JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. C3, PAGES 5733-5752, MARCH 15, 1997

Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling

John Marshall, Chris Hill, Lev Perelman, and Alistair Adcroft Center for Meteorologyand PhysicalOceanography, Department of Earth, Atmosphericand Planetary Sciences,Massachusetts Institute of Technology,Cambridge

Abstract. Ocean modelsbased on consistenthydrostatic, quasi-hydrostatic, and nonhydrostaticequation setsare formulated and discussed.The quasi-hydrostaticand nonhydrostaticsets are more accuratethan the widely used hydrostaticprimitive equations.Quasi-hydrostatic models relax the precisebalance between gravity and pressuregradient forcesby includingin a consistentmanner cosine-of-latitudeCoriolis termswhich are neglectedin primitive equation models.Nonhydrostatic models employ the full incompressibleNavier Stokesequations; they are required in the studyof small- scalephenomena in the oceanwhich are not in hydrostaticbalance. We outline a solution strategyfor the Navier Stokesmodel on the spherethat performs efficientlyacross the whole range of scalesin the ocean,from the convectivescale to the global scale,and so leadsto a model of great versatility.In the hydrostaticlimit the Navier Stokesmodel involvesno more computationaleffort than those modelswhich assumestrict hydrostatic balanceon all scales.The strategyis illustrated in simulationsof laboratory experimentsin rotating convectionon scalesof a few centimeters,simulations of convectiveand baroclinicinstability of the mixed layer on the 1- to 10-km scale,and simulationsof the global circulation of the ocean.

1. Introduction Indeed, there are many important phenomena in the ocean, for example,wind- and buoyancy-driventurbulence in the sur- The ocean is a stratified fluid on a rotating Earth driven face mixedlayers of the ocean(on scalesL < 1 km), which are from its upper surfaceby patternsof momentum and buoyancy fundamentallynonhydrostatic and so cannot be studied using fluxes.The detailed dynamicsare very accuratelydescribed by hydrostaticmodels. the Navicr Stokes equations.These equations admit, and the In the present study wc outline, discuss,and illustrate the ocean contains,a wide variety of phenomena on a plethora of use of models based on equation sets that are more accurate spacescales and timescales.Modeling of the ocean is a formi- than the HPEs: quasi-hydrostaticmodels (QH), in which the dable challenge;it is a turbulent fluid containingenergetically precisebalance betweengravity and pressuregradient forces is active scalesranging from the global down to order 1-10 km relaxed, and fully nonhydrostaticmodels (NH), in which thc horizontallyand some tens of meters vertically; see Figure 1. incompressibleNavier Stokesequations are employed.Quasi- Important scale interactionsoccur over the entire spectrum. hydrostaticmodels treat the Coriolis force exactly,by including Numerical models of the ocean circulation, and the ocean in a consistentmanner cos (latitude) Coriolis terms that are models used in climate research, are rooted in the Navier conventionallyneglected in the HPEs. These cos (latitude) Stokes equations but employ approximateforms. Most are terms can becomc significant,particularly as the equator is based on the "hydrostatic primitive equations" (HPEs) in approached,and their inclusionendows the model with a com- which the vertical momentum equation is reduced to a state- plete angularmomentum principle. Nonhydrostatic models arc ment of hydrostaticbalance and the •'traditional approxima- required in the study of the smallestscales in the ocean. In tion" is made in which the Coriolis force is treated approxi- principle,of course,NH is also applicableon the largestscales; mately and the shallow atmosphereapproximation is made; we will demonstratethat modelsbased on algorithmsrooted in see section2. On the •'large scale" the terms omitted in the the Navier Stokes equations can be made etficient and used HPEs are generallythought to be small, but on "small scales" with economy even in the hydrostaticregime, leading to a the scalingassumptions implicit in them become increasingly singlealgorithm that can be employedacross the whole range problematical. of scalesdepicted in Figure 1. The global circulationof the oceanand its great wind-driven Our considerationsin this paper are independentof a par- gyres(on scalesL •--1000 km) are very accuratelydescribed by ticular numerical rendition or discretization.The strategy set the HPEs, as are the geostrophiceddies and rings associated out here could be employed in any model. From a single algorithmic base rooted in the Navier Stokes equations, NH, with its hydrodynamicalinstability (L --• 10-100 km). The QH, and HPE models are outlined. In a companion paper HPEs presumablybegin to break down somewherebetween 10 and I km, as the horizontal scale of the motion becomes [Marshallet al., this issue]we describethe details of a finite- volume, incompressibleNavier Stokes model which imple- comparablewith its vertical scale,the "grey area" in Figure 1. ments the ideas set out here. Copyright 1997 by the American GeophysicalUnion. In section 2 we critique the HPEs and review the assump- Paper number 96JC02776. tions made in their derivation, assessingtheir validity across 0148-0227/97/96J C-02776509.00 the range of scalesin the ocean. In section3 we write down the

5733 5734 MARSHALL ET AL.' HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

L ~ 1m I km 10 km 1,000 km 10,000 km which perform efficientlyacross the whole range of scalesin the ocean, from the convectiveto the global scale. Navier Stokesmodels are specificallydesigned for the studyof small- scalephenomenon such as convection.But when deployedto study hydrostaticscales, they need be no more demanding computationallythan hydrostaticmodels. Equation setsbased on more approximateforms, QH and HPE, are readily imple- gravity convectiongeostrophic gyres global waves 'eddies' circulation mentedas special cases of NH. A comparisonof integrationsof HPE, QH, and NH at large scale(1 ø horizontalresolution) givesessentially the samenumerical solutions. The neglectof cosqb Coriolis terms is the mostquestionable assumption made by the HPEs, but we find that their inclusion(in QH and NH) HYDROSTATIC yieldsdifferences in horizontalcurrents in oceangyres of only a fewmillimeters per second.Thus it is clearthat solutions based on the HPEs are not grosslyin error. Nevertheless, modelsbased on QH (orNH) shouldbe preferred in studiesof large-scalephenomenon. NON-HYDRQ...... Finally, we have attemptedhere, as far as is possible,to present an accountwhich does not make strongassumptions Figure 1. A schematicdiagram showingthe range of scales about particular numerical choices. Details of the data- in the ocean. Global/basin-scalephenomenon are fundamen- parallel, finite-volume,incompressible Navier Stokes model tallyhydrostatic; convective processes on the kilometerscale usedhere to illustrateour ideasare givenby Marshall et al. [this are fundamentally nonhydrostatic.Somewhere between the issue].The mappingof the algorithmonto parallel machines, ge0strophicand convectivescales (10 km to 1 km) the hydro- in data-parallel FORTRAN on the Correction Machine staticapproximation breaks down: the "greyarea" in the fig- (CM5) and in the implicitlyparallel languageId on the data- ure. Modelswhich make the hydrostaticapproximation are flow machineMONSOON, is describedby Arvind et al. (A designedfor studyof large-scaleprocesses but are commonly usedat resolutionsthat encroach on thisgrey area (the left- comparisonof implicitly parallel multi-threaded and data- pointingarrow). Nonhydrostaticmodels based on the incom- parallel implementationsof an ocean model based on the pressibleNavier Stokesequations are valid acrossthe whole Navier Stokesequations, submitted to Journalof Parallel and range of scalesbut in oceanographyhave been hitherto used Distributed Computing, 1996) (hereinafter referred to as for processstudies on the convectivescale. We show in this Arvind et al., submittedmanuscript, 1996). paper that the computationaloverhead incurred by employing the unapproximatedequations is slightin the hydrostaticlimit. Thus modelsrooted in the Navier Stokesequations can alsobe 2. Critique of the Hydrostatic Primitive used with economyfor study of the large scale (the right- Equations pointingarrow in thefigure). The HPEs, which assumea precise balance between the pressureand densityfields, are almostaxiomatic to many me- teorologistsand oceanographers.They are widely used in nu- Navier Stokesequations on the sphereand discusshydrostatic, merical weather forecastingand climate simulationsof both quasi-hydrostatic,and nonhydrostaticregimes. Section 4 is the atmosphereand ocean. The terms omitted from the full concernedwith the diagnosisof the pressurefield required to Navier Stokesequations are customarilythought to be smallon ensurethat the evolvingvelocity field remainsnondivergent. In large scales(see Lorenz [1967] and Phillips[1973] for good HPE and QH a two-dimensional(2-D) elliptic equationmust discussions).However, the HPEs precludethe studyof non- be inverted for the surface pressure; in NH a three- hydrostaticsmall-scale phenomena, such as deep convection, dimensional(3-D) ellipticequation must be invertedsubject to the understandingof whichis of greatimportance to climate. Neumannboundary conditions. A strategyis developedfor the The HPEs can alsobe criticizedbecause of their approximate NH 3-D inversionwhich exploitsthe fact that in many (and treatmentof the Coriolisforce which deniesthem a full angu- pe•rhapsmost) cases of interestthe pressurefield is "close"to lar momentumprinciple. Indeed, of the assumptionsmade in one of hydrostaticbalance and canbe usedas the startingpoint their derivation,the neglectof horizontalCoriolis terms seems of an iterativeprocedure. Thus the pressurefield is separated the least comfortable.Only very recently have global atmo- into surface,hydrostatic, and nonhydrostaticcomponents, and sphericmodels been developedwhich relax the hydrostatic each part is treated separately,greatly speedingthe inversion approximation(so-called quasi-hydrostatic (OH) models)and process.In section5, NH, HPE, and QH modelsare illustrated includea full treatmentof the Coriolisforce [see Whiteand in the context of three oceanographicexamples, chosen for Bromley,1995]). We now criticallyreview, for the purposeof their interest:simulations of laboratoryexperiments in rotating designinga model for the accurateprediction and study of convectionusing NH, simulationsof convectiveand baroclinic oceancurrents from kilometer scalesup to the globalscale, the instabilityof themixed layer using NH, andsimulations of the treatment of inertial and Coriolis terms in the HPEs. global circulationof the oceanusing HPE, NH, and QH. This latter experiment was driven by analyzed winds and fluxes 2.1. Hydrostatic Approximation during the period 1985-1995 and the sea surfaceelevation of Let us inquire into the conditionfor the neglectof inertial the model comparedwith the TOPEX/POSEIDON altimeter. accelerationsin the vertical momentum equation. We thus We conclude that models can be constructed based on al- write the inviscidvertical momentum equation in Boussinesq gorithmsrooted in the incompressibleNavier Stokesequations form: MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5735

Dw 1 asp + 6=0 Dt iOref0 Z where 8p denotesa deviation from a hydrostaticallybalanced referencestate at rest, Prefis a standard(constant) value of density,b = -9(t•iO/iOref)is the buoyancy(see Appendix), and D/Dt is the total derivative.The condition for the neglect of Dw/Dt in (1) is that it shouldbe much smaller than b. For simplicitywe assumein the followingscaling that the local time derivative is of the same order as, or smaller than, the advec- tive terms. Considera phenomenonwhich has a characteristichorizon- tal scale L and vertical scale h with horizontal and vertical velocityscales U and W, respectively.The timescaleof a par- ticle of fluid movingthrough the systemis of order L/U and a considerationof the buoyancyequation

Db + N2w = 0 Dth whereN 2 = - (#/Pref)(0 p/OZ) isthe Brunt-Vaisala frequency of the ambientfluid andD/Dt h is the horizontalcomponent of the total derivative,suggests that a typicalvertical velocity will be w • bU/LN 2. Hence Dw/Dt << b if

U 2 Figure 2. Computation of the axial angular momentum of a particle of massm at a radius r from the center of the Earth. L2N2 %% 1 The latitude is 4> and the longitude X. The spherical polar velocities(u, v, w), equation (4), are also indicated. and shouldbe comparedwith the familiar condition for the validity of the hydrostaticapproximation that comparesthe frequencyof a wave motion, to,with N [see Gill, 1982, section 11.9]. In the above, U/L appears in the place of to. If the scale-10 km demandssuch high resolutions).In those places advectivetimescale is short relative to the buoyancyperiod, where the water column is weakly stratified,the condition (2) then nonhydrostaticeffects cannot be neglected.The criterion may not be adequatelysatisfied and the appropriatenessof the can also be usefully expressedin terms of the aspectratio of HPEs may be brought into question. the motion system3' = h/L and the Richardsonnumber R, = Finally, it should be mentioned that the tiPEs have been N2h2/U2. The motionwill be hydrostaticif criticized becausetheir neglect of the time derivative leads to 3'2 them being ill-posed when used with open boundaries: see n: (2) Browninget al. [1990] and Mahadevanet al. [1996a, b]. Instead, these authors recommend the use of a Nit set, but one which where we can call n the nonhydrostaticparameter. is "scaled" to alleviate demands on accuracyin numerically In hydrostaticsystems such as the HPEs, (2) is assumedto evaluatingterms in the vertical momentum equation when the be true at the outset and strict balance between gravity and flow is close to geostrophicand hydrostaticbalance. The ill- verticalpressure gradient is imposed.Then, becauseDw/Dt -= posednessof the HPEs doesnot go unchallenged,however; see 0, w cannotnow be obtainedprognostically from (1) but must Norburyand Cullen [1985]. be diagnosedfrom the continuityequation. If the stratification is strongand the flow weak (large Ri), then the hydrostatic 2.2. Traditional Approximation conditionmay still be a good one even if 3' is not small. For The HPEs make assumptionsother than hydrostaticbal- example,in laboratoryexperiments 3', dictated by the geometry ance. The derivation of the HPEs from the component equa- of the apparatus,is often of order unity; see the simulationof tions of motion involvesa series of approximationsand the the laboratoryexperiment in section5. Nevertheless,the flow neglect of various other small terms leading to what has be- can still be hydrostaticif the fluid is sufficientlystratified. In the come known as the "traditional approximation."The most main thermoclineof the oceanthe Richardsonnumber is large problematicalassumption is the neglectof the "horizontal Co- ("•102-103)and the aspectratio of the motionis small(0.1- riolis terms"(those involving 2D cos&). Their importanceand 0.01) and so (2) is well satisfied.But it clearlybreaks down in the difficultyof consistentlyincluding them in the HPEs have weakly stratifiedconditions on small horizontal scales;see, for been discussedby Phillips[1966, 1973], Vetohis[1968], and Gill example,the studyof the oceanicconvective scale (-1 km) in [1982]. White and Bromley [1995] argue that the horizontal neutral conditionsby Jonesand Marshall [1993]. There nonhy- Coriolis terms may not alwaysbe negligiblefor synoptic-scale drostaticmodels were employedin which, quite naturally,w is motionsin the atmosphere,especially in low latitudes,and they obtainedby steppingforward (1). are now included in the "Unified Model" of the United King- With the increasing power of modern computers, ocean dom Meteorological Otfice. modelsbased on the HPEs are now commonlyemployed with The important issuescan readily be understoodby consid- horizontal resolutionscomparable to the depth of the ocean ering the axial angularmomentum of a particle of fluid of mass (indeed the need to adequatelyresolve the geostrophiceddy m about the rotation axis of the Earth; see Figure 2: 5736 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

DA • gular momentumas a particle movesvertically in the absence Dt = r cos4> {net zonal force on parcel} of externalcouples; see Figure 2. Considerthe balance or Du D-T+ 2Dwcos 0 = 0 DADtx = r cøs4> {rnFx- r COSl qb Pref m03•Op } (3a) Sincew = Dr/Dr, the aboveimplies that for a particlemoving with zonally u + 2Dr cos 0 = const Ax = m{(Dr cos 0 + u)r cos 0} (3b) or the angular momentum and D/Dt the total derivative. 8u = -2DSr cos O Here 4>is the latitude, ,k is the longitude,r is the radius, D is the angular speed of rotation of the Earth, u is the zonal for smallchanges 8. Considernow the consequenceof a zonal speedof the particle, and F;, is the zonal componentof the jet at the equator rising vertically. It acquires a retrograde frictional force per unit mass.By usingthe definitions velocityof 1.2cm s- 1for every100 m it movesvertically, simply by virtue of changingits distancefrom the axis of rotation of D2t the Earth, an effect which is absent from the HPEs. u =rcos4>Dt The mostcompelling reason for the neglectof 2Dw cos4> in the zonal momentumcomponent of the HPEs is that because DO they assumestrict hydrostaticbalance and hence neglect the v = r Dt (4) 2Du cos 4>term in the vertical momentum equation, for en- ergetic consistency2•w cos 4>must also be neglectedin the Dr horizontal momentum equation (5), even though the term becomesuncomfortably large as the equator is approached. For thesereasons, White and Bromley[1995] advocatethe use (3a) and (3b) directlyyield the full zonal momentumequation of a quasi-hydrostaticequation set for global atmospheric Du uw u v tan 4> modelingthat fully representsthe Coriolis force but still ne- + - 2Dr sin 4>+ 2•w cos 4> Dt r r glectsthe Dw/Dt term in (1); this is the QH approximation; see section 3. • Op Perhapsthe key factor determiningwhether 2• cos4> terms + 9rcfrcos 0 0h- Fx (5) are important is the stratification,which can suppressvertical motion if it is strong enough.But N is rather small in large It is clear from this derivation that to ensure that a model has volumesof the ocean (mixed layers,for example,and partic- a full angularmomentum principle, all the termsin (5) mustbe ularly the very deep mixed layers created by wintertime con- retained. As is well known and argued, however, the HPEs vection).It would seemdesirable, then, that anymodel should employa carefullychosen approximation to (5) in whichthe have the facility to representthe 2D cos 4>terms. underlined terms are neglected.The term 2•w cos 4>,how- Finally, one further point must be made. It is clear that the ever, is probablynot alwaysnegligible. full representationof the Coriolis force, includingits horizon- In the tropical oceans,typical vertical velocitiesapproach tal as well as vertical components,is only dynamicallyconsis- the continuitylimit Uh/L. The 2Dw cos 4>term in (5) is then tent if one takes accountof the changingposition of a particle negligiblecompared to Du/Dt if of fluid from the axisof rotation(in (3b), for example,r should not be replacedby a constantreference value). Thus, to be 2Dh cos 4> strictlycorrect, the "shallowatmosphere" approximation must U << 1 (6) also be relaxed if horizontal Coriolis terms are to be included: Supposingh-- 100 mandU = 10 cms J,typical of equa- divisionby r in (5) retained rather than (as assumedin the torial jets, for example,the left-hand side is about 0.1 x cos 0 HPEs) replacedby a mean radiusof the Earth. Nevertheless, several authors have been content to write QH forms in which which would suggestthat its neglect in quantitative(as op- posed to theoretical) study is unacceptable.It is clear, for r is replacedby a; seethe correspondencein Journalof Atmo- example,that the neglectof cos 0 Coriolisterms in the HPEs sphericSciences between Phillips [1968], Veronis[1968], and is much more problematicalthan the neglectof Dw/Dt in the Wangsness[1970], for example.However, it canbe shownthese vertical momentum equation. shallowatmosphere forms lack a potential vorticity conserva- The cos 0 Coriolis terms are intimately related to the full tion law aswell as an angularmomentum principle [see White sphericalgeometry of the oceansand atmosphereand tran- and Bromley,1995]. scendthe approximatequasi-2-D spherical geometry of hydro- In summary,then, the quasi-hydrostaticmodel retainscos 0 staticmodels (where the shallowatmosphere approximation is Coriolis terms in the zonal and vertical momentumequations, made).An implicationof thisclose relationship is that it isvery neglectsDw/Dt in the vertical, and doesnot make the shallow difficult to investigatethe effect of the cos 0 terms on familiar atmosphereapproximation. problemswhich, in Cartesiangeometry, may be treated ana- lytically in their absence.Attempts to do so lead to nonsepa- rable differential equationseven in simple casessuch as wave 3. Incompressible Navier Stokes Equations motion in an isothermalatmosphere at rest [Eckart, 1960]. on the Sphere The physicalmeaning of the 2Dw cos0 term in (5) is readily In view of the considerations outlined in section 2 we de- understandableas representingthe conservationof axial an- velop now an ocean model in which the traditional assump- MARSHALL ET AL.' HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5737 tions implicit in the HPEs can be relaxed, if one should so G• = -v ß Vv desire. The model is rooted in the incompressibleNavier Stokesequations and employsone kernel algorithm;fully non- hydrostatic(NH), quasi-hydrostatic(QH), and hydrostatic vwu2 tan 0} (HPE) setsare outlined.The quasi-hydrostaticmodel includes a full representationof the Coriolis force and has a complete - {2f•u sin O} angular momentum principle. The nonhydrostaticmodel is +F• (15) prognosticin all three componentsof velocityand is designed for the studyof smaller-scalephenomenon. The computational Gw = -v' Vw overheadincurred in solvingthe incompressibleNavier Stokes equationson the sphereis slightprovided that the nonhydro- staticparameter, (2), is small,and so NH can alsobe usedwith + - economyin study of large-scaleflow. (u+ v2)} + 2f•u cos qb 3.1. Equations The state of the ocean at any time is characterizedby the 8p distributionof currentsv, potential temperature T, salinityS, Pref pressurep, and density9. The equationsthat governthe evo- lution of thesefields, obtained by applyingthe lawsof classical +Fw (16) mechanicsand thermodynamicsto the Boussinesqfluid, are, usingheight as the vertical coordinate, On the right-hand side of (14) the first, second,third, and fourth terms are the advection,metric, Coriolis, and forcing/ Motion dissipationterms, respectively.Note that the zonal momentum OVh equationwas derived in section2 by considerationof the axial Ot= G•h- Vhp angular momentum. The G, defined here is identical to the (7) one impliedby (5). On the right-handside of (16) the third and Ow Op : Gw fourth terms are the Coriolisand buoyancyterms, respectively. Ot Or In equations(9) and (10),

Continuity Gr = -V' (vT) V.v=0 (s) Heat +Fr (17) OT Gs= -V'(vS) Ot -Gr (9)

Salt

OS In the above,"grad" (V) and "div" (V ß) operatorsare defined

ot : by, in sphericalcoordinates, Equation of state V -= - (19) p - p(T, S, p) (11) (1rcosOOX' 0100) r OO' Or where v: w): w) V.v -- .osOl{OuO } +.1O(r2w) (20) is the velocityin the zonal, meridional,and vertical directions, Unlike the prognosticvariables u, v, w, T, and S, the pressure respectively,given by (4), field mustbe obtaineddiagnostically. Taking the divergenceof (7) and using the continuityequation (8), lead to a three- p= iOref dimensionalelliptic equationfor the pressure: where 6p is the deviationof the pressurefrom that of a resting, Vp: hydrostaticallybalanced ocean and For a given field of 9, (21) must be invertedfor p subjectto appropriate choice of boundary conditions.This method is are inertial, Coriolis, metric, gravitational, and forcing/ usuallycalled the pressuremethod [Harlow and Welch, 1965; dissipationterms in the zonal, meridional,and vertical direc- !4511iams,1969; Potter, 1976]. tions defined by 3.2. Boundary Conditions Gu = -v' Vu 3.2.1. Velocity and pressure. The configuration of the oceanbasin is definedby its depth H(X, 0) and allowsarbi-

F F trary specificationof the coastline,bottom topography,and _[uw uvtan&} connectedness.We apply the condition of no normal flow - {-2•v sin & + 2•w cos &} through all solid boundaries:the coastsand the bottom. Fur- thermore, the surfaceof the ocean is assumedto be a rigid lid +Fu (14) to filter out high-frequencysurface gravity waves. (We discuss 5738 MARSHALL ET AL.' HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

V h' (HVhP,) = SHy(k,,) where Gv is a modifiedG•: such that G•:. n = 0; the appropriate modificationto the sourcefunction adjacent to the boundaryis rigidlid readily obtainedfrom the divergenceof G•: by settingG•: nor- mal to the boundaryto zero.As is impliedby (24) the modified boundarycondition becomes

n. Vp = 0 (25) The boundary conditionson velocity are as follows. At lat- eral boundaries, v. n = 0, where n is a unit vector normal to the boundary;see Figure 3. For the tangentialcomponent, no-slip (vr = 0) or slip (Ovr/On = 0) conditionscan be employed. Finally, it shouldbe noted that, as discussedby Greshoand Sani [1987] and Dukowiczand Dvinsky[1992], the homoge- neousNeumann condition on pressure(25) is compatiblewith both slip and no-slip boundaryconditions in the continuous equations,but it may not be so when they are discretized[see Marshall et al., this issue]. 3.2.2. Fluxes of heat and salt. At the ocean bottom and V2PN/-/=Vh2P•+Oz 2-•'•vH .... * "• -'"•''"• allrigidboundaries side the diffusive flux of heat and salt is set to zero: Figure 3. A schematicdiagram of an ocean basin showing the irregular geometry:coastlines and islands.The main com- 0 putational challengein the integration of the Navier Stokes Kn•nn (T, S)= 0 (26) model forwardis the inversionof a 3-D ellipticequation for the nonhydrostaticpressure P NH with Neumann boundarycondi- where K,• is a "diffusion"coefficient normal to the boundary. tions.In the hydrostaticmodel, only a 2-D elliptic problemfor At the ocean surface, wind stressand fluxes of heat and salt the surface pressureP s has to be inverted. At the ocean's are prescribed' surface a rigid lid is employed to filter out surface gravity waves. 0 1 z=0 "v (u,v)=- Pref (27) 0 1 z=0 the rigid-lid problem here but refer the reader to Appendix2 K•• (r, s) = Pref-- (•, •) of Marshall et al. [this issue],where we describehow to treat (implicitly)the model ocean'ssurface as a free surface.)Thus where v• and K• are vertical diffusivitiesof momentum and we set heat,respectively, r is the stress applied at thesurface, and •r and•s arefluxes of heatand salt, respectively. v. n = 0 (22) on all boundingsurfaces where n is a vector of unit length 3.3. Hydrostatic, Nonhydrostatic and Quasi- normal to the boundary;see Figure 3. Hydrostatic Forms Equation(22) implies,making use of (7), that 3.3.1. Nonhydrostatic. In the nonhydrostaticmodel, all termsin the incompressibleNavier Stokesequations (7)-(18) n. Vp = n-G•, (23) are retained.A three-dimensionalelliptic equation(24) must presentinginhomogeneous Neumann boundary conditionsto be solvedwith boundaryconditions (25). It is importantto note the elliptic problem (21). that useof the full NH doesnot admit any new "fast"waves in As shown,for example,by Williams[1969], one can exploit to the system;the incompressiblecondition (8) has already classical3-D potential theory and, by introducingan appropri- filtered out acousticmodes. It does, however, ensure that the ately chosen3 function sheet of "sourcecharge," replace the gravitywaves are treated accuratelywith an exact dispersion inhomogenousboundary condition on pressureby a homoge- relation. neousone. (This mathematicaltrick cropsup in a number of It is interestingto note that Miller [1974] and Miller and interestinggeophysical contexts. For example,in potentialvor- White [1984] also "soundproof" their compressibleatmo- ticity invertibility theory it is the origin of Bretherton's[1966] sphericconvection model in thisway; they adoptpressure as a replacementof the (inhomogeneous)temperature boundary vertical coordinate and neglect various terms (which, using conditionsby isothermal(homogeneous) ones with the con- scalingarguments, they argue are small) until their equation comitant use of interior 3 function potential vorticity sheets set is isomorphicwith the incompressibleNavier Stokesequa- adjacentto the boundary.)The sourceterm • in (21) is the tions written in height coordinates:this isomorphismis dis- divergenceof the vector G•,. It can be modifiedjust interior to cussedin somedetail by Bruggeet al. [1991]. In general,atmo- the boundary so that the resulting homogeneous-boundaryspheric models employ the hydrostaticapproximation because problem, with the so-modifiedsource function, is identical to it has the beneficialproperty of eliminatingvertically propa- the original problem;with n.G v = 0 and n- Vp = 0 on the gating acousticwaves. boundary,the following self-consistentbut simpler homoge- NH has the followingenergy equation: nized elliptic problem is obtained: D 1 •d2 V2p= V. •Jv= • (24) Dt {5(u2+ + w2)+ !7z}+ V. (pv)= Q + v. Fv (28) MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5739 where v = (u, v, w) is the three-dimensionalvelocity vector, with no verticalstructure) is equal to zero and after appropri- Q is the buoyancyforcing, F is the forcing/dissipationterm in ate nondimensionalization,all other eigenvaluesgreatly ex- the momentum equations,and p = (•P/Pref'Note that the ceed unity if Az/Ax is small. The Nz - 1 Helmholtz problems pressurework term V ß(pv) vanisheswhen integrated over the are readilysolved because the ), term dominatesin (30). The oceanbasin if, as assumedhere, all boundingsurfaces, includ- 2-D Poissonequation (), = 0 in (30)) presentsthe main com- ing the upper one, are assumedrigid. putationalchallenge. This method of "projectiononto modes" NH has a complete angular momentum principle as ex- is employed,for example,by the nonhydrostaticprocess model pressedin (3). of Bruggeet al. [1991], used for studiesof convectionin flat- 3.3.2. Hydrostatic. In HPE, all the underlined terms in bottomed boxes. One cannot directly employ such a modal (7)-(18) are neglectedand r is replacedby a, the mean radius approachhere, wherewe are concernedwith a geometrywhich of the Earth. The 3-D elliptic problem reducesto a 2-D one is as complexas that of an oceanbasin, because with a nonflat sinceonce the pressureis known at one level (we choosethis bottom the 3-D problemcannot be separatedinto independent level to be the rigid lid at the surface),then it canbe computed 2-D problems;the modes"interact" throughtopography. How- at all other levels from the hydrostaticrelation. An energy ever, it stronglypoints to the advantageof separatingout, as equationanalogous to (28) is obtainedexcept that the contri- far as is possible,the depth-averagedpressure field. butionof w2 to the kineticenergy is absentand, on the right- 4.1.1. Depth-averaged pressure. For an arbitrary func- hand side, only forces on the horizontal component of the tion ½we candefine its vertical average •t" as velocity do work. The hydrostaticmodel has an angularmomentum principle analogousto (3) but with r replacedby a and the axialangular = ½(x, z) dz ) momentumdefined by A•- m{(fia cos 4>+ u)a cos 4>} (29) where H = H(A, O) is the local depth. The vertically averagedgradient operator is then Of course,(29) meansthat the hydrostaticmodel cannotrep- resentthe mechanicscontained in (3). . 1 V/,H (32) 3.3.3. Quasi-hydrostatic. In OH, only the terms under- lined twice in (7)-(18) are neglected,and, simultaneously,the The verticalintegral of (7) is then, usingthe rule (32), shallow atmosphereapproximation is relaxed. Thus all the metric terms must be retained and the full variation of the O(HV"•) •, radial positionof a particlemonitored. OH hasgood energetic 0t = HG•.,.- [V/.,(Hfi")-p(H)V•H] (33) credentials;they are the same as for HPE. Importantly, how- ever, it hasthe sameangular momentum principle as NH, (3). Since there can be no net convergenceof massover the water Again, the 3-D elliptic problem is reducedto a 2-D one. Strict column, balancebetween gravity and vertical pressuregradients is not V;,. (HV"/,) = 0 (34) imposed,however, since the 2Du cos O Coriolis term plays a role in balancingg in (16). At this point, and followingB•an [1969],ocean modelers often introduce a stream function for the depth-averagedflow. In- 4. Hydrostatic, Quasi-Hydrostatic stead, and as argued by Dukowicz et al. [1993], it is advanta- and Nonhydrostatic Algorithms geous to couch the inversionproblem in terms of pressure rather than a streamfunction. The resultingelliptic equationis In HPE and OH the vertical componentof the momentum better behaved becauseH, the local depth of the ocean, ap- equation becomesa diagnosticrelation for the hydrostatic pears in the numerator rather than the denominator of the pressure;the vertical velocityis obtainedfrom knowledgeof coefficientsthat make up the elliptic operator. Furthermore, (u, v) usingthe continuityequation. Instead, in NH, w, just the pressureequation demandsNeumann conditions(equa- like u and v, is obtained prognostically.In each model the tion (25)), whereasthe bounda• conditionson the stream main computationalchallenge lies in findingthe pressurefield. function are Dirichlet and can only be determinedby car•ing In HPE and OH a 2-D elliptic problem must be solvedfor the out line integrals around the boundary.This is a cumbersome pressureat some level surface;in NH the elliptic problem is and (on a parallel computer)costly task. In contrast,the Neu- three dimensional. mann elliptic equationfor the pressure,(24)-(25), occursnat- 4.1. Finding the Pressure Field urally in the incompressibleNavier Stokesproblem. Combining(33) and (34), we obtainour desiredequation for If the ocean had a flat bottom, then (24) could readily be the depth-averagedpressure: solvedby projecting p on to the eigenfunctionsof the 02/01'2 ...... H operatorwith boundarycondition (25) applied at the (flat) V-(Hg,,,, )+ (35) upper and lower boundaries.Then (24) can be written as a set of horizontal Helmholtz equations: where (v, - 4,): 4,) (3o) (I)(H): V. [p(H)V,,H] (36) where the Greek index labels the vertical eigcnfunctionand •, It is now clear that if the ocean has a fiat bottom (H = is the correspondingeigenvalue. We can thusseparate the 3-D constant),then q)(H) = 0 and equation(35) doesnot have problem(24) into a setof•N, independent2-D problems,if one any pressure-dependentterms on the right-handside and can truncatesat N, vertical modes. The advantageof this proce- be solvedunambiguously for •" But if the depthof the basin dure is that only one cigenvalue(corresponding to the mode varies from point to point, one cannot solve for the depth- 5740 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING averagedpressure without knowledge of •(H). We can,how- findPNH from the ellipticequation obtained by substituting ever,make progress by separatingthe pressurefield into hy- (37) in to (7) andnoting, as before, that V ßv = 0 at eachpoint drostatic,nonhydrostatic, and surfacepressure parts. in the fluid: 4.1.2. Surface, hydrostatic and nonhydrostatic pressure. O2pNH Let us write the pressurep as a sumof three terms: V2pNH= V•pN, + OZ2 = b•N, (44a) p(X, &, z) = ps(X, &) + pro-(X,ok, z) + PNH(X,d), z) (37) where The first term,P s, onlyvaries in the horizontaland is inde- b•s,= !7. P,v- 17•(Ps+ Pro,) (44b) pendentof depth.The secondterm is the hydrostaticpressure definedin terms of the weight of water in a verticalcolumn whereQ-v = (Gvh,•,•). abovethe depth z, Equations(44a) and (44b) are solvedwith the boundary conditions

pro-(X,&, z) = - • dz' (38a) 17PNHn = 0 (45) If the flow is close to hydrostaticbalance (see section4.2 where below),then the 3-D inversionconverges rapidly because PNH is thenonly a smallcorrection to the hydrostaticpressure field. The solutionPNH to (43) and (44) doesnot vanishat the • = •/ Pref•--•-P-{(u2+v2)}-211ucosck r (38b)upper surface and so refines the pressureat the lid; it is in this sensethat the Ps obtainedfrom (41) is provisional.In the Note that (38a) and (38b) are a generalizedstatement of hy- interiorof the fluid, nonhydrostaticpressure gradients, VpNH, drostaticbalance, balancing vertical pressure gradients with a drive motion. "reducedgravity" and, in QH, modifiedby Coriolisand metric terms also. The methodof solutionemployed in the HPE, QH, andNH modelsis summarizedin Figure4 below.There is no penaltyin It shouldbe notedthat the pressureat the rigidlid, p(z - 0), hasa contributionfrom both Ps andPNH (by definition,p• = 0, implementingQH overHPE except,of course,some compli- cationthat goeswith the inclusionof cos& Coriolisterms and at z = 0). In the hydrostaticlimit, PNH = 0 andPs is the the relaxationof the shallowatmosphere approximation. But pressureat the rigid lid. this leads to negligibleincrease in computation.In NH, in By substituting(37) into (35), an equationfor P s results: contrast,one additionalelliptic equation, a three-dimensional Vh' (nVhps) = •f Hy(•., (•)) one, mustbe inverted.However, we showthat this "overhead" of the NH model is essentiallynegligible in the hydrostatic + Vhß [pNn(H)VhH] - Vh(Hp/,2 --. ) (39) limit (asthe nonhydrostatic parameter, (2), n --• 0), resultingin a nonhydrostaticalgorithm that, in the hydrostaticlimit, is as where '•])HYis givenby computationallyeconomic as the HPEs. b•HV(X,&)= Vh'(HG .H)_Vh2(Hfi_•H ) 4.2. Navier Stokes Model in the Hydrostatic Limit + Vh' [pHv(H)VhH] (40) It isimportant to understandhow the NH modelperforms in the hydrostatic,geostrophic limit. Accordingly,in the Appen- In HPE and QH the (doubly)underlined terms on the right- dix we nondimensionalizeour model equationsand consider hand sideof (39), whichdepend on the nonhydrostaticpres- the balanceof terms if the flow is closeto one of hydrostatic sure, are set to zero, andP s is found by solving andgeostrophic balance. For clarityand simplicitywe neglect Vh' (HVhps) = b•m-(X,&) (41) horizontalCoriolis terms in our scaleanalysis. There are three importantnondimensional numbers: the Rossbynumber R o = The verticalvelocity is obtainedthrough integration of the U/fL, theRichardson number Ri = N2h2/U 2, andthe aspect continuityequation vertically: ratio of the motion 3/ = h/L. Quasi-geostrophicdynamics occurson the deformationscale, Nh/f, at large R i and small Ro suchthat R•Ro 2 • 1. w = - Vh'vh dz' (42) The nondimensionalform of the momentumand continuity f0 equationsmay be writtenin termsof Ro, Ri, and •, thus(see In NH, instead,w is found by prognosticintegration of the Appendix)' vertical velocity equation: ' 1 O'Vh t t t -•Ot = G'hOTHER + •OO {G' hCORI - 17h(Ps +P,r+qnpk,)} (46) OW_ Ow OpNH (43a) Ot Oz t ow'= O' - op_NH (47) where Ot' • Oz Ow= Gw+ • (43b) OW • 17h'V• + Ro•Oz = 0 (48) Note that in (43a) and (43b), the verticalgradient of the hy- drostaticpressure has been canceled out with •, (38b),render- wherethe prime symbolsindicate nondimensional quantities, ing it suitablefor prognosticintegration; see section 4.2. G•coRI are the sin& Coriolisterms, and G;,OTnE • contain all We solve(41) to givea provisionalsolution for Ps andthen other contributions, from advection, sources,sinks, etc. Here MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5741

z V h. (HV hPs) - ,.5'Hy(•, (p) Put(2,,0, z) - I -g&' 0

HPE NH

V2p•vu- V.Gv -Vh (Ps+ Put)

-- Vh(Ps + Put + P.u ) ODVh• : Gvh - Vh(Ps +Put) O•Vh• =G v•

W----oVh.¾hdZ' O,qt=Gw-• Figure 4. Outline of hydrostatic(HPE), quasi-hydrostatic(OH), and nonhydrostatic(NH) algorithms.

n = P,,vn/P,r, which comparesthe typical magnitude ofp•v, vertical variation is required. (Indeed, even in HPE a p^f, is and P,v, is given by implied and can be deducedfrom w(t) using (47), but P/v, need never be explicitlycalculated if (48) is used directly.)If q •- 0, the verticalvelocity found from (47) yieldsexactly that n--R, (49) which would have been deducedfrom the continuity equation and is just the hydrostaticparameter of the nonrotating prob- hadHPE been used if •i• • 0 (compare(46) and (47) when lem introducedin section2; see (2). Note that in (46) we have q - 0 and•;' • 0 withHPE, (46) and (48) withq : 0). We now illustrate the methods outlined above in three in- introduceda tracer parameter q; in HPE and OH, q = (), and in NH, q = 1. teresting contexts:simulation of rotating convectionin a lab- The elliptic equation for tiaopressure is, taking V/, of (46) oratory experiment(an analogueof open-oceandeep convec- and O/•,z of (47) using(48), tion), simulationof convectionand bareclinic instability in a channel(of relevanceto mixedlayers in the upper ocean),and qn , 1 , a simulationof the global circulation of the ocean in a studyof a•P;"+Oz 2 -,,R• v;,•p,•,,: R• Iv,,. •' /'('• - v;,:(p',+ the basin-scalepatterns of sea surfaceheight variability which are comparedto observationsof surfaceelevation taken by the TOPEX/POSEIDON altimeter. +R,Vj,.Gj...... } + _ az (50)

We can now more clearly identify the nonhydrostatic and 5. Oceanographic Illustrations hydrostaticregimes in this rotating systemwhere we note that The method of solution outlined in section 4 for the HPE, QH, and NH models has been implemented in data-parallel 4.2.1. Nonhydrostatic regime [nm R,,]. In the nonhydro- FORTRAN on a CM5, a massivelyparallel machinehoused at staticregime, horizontal gradientsofp,•z z are important in the the MassachusettsInstitute of Technology(MIT). The model evolution of v;,. It is still advantageousto separate out the has also been articulated in an implicitly parallel language hydrostaticpressure as in (37), but the solutionof the fully 3-D called Id and run on the MIT data-flow machine MONSOON. elliptic problem (50) will be computationallydemanding. A comparison of this implicitly parallel, multithreaded ap- 4.2.2. Transitional regime [n • Re]. The secondregime proach with the single-threadeddata-parallel model is de- is the transitionzone, the grey area in Figure 1, where nonhy- scribedin detail by Arvind et al. (submittedmanuscript, 1996). drostatic effects no longer dominate and the flow is under The numerical formulation of the model is described in detail increasinghydrostatic control. in the companionpaper, Marshall etal. [this issue],to which 4.2.3. Hydrostatic regime [n << Re]. In the hydrostatic the reader is referred for details. Here it is sufficient to note regime,P•s is found by inversionof a 2-D elliptic equation and that the model solvesthe incompressibleNavier Stokesequa- p,•, trivially,from (50) sincethe scaled3-D elliptic operator tions in sphericalgeometry; has rigid-lid and flee-surface op- •2/0Z2. Note that if the tracerparameter q = 0, then tions; has an equation of state appropriate to seawater;em- Vz,p•, vanishesfrom (46) and, accordingly,the elliptic equa- ploys height as a vertical coordinate; handles arbitrarily tion (50) doesindeed collapseto a second-orderordinary dif- complexcoastlines, islands, and bathymetry;employs a finite- ferential equation (ODE) in z. It is interesting to note that volume, predictor-correctornumerical procedure; inverts for p•,, is not zero in the hydrostaticlimit of NH, but only its the pressure field using preconditioned conjugate-gradient 5742 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

the gradientand divergenceoperators, (19) and (20), whichare requiredto step forward the fluid equations,solving the pres- sure Poissonequation at each time step is computationally expensivebecause this involvescommunication between pro- cessors.But as we shall see, the overheadincurred by NH in solving(43a) and (43b) for PN, is not severeprovided the nonhydrostaticparameter n, (49), is sufficientlysmall. This is true even in the highly irregular geometriesof an oceanbasin, providedthat P s is first obtained in a 2-D solve.Thus in the hydrostaticlimit, NH is not more demandingof computational resourcesthan HPE or OH. To invert the 3-D Poissonequa- tion for the pressurefield, a block preconditionedconjugate- Figure 5. Domain decompositionused in the data-parallel gradientalgorithm is employed.The preconditionerused is the model. The schematicdiagram shows our fluid containedin an inverseof 02/0252[see Marshall et al., thisissue], to whichthe oceanbasin decomposed into columnsdistributed over 16 pro- 3-D elliptic operator asymptotesin the hydrostaticlimit. The cessorsof a parallel computer. The thick black lines indicate regionsof the domain assignedto the sameprocessor. The thin preconditioningequations are solved using "lower/upper" lines indicate the number of "volumes" within each subdo- ("LU") decompositionwhich inverts 02/OZ 2 for eachvertical main. columnin the ocean.Thus repeatedmultiplication by the pre- conditionerdoes not involveany communicationbetween pro- cessors.Furthermore, in computingthe matrix-vectorproducts methods; has HPE, NH, and OH options; and is designed neededin each conjugate-gradient(CG) iteration, only near- specificallyto exploit parallel computers. est-neighborcommunication is required becausethe 128 do- The numerical schemeensures that the evolvingvelocities mainsare arrangedso as to take advantageof the direct links be divergencefree by solvingour Poissonequation for the in the CM5's fat-tree based communication network. The pressurewith Neumann boundary conditionsand then using model is presently achievinga sustainedperformance of 3 these pressuresto update the velocities.The equations are Gflops on the 128-nodemachine. discretized using finite-volume methods. Regular volumes The model has been employedto study a number of phe- based on a uniform discretizationof longitude,latitude, and nomenawhose scales range from centimeters(in simulations depth are employed.When they abut the bottom or coast,the of laboratoryexperiments) up to manythousands of kilometers volumesmay take on irregular shapesand be "sculptured"to (in simulationsof the evolutionof the sea surfaceelevation fit the boundary, improving our representationof coastlines over the globe). We briefly present three studiesthat have and topography;see Adcroft et al. [1996]. been recentlycarried out to demonstratethe capabilitiesof the The model lends itself very naturally to parallel computa- model: (1) simulationsof laboratoryexperiments carried out tion. Most of the parallelismis fine-graineddata parallelism, by Jack Whitehead at Woods Hole in a 1 m x 1 m x 30 cm available on the order of the total number of grid cells in the rotatingtank, in which the aspectratio 3' • 1; (2) convection computational domain. The only exception is the diagnostic and baroclinicinstability in a periodic channel,of dimension inversion for the pressurefield which involvesglobal reduc- 50 km x 20 km x 2 km, in which 3' • 0.1; and (3) circulation tions across all the cells of the domain. The model has been of the global ocean on decadal timescalesin a basin with developedin a data-parallelFORTRAN on the 128-nodeCM5 realisticgeography and bathymetry;here 3' • 10-3. Experi- available to us at MIT which enablesone to exploit the data ments 1 and 2 use NH; experiment3 has usedHPE, NH, and locality and regular structure of the model grid. In our ap- OH. The same kernel algorithm,however, is used in all cases. proach the physical domain is decomposedby allocating Model resolutionsand parametersare summarizedin Table 1. equallysized vertical columnsof the oceanto each processing unit. For example, given a typical 200 x 200 x 30 three- 5.1. Laboratory Experiment dimensionalspatial grid representingan oceanbasin, say,it is Figure 6a showsthe distributionat the surfaceof the vertical straightforwardto carve the grid in to 128 domains,each one componentof vorticity 60 s (five rotation periods) after a with roughly 12,000 points in a 20 x 20 • 30 network, each continuoussource of salinificationwas applied over a circular handlingthe computationin that sectorof the oceanfrom the patch of radius 10 cm at the upper surface of a numerical surfaceto the bottom; see Figure 5. The key to the successof simulation of rotating convectioninto a salt-stratifiedfluid. this relatively simple decompositionis its role in reducingthe The laboratoryexperiments are describedby Whiteheadet al. overheadof the potentially costlytask of diagnosingthe pres- [1996]and were usedto explorethe mechanismsat work in the sure field. convectiveoverturning of a stratifiedocean driven by buoyancy Regardlessof the specificsof any discreteapproximation to lossfrom an extendedbut confinedregion at its upper surface.

Table 1. Typical Values of StratificationN and CoriolisParameter f for the Three ExperimentsDiscussed in the Text N, s-• f, s-• L, m h, m U, m s-• Ri -- N2H2/U2 Ro = U/fL 3' = H/L Laboratory 30 6 1-5 x 10-3 10-2 10-2 10-•-10 1 1 Mixedlayer 5 X 10-4 10-4 1-5 x 103 2 X 103 10-2-10-1 10-•-10 10-•-1 0.2-1 Global currents 10 -2 10 -4 5-10 x l0 s 5 x 103 10-•-1 103 10-3 10-3

Also shownare our estimatesof the lengthscales (in the horizontalL andvertical h) and typicalspeeds U associatedwith the phenomenon observedin the numericalexperiments. R i is the Richardsonnumber, R o is the Rossbynumber, and •/is the aspectratio. MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5743

(a) (b) Vertical Vorticity b-1 Salinity ppt z=-6.75em t=60 seca x=50cm t=60 sees -1.0

75

ß -3.0 65 I...... ••"'•"•••-'"•'" ;' • •::•'•--""• g,.-.:!•-'"'-"•:....-,,,.,.,....,....,...••,,,,,.••,,.,•••..•i:•...,....,,,,,,.o•,••.,••.• ...... *'"'" '7'"'• '' '"•'•' '•' "•"" "' "'•'•••-••• •

-5.0

45 -7.0 ß

35 -9.0

25 -11.0

25 35 45 55 65 75 0 2O 4O 60 80 1O0 X (CM) Y (CM)

VerticalVorUcity s-• Salinityppt z=-6.75cm t=120 seca x=50cm t=120 seca

65 0.3 / k•0.2 •'.4.o.::o:.:.:•:'::•.¾.•,-..*:-• ¾ . F•"•':':'-ß ..::•¾"•:':':-.:.:..':-,•'.•.•.}:..,.....&<...., '-':-:-:"-':':', o •'":'•,•:•>.:.•.• •?..• ::-';'• 0

-0.1>- ...... •:-.• ½•?•.t-'i•! ..:•::-":..•ii:½i:i,•,i,::::i::-'. •'•-•:<.-,. •...'..'!..? -0.2 45 0-3 -0.4 •"'"'""•:•:••••• {•':'-'"•-<"'""•-ß-" ½"•.:-' •:•...i...-"•..*.": :'¾/"--?:• •:'•* •: •: r'"'"'"'"'""•;••-:••/ii...... = *' <"• :¾:::'•: -0.5 """'"'"'"'"'""'•'"'"'"••••--'"••iii'-"'-'"'"""'"'...... "• '"""'":•':"''••••!ii•'•'""•:"-- ß ...... ½•:-•:••..--.'.-"•'•'"'"'""•'""•"•:••:-..• - •'ø -0.6

-0.7 25 -0.8 ...... --•.:...... •i,•2,...... '-:.'.':•-•i,,,J - • •.0 25 35 45 55 65 75 0 20 40 60 80 100 x (c•) Y (CM)

VerticalVortieity s-I Salinityppt .....0.8 z=-6.75cm t=lg0 sees x=50cm t=180 seca i:•,•.:...... o:...,.•::¾:::•------;...J-'"....."•.!•.-';•....•*, .x./•... 5:•..?;• ::.q::.¾...•¾•.:....:.o• •::.o.::::::::..•.-.-,....:.-•,:•:•œ½..x:.:• ...... '-----'-'--'-'-'•-.'-:-.••••;•.•------...... ½'"'"••••••'-.:iI• ' -,.o:•...... •...... ::::!•...... - ...... -...... •:•:•:•:•:•:•...... •: •:•:•:• ...... :::sr..o.•.-'.½<.,;.:.>.'.•,.'.-::.'::::::: ...... ::::::¾..:..r:'•'••..<5•;,-: '.:.•:.-¾•g.%-'•:%-.:•' •¾s.-:::.-•[-'•:.q::::-.'-':¾.-¾s'>:-:-....•.•:...... -;•'..x.:;•.:..-•

:•.o., • I'• ...... ""*'"•"'•'"• '- •••••i "•'"'½½'"'-'"'" '•' •'•'•••':•••••iI"':'""•""'•'"•-"••1 I -•.o

>;q..,-'."..;;.J.T•''• ::::::::::::::::::::::::::::::::::: ßZ >.',•.?;•-:g• .•-:•:•'..-'.-::..•rf.-'-,.'.- :•:•-o.• "•"•"'••:•;••:•••[i•:..':::•:•:•:::::•:•:•::,.-•:•-½•"'::':'..---'--•.....'ß.•..=•...... g•.:.-.-'•,.'.-'.-•-.'.-' ';-•i...... :::...... ß...... •.-.•,..-•:::•:...... -?.-::.-ii::.'. r•,;•,..-4..+,;•.j•. •'":'":••••i•., ø.•*•-....-':._..•-'.-'2'•-*.-..-..-::•..-•.: >""'"•",,-.,//."•,...-.•....,'•.•:j•;•--'-'"•i,,.:

,,...•>••j.f,..,..:•]. •2.'..'•-'.i•,',- i,-"•. ß:-•...,>.[?,,',• r.•./...'.::•:,.-.....'$}..-'I••?,':"'"---½'•*...,...... ?•:•:••i ...... o'"$"':•'½';¾--'.-.•.'•...-'.'.."i.•--,-"•.-:-"Ji.:::•-'.-'.-:-."-.m'•-o:o>x::p.•x..-.:.:.:.:,."•-ß...... -- _:.-_'::•_.'•:: '•4•J...o:o.,½•.....:::f•.q..:::..xo:.•-..':o.'....-'•.::::.•. x¾$..::::• :,.:;j..-::.:...?..•o•.•:.-¾::•.::, .: -0.a 2s• •.'----'----"-•"••'""••':•••••••••i:'":::::/...... •-'"'•"''•'"'"'"'•'"':•'":•••••••i'"'"'"•'"'...... '"'"'"•',••*••••,•:•i'"':•-- - - ""•'"'"'"'"'•':'"••::•-"•••."'"" --'.'i.-".-,."-11.0 25 35 45 55 65 75 0 20 40 60 80 100 x (CM) ¾ (CM) Figure 6. (a) Horizontalmaps of the verticalcomponent of vorticityobtained by solutionof NH in the simulationof a laboratoryexperiment in whicha rotatingsalt-stratified fluid in a tank of dimension1 m x 1 m x 30 cm is densiftedover a discat its uppersurface by additionof saltywater. The periodof rotationis 12s, the stratification of the ambient fluid is N -- 0.94 s-1, andthe buoyancy flux is B = 0.3 cm2 s-3. Maps, shownevery five rotation periods, chart the breakup of thecolumn of convectedfluid by baroclinic instability. (b) A verticalsection through the evolvingsalt chimney. Only the top 10cm of the 30 cmdeep salt-stratified fluid is shown. 5744 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

Table 2. NumericalParameters and Measure of ComputationalEffort Required to InvertElliptic Equations in Two Dimensions and Three Dimensions Speed-up Nx,Ny, N z Ax,Ay, A z At 13 V•2p V;2ps V•2p•v. 3-D Net Laboratory 127,127, 19 0.5,0.5, 1.5 0.05 0.1-1 450 250 60 8 5 NH cm s Mixedlayer 200,119, 19 250,250, 100 120 0.1 400 250 38 11 6 NH m s Globe 360,180, 20 105,105, 100 1200 10-4 460 450 1 460 20 H/QH m s Nx,Ny, and Nz are the number ofgrid cells inthe zonal (x), meridional (y),and vertical (z)directions; Ax,Ay, Az are the grid cell dimensions in thesethree directions. In the mixed layer and global simulations the vertical grid spacing Az was, in fact,not constant, but we indicate here themean vertical spacing. The time step employed is At, andthe typical values of thenonhydrostatic parameter n computed from each experimentareindicated. The computational effortrequired toinvert the elliptic problems ismeasured bythe number ofconjugate iterations requiredtoreduce the divergence toacceptably lowvalues. The V •- 2p column shows the number ofiterations required when the elliptic problem issolved inone stage, utilizing only a single3-D inversion. The V•-2ps and V•-2pNH columns record the number ofiterations required toachieve thesame divergence when the elliptic problem issolved ina two-stageprocedure, following the right arrow in Figure 4. The 3-D speed-up isthe ratioof the number of 3-D iterations required by the inverter in the one-stage inversion tothe number of iterations required by the 3-D inverter in thetwo-stage inversion. The net speed-up takes into account the additional work arising from the 2-D inversion. Evidently, the two-stage inversionprocedure isvery effective atreducing the expense ofNH, rendering it competitive with HPE (and QH) in the hydrostatic limit.

The numericalmodel integrated forward the Cartesian form of pie, in the studyof Julienet al. [1996],who report on "direct NH (inwhich the rotation vectors II andg arealigned with one numerical simulations" of convection in a box that resolve another;see the Appendix). The domainis a flat-bottomedbox dynamicallyactive scales of motiondown to the Kolmogorov of dimension1 m x 1 m x 30 cm; in the numericalmodel the scale)because our aim is to developmethods that can be cellsare cubesof side0.5 cm. Key physicalparameters of the employedin domainsas complex as ocean basins; the method integrationare set out in Table1 andnumerical parameters in outlinedhere is equallyefficient in irregulardomains. Table 2; isotropicLaplacian diffusion of saltand momentum Finally,it shouldbe emphasizedthat there is no singlereli- was employedas a parameterizationof sub-grid-scalepro- ablemeasure of computationaleffort, particularly in viewof cesses.In Figure6a we seebaroclinic eddies forming as the the fact that we are runningon a parallelmachine in which convectivechimney breaks up into deformationscale frag- "communication"must be balancedwith "computation."But ments.Figure 6b showsa verticalsection through the evolving the one chosenhere, the number of conjugate-gradientitera- chimney;the convectiveelements can be seeneroding the tionsrequired to invertfor p, does,we feel, givea reliable verticalstratification. The deepeningof the "chimney"is ulti- guide.The number of iterationscan, of course,be reducedby matelyarrested by a modenumber 3 baroclinicinstability. usinga preconditionerwhich is a closerinverse of V2, but this "Hetonic"structures carry the saltyconvected fluid awayfrom maybe at the expenseof enhancedcommunication and so a the discof forcing;see Legg and Marshall [1993]. In thissim- reductionin speed.There is a substantialliterature on parallel ulationthe convectiveprocess is resolved(albeit coarsely) and preconditionersshowing that suitablychosen ones can have nonhydrostaticeffects are important in theoverturning of the less communicationneeds than the conjugate-gradientitera- chimney;the nonhydrostatic parameter, computed as the ratio tion andhence improve efficiency. Our designof precondition- n = PN,/Ps from the evolvingfields, is •0.1-1. ersis discussedin somedetail by Marshall et al. [thisissue]. But Table2 presentsmeasures of the computationaleffort re- the strikingrelative speed-up observed in the 3-D inversion quiredto find the pressurefield in the casewhere the 3-D demonstratedin Table 2 is independentof these consider- ellipticproblem, (24) and (25), is solveddirectly and when, ationsbecause the samepreconditioner is usedin all cases. alternatively,it is foundby a 2-D inversionfor Ps, (41), fol- lowedby a 3-D inversionfor PN--, (44a). We presentthe 5.2. Convective and BarOClinic Instability numberof conjugate-gradientiterations required to reducethe of the Mixed Layer divergencefield to onepart in 10•ø, a valuewhich was found to The Cartesianform of NH hasbeen usedto studyconvective be sufficientlysmall for numericalstability. The computation andbaroclinic instability of the oceanicmixed layer; see Plate per iterationdoes not change;the costof a 3-D iterationis 1. A periodicchannel is employed of dimension30 x 50 km in approximatelyNz timesthat of a 2-D iteration. the horizontal and 2 km in the vertical. The initially resting Table2 clearlyillustrates that the separationof the pressure oceanis uniformlystratified; details can be foundin Table 1. into its constituentparts, equation(37), is very effectiveat The motionis forcedby a steadybuoyancy loss through the sea reducingcomputation, even in thissimulation where nonhy- surface.This coolingis independentof downchannelcoordi- drostaticeffects are important.The numberof 3-D iterationsis nate but increasesacross the channelfollowing a hyperbolic reducedby almostan orderof magnitudeif the surfacepres- tangentvariation. Thus in the southernthird of the channel, sureis "takenout" of the 3-D problem.Moreover, in a hydro- there is weak surfaceforcing, in the northernthird, there is staticcalculation the surfacepressure must be found anyway, fairlyconstant densification equivalent to a heatloss of 800W andso the costin thepressure inversion of NH relativeto HPE m-2, and there is a sharptransition in between.A linear isonly a factorof 4. Instead,had we inverted forp directly,NH equationof stateis specified dependent on temperature alone. would have been more than 30 times slower than HPE. Here The resolutionis sufficientto representgross aspects of the we havenot exploitedany simple geometry (unlike, for exam- convectiveprocess. No convectiveadjustment is employed. MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5745

-f 0 f 2f 3f 11.5 11.6 11.7 11.8 11.9 12

30

10

0 10 20 30 40 50 0 10 20 30 40 50

,, •30

1,o

0 10 20 30 40 50 10 20 30 40 50

30

10

0 '--'-- 0 0 10 20 30 40 50 0 10 20 30 40 50 Alongchannel distance (kin). Alongchannel distance (Inn). Plate 1. A time sequenceof horizontal maps of the (left) vertical componentof vorticity and (right) temperatureat a depth of 65 m in a simulationof an evolvingoceanic mixed layer usingNH. The initially restingfluid (in a 50 km x 20 km x 2 km channel)is cooledvigorously to the north, initiating convection which,over time, evolvesand coexistswith barocliniceddies: (a) day 1; (b) day 6; (c) day 9.

The calculationis of particularinterest because it occursin the grey area in Figure 1, illustratingthe transitionbetween con- vective(unbalanced) and geostrophic(balanced) motion. For the first few daysof integrationa mixedlayer of depthh whereB is the buoyancyflux acrossthe surfaceat time t and N developsaccording to a simple,nonrotating, one-dimensional is the Brunt-Vfiisfilfifrequency. The vertical mixing is facili- law which predictsthe depth of mixing due to the convective tated by upright convection(modified by rotation); See Plate overturning.Namely, la where horizontal maps of vorticity and temperature are 5746 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING plotted at day 1. As a result of the developingdensity gradient the surface elevation predicted by the model for the same acrossthe channel,the flow adjuststo thermal wind balance period.The broad agreementis very encouraging,demonstrat- which becomesbaroclinically unstable. ing not only that the model has fidelity in reproducingbasin- After 6 days a mode six, finite amplitude,baroclinic insta- scalevariability induced by winds but also that the TOPEX/ bility has grown in the channelcenter and is responsiblefor POSEIDON altimeter providesus with a remarkableglobal exchangingwater laterally,from the region of deep mixingto dataset for comparingwith modelsand theory.Integrations of the unconvectedfluid and vice versa (Plate lb). In the north the model with NH instead of HPE, in which the vertical the fine plume-scaleelements can be seendrawing the buoy- velocityis obtainedby prognosticintegration of (43), insteadof ancy from the interior. At later times a field of geostrophic being diagnosedfrom continuity,give indistinguishableresults turbulence evolves to larger scale as the baroclinic waves in thislimit where n --• 0; our chosenpreconditioner is then an "break" laterally (Plate l c). Ultimately, the evolutionof the exactinverse of d2/dz2 andsolves the ODE in oneapplication; whole layer is profoundlyaffected by the lateral flux of buoy- see Table 2. Thus use of NH involvesonly marginally more ancy due to barocliniceddies. The hydrodynamicsat play in computationthan HPE. simulations such as those shown in Plate 1, and its relevance to Figure 8 plots the differencein P s and surface currents oceanic mixed layers, are discussedin detail by Haine and obtained from Pacific integrationsusing HPE and QH at 1ø Marshall [1996]. horizontal resolution. Inclusion in QH of a full treatment of Nonhydrostaticeffects are somewhatless important in this the Coriolis force only leads to small differencesin current calculationthan the laboratoryexperiment above, and, accord- speeds(---1-2 mm s -j) andsurface elevation (---0.1 cm) at this ingly, Table 2 revealsan even greater improvementin perfor- resolution. mance when the pressureseparation is made. Plate 2 repeats the mixed layer calculationbut with HPE, rather than NH. It 6. Conclusions is interestingto observethat HPE attemptsto representthe convectiveoverturning of the fluid columneven though accel- We have criticallyreviewed some of the key assumptions inherent in the hydrostaticprimitive equations(HPEs) and eration terms in the vertical momentumequation are absent. describedand implemented more accurate nonhydrostatic Staticallyunstable columns are overturnedby HPE but at the (NH) and quasi-hydrostatic(QH) formulationsfor studiesof grid scale,and the resultingfield of vertical velocityis up to ocean circulationfrom convectiveto global scale. twice as strongas in NH and much lesssmooth and coherent. Rather than assumehydrostatic balance a priori (the left- This is just as one would expectfrom linear Rayleightheory; pointing arrow in Figure 1), we have retained the material the staticinstability of a columnis more vigorousand occursat derivativeDw/Dt in the vertical momentumequation and de- smaller spatial scalesin hydrostaticcompared to nonhydro- static convection. veloped a model basedon the incompressibleNavier Stokes equations.Such models are designedprimarily for the studyof small-scalephenomena, such as mixed layer physicsand con- 5.3. Large-Scale Global Circulation vectiveprocesses in the laboratoryand the ocean.The pressure The model has been used to simulatethe variability of the field, which ensuresthat evolvingcurrents remain nondiver- surfacepressure field and currentsover the globe during the gent, is found by inversionof a three-dimensionalelliptic op- TOPEX/POSEIDON altimetric mission. The model, extend- erator, the overheadof the nonhydrostaticalgorithm. A strat- ing from 80øSto 80øN at 1ø horizontalresolution, was config- egy has been outlined and illustrated which separatesthe ured with 20 levels in the vertical, rangingfrom 20 m at the pressureinto its constituentparts: Ps, Pz•v, and PNH. As in surfaceto 500 m at the deepestlevel. Full sphericalgeometry HPE and QH a 2-D elliptic problemmust be invertedfor P s; and realistictopography were employed.The model was ini- but in NH, a further 3-D Poissonequation must be invertedfor tializedwith the "Levitus"data set and drivenby climatological PNH. A preconditioneris designedwhich, in the hydrostatic winds for a "spin-up"period of 40 years on 128 nodes of a limit, is an exactintegral of the ellipticoperator and soleads to CM5. The surfacetemperature and salinityfields were relaxed an algorithm that seamlesslymoves from nonhydrostaticto to seasonalLevitus on a monthly timescale and driven by hydrostaticlimits. Moreover, when employedin the hydrostatic analyzedwinds and surface fluxes. A convectiveadjustment limit, the nonhydrostaticmodel is fast, competitivewith the scheme(of the kind describedby Klingeret al. [1996])was used fastestocean climate models in use todaybased on the HPEs. to parameterizeconvection, and the wind stresswas appliedas It is thusideal for the studyof the wholerange of phenomena a bodyforce overthe uppermostlayer of the model.The global presentedin Figure 1 (the right-pointingarrow) and particu- oceanwas then drivenby 12-hourlyanalyzed winds and surface larly those in the grey area. Our main conclusionsare the fluxesof heat and freshwater (obtainedfrom a National Me- following: teorologicalCenter reanalysis) during the periodJanuary 1985 1. Oceanmodels based on consistentequation sets that are until January1995. The winds drive the flow toward Sverdrup more accuratethan the HPEs can be formulatedand efficiently balanceand exciteRossby and Kelvin waveswhich propagate implemented:NH and OH. at their respectivephase speeds through variable stratification, 2. Ocean modelsbased on algorithmsrooted in the incom- bathymetry,and mean flow. pressibleNavier Stokes equationscan be constructedwhich Figure 7 showssurface currentsfrom our simulationusing perform efficientlyacross the whole range of scales,from the HPE; thusonly a 2-D elliptic problemfor the surfacepressure convectiveto the global.Such models are endowedwith great was inverted.Plate 3 comparesthe surfaceelevation observed versatility;their nonhydrostaticcapability renders them suit- from the TOPEX/POSEIDON altimeter during the period able for studyof small-scalephenomenon. When deployedto January21-31, 1992(prepared by Detlef Stammerof MIT), to studyhydrostatic phenomena, they are no more demandingof MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5747

20 .• 20 lO

o 0 10 20 30 40 50 .• 0 10 20 30 40 50 Along channeldistance (km). Along channeldistance (kin).

. .

,•, 500 500

.• 1000 -•1ooo • 1500 •1500

2000 2000 10 20 30 40 50 0 10 20 30 40 50 Along channeldistance (kin). Along channeldistance (km). Vertical velocity (m/s).

-0.15 -0.1 -0.05 0 0.05

0.05 0.05 " o -0.05 -0.05 o -0.1 -0.15ß • -0.1510 20 30 40 50 0 10 20 30 40 50 Along channeldistance (kin). Along channeldistance (km).

Plate 2. On the left the verticalvelocity is plottedobtained by integrationof NH at day6 of the mixedlayer channelcalculation. On the rightthe samefields are plottedbut obtainedby integrationof HPE. The bottom two panelscompare w(x, y - 27 km, z - 400) from (left) NH and (right) HPE alsoat day 6.

computationthan hydrostaticmodels. Moreover, algorithms globalexperiment presented here). Thus it is clear that solu- rooted in NH may offer a number of advantagesover those tions basedon the HPEs are not grosslyin error at least at based on HPE. coarseresolution. Nevertheless, models based on QH (or NH) 3. In large-scaleintegrations (at 1ø horizontalresolution), ought to be preferred. HPE, QH, and NH give essentiallythe samenumerical solu- Finally,we concludeby mentioningsome future directions. tions. The neglect of cos 0 Coriolis terms is the most ques- In additionto the useof the model describedhere as a general- tionableassumption made by the HPEs, but their inclusion(in purposetool for studyof the ocean,one mightanticipate that QH and NH) yieldsdifferences in horizontalcurrents of onlya its nonhydrostaticcapability would render it particularlysuit- few millimetersper second(in the rather coarseresolution able for hydrodynamicalstudies of mixed layersand coastal 5748 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

Plate 3. (top) Surfaceelevation observed from the TOPEX/POSEIDON altimeterduring the period Jan- uary21-31, 1992,relative to a 2-yearmean December 1992 to December1994 (and in whichsteric effects have been removed).(bottom) The surfaceelevation predicted by our globalmodel, driven by 12-hourlyobserved winds over the decade 1985-1995,is also shownfor the same 10-dayperiod. The contourinterval is 4 cm of elevation;yellow regionsare raised abovethe mean, and grey areas are depressed. MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5749

40øN

0 o

40øS

80øS O•øE 100øE 160øW 60øW > 05ß rn$ -• LONGITUDE

20øN

10øN

0 o

• lOøS [.•

20ø$

30ø8

40øS

80øE 100øE 120øE 140øE 160øE > 0.5 rrts -t LONGITUDE

Figure 7. Currents at a depth of 50 m in a numerical simulation of the global ocean using HPE on a 128-node CM5. The model, which has a horizontal resolution of 1ø x 1ø and 20 levels in the vertical, was driven by monthly-meanwinds and climatologicalfluxes of heat and fresh water for a period of 40 years and initialized from the Levitus climatology.We showhere the flow during the springof the thirty-eighthyear of integration.In the global map, every other current vector is plotted; the inset showscurrent vectorsat the resolution of the model.

regionswhere nonhydrostaticprocesses play an importantrole. could be used to representupper ocean processesin conjunc- Height is used as a vertical coordinateby Marshall et al. [this tion with hydrostaticmodels (perhaps formulated isopycnally) issue]; terrain-following-coordinate,nonhydrostatic models to representthe geostrophicinterior of the ocean. could readily be implemented.Isopycnal nonhydrostatic forms One of the avenuesthat the authorsare activelypursuing is would seemto have limited application;isopycnal coordinates the constructionof a sibling atmosphericmodel based on, as are not naturally suited to the study of convectiveprocesses, outlined by Bruggeet al. [1991], the same hydrodynamicalfor- for example. But nonhydrostaticheight-coordinate models mulation as described here. 5750 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

140øE 180' 140øW 1oo,w LONGITUDE • t.50000g-03 Figure 8. Difference in surfacepressure and surfacecurrents in a 1ø x 1ø Pacificintegration resulting from the useof QH rather than HPE. The arrow shownat the bottomof the figurerepresents a currentof 1.5 mm/s; the maximum difference in surface elevation of 0.1 cm is observed in the Kurishio.

Appendix: Incompressible Navier Stokes k is a unit vector directedvertically upward, and f is the Co- in the Hydrostatic, Geostrophic Limit riolis parameter. To simplify our analysis,we have assumedan equation of We derive here the nondimensionalequations used in sec- statein whichthe densityis a linear functionof T and S (and tion 4.2 to identify hydrostaticand nonhydrostaticregimes and independentof p); b is the buoyancy: to study the behavior of the Navier Stokesmodel in the hy- drostatic,geostrophic limit. 8p

We write down the momentum and thermodynamicequa- Pref tions for an incompressibleBoussinesq fluid in Cartesian co- ordinates,nondimensionalize them, and go on to considerthe where the densityis, separatingout a constantreference value balance of terms when the flow is close to hydrostaticand and an ambientstratification po(Z) typicalof the fluid under geostrophicbalance: study:

OYh /9 -• Pref-'l- po(Z) d- ap(X, y, z, t) + Vh(Ps + Pro. + Ps,) + fk x ¾h-- 0 (A1) Ot and

D w OpN H g Opo 5 2 -- Dt + • = 0 (A2) Pref 0 Z

Dhb is the stratification. -- + N2w = 0 (A3) Ot Note that in the above,p = t•p/Prefhas been separatedinto its hydrostatic,nonhydrostatic, and surfacepressure compo- V.v = 0 (A4) nents.Furthermore, the hydrostaticpressure (which satisfies where the relation(Opi_iy/Oz) - b = 0) hasbeen canceled out with gravityin (A2). D 0 The dimensionlessequations are as follows. We scale the ---Vh.V + W Dt OZ variablesthus: % by U, w by W, x by L, z by h, Ps andp/_/y MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING 5751

by P/_/y,pN/_/ by PN/_/,f by F, b by # (p1/Pref), and N 2 by (#/h)(APo/Pref), where p• is a measureof the magnitudeof V;. v; + RoOz-•-= 0 (A4") 8p(x, y, z) and Apo is the changein Po over a depth h. where Setting D/Dt -• (U/L) (D'/Dt'), etc., where the prime symbolsindicate nondimensional parameters, (A1)-(A4) be- come

D'v• is the nonhydrostaticparameter identified in section2.1 and --+Dt' V;(ps +pro,+ npm-/)+ k x v• = 0 used in section 4.2. (Ai')

Acknowledgments. This study was supported by grants from • Dt ' + n PnY Op•vi•Ozt -- 0 (A2')ARPA, ONR, and TEPCO. The model was developedon the CM5 housedin the Laboratoryfor ComputerScience (LCS) at MIT as part of the SCOUT initiative. Much advice and encouragementon com- DID l puter scienceaspects of the projectwas givenby Arvind of LCS. Jacob (A3') Dt' Pl / White of the Departmentof ElectricalEngineering and Computer Scienceat MIT advisedon the solutionof ellipticproblems on parallel domputers.We often consultedAndy White of the UK Meteorological (A4') Otfice and our colleagueJochem Marotzke on the mathematicaland V;. v; + Oz' 0 numerical formulation of the ocean model. where P NH References Adcroft, A., C. Hill, and J. Marshall, On the representationof topog- is the nonhydrostaticparameter. raphyusing shaved cells in heightcoordinate ocean models, Mon. Now let us supposethat the flow is closeto geostrophicand WeatherRev., in press, 1996. Bretherton,F. P., Critical layer instabilityin baroclinicflows, Q. J. R. hydrostaticbalance: Meteorol. Soc., 92, 325-334, 1966. geostrophic Browning,G. L., W. R. Holland, H.-O. Kreiss, and S. J. Worley, An PHV FL 1 accuratehyperbolic system for approximatelyhydrostatic and in- : (AS) compressibleflows, Dyn. Atmos. Oceans,14, 303-332, 1990. U2 - U Ro Brugge, R., H. L. Jones, and J. C. Marshall, Non-hydrostaticocean hydrostatic modelingfor studiesof open-oceandeep convection,in Deep Con- vectionand Deep WaterFormation in the Oceans,Elsevier Oceanogr. plgh Set., vol. 57, pp. 325-340, Elsevier Sci., New York, 1991. PH¾= (A6) JOref Bryan, K., A numerical model for the study of the circulation of the world ocean,J. Cornput.Phys., 4, 347-376, 1969. Continuity(A4') togetherwith geostrophyimplies that Dukowicz, J. K., and A. S. Dvinsky, Approximate factorization as a high order splittingfor the implicit incompressibleflow equations, WL J. Cornput.Phys., 102, 336-347, 1992. U h Ro (A7) Dukowicz, J. K., R. D. Smith, and R. C. Malone, A reformulation and implementationof the Bryan-Cox-Semtnerocean model on the con- andsince (D'/Dt)b' • N'2w ' • 1, (A3') impliesthat nection machine, J. Atmos. Oceanic Technol., 10, 195-208, 1993. Eckart, C., TheHydrodynamics of Oceansand Atmospheres, Pergamon, WL 91 Tarrytown, N.Y., 1960. (A8) Gill, A., Atmosphere-OceanDynamics, Academic, San Diego, Calif., U h A9o 1982. Combining(A5), (A6), (A7), and (A8), we deducethat Gresho, P.M., and R. Sani, On pressureboundary conditions for the incompressibleNavier-Stokes equations, Int. J. Numer.Methods Flu- R,Ro2 = 1 (A9) ids, 7, 1111-1145, 1987. Haine, T., and J. Marshall, Gravitational, symmetricand baroclinic where R i is the Richardsonnumber of the flow given by instabilityof the oceanicmixed layer, J. Phys.Oceanogr., in press, 1996. N2h2 gApoh c• Harlow, F. H., and J. E. Welch, Time dependedviscous flow, Phys. Rl = U2-- •= /9refU U2 (A10) Fluids, 8, 2182-2193, 1965. Jones, H., and J. Marshall, Convection with rotation in a neutral and c is the speed of internal gravitywaves. ocean:A studyof open-oceandeep convection,J. Phys.Oceanogr., The result (A9) is well known and defines the quasi- 23, 1009-1039, 1993. geostrophicregime; if R i of the large-scaleflow is large, then Julien, K., S. Legg, J. McWilliams, and J. Werne, Rapidly rotating turbulent Raleigh-Benardconvection, J. Fluid Mech., in press,1996. the R o is small and the flow is quasi-geostrophic. Klinger, B., J. Marshall, and U. Send, Representationof convective We may now write the set (Ai')-(A4') in terms of R i, R o, plumes by vertical adjustment,J. Geophys.Res., 101(C8), 18,175- and 3, = h/L; the aspectratio of the motion thus 18,182, 1996. Legg, S., and J. Marshall, A heton model of the spreadingphase of D'v• 1 open-ocean deep convection,J. Phys. Oceanogr.,23, 1041-1056, D + •oo[V;(pk +p•/r + npk,)+ f'k x v;]= 0 (AI") 1993. Lorenz, E. N., The nature and theory of the generalcirculation of the D ' w' Op•vs atmosphere,WMO 218, TP 115, World Meteorol. Organ., Geneva, Switzerland, 1967. Dr' + Oz' = 0 (A2") Mahadevan,A., J. Oliger, and R. Street, A non-hydrostaticmesoscale D•D • oceanbasin model, I, Well-posednessand scaling,J. Phys.Oceanogr., -- + N' 2w' = 0 (A3") in press,1996a. Dr' Mahadevan,A., J. Oliger, and R. Street, A non-hydrostaticmesoscale 5752 MARSHALL ET AL.: HYDROSTATIC AND NONHYDROSTATIC OCEAN MODELING

ocean model, II, Numerical implement•ition,J. Phys.Oceanogr., in Wangsness,R. K., Commentson "Theequation of motionfor a shal- press,1996b. low rotating atmosphereand the 'traditional approximation,'"J. Marshall,J., A. Adcroft,C. Hill, L. Perelman,and C. Heisey,A Atmos. Sci., 27, 504-506, 1970. finite-volume,incompressible Navier Stokesmodel for studiesof the White, A. A., and R. A. Bromley,Dynamically consistent, quasi- oceanon parallelcomputers, J. Geophys.Res., this issue. hydrostaticequations for globalmodels with a completerepresen- Miller, M. J., On the use of pressureas vertical coordinatein modeling tation of the Coriolis foice, Q. J. R. Meteorol. Soc., 121, 399-418, convection, Q. J. R. Met•orol. Soc., 100, 155-162, 1974. i995. Miller,M. J., andA. A. White,On thenon-hydrostatic equations in Whitehead,J. A., J. Marshall,and G. Hufford,Localized convection in pressureand sigmacoordinates, Q. J. R. Meteorol.Soc., 110, 515- rotatingstratified fluid, J. Geophys.Res., 101(Cll), 25,705-25,721, 533, 1984. t996. Norbury,J., and M. J.P. Cullen,A note on the propertiesof the Williams, G. P., Numerical integration of the three-dimensional hydrostaticequations of motion, Q. J. R. Meteorol. Soc., 111, 1135- NavierStokes equations for incompressibleflow, J. Fluid'Mech., 37, 1137, 1985. 727-750,1969. Phillips, N. A., The equation of motion for a shallowrotating atmo- sphereand the 'traditional approximation,'J. Atmos. Sci., 23, 626- A. Adcroft,C. Hill, J. Marshall,and L. Perelman,Center for Me- 628, 1966. teorologyand PhysicalOceanography, Department of Earth,Atmo- Phillips,N. A., Reply to G. Veronis'scomments on Phillips(1966), J. sphericand Planetary Sciences, Massachusetts Institu te ofTechnOlo•, Atmos. Phys.,23, 626-628, 1968. Building 54, Room 1526, Cambridge, MA 02139. (e-mail: Phillips,N. A., Principlesof large-scalenumerical weather prediction, [email protected]) in DynamicMeteorology, edited by P. Morel, pp. 1-96, D. Reidel, Norwell, Mass., 1973. Potter, D., ComputationalPhysics, John Wiley, New York, 1976. Veronis, G., Commentson Phillips's(1966) proposedsimplification of the equationof motion for a shallowrotating atmosphere,J. Atmos. (ReceivedAugust 31, 1995;revised July 31, 1996; Sci., 25, 1154-1155, 1968. acceptedAugust 21, 1996.)