AD-A237 123 o2

DPTETOF OCEANOGRAPHY (~COLDEE OF SCIENCESOLD DOMINION UNIVERSITY

EC NORFOLK, V -A 23529

JUN 19 1991 Technical Report No. 91-6

D URRMITCSIDT Oor DzFrv] IOII ONI OCIL&I DEIIMCI

By

Albert D. Kirwan Jr., Principal Investigator

Final ReportFor the period 1 October 1987 to 31 January 1989

Prepared forOfficde of Naval Research800 N Quincy Street -

S- Arlington, Virginia 22217-5000

CISTATMEMN A-A~ove tM pblic ii-lczM"

UnderONR Contract N00014-88-K-0203Dr. Thomas Kinder, Scientific Officer

Submitted by theOld Dominion University Research FoundationP.O. Box 6369Norfolk, Virginia 23508-0369

1991

91-0238691 6 17 0\l\U\WIi\If'U

OCEANOGRAPHY

OLD DOMINION UNIVERSITY NTiS CP.&iDT11C TAB [

D t , o otex,, ,.,y U.annou-cc- k 3 IOli D.-.r U=ir- u -o June 5, 1991

X.isrk. lirgii=a 23529-0276 JuS 1 June .5, 1991(504) 653.4285

Fx 1504) 653.5303 By .

Chief of Naval Research DistibxionJ.c/o Dr. Thomas KinderOffice of Naval Research Avaiability Cees

800 N Quincy StreeZ Dist aiardiorArlington, VA 22217-5000 Sp

Dear Sir/Madam:

This letter constitutes the revised final report for ONR contract N00014-88-K-0203awarded to Old Dominion University for the period October 1, 1987 though January31, 1989. The original final report was submitted to the Office of Naval Research aspart of a renewal proposal.

The bulk of the research activities conducted under this contract consisted of basicresearch on ocean flow dynamics as it pertains to the prediction of ocean motion.During the contract period, three papers describing aspects of the research werepublished in the adjudicated scientific literature. These papers are:

1. "Genesis of the Gulf of Mexico Ring as Determined from Kinematic Analysis,"

J. Geophys. Res., 92(Cll), 11727-11740, 1987.

2. "Observed and Simulated Kinematic Properties of Loop Current Rings," J. Geo-phys. Res., 93(C2), 1189-1198, 1988.

3. "Notes on the Cluster Method for Interpreting Relative Motions," J. Gco-phys. Res., 93(C8), C '7-9339, 1988.

In addition to these pap..rs, the results of the research were reported at the 1988 LiegeColloquium on Ocean Hydrodynamics, Mesoscale/Synoptic Coherence in GeophysicalTurbulence..

As a part of the research effort, the funds were used to support William Indest, aPh.D. graduate student in the Department of Oceanography.

I am grateful to the Office of Naval Research for continued support for this research.

Respectfully,

A. D. Kirwan, Jr.Samuel L. and Fay M. SloverChair of Oceanography

XWXXAL OF GNWOflYWAL RESEARaLE vOL. 9L NO. CIL. PAGES 11.1.74Ml ocrosa1EI 15. 1W

Genesis of~a Gulf of Mexico Ring as Determined From Kinemnatic Analyses

JAuWs K. Lsmw~

Sdrke Irf$a-i-Mis IN-mrnaineal C-wPrion awu.Cdr Sgtaima. Trsts

A- D. KtitwAta. JR.'

Umrera 4!wk Fkhia Depm'iawvo of#MW Science St. Perrhor

The kinematics; of the Loop Current are studied using trajectories or dr~ifesun the Goff of Mezicoduring utid-June through September 15. One of the driier was; in the Loop Current proper. *Wieother drifter.% ncrc in two reccatly shed I.ootp Current imp. Thu: drifter in the Loop Current shonedsitmif anitcyamic moIti during the study period, This Loop Current ainkcdoe JIMt began off thenorthwest coast of Cuba. It rapidly moved ... hadinto the Gulf of Mexico as a ring pichied off fromthe Loop Current. Analysis of the Loop C urret drifter motion showed that the anticydone: became -nintegral part of the Loop Current; taking on many of the characteristics of the most recnly shed ring.The rcwlt% of the analynk sugget a pbocm by shidi Loop Current rings can be generated. Apparently.ibis mechaikam can cause the Loop Current to become reconfigured in 2-3 months for beginning theprcs torringsparsltion.

I. I~tU)(I(h)NThe following analysis is obtained from the movement ofThte shedding of Loop Current rings has a major impact on drifters 3354 and 3378 as well as concurrent sea surface tern-

procescs itt the central and'western Gulf of Mexico. These lieratUre (SST) data and XDT data. These. along with thelatrge atnticyclones transport a tremendous amount of momen- presence and location of Ghost Eddy. suggest a new processturn, heat. and salt across the gulf. all the way to the Mexican by which an anticyclorlic vortex is formed that ca 'n eventuallycoast Whatn. 1982: Kirwvan el at.. 1984: Lewvis and Kirwan. become 4a Loop Current ring. The original mechanism for

199-5Sj. ttt order to' c ,nidcr-ttlanccs of niotnentumit. mav.x. atid spinning tip the vortex appears to be the lateral sharinghe:,t within thc Gulf or mexico. it is itmportant to have some stresses caused by the Loop Current off the northwest coast ofideca (if the characteristics of the kinematics of the L.oop Cur- Cub;a.rent. including how often a ring may be pinched off'. A detailed description of the kinematic characteristics of the

D~uring June 1985. an attempt was made to put atn Argos Loop Current and the ring is provided by an analysis of thedrifter into the Loop Cutrrent as a ring was pinching off. How - drifter paths. This analysis gives us the time histories of theever. the ring did not totatlly disconnecct from thc Loop Cur- rotation rate, eccentricity and orientation of the ellipses of therent. and the drifter ended tip spending approximately 3 trajectories swirl velocities. and movement of the centers ofnionth% iti thc Loop Current proper. The drifter exited the rotation. A comparison is made of the kinematics of the LoopGulf of Mexico ((JONI) through the Florida Straits in Scptem- Current as deterniined by drifter 3354 during the time it co-her 1985 (Figure 1. drifter 3354). Fortunatcly, a second drifter existed with a ring. The latter is designated by drifter 3378.(3378. Figure 1) was placed in the ring in July 1985. Together, Trhe comparison of the last part of drifter 3354 (before it exitedthese two Lagrangian data sets provide a uniquc view of the the GOM) with the first month of ring 3378 provides an in-motion of thie Loop Current whtile it is in the process of shed- teresting contrast between the motion characteristics of ading a ring. They provide special insight into Loop Current northern extension of the Loop Current before and after itkinematics as this system extends northward, as well as the pinches off' to become a ring.kinematics of a Loop Current ring. Finally. wc comment onthe existence of a third anticyclone in the GONI. The eddy 2 AHDTwas discovered scermdipitously as a third drifter became en- .PTtDTtrained in its flow field and circled around the eddy all the The path data used in this study are the position data of theway to the Mexica~n coast (G. Forristall. personal communi- drifters with Argos identification numbers 3354 and 3378. Thec.Ition. 19861. Ti. eddy has beeni referred to as Ghost Eddy 60OM ring that was seeded by drifter 3378 will be referred tobecause there had been no previous indication of its existence as ring 3378. References to drifter 3354 will indicate a refer-prior it) the first of August 1985. Its general motion and extent emice to the Lbop Current proper. Drifter 3378 was drogued byduritng August and September 1985 is shown in Figure 1. a weighted 200-M line, while drifter 3354 was drogued by a

window shade drogue at 100 mn.Drifter 3354 was seeded at 25.9"N. 87.9'W on June 18, 1985.

The drifter immediately moved southeastward -525 km and

Now~ at the De-partment of Oceanography. Old Dotninion Uni. reached 23.2'N. 83.rW by June 29, 1985. At this point theversty. orflk. irgiia.drifter became entrained in a westward, anticyclonic flow field

with a center of rotation at about 24*N, 85.5'W. After twoCopyright 1987 by the American Geophysical Union. rotations (into mid-August) the drifter suddenly moved north-

Ptper number X060A6. westward and made three additional rotations centered at0141t.0227/87!007C-0606S05.00 about 25,5'N. 86.5'W. The drifter then left this flow pattern in

11.727

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I 3378I ~ all

20:s ~.

4

-~ -SL61

Eddgary..50

FiLI Tietriso ditr339 r~ i-un houhmdSetmer13 ndo ritr378fo mdJl

OhI. beinin postio- orh rfeIrjcok n ranisdnoeteedpstos

Fi-cig.be 198 Taneitedo dther 334 M dJn through theidd-estmbe a9nd reand oithere 3378l atrlat mid-Sepeler

da Straits. During that same 30-day period, ring 3378 continued movingD~rifter 3378 was secdcd in ring 3378 at 26.4 -N. 89.3 -W on westward, reaching to the 92.5'W meridian.

July 19. 1995. Bly that tiie, the ring had totally separated The oil'shorc industry launched a drifter in June 1985 whichfrom the L.oop Current. As shown in Figure 1. ring 3378 cvcntu;,llv becalie entrained in the flow field or Ghost Eddy.slowly moied westward. reaching approxiniately the 91 W At thc beginning of August 1985, this ring was eentered atmeridian by mid-August. Recall that after id(-August. the 23.5 N, 93 W 0I igure If. The ring gradually moved southwest-Loop Current (as determined hy drifter j354) ined north- ward across the deepest portion of thc GOM. By the end of

Septetmbcr 1985. Ghost Edify was still strongly rotating it23 N. 94.5 W.

TAIXt 1. tLising uor(*n~ie,; lDnring Which Xlii lDaw; WereCollected in the Centr.,t and lEu~tern Gulfrof(exicui 3. Ti~mPERATURit DATA

Vessel Dl~ae OW)t~ A number of XBT data sets were collected in- the easternGOM during May-September 1985 (Table 1). These data pro-

E. M~. Qiueelln May 7 videL indications of the location and the structure of the LoopMN Ni'.iwr I M~ay 10 11 Current and ring 3378. In addition, weekly SST contourNI V S iena Ilikpania May 17 19E. M1. Qurrn 26 27 charts arc available for the entirc gulf. Some or these SST dataKIN Sie'na I1ikpasna May 27 28 are presented here. but unfortunately little in the way of ther-M IV Siena 1i~paunta May .1-3 mal structure can be picked out since the SST gradients areE. M~. Qui'env% June 7 8

L It Q~e~n'Jun 13- 14small in trie GOM during the study period (summer and earlyMV Su'nla Hispuania, June 26.-2ltftI)M/V Siena: H~ispania June 29-30 Cruise tracks tor the period May 26-31. 1985, are shown inE. M. Quernil July 1-2 Figure 2a:. The resulting XBT data are shown in FiguresRN' Snipasler July 9 22 '11; -2. These data were collected before ring 3378 was shed.MY' Nat Co 6 July 16-19 an h de rteIopCrrn mxmmhrzna enMIV S14,111 Ilivi'aiia July 16-17 adteegso h opCret(aiu oiotltmE. Ht. Qui-enr July 31 to Aug. Iperature gradients) indicated by the XBT data are shown inNI/V Stenil Hlispan~ia Aug. 16-18 Figure 2a. (Smaller slopes of the isotherms were taken as anMYV Amobassaihir Sept. 4 5 indication that the transects were crossing the L.oop CurrentMNV Ainhassisr Sept. 13 14 edge at an angle. and some of the edges were thus drawn at a.MIV Antmlisinht Sept. 24-25 45" angle to the cruise track.)-It appears that the Loop Cur-

LIMIS AlM KIRWAN: (ekw= OA GeMr 4~ MIlwu apl

29 X 06 atI- / C7 ee

26N P .... 6 -. 2

37 1, - 27 N

263 II26 b

25 X1 25 IS 2N,

24

233 N us 23 N

S' SO. Cru .s A S

22 N Sic ' WO rs 5,3Sto HOSP00fl4 Cru-Se 65-02 C 8. 22 U70

2114~ 2136 i 20J 3M6 4@@ W1 6M 76621 9@6W 9 111asW 87 6W0 85U 84 W 83 V 82 V DISTANIM (Kl,

52A2WSTATION

STIIN' " ea-: 79z a

200 ~~200 ______

~ 300 ~300

400 -

600 79 / 00

7000

en6e1020036 400 66606 70 06 Se DISTANCE6 41 561671 16DISTANCE MITNE )f

Fig. 2. (a) Cruise tracks fr XBT data collected during Miy 26-31. 1985. The arrovs denote the flow at the edges Ofthe Loop Current based on the vertical temperature structure SuioWn in (h) temperature data from the E. M. QuIen cruise,Mayv 26-27. 198.,. (c) tempgrature data rnm the St'na Ili,1pania cruise. May 27-28. 1985. ane (d) temperature data from theSiena lfispank, cruiqe. May X)-31. 1985.

rent was flowing northward on the east side of the Yucatan indicated by the trajectory (and to some degree by the SSTStraits, swirled westward and around the region where ring data), ring 3378 had obviously not separated from the Loop3378 would be spawned, and then turned northeastward Current during this period. The return flow from the area orbefore exiting the (JOM. the northward extension appears to be mostly southeastward,

As previously mentioned, drifter 3354 was deployed in the toward thc northern coast of Cuba.northward extension of the Loop Current during mid-June Three XBT data sets were collected during June 26 to Julyi985. Trhe trajectory of 3354 during Junfe 18-24. 1985. is plot- 2, 1985 (Figure 4). and the temperature profiles are shown inted on the corresponding GOM SST chart it) Figure 3. As Figures 0h4d. The outline of ring 3378 is quite distinct in

TABLE OF CONTENTS

1.0 IDENTIFICATION OF RCC (TAB: INTRODUCTION)

2.0 GENERAL INFORMATION

2.1 FACILITY LAYOUT DRAWING

2.2 EQUIPMENT

2.3 WORKFORCE

2.4 REPAIR WORK TECHNOLOGIES

2.5 WOR;XLOAD MIX AND VOLUME

2.6 MATERIAL HANDLING

2.7 STORAGE

2.8 PROCESS FLOW CHART

3.0 80120 ANALYSIS OF RCC

3.1 VALIDATION OF 80120 ANALYSIS

4.0 DATA COLLECTION

* 4.1 DATA COLLECTION PROCESS

5.0 INPUT DATA FORMAT

5.1 PROFILE DATA SHEETS

5.2 MODEL INPUT FILES

6.0 VALIDATION OF INPUT DATA

7.0 COMPUTER SIMULATION ANALYSIS OF RCC

8.0 VALIDATION OF SIMULATION ANALYSIS

9.0 IDENTIFICATION OF TAGUCHI FACTORS (TAB: BRAINSTORMING)

10.0 EXPERIMENTATION OF TAGUCHI FACTORS

11.0 DEVELOPMENT OF QUICK FIXES (TAB: POTENTIAL IMPROVEMENTS)

12.0 DEVELOPMENT OF FOCUS STUDIES (TAB: POTENTIAL IMPROVEMENTS)

13.0 ADDITIONAL SUPPORT DATA (TAB: SUPPORTING DATA)

io

IL~LILME AN12 KSinWA9: GLNMs #*'A Guuato.~~ I i Sis

SPINME

3"w

Fig 3. Ttjeclory of drifter 3354 .:td SST data foriune 18-24. 1935.

lFietro; 41, and 4c. while tlic Loop Currcnt can bc scen to of thc ring and the Loop Current from t XBT data inextend to 25 N in Figure 4d. These data would seeni to imply Figure 4to arc also shown. and we see some of the rotationalthat ring 337M is still conuccted to tlic Loop Currcnt.iThe SST charactcristic- withE.o the Loop Current from the path of dril-data fair June 29 to July 5. 1985. along with the associated tcr 3354. Thc rotation near the Cuban coast continued'intotrajectory of drifter 3354 are, shown in Figurc 5. The outlines rnid-July.

91 W soy mu $9W 111a ? sW asu V 5 84w V a 83V 230N 30'Ar~F - wF 3

-a ~ a

0o

281 28N

?7 N--~I 27 N

I ti\

25 N- 26 N28

23 N 24-: - 2N

S Sino NrtpOn. fi.Iq a .505 A 3

22 S~n Histonso Crutse 05-*06 9 22 N/ 7HE M Quteny Ccult 85-09 ciI "N

21' M -7~'~.~I 21 N011 W V 9 8811W N 8 V81 4l NW M61 W 864 V $ V 2 DISTANCE (Kinl

1-4g 4. Its) Crui'. tracks foir X11 I dati co'llcted during June 26 to July 2. 1985. The arrows dcnote the flow at thecdc of thc Louop Current hasedl (in the vertical tempeirature structurc shown in (b) temperature data from the Stena111.5panin cruise. June 26-28. 1985. (r-I temperature data from the St cna Hispania cruise, June 29-30, 1985, and (d)less xraturc datla froma the E. Al. Quny cruiw. July 1-2. 1985.

OfAr PS 15 4Af SOUCO CePco-*Mo ci67i

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- 3M~ cc 01L.0 f I' %_-----

6" - 6W

11 in 34 t 006 q 9 2 3Ms 400 SM- 6SM 766DISTANCE (KMI DISTA1CE (M)l

Firt 4. (continued)

During July 16 19. 1985. two XBT data-scls were collected cxtension or the Loop Current is readily defined (Figure 10).tFigure 6) which distinctly show that ring 3378 had separated The trajectories for this period- of time show ring 3378froin the Loop Currcnt. The SST contours for thc same period strongly rotating but drifter 3354 leaving the rotational lea-are shown in Figure 7. There is rn o surface signature of ring ture of the Loop Current along the eastern- side of. a 28*C3378. although its rotation is well delineated by the trajectory tongue of water. The following week's SST chart (Figure If)of drifter 3378. As for the Loop Current. Figure 7 shows that shows the 28'C tongue reaching up to 27.50N, while drifterthe rotational feature within thc current had reached thc 3354 moved toward the Florida Straits along the northernnorthern edge of th 'e currenti. about 25'N. coast of Cuba. Ring 3378 was still rotating with a center at

Both thc Loop Curreunt and ring 3378 continued to rotate about 25.5 N. 92'W.anticyclonically through August -1985. An XI3T datta sct col-lcted on August 16-18. 1985 (Figure 8) indicatte,. that the 4. KINFIATic ANALYSeS

Loop Current had exitndedf northward to 26.6 N. Looking at The trajectory data were used to calculate various kin.the SST map for August 12-19. 1985 (Figure 9). Wc see thit *canatic parameters of thc flow helid. The methodology outlinedring~ 1378 and tlic Loop Currcntl Acre rotating at about the by Kirwan et at. (1984. 1987] for geophysical fluids was used;.nie lastitude, 25.5 N. Bly 1id-Sepieicr. 11we sea surface had in these calculations. (For mote details, see Kirwan et al.cooled stiffciently in thec eastern 00M~ so that thc northern f 19871.) Although Kirwan et aI.-[19g4J used a 100-hour low-

90W BO'W

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1 L731 LrwIS Asi KrnLWAi: (;r%* 41F A Gtu'k oE Mrckno RING

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221 L 23 N W

22 Steno Htsoonio Ct..,. 85-07 a ".4 e:I r 2 N 7

21 IS 1 a,22 50 in 153 233 .259 i 353*I U601 6 1 8V 87W 6 5W 8 3 W 82 W 61SIMICE (KII

SIAIJOU STATION

C ta.ii

200 - 200

400 400

600 -600

700 700

0 59 I00 150 200 250 9 t00 203 333 430 563 500, 710, seeDISTANCE (MI) DISTANCE MXI)

rig. 6. (it Crui~e tracks for XBT data collected during July 16-19. 1985, The arrows denote the flow at the edges ofthe L~oop Cutrrent and ring based on the vertical temperature structurc shown in (h) and (r) both temperature data fromthe M;V Nar Co 6 cruise. July 16- 19. 1985. and (d) temperature data from the Sta fli.sptnia cruise. July 16-17. 1985.

piss filter in making their calculation; (or-a ring in thc west- beginning or the record and during Julian days 210-225. Itcrn (3M. it was found in this study that such a filter was not was during days 220-230 that drifter 3354 moved northwest-sufficietnt to remove higher-f-requency fluctuations recorded by ward. ztrr apparent rapid northward extension of the Loopdrifter 3354. As a result, we used a 164-hour (half power point) Current. The swirl velocity tissociated with the rotation of thelow-pass filter on the velocity data of the drifters. We then Loop Current is shown in Figure 13. These magnitudes alsocalculated the period of rotation, the ellipticity, the orienta- vary from I5 to 80 cm/s.tin, ant' the velocity about the transilion ring center (swirl The extent-or water involved with the northward movement%clocity) as seen by drifters 3354 and 3378. of the Loop Current is indicated by Figure 14, which shows

the time history of the distances from the drifter to the center1.4op Current Kiinmtics of rotation. Prior to the northwest movement, the maximum

The observed (filtered) Loop Current velocities as scen by r'adius of the rotation as seen by the drifter was 85 km. Afterdrifter 3354 are shown in Figure 12. The speed fluctuates in the northwest movement the radius increased up to 100 km.magnitude from IS to 80 cm/sq. and there are minimums at the The rotational frequency of the Loop Current is shown in

UJWIS 4%13 KamiA-4 G014N A CG'U- 4 MesKII RING. 110733

95303W g.(JA 2

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95*W 9" *s4Fi.7 Taetri rdifis354(oidlrearo )ad 38(ase agearw)adST aa(C)[rJly1a31945 S oreraro~ drol te lo a te'ed e o t e oo C rrntwhledole lne idiae heloaton o L oCurn waer (bsdo30Tdt)

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30. V

24q N. S 24' N 0

27 N 22 2N 700 3 V 23 2 S32 W , 2 ; 6

200 '-00

21 N 212N 70 10203249SG6870O 0

%m 69W 88 W 87 W 86 W 85 W 64 W 83W 42 W DISTANCE (KM)

Fig. 8. tal Criw track for XBT data collected during Auguit 16-18. 1985. The arrow denotes the flow at the edge ofthe Loop Current ha~cd oin the vertical iemperaitire structure %hown in (h) temperature data from II.. Siena H~ispaiacruise. Auguit 16-18. 1985.

ILT74 is~wis Asti) KEiowAs: (r%4srs sw A Coutir ir MiriKo 01W.

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251N 22S

F~ig. 9. Trajctories gi drifters 33S4 (sold large irrow) and 3379 (dashed large arrow) and SST data (TC) tor August12 19. I9H5. Slaier arrow denoles the hlow at the-cctge or the, Loop Current. whiledotted lines indiciiie the location orLoop Current waters 'ha%ett 6n XIIT data).

As for the shape of the field of rotation, we tise eccentricity Aanc',,eulics (!f Ring 3378c. defined as the major axis length divided by the minor axislength. Also. we eaktilatte the northi'cast orientation -of tile The filtered velocitics of ring 33178 are shown in Figure 17.major axis. and both of these variahlcs arc Miowvn in Figure The magnitudes of these oscillations are relatively large, up'to16. The lOo%. field starts out rather elliptical (e -l1751 hut - 85 cm/s with a minimum of about'5O cm/s. The swirl speedsbecomes more circular by day 250 (e = IA4. The elliptical. are shown in Figure 18. and we see here an i 'nitial increaseorientation was mosly cast %%cst at first but eventually from 20 W~ 75 cm)/s followed by a decrease to about 50 to 60bcamne more iiorthwcst ':otithcast. cm/i. Thesr,,ariations in swirl magnitude coincide with vani-

95OW HOW85W

30N0

Fi. 0.Trijctri~ f ratr~.154t~li hrg aro)an 378(dshd are rrw)ad STdaa CC fr epeme

10071985

LEWas AND~ KiaMAN: G#E.N6is'4,AGuLu do. Macxiio RiwG l1;735-

05*W, WN aNOW

Fi. 3I Tae-oie o3di0r944(oi lrearw n 38(ase ag ro)adST aa( o etme

~~ations~~~ in tA sieoVh icco oain(iue ) iue1 sdrn h hp,daaiicetatt.inwaquatiie te istncs ro tc cntr f ottin, it a intillyrahe elitial Fiur 2).Witin tw rtaio

maimmraiu f 5 m prid tisccenrctyderasd oewat it,3h0rn

r Fion 11reTue cyoris 337 drftrsdualsly d crearwit thang ec(amhe lre arcul ar.STdt ' o etme

peiod. oftesz f h iceo rotation increasin from Fibout 89 dayser h tope theu 9aaidct ht'tern a

maximum radiuswoic9 pkrt ofrd hentrulfy Streae afistfomwsat diretyhro thenThottoop Cre ren ofrig334ihoniigr0 bcatang Staitsto tcirlrid Traiuts Wihin hiur flw21ed

75c ci~ii ocp Loop Current 3354) sison i

75 Loo Curn 2554

os

-50 -0.

4 -25 -75

Ju~in Das (985)Julin Dys (985

Fi, 2 Tnchsoro.sf erlerd bwvd~x.cct~rxniis Fi 3.'im isoie o hesir pedcopnet aot h

Fig. 12. the hi-o~r' rx cliirdoere de cm-et %s 23xe.a tme hiatrier of~v the wir /sed sopent aot.h

11.7*1 L.-WiA-4i, KjsiR A 4:0-?ai525 OA GoaLror Mlixic Ris~r

n o Loop Curr~ent (3354) 3

.' 60

20 -

E- %20 1* N *

-20 -X1

- 40Loop Current (3354)

-80

1 0

Julian Days (1985) Julian Days (1985)

FiL. 14. Time histories of the east/we:st (light curve) and northl/ Fig. 16. Tin'e histories of the Loop Current eccentricitysouth (darker curve) distances of drifter 3.'54-from the center of rota- (asterisks) and orientation of major axis (dashed lines). For the flowlioin in the Loup Current. .field orientation, north is in the positive e direction, and east is in the

positive time direction.

inSthbilitiCc xist which result in meandering of thc Loop Cur-rent (Ilnirlhuirt asid Thompson, 1980]. As the sizes of thc mean- ately after ring 3378 was pinched WT. Also, the trajectory ofdcrs incrcisc and reach northward, it is gcnicrally believed that drifter 3354 reveals (hat this rotational feature initially existedthie flow field wrap,. back onto itsclf and "shorts" across thc off the Coast (if Cuba. It has been shown in numerical studiesstream of' flow: part of thc How would still go northward that the flaw of the Loop Current can itself 'create an anti.around thc Loop Currcnt extension, while the remainder of cyclonic flow field off' the northwestern coast of Cubathe flow would take the more direct, southerly route to the (Tinstipsoe, 1986]. The northward flow west of' the tip ofFlorida Straits. This is analogous to other geophysical phe, Cuba along with the southeasterly flow along the north cen-nornena. such as the creattion of ox-bow lakes by meandering tral coast of Cuba will obviously produce negative vorticity offrivcr-;. Finally. a,; the curvature of the flow readies its maxi., Cuha's northwestern shore. Such an anticyclonic flow hasltiu. nic of the Loop C 'irent flow takes thie southerly [welli documented by a number of hydrographic surveys (e.g.,route, and a GOM ring is eventually pinched ofl'. Nowlin antd AtcI~ellan. 1967; Cochrane. 1972]. Using hydro.

The interesting point indicated by the data presented here is graphic data. filfiiiann and Worley (1986] estimated the depththe closed anticyclonic flow within the Loop Current inimedi-

750. 10

Rinq 3378

so

S 25

41 0

CLoop Current (3354) >C 41

.0

-50

Julian Days (1985)

Julian Days (1985) Fig. 17. Time histories of the filtered observed speed components

Fig 15. Time history of the rotational frequency within the Loop for ring 3378. The light curve is the east/west speed. and the darkerCorrellt. curve is the north/south speed.

Iu~,s . *S l3it+.'+ (;iL.. i A G;t., ei h.gmip Raqic II.r

Is

-05 .0 C'

Tf:tianm 1-19;MIl~u julian UJI.N'5 0.935)tI e- I. TimeP hi-cl fic ,f the -m idl ce com. - ms 2Km t the rile.. 2%j Tw historyo ic m moa A freq enyoftn$33TA .

C01.'ro Htiton fiirfint '%X I be lct4v gck h ia: s israfvcd.

,illd IhciftiL rCc tciglhcntrlh -.lllh : tl. An example of a modeled ri'ng pinch-off in the GOM is

thown in igures 22 and 23 [from Wirraff. 19196]. On dayo~f thi' ; C ubhan eddy to+ be - S<Ol it ui;th a cu rrent ma rnitude of" 63 Fi:'eturc 229. the mo e $.tlo -, a onfiguration similar to that

10-2.5 cm . in Sood agnre emnt with the data from th' studyv ,hn drifer" 33.-1 -As sededl. The ring is well defined as theII-ictre 12. Also,. data show that thec tempratrc ant.% :iinit - northern e"tension of the Loop Current. but it has not com-

chrctrst t of £OIrnsadteCbned r rc

charderstic ofGOMrin~ ani te Cuan 'dd arepf~ti-plecly pinched off. Note that there does exist a hint of" anti-+ally identical [rtcrl11m. 190)}: Ellio. 1921. cicon'ic motion off the northwestern coast of" Cuba_ By day

II Ii Cua Ctvip~r-t etefo edi hc rf

:I -,! 0 months later) the model shows the rng free of the LOOPtCurrent. Moreor'. the antcycione off Cuba is largr ard

,.'ttc thi hlgI~ov lcl rewedslghty nrthes whle ng much better defined. Hlowever. the model shows the northern3 78c. was still connected to the Loop 'urent. But it was only.e oput etnal to only 26N .

%borthr atr ~ rin 337 heheff that th eatyon she cdz f. opCurn xtnigtoolJ6N.hra

;u, uhafler rinc it pihed oht h ntyoXBT and dfter data presented here show an extension to

mr~r+ m~rnhwneinrliures al2tandt23..from i~Ikuref 10 9nd]I in day

f i cn edy to b out 26.5 N FigC t nt t63ure 22) the Ide I a 2nfigurt7 N in aisimilar time period 13 monh.t;lc 1th that fer this final ti flate Att 195. the rotational I i not until after 9 monohs before the Loop Current in

ficeld wencl :IU ieral comp1on:ent of the Loop Current. this sinulation pushes northward again in the process of

'lhti; the data in thkg ctr tmply that the kernels of Leop

Ctrrett rins maty come fro m waters off the northwe coast

ofl 01. 1.+; 3,

t00)

- Ring 3373

+"+ 60 2 ' - 1 °

-2+ 220 0i Rng 3378

mr mhr-4u1teiy toaot2. .=nrs1 n fad-27Ni iia iepro 3mnh

"'eta ,fc hshi+l-s (aeAgst15) h roaioa f o utl ae9monatnhefetheaLoop curent i

f n t Julian D fys th C eis) Julian Days (195)

ig. 21t Time hintic. (if h ccccnrii (aerisrke) and major

i ig. 11) Tinie hiotrie,; (ifl the e~it %.c,;I ighl ctilr~ci ind north, anxvc oricniation Idl: hed line.) for ring 3.378. For the flow field orien-.,inh d'rker crve) di,,incc of drisicr.1.17R from the L. ncr or rola- f;lion. north is in he poorith, e direction. 2nd ca.s is in he positivelion (if" ring 337X, line dircclioR,

LAYER 1 CURRENTS DATE a 063/1969* g ~..s G OF Mr(1C0 21322Z 130

31Ka a a a a a a a A

uN W MAXPL~fED PEED* 0.8 (MSECFi. 2.UperI-ier'~a~r cocta~ ro awodl r heGufofMccofo dy 6 (op ad 5 (otom Uo

II iIg a~. 9~.] 1li hp ayr ~ (a a dep

LAYER I- CURRENCTS Dl 4/u*sel c or-K -01221

tIIfsawMA PL~rD SEE - 04(N/IC

L&YE I CRRETS DlI 333196

00FMXC 23223.

,~~ ~~~~ .00 q

911T MAR PLOTTED SPEED 3.30 (U/SIC)

lig 23 Irr I~ic nCURRET Deaiisfo oe f'eCul fM~c o s23(o)Ad 333/ 1om 69roas~hvh G961 TOF MOM 20132-2 1s3.0md

11.7-M'. Ljwn-.%1 Mit IWAN: (iSMSISs to A GULl oar Matin Ra!'G

incinga oir ;antlaser ring tlFigure 2.11. fly this time. the asiti- The implicaiion ol such a. mechanismr is tihatnf timie scales.cionc: ilhat'had been %pun up by thc Loop Cuifrrt-has lost One: is not required to w~ait, until instabilities- in thie flow field,

itsidetit i;s' '?. Fa.ar 33.Thgs he rocssas ete- row. la icenu to, produce a closed rotiational'feataire. The

=1r~~ ;. !-t' :'Vr of tifr'r 3354 is diflrcent fronm this data hzere show ta ne Lon,- ciarat oaioa tr*.: .:f-a ;a. hi c itiajr ways. irst. the Loop Cur- can be well established even befbirc 1W previous-ring has to-

... :,,,! !,;Iuw~: to 27 N in 2.5 msonths tally inche o tws -cstab;iscd thl trit 3371- wait de;- ~ ~ ~ ~ ~ ~ 1 .. ., .. zhv m I) sz'ths for m.t-chmo'!cl titched front the. Loop Curecu: by wtid-Jl .415. Yet it was

~ tis~apl t~i %i as .tcctirnipatlii by seen that the new. rotatiotnal structure had-r.lived to latitude* g!circugj::zla, w'.tlex arisiilly generated off the coast 163IN by the- beginning of September 1985. thus the Loop

11 4 Ie.wvcr. wa othacr simrulations will, no dcp i-flow Current was reconligured fosr another ringseparatibua only 1.5thr~ough ilic Yucatan Straits. significant Loop Current pene- months-atr the prVious ring-separation. tlliol (19821] docu-trations can occur iit 40 days. Clearly. this is a phenonfcna menited three ring separations in a 12-month period usingChat rcqfuircs further study. -hydrographic data. The presence or Ghost Eddy along with

ring 33781 implies a similar separation rate. Model studiesI~op Crns:Ve4rsuas Ristel Ki,,e',,seics Ce.g., Hoirlbtr and Thompson, 1980; Thompson. 1986: Wall.

D~rifters 3354 and 33781 were both in thle GOM during craft. 19116] report separation rates ranging from 4 to I8Julians days 199 264. 19815. flowever, the initial variations of months, depending upon the type or instability involved.the kinematic parameters (if ring.3378 (Figures 17-21) all indi-

cateI ati;dil-tnieni periflil frontas19 2 during which .4r&nowleduinant., This work was supported by the Mineral Man.tun th drftar wth ts t8)n wighcd od oun anappo- attemetit Sersice. Gulf of Mexico Physical Oceanography Study.

priatc orbit within the ring. Thtus we comtpare the ringi and IIltsac through atcontredwt cec plctosItr act COportion and by the Office or NavalReahtruhcnratNNt4

Loop1 C~urrent kinematics front day 225 to day 264. At the 83-K-6256 with the University of South 171drida. We thank MI. Giuf-;1e-gnnting-or tIN,- tinie~periaid the Looo Cuarrent was rotating frida for her careful preparation of the figures. we also wish,-to es-

dii -I p~eriodI of a!hoaa 1(.5-days. %hile ring 3378 had A period pccially thank P. Itcinersmnar and W. Indes t for their many hours in% C2 iav:. T oricuattttrs of the elips of rotattion (Fig- performing the kinematk-calvulntlions. A. D. Kirwan, wishes to ac-

knoisledgte the support of the Shover Oceaanography EndowAnent to::ras ;ti ;wd 21) wee(hii'~~ia gne~I~ at.iet.bathe Old Dominion Uniiersity.

SCturrv~nt was sihitly mare elliptical Wc , 1.51 thanl r Ritll:c

.25)I.sed nth'rdifrvlattir t rier Co c. J, I).. Separation of :an anficyclone and suh~equent devel.:a I'~': a ;aat kylszts wee qute siilarfor ays 25- 24: 1(3 t 1 i~ts ill tIi. Lotop Cacti ent (I %Ot.-in Contariie~tiecnit ost tile I4.h11.

ail iycoli-, wreqiiie ;milr ordas 25-(4 8 Oeueires t the 6lull eql Afeskio. edited by I., H. A. Caapuriocm ior t.i le swirl n d u , 9t; kni for the radii relative tli ;andt J. L.. Reid. lip. 91-1t06. Guif Pulisihing. lloon. Tex.. 1972.

I c.*. cvant-.r. !taaere.stingly. bo'th matii:% lotws; v~ileiicced I'llti. 11, A.. Aotic.')conic rinso in tile Gauif or Mexico. j, ocvy~ Oean-

E ftzn. .,.and S. J. Worley. An inviption of the circulation'~~ed.III:I.i of te ur(ir ca -ei .1. Gaoyes, Rc's., 9110I21, 14,221-14 '6

10861Is. buwma.t S 'qteu1' 0111 lita Rit 1XI(ilta'llf litariburt. It, E.. andi J. D. Tlmonien, A~ numaeric~al study of Loop

%%' aIw% tuirn to tile kitnematics (if thle Loop Cuirrent at1 its C*urre.nt itaru~ions and edd) slialdiiai.. .1. Phys. Oeeetioqr.. 10,

:tc'a Isnatlwh'! extenlsion liulin days 2,30-264) and the kin- 1611 -, I o . I19g0.I Kirwan. N. 1). Jr.. WV. J. Meracl, Jr,. J. K. Lvvdi, . i. Whitakcr. and

of raaii3374 ~ron ttte it rok fre frot te l~ora R. Lcgeckis, A moidel for the antalysit of drifter data with an appli.

(.uarcoa (Julian days 225-235). Keeping an mind that these cation to a warns core ria ift the Gulf of Mexico, J. G;eoplus. Res.,datia represcent two different anticyclonic phetnomena. how ,A(334-43,19114.

Iyl'ictl arc thc GOM rings before and after they break off Kirwan. A. .1.. Jr.. J. K. Lewis. A. W. Indest. P. Reinersmat.. and 1,from he CurentQintero. Observed and simulated kinernatic properties of Loop

fotteLoop CurnCurrent rings. J. Getephys. Rei., in pire.. 1987.Thc similarities between the two sets of kinematics arc con- Lewis. J. K.. and A. D. Kirwan. Sons obiervations of ring topogra-

%idcr~ahle. First, die magnitudes of the swirl speeds and of the phy and,- ring-ring interactions in the Gulf of Mexico. J. Geopiivs,

radii arc practically identical. Moreover, their elliptical orien- Res.. *4C5). 9017-9028. 1985.taion, ac bthnorthwest,'southecast. The Loop Current was McLellan. ". J.. The waters of the Gulf of Mexico as observed in 1951

aatiazsarc othand 1959. Tech. Rep. 60-14T, 17 pp.,-tXp. of Occanogr.. Cotkgc:shi-ihtiy more cllipial than ring 33781. but one might expect Station. Tex.. 1960,such r 'ih~ec inelptct ccinig that an anticyclone still Nowlin. W.-D.. Jr.. anad H. J. McLellan. A characterization of tlae Gulf

,attaavlwd it, (lhe Loor -Current is a "etaptuted" phcnonmena. of Mc water% in witr .Ala,.re,. 24-39.-1967.

Mihilc a. dcutached rinig is an isolated vortex. The similarities of Thotin. J. D.. Altimeter data and-geotd error in naesoscale oceantile 111ticyclontes pilu-. thecir close periods of rotation imply that predic.'ign: Somie results front a primitive eqiuation model. J. Ga'v-

ph a's. Res.. 9ltCI. 2401 2417, 1986.the basic kinemtatic characteristics of GOM rings can be cs- Wallkraft, A. J.. Gulf of Mexico circulation modeling study, annual

tablislied as thie Loop Current pushes northwestward off the progrcss report: Year 2, report to Miner. Manage. Serv.. 94 pp..

sltorc of Cuiba. JAYCOR. Vienna. Va.. 1936.

SwamugrvA. D. Kirwan, Jr.. Department of Oceanography. Old Dominion

TheI scenario imiplied here is one it .ihIa eece of the Ulniversity, Norfolk, VA 23506.

rinig is estuablislied off (lie northwest coast of Cuba. develops in J. K. Lewis. Science Applications Intternational Corp., 1304

decpeir waters off the coastline, and then take% on its final Dao.clttSain X7M

characteristics shortly after the previous ring comapletely lie. tReveived February 26. 1987.taches from the 1.orp Current. accepted May I5. 19117.)

JOURNAL OF GEOPHYSICAL-RESEARCH. VOL. 93, NO. C, PAGES- 1189- 1198. FEBRUARY 15. 1988

Observed and Simulated Kinematic Properties of Loop Current Rings

A. D. KIRWAN, JR.,1_2 J. K. LEwis, A. W. INDEST, 4 -P. REINERSMAN, 1 AND I. QUINTEO'

Two rings, shed by the Loop Current in 1980 and 1982- wereoobserved for several months bysatellite-tracked drifters to migrate across the Gulf of Mexico. The drifter path-data-have beeui tnvertedto obtain estimates of the paths of the centers of the two rings, ring shape, and the swirl velocities. Threedrifters were deployed in the, 1980 ring, and the analysis of that data set establishes the-variability of theabove kinematic estimates for one ring. A comparison of the analysis of data from both rings providessome-idea on inter-ring variability. Both rings impacted the Mexican continenial slope at about-228'N,95.5^W. After a brief adjustment period, both rings reestablished and maintained a vortex character, forseveral months in the slope region while migrating slowly to the north. The paths of the ceniers of thetwo rings along the slope are virtually identical. The same analysis routine was applied to some simulat-ed drifter data obtained from the Hurlburt and Thompson (1980) Gulf of Mexico primitive equationmodel. In the- midgulf, the agreement between the observed rings and the simulated ring is good,although the former showed stronger interaction with the continental slope topography and/or circu-lation than was-seen in the latter. Along thi slope, the model ring kinematic characteristics were inextraordinary agreement with the observations.

I. INTRODUCTION merical simulations from the general circulation model (GCM)the subject of Hurlburt and Thompson [1980], hereinafter referred to asThe circulation of the Gulf of Mexico has been tesbet HT.

of a surprising amount of' theoretical and observational stud- HT

ies, From the turn of the century [Sweitzer, 18981 until 1973 Is the qualitative agreement between GCM calculations and,one particular ring fortuitous or the result of deterministic

[Austin, 1955;Nowlin and McClellan, 1967; Nowlin, 1972), the dyiamics? We examine this question with observations, fromemphasis was on analyses of hydrographic data. These studies two different rings along with simulated data from the HTestablished the presence of' poorly defined, but quasi- model. Specifically, we compare in detail kinematic propertiespermanent, anticyclonic signatures in the central part of the such as the movement of-ring centers, ring translation andgulf. swirl velocities, and- ring shapes from 1980-1981, 1982-1983,

Since 1973, the emphasis has shifted from qualitative de- and simulations from the HT GCM.scriptions of the hydrography to attempts at quantifying dy- The analysis routine used- to determine these kinematicnamical mechanisms associated with. the anticycl6nic struc- properties is an improved version.of that reported by Kirwantures. Sturges and Blaha [1976] and Blaha and Sturges [1978] etal. [1984b]. The routine inverts Lii'ingian path data, toproposed that the western gulf circulation was the result of the obtain the desired kinematic properties. Unlike the earlierbalance of wind stress curl and planetary vorticity, i.e., a mini- study, the present routine emphasizes ring shape and trans-western boundary current effect in the Gulf of Mexico. Elliot lation. Also, considerably better time resolution on parameter[1979, 1982) disputed this, noting that this balance was not estimates can be made with the new routine. Some details aregenerally achieved if synoptic wind data on a fine scale were given below and in the appendix. The internal consistency ofused. His work also gave more credence to the earlier idea of the analysis routine is being assessed separately with simulat-lchiye [1962) that the anticyclonic features were in fact rings ed data from the HT GCM.that had been shed by the Loop Current in the eastern gulf. The following observed data were utilized in this study.Kirwan et al. [1984a] substantiated this hypothesis by track- These come from two rings which-occurred in 1980-1981 anding the migration of a ring shed by the Loop Current across 1982-1983. We have reported previously on the former (seethe gulf to the continental shelf off of Mexico. Kirwan et al., 1984a, b] using- the more primitive analysis

In a companion paper, Kirwan et al. [1984b] assessed the routine at r ing ter rimtne andtranslation velocities, local vorticity, deformation rates, and routine. That ring had three driters in it simultaneously, andso our re-analysis provides some indication of intraring varia-

shape of the ring as it propagated across the gulf. The analyses bility along with some indication of the consistency of theof the three drifters that were in the ring simultaneously analysis routine. Only a descriptive analysis of the latter ring,showed generally good agreement of these properties. The re- which had only one drifter, has been reported before (Lewissuits also are in general qualitative agreement with the nu- and Kirwan, 1985). Comparison of thes two rings provides

some measure of intei-ring variability.These observed data are compared with simulated La-

'Department of Manne Sciences, University of South Florida, grangian data from the HT GCM. This model was a two-layer

Saint Petersburg. (lower layer active) nonlinear primitive equation model of the'Now at Department of Oceanography, Old Dominion University, Gulf of Mexico with realistic bottom topography. The grid

Norfolk. Virginia. spacing for the calculations used here was 10 km. The simulat-'Science Applications International Corporation, College Station,, ed path was obtained by advecting a parcel at each model

Texas.'Department of Oceanography, Old Dominion University, Nor- time step (4 hours) to a new position given by the product of

folk, Virginia. the velocity vector at the current position of the parcel and thetime step. Since the position of the parcel was rarely at a grid

Copyright 1988 by the American Geophysical Union. point, it was usually necessary to interpolate the velocity

Paper number 7C051 1. vector from surrounding grid points. These calculations were0148-0227/88/007C-051 1505.00 kindly performed by A. Wallcraft.

1189

1190 KIRWA. ET AL: KjnmATc Pxop~nEs of.LOOp CVanENT RINGS

Both the observations and the simulations track appropri- Flier! (1981] has pointed out that this center may not coincideate rings for many months. During this time, the real and with an "Eulerian" center. (We interpret this as a center ofgimulated rings move from the midgulf region with fairly flat mass.) Smith and Reid [1982] have developed a general for-topography in excess of 3000 m to the continental slope off of malism for studying- the centers of the distribution on anyMexico. This area of interest is depicted in Figure 1. In the property, for example, mass, potential energy, kinetic energymidgulf region, ring translations are governed essentially by or enstrophy. In general, these centers do not coincide, nor doRossby wave dynamics, and so one expects them to move their propagation velocities. Establishing the relations be-west-southwest. But along the slope, interactions of the ring tween these various centers for Gulf of Mexico rings requireswith bottom topography become important. Since the dynam- much more data than is now available.ics in these two geographic regions are so different, we have For the present model the total velocity (u, v) of a drifter isperformed separate analyses for the rings in the two regions. composed of the translation velocity (UT, VT) of the ring

center and the swirl velocity (us, vs):

30 U = U T + uS(I)

v = VT + VS

us = (d + a)x/2 + (b - c)y/2(2)

v. = (b + c)x/2 + (d - a)y/2

5 0Here x and y are the instantaneous coordinates, relative to the250 Z - ring center of a particular parcel (drifter). If (2) is regarded as a

- , '. x - Taylor expansion for the velocity field, then the parameters a- and b can be interpreted as deformation rates and c and d as

-", the vertical vorticity and horizontal divergence. It is stressed" . that these parameters are evaluated along individual paths.

\Eulerian estimates of deformation rates, vorticity, and diver-

200 ' 2 gence likely would differ from these estimates. Consequently, itis better to interpret a, b, and c as shape functions whichdetermine the orientation and ellipticity of a particular orbit

980 950 900 850 800 in a ring. See Kirwan et al. [1984b] for details.The general solution to (1) and (2) is easily obtained

Fig. 1. Depiction of the area in which the observed and simulated [Okubo, 1970]. The procedure here is to apply this solution toLoop Current rings are studied. every time interval between fixes (in practice, interpolated

positions). Thus for the interval tk ;g t 6 tk + , one obtains

uk = Un + (d + ak)x. 2 + (bk - cjy12 (3)

2. ANALYSIS PROCEDIRE vk = Vrk + (bk + ck)xd 2 + (d - a)yk/2 (4)

The observations used in this study are the positions of thePolar Research Laboratory drifters with Argos identifications where

1598, 1599, 1600, and 3374. Kirwan et al. [1984a, b] reported xA={exp Erl(t-tk)](X(lak+bk,_c 2+a)+ y(bk-ck)Jon the analysis of 1598, 1599, and 1600, which were deployed

simultaneously in November 1980. Position data from these -exp [r2 (t-tk)][Xk(ak- ,/ak2 +b, 2 -c 2 )+ Y(b-c,))}

three drifters are analyzed to establish the variability within a 2x/ b c (5)ring. Intraring variability is assessed by comparing the analy-

sis performed on the three drifters in the 1980-1981 ring with Yk = {exp [r,(t-t)][X(bk+ck)+ Y(V/a 2+bka-c,2-a_)]

that of drifter 3374, which was deployed in the 1982-1983ring. Each drifter was the standard Polar Research Labora- + exp Erk(t - Xk(bk + C) + Y(/aT+bzC,2 + a,))

tory buoy drogued by a 200-m thermistor line. No thermistor - 2,/a. + b,2 - c.2 (6)data are available from any of these units.

The analysis procedure utilizes a parametric kinematic In (5) and (6), (Xk, Y) are the coordinates of the parcel in

model discussed by Kirwan et al. [1984b] for inferring the question at time t = tk relative to the ring center. Also,

translation and swirl velocities as well as the vorticity, hori-zontal divergence, and deformation rates. This model was first r1t = (d + /a,' + b 2 - ck2)/ 2 (7)

proposed by Okubo [1970] to describe small-scale motion r21 =(d - /ai, + b " - c,')/2 (8)about flow singularities. Kirwan "1984) showed that samemodel can be applied to larger scales if the model parameters are the eigenvalues of the matrixare constant along a parcel path.

The model assumes that the swirl velocity is given as a 2M = [(d + ai) (bk - cJlinear function of the distance from a local flow center. Here M (b4 + c,) (4 - a)]the local parcel flow center will be referred to as the "ringcenter" or some times "orbit-determined center". This termin- Note that these eigenvalues are complex conjugates whenology is dictated only by the nature of the available data. c,2 > ak2 + b,2. This, naturally, produces real periodic solu-

KtIwAN r & .: KmEsiic PRoP'riis oF Loop Cl 'Tr Rmcs 1191

tions which are a characteristic of swirl velocities of drifters R, - LC-) ( 1)trapped in a ring.

To apply this model to drifter data, differentiate (3) and (4) wherethree times with respect to time. One side of these equations h 2 = a., + bt2

contains the unknown variable set Z, = (a, bA, c,, d, X, Y).The other side of these equations can be estimated from path The paradigm just outlined for analysis of Lagrangian datadata. These six equations do not contain the translation veloc- puts critical emphasis on time derivatives of path data. Incon-ity, since it is assumed a priori to be constant for a time sequential errors in the path data can become consequential ininterval. (But, of course, its value may change from one time the derivative estimates. Moreover, the nonlinear algebraicinterval to the next.) These six equations are inverted analyti- combinations of these derivatives in the paradigm may mag-cally for the six elements of Z,. The translation velocities can nify further the impact of these errors on the inversion. Thebe recovered from (3) and (4), since all other quantities are concern is that two paths which contain the same kinematicnow known. Alternativaly, the translation velocities can be information but differ by small random and/or round offobtained by differentiating the path of the ring center. Both errors, will produce vastly different kinematic estimates whenprocedures provide comparable estimates. Here we use the run through the paradigm. The internal consistency of esti-latter procedure. With all elements of Z, now determined, one mates of ring kinematics is presently being examined withthen moves to the next time interval and repeats the calcula- simulated data from the HT GCM.tions, thus obtaining a time series of Z,. Details are given inthe appendix. 3. MIDGULF

We have expended a great deal of effort in learning how tomake reliable time derivative estimates of the path data. For 3.1. Observationsthe data analyzed here, the raw position files have been edited, Here the results of the intraring and inter-ring comparisonsinterpolated to equally spaced time intervals, and low-pass are presented for the midgulf region. This includes ring kin-filtered (half power point of 100 hours). All derivative esti- ematics as inferred from the three drifters in the 1980-1981mates were obtained by fourth-order accurate, centered differ- ring and the single drifter in the 1982-1983 ring. While in theence schemes. Each derivative file was edited and low-pass midgulf region, both rings began to interact with the conti-filtered. The calculations. as described in the appendix for Z, nental slope topography and/or circulation. This period is iso-were performed with the latter derivative files. The calculated lated in the analysis. The same analysis routine is then appliedZ, in turn, were edited and low-pass filtered. to simulated midgulf ring data from the HT GCM.

The analysis procedure just outlined is an improvement The 1980-1981 ring. The data for this portion of the studyover that used by Kirwan et al. (1984b]. Their method re- come from three drifters (1598, 1599, and 1600) which werequired an a priori assignment of a solution form to (4) and (5); seeded in the 1980-1981 ring. Kirwan et al. [1984a, b] havei.e., (7) and (8) were assumed to be complex conjugates, and already reported on this. The data have been reanalyzed usingvalues of Z. were assumed to be constant over one ring revo- the general algorithm discussed in the preceding section andlution (approximately 15 days). The procedure used here does in the appendix.not require an a priori assignment of the solution form; it lets Figure 2 shows the paths of these three drifters in thethe data determine this. Thus it should have wider utility. In midgulf region along with the positions of the centers and theaddition, the Zk are now required to be constant over seven orbit ellipses. Arrows on the drifter and center paths marktime intervals (42 hours in the present case) rather than 2 10-day intervals, while the ellipses are presented at 15-dayweeks, intervals. The paths for each drifter are for a common time

As was indicated above, a, b, and c are not especially useful interval (day 340, 1980, to day 118, 1981). This figure es-for comparison with Eulerian data. For rings a more appro- tablishes three points. First, all three make the same swathpriate description would be the elliptical structure of the orbit through this portion of the gulf, suggesting that they generallytraversed by a particular drifter. As was shown by Kirwan et followed the ring. The overall path characteristics of driftersal. [1984bj, the characteristic equation is 1598 and 1600 (Figures 2a and 2c) are remarkably similar,

suggesting that they were in very similar orbits. Second, the(ck + bk)xA2 + (c. - b)yk' - 2aaxtYA paths for the center of the ring, as determined independently

= LA exp [-dA(t - t,)) (9) from the three drifters, is in good agreement as far as about94°W (the first 2 months of the record). Discounting the very

Here Lk is the angular momentum per unit mass relative to beginning where there are numerical problems, the typical dif-the ring center. This can be calculated directly (see equation ference in the location of the center at any one time is 30 km,(A18)). In (9) the argument in the exponential term can be about 20% of the diameter of the ring as determined frommade zero by making the evaluation at the beginning of the satellite imagery (Kirwan et al., 1984b]. This is about 10 kmtime interval t = t.. Note that a requirement for (9) to describe larger than the rms displacement between the maximum sur-an ellipse is that c. 2 > ak' + bk 2. face and interface displacement anomalies of the HT GCM.

From analytic geometry it is ! nown that for anticyclonic The 30-km variability is considerably larger than that report-motion, cA < 0, the major axis of the ellipse makes an angle ed by Hooker and Olson [1984] using a variant of the tech-with east of nique employed here under ideal conditions. In view of our

results along the continental slope, described below, it is notk= tan-' (-a/b) (10) clear whether the large variability is due to data, technique, or

rapid evolution of the ring.In evaluating (10), care must be exercised in the quadrant Finally, the ellipses also show generally good agreementassignment. Also, the semilength of the major axis is east of 94°W. Note the tendency to develop a northwest elon-

1192 KIRWA E AL: KINMATIc PROPERIEs.OF Loop CURRE NT RINGS

gation- as the ring approaches 94'W. West of 94*W, all three Figure 3 shows the-time series for the swirl-and translationorbits suggest that the ring begins to interact with the conti- velocities. Allthree records indicate swirl velocities in excess ofnental slope circulation and/or topography. This is discussed 50 cm/s throughout most-of the record. The dominant periodlater, is about 14 days with a variation of about 2 days between the

records. This is recognized as the ring's rotation period. How-270 ever. from about,day 30 to 60, 1981. all three drifters show a

POSITION- smaller-period component (about 5 days) in the time seriesELLIPSE - and a decrease in swirl amplitude. As will be discussed below,

26' PATH -

CENTER

25050252

~ > .

If- I/•

2 '

-oo .....I'

> °j

tl I

\ -J

220 ,

-in

2 1 6 L -_ L 3 :*

V S W IR L -' -

98* 970 96 ° 950 940 930 920 910 90

o U SWIRLV TRANS

LONGITUDE -100 "AN50

Fig. 2a 31.0 01.6 1181980 1981

270 TIME (JULIAN DAYS)POSITION - Fig. 3aELLIPSE

260 PATH' 100 50CENTER

250 U >-

W

I. :.lU

SWIRL;0

V TRANS

210

U TRANS98' 970 960 95 o 940 930 92" 910 90 -100

0 0-'5

LONGITUDE 31.0 01.6 1181980 1981

Fig. 2bTIME (JULIAN DAYS}27 '

Fig. 3bELLIPSE --

000

25' PATH ---CENTER -I

2 *

:j

0

ci....._,I.o23

V /WR

LA U SWIRL

201

V TRANS

%A

U TRANS21 0

.0.L.__

0-100i ,5

98* 970 960 950 940 93' 920 910 900 310 06111980

1981Fig 2b TIME (JU.LIAN DAYS)

Fig. 2c Fig. 3c

Fig. 2. Drifter paths, inferred center path (red) and orbit ellipse(blue) for the three drifters in the 1980-1981 ring: (a) 1598, (b) 1599, Fig. 3. Swirl (left scale) and translation velocites (right scale) in-and (c) 1600. Arrows on the paths are at 1-day intervals, while the ferred from the three drifters in the 1980-1981 ring. (a) 1598, (b) 1599,ellipses are shown at 1-day intervals. Here S and F refer to start and and (c) 1600. The east and north components of the translation veloc-finish. ity are depicted in red and blue, respectively.

KIRWA, Er AL.: Kimm-Aiic PROP'ntES OF Loop CukENT RiwGS 1193

it is speculated that during this period the drifters were re- swirl velocities decrease significantly and show some evidencesponding to effects produced by the continental slope and/or of higher-frequency components. This is the period when thecirculation further to the west. All the translation velocities drifter moves west of, 94°W and executes the deformed,show a consistent westward component of about 4 cm/s up northwest-oriented loop. As with the earlier ring, this is inter-until about day 30, 1981. For the reason just given, it is not preted as evidence for, topographic and/or slope circulationclear that the analysis applies to this ring for the period from interactions.day 30. 1981. to day 60. 1981. After day 60, 1981, the ring Figure 2 indicates that west of 94*W, the three drifters incenters appear to be stationary. the 1980-1981 ring show significant divergence in the calcula-

The 1982-1983 ring and comparison with the 1980-1981 tion of the inferred center. Furthermore, drifters from bothring. The same characteristics are seen in the analysis for the rings showed anomalous path characteristics in this region. In1982-1983 ring. Figure 4 shows the drifter path, inferred particular, 1598 and 1600 appeared to become entrained incenter path, and the orbit ellipses for drifter 3374. As in Figure smaller, anticyclonic eddies, while 1599 and 3374 exhibited2, there is significant distortion of the orbits west of 94 0W. warped and elongated ellipses. This has been attributed toUnlike the earlier ring, no significant northwest ellipticity is interaction between the ring and the continental slope and/ordeveloped as the ring approaches 94'W. Comparison with slope circulation further to the west.Figure 2 indicates that the 1982-1983 ring center followed Figure 6 focuses on this time interval. Figure 6a shows thenearly the same path as that of the 1980-1981 ring. paths of all four drifters during the period in question. It is

clear from this that all four drifters were under the influence ofdifferent dynamical processes here than those they experienced

POSITION - to the east.26- ELLIPSE -

PATH -- 27'CENTER 270,.

25' 15.. - --l1599

w 260 :60o..% 3374

S24'- 25' -

23' . "" J'\

I- ' V4

220 230

97' 960 950 940 930 92' 91' 900 890 22'

LONGITUDE

Fig. 4. Drifter path, inferred center path (red), and orbit ellipse 21' ' ' 9 9 I

(blue) for the dnfter in the 1982-1983 ring. Arrows on the paths are at 98' 970 960 954 94' 93' 92- 91' 90'tO-day intervals, while the ellipses are shown at 15-day intervals. LONGITUDE

Fig. 6a

270100 50 1598 ---

1599

260 '600 ......U 337-uJ I-

I"flfl SW

>. 4

0 0 - 2F"

> V ' 230 i

4 F

. MR. %" 22'1 TRANS1

- UO I._ 50 ,A Ni

293 34,7 036 98' 970 960 95' 9, 930 920 910 90'1982 1983

LONGITUDETIME tJULIAN DAYS) Fig. 6b

Fig. 5. Swirl (left scale) and translation velocities (right scale) in- Fig. 6. Paths of drifters 1598, 1599 and 1600 in the 1980-1981ferred from the three dnfters in the 1982-1983 ring. The east and ring and 3374 in the 1982-1983 ring for the period when they werenorth components of the translation velocity are depicted in red and interacting with the slope topography and/or circulation. The arrowsblue. respectively. are at 10-day intervals. (a) Drifter paths. (b) Inferred centers of flow.

Figure 5 shows the translation and swirl velocities for the Figure 6b compares the movement of the inferred centers.1982-1983 ring. The swirls are slightly less than 50 cm/s, In interpreting this figure, it should be emphasized that thewhich could reflect a smaller orbit of the drifter. The dominant center calculation is greatly complicated by the changes inperiod is still about 14 days. Note that around day 1, 1983 the curvature of the paths. The paradigm interprets this as a

i i94 KIRWAN Er AL: KINEMATIC PROPERTIES OF Loop CuRRENT RINGs

change from an anticyclonic to cyclonic structure with a Figure 7 shows the path of one of several simulated driftersconsequent change in the location of the center. No doubt the from the HT GCM along with the orbit ellipses. Figure 7areal dynamics during this period are much more complicated depicts the results using the ring center as inferred from thethan the rather simple scenario available from the kinematic drifter orbit, while Figure 7b repeats the ellipse calculationmodel. The geographic variation in the paths clearly is much using the position of the maximum displacement anomaly oflarger than that seen east of 94°W or, as will be seen shortly, the interface. The choice of the interface displacement as rep-than that found along the slope. This suggests that each of resentative of the ring center is arbitrary. Presently, we arethese drifters was either briefly pulled out of the parent ring or comparing the maximum interface displacement, the maxi-its orbits deformed through interaction with the continental mum surface elevation displacement and orbit-determinedslope topography or circulation. This hypothesis accounts for centers. In general, all are within 30 km of each other.the deformed orbits and the brief occurrence of high fre- The most significant difference between the orbit-quencies in the swirl velocities, determined centers and interface displacement anomaly occurs

in the interval between 91°W and 930W where the orbit-3.2. Simulations inferred ring center shows a significant southern loop. This is

Attention is now turned to the simulated data. There are not seen in the path of the maximum interface displacementtwo issues involved in the utilization and interpretation of this anomaly. At 930W the orbit-inferred center executes an anti-data. First, how consistent is the paradigm in recovering ring cyclonic loop that, again, is not seen in the displacementkinematics which should be independent of the orbit? This anomaly path. Elsewhere, the orbitwdetermined path is consis-

issue is presently being addressed. Addressed here is the issue tently 5 to 25 km south of the displacement anomaly path.of establishing how well the GCM agrees with the observa- A comparison of Figures 2, 4, and 7 shows that the simulat-tions. Specifically, the center path, orbit ellipses, and rans- ed and observed centers follow the same general path through

this part of the midgulf. In general, the simulated center pathlation velocity as inferred from the simulated data ate - is 10 to 35 km south of the observed path, depending uponpared with the same properties as determined from the obser- whcoftewosmledenrpasisud.Tees

vations. which of the two simulated center paths is used. There isvirtually no evidence of the interaction with the continentalslope topography and circulation in the model results as was

POSITION - seen in the observations. None of the simulated orbits were26" ELLIPSE - deformed or appeared to be pulled out temporarily from thePATH - -mCENTER - ring. Somewhat" surprisingly, we have found less consistency in

25' the ellipse calculations for the simulations than in the observa-

W- tions. A number of the ellipses from the simulated data areC3 _-quite deformed. They also show frequent reversals of the1 24" *. 11 major and minor axes. Overall, however, as the ring nears the

slope region, a northeast orientation appears to develop in the" .. orbit ellipses.

23" , . Figure 8 shows the translation and swirl velocities for the, simulation. Figure 8a gives the results using the orbit center,

22" . ,and Figure 8b uses the maximum displacement anomaly as* *the ring center. The dominant period in the swirl velocities is

970 960 950 940 93' 920 91. 90' 89' about 24 days, or 10 days more than was found in the obser-vations. Note that the swirl velocities, as calculated fromFig. 7a either center, are of comparable magnitude, about 30 cm/s.

Fig. 7a

POSITION -26* ELLIPSE 100 50

PATH --

CENTER -

25' - W

% ,..

w I,

L- 24' % A-J

23* - > V)

,# : V SWIRL =* Io.* 0 U SWIRL

220 o4v TRANSF- U TRANS

-._ 1000 r i _50970 960 950 940 936 920 910 900 890 t0 8 120 160

LONGITUDE TIME (JULIAN DAYS)

Fig. 7b Fig. 8a

Fig. 7 Path and orbit 0!ipses (blue) of a simuiated drifter from Fig. 8. Translation and swirl velocities as inferred from the HTthe HT GCM. Arrows on tho path are at 10-day intervals, while GCM. The east and north components of the translation velocity areellipses are shown at 15-day ir, tervals. Figure 7a shows the ellipes depicted in red and blue, respectively. The scale for the translationusing the orbit-inferred center ,., the flow center, and Figure 7b uses velocity is on the right. Figure 8a shows results using the orbit-the maximum interface displacement anomaly. Both centers are inferred center. while Figure 8b (next page) shows results using theshown in red. maximum displacement anomaly as the flow center.

Ps m • rm • • a a • u

K|RWAN ET AL.: KINEMATIC PROPERTIES OF Loop CURRENT RINGS 1195

100 so 100 km, which is only two-thirds thediameter of a typicalring. For most of the paths, the orbit-determined centers are

L) somewhat south of the displacement anomaly path. However,Z; as the paths near 94'W, the observed centers move north of0_j the displacement anomaly. All the orbit-detiermined centerwr '> paths show considerably more structure than does that of the

E3- 0 0 0 displacement anomaly.

4. CONTINENTAL SLOPE>-J

. < It is remarkable that both observed rings and the simulatedcr

3m V SWIRL- ring impacted the continental slope in the same area. This is€ U SWIRL

V TRANS- seen in Figure 10, which shows the drifter paths, inferredU TRANS-100o ___ "0 .o .o .... o5 center paths, and orbit ellipses for all three cases. In the case

TIME (JULIAN DAYS) ~Fig. 8b

This is somewhat less than the observed swirls, but this could ELLIPSE

be rectified by picking a simulated drifter in a greater orbit. PATH

There seems to be more difference in the translation velocities. CENTER

Because its path is straighter, the displacement anomaly path 2"has less variance. The mean westward component in both ' ofcases of about 5 cm/s is consistent with the observations.

230

270

220 -

260 980 97* 960 95' 9/,

LONGITUDE

Fig. 10a

250

Wo .SIO -o-

* CENTER: 220I

230 •.I..ORBIT 13374 me..-

1598 * ' 23 "1%

220 1599 %..

1600 -m

PRESSURE 22 10 I I98o 97' 96 950 940

940 930 920 910 900

LONGITUDE

LONGITUDE Fig. 10b

Fig. 9. The centers for the 1980-1981 ring, the 1982-1983 ring, Fig. 10. Drifter paths, inferred centers and orbit ellipses for (a) theand the HT GCM simulation. Orbit 1 (red) refers to the orbit- 1981-1982 ring, (b) the 1982-1983 ring (above); (c) the simulated ring,determined center, and pressure (blue) refers to the maximum dis- and (d) center paths along the continental slope (next page). Arrowsplacement anomaly path. Arrows depict 10-day intervals, on the paths depict 10-day intervals, while the ellipses are at 15-day

intervals.

Figure 9 compares the inferred center movement for all four of the simulated data (Figure 10c), the orbit-inferred centerdrifters along with the inferred center of the simulated drifter path and the displacement anomaly path are virtually identi-and the displacement anomaly path east of 94°W. This is the cal. Consequently, it is only necessary to display one set ofregion where there is no apparent effect of slope topography calculations. The orbit-determined centers were used here.or circulation in the observations. Of course, details in the Figure 10d illustrates how close the movements of the twopaths vary, but overall the agreement is quite good. The maxi- observed and simulated rings are. There the paths of the in-mum deviation at any one time between any of the paths is ferred centers along with the displacement anomaly path have

1196 KIRWAN ET AL.: KINEMATIC PROPERIES OF LOOP CURRENT RINGS

been superimposed. Except for the beginning and end of the a decrease in the dominant- period to about 15 days. Thepaths, the maximum deviation is 70 km with a rms deviation magnitudes of the observed ring's swirl velocities are about 40of 12 km. Moreover, all paths show the same general charac- to 50 cm/s, which is close to that found in the midgulf. Theteiristics. Both the observations and simulation show a north- swirl velocity for the simulated ring is about 80 cm/s, substan-ward migration along the isobaths but with some eastward or tially more than what was obtained for the midgulf. All trans-looping component. lation velocities indicate a fairly steady northward movement

of about 4 cm/s.

0~ s

25* - POSITION -

ELLIPSEPATH .

0 0

23* ;

,-"Y SWIRL ..U SWIRLr RANS-

TRANS

-100 -5022" S 027 117 157980 970 96. 950 91..

TIME (JULIAN DAYS)LONGITUDE

Fg lFig. 10c Fig. a

250 - 60181 RING - -$2183 RING *....*PRESSURE W*.tANOMALY >

-- SIMULATEO C 0 ZRING

V ~ SWIL I.I'0 _

-jI-r

23" V SWIRL.V TRANSU TRANS

-100 -50s 73 123 1721983 1983

22', TIME (JULIAN DAYS)98' 970 96 950 9. Fig. lb

LONGITUDE U SWIRL V SWIRL U TRANS V TRANS sFig. 1Od 100

Fig. 10. (cont.) Drifter paths, inferred centers and orbit ellipses for(c) tesmltdring. n (d) center paths along the continental slope. ~aArrows on the paths depict 10-day intervals, while the ellipses areshown at IS-day intervals. W j

The northward migration along the slope is puzzling, as 00 zconventional theory (Smith and OBrien, 1983] would have:topographic beta drive the rings to the south. Recently, Smith Z[1986] has suggested that the northward migration may be Edue to boundary effects. An alternate explanation was offered (Aby Nakamoto [1986], who found northward propagating sol-ton solutions in a two-layer, nonlinear, quasi-geostrophic -100 .- 5Model. 160 180 200 220 240 260 280

The swirl and translation velocity time series are shown in TIME (JULIAN DAYS)Figure 11. In the 1980-1981 ring the dominant period is 15 Fig. licdays, almost exactly that found in the midgulf For the 1982- Fig. 11. Swirl (left scale) and translation velocities (right scale) for1983 ring the dominant period is about 11 days, slightly less the (a) 1980-1981, (b) 1982-1983, and (c) simulated rings along thethan that found in the midgulL The simulated ring also shows continental slope.

KIRWAN ET AL.: KiNDATic PROPERTIES OF Loop CU.taEN RINGS 1197

5. CONCLUSIONS along the satellite path but give no information on the actual

The paths, translation and swirl velocities, and orbit ellipses motion. Supplementing the latter data with the former data inhave been calculated from Lagrangian observations for two eddy-resolving general circulation models could significantlylarge anticyclonic, Gulf of Mexico rings and one simulated upgrade their predictive capabilities.

ring. Within the capabilities of the data base and the analysisroutine, the following are now established. APPENDIX

1. The paths of the three rings across the central Gulf of The purpose here is to provide some details on the inver-Mexico to the continental slope off of Mexico are virtually sion of (3) and (4) to obtain the time series for the elements ofidentical. However, not all Loop Current rings will follow this Z. The basis for this inversion is a Taylor expansion in timepath. See Lewis and Kirwan (1987]. about the instant tk of the velocity vector. From the left-hand

2. The intraring variability of translation and swirl veloci- side of (3) and (4) one obtainsties and the inferred center paths for the 1980-1981 ring areessentially the same as the inter-ring variability between the u(t) z (t ) + u'(tXt - t,) + u"(tkXt -

1980-1981 and 1982-1983 rings. + u.'(rXt - tk)3/6 (Al)3. For the observed rings, there was evidence of strong

ellipticity developing as these rings approached the conti- V(t) =t) -, ,tkXt - t) + v"(tXt - tk)2/2nental slope. The axis of orientation was approximately + v.'(tkXt - t) 3/6 (A2)NE-SW. No independent data on ellipticity were available forthe 1982-1983 ring, so it cannot be ruled out that this ring for the interval th -_ t :_ tk + I. The primes represent time de-was so orientated. Similar but less pronounced ellipticity was rivatives. Now each of the terms on the right-hand side of(Al)seen in the simulated ring. and (A2) can be estimated from the velocity record. For exam-

4. Both observed rings showed significant distortion of the ple u(th) and Ljtk) are merely the velocities at time tI. Thedrifter orbits west of 94°W. This is attributed to interaction of derivatives can be estimated by a variety of techniques; herethese rings with the continental slope topography and/o slope we have employed centered finite differences.circulation features. Such interaction was not observed in the Now the analytic solution, (5) and (6), can be expanded in asimulated ring. Taylor series as well. When this is done and coefficients of the

5. After encountering the continental slope off of Mexico, appropriate powers in this expansion and (Al) and (A2) areboth observed and simulated rings retained their integrity for equated, a system of simultaneous nonlinear equations is ob-at least several months while they migrated to the north. The tained for each time interval. With the subscript k suppressed,track of the center of both was very close to the 1200-in these areisobath. The observed swirl speeds were slightly changed from X(a + d) + Yb - ) + 2Ur - 2u (A3)the values obtained from the midgulf. The simulated ring ex-hibited a substantial increase in swirl velocity. For both the X(b + c) + Y(d - a) + 2Vr = 2v (A4)observed and simulated data, the translation speeds were com- X((d + a)2 + b2 - c2] + 2Yd(b - c) = 4u' (AS)parable to those obtained for the midgulf.

There are a couple of broader implications of this study 2Xd(b + c) + Y(d - a)? + b2 - c2] = 4v' (A6)pertinent to eddy-resolving GCM's. Both the volume of dataas well as the time and space scale resolution of simulated X[(d + a)' + (bl - c2X3d + a)]

data available from these models far surpasses the typical ob- + Y(b - cX3d 2 + a + b' - c') = 8u" (A7)servational data bases used for verification. Because of thedifferent time and space scales of resolution in the simulated X(b + cX3d 2 + a2 + b2 - c2 )

and observed data bases, it is not clear how to establish refi- + Y((d - a)' + (ba - c2X3d - a)] = 8v" (A8)able criteria for statistical comparisons. This study suggeststhat a potent alternative is to compare model and observed X[(a2 + b2 - c2 + d2)2 + 4d2(a2 + b' - c2)Lagrangian kinematics. Both simulated andl observed La- + tad(a2 + b1 - c2 + d2)]grangian data sets provide comparable space-time resolutionon the evolution of specific circulation features. Moreover, the + 4Yd(b - cXa2 + b2 - c2 + d2)] = 16u.' (A9)assessment of the prediction of such features is a more strin- 4Xd(b + cXa 2 + b" - C2 + d2 ) + YC(a 2 + P - c2 + d2)2

gent test of a model's predictive capability than are generalstatistics. The Lagrangian data also provide a means of fine- +4d2(a2 +b -c 2 )-4ad(a2 +b2 -c2 +d)] = 16v'" (A10)tuning model parameters such as layer depth, density differ- These eight equations are inverted at each time step for X, Y,ences, etc., so that observed and simulated swirl velocities can U V,, a, b, c, and d. To our surprise it seems that this can bebe matched. As to the HT model, this study has documented done analytically. The key to this is the observation that the

the interesting situation that the prediction improves in time, geometric invariants of the matrix M, i.e., Tr () and det (h),

at least as far as the movement of the ring is concerned.

The second and related issue is the use of Lagrangian obser- can be calculated from observations without knowing a, b, or

vations for updating prognostic models. In character, such c, a priori. After 3 years of inspection it was seen that

data are similar to sea surface topographic data derived from det (M) = 2(u"v"' - u.'")/(u'v" - vu") = M" (All)satellites (see Hurlburt, 1986; Thompson, 1986; Kindle, 1986].The Lagrangian data contain information on the motion of Tr (M) = (u'v"' - v'u")/(u'v" - v'u") = d (A12)representative parcels but no information on what is happen- Insertion of M and d, calculated from (All) and (Al2),,ng nearby Satellite topographic data return information just respectively, into (A5HAI0) significantly simvlifies the latter

1198 KIRWAEr AL.: KNEmATiC PiOtPmvi. OF L6O CuRRENT RINGS

expressions. Considerable routine algebra then yields Huribuit. H. E.. Dynamic transfer of-simulated-altimetir data into-subsurface information-by a numerical oean model; J_/Geophys.

X = 8[u'(M 2 + 2i1) 2u"'J/M' (A13) Res.. 91(C2), 2372-2400, 1986.Huriburt, H. E., and -J. D.- Thompson, A- numerical study of Loop

Y = 8[v'(M 2 + 2d2) - 2v'"J/M' (A14) Current intrusions. and eddy shedding, J. Phys. Oceanogr., 10.1611-1651, 1980.

a= -(K 1 M 2 + K2 + K3(d12)}/N (A15) lchiye, T.. Circulation and water mass distribution in the=Gulf ofMexico, Geofis. Int.. 2, 47-76, 1962.

b = {HtM 2 + H2 + H3(d/2)}/N (A16) Kindle. J. C., Sampling strategies and model assimilation of altimeiricdata for ocean, modeling and, prediction, J. Geophys. Res., 91(C2),c = M2 - G1 + G3(d/2)}/N (A17) 2418-2432, 1986.

where Kirwan, A. D., Jr.. A note on a geophysical fluid dynamics variationalprinciple, Tellus, Ser. A, 36, 211-215, 1984.

N = 8(u'v" - i"V); Kirwan, A. D., Jr., W. J. Merrell, Jr., J. K. Lewis, and R. E. Whitaker,Lagrangian observations of an anticyclonic ring in the Western

GI - 2(u'1 + v'2); Gulf of Mexico, J. Geophys. Res.,89, 3417-3424. 1984a.Kirwan, A. D., Jr.. W. J. Merrell, Jr., J. K. Lewis, R. E. Whitaker, and

G_ = 8(u' 2 + v"2); R. Legecks%,-A model for the analysis of drifter data with an appli-cation to a warm core ring in the Gulf of Mexico, J. Geophys. Res.,

G3 = 16(u'u" + v'v"); 89, 3425-3428, 1984b.Lewis, J. K,, and A. D. Kirwan, Jr.. Some observations of ring-

HI - ' - v'2); topography and ring-ring interactions in the Gulf of Mexico, J.

H 2 = 8(u" - v 2); Geophys. Res., 90(C5), 9017-9028, 1985.Lewis, J. K., and A. D. Kirwan, Jr.. Genesis of a Gulf of Mexico ringH3 = 16(u'u" - v"v'); as determined from kinematic analyses, J. Geophys. Res., 92(C1 1),

11,727-11,740,1987.K = 2u'v'; Nakamoto, S., Application of solitary wave theory to mesoscale

eddies in the Gulf of Mexico, Ph.D. dissertation, 50 pp., Dep. ofK2 = 8u"v"; Oceanogr., Tex, A&M Univ., College Station, 1986.

Nowlin, W. D., Winter Circulation Patterns and Property Distri.K3 = 8(u'v" + u"v'). butions, Tex. A&M Univ. Oceanogr. Stud., vol. 2, edited by L. R. A.

Capurro and J. L. Reid, pp, 3-53, Gulf, Houston, Tex., 1972.Incidentally, the parameter N is directly related to the angu- Nowlin, W. D., and H. J. McClellan, A characterization of the Gulf of

lar momentum per unit mass of the orbit L: Mexico waters in winter, J. Mar. Res., 25,29-59, 1967.Okuho, A., Horizontal dispersion of floatable particles in the vicinity

L = -4N/M 4 (A18) of velocity singularities such as convergences, Deep Sea Res., 17,445-454, 1970.

The final step in the calculations is to determine the trans- Smith, D. C., IV, A numerical study of Loop Current eddy interactionlation velocities from (A3) and (A4). This is trivial, since all with bottom topography in the western Gulf of Mexico, J. Phys.other terms are now known, Oceanogr., 16(7), 1260-1272, 1986.

Smith, D. C., IV, and J. J. O'Brien, The interaction of a two layer

Acknowledgments We acknowledge support for this research from isolated mesoscale eddy with topography, J. Phys. Oceanogr., 13,

the Office of Naval Research under contract N-or 483-K-0256 to 1681-1697, 1983.Smith, D. C., IV, and R. 0. Reid, A numerical study of non-friction

the University of South Florida. Support for J.K.L. was supplied by decay of mesoscale eddies, J. Phys. Oceanogr., 12(3), 244-255, 1982.the Minerals Management Service, Gulf of Mexico Physical Ocean- Sturges, W., and J. P. Blaha, A western boundary current in the Gulfography Study, through contract with Science Applications Interna- of Mexico, Science, 192, 367-369, 1976.tional Corporation. A.D.K. acknowledges the support of the Slover Sweitzer, N. B., Jr., Origin of the Gulf Stream and circulation ofOceanography Endowment to Old Dominion University. A. Wall- waters in the Gulf of Mexico with special reference to the effect oncraft very kindly supplied the simulated data from the HT GCM. jetty construction, f rans. Am. Soc. Civ. Eng., 40, 86-98, 1898.

Thompson, J. D., Altimeter data and geoid error in mesoscale oceanREFERENCES prediction: Some results from a primitive equation model, J. Geo-

Austin, G. B., Some recent oceanographic surveys of the Gulf of phys. Res., 91(C2), 2401-2417, 1986.Mexico, EOS Trans. AGU, 36(5), 885-892, 1955.

Blaha, J. P., and W. Sturges, Evidence for wind forced circulation inthe Gulf of Mexico, technical report, Dep. of Oceanogr., Fla. StateUniv., Tallahassee, 1978. A. W. Indest and A. D. Kirwan, Department of Oceanography, Old

Elliot, B. A., Anticyclonic rings and the energetics of the circulation of Dominion University, Norfolk, VA 23508.the Gulf of Mexico, Ph.D. dissertation, Tex. A&M Univ., College J. K. Lewis, Science Applications International Corporation, 1304Station, 1979. Deacon, College Station, TX 77480.

Elliot, B. A., Anticyclonic rings in the Gulf of Mexico, J. Phys. Ocean- 1. Quintero and P. Reinersman, Department of Marine Science,ogr., 12, 1293-1309, 1982. University of South Florida, 140 Seventh Avenue South, Saint Peters-

FlierI, G. R., Particle motions in large amrlitude wave fields, Geophys. burg, FL 33701.Astrophys. Fluid Dyn.. 18,39-74,1981.

Hooker, S. B., and D. B. Olson, Center of mass estimation in closedvortices: A verification in principle and practice, J. Atmos. Oceanic (Received March 20, 1987;Technol., 1(3), 247-255, 1984. accepted May 18, 1987.)

JOURNAL OF GWPIIYSICAL RESEARCH. VOL 9.NO. M~ PAGES AU-9 G.MrG3 15. Im

Notes on the Cluster Method for Interpreting Relative Motions

A. D. KmwA.,. JR.

It is A oun tha te so-afdl dstr msbod for detemig sAk hsmsl Flfsx of he hwIrwwsvelmy has somw basic fmiuasos caused by rcsL-a rmsy" ausxmWniosF. For brwc-aec cias:here Es aliasing of these estiates by brge-scalc shear. uhe for 9maI-aara coom. the dispbacamn otthec staistiCao" teenter from a phlsicassifliicus fow em" Cast produce seriowsly 3amasd C100"modd paramete esasasm These efects ae Ladepede of sijkal crqpnu proloems so* asmasurm t errot and tvaisical riab~a. AA alianadie to she dos" r a ara is proosL Thi.hovs-r. requires solsing a sonhnar ierse prodlenm to ecoser ses da e d zoul sdouygladieats.

l%'ruooCno% duce the duster average of property V

Estimates of the horizontal spatial derivatives of the near- (1<P> IN/Nsurface, horizontal velocity are of obvious importance in awide range of scales in ocean dynamics. They are notoriously Miere N is the statistical degrees of freedom in the dusterdifficult to obtain. Most estimates have come from relative average and the sum is over all drifters in the duster [seemotion analysis of drifter duster For an older summary of Sanderson et aL. 1988]. Note that by definitionresults. see Kirwan [1975]. Fahrbach er aL [1986] have recent- (7> 0 (3)ly applied this technique to drifter data from the equatorialAtlantic. This is the second part of the duster paradigm- It is stressed

Sanderson el aL (1988] have pointed out a statistical bias in that (3) significantly constrains the basic assumption and va-using this technique if the number of drifters in the duster is lidity of a two-term Taylor expansion.small. Of course, the statistical significance of model parame- By standard minimization augmented with (3). the leastter estimates, as well as the effect of measurement errors in squares estimates of U.' and H,, areestimating the velocity gradient tensor, is important. There is,however, another problem with this technique. This is the tacit U., = <u.") (4)assumption that the center of mass of the cluster corresponds H = R(u (5)to the center of the flow field about which a Taylor's ex-pansion of the velocity field can be made. where

The purpose here is twofold. The first is to show that withthe cluster paradigm, a deviation of the cluster center from a Rol= (r'r 7')> - (6)flow center can result in serious biases in gradient estimates. is the inverse of the statistical moment of inertia matrix of theThe other is to propose an alternative to the cluster model cluster Details and variations of(4) and(5) are given by Moim-that avoids the center bias problem. The new approach pro- ari and Kirwan [1975], Okubo and Ebbesmeyer (1976], andvides independent estimates of the velocity gradient estimates; Okubo et aL £1976].however, it requires solving a nonlinear inverse problem. This model is critically dependent upon (1) a translation

velocity U.' common to all drifters. (2) a velocity gradientTtzogy tensor Hb common to all drifters, i.e., a homogeneous defor-

The first part of the cluster paradigm is given by mation field, and (3) equation (3).In order to assess the role these assumptions play. consider

(r.P - r,1)/A = U.P = U.0 + Y H ob r.b + r.'/A (1) a more general kinema:ic flow model:b

Here A is the time difference between fixes, r' is the a compo- u.' = V' + XG.b'xh + r,'/A (7)

,ent of the position vector from the cluster center of mass tothe drifter path, r. is its original or previous position vector,H., is the velocity gradient, and U.* is a steady mean flow. physically relevant local origin. This is called the flow center

Figure la is a cartoon of the general geometry. As has been to distinguish it from the cluter center. See Figure la for thenoted by many previous investigators, (1) can be obtained by a geometric relation between xb' and rb'. The latter dependsTaylor's expansion of the velocity field, upon the geometric characteristics of the drifter array; the

The first term on the right-hand side of (1) is a uniform former depends on true flow field characteristics. This modeltranslation of the center, and the second term is a local swirl differs from (1) in that it allows for a separate Taylor's ex-induced by a homogeneous deformation field. The last term in pansion for each drifter. Thus there are distinct translation

(1) is a residual, the minimization of which yields estimates of velocites Vi', as well as distinct velocity gradients G,,P, forU. 0 and H.b For the latter operation it is necessary to intro- each drifter. Also in contrast to (3)

(xb') 0 (8)

Copyright 1988 by the American Geophysical Union. It should be clear, however, that if the three assumptions listed

Paper number 8C0347. in the preceding paragraph are imposed on (7), it will reduce0148-022788/008C-0347S5.00 to (1).

9337

9~38 KIMWA.: CUSIV )EemOD MRn LYmMVE1VV RVLATME MU4TuoeS

A,_ E velocity and is of the order of the true velocity gradient times

El the distance between the two center, For small area dusters,I this displacemet could well be of the duster horizontal scakand so the duster determined translation velocity could be of

o the order of the swirl velocity. In the case of the velocitygradient estimate, there is no a-priori reason to expect

o <zr to be small: henc the correction- tem in (13) couldbe of the order of the correct value. As an aside. (13) is recog-

. 3" A7 nized as an extension of Pappas theomem from mechanics.2 arfor Case IL The translation velocity varies from drifter to

pa'sit drifter:. however, assumptions 2 and 3 hold (see Figure Ic).0 0 RZ This case would include the situation where the drifters are

ce er deployed in a jet such as the Gulf Stream. Perhaps. fortu-

itously. the cluster center initially coincides with the flow* X cluster center, say. the axis of the jet. The duster parameter estimates

Centernow are

C. A - A U., = (. > (14)

If the flow center is indeed the jet axis, then the clustermodel will yield a translation velocity for the center somewhatless than the maximum velocity of the jet as indicated by (14).Moreover, if the jet velocity is symmetric about the flow axis.

Fig. I. Cartoon showing the esseatial geometry. (a) Basic vectors then the duster gradient estimates of the cluster scale velocity7- r. and x for the present configuration. (b) Displacements for case L gradients will include an alias from the larger-scale shearThis could result from equal shear and normal deformation and anti- across the jet as exemplified in the inhomogeneous nature ofc clonic rotation. The solid line indicates the present configuration,and the dashed line indicates the previous configuration. ( - V' (the second right-hand side term in (15)). In general, the

ments for case I. Initially. the duster and flow centers coincide, but duster model may not be able to distinguish between small-in the present (solid line) configuration the flow center has moved scale and large-scale shears.a%y from the duster center. All panels depict the movement and These two cases highlight a basic difficulty of the clusterdeformation of the material lines connecting the drifters as well as a method. If the area covered by the cluster is small, there is ah t]othtical array initially symmetric about the flow center. high likelihood that there will be a significant disilacement

between the cluster and flow centers. This is where case I

Given only cluster data, the observables are U.' and r.'. applies. This can be alleviated by increasing the area extent of

We now examine the effect of naively applying the cluster the cluster (with or without additional drifters), but then the

paradigm to this more general situation. Taking the cluster large-scale shear may defeat the cluster model, as is indicated

center as the local origin and letting by the analysis in case II.

r., V (9) Case III. Only assumption 2 holds. In this case, the clus-ter parameter estimates will superpose (12) and (14) for U,*

be the displacement vector from the cluster center to flow and (13) and (15) for H,,.

center, it is deduced from (4), (5), (7), and (9) that It is emphasized that the problems indicated above cannotbe rectified by increasing the number of drifters and/or the

(V'> + (GP(r,' - z,')> (10) cluster area and/or the position fix accuracy and frequency.

4 RThe basic issue here is the weakness of the physics implied byH,1 = Rb((Vr,> + (G,&(r,' - Zr')YJ) (1 1) the approach, not the accuracy or statistical significance of the

7 Jmeasurements.

These last two equations show that in general, the cluster

model parameters are weighted averages of the flow model. Inorder for (10) and (11) to reduce to (4) and (5), all three as- DIscusstoNsumptions given above must be made. The preceding analysis suggests that the cluster method for

The following special cases illustrate the tenuous character estimating velocity gradients should be used with extreme cau-

of he cluster model. tion. Other observations should be used to establish its appli-

Case 1. There is no translation (VP = 0), and the defor- cability in specific experiments.

mation field is homogeneous; i.e., G,' P is the same for all There is an alternative to the cluster paradigm for extract-

drifters (G P = G.). An example of this situation would be ing velocity gradient information from path data. The alter-

when the drifters are embedded in a flow field characterized by native makes use of a kinematic model first proposed by

no translation and homogeneous vorticity and deformation Okubo [1970] and recently discussed by Perry and Chong

fields (see Figure 1b). It is seen immediately from (10) and (11) [1987]. This approach makes use of the fact that (7) (and

that the cluster model will yield bogus parameter estimates of hence (1)) can be readily integrated to obtain particle paths. Ifthe deformation field is homogeneous and stationary, then a

U. 0 = - " G,(z.) (12) single particle path contains the same information on the de-

b formation as does a hypothetical cluster of nearby particles.

H = G., - ' R,, G<zjr,'> (13) The only characteristic difference in the paths is their startingy 6 positions.

The bogus translation velocity has the structure of a swirl To illustrate how the deformation information is contained

KutwA.M: CwLsu MEmo mOit-retixnt REtIAZ1V MolioNs 039

in the path data, it is necessary to calculate the path. Neglect- are truly Lagrangian. and so there may -be some difficultying the last term in (7) and assuming that V,' is steady in time. relating them to Eulerian estimates. Second, there are certainstandard techniques yield pathological- situations where the deformation field will pro-

duce variability in only one dt the velocity components. Inwt'() - WjM(r.) = V1'(t - ) + x,'t) (16) these cases it may not be possible to recover all the gradient

- W,(t) = V2 '(t - to) + x 1 (1t) (17) information. See Okubo (1970] and Perry and Chong [1987]for a complete discussion of pathological situations in critical

Here wa'(t) is the absolute position at time t, and W*(to) is points in flow fields.the initial position of the flow origin at time t, of drifter p.

The other quantities in (16) and (17) are Acknowledgments. This research was supported by. the Office ofNaval Research under contr-ct N00014-8g-K-O101 to Old Dominion

x,,(t) = {exp (m.'(t - t,)][XP(1P + GP) + X 2'G12 ']} University. The author also acknowledges the support of the Sloverendowment to Old Dominion University.

- {exp im _( - 0 )JXi(G'- )+ X'G 12 J}121'RUEMNCES

(18) Fahrbach, E. C. Brockmann, and-J. Meincke, Horizontal mixing in

x(t)= exp in. '(t - t)]X 1P G21' - X 2 '(i' - G')]} the Atlantic Equatorial-Undercurrent estimated from drifting buoy

clusters, J. Geophys. Res., 91(C9), 10,557-10,565, 1986.

+ {exp [m-'(t-t,)] -X'G 2 1 '+X'('+G')}/21 Kirwan, A. D., Jr., Oceanic velocity gradients, J. Phyi. Oceanogr., 5,729-735. 1975.

(19) Kirwan, A. D, Jr, J. K. Lewis, A. W. Indest, P. Reinersman, and I.Quintero. Observed and simulated kinematic properties of loop

m..-= [Tr(G.&') ± 11/2 (20) current rings, J. Geophys. Res., 93(C2), 1189-1198, 1988.Molinari, R., and A. D. Kirwan, Jr., Calculations of differential kin-

G' G11 - G2 2 " (21) ematic properties from Lagrangian observations in the WesternCaribbean Sea, J. Phys. Oceanogr., 5, 361-368, 1975.

P -{(GP)z + (G, 2P + GztJ1)- (G 2 D-G 2-P)z}m (22) Okubo. A., Horizontal dispersion of floatable particles in the vicinityof velocity singularities such as convergences, Deep Sea Res., 17,

X., = X 4'(t0 ) (23) 445-454. 1970.Okubo, A, and C. C. Ebbesmeyer, Determination of vorticity, diver-

Given the path data, i.e., the left-hand side of (16) and (17) gence and deformation rates from analysis of drogue observations,for a few intervals of time, the velocity gradient G1V and trans- Deep Sea Res., 23, 349-352, 1976.lation velocity V*P can be estimated. This, however, is a prob- Okubo, A., C. C. Ebbesmeyer, and 1. M. Helseth, Determination of

Lagrangian deformation from analysis of current followers, J. Phys.lem in nonlinear parameter estimation and hence of more Oceanogr., 6, 524-527, 1976.fundamental mathematical interest than purely linear statis- Perry, A. E., and M. S. Chong, A description of eddying motions andtical models. Kirwan et al. [1988] have provided one method flow patterns using critical-point concepts, Annu. Rev. Fluid Mech.,

of obtaining these estimates. Presumably, more efficient meth- 19, 125-155, 1987.Sanderson, B. G., B. K. Pal, and A. Goulding, Calculations of unbi-

ods are available. ased estimates of the magnitude of residual velocities from a smallIt is stressed that the estimates of the velocity gradient and number of drogue trajectories, J. Geophys. Res., 93, 8161-8162,

translation velocity apply only to drifter p. Other drifters, if 1988.present, would supply independent estimates. This wouldpresentowostatstica l y indesens oh e imtes oThis o ld A. D. Kirwan, Jr., Department of Oceanography, Old Dominionallow statistical assessments of the uniformity of the flow field University, Norfolk, VA 23529.parameters.

A cautionary note on application of the proposed technique (Received April 9, 1988;should be made. First, gradient estimates from this approach accepted April 15, 1988.)