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 approximate forms. 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