where an upper dot represents a Lagrangian time derivative. (at constant X). ... split interface have the same X value but now different posi- ... dicular to the x axis of the one-dimensional model (so that ... one now assumes that two frontal lips have formed and that ..... positive constants, implies the following upper bound on d.
JOURNALOF GEOPHYSICALRESEARCH,VOL. 90, NO. C1, PAGES889-894,JANUARY20, 1985
Conditionsfor InterfaceSurfacing, Upper Boundson Extent of Ventilation,
and Formationof BottomLensesAboveTopography BENOIT CUSHMAN-ROISIN
Mesoscale Air-SeaInteraction Group,TheFloridaStateUniversity, Tallahassee
Conditions underwhichtheinterface of a rotating, layered modelcansurface andformgeostrophic frontsare investigated. It is shownhow a simpleone-dimensional, reduced-gravity model,despiteits oversimplicity,can lead to surfacingcriteria and upper boundson the extentof the ventilationarea
createdby the separation of two newlyformedfronts.Firsta no-surfacing theoremis established under restrictive conditions. Then,by relaxingconditions oneby one,wind,coast,andtopography effectsare investigated andfront propertiesderived.The presentsearchfor frontogenetical criteriaand ventilation upperbounds, whichperseis a novelapproach to frontalstudies, isaimedat complementing theexisting frontalstabilitystudies by offeringadditionalinformation on frontproperties. Extension to multilayered and two-dimensional modelsis brieflyanticipated, callingfor additionalwork beforemoregeneral criteria can be formulated.
1.
INTRODUCTION
Reduced-gravity modelscontinueto play an importantrole inphysical oceanography as a practica]too] for the studyof near-surface dynamics, ranging from small-scale (coastal
plumes, Garvine[1984]),to mesoscale (isolatededdies,Nof [1983]),and to large-scaleproblems(Indian Ocean,Luther andO'Brien[1984]). Despite their restrictedvertical representation, thesemodelshave the advantageof combiningadequate horizontaldescriptionwith simpleequations. Quiterecently,reduced-gravitymodelshave beenappliedto studies of geostrophicfronts, for which the f plane dynamics havea lengthscaleon the order of the radius of deformation {typically 10-50 km). In such studiesthe interface at the foot ofthe active layer is allowed to surface at one or more locations. At eachsurfacinglocation the horizontal densitydiscontinuity representsa front. Stern [1980'1,and later Stern et al.[1982],showedthat the coastalfront formingthe offshore
"limit of a coastalintrusioncan undergoa varietyof regimes, suchas dispersion, wave steepening,and blocking.On the otherhandthe open-oceanfront wherethe interfaceextends fromthefront to infinity,a modelof the Gulf Streamand Kuroshio fronts,is stableto all small-amplitude perturbations
[Pddor,1983a].The double-frontstructure(band of light water) is unstable to small-amplitude, longwaveperturbations
[Grijfiths etal.,1982],whereas Paldor[1983b]concluded that .the single-front alonga coastis stableto the sameperturhations, arguing thatthedouble-front instability is caused by
fronts, the problem of frontal formation remains of interest and requiresfurther investigation.In reduced-gravitymodels, frontscan appear spontaneouslyas the interfacesurfacesand disappearwhen two pools of light water collide.Oceanicexamplesincludethe formationof a coastalupwellingfront; the questionof surfacingunder intense,interfacial, internal wave activity;and the eddysheddingand reabsorptionin the vicinity of a front-current structuresuch as the Gulf Stream. The presentwork is intendedto lay a new framework adapted to such investigations and to answer partially some of the questionsrelated to frontal appearancesand resappearances. It is thus a complementto existingevolution and stability studies.
As a first step a one-dimensionalframework (cross-frontal coordinateand time only) is retained. The approach focuses on the surfacingproblemsand on the propertiesof the newly formedfronts as a resultfrom their origin. First, the question of surfacing far away from a coast and in the absenceof forcingis examined.The answershowsthat, under quite general conditions,suchsurfacingcannot occur.Then, by relaxing theseconditionsone by one, various surfacingproblems are investigated,and ventilationconsiderationsare developed.In particular an upper bound on the extent of ventilation (contact of the bottom layer with the atmosphere)is derived in some
cases.
2.
No-SURFACING
THEOREM
In this section a theorem establishesthat, in the absenceof
monanee. While thesestudiesmentionedabove rely on the
hypothesis ofuniform potential vorticity, Killworth andStern boundaries,of forcing,and of prerequisitediscontinuities,sur[t982] demonstrated thatthesingle frontin theproximity ofa facing cannot occur spontaneouslyin the one-dimensional mastis unstableto infinitesimal-amplitude perturbations context.Beyond standingas a warning,this theoremis useful
when thepotential vorticityincreases towardthe coast.All
in determiningunder which conditionsfront formation can
these studies demonstrate thatthereduced-gravity model isan occur.To exploresuchpossibilities,conditionswill be relaxed adequate toolforfrontalproblems. It simplifies thedynamics one by one, and a seriesof frontal formation problemswill toaminimum, whileit stillallowsformanyfrontalbehaviors follow.
In the time-dependentstudy of a one-dimensionalreducedgravity system,one may imaginethat, due to remotelyforced, Beside studies of theevolution of oneor several existing internalgravity wave activity on the interface,the latter may riseand eventuallyhit the surface,thus forming a pair of new frontsthat subsequently separate.The questionaddressedby Copyright 1985 bytheAmerican Geophysical Union. the presenttheoremis whethersuchspontaneoussurfacingis possible.The answer is best obtained by elementary LaPaper number 4C0989. 0148.0227/85/004C.0989505.00 grangianconsiderations.
ofpractical concern.
889
890
CUSHMAN-ROISIN: INTERFACE SURFACING
The one-dimensional, nonlinear,inviscid,rotating,reducedgravity model is
u, + uux-fv = -ghx
(1)
vt + uvx+ fu = 0
(2)
h• + uh•,+ hu• = 0
(3)
In summary, two adjacentwatercolumnsin a regionof continuous flowproperties canneverbe separated by a finite
amount, andnonewsetof frontscanspontaneously emerge.
The negativeaspectof thisconclusion turnsinto positive conclusionsas one realizesthat a necessary conditionfor fronto-
genesis is theviolation of at leastonecondition required by the abovetheorem.Relaxationof conditionsone by onewill
of frontogenetical cases, forwhichthesame apwhere g is the reduced-gravity,and h the local water column yielda series height (h = 0 at a front edge,h > 0 elsewhere).From (2) it proach as used in this theorem will provide variousconstraintsfor the new front(s). immediatelyfollowsthat the semigeostrophic coordinate 3.
x = x +-
f
(4)
is conservedby particles.Its x derivativeis proportionalto the total vorticity. Since, in the most general case,this quantity may locally vanishand changesign(althoughmostlikely not), X cannot be chosenas a particle tag. However, it is obvious that, in any region of continuousflow properties,X is a smooth function, and thus two neighboring water columns have, in the limit of their distancegoing to zero, the sameX value. Expressingv asf(Xx) from (4) and replacingin (1), one obtains
SURFACING UNDER WIND STRESSCURL
A first example that does not fall under the conditionsof the above no-surfacingtheorem is the caseof an initial dis-
continuity in theLagrangian coordinate X. Suchdiscontinuity will, of course,occur if two initially separatedlight-water pools(with independent X distributions) havecollided.TheX discontinuityresultingfrom the collisionis the seedfor future separationandfront reformation.Anotherdiscontinuity in the particle-conserved quantitycan arisefrom the pastactionofa surfacestresswith a discontinuity.Examplesare the stress discontinuities at the water surface when this latter is covered
on one sideby ice or an oil slick and exposedon the other.
• + f2(x- X)= -ah•
(5)
where an upper dot representsa Lagrangian time derivative (at constantX). Equation (5) is hybrid with a Lagrangianleft side and a Eulerian right side.The exact Lagrangianequation
In the presenceof a surfacestressin the directionperpendicular to the x axis of the one-dimensionalmodel (so that Ekman currentsflow in the x direction),the governingequation (2) becomes
is
v• + uv,,+ fu =-
5i+f2(x--X)= --gXx x 1(•xx)
where z is the surface stressin the v direction and p is the
upper-layerdensity.The time integrationof the equationfollowing a particleyields
+f2(X-x)=g(HX•,),,(7)
whereH(X) is suchthat hdx= HdX and alsosuchthatf/H is the particle-dependent potentialvorticity. In the eventthat, in a region of previouslycontinuousflow properties, the interfacehassurfaced, the two lipsof the new, split interfacehavethe sameX valuebut now differentpositions, say x_ and x+ for the particle on the left (smallerx value)and on the right (greaterx value),respectively (Figure 1). Moreover,on the left the interfacerisesto the surface, while on the right it deepenswith increasingx' hxl- -< 0
h•i+ > 0
(I2)
(6)
while the correspondingEulerianequationis
• ,--•
ph
(8)
and thus
.i•++ f2(x+ -- X) • 0
(9)
5i_ + f2(x_ -- X) >__ 0
(10)
[l'q- f2d 0)at d = 0,whichviolates distance d.Theinequalities governing theseparation dare (11). For a more rigorousproof, see the appendix(for = 0).
•'+ f2d< f2AX
d>_0
(16!:
CUSI-!MAN-ROISlN:INTERFACESURFACING
891
magnitudeof the upwelled-frontexcursion.Suchupperbound on the extent of ventilation in coastal upwelling has never beenformulated.While this upper-boundcriterion appliesto the time-dependentproblemin general(includinginertial oscillations),it will be verified that the particular steady state solutionensuingwind relaxationmeetsthe criterion.Notwithstandingits dependenceon various additional physicalprocesses, the time-dependentnumericalsolution of de Szoeke ventilated area and Richmanalso closelymeetsthe upper-boundcriterion. This suggeststhat the presentanalysisretains the dynamics Fig.1. Sketch offrontal lipsafterseparation. Thetwoparticles at x+andx_wereonce adjacent particles ofa continuous upper layer. controllingthe frontal excursion. X•
X+
X
Once winds have ceased,the water column initially along the coastal wall has been subjectedto a total, wind stress impulsenoted I. Sincethis particlewas initially at the coast This system admits no solution for AX < 0. Henceif thesur- and at rest(x -- 0, v = 0) (referto Figure 2), X0 is zero and X face stress actingon therightparticleis smallerthanthe stress equalsl/f by virtue of (13). Its position (x) after the event acting ontheleftparticle(AI < 0), thetwo particles cannot eitheris x = 0 (still alongthe wall) or is governedby
The region between thetwofrontsisa ventilation area.
separate, andnofrontpairisformed. Thisisobvious, since the
resulting Ekmancurrentsare convergent, and the interface
5i+ f2(x- X)= -•7•xx x> 0
(19)
tendsto subsiderather than rise. On the other hand, for &X'>_0 themathematical development in the appendixshows In thislatter, surfacingcase,c3h/t•x is positivefor that particle, and therefore thatfinitevaluescanexistbut that theseareboundedby
5!+ f2x __0), one can apply the result of the appendixand
write the upper bound
R is the radiusof deformation (gH)•/2/f,and A is a constantof integrationto be determinedby imposingthe boundarycondition on the particle closestto shore. For this particle the quantity v +fx, initially zero, has been augmentedby the wind impulseI during the stressevent and has remainedconstant thereafter,so that in its final position the particle is marked by
a
all.,, +-P
-
-
+-p
+
(28)
provided that the bracketedquantity is positive,otherwise d = 0, and no frontal pair can be formed.As a conclusion the interfacecan intersectthe bottom only if the topographic slopeand/or the pressuregradient above the layer decrease as v + fx = I (24) x increases. To renderthis generalconclusionmore physical, it Two casesare possible.If the particle is still along the coast, is bestto separatethe topographyand pressureeffects. In the absenceof an ambient pressuregradientvariation x = O,v(x = O)= I by (24),A = l(H/g) •/'- by (23),and h(x= O)= H- l(H/g)TMalsoby (23).Sinceh > 0 for all particles is (P•,- = Px+), fronts can be formed only if the topographyis a physicalrequirement,the particle is still alongthe coastonly convex(H•,• < 0, as a bump) and, moreover,is of suffident if I < (gH)•/•. If theparticleis no longeralongthecoastbut at curvature.Indeed, (28) implies that the slope differenceover d oughtto be not lessthand multiplied an offshore distance, say a, surfacing implies h(x = a)= 0 or thefrontalseparation
A = H exp (a/R) by (23),v(x= a) -- (gH)•/• alsoby (23),and (gH)•/2 -{-fa-= I by (24). Sinceonly positivevaluesof a are
by f2/2g, i.e.,that the curvaturenot onlymustbe downward butin excess of f2/2g (Figure3).Finally,if thebottomslope is physicallyacceptable,this secondpossibilityis realizedonly if discontinuous (a good approximationto the shelfbreak, [1983]),the rightsideof (28)isnot I > (gH)•/2. Overall,the extentof separationfrom the coast, HsuehandCushman-Roisin and thus the extent of ventilation between the coast and the implicitlydependenton d, and an absoluteupperboundo.n upwelledfront,will not exceedI/f- R, a valuelessthan 2I/f, the extentof separationcan be stated which thus meetsthe previouslystated upper bound. Finally, it is worth noting in passingthat as the wind impulse I in-
2g
creases fromzeroto overits criticalvalue(gH)•/2,the steady saturation value. In other words, for strong wind eventsI-I >
whereZXs isthediscontinuity of bottomslopeat thebreak. In the last casetreatedby Hsuehand Cushman-Roisin [1983, Figure12],in whicha detached bottomlenshasformed, the
(gH)•/2] a firstportionof theimpulseis usedto accelerate the
distanceseparatingthis lensfrom the body of coldwaterre-
state longshorevelocity maximum (for the particle closestto
shore)increases fromzeroto (gH)TMand thenremainsat that
maining ontheshelfamounts to onefifthoftheupper bound (29).Sincetwolayersareactivein themodelof Hsueh aM thecriterion (29)derived froma reduceddisplace the upwelledfrontoffshorein an Ekmanfashion.This Cushman-Roisin,
fluid in the wind direction but only up to a saturation speed, while the remaining contribution of the impulse is used to
is another reason why no anticipation of the displacement gravitymodel,althoughnot strictlyapplicable,seemsto proscalecould be statedat the onsetof the problem. videa correctorderof magnitude of separation thatonemay 5.
INTERSECTION WITH BOTTOM TOPOGRAPHY
expect.
In thepresence ofauniform bottom slope (H•_= H•+)and of a varying upperpressure gradient (p,,_• Px+),inequality thattwobottom fronts canformif ananticyclonic surfacingtheoremof section2 is the one of bottom topogra- (28)implies Another case which does not fall in the scope of the no-
phy. Indeed,the reduced-gravity modelcan be turnedupside shear existsin the water column above the bottom layer,
CUSH:MAN-ROISIN: INTERFACE SURFACING
893
leastoneroot will havea positivereal part, and the particles can becornerapidlyseparatedby a finite distance.One concludesthat a necessary but not sufficient conditionfor frontal formationor eddyshedding is theviolationof at leastoneof the inequalities in (33) and (34). At one dimension, (34) is alwaysmet,and violationof (33) [h=,< _f2/g] is required. However,this corresponds to the presence of an interfacial trough,whichis obviously not suitableto frontogenesis. One • X_ X+ hasthusrecoveredthe no-surfacing theoremof section2. At two dimensions a domefeaturemeetsall threeinequalitiesin Fig.3. Sketch offrontal separation along a concave-downward topography. Separation canoccur only fora downward curvature in (33)and(34),whilea troughfeatureis obviouslyunsuitable to excess ofa criticalvalue. frontogenesis. It remainsthat,withinthe presentassumptions, onlya saddlefeature,whichalwaysviolatesthe secondinwhereas anuppercyclonic shearprevents theinterface from equalityin (34),maygiveriseto finiteseparationbetweentwo reaching thebottom. adjacentparticles.Eddy shedding, whichtakesplacewhen a saddle point in the height field surfaces, i.e.,whena frontalline 6. RECAPITULATION ANI5EXTENSION meetsitself,is possible, while ventilationby domesurfacing ,
Theabovemathematical developments weredevotedto and formation of a new,closedfrontal line is not. frontal formation by interface surfacing in one-dimensional,Furtherinvestigations are necessary to determinewhether
reduced-gravity models. Themainresults canbesummarized upperboundson the separation distance between eddyand
asfollows. Underquitegeneral conditions, butin theabsence motherfront can be establishedand whetherthe presenceof ofcoast, forcing, andtopography, it is demonstrated that forcing, coast,or topography canproduceventilation through
frontogenesis cannot occur, regardless of theintensity of any
formation of a new frontal line. It also remains to recast the
possible, such asin thecase ofmorethanoneactive layerin
conservationarguments.
'internal-wave activityon theinterface. A necessary condition abovesketchyargument into a rigorousmathematical frameforfrontogenesis is thustheviolation of at leastoneof the work. coMitions required by the theorem. In the presence of a Finally,it is worthnotingthat in thisentireworkno concoastal wall,spatia!ly varyingsurface forcing, andconvexto- siderationwasbasedon the continuityequation,whichironipography, fronts canbeformed, andthetheory predicts upper callyenoughis theonethatpredicts interface heightchanges. bounds ontheextentof separation between frontandwall or In other words,surfacing, or the absenceof it, are strongly between two fronts.More one-dimensional applications are regulatedby dynamical constraints rather than by massthevertical.
APPENDIX Although theresults gathered herecanbe directly useful, criteria of widespread applicability shouldoriginate from a In this appendix, it is demonstrated that the inequality no-dimensional theory.The generalization of the present problem mathematical developments fromoneto twodimensions isfar from obvious, butsome anticipative andsketchy arguments d'+ f2d _ 0 (A1)
arepresented briefly.
a(t)>_0
At twodimensions, if two neighboringparticlesare separat-
(A2)
edbythedistance x alongthex axisandy alongthey axis, where(A1)and(A2)holdat all times,wherean upperdot andif theparticles aresufficiently closesuchthatx andy are represents a timederivative, andwheref2 andAX aretwo infinitesimal quantities, thesedistances aregoverned by positive constants, implies thefollowing upperbound ond 5i--f• = -gh•x - gh•y (30) d(t)< 2AX (A3)
y +f• = -ghx,x- gh•,y
(31) Afterthemathematical proof,a physical interpretation based onananalogy withtheharmonic oscillator isalsoproposed. where thehx•,h,,•,andh•yrepresent thesecond derivatives of For the purpose of the mathematical demonstration it is theheight fieldbetween the two particles and whereall the conditions of the no-surfacing theorem(section2) are met. Assuming for the sakeof the followingargumentthat these
usefulto introducethe unknowntime-dependent functionq(t)
such that
derivatives which characterizethe ambient state remain con-
a'+ f2d = faAX - q(t) (A4) stant asoneperturbsthe particles, onecansolvethe system unknown, thefunction q(t)isconstrained by(A1)to (30)-t31). Thesolutionin x andy behaves in timeasexp(20 Although be nonnegative with2 givenby ' q(t)>_0 (A5)
• -- -«(f2 + ah• + ah•) _+«[(f. + ah=+ ah.)2+
_
by (A2).The general (32) andis furtherbut indirectlyconstrained
Allroots willbepurelyimaginary if andonlyif thetwofollowing inequalities aremet
f2 + gh•,•+ gh• >_. 0
(33)
_(f2 + gh• + ah.)2< 4g2(h• 2- h•xhyy)_< 0 (34) If either or bothof theaboveinequalities is not satisfied, at
solutionof (A4)for arbitraryq(t)is
d(t) =AX +Acos fi+Bsin fi-f•q(z) sin f(t-z)dz(A6) whereA and B are constantsof integration.
Undertheassumption that,at sometimetx,d exceeds the value2AX, (A6) provides
894
Cos•N-RoIsIN:I•^cœ
A cosft•+ B sinill --
q(x)sinf(t• - •) dx> AX
(A7)
and the earliervalueat t x - x/f is givenby
SUm•,CING
amplitudeof the oscillations if it actsprimarilyduringthe decreasing half of the cycle(decreasing d). But sinced must remainnonnegative at all times,the amplitudecannotexceed AX, and evenif the forcewere turnedoff duringthe next increasing half of the cycle,the oscillatorcouldnot bounce backbeyond2AX.
d(t•-•) =AX +Acos (ft•-•c)+B sin (ft•-•c) -
q0:)sin [f(tl - z)- •r] dr
Acknowledgments. The authoris indebtedto RobertO. Reidfor keysuggestions onpapercontent andemphasis. He is alsogreatful to
Jame• J. O'Brien and Germana Peggionfor stimulatingdiscussion
Thisresearch wassponsored by the Officeof Naval Research andis
contribution205 of the GeophysicalFluid DynamicsInstituteat The
=AX--IAcosfi•+bsinfi• deSzoeke, R.A.,andJ.G. Richman, Onwind-driven mixed layers --•/•q(•) sin f(t• -z)dt] tl Florida State University.
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--
q(z)sin f(t• -- •) dt
• - •lf
Useof(A7)anda change ofvariable in thelastintegral yield
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current alongthecoastof a rotating fluid,J. FluidMech., 123,
certain amplitude prescribed bytheinitialconditions, butnot
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M. E.,J.Whitehead, andB.L. Hua,Theintrusion ofa density . (A1)would correspond to the•imple harmonic oscillator, and Stern,
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inequality signin (A1)is'physically equivalent to adding a forceacting ontheoscillator thatisarbitrary butpushing the oscillatortowardsmalld values.This forcecanincrease the
(Received May 24,!984; accepted June20, 1984.)
