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4 a Generalization of a Sigma Coordinate Ocean Model and an Intercomparison of Model Vertical Grids

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4 a Generalization of a Sigma Coordinate Ocean Model and an Intercomparison of Model Vertical Grids 4 A Generalization of a Sigma Coordinate Ocean Model and an Intercomparison of Model Vertical Grids GEORGE MELLOR1, SIRPA HÄKKINEN2, TAL EZER1 AND RICHARD PATCHEN3 1Atmospheric and Oceanic Sciences Program Princeton University, Princeton, NJ 2NASA Goddard Space Flight Center, Greenbelt, MD 3Dynalysis of Princeton, Princeton N.J. 4.1 Introduction Numerical ocean models increasingly make use of s - coordinate systems. A paper by Gerdes (1993) shows that these coordinate systems can be more general; he termed the generalized form an "s - coordinate" system. The main advantage of the s or s - system is that, when cast in a finite difference form, a smooth represen- tation of the bottom topography is obtained; one can also easily incorporate a bot- tom boundary layer as well as a surface boundary layer in those coordinate systems. This is intuitively appealing and Gerdes has shown that superior numeri- cal results are obtained relative to a z - level system. However, in regions of steep topography and crude resolution - a limiting case would be a seamount represented by a single grid point surrounded by a flat bottom - the so-called sigma coordinate pressure gradient error exists (Haney 1991, Mellor et al. 1994, 1998) and at least locally a z - level coordinate system might be preferred. On the other hand, in a recent study, Bell (1997) has shown that the step structure of z - level models lead to vorticity errors and consequent errors in the barotropic component of the flow which, he reports, cause rather large temperature errors (3 to 4° C) on a 1° x 1° grid of an Atlantic Ocean model after 3 months of integration. And it is difficult to model bottom boundary layers in a z - level model (Winton et al. 1998). The pioneer Bryan-Cox model (Bryan 1969, Cox 1984) is a z - level model. A modification by Spall and Robinson (1993) is termed a "hybrid" coordinate system; they describe it as a z - level system in the region, 0 > z > - zc = constant, and a s system when zc > z > - H(x, y) where the transformed sigma equations apply. Pre- sumably, this system is adopted so that surface mixed layers, which do not scale on depth, may be best represented. However, the hybrid system would appear to require separate numerical implementations for the two regions; their objectives can be realized more simply with the s - coordinate system described here. For the sigma coordinate system of the Princeton Ocean Model (henceforth POM), the top numerical level, k = 1, follows the free sea surface and the lowest numerical level, k = kb, follows the bottom depth; for 1 < k < kb, the distance 56 George Mellor, Sirpa Hakkinen, Tal Ezer and Richard Patchen between levels are in fixed proportion to each other independent of elevation or depth. In this paper we revise POM so that it is basically an s - coordinate system (although the variable, s, is, for very good reason, different from that used by Ger- des) and the proportionality constraint is removed. However, unlike the Gerdes for- mulation, the derivation and implementation permit a free surface and the distance between levels can change in time. Also the structure of the numerical algorithm is designed to support a wide variety of vertical coordinate systems including a z - level system. The present model generalization permits a z - level representation everywhere in the model domain or locally in selected regions of the domain. Song and Haidvogel (1994) have adopted many of the characteristics of POM but have also generalized to an s - coordinate system. Our formulation and imple- mentation strategy, as described below, differs from theirs and we include the z - level option. Intercomparisons between different models have been the focus of recent projects (DYNAMO 1997, Willems et al. 1994). The intercomparisons of this paper isolate the effect of different vertical grids; otherwise model physics and numerics are identical. 4.2 The Governing Equations We first restrict attention to the analytic description of the basic equations although it is our intention that they be cast in finite difference form and we condi- tion the description with that in mind. The basic equations described in an (x,y,z) Cartesian coordinate system for the velocity components, U, V and W, and for potential temperature and salinity, T and S, are Á()1 = 0 (12) ¶h g h ¶r' ¶ ¶ U Á()U – fV ++g------ ----- ------- dz' = KM------- + Fx (13) ¶ x roòz ¶ x ¶z ¶z ¶h g h ¶r' ¶ ¶V Á()V +fUg++------ ----- ------- dz' = KM ------ + Fy (14) ¶y roòz ¶y ¶ z ¶ z ¶ ¶T ¶ R Á()T = K ------ + F – ------ (15) ¶ z H ¶z T ¶z ¶ ¶S Á()S = K ------ + F (16) ¶ z H ¶ z S and for twice the turbulence kinetic energy, q2 and length scale, l , we have A Generalization of a Sigma Coordinate Ocean Model 57 q2 Á()q2 = ¶ K --------¶ ¶ z q ¶ z (17) ¶ U 2 ¶ V 2 2g ¶r˜ 2q3 + 2K æö------- + æö------ ++------ K ------ – -------- F M èø èø H q ¶z ¶z ro ¶ z B1l ¶ ¶q2l Á()q2l = K ---------- ¶z q ¶z (18) U 2 V 2 g 2 q3 E læöæö¶ æö¶ E K ¶r W˜ F + 1 èøKM èø------- + èø------ + 3----- H --------- – ----- + l ¶z ¶ z ro ¶z B1 ¶j ¶ Uj ¶Vj ¶Wj Áj()º ------ +++----------- ---------- ------------ (19) ¶ t ¶x ¶ y ¶ z In the above, x and y are horizontal coordinates (to be cast in a curvilinear, orthogonal coordinate system in section 4.4.) and z is the vertical coordinate; g is the gravity constant and f is the Coriolis parameter. The density is r whereas ¶r˜ ¤ ¶z is the vertical density gradient corrected for adiabatic lapse rate. ¶R˜ ¤ ¶z on the right hand side of (4) is the divergence of radiation flux, R. The vertical mixing coefficients are KM , KH , and Kq and are functions of a Richardson number depen- dent on density stratification (Mellor and Yamada, 1982, Galperin et al, 1988 and Mellor, 1996). W˜ in (7) is a "wall proximity function". Horizontal diffusion fluxes are represented by Fa ( a= x, y,T,S,q,l) . 4.3 Transformation to the s-coordinate system The Cartesian coordinate system, (x, y, z, t), is transformed to the s - coordinate system, (x*, y*, k, t*), according to xx= * (20a) yy= * (9b) tt= * (9c) z = h()x*,,y* t* + sx()*,y*,,kt* (9d) where k is a continuous variable in the range, 1 ££kkb , but when the differen- tial equations are discretized, k will also be discrete and will be the label of the numerical level. This usage is quite convenient and obviates the need for an inter- mediate variable [which Gerdes called s, and is not the usage in (9d)]. We define h 58 George Mellor, Sirpa Hakkinen, Tal Ezer and Richard Patchen to be the surface elevation so that we will require s = 0 at k = 1. The key attribute of the transformation is (9a,b); i.e., the horizontal coordinates in the Cartesian coordinate system are equated to the horizontal coordinates in the sigma system. Otherwise, the transformation is fairly general and includes, for example, a z - level system, s = s(k)(Hmax + h(x,y,t)) , and the sigma system, s = s(k)(H(x,y) + h(x,y,t)); H(x,y) is the bathymetry and Hmax is its maximum value. In both cases, s(1) = 0 and s(kb) = -1. In the case of the z - level system (really a quasi z - level system because s is a function of time, but it is a weak function of time since, gen- erally, h/H <<1), levels near the bottom are masked to approximate the bathymetry and, thus, kb is a function of the horizontal coordinates. The distinguishing feature of both of these limiting cases is that k is functionally separated from x, y and t. In general they need not be separable as we shall demonstrate. There are two ways of deriving the s - coordinate equations. The first way is to apply integral (control volume) equations to a volume element bounded by x* ,y* , k and x* + Dx*, y* + Dy*, k + Dk. The second way is to formally transform equa- tions (1)-(5). Thus, if f is any dependent variable such that f(x, y, z, t) = f* (x*, y*, k,t*) then * * * * ------¶f =---------¶f + ---------¶f -----,¶ k ------¶f =---------¶f + ---------¶f -----¶k (10a,b) ¶x ¶x* ¶k ¶ x ¶ y ¶ y* ¶k ¶y ¶f ¶f*¶k ¶f ¶f ¶f*¶k ------ = --------- -----, ------ =------ + --------- ----- (10c,d) ¶z ¶ k ¶ z ¶t ¶ t ¶k ¶t From the fact that ¶z ¤ ¶ x = 0 , we obtain ¶k ¤h¶x = –()x + sx ¤ sk where sx º ¶s ¤ ¶x* and hx º ¶h¶¤ x*. Similarly, ¶k ¤h¶y = –()y + sy ¤ sk , ¶ k ¤ ¶ z = –1 ¤ sk and ¶k ¤h¶t = –()t + st ¤ sk . (In discretized form, sk º ds ¤ d k is simply the distance, ds, between levels since dk = 1. ) Therefore, ¶f ¶f* ¶f*sx + hx ¶f ¶f* ¶f*sy + hy ------ = --------- –--------- ----------------, ------ = --------- –--------- ---------------- (11a,b) ¶x ¶ x* ¶ k sk ¶y ¶ y* ¶k sk * * *s + h ------¶f = ---------¶f ----,1 ------¶f = ---------¶f –---------¶f ---------------t t (11c,d) ¶ z ¶k sk ¶t ¶ t* ¶k sk We define a new variable, w , such that w º Ws– ()x + hx U – ()sy + hy V – ()st + ht (12) A Generalization of a Sigma Coordinate Ocean Model 59 Note that the surface kinematic condition, W = hxU ++hyV ht , is satisfied if we require that w = 0 where s = 0. Similarly, at the bottom, WH= – xU – HyV , and we set w = 0 where sH= –()+ h . If we use (11a, b, c, d) and (12) in equations (1) through (9) and drop all of the asterisks from independent and dependent variables, we obtain, Á()1 = 0 (13) 0 ¶h sk ¶r' ¶r' Á()U – fVsk ++gsk------ g----- sk------- – ()sx + hx ------- dk' ¶ x roòk ¶ x ¶ k' (14) K ¶s t ¶s t = ¶ -------M -------¶ U ++æ--------------k xx --------------k xyö è ø ¶ k sk ¶k ¶x ¶y 0 ¶h sk ¶r' ¶r' Á()V +fUsk ++gsk ------ g----- sk------- – ()sy + hy ------- dk' ¶y roòk ¶y ¶k' (15) ¶ KM ¶V æ¶ sktxy ¶ sktyyö = ------- ------ ++è-------------- -------------- ø ¶ k sk ¶k ¶ x ¶y K T ¶s q ¶s q R T ¶ H ¶ æök T
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