonlinear instabilityin the continuous time case (sometimes referred to as thesemi-discrete equations).

2 . Fourth Order SchemeFourth order space differencing which takes into

account a greater number of grid points and reduces trunca-tion error approximates uh tendencies by

2f^l.3 - */3(,1J )uh -H- [


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AUAU 3 = At F[Un + , (n + p At]AU 4 = At F[U + AU , (n + 1) At]

Un+1 = Un + rt^U- + 2AU + 2AU_ + AU.]O 1 l J 43. Euler Backward

U* - Un = At Fn

Un+1 -Un = At F*

Denotes trial values.

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Ill . BOUNDARY CONDITIONS

A. A PARTICULAR STREAM FUNCTIONThe boundary conditions in this case are applied to an

initial stream function which is an analytic solution to thenondivergent barotropic vorticity equation and after eachtime step during the forecast.

1 North-South Boundary ConditionsFor the 63x63 grid a wall exists between y=l and

y=2 and also between y=63 and y=62 (See Fig. 1). No fluxis permitted across this rigid boundary. This is accom-plished by setting v , = -v ,_ and v = -v therebyx,63 x,62 x,l x,2making the average v =0 alone the wall. The other para-x , ymeters are set equal across the wall. Summarizing, theboundary conditions are

North Southv = - v ,, V n = - V ,x, 63 x , 62 x,l x,2U (LIx , 63 = u , u n =u, 62 x , 1 x ,*x,63 = *x,62 *x,l = *x,2

where (j) is the geopo tent ial2 East-West Boundary Conditions

The East-West boundary conditions are cyclic assuggested by the function plotted in Fig. 1. Summarizing,the boundary conditions are

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East Westl,y 62, 63, y 2,y

where A represents parameters v , u and d>x,y x,y Y x,y3 . Variation for Fourth Order Space Difference

In order to apply fourth order space differencingand still maintain the boundary conditions of the originalgrid, it was convenient to create an additional outside rowof points and relabel as a 65x65 grid. (See Fig. 2) Theboundary conditions are summarized as follows:

North Southx,65 x,62 x,l x,4

A represents u ,x >y j a x,yv ,. =-v ,_ v = -v J and mx,64 x,63 x,2 x,3 x,yx,65 x,62 x,l x,4x , 54 x,63 x,2 x,3

East WestA = A A = Al,y 62, y 65, y 4,y A represents u ,x > y x > yA = A- A,. A_ v , and (f>2,y 63, y 64, y 3,y x,y Yx,y

B, ACTUAL 500 MILLIBAR DATAWhen the equations were integrated using real data, the

"spongy" boundary conditions of the FNWC model were appliedAfter each time step, the prognostic results were modifiedby setting t f . ,uh = (1-k )uh_ + k uh b

vh = (l-k')vh + k' vhh = (l-k')h + k' h

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where the subscript indicates initial values. Below lat-itude four degrees north set k'=4. Above 17 degrees northset k'=0. Between four degrees north and 17 degrees north,there is linear variation in k ' from zero to one. This re-stores the variables to their initial values south of fourdegrees north, while north of 17 degrees north, full varia-bility is permitted. There is no restoration north of 17degrees north. Between, there is partial restoration.

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^_ y = 3 >/ / 1 1 /it/ 1 it/ zrnxz&L *wfl[* "^xxaxxxzznxzn zzzzzzzzzzzgafe

ytf2 ^r

Xl *2

y-2.iu/i/i//i//ii ///tuff** ^JJft tg./i imui/ uiiiinii wtr/u,//

Illustrating the 63x63 square grid andboundaries as applied to a particularstream function. The dimens ionless sym-bols x and y are used for notation con-venience .Figure 1

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~- H=GS

/VORTHrn/i >//> jf/t j j/j ///jrfZ> WfiU ?&/)} // / / j/f sit > > // / fti-r?j--63

y--62

/ /JtJj/7

X=l X2 *3 Xsif1/1 i fi \/\

X-S2 X'G3 Xs C X'fer

!/ I y=*

y-3

$

gzzzzzgzzzzzzzz zzzzzzzzzz^c 5$JJ ^zazzzzzzzzzzzzzzzzzzzzzzzzgzzzzzzzyr2.

J

Illustrating the 65x65 expansion of the grid inFig. 1 in order to apply fourth order space dif-ferencing over the same area. The dimens ionlessymbols x and y are used for notation convenienceFigure 2

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IV. GRID

A. A PARTICULAR STREAM FUNCTIONA 63x63 square grid as in Fig. 1 and as modified in

Fig. 2 for fourth order space differencing is used tominimize the number of changes required to adapt programsto the FNWC 63x63 hemispheric grid. The boundaries inlatitude extend from 30 to 90 degrees. Grid length isdetermined by dividing the geographic distance representedby this latitude difference into 62 equal parts.

B. FNWC 63x63 HEMISPHERICIn conjunction with the actual 500 millibar data, the

FNWC 63x63 hemispheric grid is used. This is a square re-presentation of a hemispheric polar stereographic grid inwhich the grid distance d is determined by

, d 1 , . . ,1 + sin 60 *d = "m- = d ' ( 1 + sin Y }where m is the map factor, d' is the grid distance at 60latitude (381 kilometers), and sin y is defined by

973.71 - R 2 , D 2 ,_ _ .2, ,_ _ ,2sxn y = 2 wnere R = (I-I ) + (J-J )973.71 + R P P

I, J = arbitrary grid point.I ,J = grid point for the north pole = 32,32.P P

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V. INITIAL CONDITIONS

A. A PARTICULAR STREAM FUNCTIONIn order to obtain a smooth initial stream function

field, an analytic solution to the nondivergent barotropicvorticity equation is used, namely

]b - lb sin - (y - -r) cos - (x-ct) L I Ly xwhere c is the phase speed and the wavelength in the x dir-ection and the half wavelength in the y direction are equalto 61d (L = L = 61d) . The initial stream function isa y

i i T / d> 2tty = i> sin - (y - -r-) cos - x .O Li Z Ly XHere, \p is determined by using the relation

v LI z o X^o 2tt

where v is the initial specified amplitude of the meri-odional wind speed derived from

drp 2tt, r . IT , d N 2 tt ,V = 9^ = L~ ^o [sin L~ (y " 2> COS Z~ X] *x y x

Using the initial field thus obtained, the linear balanceequation

V 2 = fV 2 ijj + Vf ViJ;which in finite difference form may be written? 2 . .= f . . V 2 iK. +t[^ f v ^+ v f V i>] = F. .

1J 1JY1J 4 X X y y

yj ij

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where* 2 < > - < > i+i (j+ < > t-i. i+ < >i, j+i+ < >i.j-i- 4 < >u\- < >i+i.3 - < h-l,i =The preceding equation may now be solved as a Poisson

equation for the geopotential field. Thus,V 2 /g, computations for the initial height fieldare completed. Additionally, a constant height is addedto each grid point value in order to eliminate negativeheights, which would otherwise cause computational insta-bility. (See Fig. 3)B. ACTUAL 500 MILLIBAR DATA

The initial height field is obtained from the FNWC 500millibar analysis. The linear balance equation is solvedfor the stream function as a Poisson equation using ip=


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DifferencingSchemeLeap FrogLeap Frog FourthOrder SpaceRunge KuttaEuler Backward

RMSE For Position(Nautical Miles) RMSE For CentralHe i ght (Me t e r s97 3076 31

93 3188 25

Table of root mean square errors in position and centralheight of the three low centers at 24, 48, and 72 hoursas forecast by various differencing schemes (on actual data)in comparison to the corresponding analysis.

TABLE 1

Latitude = 90

{ Goo 580 WO tOO 520 600 J-j

Latitude = 30

Initial height field as produced bycular analytic stream function on agrid. Only the central contours offeatures are illustrated.Figure 3

a par t i-63x63the three

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VI. RESULTS

The results shown in the following figures were obtainedfrom the computer subroutine mapping of the 63x63 heightfield obtained by the various differencing schemes using 10minute time steps. One grid squares are used in the figuressimply as a convenience to provide a ready reference scaleto compare positions of low centers.

A. PARTICULAR STREAM FUNCTIONResults for the forecasts made using a particular

stream function for the initial values are summarized inFig. 4 for central height and relative position. The fore-cast results for the individual differencing schemes arepresented in greater detail in Fig. 8 through Fig. 12.

B. ACTUAL 500 MILLIBAR DATAThe results of forecasts made on three low systems from

the 500 millibar FNWC analysis are summarized in Fig. 5,Fig. 6 and Fig. 7. Although only low systems are illustra-ted in this study, similar results were obtained from themovement and intensification of highs. The time frame ofthis study begins with the 1200Z March 24, 1966 500 milli-bar heights and terminates with the 1200Z March 27, 1966forecast. Low number one (See Fig. 5) was located overcentral Siberia. Low number two (See Fig. 6) and low num-ber three (See Fig, 7) were located over Southwestern

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Canada and Labrador respectively. Root mean square errorcomputations for position and central height as forecastby each differencing scheme when compared to the correspond-ing analysis are contained in Table 1.C. COMPUTER REQUIREMENTS

Although peak program efficiency was not attempted dur-ing the application of the various numerical schemes, con-sistency in application was carefully maintained. Usingthe leap frog scheme as a standard, the leap frog fourthorder space differencing and Euler backward schemes requireapproximately a 35 percent increase in run time. The RungeKutta scheme requires approximately a 170 percent increasein run time and a 25 percent increase in computer storage.

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72hrf""I

J

72hr

r - 17*

48hr| 1I x ,I II I

i i

Nn

24hri 1

I & II I

l:::J

58 Nautical MilesijL.

One grid square

Differencing SchemeForecas t Central Height (Meters)24 hr. 48 hr. 72 hr.x Leap FrogoLeap Frog FourthOrder SpaceiRunge KuttaEuler Backward

382 383 383

383 384 383382 383 380383 384 385

Initial Central Height 383 MetersRelative position and central height of aparticular stream function feature as forecast by various differencing schemes.

Figure 4

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NA

72hr

L j48 hr

I

~A 5 n,T !L24 hrIT! 206 Nautical Miles

i 1.One. grid square

Differencing Scheme Central Height (Meters)Forecast 24 hr. 48 hr . 72 hr.x Leap FrogoLeap Frog FourthOrder SpacevRunge KuttaEuler BackwardCorrespondingFNWC Analysis

Initial HeightFNWC Analysis

4930

494049304940

4940

496049404940

4990

503049904990

5020 5090


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N

72hro

O/ll 48hr1 I

I\oSL-L-J 72 hr

( ri

L.__J

0)

i i-xl I

"I 206 Nautical MilesOne grid square

Differencing Scheme Central Height (Meters)Forecast 24 hr. 48 hr . 72 hr.* Leap FrogoLeap Frog Fourth

5020 5070 5080

Order Space 5060 5080 5050ARunge Kutta 5020 5080 5060Euler Backward 5020 5080 5080CorrespondingFNWC Analysis

Initial HeightFNWC Analysis5020 5120 5160

5100Forecast and analyzed relative positions andcentral heights for an actual data 500 milli-bar feature designated low number two.

Figure 6

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N72hr

( 1

A

24hr]o

! I

48hr| *A OI

-

(2)

i IT|

',206 Nautical Miesi i J,

One grid squareDifferencing Scheme Central Height (Meters)Forecas t

x Leap Frogo Leap Frog FourthOrder SpaceARunge KuttaEuler BackwardCorrespondingFNWC Analysis

Initial HeightFNWC Analysis

24 hr4960

496049604960

48 hr. 72 hr.4960

498049504960

4910

496049204930

4960 5110 Filled4990

Forecast and analyzed relative positions andcentral heights for an actual data 500 milli-bar feature designated low number three.Figure 7

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VII. CONCLUSIONS

The various differencing schemes were applied to thesmooth analytic solution in order that differences in fore-casts produced by these schemes would be more readily dis-tinguishable than with actual data. However, the choiceas to resolution was probably too high in this study (61grid lengths per wavelength) and shorter wavelengths areyet to be tested. Differences produced in location andcentral values were not conclusive when compared to theresults produced using actual data. A reduction in resolu-tion for the analytic initial field (approximately six gridlengths per wavelength) would possibly produce a more mean-ingful separation in the accuracy produced by the variousschemes

The negligible increase in accuracy (four percent basedon a root mean square error comparison) of the Runge Kuttascheme over the leap frog scheme does not justify the ex-

tremely large increase in computer time and moderate in-crease in computer storage. Both the leap frog fourthorder space and Euler backward differencing schemes showpromise for improved accuracy with a moderate increase incomputer run time (approximately 35 percent) . The leapfrog fourth order space differencing scheme gave a 24 per-cent increase in position accuracy. The Euler backwardscheme produced a nine percent increase in position accur-

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acy and a 17 percent increase in central height. With boththe Euler backward and the leap frog fourth order spaceschemes, the 35 percent increase in computer time could bereduced somewhat by more efficient programs. The smallstatistical sample of pressure systems examined in thisstudy does not permit firm conclusions to be drawn regardingthe various schemes tested. However, there is reason tobelieve that the fourth order space differencing of thenonlinear advection terms will reduce the phase error in theprediction of pressure systems and this method should betested further. Williams [2] has shown that the pressureforce need only be differenced with second order accuracywhich permits a somewhat larger time step without computa-tional instability than when fourth order differences areused throughout.

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390meters

206N. Mi.

One grid squareOne feature in an initial height field produced from a particular stream functionillustrating the central height and 390me ter contour

Figure 8

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\ V

L206 N. Mi.

24 hr.48 hr.72 hr.

One grid square

Forecast height in meters of central values andpositions of centers and 390 meter contours fora particular stream function by the leap frogscheme Figure 9

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One grid square24 hr.48 hr.72 hr.

Forecast height in meters of central values andpositions of centers and 390 meter contours fora particular stream function by the Runge Kuttas cheme

Figure 10

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/N

t/

\

L 24 hr.48 hr.72 hr. One grid "squareForecast height in meters of central values andpositions of centers and 390 meter contours fora particular stream function by the Euler back-ward scheme

Figure 11

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N

t

//

Immmm

V*. %,

24 hr.48 hr.72 hr. N. Mi.k.

One grid square

Forecast height in meters of central values andpositions of centers and 390 meter contours fora particular stream function by the leap frogfourth order space differencing.

Figure 1234


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BIBLIOGRAPHY

1. Haltiner, G. J., Numerical Weather Prediction , Wiley,1971.2. Naval Postgraduate School Report 5lWu72031A, PhaseSpeed Errors with Second and Fourth Order SpaceDifferences with Staggered and Unstaggered Grids ,by R. T. Williams, March 1972.

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INITIAL DISTRIBUTION LISTNo . Copies

1. Defense Documentation Center 2Cameron StationAlexandria, Virginia 223142. Library, Code 0212 2Naval Postgraduate SchoolMonterey, California 939403. Professor G. J. Haltiner, Code 51Ha 5Department of Meteorology

Naval Postgraduate SchoolMonterey, California 939404. Lieutenant Commander J. R. Slaughter 2Carrier Airborne Early Warning TrainingSquadron 110 (RVAW-110)Naval Air Station, North IslandSan Diego, California 921355. Department of Meteorology 3

Naval Postgraduate SchoolMonterey, California 939406. Naval Weather Service Command 1

Washington Navy YardWashington, D. C. 20390

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Security ClassificationDOCUMENT CONTROL DATA -R&D

(Security classification of title, body of abstract and indexing annotation mux/ be entered when the overall report is classitied)ORIGINATING ACTIVITY (Corporate author)Naval Postgraduate SchoolMonterey, California 93940

2a. REPORT SECURITY CLASSIFICATIONUnc lassified

2b. GROUP

REPORT TITLENumerical Integration Experiments with aBarotropic Primitive Equation Model

4. DESCRIPTIVE NOTES (Type of report and.inclusive dates)Master's Thesis, March 1972

5- AUTMORiS) ffirsl name, middle initial, last name)Jimmy R. Slaughter

6. REPORT DATEMarch 1972

7a. TOTAL NO. OF PAGES38

7b. NO. OF REFS

8a. CONTRACT OR GRANT NO.

6. PROJEC T NO.

9a. ORIGINATOR'S REPORT NUMBER(S)

96. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)

10. DISTRIBUTION STATEMENTApproved for public release, distribution unlimited

11. SUPPLEMENTARY NOTES 12. SPONSORING MILI TAR Y ACTIVITYNaval Postgraduate SchoolMonterey, California 93940

13. ABSTR AC TExperiments are carried out with various time and spacedifferencing schemes applied to the barotropic primitiveequations using both real data and a particular stream func-tion which is an analytic solution to the nondivergent baro-tropic vorticity equation. With both types of data therewere some significant differences in the forecasts produced

by the various schemes. Replacement of the widely used leapfrog (centered) scheme by others which eliminate or lessensome of its inherent errors at the expense of more computertime or storage appears to be justified at auch time whencomputer capacity no longer restricts operational use ofthese more time consuming schemes.

FORM1 NOV 65

S/N 0101 -807-681 11473 (PAGE 1) 37

Security Classification A-31408


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Security Classificationkey wo R DS

barotropic primitive equationsnumerical forecast

,Fo1".,1473


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Sl*9hter 1 3427n. ^erf Ca/ . 4U**pef-iment! ' nte9rat!

5 2 5 * .

Thesis 134270S57025 Slaughterc- 1 Numerical integrationexperiments with a baro-tropic primitive equa-tion mode?


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thesS57025N^rical integration experiments with

III Hill

3 2768 002 01141 3DUDLEY KNOX LIBRARY

Numerical Integra 00 s Lau

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