A General Pressure Gradient Formulation for Ocean Models. Part I: Scheme Design and Diagnostic Analysis

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1 DECEMBER 1998 SONG 313 A General Preure Gradient Formulation for Oean Model. Part I: Sheme Deign and Diagnoti Analyi Y. TONY SONG Earth and Spae Siene Diviion, Jet Propulion Laboratory, California Intitute of Tehnology, Paadena, California (Manuript reeived 30 September 1997, in final form 1 Deember 1997) ABSTRACT A Jaobian formulation of the preure gradient fore for ue in model with topography-following oordinate i propoed. It an be ued in onjuntion with any vertial oordinate ytem and i eaily implemented. Vertial variation in the preure gradient are epreed in term of a vertial integral of the Jaobian of denity and depth with repet to the vertial omputational oordinate. Finite differene approimation are made on the denity field, onitent with pieewie linear and ontinuou field, and aurate preure gradient are obtained by vertially integrating the direte Jaobian from ea urfae. Two direte heme are derived and eamined in detail: the firt uing tandard entered differening in the generalized vertial oordinate and the eond uing a vertial weighting uh that the finite differene are entered with repet to the Carteian z oordinate. Both heme ahieve eond-order auray for any vertial oordinate ytem and are ignifiantly more aurate than onventional heme baed on etimating the preure gradient by finite differening a previouly determined preure field. The tandard Jaobian formulation i ontruted to give eat preure gradient reult, independent of the bottom topography, if the buoyany field varie bilinearly with horizontal poition,, and the generalized vertial oordinate,, over eah grid ell. Similarly, the weighted Jaobian heme i deigned to ahieve eat reult, when the buoyany field varie linearly with z and arbitrarily with, that i, b (,z) b 0 () b 1 ()z. When horizontal reolution annot be made fine enough to avoid hydrotati inoniteny, error an be ubtantially redued by the hoie of an appropriate vertial oordinate. Tet with horizontally uniform, vertially varying, and with horizontally and vertially varying buoyany field how that the tandard Jaobian formulation ahieve uperior reult when the ondition for hydrotati oniteny i atified, but when oare horizontal reolution aue thi ondition to be trongly violated, the weighted Jaobian may give uperior reult. 1. Introdution Topography-following oordinate ytem are gaining popularity in numerial model beaue they implify apet of the omputation by mapping the varying topography into a regular domain, and they an be ued to better reolve the urfae and bottom layer of the oean. For thee reaon, the igma () oordinate ytem, whih i the linear funtion of bottom topography propoed by Phillip (1957) (ee appendi A), i now ued widely in both atmopheri and oeani modeling. Several generalized topography-following oordinate ytem have reently been propoed for improving reolution in the urfae and bottom boundary layer of oean model. Eample inlude the hybrid oordinate of Gerde (1993) and the -oordinate of Song and Haidvogel (1994). Numerou oean irulation model have Correponding author addre: Dr. Y. Tony Song, Earth and Spae Siene Diviion, Jet Propulion Laboratory, California Intitute of Tehnology, Paadena, CA ong@paifi.jpl.naa.gov been developed baed on topography-following oordinate, inluding the Prineton Oean Model (Blumberg and Mellor 1987), the Lamont-Doherty Oean Atmophere Model (Zebiak and Cane 1987), and the S- Coordinate Rutger Univerity Model (Song and Haidvogel 1994). Appliation have been made to a broad range of oeanographi problem, inluding etuarine, oatal, and large-ale oean irulation problem (e.g., Oey et al. 1985; Glenn et al. 1996; Spall and Robinon 1990; Mellor and Ezer 1991). Unfortunately, model uing topography-following oordinate have uffered from error in the horizontal omponent of the preure gradient over teep topography. The oure of thi error i eaily undertood. For eample, in oordinate, the -omponent of the preure gradient fore i determined by the um of two term; that i, p p p h, (1.1) h z where z/h. The firt term on the right involve the variation of preure along a ontant -urfae and the eond involve the uual vertial variation of preure Amerian Meteorologial Soiety

2 314 MONTHLY WEATHER REVIEW VOLUME 16 Near teep topography thee term are large, omparable in magnitude, and typially oppoite in ign. In uh ae, a mall error in omputing either term an reult in a large error in the total horizontal preure gradient fore. Thi problem wa firt realized by Smagorinky et al. (1967). Later, Janjić (1977) and Meinger (198) pointed out an undeirable feature of the preure gradient alulation in -oordinate, whih i often referred to a hydrotati inoniteny (ee etion 3a for further diuion). More reently, Haney (1991) ha foued the attention of the oean modeling ommunity on the preure gradient error aoiated with the ue of topography-following oordinate. Steep topography play an important role in oean dynami and partiular are i required to redue the poibility of eriou error when uing topography-following oordinate in numerial model. Over the pat three deade, meteorologit and oeanographer have put a great deal of effort into the development of aurate and effiient numerial method for ue in uh model. Thee effort an be divided into the following four ategorie. 1) Vertial interpolation method: Thi i a method of interpolating denity bak to z level to alulate the preure gradient fore. Speial are i required to avoid numerial error, whih an aue eriou problem ine integral propertie are not guaranteed. One partiular problem i that etrapolation i often required when dealing with the highet and lowet level of the model over teep topography. In addition, thi method would be very otly if the interpolation were required at every time tep and every grid point, partiularly if the level were allowed to be time dependent a in free urfae model. Diuion of thi method an be found in Mahrer (1984) and Fortunato and Baptita (1996). ) Subtrat referene tate: Another tehnique i to formulate the preure gradient fore in term of deviation from a uitably hoen referene tate (z) (Gary 1973). Thi tehnique i imple to implement and ha proven ueful in limited-area oean model, where the departure of the denity from the referene tate i relatively mall. However, it may be of le help in large-ale model or in long-time integration where the departure may not be mall. Referene to other paper diuing thi method an be found in Batteen (1988). 3) Higher-order method: The ue of higher-order numerial heme to etimate the preure gradient term ha alo been propoed to minimize error. Bekmann and Haidvogel (1993) ue the petral method to redue trunation error baed on a Taylor erie epanion of z-bae funtion, and MCalpin (1994) introdue a fourth-order approimation for the horizontal derivative of preure along level [the firt term on the right ide of Eq. (1.1)]. Suh high-order method hould yield a more aurate preure gradient fore. However, thi approah fail to ahieve ignifiant improvement in ome ae, uh a the ae of trongly tratified flow reported by Bekmann and Haidvogel (1993). 4) Retaining integral propertie: Arakawa and Suarez (1983) and Arakawa and Konor (1996) emphaize the requirement for the direte formulation to retain important integral propertie of the ontinuou equation. Sine error in the direte equation annot be eliminated ompletely, ertain integral propertie hould be atified to avoid the gradual development of large error, perhap due to puriou oure and/ or ink of total ma, energy, or vortiity. Baed on the formulation (1.1), they deign ome direte heme that onerve a variety of important integral propertie. Clearly, the bet preure gradient formulation hould minimize trunation error while imultaneouly retaining important integral propertie. From thi brief review of the work aoiated with the -oordinate ytem, it i lear that there i great onern about the preure gradient formulation. A more generalized vertial oordinate model are developed (e.g., Zhu et al. 199; Gerde 1993; Song and Haidvogel 1994; Arakawa and Konor 1996), two quetion mut be addreed: 1) I there a generalized preure gradient formulation that i uitable for all uh model, and ) How well doe any partiular formulation deal with the interation of buoyany variation and teep topography? The primary goal of thi tudy i to introdue a preure gradient formulation that i uitable for ue with generalized topography-following oordinate ytem. Mot effort to improve the preure gradient formulation have been baed on (1.1) or imilar epreion where the gradient i applied after the preure ha been determined on urfae. One eample i the Arakawa and Suarez (1983) formulation for atmopheri modeling and it numerou etenion in oean modeling. A novel feature of the preent formulation i that it i baed on integrating a Jaobian of and the vertial oordinate z with repet to the vertial omputational oordinate; that i, it i baed on the relation 0 p p g z z d, (1.) z z 0 where, are the generalized topography-following oordinate and z i the poition of the free urfae. Note that the gradient i applied on the denity before the integration i done. Equation (1.1) and (1.) are analytially equivalent, but their diretized form may differ. Baed on the new formulation, we deign numerial heme that minimize trunation error and retain ome important integral propertie of the ontinuou equation. Conervation of momentum, total energy, and bottom preure torque are onidered in Song and Wright (1998, hereafter Part II). The eondary goal of thi tudy are to eamine the

3 DECEMBER 1998 SONG 315 performane of the propoed heme in oean modeling appliation and to provide guidane for their ue. In thi paper, we eamine the formulation of the preure gradient fore in two tep. Firt, we analytially eamine the trunation error and onider the impliation for hydrotati inoniteny. Seond, we eamine the preure gradient error in two et of diagnoti alulation, the firt with realiti vertial variation but horizontally uniform denity, and the eond with both vertial and horizontal variation in denity. Reult are ompared with analytial olution in order to evaluate the effetivene of the vertial weighting heme and the ue of nonuniform vertial reolution. In a ompanion paper (Part II), we onentrate on direte oniteny and the performane of the new formulation in realiti, prognoti oean modeling appliation.. The preure gradient formulation In thi etion, we derive the Jaobian form of the preure gradient fore in both analytial and direte form. Without lo of generality, we retrit attention to two dimenion, and z. 0 p p z b z b d. (.3) * * z z Clearly, vertial variation in the horizontal preure gradient are imply given by an integral of the Jaobian, z b z b J(z, b). (.4) It hould be emphaized that the formulation in term of a Jaobian i ignifiant ine it i learly independent of the partiular form of the vertial oordinate. A eample, in Carteian oordinate z and J(z, b) b/) b/*) z ;inoordinate (z )/ H and b b H J(z, b) H, where H h ; and in iopynal oordinate and g z J(z, b). 0 a. Analytial formulation Let *, z, t be the Carteian oordinate ytem (or z ytem) and let,, t be the generalized topographyfollowing oordinate ytem (or ytem). A inglevalued monotoni relationhip between z and i aumed uh that z z(,, t) 1 0. Some partiular eample, inluding Phillip (1957) ytem, Song and Haidvogel (1994) -ytem, and the iopynal oordinate ytem, are given in the appendi. With the hydrotati approimation, and uing p to repreent the preure divided by the referene denity, 0, the horizontal preure gradient in the * diretion of the momentum equation i given by p p b dz, (.1) * * * z z z z where (, y, t) i the ea urfae elevation and b g/ 0 i the buoyany. The ) z -ymbol emphaize that the derivative i arried out with z held ontant, otherwie with held ontant. Uing the hain rule we have b b b z, * z z and fatoring out z/, then rearranging term give b z b z b. (.) * z z Subtituting into Eq. (.1) we obtain the Jaobian form of the preure gradient: b. Direte formulation The preure gradient formulation diued here an be ued with eentially any numerial heme. We have hoen to make ue of an Arakawa C-grid (Arakawa and Lamb 1977) in the horizontal and a taggered grid [grid 4 a in Lelie and Purer (199)] in the vertial (Fig. 1). The taggered vertial grid, a reviewed by Lelie and Purer (199), ha the attrative feature that the variable u,, and are taggered relative to p and (or ṡ, the vertial veloity in the oordinate) o that integral quadrature of the ontinuity and hydrotati equation i eaily implemented uing the natural ontrol volume approah. To derive direte heme for the horizontal omponent of the preure gradient, we will ue the following differening and averaging operator: b b b, i1 i b b b, k1 k 1 b (bi b i1), 1 b (bk b k1), b bk b k1, 1 k k1 b b b ), b b k b k1, (.5) where indie i and k repreent direte loation in the horizontal,, and vertial,, diretion, repetively.

4 316 MONTHLY WEATHER REVIEW VOLUME 16 K k i1/,k1/ kk PX PX J, (.6) where J i the direte form of J(z, b)d d, whih will be diued net. Here PX i the value of the preure differene at the urfae, and k 1 and K orrepond to the bottom and top level, repetively. If a taggered vertial grid i ued a hown in Fig. 1, the top level k K i a half-level and the Jaobian i omputed imilarly to the other level (ee below), but uing appropriate urfae boundary ondition to etimate derivative at the upper level. FIG. 1. The omputational tenil of the Arakawa C-grid in the horizontal, and a taggered grid in the vertial. The open irle are the loation of and, and the quare are the loation of u. The olid dot indiate the evaluation poition for J i1/,k1/. The uperript and indiate averaging in thee diretion, and the ubript indiate averaging over the loal tenil to give a mean value, a illutrated in Fig. 1. An overbar alway orrepond to a imple-entered mean. The weighting fator, and are inluded to allow u to hooe the vertial level at whih the Jaobian i evaluated. A tilde i ued to indiate an optimal hoie for thee oeffiient (diued below). If the Jaobian i omputed at the tenil enter with repet to and, the preure differene aro the grid ell at the kth level an be obtained by integrating from the urfae: 1) THE STANDARD JACOBIAN The implet direte analog of the Jaobian i the eond-order entral differene heme, J z b z b. (.7) We refer to thi heme a the tandard Jaobian ine it ue tandard entered differene in the -oordinate ytem. It i noteworthy that thi heme orrepond to fitting the bilinear form b b b b ( ) ( ) b ( )( ) (.8) over eah grid ell of the model, reulting in a welldefined, ontinuou approimation to the denity field over the entire domain. In pratie, the finite-differene approimation for and it derivative are alo onitent with being onitently repreented by a bilinear funtion of and z over eah grid ell [ee (.16)]. Thu, if the denity ould atually be repreented by a bilinear funtion of and over eah grid ell, then eah term in (.7), and hene alo the preure gradient reult, would be eat. Thi heme an be eaily implemented in any numerial model with any deired trething of the vertial oordinate. In the ae of the z ytem, the eond term in (.7) vanihe and it lead to the uual Carteian formulation. In the ae of oordinate, it give a heme equivalent to that diued by Mellor et al. (1994) with a uniform grid. In the ae of an iopynal-ytem, the firt term in (.7) vanihe and it give the preure gradient on iopynal urfae. A bilinear approimation in and i probably near optimal for fitting buoyany variation aoiated with plume on ontinental lope, but thi truture will not be optimal in all ae. Indeed, gravity and rotational effet ombine to aue mot feature (in partiular, thoe aoiated with eddie and large-ale urrent) in the oean to have a mall vertial to horizontal apet ratio. Hene, even after removing the domain-averaged vertial buoyany profile mot of the remaining buoy-

5 DECEMBER 1998 SONG 317 any anomalie will be approimately aligned with the horizontal. To better deal with uh buoyany anomalie, we now onider an alternative formulation deigned to give improved reult when the buoyany variation are well approimated by a bilinear funtion of and z over eah grid ell. ) THE WEIGHTED JACOBIAN In thi etion, we determine the optimal hoie for the weighting fator for the partiular ae when the buoyany varie bilinearly with and the depth oordinate, z. Before doing o we reord the following relationhip ine they prove ueful in the algebrai manipulation: 1 b b b, i1 1 b b b, i 1 b b b, k1 1 b b b, k b b b, k1 b b b. (.9) k In general, the vertially weighted Jaobian i given by: J z (bk b k1) ( z z ) b, (.10) k k1 where and (1 ) are the weighting fator. The tandard heme, entered in the trethed oordinate, orrepond to ½. We now determine uh that eat preure differene reult are obtained for a buoyany diturbane of the general form b(, z) b 0 () b 1 ()z. In thi ae, the eat preure differene at depth z i given by: K PXk PX [b0 z b 1]z, (.11) kk where z (z 1 z z 3 z 4 )/4, and the z i are the z level of the orner of the ontrol volume. Note that the vertial variation in the preure differene aro a ell depend only on the denitie along the idewall of the olumn onidered, and not on the interior value. Thu, preure differene will depend on the differene in b 0 and b 1 aro the ell and not on detail of the horizontal buoyany variation within the ell, eatly a for the uual Carteian oordinate ytem. Firt, onider b b 0 (). In thi ae, b 0 () 0, and J z / b (), 0 whih give the eat preure differene a required in (.11). Note that thi plae no retrition on the weighting fator. Now onider b b 1 ()z. In thi ae, J z { [b 1()z] k [b 1()z] k1} { z z } b ()z k k1 1 [ ] 1 1 z b b z b b z 1 b 1 b1 zi1,k1 [ 1 1 i1,k 1 1 i,k 1 1 i,k1 1 b b z 1 [(zi1,k z i,k) (zi1,k1 z i,k1)] [ ] 1 1 b 1 b1 zi1 b 1 b1 z i, whih, after ome traightforward algebra, may be rewritten a [ ] 1 J z z (z)z b 1. (.1) 4 For oniteny with (.11), we require that [ ] 1 z z z z b1 z z b 1. (.13) 4 Clearly thi ondition plae no retrition on the value of if b 1 0. For a buoyany field that varie linearly with z, but i independent of horizontal poition, preure gradient will be aurately repreented for any weighting fator, inluding the uniform weighting, 1. Thu, there i no obviou advantage of any partiular hoie of weighting for the ae of a horizontally uniform buoyany field, whih i often ued a a tet ae. Equation (.13) how that our weighting i determined to properly allow for horizontal hange in the vertial variation of the buoyany field. Uing thi ondition and the relation (.5), ome algebra give where (z ) (z)zk1 4(z ) (z) 1, (.14) ]

6 318 MONTHLY WEATHER REVIEW VOLUME 16 z (z). (.15) 4 z z i1 i Clearly, our weighted Jaobian formulation orrepond to evaluation of the Jaobian at and k k1 ( k k1 )/ ( k1 k ). Below, we how that thi orrepond to the poition, z z in the, z oordinate ytem that i, the Jaobian i entered in the z-oordinate ytem. Firt, we epre in term of the Carteian oordinate and z, a ( ) (z z ) z ( )(z z ). (.16) z Note that the point (, z )in, z pae i equivalent to the point (, )in, pae. Defining i uh that ( i, i ) orrepond to the point ( i, z i ), we have the relation (z z ) (z z ), z z (z z ) (z z ), z z (z z ) (z z ), z z (z z ) (z z ). (.17) z z Averaging thee equation give z, (.18) 4 z where (.19) 4 Now, from (.15) we have z (z) 4 z i1 z i z 4 z z i i1 z, (.0) 4 z o that (.18) an be rewritten a. (.1) That i, give the frational hange in required to hift from the midpoint in pae to the midpoint in z pae. Sine the weighted Jaobian orrepond to evaluation of the Jaobian at (i.e., z z ), thi heme i effetively entered in z pae rather than in pae, a for the tandard Jaobian. Thi i learly onitent with the fat that the heme ha been deigned to give eat reult when the buoyany field varie linearly with z (rather than ) over eah grid ell. Thi i the only differene between the tandard and weighted Jaobian formulation. Note that the hoie of given by (.15) i independent of the buoyany field: if the rigid-lid approimation i made, it need to be alulated only at the beginning of the model run. Even with a free urfae, the rigid-lid approimation to might be ued in ome appliation with little lo of auray. Alo note that for any uniform grid of parallelogram (inluding retangle), ( z) 0, o ½ in thi ae, and the weighted Jaobian redue to the tandard Jaobian, a epeted. The two heme differ only in the ae of a nonuniform grid. 3. Analytial error analyi Haney (1991) ytematially analyze the error aoiated with omputing the preure gradient fore over teep topography in -oordinate oean model. Hi analyi i baed on the fat that a preure gradient heme hould not generate any fore if the urfae preure i ontant and iopynal are horizontal that i, b b(z). In thi ae, any nonzero value omputed by the heme i due to trunation error. In pratie, a referene tate b(z) i often removed from the buoyany

7 DECEMBER 1998 SONG 319 field and only the perturbation b b(, y, z, t) b(z) i ued in omputing the preure gradient. The perturbation field b(, y, z, t) learly doe not atify the above ondition of horizontal uniformity. Further, we have already pointed out that the weighting fator in the weighted Jaobian heme i determined by horizontal variation in the vertial tratifiation, and thi effet i learly not addreed through the onideration of a horizontally uniform buoyany field. Neverthele, thi peial ae i of interet ine large-ale buoyany variation will generally reult in there being vertial truture in the horizontal mean buoyany perturbation over the loal tenil. Further, reult for thi peial ae help to reveal potential problem with topography-following oordinate ytem and partiular advantage of different numerial heme. Haney diue four apet of the error that our in the onventional -oordinate formulation of (1.1): R Error aoiated with a buoyany field that i independent of z. In thi ae, he how that the numerial form ommonly ued i eat. R Error aoiated with iopynal perturbation that are a linear funtion of z that i, b b 0 N 0 z. For thi ae, he find that the error i proportional to K k k k k. k1 Thi error i zero only for a uniform vertial grid. R The (o-alled) hydrotati inoniteny error. R Error aoiated with the buoyany profile orreponding to the firt three barolini Roby mode. The firt two kind of diturbane error hould vanih eatly in model of eond-order auray. Unfortunately, for the ae of nonuniform vertial grid paing the auray of the heme will typially redue to firt order in the grid paing (Chen and Beardley 1995). A noted by Haney (1991), if the igma grid paing i maller near the urfae and larger at depth, k will be negative in the upper oean and poitive in the deep oean. The reulting erroneou preure gradient will tend to produe a geotrophi flow along the iobath with hallow water to the right in the upper layer and to the left in the lower layer. We know from the previou etion [ee (.13) and the aoiated diuion] that the new formulation introdued here give eat repreentation of the preure gradient fore if the buoyany i horizontally uniform and varie linearly with z in the vertial diretion. In addition, by hooing to be an appropriately trethed oordinate, urfae and bottom layer an be well reolved without introduing nonuniform level, o eond-order auray in the vertial grid paing i retained. Thi i an important advantage of uing the preent formulation in ombination with generalized oordinate. We net analytially onider trunation error and hydrotati inoniteny. a. Trunation error and hydrotati inoniteny Reearher have known about the o-alled hydrotati inoniteny aoiated with the -oordinate ytem for a long time. In meteorology, Roueau and Pham (1971), Janji (1977), and Meinger (198) diu problem aoiated with violation of the ondition h. (3.1) h If thi inequality i not atified, then an error our in the etimation of the horizontal preure gradient that doe not tend to zero a the vertial reolution i inreaed, a problem whih i ommonly referred to a hydrotati inoniteny. Haney (1991) invetigate the problem of hydrotati inoniteny in ome detail within an oeanographi ontet. Mellor et al. (1994) emphaize that the preure gradient error in their formulation i not numerially divergent: intead, it i proportional to the differene between two term that dereae a the quare of the vertial and horizontal grid element ize, repetively. Their Eq. (7) learly reveal the oure of the error minimum a a funtion of vertial reolution, whih Haney (1991) diue. Thi minimum i the reult of anellation of trunation error aoiated with finite vertial and horizontal reolution: neither of thee error atually inreae beyond the point where the minimum error our. From the derivation given in the previou etion, we know that the tandard formulation give eat reult if the buoyany varie bilinearly with and over eah grid ell. If the buoyany field atually varie linearly with, but nonlinearly with, then a bilinear funtion of and over eah grid ell beome a better approimation to the real buoyany field a the vertial reolution i improved, and we epet thi heme to onverge to the orret anwer a the vertial reolution i improved. Thu, for thee peial buoyany profile, the tandard Jaobian formulation ha a lear advantage. On the other hand, if the real buoyany field varie linearly with and nonlinearly with z, then inreaing the vertial reolution in the -oordinate ytem will not neearily give improved reult uing either the tandard or weighted Jaobian formulation. Thi i imply a onequene of the fat that reduing the inrement in will not ignifiantly redue the range of z value within a ell if the ell i near the bottom over teep bottom topography. To give quantitative reult on hydrotati inoniteny, we now derive the trunation error aoiated with our two heme for the peial ae of a buoyany field, whih i a quadrati funtion of z alone. For the peial ae of horizontal iopynal, any vertial variation in the preure differene aro a ell mut be due to trunation error. Epanding b(z) around an arbitrary vertial level, z, give

8 30 MONTHLY WEATHER REVIEW VOLUME 16 b(z) b b b [z z ] [z z ] z z higher order term, (3.) and inerting thi epreion into the Jaobian heme, we obtain (after oniderable algebra) the trunation error in the horizontal preure differene hange over the vertial etent of a ingle ell: 1 Error (z)[(z ) ( z )] 4 b z [zi1z i] z higher order term, (3.3) For the tandard Jaobian, 0 and the eond term vanihe. Negleting the higher-order term, the trunation error beome 1 b SJ E ( z)[( z ) ( z ) ]. (3.4) 4 z For our weighted Jaobian, i given by (.14), and (3.3) redue to 1 b WJ E ( z)( z ). (3.5) 4 z A epeted, in either ae the trunation error vanihe if b i olely a linear funtion of z. The weighted Jaobian offer no obviou advantage in thi ae. However, if b(z) varie nonlinearly over the depth of a ell, the two heme have ditintly different propertie. For the tandard Jaobian ( 0), (3.4) i the generalization to arbitrary vertial oordinate of Eq. (7) in Mellor et al. (1994). For the peial ae of oordinate, z h and z h/ and our formula give their equation. It i alo lear that the generalization of the hydrotati oniteny ondition (3.1) i z z, (3.6) whih i reputed to be required to obtain optimal auray with topography following oordinate. In fat, from (3.4), we ee that optimal auray i atually ahieved at the point of equality that i, when equality i atified in (3.6). Thi error minimum for the tandard Jaobian formulation i eaily eplained. If the vertial reolution i further redued without reduing the horizontal reolution, then interpolation in or etrapolation in z i required to etimate the horizontal buoyany differene aro the grid ell. However, value lightly eeeding the point of equality will not be muh wore than lightly maller value. Thi i true for any -oordinate ytem, inluding oordinate. From (3.5) we ee that, for the weighted Jaobian, the term with ( z ) i anelled from the trunation FIG.. Theoretial etimate of the bottom-level geotrophi urrent aued by the preure gradient fore error a a funtion of the number of vertial level (K) for a denity profile that varie quadratially in the vertial. The olid line how reult predited for the tandard Jaobian and the broken line how the reult for the weighted Jaobian. error. Thu, for a buoyany profile that i quadrati in the vertial, the error onverge monotonially to zero a the number of vertial level inreae (i.e., with dereaing z h in the ytem). Note that the error anellation, whih the tandard Jaobian benefit from, doe not benefit the weighted Jaobian beaue it require the denity differene at the mean z level rather than at the mean level at whih the buoyany differene i known (when the oppoite orner of a ell are at the ame z level). To illutrate the effet on veloity of the differene between the tandard and weighted Jaobian heme, we onider a denity perturbation field that varie quadratially in the vertial. Thu, we onider a denity perturbation of the form z h (z) ma. (3.7) h ma The bottom preure gradient error obtained with the tandard Jaobian uing the uniformly paed ytem an be etimated by umming up (3.4) over the depth of the water olumn. The orreponding geotrophi urrent error, aumulated over the full depth of the water olumn, i given by: g V, (3.8) f 100 K h 3 K 6K 0 where h/ i the bottom lope and K i the number of vertial level. It an be een that the eond term dereae with, but doe not onverge to zero with inreaing K. For the weighted Jaobian heme, the term proportional to i ompletely abent. Figure how the reult for h 000 m, 0.05 and

9 DECEMBER 1998 SONG 31 horizontal reolution of 10 km and 0 km. The tandard Jaobian heme (olid line) outperform the weighted Jaobian heme when the hydrotati oniteny ondition i atified, but perform le well when the ondition i ignifiantly violated. For large-ale problem, the horizontal reolution i generally dereaed for pratial reaon and the ondition for hydrotati oniteny may be trongly violated. In thi ae, the weighted Jaobian may give uperior reult to the tandard Jaobian, with the gain in auray inreaing a ( h/h). Comparion of Fig. a and b onfirm that the error aoiated with the tandard Jaobian heme dereae like ( h/h) () when the hydrotati oniteny ondition i violated, wherea the error in the weighted Jaobian heme (dahed line) onverge to zero with inreaing K. We emphaize that the tandard Jaobian benefit from there being no need for vertial interpolation when the upper orner on one ide of a ell i at the ame z level a the lower orner on the other ide of the ell. The weighted Jaobian doe not benefit from thi anellation beaue thi onfiguration give the denity differene at the midlevel in pae, wherea the weighted Jaobian require thi differene at the midlevel in z pae. Thu, the tandard Jaobian will be more aurate when equality i approimately atified in (3.6). Thi might, for eample, be ued to advantage in numerial tudie of plume over teep bottom topography. However, model often trongly violate thi ondition. A pointed out by Mellor et al. (1994), with 0 evenly paed level in the vertial, thi ondition require h/h 0.05, whih i a very evere ontraint in region of teep topography. The weighted Jaobian i intereting with regard to thi point ine, for a horizontally uniform buoyany field with quadrati dependene on z, the hydrotati oniteny problem i eliminated. That i, when the buoyany field i horizontally uniform and varie a a quadrati funtion of z over the full vertial etent of eah grid ell, the model auray an be improved by imply inreaing the vertial reolution without onern about hydrotati inoniteny. 4. Diagnoti eample To quantitatively eamine trunation error and hydrotati inoniteny, we onider ome peifi diagnoti eample. In thee eample, we firt follow Haney and eamine buoyany variation, whih would be aoiated with the firt three vertial Roby wave mode. We then onider a diagnoti eample that onider the error aoiated with perturbation, whih inlude both horizontal and vertial variation in the buoyany field. a. Modal buoyany profile To eamine the error aoiated with realiti vertial variation in the buoyany field, we onider the loal buoyany perturbation due to the firt three Roby wave mode in the preene of a preribed mean buoyany frequeny. Note that thee reult imply onider the effet of different vertial variation in the buoyany field: iopynal are till horizontal o the analytial olution ha no flow. The vertial mode are firt determined by olving the eigenvalue problem: qf d qf dt m mt m, (4.1) N(z) dz N(z) dz with boundary ondition T(0) T() 0. (4.) The analytial olution i known for N(z) 0.01 z/h 1 e, where, h i a referene depth and q i a di- menionle ontant. The eigenfuntion and eigenvalue are and m z/h T (1) o(me m ) (K) (4.3) mq. (4.4) m We onider reult orreponding to h 400 m, h 000 m, and q 100. The buoyany frequeny and the firt three mode are plotted in Fig. 3 and are imilar to the barolini Roby mode alulated by Rieneker et al. (1987) for the wet oat of North Ameria, eept for the urfae mied layer. Thee mode have been ued by Rieneker et al. (1987) to deompoe and analyze dynami variable. For eample, the preure an be epanded a p m m0 h p (, y, t)t (z). The barotropi mode, p 0 T 0, aount for about 91% of the variane in the preure field for the region off Oregon, and the firt three barolini mode aount for about 7%, 1.%, and 0.5%, repetively. Thee mode are thu relevant to the vertial truture of oeani flow and will be ued to invetigate the preure gradient error. It hould be pointed out that the profile ued here are lightly different from thoe ued by Haney (1991), but there i no ignifiant differene in numerial error if Haney profile are ued. We ue the preent profile beaue the eitene of analytial olution offer ome obviou onveniene. Let b m (z) gt m (z) repreent the buoyany diturbane, where K 1 i the thermal epanion oeffiient and g i the aeleration due to gravity. To ompute the direte preure gradient fore, we ue a typial ontinental lope of 0.05 over an average m

10 3 MONTHLY WEATHER REVIEW VOLUME 16 FIG. 3. The vertial profile of the buoyany frequeny (ph 1000) and the firt three barolini Roby mode (1 K) omputed for a 000 m deep oean with tratifiation repreentative of the California Current region. depth of 000 m. Sine the atual preure gradient fore i known to be zero in thi ae, any nonzero value omputed by the differene heme i due to trunation error. The fale preure gradient an be epreed a an erroneou geotrophi urrent parallel to the iobath, V PX k / f, where f i the Corioli parameter. 1) CASE A: NO HYDROSTATIC INCONSISTENCY ( 1 KM) We firt onider reult obtained with 1kmo that potential problem aoiated with hydrotati inoniteny are avoided. The error orreponding to the firt three Roby mode are hown in Fig. 4 for three different vertial reolution: K 10, 0, and 30. In eah ae, reult are hown for the tandard Jaobian heme with uniformly paed oordinate (olid line) and with the oordinate ytem introdued by Song and Haidvogel (1994), with 3 to improve the reolution in the region of trong vertial tratifiation (dahed line). The reult orreponding to the uniformly paed oordinate ytem were obtained by taking 0 in the latter oordinate ytem. The error obtained with 0 uing the new preure gradient formulation are about half thoe omputed by Haney baed on the tandard formulation. Thi improvement i ignifiant, but not partiularly dramati. Ue of the weighted Jaobian heme (reult not hown) doe not ignifiantly redue the error for thee alulation. Thi i not unepeted ine our analytial reult have uggeted that the weighted Jaobian give improved performane when there are horizontal variation in the vertial tratifiation in ombination with a nonuniform vertial grid or when there are problem with hydrotati oniteny, neither of whih are preent in thi eample. A noted above, a major advantage of uing generalized oordinate i that nonuniform vertial grid an be ued without giving up the eond-order auray of the numerial heme. The partiular oordinate ytem diued by Song and Haidvogel (1994) provide an eample of how thi may be ued to advantage. Thi oordinate ytem inlude three parameter h, b, and whih an be ued to adjut the vertial variation in the vertial reolution. Here 0 orrepond to the uniformly paed -oordinate ytem and larger value of orrepond to larger variation in the vertial reolution over the depth of the water olumn. Here h and b have no effet when 0, but for nonzero, h determine a vertial poition above whih the vertial reolution i nearly ontant and maimized, and the value of b determine where inreaed reolution hould be onentrated; h i alway le than or equal to the minimum water depth within the region being modeled. Chooing b 0 maimize the reolution in the upper portion of the water olumn and b 1 give imilar reolution near the urfae and bottom. At the urfae, the -oordinate ytem nearly oinide with the z-oordinate ytem ( 0 orrepond to the poition of the free urfae), o problem with the preure gradient error are redued in thi region. Sine we know that our diretization heme i well uited to the ae where the buoyany varie linearly over eah individual grid ell, it i likely that reult an be improved by hooing h, b, and uh that N(z) i roughly ontant over eah grid ell. We thu et b 0 and h 400 m to fore any inreae in reolution to be onentrated in the near urfae region where N(z) varie mot rapidly. We then eperiment to determine that 3 give a reaonable repreentation of N(z) over the entire water olumn. With thi hoie of parameter, we find that with the tandard Jaobian formulation the error i about a fator of 10 maller than that in the -oordinate ytem for eah mode and for eah hoie of K (dahed line in Fig. 4). Reult are not ignifiantly different for the weighted Jaobian formulation. Clearly, an objetive proedure ould be developed to determine optimal parameter value in any -oor-

11 DECEMBER 1998 SONG 33 FIG. 4. Vertial profile of the preure gradient fore error uing three different vertial reolution (K 10, 0, 30) diturbed by the firt three barolini Roby mode. Solid line indiate reult omputed with a uniformly paed ytem and dahed line orrepond to reult obtained with Song and Haidvogel ytem with 3. dinate ytem baed on atifying peifi reolution riterion. The preent eample ugget that thi may be worthwhile, but it ha not yet been invetigated. The main point of the preent eerie i that error an be dramatially redued through an appropriate hoie of the oordinate ytem and, by uing an appropriately trethed grid, there need not be any redution in the order of auray. ) CASE B: WITH HYDROSTATIC INCONSISTENCY ( 10 KM) Net we onider eample that are intended to onfirm our analytial predition regarding hydrotati inoniteny and to further eamine it qualitative effet. We again follow Haney (1991) and onider a peifi eample in whih the horizontal and vertial reolution violate the ondition (3.1) or (3.6). In partiular, we onider horizontal grid with 10 km, a oppoed to the 1-km grid ued above. With thi value of, the oniteny ondition i violated for K 1 with uniformly paed level. The larget trunation error in the water olumn are hown in Fig. 5 a a funtion of K for the three different Roby wave mode, uing three different heme: (a) the tandard Jaobian in the ytem; (b) the tandard Jaobian in the ytem of Song and Haidvogel (1994) with h 1000 m and optimal (K) by taking advantage of the error anellation diued in the previou etion (ee below); () the weighted Jaobian in the ytem with 3. In eah ae, the error inreae with mode number a epeted. The error in Fig. 5a orreponding to the tandard Jaobian in the ytem are imilar to thoe preented in Fig. 7 of Haney (1991). Conitent with Haney reult, the error inreae for K 10. Thee reult are imilar to the reult hown in Fig. for a quadrati vertial profile and further onfirm that the error minimum that ha been aoiated with the hydrotati oniteny boundary i atually due to anellation of the trunation error aoiated with finite horizontal and vertial reolution. Reult obtained with the tandard Jaobian heme in the ytem with optimal (Fig. 5b) how that error are redued ubtantially for thi typial hoie baed on aurately repreenting the vertial tratifiation. In the preent ae, the improved repreentation of vertial tratifiation i aompanied by redued hydrotati inoniteny at depth due to oarer reolution there. Figure 5b how that, at leat in ome ae, problem with hydrotati inoniteny an be ubtantially redued with the tandard Jaobian heme if the vertial oordinate ytem i arefully hoen. In thi ae, the error ontinue to dereae with inreaing K beaue the value of ha been hoen uh that the larget trunation error aoiated with finite horizontal and vertial reolution anel. That i, we have hoen uh that equality in (3.6) i atified at the bottom. For

12 34 MONTHLY WEATHER REVIEW VOLUME 16 FIG. 5. Maimum value of the preure gradient error a a funtion of the number of vertial level (K) for vertial denity profile orreponding to the firt three Roby wave mode. Reult for the three mode are hown in eah frame. The larger error orrepond to the higher mode, whih generally ontain le of the variane under realiti ondition. The three frame how reult orreponding to three different approahe: (a) the tandard Jaobian heme in ytem, (b) the tandard Jaobian heme in the ytem with hoen to minimize trunation error, and () the weighted Jaobian heme in the ytem with 3. b 0inthe-oordinate ytem of Song and Haidvogel (1994) (ee the appendi), thi give: h h K. (4.5) h h h h In Fig. 5b, we have ued h 0.05, 10 km, h 000 m, and h 1000 m, whih give 0.5K 1, and we ee that thi hoie doe indeed ubtantially redue the model error. However, from Fig. 5 we ee that the minimum error a a funtion of K i not preent, a epeted from our analytial onideration. Comparing Fig. 5b and 5 reveal that the error are atually inreaed in the weighted Jaobian heme. Thi i a onequene of the error anelation, whih the tandard Jaobian formulation benefit from, but the weighted Jaobian formulation doe not. We alo note, in ontrat to the reult of Fig., that the error in the weighted Jaobian formulation do not ontinue to dereae a K i inreaed. Thi i a refletion of the fat that hydrotati inoniteny i eliminated only for buoyany profile, whih vary quadratially with z over the full depth range of eah grid ell. Clearly, thee barolini modal buoyany profile do not atify thi ondition. The reult hown in Fig. 5 eem to ugget that the tandard Jaobian formulation ha a lear advantage over the weighted Jaobian formulation. Thi uggetion i omewhat mileading. A we noted following (.13) the weighting ued in the weighted Jaobian formulation i determined by the ombined effet of horizontal and vertial variation in the buoyany field. Sine the above eample doe not inlude horizontal variation in the buoyany field, the poible advantage of the weighted Jaobian formulation i not onidered by thee tet. We now onider an eample that inlude both horizontal and vertial buoyany variation in order to tet the formulation under omewhat more realiti ondition. b. Coatal front In the above two ubetion, we have eamined trunation error due to iopynal diturbane that are horizontally uniform that i, b i a funtion of z, only. In reality, oean flow an be trongly tratified in both horizontal and vertial diretion. Suh ae are often een in oatal oean where the ombination of omple oatal geometry and teep topography offer great hallenge for numerial modeler. Here, we onider a ae with both vertial and horizontal variation in denity, imilar to the helf-break front in the Middle Atlanti Bight. During the winter,

13 DECEMBER 1998 SONG 35 FIG. 6. Cro-helf etion of (a) potential denity, (b) eat geotrophi urrent, () preure gradient fore error with the tandard Jaobian heme, and (d) preure gradient fore error with the weighted Jaobian heme. Horizontal reolution i 5 km and there are 0 level. For the weighted Jaobian, the error i zero, but even for the tandard Jaobian, the error are generally le than 0.05% of the maimum geotrophi urrent. Reult above 1500-m depth are hown. The horizontal ditane i 00 km. the water olumn over the helf and the upper lope i nearly linearly tratified in both temperature and alinity. However, a peritent, harp helf-break front i found near the helf edge where the freher, older helf water meet the altier, warmer lope water. The aurate omputation of the preure gradient fore over uh a region of teep topography i potentially important for the tudy of frontal dynami in the oatal oean. Following Chapman and Gawarkiewiz (1993), the denity field i approimated by: (, z) 4 { (T 10.5) 0.779(S 34)} Z(z), (4.6) where z Z(z) 1, and the urfae temperature and alinity are given by h T tanh[( L)/0] S 34 tanh[( L)/0] (C) (pu). The orreponding denity field i plotted in Fig. 6a, uperimpoed on the model topography from Gawarkiewiz and Chapman (199). The topography inlude a gently loping ontinental helf from 50-m depth at the oat to about 180-m depth at the helfbreak, an adjaent teeply loping region etending offhore to

14 36 MONTHLY WEATHER REVIEW VOLUME 16 about 000-m depth with a lope of 3%, and an abyal plain. The width of the helf, lope, and abyal plain are 50 km, 50 km, and 100 km, repetively. The ditane of the front from the oat i L 50 km, the depth parameter h 150 m, and 0.01 determine the vertial gradient in the denity field. Even in thi ae, where horizontal gradient are trong, removing the referene tate i till worthwhile and thi ha been done in the eample preented below. The eat preure gradient fore an be alulated analytially and the geotrophi etimate of the aoiated urrent, relative to the urfae, i plotted in Fig. 6b. Although thi eample doe not onider the dynami that determine the urfae preure field, it i well uited to revealing problem with the repreentation of the preure gradient fore aoiated with the ombination of horizontal and vertial variation in the buoyany field. The error in the geotrophi urrent etimated with the tandard Jaobian and 3 are plotted in Fig. 6, and the orreponding error obtained with the ame value of, but uing the weighted Jaobian heme, are plotted in Fig. 6d. The error i mall in Fig. 6 ( m 1 ), but the weighted Jaobian heme (Fig. 6d) give the eat preure gradient fore even over thi teep topography. Thi i a epeted ine our vertial weighting fator wa hoen to give zero error for a denity field, whih varie linearly with z. Of oure, we epet ome error to be introdued in the weighted Jaobian heme if the vertial variation in denity i not linear. To invetigate the error aoiated with horizontal variation in ombination with nonlinear vertial denity profile, we now onider reult for a denity profile that varie eponentially with z. In partiular, we onider a denity field of the form (4.6), but with z Z(z) ep, (4.7) where Reult orreponding to thi denity profile, with all other parameter eatly a in Fig. 6 are hown in Fig. 7. Maimum error are inreaed ubtantially to m 1 and m 1 for the tandard and weighted Jaobian heme, repetively. However, eah of thee error remain mall in omparion with the maimum urrent peed, whih i approimately 0.55 m 1 (Fig. 7b). The effet of inreaing the horizontal grid ize from 5 to 0 km i hown in Fig. 8. For eah formulation, the maimum error are inreaed by about a fator of 10, omparable with the epeted fator of 16 baed on the heme being eondorder aurate in the grid reolution. Even at thi oare reolution, the error would be tolerable for mot appliation: the maimum error i about.6% of the maimum urrent for the tandard Jaobian and about 0.1% for the weighted Jaobian. h 5. Summary and onluion The preure gradient formulation remain one of the mot important iue in the deign of numerial model with topography-following oordinate ytem. In thi paper, we have introdued a generalized method to ompute the preure gradient fore baed on integrating the Jaobian of denity and vertial oordinate. The Jaobian formulation allow u to deign a heme that an be ued with any vertial oordinate ytem. Two different heme are onidered in ome detail: the tandard Jaobian heme, whih i eat for any topography provided the buoyany field varie bilinearly with and over eah grid ell; and the weighted Jaobian heme, whih i eat for any topography provided the buoyany field varie bilinearly with and z over eah grid ell. Both heme retain eond-order auray in the preene of horizontal and vertial variation in both the denity field and the vertial grid paing. For eah of the heme diued here, finite differene are applied to the denity field to etimate horizontal denity gradient prior to integrating to determine the horizontal preure gradient. Sine the denity i alway approimated by a linear funtion of, the preure field i onitently approimated by a quadrati in the vertial oordinate over eah grid ell. Thi ontrat with the onventional approah in whih the denity field i firt integrated to determine the preure, whih i then finite differened to etimate horizontal preure gradient. In thi ae, the preure i effetively quadrati in the vertial oordinate over eah grid ell ine it i obtained by integrating a linear funtion of denity, but ubequent finite differene of the preure field may not properly aount for thi quadrati dependene. By differentiating the denity prior to integrating, the preent numerial heme avoid thi inoniteny in a natural way. The trunation error i eamined analytially for horizontally uniform iopynal diturbane, and quantitatively for partiular horizontally and vertially varying iopynal diturbane. The auray of the Jaobian heme are improved ignifiantly over onventional formulation baed on finite differening the preure field. Conitent with the reult of Mellor et al. (1994), we find that, for a horizontally uniform buoyany field that varie quadratially in the vertial, the tandard Jaobian heme onverge to the eat olution with the quare of the horizontal and vertial grid ize. In thi ae, we find that the weighted Jaobian onverge quadratially with the vertial grid ize, independent of the horizontal grid paing. However, it hould be emphaized that the latter reult hold only for a buoyany field that i horizontally uniform and varie quadratially with z over the full vertial etent of eah grid ell. A generalized hydrotati oniteny ondition i derived that applie for an arbitrary oordinate ytem when the tandard Jaobian formulation i ued. Un-