A Global Isopycnal OGCM: Validations Using Observed Upper-Ocean Variabilities During 1992–93

706 MONTHLY WEATHER REVIEW VOLUME 127

A Global Isopycnal OGCM: Validations Using Observed Upper-Ocean Variabilities during 1992±93

DINGMING HU Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, Washington

YI CHAO Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

(Manuscript received 9 April 1997, in ®nal form 13 May 1998)

ABSTRACT In this study, a global isopycnal ocean model (GIM) is described and used for a simulation of variabilities of the global upper ocean during 1992±93. The GIM simulations are compared and validated with both the available observations and simulations with the Geophysical Fluid Dynamics Laboratory Modular Ocean Model (MOM). The observations include sea surface height from TOPEX/Poseidon (T/P), sea surface temperature (SST) from weekly National Centers for Environmental Prediction analysis, and vertical temperature pro®les from gridded expandable bathythermographs (XBTs) data. The major differences between the GIM and MOM used in this study are the vertical coordinates, a Kraus±Turner mixed layer, and a tracer-transport velocity associated with an isopycnal-depth diffusion. Otherwise, the two models are formulated in the same parameter space, model con®guration, and boundary conditions. The effects of these differences in model formulation on the model simulations are investigated. Due to the difference in the orientation of interior ¯ow and mixing, SST and the thermocline strati®cation in the eastern equatorial Paci®c in GIM are more sensitive to the wind-driven upwelling than they are in MOM. In GIM there is no effective means to transfer heat between the upwelling cold water and the surrounding warm water since subsurface ¯ow and mixing predominantly occur along isopycnic layers. As a result, the SST tends to be cold and the front tends to be sharp compared with the observations in the wind-driven upwelling region. The sharp front could potentially cause numerical instability in GIM. Thus, a large isopycnal-depth diffusivity has to be used to maintain the model stability since the isopycnal-depth diffusion is the most effective way to reduce the steep slope of isopycnals and the strength of the front associated with the cold upwelling in GIM. But the large isopycnal-depth diffusion results in excessive smoothing in the meridional isotherm doming in the equatorial and tropical thermocline. The trade-off between the numerical instability and the excessive isopycnal smoothing points to the necessity of improvement in the isopycnal-depth diffusion. Sea level variabilities during 1992±93 simulated with both GIM and MOM are in good agreement with T/P observations. However, MOM poorly simulates the vertical distribution of the seasonal temperature anomalies in the upper ocean (the baroclinic component of the sea level variability) during 1992±93. Due to the lack of a realistic surface mixed layer, the MOM-simulated temperature pro®les have a sharp subsurface gradient, which is not evident in both the GIM simulation and the XBT observation. As a result, the region below the subsurface gradient is almost insulated from the in¯uence of the seasonal temperature variation. The Kraus±Turner mixed layer used in GIM helps to improve the model-simulated seasonal variations of the upper-ocean temperature and the background sea level variability. Implications of de®ciencies in both GIM and MOM on the altimetric sea level data assimilation and transient tracer simulations are discussed.

1. Introduction OGCM development and improvement. The most widely used OGCM is that developed at the Geophysical The past decade has witnessed increasing applications Fluid Dynamics Laboratory (Bryan 1969; Cox 1984; of ocean general circulation models (OGCMs) to the Pacanowski et al. 1991; Pacanowski 1995); this model study of the ocean's role in the climate over a wide is also known as the Modular Ocean Model (MOM). range of time- and spatial scales. With the increasing MOM is a level model since the prognostic variables demand, tremendous efforts have been devoted to in the model are carried at constant-depth levels. In a standard version of MOM, mixing of tracers occurs predominantly along constant-depth levels, which is in- Corresponding author address: Dr. Dingming Hu, JISAO, Uni- consistent with our knowledge that tracer mixing occurs versity of Washington, Box 354235, Seattle, WA 98195. predominantly along isopycnal/neutral surfaces (Mont- E-mail: [email protected] gomery 1938; Iselin 1939; McDougall 1987a). Recent

᭧ 1999 American Meteorological Society

Unauthenticated | Downloaded 09/26/21 10:49 PM UTC MAY 1999 HU AND CHAO 707 progress has been made in parameterizing isopycnal there is no data. Although a Kraus±Turner mixed-layer eddy mixing processes in level models (Redi 1982; Cox model can be introduced into MOM, it is beyond the 1987; Gent and McWilliams 1990; Gent et al. 1995); scope of this paper. Therefore, this study is not intended this has signi®cantly improved the ability of MOM to for a strict examination on the effect of the the different simulate the global ocean strati®cation and meridional vertical coordinate system on the simulation. In a model heat transport (England 1993; Hirst and Cai 1994; Dan- comparison, more differences in model formulation abasoglu et al. 1994). make it more dif®cult to understand the differences in Isopycnal mixing can be naturally incorporated in iso- model simulations. Nevertheless, understanding can still pycnic coordinate models. In these models, the prog- be gained on how each type of the models can be im- nostic variables are carried at layers of constant potential proved. Two of such examples are Chassignet et al. density (isopycnic layers), and thus both lateral mixing (1996) and Roberts et al. (1996). In the present study, and ¯ow occur strictly along isopycnic layers. Since the the major difference in model formulation between GIM early 1980s, many isopycnal ocean models have been developed, for example, the Miami Isopycnic Coordi- and MOM are vertical coordinates, a Kraus±Turner nate Ocean Model [MICOM hereafter, see Bleck et al. mixed layer, and an isopycnal depth diffusion. Effort is (1992) and Bleck and Chassignet (1994) for a complete made to investigate the effects of these differences on description], the isopycnal model developed by Ober- the model simulations. huber (1993), and other layer models (Schopf and This article is structured as follows. Section 2 de- Loughe 1995; Murtugudde et al. 1995; Gent and Cane scribes the isopycnal model. Section 3 describes the 1989; McCreary and Kundu 1988; Luther and O'Brien design of real-time simulations of the large-scale sea 1985). As more OGCMs are developed and improved, surface height and upper-ocean temperature for the pe- it has become apparent that the similarities and differ- riod of 1992±1993 with the GIM and MOM. Section 4 ences between these OGCMs need to be understood. describes several types of observations used for vali- However, a direct intercomparison between level and dation of the model intercomparison. Section 5 gives isopycnal ocean models is not as simple as it seems, the intercomparisons between the simulations with the because each OGCM is usually formulated with its own two global models and validations with the observa- unique con®guration, turbulence closure parameteriza- tions, and section 6 summarizes the results and conclu- tion, and primitive equation simpli®cation. This is par- sions of this study. ticularly true for existing isopycnal models. For instance, in the isopycnal models of Schopf and Loughe (1995) and Murtugudde et al. (1995), reduced-gravity approximation is assumed. In Oberhuber's model, ver- 2. Model description tical mixing is parameterized in a way analogous to that of the bulk mixed-layer parameterization, which differs a. The governing equations from the vertical eddy diffusivity parameterization ex- tensively used in OGCMs. In MICOM, the model is laid out on a Mercator projection, whereas most of the In the GIM used in this study, the interior ocean is OGCMs are con®gured on the earth's spherical surface. represented by a stack of layers of prescribed potential Note that for a given degree of horizontal resolution, density and the oceanic surface boundary layer is rep- the meridional grid size decreases poleward on a Mer- resented by a Kraus±Turner type of mixed-layer model cator projection and remains invariant in the spherical (Kraus and Turner 1967). The model governing±equa- coordinate. A fair model intercomparison has to be con- tions are the primitive conservation equations of mo- ducted on the same model mesh. mentum, mass, heat, and salt, with the Boussinesq and Such a direct intercomparison between isopycnal and hydrostatic approximations. Written in the generalized level models was achieved by Hu (1997), in which Hu's vertical coordinate s (Bleck 1978), the equations read global isopycnal model (GIM) outperformed MOM in simulating the climatology of the World Ocean. How- vץ vv2 eץ ever, Hu did not use a realistic surface mixed layer for ϩ ( f ϩ ␨)k ϫ v ϩ ١ ϩ sץ zץ t s 2ץ the sake of direct model comparison. In this study, a Kraus±Turner mixed layer is introduced to the GIM. ΂΃s sץ Also the constant vertical mixing parameterization used in Hu (1997) is replaced by a Richardson number±de- 1 (␾ p ϩ ␳١ ١) ϭϪg١␩ Ϫ pendent mixing parameterization. Our object is to val- ␳ sd s idate the ability of the GIM in simulating transient var- 0 iabilities of the world upper oceans. Observational data vץץ zץ 1 (v ϩ A MV, (1 ١ ١sMHs´ A during 1992±93 are used for the validation. MOM sim- ϩ zץ sץ sץz[]΂΃΂΃ץ ulation with the same con®guration and forcing is also sץ used to provide more consistent test in those areas where

Unauthenticated | Downloaded 09/26/21 10:49 PM UTC 708 MONTHLY WEATHER REVIEW VOLUME 127 e c. Mixed-layer entrainment and detrainmentץ zץ zץץ (v ϩ v*) Ϫ , (2)´ ϭϪ١ -s In GIM, e is a vertical velocity associated with inץ sץ s sץ tץ ΂΃s [] terlayer mass exchange. At the bottom of the mixed z layer, e is mainly caused by the mixed-layer entrainmentץ pץ ϭϪ␳g , (3) -s and detrainment predicted by a Kraus±Turner mixedץ sץ layer model. The mixed-layer entrainment and detrain- e␪) ment are determined by the net turbulent kinetic energy)ץ zץ zץץ v ϩ v*)␪ Ϫ)´ ␪ ϭϪ١ -s generation rate, ⌬E, which arises from mechanical stirץ sץ s sץ tץ ΂΃s [] ring of the surface wind stress and the destabilizing effect of upward surface buoyancy ¯ux and is param- ␪ץץ zץ ␪ ϩ A , (4) eterized as ١ A ´ ١ ϩ zץs HVץ sץsHHs΂΃΂΃ 2mu3 B Ϫ |B | B Ϫ |B | ⌬E ϭϩ* 00 ϩn 00, (8) eS) h 22)ץ zץ zץץ S ϭϪ١s ´(v ϩ v*)S Ϫ sץ sץ s sץt΂΃ץ [] where u is the frictional velocity, B 0 is the surface * buoyancy ¯ux (positive upward), h is the mixed-layer S depth, and m and n are two empirical parameters. Whenץץ zץ (S ϩ A . (5 ١ A ´ ١ ϩ z ⌬E is positive, it is assumed to be spent on entrainingץs HVץ sץsHHs΂΃΂΃ heavier water from below the bottom of the mixed layer. Here v [ϭ(u, ␷)] is the horizontal velocity vector; e The entrainment rate is given as s) is the generalized vertical velocity; ␳ 0 is aץ/zץϭsÇ) reference density; g is the gravitational acceleration; eϪh/hc⌬E wen ϭ , (9) s is the layer thickness; ␩ is the surface elevation; bmixϪ b subץ/zץ p (ϭ∫0 ␳gdz) is the depth-dependent hydrostatic pres- d z where b and b are, respectively, buoyancies in the ϫ v)isthe mix sub sure; ␾ (ϭgz) is the geopotential; ␨ (ϭ١ s mixed layer and immediately below the mixed layer; ١s is the horizontal gradient operator ;relative vorticity eϪh/hc is an exponential decay factor; and h is the thick- along a constant s surface; f is the Coriolis parameter; c A and A are, respectively, lateral viscosity and dif- ness of a turbulent boundary layer. A reasonable esti- MH HH mate of h is given in Oberhuber (1993). In this study, fusivity along s surfaces; A and A are, respectively, c MV HV h is simply set to 50 m. When E is zero or negative, vertical viscosity and diffusivity; c ⌬ the mixed layer retreats to a depth at which wind stirring and surface heating reach an equilibrium. The new depth ץ (z) (6 ١ v* ϭϪ (K z Is is thus obtained by setting ⌬E to zero in Eq. (8), whichץ yields is a transport velocity associated with a layer-interface 2mu3 depth diffusion; and KI is lateral diffusivity for the in- h ϭ *, (10) terface depth. Numerically, KI introduces an explicit ϪB0 damping in the continuity equation and helps with the where n ϭ 1 is assumed for the mixed-layer detrainment. model numerical stability. Interface-depth diffusion was For m ϭ 1.25, the mixed-layer depth given by Eq. (10) ®rst used by Bleck et al. (1989) and was later docu- is equal to the Monin±Obukhov length. The mixed-layer mented in MICOM (Bleck et al. 1992). In the isopycnic entrainment algorithm used in the present model is ba- interior, the transport velocity can be written as sically the same as those of Bleck et al. (1989) and Hu (1991). The mixed-layer detrainment algorithm is de- .١␳ scribed in appendix Aץץ (v* ϭϪ (KI١␳ z) ϭ KI , (7 z΂΃␳zץ zץ which is identical to the eddy-induced transport velocity d. Interior diapycnal velocity formulated by Gent and McWilliams (1990) and Gent In the region underlying the top model layer, s ϭ ␳ et al. (1995). and e represents diapycnal velocity, which is given as (Hu 1991; Hu 1996a) b. Equation of state ␪ץץ g ␪) ϩ A ١ A)´ ١ ␣ e ϭ zץ z΂΃HVץ In GIM, ␳ can be any one of the globally de®ned N 2 Ά []␳ HH ␳ potential densities used in the oceanography practice, Sץץ such as ␴ 0, ␴ 2, and ␴ 4 [see a discussion of ␳ used in (١␳ ´(AHH١␳S) ϩ AHV , (11 ␤ isopycnal OGCMs in Hu (1996a)]. In this study, ␴ is Ϫ zץ zץ 0 used as the model vertical coordinate and is calculated []΂΃·

,S] areץ/␳ץ(␪] and ␤ [ϭ(1/␳ 0ץ/␳ץ(by the UNESCO equation of state. where ␣ [ϭϪ(1/␳ 0

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respectively, thermal expansion and haline contraction f. Comparison with MICOM formulation z)] is theץ/Sץ␤ z ϩץ/␪ץ␣coef®cients, and N 2 [ϭg(Ϫ buoyancy frequency. Equation (11) states that diapycnal The GIM used in this study was developed on the velocity in the present model is the result of isopycnal basis of the box-basin isopycnal models developed at and diapycnal mixing of potential temperature and sa- the University of Miami (Hu 1991; Bleck et al. 1989), linity. which were early versions of MICOM. Thus, a better understanding of GIM can be gained by a comparison of model formulation with MICOM, which is given as e. Boundary conditions follows. The model boundary conditions are speci®ed as fol- 1) In MICOM, the Boussinesq approximation is not lows. Let n and t be unit vectors perpendicular and assumed. Thus, the model pressure-gradient term is given by M for the isopycnic interior ( is the speci®c ␣ ١o parallel, respectively, to the coastline. The lateral boundary conditions at the coast are volume and M ϵ ␣p ϩ gz is the Montgomery potential). In GIM the Boussinesq approximation is assumed. Ac- sz ´ n ϭ 0, (12) cordingly, the pressure-gradient term is expressed in its ١ S ´ n ϭ ١ ١ss␪ ´ n ϭ general form in s coordinates under the Boussinesq ap- v ´ n ϭ v ´ t ϭ 0. (13) proximation. At the bottom boundary where s ϭ s , we have 2) In both MICOM and GIM, there exists an implicit b vertical viscosity associated with the isopycnal-depth .(S diffusion (Greatbatch and Lamb 1990; Gent et al. 1995ץ ␪ץ ϭϭ0, (14) z But MICOM does not include explicit vertical viscosityץ zץ terms in the momentum equations and the GIM does. e ϭ 0, (15) In particular, the GIM incorporates a Richardson num- -v ber±dependent vertical viscosity (Pacanowski and Phiץ AMVϭ ␶ b, (16) lander 1981), which is widely used in OGCM simula- zץ tions in the equatorial region.

where ␶ b is a bottom drag. At the sea surface s ϭ st, 3) The prognostic variable in the continuity equation s for MICOM and true layerץ/pץ ''boundary conditions are given as is pressure ``thickness -s for GIM. Correspondingly, the interfaceץ/zץ thickness v depth diffusion is applied to the interface pressure p inץ AMV ϭ ␶, (17) z MICOM and to the interface depth z in GIM. Sinceץ -␪ 1 pressure p is not linearly proportional to z, the interfaceץ ␪ AHV ϭ Q , (18) depth diffusion is not exactly the same in MICOM and .z ␳0cw GIMץ e ϭ 0, (19) 4) In MICOM, since p is predicted in the continuity equation, and pressure gradient is given by the Mont- Sץ A ϭ QS, (20) gomery potential M, the hydrostatic equation is written ϭ p. Thus, M can be obtained by integrating ␣ץ/Mץ z asץ HV ϭ p with respect to ␣. In GIM, p is integrated ␣ץ/Mץ ␪ where ␶ is the surface wind stress, Q is the surface from Eq. (3) and is directly used for pressure gradient S heat ¯ux, cw is the speci®c heat of sea water, and Q is computation. a surface salinity ¯ux. Since the model is a free-surface 5) In the present model, Hu's (1996a) algorithm for model, the model surface salinity can also be forced computing interior diapycnal mixing and diapycnal ve- with the following natural surface boundary conditions locity is used (see appendix B for a summary). This is (Huang 1993): the only algorithm that has been tested and veri®ed in e ϭ E Ϫ P, (21) both extensive one-dimensional vertical diffusion problems (Hu 1991, 1996a) and three-dimensional isopycnal Sץ A ϭ 0. (22) model experiments of vertical-diffusivity sensitivity (Hu z 1991, 1996b). The computed diapycnal mass exchangesץ HV are consistent with the de®nition of diapycnal velocity, In the model, the interface-depth diffusion is applied and at the same time, maintain density coordinates at not only in the interior ocean but also on the sea surface each time step. The algorithm can incorporate any equa- to remove the checker board noise in the sea surface height ®eld. At the sea¯oor, on the other hand, the bot- tion of state, including the UNESCO equation of state, tom depth z should not be changed by the interface while maintaining reasonable computational ef®ciency b (Hu 1996a). Diapycnal mixing algorithm used in MI- diffusion. Thus, KI is set to zero at z ϭ zb. This imposes a boundary condition on (6); that is, COM is described by Bleck and Chassignet (1994), which is a simpli®ed version of Hu's (1991) algorithm.

KI١szb ϭ 0. (23) In practice, MICOM's algorithm has been used only

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with an equation of state from which temperature can CF hereafter). As in CF, the model computational do- be analytically solved as a function of salinity and po- main covers a global ocean spanning 80ЊSto80ЊN with tential density. realistic coastal geometry and bottom topography. The 6) In addition to the Kraus±Turner mixed layer, the horizontal resolution is 2Њ long and 1Њ lat in spherical present model has another option of the model surface coordinates. In the vertical, the model has 12 layers with layer. In this option, the model surface layer is a con- prescribed potential densities of 22.0, 23.0, 24.0, 25.0, stant-depth level (Hu 1997) as that in MOM, and thus 25.8, 26.2, 26.8, 27.2, 27.4, 27.6, 27.85 in ␴ units for no parameters associated with the Kraus±Turner mixed- the 11 isopycnic layers underlying the mixed layer. This layer model are introduced. vertical layout provides about nine layers in the top 550 7) Both MICOM and GIM use explicit mode-split- m, compared to about 12 levels in the same depth range ting techniques to allow prediction of sea surface height. in CF.

In MICOM, however, mode splitting involves an as- As in CF, the lateral viscosity AMH and diffusivity AHH sumption that layer thickness perturbations caused by are, respectively, 2 ϫ 108 cm2 sϪ1 and 2 ϫ 107 cm2 sϪ1,

barotropic ¯ux divergences are proportional to the layer and the vertical viscosity AMV and diffusivity AHV are thickness itself. In GIM, this assumption is relaxed, and Richardson number dependent (Pacanowski and Philan- the total layer thickness is directly predicted from Eq. der 1981); that is, (2) (see Hu 1997). ␯ 8) In MICOM the computational domain is de®ned A ϭϩ0 ␯ , (23) MVn b on a Mercator projection, whereas in the present model, (1 ϩ aRi) the domain is de®ned directly in spherical coordinates. A A ϭϩMV ␬ , (24) 9) In ®nite difference formulation, MICOM uses the HV(1 ϩ aR )n b potential-enstrophy conserving ®nite-difference equa- i 2 22 tions in s coordinates derived by Bleck (1979) on a C where Ri ϭ N /(uzzϩ ␷ ) is the Richardson number, ␯b 2 Ϫ1 2 Ϫ1 grid (Arakawa and Lamb 1977), whereas GIM use an (ϭ1cm s ) and ␬b (ϭ0.1 cm s ) are the background 2 energy-conserving ®nite-difference equation in s coor- dissipation parameters, n ϭ 2, a ϭ 5, and ␯ 0 ϭ 50 cm Ϫ1 2 dinates derived by Hu (1997) on a B grid. s . Note that AMV and AHV can be as large as 50 cm sϪ1, which poses a constraint on the model time step in that the algorithms for computing diapycnal diffusive g. Comparison with the MOM formulation and advective ¯uxes (Hu 1996a) are numerically ex- The GIM was designed to facilitate a direct compar- plicit. It turns out that the large vertical diffusivity con- ison with MOM, and thus it bears many resemblances strains the model's time step. To get around this prob- 2 Ϫ1 to MOM in model formulation. Both MOM and the lem, we set an upper limit 5 cm s for AHV computed present model use primitive equations with the Bous- from Eq. (24). sinesq and hydrostatic approximations. In both models, The Richardson number±dependent mixing imple- the ®nite-difference equations are formulated on a B mentation is very expensive in GIM. One reason is that grid in the spherical coordinates and conserve momen- the momentum equations in GIM are solved after solv- tum, mass, heat, and salt. Furthermore, the quadratic ing the continuity and tracer equations (including quantity conserved by the ®nite-difference equations is mixed-layer entrainment/detrainment), so the model energy in both models. The basic parameters in MOM strati®cation is different from that for solving the tracer

are AMH, AMV, AHH, and AHV. In comparison, the present equations. Because AMV is needed in computing AHV [see model has one more basic parameter KI, in that the (24)], the expensive AMV is computed twice in the pre- model usually becomes unstable without this explicit sent implementation. As a result, GIM costs about the damping in the continuity equation. But in the latest same cpu time as MOM in this study. Both models version of MOM (Pacanowski 1995), the isopycnal- needed about 24 cpu hours of Cray/Y-MP per model depth diffusion parameterization of eddy-induced tracer year. Although the MOM version used in this study has transport by Gent and McWilliams (1990) and Gent et higher resolution, it does not contain the Gent±Mc- al. (1995) is incorporated into the model. Thus, it is Williams (1990) isopycnal depth diffusion scheme as possible to compare the two models in the same param- GIM does. Thus, the computational cost required by eter space, boundary conditions, and con®guration (ex- GIM and MOM should be comparable. cept the vertical coordinates) when the constant-depth In CF, the standard version of the GFDL MOM with surface layer option is used in GIM (see Hu 1997). horizontal/vertical mixing parameterization was used. As mentioned in section 2g, the present model has one

more basic parameter, KI. For the reason to be discussed 3. Design of the real-time simulation 7 2 Ϫ1 in section 5a, a large KI (ϭ3 ϫ 10 cm s ) is used in In this study, a real-time simulation during 1992±93 the isopycnal model simulation. is carried out with the GIM described in the preceding As in CF, the model was spun up from the Levitus section. The model simulation is designed according to (1982) January climatology with the Hellerman and Ro- the MOM simulation published by Chao and Fu (1995; senstein (1983) climatological monthly mean surface

Unauthenticated | Downloaded 09/26/21 10:49 PM UTC MAY 1999 HU AND CHAO 711 wind stress. The surface temperature and salinity are 3-day intervals. A Gaussian weighting function was relaxed to the Levitus monthly mean values. In CF, the used in both space and time with e-folding scales of relaxation timescale ␶ R is 30 days for a 10-m top level. 500 km and 5 days, respectively. The resulting dataset In other words, the surface anomaly damping rate suppresses signi®cant portions of mesoscale motions Ϫ1 ⌬Z1/␶ R is ⅓ m day in CF (⌬Z1 is the top-level depth). and retains mainly the large-scale sea level variabilities. In the present model, the mixed-layer depth is variable; The SST data were taken from NCEP's weekly anal- we can prescribe either ␶ R or ⌬Z1/␶ R, not both. We ysis. The data combine the in situ and satellite (Ad- Ϫ1 decided to set ⌬Z1/␶ R to ⅓ m day in that the prescribed vanced Very High Resolution Radiometer) observations damping rate assures that the model surface heat ¯ux (Reynolds and Smith 1995). The resolution is 1Њϫ1Њ. will be the same as that of CF when the difference The subsurface temperature pro®les are provided by Dr. between the model and the Levitus SST is the same. Warren White at Scripps Institution of Oceanography After 10 yr of spinup, the model was driven with the based on the ship of opportunity expandable bathy- surface wind stress derived from the daily wind from thermographs (XBTs). The XBT data were gridded with the National Centers for Environmental Prediction a resolution of 5Њ long ϫ 2Њ lat. Both the SST and XBT (NCEP) 1000-mb analysis data for January 1992±De- data were obtained during the same period as the TO- cember 1993. The wind speed was also used to compute PEX/Poseidon data and the OGCM simulations (1992± the real-time surface sensible heat (SH) and latent heat 93). (LH) ¯uxes, which are, respectively, given as (see CF)

SH ϭ ␳aDpaCCU(T0 Ϫ Ta), (25) 5. Model±data intercomparisons In this section, we shall present the sea level and LH ϭ ␳aDCLU a[E s(T0) Ϫ ␥Esa(T )](0.622/P a), (26) upper-ocean temperature simulated with MOM and GIM where T 0 is the model sea surface temperature, Ta is the for the period of September 1992±September 1993. The atmospheric surface temperature, Ua is the wind speed, data for MOM is from the CF simulation. The obser- Ϫ1 ␥ is the mixing ratio, CD ϭ 0.0012, Cp ϭ 0.24 cal g vational datasets described in section 4 are used for the Ϫ1 Ϫ1 Ϫ3 ЊC , L ϭ 595 cal g , ␳a ϭ 0.0012 g cm , Pa ϭ 1013 validation of the model comparison. Therefore, we in mb, and the saturation vapor pressure Es is given as fact present a three-way intercomparison between the (9.4Ϫ2353/T) data and the two model simulations. Es(T) ϭ 10 . (28) As in CF, the total surface heat ¯ux (Q) for the real- time simulation with GIM is a. The annual-mean conditions Q ϭ SW Ϫ LW Ϫ LH Ϫ SH, (29) Figure 1 shows maps of annual-mean sea surface height during September 1992±September 1993 for TO- where SW and LW are the shortwave and longwave PEX/Poseidon (T/P hereafter), MOM, and GIM. Both radiative heat ¯uxes, which are assumed to be only a GIM and MOM are able to simulate the main features function of latitude (Chao and Philander 1993; Chao et of the observed annual-mean sea surface height such as al. 1993). 1) highs associated with subtropical gyres in the Paci®c and Atlantic Oceans, 2) lows associated with subpolar 4. Observational datasets gyres in the North Paci®c Ocean and North Atlantic Ocean, and 3) the low belt associated with the Antarctic The altimeter data used in this study were collected Circumpolar Current. The major qualitative differences from the National Aeronautics and Space Administra- between the model simulations and the observations are tion dual-frequency radar altimeter and the Centre Na- found 1) along the coasts of Peru and northern Chile, tional d'Etudes Spatiales single-frequency solid-state ra- where local highs in sea level are observed in T/P but dar altimeter (Fu et al. 1994) during the ®rst 360 days absent in both GIM and MOM; 2) in the equatorial of the TOPEX/Poseidon mission (23 September 1992± maritime region in the eastern Indian and western Paci®c 18 September 1993). Standard correlations and editing Oceans, where the lows observed are absent in both procedures were applied as suggested in the Geophys- GIM and MOM simulations; and 3) in the eastern equa- ical Data Record Users Handbook (Callahan 1993). Ad- torial and tropical Atlantic, where the observed sea sur- ditional corrections were applied for the effects of the face height is positive, whereas the opposite is true in ocean tides, the solid earth tides, the inverted barometer both MOM and GIM. Because the mean sea surface correction, and the pole tide. The altimeter data were height in both MOM and GIM is almost the same in ®rst interpolated to ®xed grids 6.2 km apart along each these three regions, the qualitative differences must be satellite track for collinear analysis. The time average associated with either de®ciencies common in both of sea level at each grid was then calculated and re- models, such as the model forcing and resolution, or moved. To create arrays of time series on a regular errors in computing the Geoid using the T/P observa- space±time grid for the analysis, the sea level data were tion. Quantitatively, the amplitude of the spatial vari- interpolated onto a space±time grid of 1Њϫ1Њ boxes at ation of the mean sea surface height in GIM is somewhat

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FIG. 1. Annual mean sea surface height for 23 Sep 1992±23 Sep 1993 for (a) the TOPEX/Poseidon observation, (b) the MOM simulation, and (c) the GIM simulation. FIG. 2. Annual-mean SST for 23 Sep 1992±23 Sep 1993, for (a) the Reynolds and Smith observation, (b) the MOM simulation, and (c) the GIM simulation. smaller than those in MOM and T/P. This is probably due to the fact that the interface smoothing is also ap- Figure 3 shows the observed and model-simulated plied on the sea surface in GIM. annual-mean potential temperature for the upper 400 m Figure 2 shows the maps of annual-mean SST for the of ocean on the equator. In the central and eastern equa- observations and simulations of MOM and GIM. In gen- torial Paci®c, the simulated main thermocline in both eral, the annual-mean SST distributions in MOM and MOM and GIM is in general not as sharp as observed. GIM agree well with the observations. But SST in both Between 90ЊW and the east Paci®c coast, however, the models is about 1Њ to 2ЊC warmer than that observed GIM-simulated thermocline in the upper 50 m is much in the equatorial and tropical regions, except the eastern sharper than observed, whereas that in MOM looks more equatorial Paci®c and Atlantic. This warm SST bias is reasonable. The SST in GIM is also cooler than MOM probably associated with the lack of solar penetration in this region. Why? In this near-shore region, the strong in both models. In MOM, the effective solar penetration wind-driven divergence in the surface layer causes a depth is 20 m, whereas that in GIM is the mixed-layer subsurface convergence and upwelling in both models. depth, which is only a little more than 20 m in the It should be pointed out that the mixed layer gets shal- equatorial and tropical region in annual average. Thus low in response to the surface divergence in GIM. But the solar penetration depth seems too shallow in both the surface divergence in this region is so strong that models, and inclusion of solar penetration would help the mixed-layer depth would be driven to zero. As in improve the warm SST bias in this region. MICOM, we set a minimum mixed-layer depth in GIM,

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FIG. 4. Annual-mean zonally averaged potential temperature for FIG. 3. Annual-mean potential temperature for the upper 400 m of the upper 400 m of global ocean for (a) the Scripps observation, (b) equatorial ocean for (a) the Scripps observation, (b) the GFDL MOM the MOM simulation, and (c) the GIM simulation. simulation, and (c) the GIM simulation.

is sharper in GIM than MOM due to the lack of sub- which is 10 m in this study. Since both GIM and MOM surface lateral heat exchange. The sharp front could have the same surface-layer depth in this eastern equa- potentially cause numerical instability in GIM. The most torial Paci®c near-shore region, the vertical velocity at effective way to diffuse the sharp thermocline in GIM the bottom of the surface layer should be roughly the is isopycnal-depth diffusion. It reduces the steep iso- same in both models. So the SST in this region is de- pycnal slope, in¯ates the outcropping layers, and thus pendent on the temperature of the water upwelling into helps with the model stability. For this reason, the large 7 2 Ϫ1 the surface layer. isopycnal-depth diffusivity KI (ϭ3 ϫ 10 cm s )is In MOM subsurface convergence and mixing occur used. Despite the large KI, the thermocline slope in GIM horizontally, which effectively transport heat from the is still steeper than that of MOM in the eastern equatorial surrounding warm waters to the region of upwelling. Paci®c.

As a result, the cold water upwelling from the depth is Although the large KI is not large enough for the warmed up, resulting in a less sharp thermocline. This eastern equatorial Paci®c, it is too high for other regions. mechanism, however, does not exist in GIM, in which Figure 4 shows the zonally averaged potential temper- subsurface advection and mixing predominantly occur ature for the upper 400 m of global ocean. Due to the along isopycnic layers. Because temperature is almost large interface diffusivity, the isotherm doming in the homogeneous within an isopycnic layer, isopycnal ad- equatorial and tropical thermocline is excessively vection and mixing do not result in effective lateral heat smoothed in GIM. The trade-off between the numerical exchange. Therefore, the SST is cooler and thermocline stability and excessive isopycnal smoothing in GIM in-

Unauthenticated | Downloaded 09/26/21 10:49 PM UTC 714 MONTHLY WEATHER REVIEW VOLUME 127 dicates the necessity of improvement in the isopycnal- the Brazil/Malvinas con¯uence, respectively (Figs. 5a depth diffusion used in the GIM simulation. Note that and 6a). This dipole structure is qualitatively simulated the grid size in meridional direction is only half that in in both GIM and MOM (Figs. 5b,c and 6b,c). The major the zonal direction. The grid-size-dependent interface model data difference is that the amplitude of the model- diffusivity used in Bleck et al. (1992) seems to be more simulated sea level variability is much larger than ob- reasonable; this would reduce the smoothing of ther- served on the equator and in the Tropics, but is much mocline doming in the meridional direction, and in the smaller than observed in the areas of the Gulf Stream mean time, maintain reasonable zonal slope of the equa- and the Brazil/Malvinas con¯uence. In comparison, it torial thermocline. seems that MOM is better in simulating the sea level Note that the MOM-simulated annual- and zonal- variabilities in the western boundary current out¯ow mean vertical temperature gradients (Fig. 4b) are re- regions and GIM is better in simulating the background markably sharper than those of the GIM simulation (Fig. variabilities in the North and South Atlantic open 4c) and the XBT observation (Fig. 4a) at about 20 m oceans. of depth. The sharp temperature gradient indicates a In the Indian Ocean, MOM simulates the seasonal poor vertical heat exchange between the surface bound- variation of sea level reasonably well despite somewhat ary layer and the interior in the MOM simulation. As strong amplitude in the model. This can be seen in Figs. will be discussed in section 5c, the sharp subsurface 5 and 6. The GIM-simulated seasonal variation in the temperature gradient in MOM is due mainly to the lack Indian Ocean is similar to that of MOM, but the am- of a realistic mixed-layer model. plitude is much larger, indicating waves are more active in the free-surface GIM than in the rigid-lid MOM in b. Seasonal variation of sea level this small basin with monsoon forcing. The major dis- crepancy between the GIM simulation and the T/P ob- Large-scale sea level variability in T/P observation servation in the Indian Ocean occurs around Madagascar and MOM simulation was discussed in CF by means of Island, where the GIM-simulated and T/P-observed sea standard deviation and spectrum analysis. In this section level variabilities seem to be out of phase. In the open we shall use maps of sea surface height anomaly in the ocean of the south Indian Ocean, we note again that the boreal winter (December±February) and boreal summer GIM-simulated background variabilities are better than (June±August) to describe and compare the observed those of MOM. and simulated seasonal variations of sea level during In summary, the seasonal variations of sea surface the years 1992±93 (Figs. 5 and 6). height simulated with both MOM and GIM are in rea- In the boreal winter of 1992±93, the Paci®c was dom- sonable agreement with the T/P observations. Although inated by negative sea level anomalies north of 7ЊN and it is hard to determine which model has better overall with positive anomalies south of 7ЊN (Fig. 5a). The opposite is true in the boreal summer of 1993 (Fig. 6a). performance, it seems that MOM performs better in the Another observed feature of sea level variability is that western boundary current out¯ow regions and GIM is the amplitude of seasonal variability is larger in the better in simulating the background variability in the North Paci®c than in the South Paci®c. Seasonal vari- open ocean. This may be explained as follows. As noted ations of sea level simulated with both MOM and GIM in CF, the MOM-simulated sea level variability in the are in reasonable agreement with the observations in region of the Gulf Steam is closely related to the current the Paci®c Ocean (Figs. 5b,c and 6b,c). In the eastern strength. Weak western boundary current out¯ows due equatorial Paci®c between 20ЊS and 10ЊN, however, the to the coarse resolution is believed to be a major reason seasonal variabilities of sea level in the two models are for the weak sea level variabilities simulated with MOM much stronger than the T/P observations. In both mod- in the region. Note that the surface mixed layer in GIM els, maximum variabilities occur not only on the two is much deeper than the top level of MOM in the mid- sides of the boreal intertropical convergence zone, but latitudes, and thus the western boundary current out- also on the two sides of 7ЊS. This suggests there may ¯ows are much weaker in GIM. As a result, sea level be a stronger than normal South Paci®c convergence variability in GIM is even weaker than that in MOM. zone in the NCEP analysis (a common problem in most The background sea level variability in the open ocean, atmospheric models), which produces a symmetric sea on the other hand, is closely related to that of heat level response about the equator. Furthermore, the back- content in the upper ocean (see the discussion in section ground variabilities in both GIM and MOM are weaker 5b). Due to the Kraus±Turner mixed layer, seasonal var- in the North Paci®c and stronger in the South Paci®c iations of heat content simulated with GIM agree better than those in T/P. In comparison, the background var- with the observations than those simulated with MOM, iabilities simulated with GIM agree with the observa- and so do the background variabilities simulated with tions slightly better than those simulated with MOM. GIM. As noted in CF, MOM produced too much vari- In the Atlantic Ocean, seasonal variation of sea level ability in the equatorial and tropical regions. This is also displays a north±south dipole structure with maximum true for GIM. This further con®rms CF's conjecture that variabilities located in the areas of the Gulf Stream and wind stress is overestimated in these regions.

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FIG. 5. Sea surface height anomaly averaged over Dec±Feb 1993 for (a) the TOPEX/Poseidon observation, (b) the MOM simulation, and (c) the GIM simulation. c. Seasonal variation of the upper-ocean temperature and summer months. The temperature anomaly sections The seasonal variation of upper-ocean temperature is derived from the XBT observations and simulations studied by plotting latitude-depth sections of zonally with MOM and GIM are shown in Figs. 7 and 8 for averaged temperature anomalies for the extreme winter March and September of 1993. The observed seasonal

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FIG. 6. As in Fig. 5 but for Jun±Aug 1993. variation of temperature is pronounced in the top 100 cooling (southern summer warming). In contrast, we see m (Figs. 7a and 8a). Just like the seasonal variation in boreal summer warming (southern winter cooling) in sea surface height, the seasonal temperature variation is September. Furthermore, the amplitude of the seasonal not symmetric about the equator. In March, the cold temperature variation north of 7ЊN is larger than that (warm) anomalies north of 7ЊN indicate boreal winter south of 7ЊN. As noted in section 5a, this is also true

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FIG. 7. Zonally averaged temperature anomaly for the upper 400 m of global ocean in March 1993 for (a) the Scripps observation, (b) FIG. 8. As in Fig. 7 but for Sep 1993. the MOM simulation, and (c) the GIM simulation.

in MOM is much less than observed. In GIM, the sharp for the seasonal variation of sea surface height. The vertical gradient is gone and downward penetration of similarities of the seasonal variations of the upper-ocean the seasonal variation is much improved. temperature to those of sea surface height indicate a The difference in the downward penetration of sea- close relationship between the upper-ocean heat content sonal temperature variation between MOM and GIM and sea surface height: sea level is high when the water seems to arise from different surface boundary-layer column is heated and low when the water column is formulations in the two models. In MOM, the wind- cooled. stirring effect is represented by a large vertical diffu- Qualitatively, the above observed features of the sea- sivity and viscosity at the bottom of the top level, and sonal variation in the upper-ocean temperature can be thus the wind-induced mixing can directly reach only seen in both MOM and GIM simulations (Figs. 7b,c and the second model level. Due to the high vertical reso- 8b,c). Quantitatively, however, there exist obvious mod- lution, the total depth of the top two levels in MOM is el±data differences. In MOM, there is a sharp vertical only 20 m. Since this depth is coincident with the depth gradient of temperature anomaly at about 20-m depth, of the sharp gradient of the temperature anomaly in Figs. which is consistent with the sharp mean temperature 7b and 8b, the weak penetration of the seasonal tem- gradient shown in Fig. 4b. Due to the sharp mean sub- perature variation seems to be associated with the lack surface gradient, the region below it is almost insulated of wind mixing below 20 m. In GIM, the wind-stirring from the seasonal temperature variation. As a result, the effect is represented by a Kraus±Turner mixed-layer penetration depth of the seasonal temperature variation model, and the penetration depth of the seasonal vari-

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erhuber's isopycnal model, which has a Kraus±Turner mixed layer. The results shown in Figs. 7 and 8 are consistent with Duffy et al.'s reasoning that low uptake of bomb 14C with MOM would get worse when the model vertical resolution increases. Therefore, a realistic surface mixed layer is needed in simulating the ocean uptake of transient tracers particularly at high resolutions. As noted at the beginning of this subsection, sea level variability is closely related to the variability of heat content. Note that the temperature anomalies shown in Figs. 7 and 8 are zonally averaged. So the associated heat content variability is basically related to the background sea level variability in the open ocean. The weaker background sea level variabilities in MOM and GIM, therefore, seem to be associated with the weaker heat content variabilities in the two models. Heat content variability is determined by both the amplitude and the penetration depth of seasonal temperature variation in FIG. 9. Zonally averaged mixed-layer depth (unit: m) in Mar (solid the upper ocean. In MOM, the amplitude of seasonal curve) and Sep (dashed curve) in GIM. temperature variation is as big as that observed in the boreal ocean and is larger than observed in the southern ocean. However, the penetration depth is too shallow. ation is determined by the annual maximum depth of In GIM, the penetration depth is much deeper than that the mixed layer. Figure 9 shows the zonally averaged in MOM but remains shallower than that observed. In mixed-layer depth in GIM in March and September the meantime, the amplitude of the temperature variation when the model mixed layer reaches roughly its max- in GIM is much smaller than observed in the boreal imum in the boreal and southern oceans, respectively. ocean. Therefore, heat content variabilities in both mod- The dashed curve south of 7ЊN and the solid curve north els are weaker than the observations. In comparison, of 7ЊN in Fig. 9 approximately represent the depth that heat content variability in GIM is somewhat better than the seasonal variation can reach in an annual cycle, that in MOM, especially in the southern ocean. This is which roughly coincides with the 0.5ЊC/Ϫ0.5ЊC contour consistent with our analyses in section 5a that show that lines in Figs. 7c and 8c. As shown in Figs. 7 and 8, the background sea level variabilities in GIM seem to be 0.5ЊC/Ϫ0.5ЊC contour lines in GIM are much deeper in better agreement with the observations than those in than those in MOM. MOM. Note that the smaller-than-observed penetration Figures 7 and 8 show that although the penetration depth in GIM may be associated with the fact that an depth in GIM is larger, the amplitude of the temperature e-folding depth of 50 m is set to the available surface variability is smaller. The temperature anomaly ampli- turbulent kinetic energy. We expect that the penetration tude in GIM agrees with the observations in the southern depth will increase as the e-folding depth increases. On ocean, but it is too small in the midlatitudes of the boreal the other hand, the small amplitude of temperature var- ocean. This implies that the surface heat ¯ux forcing in iation in GIM in the boreal ocean indicates that thermal the boreal midlatitudes is too weak. Besides, the pen- forcing in the boreal midlatitudes is too weak. etration depth in GIM is still small in low latitudes compared with the observations. According to Ober- huber (1993), the downward mixed-layer penetration in d. Intraseasonal sea level variability low latitudes can be improved when a larger m and a realistic estimate of hc are used in (8) and (9). In addition to the annual cycle, the T/P data also The better downward penetration of temperature exhibit pronounced intraseasonal sea level ¯uctuations anomaly in GIM also indicates the importance of a re- over many parts of the World Ocean (CF). It is a good alistic mixed layer in simulation of uptake of anthro- test for GIM that to what extent the model can reproduce pogenic geochemical tracers. The mechanisms that drive these intraseasonal variabilities. Here we show time se- the downward heat penetration in OGCMs also play a ries of sea surface height anomaly averaged over four predominant role in driving the downward penetration areas for T/P, MOM, and GIM in Fig. 10. The four areas of transient tracers in these models. In simulations of are, respectively, in the North Paci®c Ocean (40Њ±50ЊN, uptake of bomb 14C, Duffy et al. (1995) showed that 160ЊE±160ЊW), the southeast of the Indian Ocean compared to the GEOSECS observations, bomb 14C in- (40ЊS±60ЊS, 80Њ±120ЊE), the southeast Paci®c Ocean ventory and penetration depth tend to be small in sim- (40Њ±60ЊS, 120Њ±80ЊW), and the southwest Atlantic ulations with MOM and large in simulations with Ob- Ocean (40Њ±50ЊS, 30Њ±50ЊW), in which, as noted in CF,

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FIG. 10. The sea surface height time series averaged over areas in the North Paci®c Ocean (40Њ±50ЊN, 160ЊE±160ЊW), the southeast of the Indian Ocean (40Њ±60ЊS, 80Њ±120ЊE), the southeast Paci®c Ocean (40Њ±60ЊS, 120Њ±80ЊW), and the southwest Atlantic Ocean (40Њ±50ЊS, 30Њ±50ЊW) for the TOPEX/Poseidon observation (solid curves), the MOM (dotted curves), and GIM (dashed curves) simulations. intraseasonal variabilities of sea level are signi®cant in Paci®c Ocean where the coherence values are compa- both T/P and MOM. rable. Overall, the GIM-simulated sea level agrees well with the MOM simulation and observations. As noted in CF, 6. Summary and discussions intraseasonal variations are superimposed on the seasonal variations in the North Paci®c and South Atlantic In this study, the upper-ocean variabilities during and are predominant in the South Indian and Paci®c 1992±93 are simulated with a GIM and MOM at non- Oceans. All these features are well simulated by both eddy-resolving resolutions. Results from the model sim- MOM and GIM. The GIM-simulated time series appear ulations were compared with the available observations to agree better with the observations than the MOM for the same time period; the observations include sea simulated. Table 1 shows the coherence amplitude for surface height from TOPEX/Poseidon, SST from week- MOM and T/P and for GIM and T/P. The coherence ly NCEP analysis, and vertical temperature pro®les from amplitude values for GIM and T/P are larger than those gridded XBT data. The major differences between GIM for MOM and T/P in the four areas except in the North and MOM used in this study are the vertical coordinates,

TABLE 1. Coherence amplitude. Coherence N. Paci®c S. E. Indian S. E. Paci®c S. W. Atlantic amplitude (40Њ±50ЊN, 160ЊE±160ЊW) (40Њ±60ЊS, 80Њ±120ЊE) (40Њ±60ЊS, 120Њ±80ЊW) (40Њ±50ЊS, 30ЊE±50ЊW) T/P and MOM 0.58 0.67 0.61 0.53 T/P and GIM 0.57 0.73 0.64 0.55

Unauthenticated | Downloaded 09/26/21 10:49 PM UTC 720 MONTHLY WEATHER REVIEW VOLUME 127 a Kraus±Turner mixed layer, and a tracer-transport ve- show that the background sea level variability in the locity associated with interface-depth diffusion. The ef- open ocean is closely related to that of heat content in fects of these differences in model formulation on the the seasonal thermocline, which depends on both the model simulations are investigated. amplitude and the penetration depth of the seasonal tem- Due to the difference in the orientation of interior perature variation in the upper ocean. In GIM, the ¯ow and mixing, SST and the thermocline strati®cation Kraus±Turner mixed-layer model enhances the down- in the eastern equatorial Paci®c in GIM are more sen- ward penetration of the seasonal temperature variation, sitive to the surface wind stress forcing than they are and thus the model-simulated background sea level var- in MOM. In MOM, subsurface horizontal advection and iabilities tend to be better than those in MOM. mixing effectively transfer heat to the wind-driven up- The Kraus±Turner mixed layer used in this study also welling region and warm up the water upwelling into has de®ciencies. It makes the western boundary current the surface boundary layer. In comparison, without is- out¯ows in GIM weaker than those in MOM. In these opycnal-depth diffusion, there is no effective means in regions, the sea level variability is closely related to the GIM to transfer heat between the upwelling cold water out¯ow strength. As a result, GIM does not simulate and the surrounding warm water since subsurface ¯ow sea level variabilities as well as MOM because of the and mixing predominantly occur along isopycnic layers. weaker currents associated with the deeper top model In GIM, therefore, the SST tends to be cold and the layer. Therefore, the single-layer Kraus±Turner mixed front tends to be sharp in the wind-driven upwelling layer used in the isopycnal model needs to be improved. region. The sharp front could potentially cause numer- A better alternative seems to be a shallow constant-depth ical instability in GIM. The isopycnal-depth diffusion Ekman layer embedded in a deeper and variable Kraus± is the most effective way to reduce the steep slope of Turner mixed layer, which maintains the strength of the isopycnals and the strength of the front associated with surface currents, and in the meantime allows deeper the cold upwelling in GIM. Thus, a large isopycnal- penetration of wind stirring. depth diffusivity KI has to be used to maintain the model In the equatorial and tropical regions, the strong sea stability in the eastern equatorial Paci®c. But the large surface height variabilities simulated with the isopycnal

KI results in excessive smoothing in the meridional iso- model supports CF's conclusion that wind forcing used therm doming in the equatorial and tropical thermocline. in the real-time simulation is too strong in these regions.

A grid-size dependent KI (Bleck et al. 1992) seems help- In this region, both models have a warm SST bias of ful in reducing the excessive isopycnal smoothing and 1Њ to 2ЊC. It seems that inclusion of solar penetration is recommended for the follow-up study. would help to reduce this warm bias. The sea surface height simulations with both models are in reasonable agreement with the T/P observations. Acknowledgments. The GIM used in this study was However, MOM poorly simulates the vertical distribu- developed by the ®rst author at the Geophysical Fluid tion of the seasonal temperature anomalies in the upper Dynamics Laboratory (GFDL) and the Joint Institute ocean (the baroclinic component of the upper-ocean var- for the Study of the Atmosphere and Ocean (JISAO). iability) during 1992±93. Due to the lack of a realistic The ®rst author would like to thank the UCAR Ocean surface mixed-layer formulation, the MOM-simulated Modeling Postdoctoral Program, GFDL, K. Bryan, and vertical temperature pro®les have a sharp gradient at P. Rhines for their support for the primary development about 20 m of depth. As a result, the region below the of GIM during his visit to GFDL in 1992 and 1993, sharp temperature gradient is almost insulated from the and thank E. Sarachik for his support for further de- in¯uence of the seasonal temperature variation. This velopment and improvement of GIM at JISAO. The de®ciency in MOM has potential problems when used authors are grateful to GFDL for providing the Modular to assimilate the altimetric sea level data. Although the Ocean Model, to L. Fu [Jet Propulsion Laboratory simulated sea level agrees well with the altimetric ob- (JPL)], and C. K. Shum (University of Texas, Austin) servations, the vertical temperature (and salinity) pro- for providing the gridded TOPEX/Poseidon data, and ®les are poorly simulated. When sea level is projected to Warren White (Scripps Institute of Oceanography) onto the subsurface temperature and salinity according for providing the gridded XBT data. An anonymous to the model statistics (e.g., Mellor and Ezer 1991), it reviewer suggested that the warm SST bias in equatorial is very dif®cult, if not impossible, to recover the bar- and tropical region in both GIM and MOM can be im- oclinic part of the sea level variations regardless the proved by taking into account the effect of penetration assimilation schemes being used. In comparison, the of solar radiation. In this study, the ®rst author was GIM-simulated vertical temperature pro®les are free of supported by grants from the Department of Energy the sharp subsurface gradient and are in good agreement National Institute for Global Environmental Change and with the XBT observations. The results indicate that a the NOAA Of®ce of Global Programs to the Hayes Cen- realistic mixed-layer model is needed in order to assim- ter at JISAO. This study was performed in part by the ilate altimetric sea level observations and to simulate JPL under contract with National Aeronautics and Space the uptake of transient tracers. Administration. Computations were performed on the Both observational data and the model simulations Cray/Y-MP through the JPL Supercomputing Project.

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APPENDIX A Mixed-Layer Detrainment During the mixed-layer detraining season, the turbulent mixed layer (ML) recedes upward and leaves old old water in a fossil ML below it. Let␳11 and⌬Z be the ML density and depth at the beginning of time interval ⌬t. If ML detrainment occurs, the new ML depth at the end of ⌬t is given by the Monin±Obukhov length LMO new [see Eq. (10)]. The new ML temperature␪ 1 and sa- new linityS 1 can be determined by distributing the incom- ing heat and salinity over any depth between the old old ML depth⌬Z1 and the Monin±Obukhov length LMO. In the present model, the new ML temperature and salinity are determined by LMO, and the new ML density new ␳1 is then obtained from the equation of state. Although density in the ML is allowed to vary con- tinuously, density in the interior assumes certain discrete values only because of its role as the vertical coordinate. As a result, the density of the ¯uid detrained from the ML usually does not match any of the discrete density FIG. A1. A schematic diagram of the mixed-layer detrainment. The solid curve indicates the initial density pro®le and the short dashed values of the interior, and therefore must be modi®ed curve indicates the mixed-layer density and depth predicted by the to allow detrainment into the isopycnic layers. Let us Kraus±Turner mixed-layer model. The shaded area shows that the assume that the density of the detrained water column fossil mixed-layer water is partitioned into two parts. One is distributed into layer k and another is returned to the mixed layer. falls between two coordinate density values, ␳k and ␳kϪ1 (k increasing with ␳). In general, the detrained ¯uid can be distributed into layer k and those isopycnic layers whose reference densities are less than ␳k, but larger ␳newϪ ␳ old than or equal to␳new . In the detrainment algorithms de- ␦zbot ϭ 11d, (A1) 1 1 new veloped in Bleck et al. (1989, 1992), layers k and k Ϫ ␳1 Ϫ ␳k 1 are considered as possible layers that receive detrained old where d ϭ⌬Z 1 ϪLMO. Since water column d Ϫ ¯uid. Certainly one can consider more possible receiv- bot ␦z1 is retained in the new mixed layer, we match its ing layers. For the most time during the ML detraining new new temperature and salinity to␪ 11 andS (calculation of season, however, the old ML ¯uid can be detrained only new new ␪ 11,S is described in the ®rst paragraph of this ap- into layer k. This is particularly true when the time step bot new old pendix). This requires that (d Ϫ ␦z11)(␪␪Ϫ 1) is small (e.g., a couple of hours) and the vertical res- bot new old amount of heat and (d Ϫ ␦zS11)( Ϫ S 1) amount of olution is not very high (e.g., about a dozen layers). bot salt, respectively, be transferred from ␦z1 . Temperature This is the case of the present model. Therefore, we and salinity in layer k are then obtained by mixing consider the case that the old ML ¯uid is detrained into bot bot new ␦zz111with layer k, and subtracting (d Ϫ ␦ )(␪ Ϫ layer k only for computational economy. A simple way old bot new old ␪ 1111) and (d Ϫ ␦zS)( Ϫ S), respectively. The re- of doing this is to pump all the detrained ¯uid into layer sulting equations are k, and then to entrain an appropriate amount of denser ␦zbot(␪ newϪ ␪ old) Ϫ d(␪ newϪ ␪ old) ¯uid from isopycnic layers below layer k so that the ␪ ϭ ␪ old ϩ 11 k 11, (A2) kk old bot density coordinate in layer k is restored. However, we ⌬Z k ϩ ␦z1 found that this scheme results in a very diffusive ther- ␦zbot(S newϪ S old) Ϫ d(S newϪ S old) mocline because of the arti®cial mixing with the un- S ϭ S old ϩ 11 k 11, (A3) kk old bot derlying denser water during the ML detrainment. Fur- ⌬Z k ϩ ␦z1 thermore, ML detrainment in the north usually begins where the superscript ``old'' denotes the quantities be- with the bottom layer, and therefore there is no denser fore the input of the surface heat and salinity ¯uxes. In water available at all for restoring the bottom layer co- general, ␪ and S do not yield the reference density ␳ ordinate. k k k due to the nonlinearity of the equation of state. The In the present model, the detrained water column is problem is that ␦zbot given in Eq. (A1) is based on a partitioned into two parts and the buoyancy in the lower 1 linear density budget. According to Hu (1991) and Bleck part is transferred to the the upper part so that the density bot new et al. (1992), an exact ␦z1 can be obtained by substi- in the upper part matches␳1 and the density in the tuting ␪ k and S k in Eqs. (A2) and (A3) as well as the lower part matches ␳k. The depth of the water column bot reference density ␳k into the equation of state; that is, detrained into layer k, ␦z1 , is then given as (see Fig. A1) ␳k ϭ ␳(␪ k, S k). (A4)

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Because ␳ is a nonlinear function of ␪ and S, Eq. (A4) spectively, diapycnal advective ¯uxes of volume, heat, bot is a nonlinear equation for ␦z1 according to Eqs. (A2) and salt. Their ®nite-difference expressions are not giv- and (A3). When the UNESCO equation of state is used, en yet. The last two terms on the rhs of (B2) and (B3) bot Eq. (A4) is a very complicated equation of ␦z1 and are, respectively, diapycnal diffusion terms of ␪ and S. very expensive to solve numerically. Following Hu Except these terms, there is no other diapycnal diffusion (1996a), we use a simpler and less expensive approach computation involved. In other words, these are the only in the present model. First of all, we compute ␪ k and terms for diapycnal diffusion. bot S k from Eqs. (A2) and (A3) using ␦z1 given by Eq. To solve equations (B2)±(B3), all terms on the rhs of (A1). If the density yielded from ␪ k and S k is larger (B2) and (B3) are ®rst computed and integrated for one than the reference density ␳k, we then entrain a slab of time step. The resulting layer temperature and salinity diff diff ¯uid from the ML to layer k. If the opposite is true, we are, respectively, denoted by␪ kk andS . In general, diff diff detrain a slab of ¯uid from layer k to the ML, and match there is a density drift associated with␪ kk andS . This new the temperature and salinity in the slab to␪ 1 and density drift is caused by isopycnal and diapycnal dif- new S 1 by transferring the excess buoyancy in layer k to fusion of ␪ and S. Theoretically, isopycnal advection the slab. The depth of the slab is determined from the does not cause any density change, but numerically, it linear density budget. If the new temperature and salin- does. Among these three causes of density drift, the ity do not yield ␳k, we repeat the procedure until the major cause is vertical diffusive ¯uxes across the layer density drift in layer k is negligibly small. In our mod- interfaces. This density drift can be removed only by eling practice, satisfactory results can be obtained by diapycnal advective ¯uxes of ␪ and S across the layer about four iterations. interfaces because these are the only terms not computed in Eqs. (B2) and (B3). Hu (1991, 1996a) showed that diapycnal velocity, e, APPENDIX B at an interface has to be determined by the vertical Summary of Algorithms for Diapycnal Mixing and diffusive ¯uxes at the mass points of the two bounding Advection Computation isopycnic layers. Because a mass point is an inner (cen- ter) point of the isopycnic layer, the diffusive ¯uxes at In three-dimensional isopycnal models, the ®nite-dif- this point do not change the layer heat and salt content ference equations for diapycnal mixing and advection at all. However, they do serve to redistribute heat and can be given as (Hu 1997) salt within the layer, making the diagnosed density drift in the top half of the layer different from that of the ץ (⌬Zkx) ϩ ␦ (U)␳ ϩ ␦y(V)␳ ϩ (e)kϪ1/2 Ϫ (e)kϩ1/2 bottom half. Hu showed that it is these density drifts .t that should be used in diapycnal velocity computationץ ϭ 0, (B1) Based on the de®nition of diapycnal velocity, ®nite- difference representation of e at an interface, say Z kϩ1/2 ץ (␪⌬Z ) ϩ (e␪) Ϫ (e␪) (this is the interface between layer k and k ϩ 1), can t kkϪ1/2 kϩ1/2 be given as [see Eq. (3.47) in Hu (1991); Eq. (27) inץ xyx Hu (1996a)] ϭϪ␦x(U␪ )␳ Ϫ ␦y(V␪ )␳ ϩ ␦xkIx[⌬ZK␦␪]␳ (␳ botϪ ␳ )⌬Z (␳ top Ϫ ␳ )⌬Z e ϭϩkkkkϩ1 kϩ1 kϩ1 , (B4) y ␦␪ ␦␪ kϩ1/2 2⌬t(␳ Ϫ ␳ )2⌬t(␳ Ϫ ␳ ) ϩ ␦ykIy[⌬ZK␦␪]␳ ϩ KddϪ K , kkϩ1 kkϩ1 ΂΃␦z k 1/2 ΂΃␦z k 1/2 Ϫ ϩ bot top where ␳␳kkϪ ␳k and ϩ1 Ϫ ␳kϩ1 are, respectively, den- (B2) sity drifts in the top half of layer k and bottom half of bot layer k ϩ 1. Note that the diagnostic quantities␳ k and ץ (S⌬Zkk) ϩ (eS) Ϫ1/2 Ϫ (eS)kϩ1/2 top -t ␳ kϩ1 in Eq. (B4) do not mean that the algorithm introץ xyx duces prognostic variables de®ned for top/bottom half ϭϪ␦x(US ) Ϫ ␦y(VS ) ϩ ␦xkIx[⌬ZK␦ S] bot bot top ␳ ␳ ␳ of isopycnic layers. In fact, substituting␪ kkk ,S ,␪ ϩ1 , top andS kϩ1 [see Eqs. (3.52)±(3.55) in Hu (1991); Eqs. (31) y ␦S ␦S ϩ ␦ [⌬ZK␦ S] ϩ K Ϫ K , and (32) in Hu (1996a)] into the equation of state ␳ ϭ ykIy␳ dd␳(␪,S) we have ΂΃␦z k 1/2 ΂΃␦z k 1/2 Ϫ ϩ (B3) ⌬t ␦␪ ␦␪ ␳ bot ϭ ␳␪iso ϩ K Ϫ K , kk⌬Z d␦z d␦z where U and V are, respectively, advective-volume ¯ux- Ά k []΂΃΂΃kkϩ1/2 es in zonal and meridional directions. Equations (B1)± (B3) are straightforward ®nite-difference representa- ⌬t ␦S ␦S S iso ϩ K Ϫ K , (B5) tions of Eqs. (2), (4), and (5) without including iso- kdd⌬Z ␦z ␦z k []΂΃΂΃kkϩ1/2 · pycnal depth diffusion. The e terms on the lhs of Eqs. (B1)±(B3) are, re- where

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the bottom half of layer k are obviously␦zbot␪* and ⌬t xy kkϩ1 ␪ iso ϭ ␪ NϪ1 ϩϪ␦ (U␪ ) Ϫ ␦ (V␪ ) ␦zSbot * , respectively. Here␪* andS* are, respec- kk⌬Z x ␳ y ␳ kkϩ1 kϩ1 kϩ1 k [ tively, actual layer (not half layer) temperature and sa- bot linity of layer k ϩ 1, and the sign of␦zk determines xy bot if the water slab is moving upward into layer k (␦zk Ͼ ϩ ␦xkIx(⌬ZK␦␪)␳ ϩ ␦ykIy(⌬ZK␦␪),␳ bot ] 0) or downward into layer k ϩ 1(␦zk Ͻ 0). Similarly (B6) those due to coordinate restoration in the top half of top top layer k ϩ 1 are, respectively,␦zkϩ1␪*kk and␦zSϩ1 *k . ⌬t xy Therefore, the total diapycnal advective heat and salt iso NϪ1 S kkϭ S ϩϪ␦ x(US )␳ Ϫ ␦y(VS )␳ ⌬Z ¯uxes across Zkϩ1/2 are, respectively, k [

xy ϩ ␦xkIx(⌬ZK␦ S)␳ ϩ ␦ykIy(⌬ZK␦ S),␳ bot** top ] ␦zkk␪␦ϩ1 zkϩ1␪ kϩ1 (e␪) ϭϩ (B7) kϩ1/2 ⌬t ⌬t

bot are, respectively, layer potential temperature and salinity (␳ kkkϪ ␳ )⌬Z * ϭ ␪ kϩ1 after isopycnal advection and diffusion. One can simi- 2⌬t(␳kkϪ ␳ ϩ1) top larly compute␳ k 1 . Note that ␪ and S in (B5)±(B7) are ϩ top both variables de®ned for the layer (not half layer). (␳ kϩ1 Ϫ ␳kϩ1)⌬Zkϩ1 * bot top ϩ ␪ k , (B8) Substituting␳␳kk in (B5) and similarly computed ϩ1 2⌬t(␳kkϪ ␳ ϩ1) into (B4) yields a ®nite-difference expression with layer variables. Note that neglecting the diapycnal diffusion ␦zSbot**␦zS top (eS) ϭϩkkϩ1 kϩ1 k terms, the rhs of Eqs. (B2) and (B3) are, respectively, kϩ1/2 ⌬t ⌬t identical to the terms in the square brackets of (B6) and iso iso bot (B7). Therefore␪ andS have already been obtained (␳ kkkϪ ␳ )⌬Z kk ϭ S * as a result of solving (B2) and (B3) fordiff andS diff . kϩ1 ␪ kk 2⌬t(␳kkϪ ␳ ϩ1) Equations (B6) and (B7) are given here only to show top a complete ®nite-difference expression for diapycnal ve- (␳ kϩ1 Ϫ ␳kϩ1)⌬Zkϩ1 * ϩ S k . (B9) locity. 2⌬t(␳kkϪ ␳ ϩ1) Equation (B4) gives only the diapycnal volume ¯ux based on the de®nition of diapycnal velocity. It is the diapycnal advective heat and salt ¯uxes (e␪), (eS)in bot top Eqs. (B2) and (B3) that remove the density drifts and Again, because␳␳kk andϩ1 are given by layer vari- restores the layer density coordinates. The ®nite-differ- ables, Eqs. (B8) and (B9) are also given by layer var- ence expressions for the heat and salt ¯uxes in Hu's iables. algorithm can be readily derived from Hu's density- The ®nite-difference equations (B4), (B8), and (B9) coordinate restoration scheme as follows. From Eqs. can be readily expressed in the so-called dual-entrain- (25)±(26) of Hu (1996a) the two terms on the rhs of ment form (Wunsch 1988; Oberhuber 1993). In Eq. bot top (B4), the ®rst term on the rhs is the diapycnal advective (B4) can be written as␦zkk /⌬t and␦z ϩ1 /⌬t, respectively. bot volume ¯ux between layer k and k ϩ 1 caused by the The absolute value of␦zk is the thickness of a slab of density drift in (the bottom half of) layer k, and is de- water transferred across the isopycnal interface Zkϩ1/2 k due to restoring density coordinate in the bottom half noted byE kϩ1 . The second term on the rhs of (B4) is top the volume ¯ux between layer k and k ϩ 1 caused by of layer k, whereas that of␦zkϩ1 is the water slab thick- the density drift in (the top half of) layer k ϩ 1, and is ness transferred across Zkϩ1/2 due to restoring density kϩ1 coordinate in the top half of layer k ϩ 1. Hu's algorithm denoted byE k . Therefore, (B4), (B8), and (B9) can for density coordinate restoration is designed to ensure be rewritten as that density coordinate restoration in a layer never alters temperature, salinity, and density in its adjacent layers. e ϭ E kkϩ E ϩ1, (B10) To restore density coordinate in the bottom half of layer kϩ1/2 kϩ1 k bot k, for instance, one can either entrain (when ␦zk Ͼ 0) (e␪) ϭ E kk␪ **ϩ E ϩ1␪ , (B11) bot kϩ1/2 kϩ1 kϩ1 kk to layer k or detrain (when ␦zk Ͻ 0) to layer k ϩ 1a water slab of thickness |zbot | that has the same temper- ␦ k (eS) ϭ ESkk**ϩ ESϩ1 . (B12) ature and salinity as those of layer k ϩ 1. In this way, kϩ1/2 kϩ1 kϩ1 kk density restoration in the bottom half of layer k changes only the volume of layer k ϩ 1, but never changes its temperature and salinity. The amounts of heat and salt Substituting (B10)±(B12), respectively, into (B1)±(B3) transferred across Zkϩ1/2 due to coordinate restoration in yields

Unauthenticated | Downloaded 09/26/21 10:49 PM UTC 724 MONTHLY WEATHER REVIEW VOLUME 127 always take the latest updated values. This guarantees ץ (⌬Z ) ϩ ␦ (U) ϩ ␦ (V) ϩ E kϪ1 ϩ E kkkϪ E Ϫ E ϩ1 -t kx␳ y ␳ kkϪ1 kϩ1 k that density restorations in one layer never change temץ perature and salinity of the adjacent layers. Because of ϭ 0, (B13) this property, it does not matter if one chooses to remove kk , density drifts layer by layer (applyE kϪ1␪*kϪ1 ,ESkϪ1 *kϪ1 ץ kϪ1 **kkk *ϩ1 *kk (␪⌬Zkkkk) ϩ E ␪ ϩ E Ϫ1␪ kϪ1 Ϫ E kϩ1␪ kϩ1 Ϫ E kk␪ E ␪* , andES* layer by layer), or choose to t kϩ1 kϩ1 kϩ1 kϩ1ץ xyx remove density drifts in the top half of each layer (apply kk ϭϪ␦x(U␪ )␳ Ϫ ␦y(V␪ )␳␦xkIx[⌬ZK␦␪]␳ E kϪ1␪*kϪ1 andESkϪ1 *kϪ1 layer by layer) ®rst and then re- k move those in the bottom halves (applyE kϩ1␪*kϩ1 and y ␦␪ ␦␪ ESk * ). Of course, one can also choose to remove [ ZK ] K K , kϩ1 kϩ1 ϩ ␦ykIy⌬ ␦␪␳ ϩ ddϪ density drifts in the bottom halves ®rst and then the top ΂΃␦z k 1/2 ΂΃␦z k 1/2 Ϫ ϩ halves. In general, the diapycnal advective ¯ux terms (B14) do not 100% remove the density drifts in the isopycnic ,layers due to the nonlinearity of the equation of state ץ (S⌬Z ) ϩ ESkϪ1 **ϩ ESkkkϪ ES *Ϫ ESϩ1 * t kkkkϪ1 kϪ1 kϩ1 kϩ1 kkbut the remaining density drifts are negligibly small [seeץ xyx Hu (1996a) for an explanation]. ϭϪ␦x(US )␳ Ϫ ␦y(VS )␳␦xkIx[⌬ZK␦ S]␳ According to the above summary, it should be clear that computing the density drifts in the top and bottom y ␦S ␦S ϩ ␦ [⌬ZK␦ S] ϩ K Ϫ K . halves of each isopycnic layer is only for diagnosing ykIy␳ ddthe diapycnal advective volume ¯ux terms [the E terms ΂΃␦z k 1/2 ΂΃␦z k 1/2 Ϫ ϩ in (B13)], which are used to construct the diapycnal (B15) advective heat and salt ¯uxes. As shown in (B14) and (B15), these diapycnal ¯uxes are used to modify the Equations (B13)±(B15) are complete ®nite-difference bulk temperature and salinity for the layer, not those for equations (B1)±(B3) with diapycnal advective ¯uxes in the half layer. From the ®nite-difference expressions dual-entrainment form. All dual-entrainment terms in (B14) and (B15) are associated with the diapycnal ad- given in this appendix, it should also be clear that the vective volume ¯uxes in the continuity equation (B13), computed diapycnal diffusive and advective ¯uxes are and thus represent the diapycnal advective heat and salt all based on local ␪±S properties. Hu (1991, 1996a) also ¯uxes. That is why Eqs. (B14) and (B15) still explicitly addressed the dif®culties associated with massless lay- include both diapycnal and isopycnal diffusive ¯ux ers. The algorithms to handle massless layers in com- terms, because they are the cause of all dual-entrainment puting diapycnal diffusive and advective property ¯uxes ¯uxes. It is inconsistent to include diapycnal advection are based on locally averaged ␪±S properties. It should without including diapycnal diffusion in the conserva- not be dif®cult for the reader to ®gure this out in Hu tion equations (McDougall 1987b; Hu 1996a). (1991, 1996a). Summarizing, solving three-dimensional diapycnal mixing and advection equations can be done in the fol- REFERENCES lowing steps. 1) Compute the rhs terms of Eqs. (B14)±(B15), and Arakawa, A., and V. R. Lamb, 1977: Computational design of the diff diff integrate for␪ kk andS , which are the temperature basic dynamical processes of the UCLA general circulation mod- and salinity that have to be modi®ed by the diapycnal el. Methods Comput. 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