Elliptic warm-core theory: The pulsrodon

Volume 138, number6,7 PHYSICSLETTERSA 3 July 1989

ELLIPTIC WARM-CORE THEORY: THE PULSRODON

CohnROGERSDepartmentof MathematicalSciences,LoughboroughUniversityof Technology,LeicestershireLEJJ 3TU, UKandDepartmentofAppliedMathematics,Universityof Waterloo,Canada

Received15 February1989; acceptedfor publication3 May 1989Communicatedby A.P. Fordy

A systemdescriptiveof the evolution of warm-corerings is shownto be analytically tractable.In particular,thenotion ofpulsrodon is introducedcorrespondingto an important subclassof solutionsinvolving pulsatingelliptic warm-corerings. Theresultsareapplicable“mutatis mutandis”to theanalysisof rotatingshallowwatermotion in circular paraboloidalbasins.

1. Introduction to a pulsatingcircular eddy.This hasbeentermedapulson [12].

Evidenceof warm-corering generationin theocean The work of Cushman-Roisinandco-workershasis well-documented[1—3].Theedgeof suchrings is stimulatedconsiderableinterest in the area. Thus,commonlyelliptical [4—6]. Satelliteobservationof Young [13] hasexploitedintegralsof motion of thesuch rotating ellipitic rings has been recently re- systemwhile Ripa [14] hasstudiedthe stability ofcordedby Evanset al. [7]. rodon-typesolutions.A systematicnumericalanal-

The analyticstudyof the time-dependentbehav- ysis of a reformulationof the nonlinearsystemgen-iour of upper-oceanelliptic eddieswassubsequently eratedby the ansatzof ref. [8] hasbeenconductedintroducedby Cushman-Roisinet al. [8]. The re- recentlyby Kirwan and Liu [121. However, up toducedgravity model introducedthereinis in direct now, no new analytic solutionsof the systemofanalogywith a rotating shallow water systemwith Cushman-Roisinand co-workers have been ob-planebottom topography.This system is, in turn, tamed.Here,it is shown that,in fact, the nonlinearanalogousto onedescriptiveof certainplasmaflows systemis reducibleto a singlequadrature.In partic-in thepresenceof a strongmagneticfield [91.In the ular, an importantnew subclassof solutionsis pre-latter correspondence,the Coriolis force is analo- sentedwhich correspondto pulsatingelliptical ed-gous to the Lorentz force, while the depthpertur- dies.These,weterm pulsrodonssincethey combinebationsof the liquid are analogousto electro-static thepulsatingcharacteristicof thecircularpulsonwithpotential perturbationsin the plasma. It was ob- the moregeneralelliptic geometryof the rodon.It isservedby Petviashvihithat the analogycan be ex- importantto note,that thenewresultsobtainedhereploited for the laboratorysimulationof plasmaphe- apply “mutatis mutandis” to the nonlinearsystemnomenaby usingrotating shallow water models:a governingrotating shallowwater motion in circularsequenceof such simulationshasbeencarriedout paraboloidalbasinsasoriginally investigatedby Ballwith successin experimentscited in ref. [9]. [15,16] andsubsequentlyby Thacker[17]. Indeed,

Cushman-Roisinandco-workers[8,10,11] have the ansatzemployedin ref. [8] in the warm-corepresentedtwo analytic solutions of their reduced modelcontextis thesameasthatadoptedin ref.[171gravity model which haveapplicationsin oceanog- in the investigationof oscillatory modesin shallowraphy. One solutioncorrespondsto a steadyclock- water basins.It is remarkedthat the new solutionswise rotation of an unchangingelliptical eddy. This presentedmaybeboostedby theapplicationofgroupwastermeda rodon. The othersolutioncorresponds invariancestothoserecentlyderivedby Rogers[18]

0375-9601/89/s03.50© Elsevier SciencePublishersB.V. 267(North-HollandPhysicsPublishingDivision)


andLevi et al. [19] in shallow water theory. i~2 = — U1 U2 — U2 02 + V., — 2b,

v2=—u2v,—v~,—u2—2c,

2. The governingequations â=—(3u, +v2)a—2v,b,

b=—u2a—2(u,+v2)b—v,c,In thepresentcontext,therelevanthydrodynamic

f-planeequationsare [8] ô= —2u2b—(u, +3v7)c,

U, +uu~+vu~+h~—v=0, = —(ui +v2)ho. (2.4)

v,+ uv ~. + tv~,+ h,, + u = 0, At this point, it is convenientto introducenewvari-ablesaccordingto ref. [12],

h~+(uh)~+(vh)~=0. (2.1)G=u, +02, GR=~(v, —u2),

Here,the upper-layerdepthh hasbeenscaledby areferencedepthH such as the maximum depth at G~=~(v,+u2), GN=~(u,—02),

some specified time. The velocity componentsu B=a+c, B~_—b,BN=~(a—c). (2.5)and v havebeenscaledby the gravity wave speed(g’H)”

2 whereg’ is thereducedgravity. The co-or- Thus,G andGR representin turn thedivergenceanddinatesx, y havebeenscaledby the radius of de- spin of thevelocity field, while G~andGN representformation (g’ H)’ ‘2/f wheref is the constantCor- shearandnormal deformationrates [20]. The ge-iolis parameter:the time has been scaledby the ometryof the lens boundaryis determinedby B, B~inertial time 1/f and BN togetherwith h

0.The eddy region is determinedby h> 0 while its Two relationscontainedin ref. [8] which arekey

edgeis theclosedcurveh(x,y, t)=0. As in ref. [8], to the subsequentdevelopmentare now recorded,we shall be concernedwith lens-likeeddiesembed- ViZ.,

ded in quiescentoceans.The kinematic and dy- A1=—3GM, (2.6)namicboundaryconditionson h= 0 havebeenin-corporatedin the system[8]. In a similar way the Qr_3GQ, (2.7)correspondingshallowwatersystemincorporatesthe whereprevailing boundaryconditions in a mannerde-scribedin detail by Rogers [18]. M= a ( u2 — ~ ) + b(v2 — u, ) — c( v, + ~)

As in ref. [8], the ansatz=B(—~—GR)+2(BNGs—BsGN) (2.8)

u=u,(t)x+u2(t)y, v=v,(t)x+v2(t)y,and

h=a(t)x2+2b(t)xy+c(t)y2+h

0(t) (2.2)Q= — a ( u ~+ v~)+ 2b(u, u2+ ,v2)

is introduced.The requirementthat the eddy edgeh=0 be closedyields

=—B(G~+G~+G~+~G2)

4=ac—b2>0. (2.3)+4GR(BNGS —BSGN)

In general,the eddy’s rim is an ellipse. Thisdegen-eratesto a circle if a=c andb=0. +2G(BsGs+BNGN)+4A. (2.9)

Substitutionof the expressions(2.2) into the~ Theseresultsare readilyestablishedon appealto theplaneequations(2.1) producesthe nonlinearcou-pledsystem system(2.4).

On substitutionof the variables(2.5) into (2.4)= —u~—u

2v,+v, —2a, we obtain the equivalentKirwan—Liu system

t~ =—u,v —vv2—u,—2b, h0+h0G=0, (2.10)

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h+2[BG+2(BNGN +B~G~)]=0, (2.11) It is thecouplednonlinearsystem(2.23)—(2.28)thatwe now seekto solve.

BS+2BSG+GSB—2BNGR=0, (2.12)

BN+2BNG+GNB+2BSGR=0, (2.13)

6+~G2+2(G~ +G~—G?~) 3. The parametrisation

—2GR+2B=0, (2.14) Theconditions(2.24), (2.25) and (2.27), (2.28)

GR+GGR+~G=0, (2.15) combinein turn to provide the integrals

GN+GGN—GS+2BN=0, (2.16) B~+B?4—~B

2=C11, (3.1)

Os+GGs+GN+2Bs=0. (2.17) and

It is notedthat in the caseof the analogousThacker + — B= C111, (3.2)

ansatzin shallow watertheory the presenceof a cir- whereC11, C,,, areconstantsof integration.It is notedcularparaboloidalbottomtopographymerelyresults

that (3.1) yieldsin the additionof a constantto (2.14) with the re-mainingequationsunchanged.Accordingly,modulo 4 = — CH/Q

8, (3.3)thatchange,the subsequentprocedureappliesto the

whencetheclosededdycondition(2.3) requiresthatThackersystem.

We now set C~,<0.The relations (3.1) and (3.2) may now be para-

Q= (GR + ~ ) — 1/2 (2.18) metrisedin turn accordingto

so that ~= ±~/~i~cosØ(t),

GR=1/Q2—~, (2.19) BN = ±\/~+~2 sinø(t),

while (2.10) and (2.15) yield, in turn, G~/~,+~sin0(t),

h0 =C~/Q

2, (2.20) GN = ±~fc11~+.~cos0(t). (3.4)

G=2Q/Q, (2.21) In order to illustrate the subsequentprocedureweconsideraparticularparametrisationwhich will lead

whereC, is a constantof integration, to the pulsrodon,viz.,New variablesarenow introducedaccordingto

~s~J~T~2 cosØ(t),~=Q

4B, ~~=Q4B~, BN=QBN, BN=—\/~~BsinØ(t),

6~=Q2G~,GN=QGN, (2.22) Gs\/sin0(t),

where the remainingequationsin the Kirwan—Liu 6 = ~ 0(1). (3.5)systemreduceto

In termsof the presentparametrisation,it is noted~+ 4(.~GN +1~U~)/Q2 = 0, (2.23) thatthesemi-majorandsemi-minoraxesof theeddy

B~+ (~G~—2E~)/Q2+E~=0, (2.24) edge h=0 havelengths

h0BN + (~GN+2~~)/Q

2—~~=0, (2.25) ( ~BI — (~(2.26)

QGN—GS+2BN/Q=0, (2.27) = (3.6)

[I ~BI — (~B2+C11)’’

2]1/2

Gs+GN+2~s/Q2=0. (2.28)and

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1/2 obtainedby useof (3.12) in therelations(3.10) and(II ~ + (~2 +~2) 1/2 (3.11) respectively.

It is readily established,on use of (2.7) thatQ

(3.7) Q=C~Q6 (3.15)[I~I—(~B2+C,,)”21 ‘/2

and,in turn, on useof (2.25) thatrespectively.The instantaneousangleof orientationof the majoraxis of the elliptical eddyedge,relative (Q2~)“+Q2B= ...2(Q+M)Q6to the east,is given by

— 2(C,~+C~). (3.16)(3.8) ——

Thelatterrelationis are-statementofthewell-knownSubstitutionof theparametrisation(3.5) into (2.23)

result [8,15] thatyields

~+ ~/~B2+C,, \/~+C,,, sin(0—Ø)=0. 43/2h0+M), (3.17)

whereI is the momentof inertiagiven by(3.9)

Conditions (2.24), (2.25) reduceto a single con- ~= JJ(x2+y2)h~dy=—A312h~(a+c)

dition, viz.,C, ~\

(3.18)—~J~,,cos(0—Ø)

+ C,, [~— (1 —2/Q2)]= 0. (3.10) with thedoubleintegral takenoverthe movingeddy~. Integrationof (3.10) yields

Likewise, conditions(2.27), (2.28) reduceto asin-gle condition,namely Q2B=C~~cost+C~,,,,sin t—2(C,~+C~), (3.19)

________ where Cs,, andCv,, are integrationconstants.— 2~1~2+C cos(0—Ø) Elimination of 0—0andQ in (3.9) via the rela-

______ tions (3.12) and (3.19) showsthat B is givenby+~fE+C,,,(1—O)=0. (3.11)

B

At thispoint,we returntotherelations(2.6), (2.7). J ~—‘[a3+(C,,, —1)o2+2(2C,,—C,~)aThus, (2.6) showsthat

M [—.~+2(BNGs ‘BSGN)]QCIVQ, +4C,1C,,,—C~v]~

2da

_____ ____ = —2whence,on useof the parametrisation(3.5), di— C,~,+ 2 \/~B+C,, \/~~+C,,, cos(0—Ø), C’~,cos t+ Cv,,sin i—2(C,~+C~)

(3.12)B�const, (3.20)

whereC,~is a constantof integration.It is readilyshown that (3.12) satisfies (3.9). In that connec- whereC~,,,=B,

0.The elliptic integral involving Btion, use is madeof the relations rn (3.20) maybe treatedby classicalmethodsasde-

scribedin ref. [21].1—0= [l—.~(B+C,~)/(B

2+4C,,)], (3.13) The natureof the B(t) dependencedeliveredby(3.20) dependscritically on the natureof the con-stantscontainedtherein. Physicalinvariantsof the

[(~+C,~)/(~+C,,,)], (3.14) motion havebeennoted in ref. [8]. Their relationto the C

1 is notedbelow:

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r I withthe relations(3.10), (3.11) reducingtothe sin-(i) volume= Ii hdxdy

j gle condition

it C2 O=1—2/J/aQ2, (4.3)_I A—I/21,2_ — I—2 0.’~ ~ c’ ‘~—1/2’~. ~.~‘—1I) where

(ii) angularmomentum a=\/k,,,, (4.4)

= JJ [xv—yu+~(x2+y2)]h~dy ~~K2+C,, (4.5)

3 Consistencyrequiresthat

=~it4312h~Mj~C~v((.~)1/2)~ K=2/3(a—/3)/a2. (4.6)

v —u + 1 Thisrelationship,togetherwith (4.4), (4.5) serves(iii) potentialvorticity= JJ h h dx dy to determineK in termsof C,, andC,,,. Accordingly,

/ C \ Bs=_4cos(t_~~~JQ2(r)di+~=it4_1~~2ho(2GR+l)=2it~(CI),/

2), Q a

(iv) energy=JJ [~h+~(u2+v2)Jhdxdy BN = — ~sin(t_ ~JQ2(i)

- =~it4312h~Q=~ ~ ~ G~= — ~sin(t_ ~JQ2(i)

OnceB hasbeengeneratedvia (3.20),Q is thende- /

terminedby(3.19) while Oand0 aredeterminedby GN= -cos( t— Q2(t) di+~, (4.7)integrationof (3.13), (3.14). It is recalledthat the ~ \ ‘~ Ioriginal system(2.4) consistsof eight equationsso -

that the generalsolution set consistsof eight inde- where 0 is a constantphaseangleandpendentconstants.In thisconnection,it is observedthat the constantsof integration introducedby the JQ_2(~) diintegrationof (3. 13) are relatedto C,,, C,,,, C,~and ~Cv,,,, through (3.12). 1

We concludethis sectionby remarkingthat a so- f Kdi (4 8)lution set may be subjectedto boostsgeneratedby ~ Cv, cos i+C~,,sinz—2(C,~+C~)’Lie group invariancesanalogousto thosederivedinshallow water theoryin refs. [18,19]. It is interest- It is notedat thispoint, by direct appealto (2.26)ing to note that the moment of inertia equation thatQ is governedby a Pinney-typeequation[22](3.17) arisesnaturally in that context. ~+~Q=(l—c,,,—2K)Q—~. (4.9)

The solution of the nonlinearoscillator equation

4. The pulsrodon (4.9) subjectto the initial data

QI,=o=wo, (4.10)In this section,werestrict our attention to the case

Q11_o=vo, (4.11)

~=const=K~—C1,,#2~/ië~. (4.1) . - . . . .

is a nonlinearsuperpositionof linearly independentBy virtue of (3.9) this requires solutionsof the associatedlinear oscillatorequation

0=0, (4.2) ~+~2=0. (4.12)

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Thus,if Q, andQ2 arelinearly independentsolutions by thesolutionQofthenonlinearoscillatorequationof (4.12) suchthat (4.9). Here,thesolutioncorrespondingtogiven ini-

tial elliptical eddygeometryandaxis rateof changeQ, t=0=l:1)o, Q, 1=0=w0, hasbeenpresented.It correspondsto a pulsatingel-

Q2 L=o=O, ~2 ~ (4.13) liptical eddy-edge geometry. The special caseQ=constdeliversthe rodon solution of ref. [8]. In

then the solution of the Pinneyequation(4.9) sub- view of the incorporationof pulsationin the ellipt-ject to the initial data (4.10), (4.11) is

icaleddy-edgewe termour newsolution apuisrodon.Q [Q~+ W—

2(1—C,,,—2K)Qfl~/2 (4.14) The circular eddy-edgecase requiresBN=BS=0

whereWis theWronskianW=Q,£~2—Q2~,.Hence, whence, on direct appeal to the system (2.23)—(2.28), it isseenthat

6N= 6s= 0 togetherwith ~= K.since

The residualmember (2.25) of the systemis theQ, =w

0 cos ~t+2v~sin ~t, Pinneyequation

Q2=csin~t, c~0, (4.15) ~+~Q=(l—K)Q—3,

it is seenthat, in termsof its initial data, which, with initial data (4.10), (4.11) has the

Q= [(cot,cos ~t+2v0 sin ~t)2 solution

+4w,52(1—C,,,—2K)sin2~t]”2. (4.16) Q= [(co~coslt+2v

0 sin ~t)2

Accordingly, +4(1 —K)co62sin2~t]”2 (4.20)

and h0, B, G andGR againgiven in terms of Q via$ Q

2(’r) dr the relations (4.18). This is the pu/sonsolution ofref. [8]. Here,the radiusof the circular eddy-edgeis (—2C,/K)”2Q.

= 2 tan’ (4 [v~ + (1 — C,,, — 2K) /w~I T+ 2v0w0} Ripa [14] hasinvestigatedthe stability of the ro-

—2 tan’ (2v0w0), (4.17) don.The examinationof the stability of the puls-rodonremains.whereT=tan ~t.

Substitutionof theexpressions(4.16) and(4.17)into (4.7) now gives the temporalevolutionof B5,BN, G5, GN. Moreover,the remainingquantitiesh0, AcknowledgementB, G andGR of the system(2.l0)—(2.17) aregivenin terms oft via (4.16) andthe representations The authoracknowledgeswith gratitudesupport

underNatural Sciencesand EngineeringResearch— — Council of CanadaGrantNo. A0879.B K— ,

G G 1 —2. (4.18)= —, R= Q2 References

The original variables u,, v,, 1= 1, 2 anda, b, c are[I JG.T.Csanady,J.Geophys.Res.84 (1979)777.given by [2] TM. Joyce,J.Phys.Oceanogr.14 (1984)936.

= ~G+GN, V~=G~+GR, [3] T.M. Joyceetal.,Nature308 (1984)837.[4] O.B. Brown,D.B. Olson,J.W.BrownandR.H.Evans,Austr.

u2 =G5—GR, 02 =~G—GN, J.Mar. FreshwaterRes. 34(1984)538.

[5J AD. KirwanJr.,W.J.Merrell Jr.,J.K. Lewis, RE. Whitakera=~B+BN, b=B5, c=~B—BN. (4.19) andR. Legeckis,J. Geophys.Res. 89 (1984)3425.

[6] K. Kitano,J.Phys.Oceanogr.4 (1974)670.The expressions (3.6), (3.7) show that the length of [7]R.K. Evanset a!., ReportWoods Hole Oceanogr.Inst.,the semi-axes of the elliptical eddy-edge are driven WoodsHole,MA, USA (1984).

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Volume138, number6,7 PHYSICSLETTERSA 3 July 1989

[81 B. Cushman-Roisin,W.H. Heil andD.Hof, J.Geophys.Res. [15] F.K. Ball, J. Fluid Mech. 17 (1963) 240.90(1985)11756. [16]F.K. Ball,J. FluidMech. 22(1965)529.

[9] S.V. Bazdenkov,N.N. Morosov and OP. Pogutse, Soy. Phys. [171W.C. Thacker,J.FluidMech. 107 (1981)499.Dokl. 32 (1987) 262. [18] C. Rogers,in: Proc.IMACS Meeting,Paris (1988).

[101 B. Cushman-Roisin,J. Phys.Oceanogr.16 (1986) 132. [19] D. Levi, MC. Nucci, C. RogersandP.Winternitz, Preprint,[11] B. Cushman-Roisin,TellusA 39 (1987) 235. CRM UniversitédeMontreal (1988).[12] A.D. KirwanJr.andJ. Liu, in: Proc.mt. Schoolof Physics [20] AD. KirwanJr.,J.Phys.Oceanogr.5(1975) 729.

Enrico Fermi, Varenna,Italy (1988). [21] F. Bowman,Elliptic functions(EnglishUniv. Press,London,[13]W.R.Young,J.FluidMech. 171 (1986) 101. 1953).[14] P. Ripa, J. FluidMech. 183 (1987) 343. [22] E. Pinney,Proc.Am. Math.Soc. 1 (1950)68!.

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Elliptic warm-core theory: The pulsrodon

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