1. Introduction
The numerical modeling studies carried out for the North and equatorial Atlantic basin over the past several decades have demonstrated advances both in model architecture and in the availability of computational resources for the scientific community. Perhaps the most noticeable aspect of this progression has been the evolution from simulations on coarse-resolution horizontal/vertical grids outlining basins of simplified geometry and bathymetry and forced by idealized stresses, to fine-resolution simulations incorporating realistic coastal definition and bottom topography, forced by observational data on relatively short timescales (Smith et al. 2000b; Chassignet and Garraffo 2001).
A remaining issue of importance for the design of numerical models is that of the choice of vertical coordinate. Recent model comparison exercises performed in Europe [Dynamics of North Atlantic Models (DYNAMO); Willebrand et al. 2001] and in the United States [Data Assimilation and Model Evaluation Experiment (DAMÉE); Chassignet et al. 2000] have shown that no single vertical coordinate—depth, density, or the terrain-following sigma—can be by itself optimal everywhere in the ocean. These and earlier comparison studies (Chassignet et al. 1996, hereinafter CSBB; Roberts et al. 1996, Marsh et al. 1996) have shown that the models considered are able to simulate the large-scale characteristics of the North Atlantic circulation reasonably well but that the interior water mass distribution and associated thermohaline circulation are strongly influenced by localized processes that are not represented equally by each model's vertical discretization. Ideally, a numerical ocean model should retain its water mass characteristics for centuries (a characteristic of isopycnic coordinates), have high vertical resolution in the surface mixed layer and maintain sufficient vertical resolution in unstratified or weakly stratified regions of the ocean (a characteristic of depth coordinates), and have high vertical resolution in coastal regions (a characteristic of terrain-following coordinates).
The realization that there is no optimal vertical coordinate has led to the development of a generalized-coordinate ocean model, the Hybrid Coordinate Ocean Model (HYCOM) (Bleck 2002; see online at http://hycom.rsmas.miami.edu). The implementation of the generalized coordinate in HYCOM follows the theoretical foundation set forth in Bleck and Boudra (1981) and Bleck and Benjamin (1993); that is, each coordinate surface is assigned a reference isopycnal. The model continually checks whether grid points lie on their reference isopycnals and, if not, attempts to move them vertically toward the reference position. However, the grid points are not allowed to migrate when this would lead to excessive crowding of coordinate surfaces. Thus, vertical grid points can be geometrically constrained to remain at a fixed depth while being allowed to join and follow their reference isopycnals in adjacent areas (Bleck 2002). The default configuration in HYCOM is one that is isopycnal in the open stratified ocean but smoothly reverts to a terrain-following coordinate in shallow coastal regions and to fixed pressure-level coordinates (hereinafter referred to as p) in the surface mixed layer and/or unstratified seas. In doing so, the model combines the advantages of the different types of coordinates in optimally simulating coastal and open-ocean circulation features. It is left to the user to define the coordinate separation constraints that control regional transitions among the three coordinate choices.
HYCOM is a generalized-coordinate mass-conserving ocean numerical model and can therefore be configured for and integrated with only one coordinate, that is, isopycnic, pressure-level, or terrain-following. Some single-coordinate experiments will be discussed later on, not only to illustrate the flexibility of the model but also to bring forward some of the limitations associated with such a choice.
The first goal of this paper is to demonstrate the viability of the generalized (hybrid) coordinate ocean model by validating it against observations and previous simulations configured for the North and equatorial Atlantic. Building on past studies with the Community Modeling Experiment (CME) configuration (Holland and Bryan 1994; CSBB; Smith et al. 2000a, hereinafter SCB), we take advantage of the built-in vertical coordinate flexibility of the hybrid model in order to examine the effect of the coordinate choice(s) on the model's ability to accurately represent the water mass distribution and three-dimensional circulation of the Atlantic. The second goal is to document the importance of thermobaricity (i.e., the modulation of seawater compressibility by potential temperature anomalies) and the choice of reference pressure. Specifically, we address an issue that is present in models such as the Miami Isopycnic Coordinate Ocean Model (MICOM) or HYCOM that employ potential density either as the sole vertical coordinate or as one component of the hybrid framework, namely, the question of how best to infer local buoyancy contrasts from a given potential density distribution based on a uniform reference pressure.
Experiments with potential density referenced to the surface (σθ) and to 20 MPa (∼2000 m) (σ2) illustrate the increased influence of pressure errors with increasing distance from the reference pressure. The σθ hybrid experiment does not properly reproduce the northward flow of Antarctic Bottom Water (AABW), while large errors in near-surface pressure gradients in the σ2 experiment produce a wind-driven gyre circulation that is too strong, when compared with observations, and a North Atlantic Current that follows an unrealistic path. These near-surface and near-bottom pressure errors are removed when thermobaric effects are included.
The paper is organized as follows: Section 2 describes the generalized-coordinate (hybrid) model and the configuration of the experiments. Section 3 first compares a HYCOM integration with a Kraus–Turner mixed layer with simulations performed with MICOM (the basis of the hybrid code). We then introduce the baseline HYCOM simulation with the K-profile parameterization (KPP) vertical mixing model and compare the results with simulations performed in pure isopycnic and pressure-level modes. We continue through a series of HYCOM experiments that vary the choice of reference pressure for the vertical coordinate, focusing on differences in the surface circulation, the representation of water masses, the overturning circulation and meridional heat transport, and the interior deep circulation. Section 4 summarizes and discusses the results.
2. The numerical model and its configuration
The generalized-coordinate HYCOM used in this study retains many of the characteristics of MICOM while allowing coordinate surfaces to locally deviate from isopycnals wherever the latter may fold, outcrop, or generally provide inadequate vertical resolution in portions of the model domain. The freedom to adjust the vertical spacing of the coordinate surfaces in HYCOM simplifies the numerical implementation of several physical processes (mixed layer detrainment, convective adjustment, sea ice modeling, …) without robbing the model of the basic and numerically efficient layer architecture that is characteristic of MICOM throughout most of the ocean's volume. Documentation of the hybrid-model numerics, including details of the grid generator that allows coordinate layers to maintain finite thickness (therefore becoming nonisopycnic), can be found in Bleck (2002) and online at http://hycom.rsmas.miami.edu.
Several vertical mixing parameterizations have been implemented in HYCOM. The model can be integrated with a Kraus–Turner (K–T; Kraus and Turner 1967) slab mixed layer, a KPP (Large et al. 1994, 1997) nonslab vertical mixing formulation, a Price–Weller–Pinkel (PWP; Price et al. 1986) slab mixed layer, a Mellor–Yamada level-2.5 turbulence closure model (Mellor and Yamada 1982), or a Goddard Institute for Space Studies (GISS) level-2.0 turbulence closure model (Canuto et al. 2001, 2002). A detailed discussion of the relative performance of these five mixed layers can be found in Halliwell (2003). In this paper, we will first use the slab Kraus–Turner mixed layer for comparison with earlier MICOM experiments and then the KPP vertical mixing scheme (widely used in large-scale ocean simulations) for the remaining experiments.
The semi-implicit KPP closure scheme is easily implemented in HYCOM because of the model's ability to assign several coordinate surfaces to the oceanic mixed layer. In the KPP scheme, the surface boundary layer diffusivity and viscosity profiles are smoothly matched to the relatively weak diapycnal diffusivity and viscosity profiles of the interior ocean, to provide mixing throughout the water column. The contribution of background internal wave breaking, shear instability mixing, and double diffusion are parameterized for the ocean interior, while for the surface boundary layer, the influences of wind-driven mixing, surface buoyancy fluxes, and convective instability are accounted for (Large et al. 1994, 1997).
The computational domain for these experiments is identical to that in CSBB and SCB, namely, the domain of the World Ocean Circulation Experiment (WOCE) Community Modeling Experiment (CME) (Bryan and Holland 1989; U.S. WOCE 1993). The model basin comprises the North and equatorial Atlantic Ocean from 15°S to 65°N, including the Caribbean Sea and Gulf of Mexico but excluding the Mediterranean Sea. Bottom topography is derived from a digital terrain dataset with 5′ resolution in latitude and longitude (ETOPO5). All surface boundary conditions are based on seasonal climatological datasets. The wind stress and wind work values are taken from the Hellerman and Rosenstein (1983) climatology. The surface thermal boundary conditions are specified by a linear bulk formula described by Han (1984), while those for freshwater flux are implemented in traditional fashion as a linear relaxation of the surface salinity toward the Levitus (1982) climatology. Open ocean boundaries are treated as closed but are outfitted with buffer zones in which the temperature T and salinity S are linearly relaxed toward their seasonally varying climatological values. These buffer zones restore the T and S fields to the Levitus climatology in order to approximately recover the vertical shear of the currents through geostrophic adjustment. The four buffer zones are located at the northern boundary and at the southern boundary (4° latitude band), in the northwestern corner (representing the Labrador Sea), and in the Gulf of Cádiz (representing the Mediterranean Sea). The restoring timescale for the boundaries varies from 25 days at the inner edge linearly decreasing to 5 days at the walls. The timescale for the Labrador Sea region is 25 days and is 1 yr for the Mediterranean. For more details, the reader is referred to Bryan and Holland (1989) and Holland and Bryan (1994).
The horizontal grid is on a Mercator projection with resolution of 0.9° longitude. The coefficients for the along-coordinate mixing of momentum and diffusivity of tracers are 1 × 104 and 2 × 103 m2 s−1, respectively. The coefficient for vertical mixing of tracers is 3 × 10−5 m2 s−1, and the coefficient for smoothing of the isopycnals (equivalent to Gent and McWilliams 1990) is 1 × 103 m2 s−1. The quadratic bottom drag coefficient is 1.3 × 10−3.
The present experiments (summarized in Table 1) are as follows,
EXP-M′ and EXP-M: MICOM with slab K–T mixed layer, σθ coordinate,
EXP-HKT: HYCOM with slab K–T mixed layer, hybrid σθ/p coordinate,
EXP-H: HYCOM with KPP vertical mixing, hybrid σθ/p coordinate,
EXP-HISO: HYCOM with KPP vertical mixing, predominantly σθ coordinate,
EXP-HP: HYCOM with KPP vertical mixing, pressure-level coordinate only, with σθ initial conditions,
EXP-Hσ2: HYCOM with KPP vertical mixing, hybrid σ2/p coordinate, and
EXP-H
σ*2
The MICOM EXP-M′ was integrated using the invertible Friedrich and Levitus (1972) equation of state with reference pressure at the surface, for comparison with the experiments of CSBB and SCB. The equation of state used for the remaining MICOM and HYCOM experiments is that of Brydon et al. (1999), with reference pressure at the surface for EXP-M, EXP-HKT, EXP-HISO, EXP-HP, and EXP-H (σθ coordinate), and at 20 MPa (∼2000 m) for EXP-Hσ2 and EXP-H
In all experiments but one [EXP-HP, which has 30 fixed levels as in the Geophysical Fluid Dynamics Laboratory (GFDL)–CME model (Bryan and Holland 1989)], there are 22 coordinate surfaces in the vertical with σθ and σ2 values as listed in Table 2. The heaviest 16 σθ coordinate densities are identical to those used in the 16-layer MICOM experiments of CSBB and SCB. In the σ2 experiments, the densities were chosen so that the basinwide-averaged layer thicknesses for the initial state of the experiments, derived from the January climatology of Levitus (1982), would closely match those for the σθ experiments in the approximate depth range from 400 to 2000 m. Above 400 m, the σ2 densities were chosen so as to create three fewer layers in that depth range than exist in the σθ case, making it possible to increase by three the number of σ2 layers below 2000 m. These additional deep layers permit resolution of the densest water masses (in particular, the Antarctic Bottom Water) that are not present in the σθ discretization because of the folding of σθ isopycnals in the AABW density range.
In all hybrid cases (σθ and σ2), 6 layers are added above the 16 heaviest layers with densities outside the range of those found in the North and equatorial Atlantic. This ensures the existence of nonisopycnal coordinate surfaces (minimum separation of 10 m) that will be maintained in a finite-thickness state by HYCOM's grid generator (Bleck 2002).
EXP-HISO differs from the other experiments in the sense that the 22 layers are active isopycnals, with the 6 lightest layers added to increase the vertical discretization near the surface above σθ = 25.0 (see Table 2 for details). The minimum distance between two isopycnals in EXP-HISO is required to be 1 m, in order for the tridiagonal matrix solver in the KPP vertical mixing scheme to accurately estimate the K profile. This configuration is therefore isopycnic over all but a few meters of the entire water column.
3. The experiments
In this section, we document the differences in model response to the choices of vertical discretization and reference pressure. A discussion is provided in section 4. All experiments (listed in section 2) are initialized with temperature and salinity for January conditions from the Levitus (1982) climatology and then integrated for 20 years. Unless otherwise noted, the results presented in this section are for the average of the final 3 years of integration.
a. Comparison to earlier studies: EXP-M′ and EXP-M
Our reference set of experiments is as described in the comparison exercise of CSBB between the MICOM (version 2.6) and GFDL models. The purpose of this section is to briefly document some of the differences that are seen between the earlier MICOM simulations and those performed with the latest version of the code (version 2.8).
In addition to improvements in the numerics, MICOM version 2.8 employs a different equation of state from that used in version 2.6. Therefore, in order to assess the impact of the version-2.8 numerics on the solution, EXP-M′ was integrated with the version-2.8 code and the Friedrich and Levitus (1972) equation of state used in 2.6. The differences between the EXP-M′ solution and the version-2.6 solution (EXP-MW; CSBB) are relatively small, with a maximum heat transport in EXP-M′ of 1.0 PW, versus 1.15 PW in EXP-MW, and a maximum overturning in EXP-M′ of 17 Sv, versus 19 Sv in EXP-MW (Sv ≡ 106 m3 s−1). The largest difference is seen in the 3-yr average March mixed layer depth. EXP-MW has two distinct areas of deep convection in the Labrador and Irminger Basins with an average March mixed layer depth of 3000 m (Fig. 10a in SCB). The March mixed layer depths in EXP-M′ (not illustrated), on the other hand, are shallower in both the Labrador and Irminger Basins, with an average of 900 m. Deepening of the mixed layer in winter depends on preconditioning, which is determined by lateral advection of temperature and salinity. Since the forcing is the same in the two experiments, we surmise that the differences in mixed layer depth result from the time evolution of the temperature and salinity profiles, which differs in the two experiments because of changes in the mixed layer detrainment scheme and to a lesser extent numerical improvements in the tracer advection routine.
EXP-M differs from EXP-M′ only in the equation of state used. The Brydon et al. (1999) polynomial expression allows for an invertible equation of state in θ, S, and p [Friedrich and Levitus (1972) used the in situ temperature T]. The differences after 20 years (Figs. 1a, b) are small and are, for the most part, due to differences in the initial conditions (e.g., the initial interface depths, Fig. 2) that arise from the use of in situ versus potential temperature.
b. Surface circulation, as represented by the sea surface height (SSH)
Figure 1 displays the mean SSH for eight experiments (M′, M, HKT, H, HP, HISO, Hσ2, and H
The strength of the surface circulation, as measured by the difference between the subtropical SSH maximum and subpolar SSH minimum, is of the same order of magnitude in all experiments except EXP-Hσ2 (Fig. 1g), which is substantially stronger. Of the seven experiments with similar circulation strength, EXP-HP (Fig. 1e), with fixed coordinate levels, develops a much more diffuse flow between the gyres after the Gulf Stream separates from the western boundary. The SSH field of EXP-HP is more diffuse and exhibits weaker gradients across the basin than seen in its hybrid and isopycnic counterparts (see SSH profile along 65°W in Fig. 3). The σ2-coordinate EXP-Hσ2 (Fig. 1g), on the other hand, exhibits a much stronger SSH gradient across both lobes of the subpolar gyre, across the North Atlantic Current (NAC) along the southern and eastern edges of the subpolar gyre, and across the Gulf Stream along 65°W (Fig. 3). The stronger surface circulation in EXP-Hσ2 is the result of inaccurate pressure gradients at the surface, because the reference pressure in that experiment is at 20 MPa (∼2000 m) and the density variations along potential isopycnals caused by thermobaricity are not taken into account. The use of the virtual potential density,
The other notable difference among the experiments illustrated by Fig. 1 is in the path of the NAC. The path can vary significantly from one experiment to the other and is, to a large extent, dependent upon the strength of the subpolar gyre. Experiments with weaker subpolar gyres (EXP-M′, M, HKT, H, HP; Figs. 1a–e) have a NAC that turns north at Flemish Cap at 45°W before turning east at between 50° and 55°N. The eastward turn is more sharply defined in EXP-M′, M, and H than in EXP-HKT and HP, and is thus in better agreement with observations (Dietrich et al. 1980; Lazier and Wright 1993). Both experiments with 20-MPa reference pressure (EXP-Hσ2 and EXP-H
Proper model representation of the subpolar gyre strength and of the NAC pathway is the result of a fine balance between mixed layer response and preconditioning of the deep water masses. For example, a small change in model formulation, namely, the use of biharmonic instead of harmonic (Laplacian) viscosity, led to a shutdown of convective events in the Labrador Sea in the German Family of Linked Atlantic Model Experiments (FLAME; C. Böning 2002, personal communication). In EXP-HISO (Fig. 1f), the eastward extension of the NAC almost to Ireland is a direct consequence of the poor representation of mixed layer processes in the subpolar gyre. Only three of the EXP-HISO isopycnic layers are active in that region (not illustrated), which is an insufficient number for the KPP closure to be properly implemented (Large et al. 1997; Halliwell 2003). In an additional experiment (not illustrated), 14 isopycnal layers were added in EXP-HISO (for a total of 36 layers) in order to increase the vertical discretization of the subpolar gyre. This allowed a proper representation of the mixed layer physics in the subpolar region and hence a good representation of the NAC pathway, but at a greater computational cost than that of the hybrid approach.
c. Mixed layer depth and ventilation patterns
1) Mixed layer depth
Halliwell (2003) provides an extensive review of the performance of the mixed layer parameterizations available for use in HYCOM. In this section, we investigate the model mixed layer's sensitivity to the vertical coordinate choice. The 3-yr average March mixed layer thicknesses are displayed in Fig. 4 for experiments M, HKT, H, HP, Hσ2, and H
Large differences in maximum winter mixed layer depth are also evident over the subpolar gyre among the remaining experiments (Fig. 4). Mixed layer response varies among the experiments since its deepening in winter depends on preconditioning, which is determined by lateral advection of temperature and salinity. In weakly stratified regions, large changes in mixed layer depths can result from small changes in temperature and salinity that arise from the different vertical discretizations (isopycnic, pressure level, hybrid) as well as the different reference pressure choices. According to observations, outside of the northern buffer zone (i.e., south of 61°N) there are two areas where deep convection occurs: in the Labrador Sea and in the southwest Irminger Sea (Pickart et al. 2003). The maximum mixed layer depth in the Labrador Sea varies from 700–800 m in the Kraus–Turner EXP-M and EXP-HKT (Figs. 4a,b) to 1600–2000 m in the KPP cases EXP-H, HP, Hσ2, and H
2) Ventilation patterns of the subtropical gyre
The ventilation of the subtropical gyre is governed by the existence of areas of strong subduction rates along the southern flank of the deep winter mixing region, where the southward Sverdrup flow carries water away from the base of the winter mixed layer. The subtropical gyre primarily ventilates on density surfaces between σθ = 26.5 and 27.2 (New et al. 2001). In this section, we investigate the ventilation pathways in the model subtropical gyre on two surfaces, σθ = 26.52 and 27.03, in two experiments, EXP-H and EXP-H
On the 26.52 σθ surface, the mode water is observed to exist primarily in the Sargasso Sea (Fig. 5a) with q values of less than 10. In both EXP-H and EXP-H
Both EXP-H and EXP-H
d. Cross sections
1) At 31.8°W
In this subsection, we discuss the interior water mass distribution at the end of the 20-yr integration by examining meridional cross sections at 31.8°W for four experiments (EXP-H, EXP-HP, EXP-Hσ2, and EXP-H
Figure 6 illustrates two of the main differences between the fixed coordinate EXP-HP (Fig. 6b) and the hybrid experiment EXP-H (Fig. 6a): (i) most of the dense water belonging to layer 22 (σθ ≃ 27.92) has disappeared in EXP-HP after 20 years (see Fig. 10 in section 3d for a discussion of its time evolution), and (ii) density gradients are weaker in EXP-HP than in EXP-H. The divergence in deep-water mass structures over time between model results in level and isopycnic coordinates, first documented by CSBB and Roberts et al. (1996), is also seen here when comparing EXP-H and EXP-HP [see Griffies et al. (2000a) for a review of this topic]. The weaker density gradients of EXP-HP as compared with EXP-H are also expressed in the surface fields (section 3b, Fig. 1 and section 3c, Fig. 4) and are in part due to the fact that fronts are better represented in the hybrid and isopycnic coordinate experiments, in which isopycnals can congregate to provide high resolution in regions of high density contrast. In addition, no rotation of the diffusive tensor along isopycnals is included in EXP-HP, a feature which has been shown to significantly reduce numerically induced diapycnal diffusion (Griffies et al. 1998, 2000b). The main purpose of EXP-HP is to illustrate that the generalized coordinate aspect of HYCOM works well by comparing EXP-HP with similar experiments such as the GFDL–CME experiment discussed in CSBB, which also did not include any tensor rotation. In the hybrid experiments, during most months of the year, the fixed coordinates are confined to the mixed layer, where there is no need for tensor rotation. During summer, fixed coordinates extending at most 20–30 m below the mixed layer depth in the subtropical gyre can indeed be found in the hybrid model results.
The density front associated with the northeastward-flowing North Atlantic Current is well defined in the hybrid experiments by isopycnals rising sharply from about 1000 m to the surface (Figs. 6a,c,d). As discussed in reference to the surface circulation (section 3b), the actual location of the front differs among the hybrid experiments. In EXP-Hσ2 (Fig. 6c), the isopycnals are relatively flat in the 50°–55°N latitude range, reflecting the due-north path of the NAC along 35°W in that experiment (Fig. 1). EXP-H
We now further investigate the impact of the reference pressure choice and of thermobaricity by plotting the density difference between pairs of experiments displayed with the same reference pressure. In Fig. 6, the cross sections are displayed in the “native” coordinates for each experiment, that is, σθ for EXP-H and σ2 for EXP-Hσ2, H
Conversion to σ2 coordinates for EXP-H, or to σθ for EXP-Hσ2, does not significantly alter the density distribution in the upper 1000 m, south of 20°N (Figs. 7a,b). North of 55°N, both figures indicate lighter water present in EXP-Hσ2, where the NAC has carried relatively warm water northward before turning east (Fig. 1). In EXP-H, however, the NAC turns north at a point farther to the west than in EXP-Hσ2 and then turns sharply to the east, confining cold subpolar water to higher latitudes along this section. From ∼25–50°N, denser water is present between 500 and 1000 m in EXP-Hσ2 relative to EXP-H due to the deeper subtropical gyre thermocline of EXP-H as seen in Fig. 6a. Water below ∼2000 m appearing heavier in EXP-Hσ2 relative to EXP-H when viewed in σ2 (Fig. 7b) (and the opposite when viewed in σθ; Fig. 7a) arises from the different initial conditions between the two experiments.
Figure 7c shows the density difference between EXP-H
2) At the equator
Proper representation of the dynamics of the equatorial region in a numerical ocean model, in particular the strength and extent of the modeled Equatorial Undercurrent (EUC), also depends strongly on the details of the initial vertical stratification and on the vertical mixing scheme employed (Schott and Böning 1991; Blanke and Delecluse 1993). The EUC is a robust feature in all of the present experiments. Cross sections of the upper 250 m are presented in Fig. 8 (along 31.8°W, crossing the equator) and in Fig. 9 (along the equator) for three of the experiments, EXP-H, EXP-HP, and EXP-H
Increased vertical discretization in the upper 300 m, by a factor of 2, is evident in the hybrid experiments EXP-H and EXP-H
e. Time evolution of water masses
Analysis of the cross sections along 31.8°W in the previous section allowed us to document some of the differences in model results that arise from the choice of model configuration. That analysis does not, however, provide any indication of the time evolution of the different water masses initially present in the simulations. For that purpose, we monitor in time the amount of mass in 16 density classes that approximately correspond to the 16 heaviest coordinate densities in the σθ and σ2 experiments (see Table 2 for details).
The layer thickness trends shown in Figs. 10a–c for the σθ experiments [EXP-M (MICOM), EXP-HKT (HYCOM with K–T mixed layer), and EXP-H (HYCOM with KPP vertical mixing)] are highly similar, indicating that the interior mass distribution is not unduly dependent upon either the mixed layer architecture (which can lead to large temporal changes in basin-averaged layer thickness) or the vertical coordinate choice (hybrid versus isopycnic). In the deepest layer of these experiments, the volume remains nearly constant over time. In contrast, EXP-HP (Fig. 10d), configured with fixed coordinate levels, shows the loss of nearly all of the densest (and coldest) water in the domain over the 20-yr period, a loss that is reflected in the greater increase in heat content seen for this experiment. These results again illustrate the divergence over time in deep-water mass structures between models formulated in depth and isopycnic coordinates (CSBB; Roberts et al. 1996; Griffies et al. 2000a; Willebrand et al. 2001).
The changes over time in water mass distribution in the two σ2 experiments are much less than in the σθ experiments. The five deepest layers remain relatively constant in basin-averaged thickness over 20 years in both EXP-Hσ2 and EXP-H
f. Overturning and heat transport
The meridional overturning cell and associated heat transport, which are driven by the water mass transformations that take place both inside and outside the buffer zones, serve as good indicators of the model's performance. Comparison of the meridional overturning streamfunctions for experiments EXP-H (with hybrid pressure-isopycnal coordinate surfaces) and EXP-HP (with fixed levels) (Figs. 11a,b) shows the strong impact of the coordinate choice. EXP-HP exhibits a maximum overturning of 13 Sv, centered north of 45°N, in contrast with a maximum 17 Sv for EXP-H reaching the southern boundary. Below 3000 m, the southward flow of EXP-HP is much weaker (maximum 5 Sv) than the 9 Sv reaching the southern boundary at depth in EXP-H. A similar result was obtained by CSBB when comparing the GFDL and MICOM experiments. As noted in section 3d(1), there is no rotation of the diffusive tensor along isopycnals in EXP-HP. Rotation of the tensor, together with the inclusion of a parameterization of the effects of mesoscale eddies on the transport of tracers (Gent and McWilliams 1990; Gent et al. 1995; McDougall and McIntosh 2001), increases the strength and extent of the overturning circulation in depth-coordinate models (CSBB; Böning et al. 1995).
The overturning streamfunctions for experiments in σ2 coordinates, EXP-Hσ2 and EXP-H
These differences in overturning circulation can be seen more clearly by examining the zonally averaged meridional transport below 2000 m for EXP-H, HP, Hσ2, and H
The mean model heat transport for years 18–20 (Fig. 13a) clearly shows the relatively low transport values for EXP-HP, with a maximum of 0.6 PW, similar to that found in the GFDL CME experiment without parameterization of the eddy-induced “bolus” transport (Gent and McWilliams 1990; Gent et al. 1995; Böning et al., 1995; CSBB; McDougall and McIntosh 2001). For the remaining experiments, the order from lowest (1.0 PW) to highest (1.3 PW) maximum heat transport is EXP-M, HKT, H
The January and July variations in heat transport as a function of latitude, for the average of model years 18–20 (Fig. 13b), show the same order of lowest to highest transport values among the experiments as was seen for the annual mean transport. In both months, the hybrid KPP vertical mixing cases EXP-H, Hσ2, H
The order of lowest to highest heat transport values among the KPP experiments (EXP-HP, H
g. Deep-water mass transport
A significant difference between results of the σθ and σ2 coordinate experiments is the model representation of AABW. This is not surprising since there is no distinct coordinate layer representing AABW in a σθ discretization: one cannot have a monotonically increasing σθ distribution in the vertical in the presence of AABW due to the folding of σθ isopycnals in the density range of that water mass. However, this is not the case when the reference pressure is set to 20 MPa (∼2000 m).
The meridional overturning streamfunctions for the three experiments—EXP-H (σθ coordinate), EXP-Hσ2 (σ2 coordinate), and EXP-H
In this section, we further investigate the impact of the reference pressure choice and thermobaricity on the deep ocean by examining the horizontal flow patterns of the “lower deep” density class [as defined by Schmitz (1996): σθ > 27.8; roughly corresponding to σ2 > 36.7] for experiments H, Hσ2, and H
With a reference pressure at 20 MPa (∼2000 m) and no correction for thermobaric effects, EXP-Hσ2 exhibits a stronger circulation in the deep ocean than EXP-H, just as previously seen at the surface (Fig. 1; section 3b). The total southward deep transport is a maximum in EXP-Hσ2, relative to EXP-H and EXP-H
4. Summary and discussion
This study has two primary goals: (i) to show the viability of a generalized (hybrid) coordinate ocean model by validating it against observations and previous simulations configured for the North and equatorial Atlantic and (ii) to document the importance of thermobaricity and the choice of reference pressure.
The first goal was addressed by analyzing a series of experiments performed with HYCOM under a variety of conditions. The generalized aspect of the model was documented by performing a few single-coordinate experiments (isopycnic, pressure level) and the hybrid aspect by performing experiments with two mixed layer parameterizations (Kraus–Turner and KPP). These experiments do indeed illustrate that the concept of a generalized (hybrid) coordinate is viable.
While the single-coordinate experiments illustrate the flexibility of the model, they bring forward some of the limitations of choosing only one coordinate. Using KPP, it is possible to run HYCOM using only isopycnic coordinates (i.e., no bulk mixed layer as in MICOM). However, the model representation of the subpolar gyre strength and of the NAC is very sensitive to the number of nonmassless (and thus active) layers in that unstratified region. The hybrid configuration is more efficient in the sense that a proper vertical discretization of the subpolar gyre can be achieved with 10–20 layers less than the number that would be required for the pure isopycnic configuration. The hybrid configuration also provides high vertical resolution in the mixed layer throughout the basin. When only pressure-level coordinates are used, the simulation exhibits the limitations (pointed out by previous model comparison exercises) of fixed versus isopycnic coordinates, that is, loss of the dense water masses and lack of vertical resolution in regions of high density contrast when fixed coordinates are used [see Griffies et al. (2000a) and Willebrand et al. (2001) for a review].
The second goal addresses a specific issue which is present in models such as HYCOM that employ potential density either as the sole vertical coordinate or as one component of the hybrid framework: the question of how best to represent buoyancy contrasts given a potential density distribution based on a uniform reference pressure. The thermobaric effect, that is, the modulation of seawater compressibility by potential temperature anomalies, must be considered in order to account for the difference between buoyancy and potential density. The questions of representation of the potential density distribution for a given reference pressure and of the importance of accounting for the thermobaric effect have been addressed in this work principally by reference to the results of three experiments—one with reference pressure at the surface (σθ coordinate), one with reference pressure at 20 MPa (∼2000 m; σ2 coordinate), and one with reference pressure at 20 MPa and with thermobaric effects included (
The main difference between the σθ and σ2 experiments is the model's representation of AABW: there is no distinct water mass representing AABW in the σθ discretization since one cannot have a monotonically increasing σθ distribution in the vertical in the presence of AABW because of the folding of σθ isopycnals in the density range of that water mass. The density structure of the AABW is well represented when a reference pressure of 20 MPa (∼2000 m) is selected. It is important to note that there is no single reference pressure value that would not create coordinate folding somewhere in the World Ocean. However, it has been found that a reference pressure of 20 MPa is near optimal in that regard. Furthermore, in the few places where folding might occur, the generalized-coordinate aspect of HYCOM is such that it allows interfaces to not necessarily coincide with potential isopycnals.
The importance of thermobaricity is illustrated by the differences between the σ2 and
The experiments discussed in this paper not only demonstrate the versatility of the hybrid approach, but also the importance of the reference pressure choice and of the need to include thermobaricity. Further investigation of the model's performance is under way in a fine mesh (1/12°) North and equatorial Atlantic configuration. HYCOM is at the center of an effort entitled “The Hybrid Coordinate Ocean Model (HYCOM) Consortium for Data-Assimilative Ocean Modeling” funded by the National Ocean Partnership Program (NOPP). The primary purpose of the consortium (see http://hycom.rsmas.miami.edu) is the establishment of a global real-time ocean forecast system based on HYCOM with sophisticated data assimilation techniques.
Acknowledgments
This research was supported by the National Ocean Partnership Program (NOPP) and the Office of Naval Research (ONR) under awards N00014-99-1-1048 and N00014-99-1-1066. The authors thank Adrian New for encouraging us to look at the modeled ventilation pathways and Jean Carpenter for her help in redrafting the figures.
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APPENDIX
Inclusion of Thermobaricity in Isopycnic-Coordinate Ocean Models
In order to isolate the components of density variability that are capable of creating circulation-inducing buoyancy contrasts from those that are dynamically passive, κ is split into a pressure-dependent component κ(p) and a residual thermobaric component κ(θ). The latter component has some salinity and pressure dependence as well.
This implies that estimates of the horizontal pressure gradient force are insensitive to whether the hydrostatic equation has been integrated using the in situ or the virtual potential density, that is, whether the oceanic density field is subject to compression by the pressure-related part κ(p) of the compressibility coefficient. Hence, the frequency of buoyancy oscillations remains unchanged if the specific volume of seawater in the model is changed from its in situ value to α*. For details, the reader is referred to Sun et al. (1999).
Sea surface height (cm), mean years 18–20. Contour interval is 10 cm. (a) EXP-M′, (b) EXP-M, (c) EXP-HKT, and (d) EXP-H. Vertical line is along 31.8°W
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
(Continued) Sea surface height (cm), mean years 18–20. Contour interval is 10 cm. (e) EXP-HP, (f) EXP-HISO, (g) EXP-Hσ2, and (h) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Initial conditions along 31.8°W: (a) EXP-M′ and (b) EXP-M. Contour interval is 0.1 for density (σθ). Solid lines are layer interfaces. Integers are layer numbers. Depths are shown in meters on vertical axis
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Sea surface height along 65°W (cm), mean years 18–20. Latitude is shown on horizontal axis
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Mixed layer depth (m), Mar mean years 18–20. Contour interval is 100 m. (a) EXP-M, (b) EXP-HKT, (c) EXP-H, (d) EXP-HP, (e) EXP-Hσ2, and (f) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Potential vorticity (q × 1011 m−1 s−1) for (left) σθ = 26.52 and (right) σθ = 27.03. (a),(d) Initial conditions (Levitus); (b),(e) EXP-H (Mar mean years 18–20); (c),(f) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Vertical cross sections along 31.8°W, mean years 18–20: (a) EXP-H, (b) EXP-HP, (c) EXP-Hσ2, and (d) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Vertical cross sections along 31.8°W, mean years 18–20, showing density difference between two experiments: (a) EXP-Hσ2 − EXP-H, displayed in σθ; (b) EXP-Hσ2 − EXP-H, displayed in σ2; and (c) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Vertical cross sections along 31.8°W for ∼7°S–7°N, Mar year 20, showing contours of density (left: dashed contours, contour interval 0.1) and zonal velocity [right: solid (dashed) contours are positive (negative) velocity, contour interval 5 cm s−1]. (a),(b) EXP-H (σθ); (c),(d) EXP-HP (σθ); (e),(f) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Vertical cross sections along the equator across the model basin for Mar year 20 showing contours of density (left: dashed contours, contour interval 0.1) and zonal velocity [right: solid (dashed) contours are positive (negative) velocity, contour interval 5 cm s−1]. (a),(b) EXP-H (σθ); (c),(d) EXP-HP (σθ); (e),(f)EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
20-yr time evolution of the domain-averaged layer thicknesses in meters for 16 density classes chosen as described in the text and of the domain-averaged heat content (bottom curve), computed for the month of September for (a) EXP-M, (b) EXP-HKT, (c) EXP-H, (d) EXP-HP, (e) EXP-Hσ2, and (f) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Meridional overturning streamfunction, mean years 18–20, for (a) EXP-H, (b) EXP-HP, (c) EXP-Hσ2, and (d) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Zonally averaged meridional overturning transport below 2000 m, mean years 18–20. Vertical axis is in Sverdrups. Latitude is shown on horizontal axis
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
(a) Meridional heat transport, mean years 18–20; (b) meridional heat transport, Jan mean and Jul mean, years 18–20; extrema highlighted. Vertical axis is in petawatts. Latitude is shown on horizontal axis
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Annual cycle of meridional heat transport for model years 18–20 at 25° and 40°N. Vertical axis is in petawatts
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Transport function for “lower deep” water masses (σθ > 27.8), mean years 18–20: (a) EXP-H, (b) EXP-Hσ2, and (c) EXP-H
Citation: Journal of Physical Oceanography 33, 12; 10.1175/1520-0485(2003)033<2504:NASWTH>2.0.CO;2
Summary of the model experiments
Layer densities for EXP-M′, M, HKT , H (σθ ), EXP-HISO (σθ ), and EXP-Hσ 2 ,