U.S. DEPARTMENT OF COMMERCENATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION

NATIONAL WEATHER SERVICENATIONAL METEOROLOGICAL CENTER

OFFICE NOTE 180

A Method for Solving the Helmholtz Equation in aRectangular Region Using Homogeneous Neuman Boundary Conditions

Joseph P. Gerrity, Jr.Development Division

JUNE 1978

This is an unreviewed manuscript, primarilyintended for informal exchange of informationamong NMC staff members.

1. Introduction

The Helmholtz equation is

V24 - X(x,y)4 = G(x,y) (1)

where X is a real, non-negative function, V2 is the Laplacian, ~ is the

unknown, and G is a forcing function. The boundary conditions to be

satisfied are -bn= 0, where - is the outward normal derivative.

The numerical algorithm presented here for the solution of this

equation is a split-pseudo-direct method. We approximate the equation

(1) using finite differences. A recurrence, or relaxation, equation is

written to develop a sequence of solutions. Unlike the S0R method, we

use implicit approximations in the recurrence equation in one coordinate

direction. The implicit equation is inverted using Gaussian elimination.

After completing the generation of the new approximate solution the

process is repeated, but the other coordinate direction is now approxi-

mated implicitly in the recurrence equation.

Just two such passes have been found necessary for a test problem.

2. The finite difference equation

Assume that the region is rectangular of size ((P+l)Ax,(Q+l)Ax)-.

Equation (1) is written,

Oi+l,j + Oi-l,j + oi,j+l + i,j- (4 +ij) ij = Gij (2)

whereij = (Ax)2 X((i-l)Ax,(j-1)Ax)

andGij = (Ax)2 G((i-l)Ax,(j-l)Ax)

-2-

The recurrence (or relaxation) equation is written in step 1.

n+l n rn+l n+l n n+l n+l~ ~4i+~,.j + n + n+l~ - (4+Xij)4.' - G.)(3)~ij = ~ij [i+l,j + ~-l,j+ +~ ij + T ij-1 _j lj

In this step the i coordinate is done implicitly, The boundary con-

ditions are evoked wherever they can be used.

The equation (3) is rewritten

n~~l 1),~n+l n+l- an + (l+a(4+Xij)),j+l - a 4i+l = R.. (4)

where R.. n + a, (O n + on~~~~~~~~~~l (5)ij 1 J +l ijl ij- 1 ij)

The solution of (4) by Gaussian Elimination (cf. Richtmyer and

Morton, p. 198) is carried out, augmenting j after each solution. On the

row j=2, the variable ,jn+l is equal to n+l because of the boundary con-

dition, and this fact is used to modify (4). Similarly when j=Q, the

parameter ij+l is replaced by ni+l because of the boundary condition.

In step 2, the recurrence equation is modified to

n+lj = n + a (in+l + 4ij l + i n+l - (4+Xi) n+l -_ G.) (6)= 4. + i+l,j i-lj J ij j

The j-coordinate direction is now done implicitly. The modifications of

the system are similar to those outlined in step 1.

3. A test problem

In Appendix A, the Fortran code for a test problem is given. A 'true'

solution was generated using an IBM random number generator. The Helmholtz

term was similarly generated. Using these fields, the forcing function was

-3-

calculated for P=52, Q=46. Employing a first guess solution that was

everywhere equal to the mean value of the 'true' solution, we ran the

code with a=k. At no point did the numerical solution differ from the

'true' solution by 1 part in 1000.

The average value of the 'true' solution is 5496.098, that of the

numerical solution is 5496.094. The Bias was -.321E-2, and the Root-

mean-square error of the numerical solution was 0.365E-2.

4. Further work

The method will be explored further using larger domains and different

values of X, including zero.

Reference

Richtmyer and Morton, Difference methods for initial value problems.

Interscience, New York, N.Y., 1967.

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C. A tMl PROGRAM FOR TtIF SOLUTIOM OF THE HELMHOLTZ EQUATIONC

DItiENS[Tn Z7(53947)]TSOL(53,47),R(5347),HLN(53147)CC Z IS'THF 'I.PMERICAL RnLUITIOnM OF THF DTFFFRF.ICE Ent.IATIOTNC :)TSOL TS THF TP'IF SlL.jTI[ON OF THE DIFFERENCE EIUATIOr, IC . IS THF FnPCIIjG F!.F:tcTTON FOR TS13L,C HLM IS :THE HELrHOLTZ COEFFICIFNT.CC ::AL-'HA IS TIlE COEFFICIENT OF RELAXATION.

DATA AL.PHA/.25/C

*CALL I!!ITL(PTSOL·HLM)CC SET GIUF.SS.TO MEAN VALUE OF TSOL.C

FACT = 51*4- 5 .TrEAr = O.0 D 1 T=2,520- 1 J=2.46TrEAI = TMIEAN + TSOL(IJ)/FACT .

I COTI'JTTNUE- ::DC 2 1=1,53- .c J=11.47ZtT,.t) = TMEAN

2 CO']ITI1:7

PRTNT 3, Tr'EAN3 FORr"AT(1HO0,'AVERAGE VALUE OF TSOL=',F1lO3)

CC STEpI SOLVES THE ITMpLICTT ITEpATIVE En. IN THE X DITRECTIC

00ono CCrITT.IIjECALL STFI (7,R.HlLMALPHA',IER): .CALL rTFOI.!T4TSOL,Z,1 ) CALL STFP2( ,RHLMALPHA·IER)CALL DTFOUT(TSOLZ,2)IFTFER Eo O0) GO0 4PR I TNT 9

5 FORMAT(ItO0,HAVE TO STOP PRORLF'r IN STEP1')SToP

4 CONTNIUEC

C CALCIILATE STATISTICS OF SOLUTION

CALL STAT(Z7TSOL) STOP . .Ern N..- '

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o0 13 J=1,47CAtLL RfrIGTT(Iy,Tyy) ..X.(lT,J) = Y * 10. * **2): IX TY

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SURROUTIriF: PrIGIT(IX,IYgY)IY TY * 65539IF(TY) 5,6,,:

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. sE' Ur' RE^LAXATION Co.J = 2 .... STAPT- <ln I=2,52 L :1F(1-1) = Z7,2)1 +.*ALPHA *( Z(I,3). - R(I2) )

10 COHT!JI.UED.. -- 11-=1,51 - :. . .:D(T) .= 1. + ALPHA k*- 3. HLMfI.+,2) ) 'C(U) = -- ALPHA-ET)= -ALPHAIF( I .EO. 1 , OR.:: I E. 51) Dn).= DtI) + CIT)

11 CONiTINUIJE . :C SOL>'E EO*

CALl -S :I. VICo (,DEFi.AmS I, TER'i951)T F(IFR ."!r. 0) GO TO 9no000.C, 12 I=2,5?Z(:T2) = tr:'S(T-1) ' :" .

12 £.OhlT!~HUJE . -Z(1,2) ='Z('2*2? ' · . :Z(53,2) = 7(52,2) -DO 13 T:1,937(Tl) :Zf 7I 2)

13 CCNTIPJINUE . :C J =2 <( .... FND

C LOr,P TO Dn J=3 THIROuGH 45 ...-STAPTC

DO D0 J=3,-45.C : SFT -UP EQUATIOF! PARAMETERS

DC) 30 1=1,51C(1.. - ALPHA ' :fl(T) = 1. + ALPHA * 4. + HLM(I+lJ) :-F(T-) = - ALPHA ..-IF( I EO. 1 .OR. T .EO. 51 DiI) = D(I) + C(T)'aF(T)' = Z(I+l1J) + ALPHA * ( Z(T+IJ+l) + Z(T++,J-1 -+(I+1

33 CONPTIMIIF so' , Vr Eo

' -CALl.- [FELI.F(C,,0EF, Ar!S, TER 951) -IFi{ER.rcE.n) GO TO -o-00[0 31 T=2,52: 7(,J) = AFISII-1I.) -. . .

31. CONTTNUE .. .Z(1'J) = ?2,J)Z(53,J) = 7(52,J)

C 20 CONTINUEC

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:

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:C ENn OF LOOP C: C ' ' " ' '

C J =46 ..... STAPTC SET iUP Etl')ATIOrfJ

DO 40 T=1 51C') = - ALPHA.D(T)= I. + ALPHA * 3. + HLftI+1,46) ): (I) = - ALPHAIF( T .Fq..1 .OR. I .F. 51) n(I)r= (i) + CIT)

40 COr4JTTN!i :DJO 41 I=1,51FtT) = 7(T+1,4(,) + · ALPHA ' I Z(I+,45) -- R(T+l946) )

41 COb!TIm'll: ;[C SnLVr FO.

CALL GFl. WECnEF, Ars, TER,51) -TIF(IFR.Frl.n) GO TO s9000 DO 42 I=2,:2Z(T476) = Z'(I-46) : : .

42 CONTTI!NJ UE: ;0 : (1:.14A) := Z(P*46): ;0. :: : : ; V

Z(53,4A) = 7(52,46);:.E S ; - iZII ,47 )- = ' ( 2,qr 1 2 0 -: :

' 7(- ,l47) = Z(52,46)

C COMPLETES SOLUTOPI ... ALL pTSC

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SLJ3ROUTINE STEP2(7,PHLMALPHAf,]RIR)'DI.PIEM'I-T N 7 (5?, 4 7),fR(53,47) ,L"(93,U7)DOURLE PRECISTON C(9:1),D1I,/r51),F(~1IhANS(51)r

C SET Up REL.XATTON, E.-C 1=2 .,. START

P0 1.0 .d=2,q6 '~F(J-1) = Z(2,dJ .. + ALPHA 7 Z(. J) - R(2,J)

lo CO.ITIrNUIJDO 11 J=1,45p(J)-= 1; + ALPHA * (.3. + HLM(2,,J+1)C(J) =- ALPHA.;' E (J) - ALPHAIF( J .Eq. 1 (OP. j *EO. 45! D(J) = D(J) + C(J)

11.. COTTJ.UEC SOLVE EQoCALL GCLITM(CfL[,EFA ,S',TER,45).

IFirFUP .Hr!r. ) GO: TO 9000anO 19 J='2t'f,7(2,J# = ANq(J-1)

12 CONTTNUE7¢2,1) = Z(2,2)Z·2*7) = *6'I3 J=i, ...

7(1,.J) = ZI2. J)13 CONTINIIE

C I =2 X.. ENDC

C. LOOp TO+ DO 1=3 THRU 51 ... START'C

[30 20-T=3,51C:' SFT UP EQUATION PARAM£ETERS

PC 30 J=],45.C(J) - ALPHAEl (J) = 1. + ALPHA * ( 4.. + HLM¢],+IE(j) = - ALPHAIF( J *tI. 1 'OR. J Eon. 45 1 0(J) = 1n(j) + C(J)FIJI 7=T?,+l) + ALPHA * (.Z(T+1e.J+) + Z(I-2,J+l) -R(TJ+i)

30 COITINHIESOL'E FO.CALl- GL'LFM(CD,FF,FA'S, TEP,45)Tr(IpR f'tFr. 0) GO TO 9000[)0 'zl J=2,46·'Z([,UJ : AI'SIJ-1)

31 CO.ITTNJUF?lI19I) = Z(IT2)Z(T.q47).= Z(I,4-6)

C20 CONTINUE

C

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LEVEL 2.2'

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DC, 4 n J=1, 4 5. . V f) = - AI_*FV :t : :. PH86(JrJ) = 1. + ALpH!A * ( ̂ . + H4LM 5.,J+) !F(J) = - ALPHAIF.( J .rO. 1 OP.J .EQ. 45) D(J) : f(J) + CrJ

u4n CCNT lIlJF Do 41 J=1,45F (J) Z(52,J+l + AL P4A * (Z51,J+11 - R

421 CONTTNUEC-C SOLVE ECO. -

C:,"PcLL C.LIr(C,fEF,AmS,TER, 45 )TIF(IEP .!"E. 0 ) GO TO qOO 0;30 42 J=2,46

Z(52,J1 = AN$S J-l l I - -Z(53,J) = Z(52,J)

42 CONT IJCE:Z(5?,tI) = Z(52 ,2)7 ( 5?, 47)= 7(52 46) - .

.(f531) = Z(,2 " : 2 . .7Z(5947) = 7(52,46)

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T -w 00122t7 s 0023?g-I 002 ~1 10' i 00251;-I 0026 21 : 1 0on27T~t~ 0028:: - T ]N ri o. '2 8 : ::-1:. -00o29T r! 0 030'.i- ;r~ n wj~i

.SUiVRnUTTtIE SELIM(Cn,E,F,AlNSvI-R*N)pPrtPLF P~r.CSTO' 1t4(lA),M(5)S51)rOUF;LE PRECTSTON0 C( fl!'(01),'(53 (TUSE cSSIAlL FLIITF1ATT

In M! PROCC 'S TO nnl VE THE , ATRTY nCOUT TON

T * t/VIS) (F), tERE T IS A TRI-DTAGOPlAL MATRIX OF ORDER Ng

An AE 'AA'!D F AE !", DT. ENSTIOMAL vr'CTOr'Sb CDA'TI;TIDE ARE THF tELEIL'TS OF THE [E.ATRIX T, NOTE Ctl&E(N) DO

APPEAR.

INTTIALTZE

: D(l)TF ( r .EO. 0,. GO TO 9000 .

SW = F{) / G

EL.TIT?,:ATION (FORWARD pROCESS)

G.=(V)'- l(K-1) *oC(K)IF( .frn. 0.) Go0 T_ 9000 ·It{K) = F(K) / cS(K) = I F(K) C(K) * S(K-1) ) / G

o CONITTNtIIE

LAST O!'FRS .. NOTE HN} = o0

= r ('i) -.(l) * H -:'-I)IF( r- .F. 0f.) GO TO 9n0-S{ql = ( F~t6) -CmiN * S(N-!} ) / r'

ncA.CK SluFSTITUTIONl

ro 'O K=1 ,IrdI' n n -T .k'

7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I'.tIT1 =-MI) t It (T) * AS {I+1) -o or4TTI.l IE

iER = 0RFTIURkN

)0*I[ ~ = 1RIETURNENMD

o::T I orJs TN .C NrFFEC *NArF( ',.",) NlooPTTMIZE LTNr..OUI!T ( 60 ) SIE (MAX) AIITODBL !ONF )

*nITItIJS IN EFFEC.,,S~tRCE FrCDIC rJOLIST M.P, ECK ORJECT NOMAP rJOFOPMlAT GOST-T tlIOYREF

*,OTIONS IN.EFFEC.* FUlCTInr'IS IrlLINE APE: NONE

A!

I

b Z

rr _-ON- . f ,If . :

I

??9 .

'arbI t : . , . . :.

LEVEL 2.2 , (SEPT 76)D OS/36 0 FORTRAN H FrYTENDEn PLusRE2UESTEn OPTIONS. 'NODEcKOr.O-ISTOPTTMIMrnn ,EMApSTZE (mAX) MNILFoyRFF ,MnTrRM'LCOPTION,]S rN EFFECT. NAr;F MAAT ,iN fionPTr'iTF TIrFCOUrIT(6n) ST7lr(MAXJ AITOJrLFnN :

SOURCE ErCDTC NrOLIST 'nnEirDEoK 0.RJC '10.rAP , FPO IT GOSTMT 'JXREF

'FU9CTI0ItS NIILTiE ARE: NONOF

/s NI SN

* Isr.JJsJIstrII Sf'J

t*0 :000 0 00 000!S,Ft4

T SrJisrjTSrl

· f. TSIJTSr.

I S rI ZSf

ISl'~

· rsrJTS!J

0002n0003

::oon4OO0400O50006000700080009001000110012001300141001500160017(4)18

00190020

. . *O=TrTr-NS INr. 4lOPTIONS Ir]

._ *O.PTIONS IN

·0T]OPJS INS :-*STATISTICS*

*TASr.TISTICS*

0 ***** END C

:SUWPOUTI.-,r STAT(,*TSrnL) ..DIrMEMSTOIN 7(;53,47),TSOL (53,47)

C CALCUILATE 'PTAS A!D rnnT UEANI SOUARE ERROR ALSO THE MEAN OF ZFACT - 51 *. '45ZflFAN = a.RTAS . 'RrSr' o.DO i0 -=2,952 .:$'; f- ·rs- 3:X0C f :;t;f~f ,; ;,S0 :n O 10 J=> 46 - ' ' :RZMEA = 7r'EA.! -+ 7.(Tj) / FACT7 P = Z(T'.J) - TSnL(T,J)D2 =D*PR11S = RA +. n/FACTr.".,SF = RSE +. .D/FACT

10 COr-!'"T ,Jl F: r 's = SORT(RMSF) ' -P PTRr.T 20,7.FAf',PIASRigSE L. ./ -TA Z

2n FOR.5Y.AT At4HSTt.TIRTTCS OF SOLUTZDN,/lIO ,fEAMN Z=ttFl0.3,3X2IO NPTAS=' E12135X , 2Pi;E= .El2.3)RE TURN. :

EFFECi*JANFM.:. 1fNI) NfOOTIMIZE LTYECOU"IT(t60) ST7EftAx) AqJTODRL(NOPJE)

EFFEC *SOl!RCE..E.r!CDTIC 'OLIST m0tODFCK OBJECT MO40MAP 1'OFOPMAT 0GOST'T MtxRFrF

EFFEC,* FUICTIONS I NLINE ARe: NON.

EFFEC .*.

'F

.,OURCE STATFrTEPTS

NO .UTAGNOSTICS CGENERATED

COMrILATION ******

19' PROr)GRAM SIZE

ff' ' ' '$ t.S 0 0 -e f .,: . , 0 0 f: f f f 0 0 .~~~~~~~~~~~~~~~~

75T, SUPOROGRAM N

70K PYTES 0

ii~~~ i. - · ... I .: I. ...

-1 i , r'; f

.0G_ -LEVEL 2.2 (SEPT 76) oS/360 roPTRA Hf FXTErinr PLUS.REVUESTEn OPTICNS. NODECK,IOLTST,(PT;T.T?r ( ,fNOMAPSTZE(MAi) tnILMtOXREF ,NnTFrr.,LC {-. OPTIONS IN EFFECT fWAT V,'{'lIATl' 'RJOOPT rf'j7F- I.TrlFCrrUlIT(6n). STZFIIAX .AuTO'. I F

S-OU CE-Epc rIc rNOLT T ,,nr)ECKI ORJECT 0"JIt!A NO FOF MAT-GO..TfT ' OXPTF

FUNCTIONS INLTE An'E. NONEI._

TSN OOG2-isrI 0003

I SN 0006*I~- 0007ISM 0007T Ysri 0009

- J~~~srJ oollo.ISr 0011.ISN 0012

-3 -. -Isra o013-TsIJ oo0014ISN 0015- Sr 0016ISrl 0017

-: L--S N 018

TSN o0019ISN 002n-;:_ -' TSP! 0021

.,C, ISiH 0022TSJ 0023

D 0 - : I S-J 0 024 ISN 0025IS. 0026

'stirRnOJTT.IE [TIFOlJTi(TqOL,7,i.S:CAM): ;DI~HrgSTON T.~OL(5,~ttl7),7(53,~.7),rlIF(;~?47)

QVI'-;.^S'TOM KTLI'.~3).,'TARq9)l ,COr{q4),KrH§M(4::)PATA r.iTAp/-1 ,53,47,4?7,47,1tO953/DATA COP'/O.,1.v60o.,O./DA A KT[L/uH ,.'T.MPL,' HLM*,IHLTZ.,' EQ ',lSOL .,27*#H DATA fe, lrl/v- ST', ;2 P.Ds,$$SCA,qv v!,/DATA K TSO: /'T :L/n p.,r z, 71* Z,'DATA KnTF/'"IF ' i- :KT.(9): I='MrlU¢T ISCAN)iKTL(1O) = KUtM(3)FKTL(11) = KNUM14)i -- : : DO 1 I=l1,3

' nO 1 J=1,47 DIF(IJ) = t(IJ):- TSOL(IjJ)

1 CONTINUECC PRINT OUT SOLUTTION- ERROR FIELn . :' -: C

KTL(13) := TSnL . : CALl GRDPPT (TSOLNTArnCO.!,KTL 1,1) FTL¢13)-= KZ

CALL GRnmPT7fFZ HTASC,',KTL,1, : -.; -:KTL[1) : ~DIF ' . -:CALl.. GP PPRT(DIF,AITAR,CON,KTL , ,1)RETURNErND

----OPTIONS IN" EFFEC '*NAFE(MAT N ) MOOPTTM¶IZE LTNFCOUNT(6n) 'STZE(MAX) A'lTOOBLfLtlMNF)'CO _'O'TIONS IN EFFECs*SnuRcE ERCTTC r:OLIST !.InDECK OEJECT NOM/P MOFnRMAT GOSTMT' NOXRFF..

O= t TIONS IN EFFEC.* FUCTI)NtS TNLINE ARF_: -,O'JEJ*O1)TIOP!S IN EFFEC,* -.-*STATISTICS* ,UOURCE STATEMENT. = 5 , PPOGRA,¶ SIZE = 20882e SUPDROGRAW N/

'*.STATISTICS. NO. : AGN.!OSTICS GENERATED*:***** ErND OF CO'-ILATTON****** ' ' " 6.K RYTES OF

, .! . d~~~~~I

. .

SST MAP CLIMAT

Fig. 11. October average sea surface temperatures 0 K: Northern He.misphere.

The 270K isotherm approximates the ice edge.

-------T--,--,---. .. . .. d .. , ^ f - V . C ... . , , , ......

SST HAP CLIMAT

Fig. 12. November average sea surface temperatures OK: Northern HemiL

The 270K isotherm approximates the ice edge.

SST AP CLINAT

Fig. 13. December average sea surface temperatures K:The 270K isotherm approximates the ice edge.

Northern Hemisphere.

SST MAP CLIMAT

Fig. 14. January average sea surface temperatures oK: SouthernThe 270K isotherm approximates the ice edge.

-emisphere.Hemisph~ere.;0

SST MAP CLtIMAT

Fig. 15. February average sea surface temperatures K: Southern HemisphereThe 270K isotherm approximates the ice edge. ·

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

-, /0,,,,

0 I 0

- - .... ................ MAR WMCtINMC WASHINGTO

SST MAP CLIMAT

Fig. 16. March average sea surface temperatures OK: Southern HemisphEThe 270K isotherm approximates the ice edge.

SST MAP CLIMAT

Fig. 17: April average sea surface temperatures K: Southern Hemisphere.The 270K isotherm approximates the ice edge.

/.:.. ..... .... I - -

SST MAP CLIHAT

.g. 18. May average sea surface temperatures 0K: Southern Hemisphere.The 270K isotherm approximates the ice edge.

Fi

1, o :� � �

SST MAP CLIHAT

Fig. 19. J-ne average sea surface temperatures OK: Southern Hemisphere.

The 270K isotherm approximates the ice edge.

· 0

0~~~~~~~~~~~~~~1

y~~~~~~~~~~~~~~~~~~~~ eThe. 2

2/

9

0. -. ~~~~~~~~~~~~~~~~~~~~~~~.

0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ·~

SST HlAP CLIi'IAT

* * Fig. 20. July average sea surface temperatures OK: Southern Hemisphere.The 270K isotherm approximates. the ice edge .__

SST MAP CLIMAT

0Fig. 21. August average sea surface temperatures K: Southern Hemisphere.

The 270K isotherm approximates the ice edge. · - - E-- - X L - -.- · . ..Df

g :: : \ f X :: :~~~~ .. \

~~.0 Ct~e: u f : : a~f: D 7 f ::: f 0:: 0 0 : \?:: :. ·.a.

1~~ w

SST MAP CLIMAT

~. 22. September average sea surface temperatures K- Southern Hemispheres.'

The 270K isotherm approximates the ice edge., aFig

- i'01-. � � �, �,.: 7 �; i

SST 'MAP rCL,!ATL ,I: ..

Fig. 23. October average sea surface temperatures oK: Southern Hemisphere.

·;: :The 270K isotherm approximates the ice edge.

n ,,, _ Q | .7 , R S , , , , A ~ ~ ~~~~ , ., X,:!! .: ·., .:::: :i !jJ.i : -! : .- S :XX .

SST HAP CLIMAT

Fig. 24. November average sea surface temperatures K: Southern Hemisphere.

The 270K isotherm approximates the ice edge. _:

, .:� �� : � I , i

� I : � .� �, � I �.�

� I �- ��,�L �; !.~ -:: I t ,. '~ Ti': ff lf- .1''0 0 .'7 ' S'.S L i '' a:, ['d! I In' t., .0 'E: I

;00

SST HAP CLIMAT

Fig. 25. December average sea surface temperatures OK: Southern

The 270K isotherm approximates the ice edge.

Hemisphere