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Vorticity Balance in Coarse-Resolution Global Ocean Simulations YOUYU LU*
AND
DETLEF STAMMER
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California (Manuscript received 1 July 2002, in final form 12 August 2003) ABSTRACT The vorticity budget of the vertically integrated circulation from two global ocean simulations is analyzed using a horizontal spacing of 28 3 28 in longitude/latitude. The two simulations differ in their initial hydrographic conditions and surface wind and buoyancy forcing. The constrained simulation obtains optimal initial condition and surface forcing through assimilating observational data using the model’s adjoint, whereas the unconstrained simulation uses Levitus climatological conditions for initialization and is driven by NCEP–NCAR reanalysis forcing, plus restoring to the monthly surface temperature and salinity climatological conditions. The goal is to examine the dynamics that sets the time-mean circulation and to understand the differences between the constrained and unconstrained simulations. It is found that, similar to eddy-permitting simulations, the bottom pressure torque (BPT) in coarse-resolution models plays an important role in the western boundary currents and in the Southern Ocean, and largely balances the difference between wind stress curl and bV for the depthintegrated flow. BPT is a controlling factor of the interior abyssal flow. The geostrophic vorticity relation holds in the interior basins in intermediate and deep layers and breaks down in the upper ocean toward the surface. In the upper layer of the interior basins, the model simulations show statistically significant deviation from the Sverdrup balance. In the subtropical gyre regions, the deviation from Sverdrup balance is confined to zonal bands that are balanced by the curls of lateral friction and nonlinear advection. The differences between the constrained and unconstrained simulations are significant in vorticity terms. The adjustment to Levitus hydrographic climatological data as the model’s initial condition causes the most significant changes in BPT, which is the main reason for changes in abyssal flow. The analysis also points to needs for further improvement of models and controlling the influence of data errors in ocean state estimation.
1. Introduction Our ability to observe and model the large-scale circulation of the oceans advanced significantly during the World Ocean Circulation Experiment (WOCE). Although the synthesis of WOCE observations will improve during the years to come, their assimilation into ocean models began to provide a quantitatively more accurate description of the ocean state than is available based on either observations or models alone. Stammer et al. (2002) summarized such an attempt of a global synthesis by the Estimating the Circulation and Climate of the Ocean (ECCO) consortium. The synthesis utilizes a mathematically rigorous approach to constrain a coarse-resolution global ocean model by much of the WOCE data and surface fluxes from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis, us* Current affiliation: Ocean Sciences Division, Bedford Institute of Oceanography, Dartmouth, Nova Scotia, Canada. Corresponding author address: Detlef Stammer, Scripps Institution of Oceanography, 9500 Gilman Dr., MS 0230, La Jolla, CA 920930230. E-mail:
[email protected]
q 2004 American Meteorological Society
ing the model’s adjoint. The result is an estimate of the global ocean state on 28 longitude 3 28 latitude grids from 1992 through 2000. The adjusted parameters include initial temperature and salinity fields and daily surface forcing over the entire period. To illustrate the benefit and quality of data assimilation, one usually compares the results from the constrained model run with the unconstrained reference run (hereinafter the control run) and with independent information available from data that have not been assimilated. The previous analyses by Stammer et al. (2002) illustrate that the state estimation displays considerable skill in simulating the large-scale and time variations of withheld data. Stammer et al. (2003a) subsequently computed estimates of the horizontal transports of mass, heat, and freshwater and the adjustment to surface forcing fields. Both papers clearly indicate that the state estimation improves the estimates of the time variation of the circulation and its transport properties. The analyses also indicate that several aspects of the time-mean state have been improved as well. An important aspect for this study is that the constrained solution shows significantly fewer differences from the climatological hydrographic conditions as compared with the unconstrained solution, implying that the pressure field and vorticity torque terms may have been improved.
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Because of the tremendous computational burden associated with the constrained nonlinear optimization problem, this first global ocean state estimation was limited by using a coarse-resolution model and a relatively short (9 yr) estimation period. In contrast, process-oriented forward simulations without rigorous data assimilation can relax the limitation on either integration period or on model resolution. It is important therefore to test, now, before full eddy-resolving assimilation runs become available, whether a constrained model shows common or robust dynamics with those models using finer resolutions or integrated to longer periods. Such analyses will also be helpful to assess the strength and deficiency of the state estimation procedure and product, which motivates this study. Our focus will be on the time-mean circulation and vorticity balance. Our goal is to understand the dynamical difference between the constrained and control runs, and to understand how these differences relate to the estimated adjustments in surface forcing, initial conditions, and other parameters. It is also important to check how different the coarseresolution simulations are from recent analyses on eddypermitting simulations. The structure of the remainder of the paper is as follows: In section 2 we summarize the model setup and estimation conditions. Section 3 briefly introduces the analysis procedure. In sections 4, 6, 7, and 8 we discuss the circulation and vorticity balance for the depth-integrated flow and in the upper, intermediate, and deep layers, separately. In each of these sections we pay particular attention to the differences between the solutions from the constrained and unconstrained runs. The common and different aspects with theories, other models, and observational evidence will also be discussed. Section 5 discusses in particular the possible deficiency and improvement of the wind fields obtained through data assimilation. In section 9 we expand the discussion on upper-ocean Sverdrup balance and offer a test of geostrophic vorticity relation. The analyses are summarized in section 10. 2. The model setup Stammer et al. (2002) provide a full account of the state estimation method (Lagrange multipliers, or adjoint) and the model configuration. The estimate is based on the Massachusetts Institute of Technology ocean general circulation model (Marshall et al. 1997a,b) and its adjoint (Marotzke et al. 1999). This model uses the primitive equations on a staggered ‘‘C-grid’’ (Arakawa and Lamb 1977) under the Boussinesq approximation. Spatial coordinates are longitude, latitude, and height. In these simulations, we use a hydrostatic version with an implicit free-surface scheme. The surface-layer mixing is parameterized using the K-profile parameterization (KPP) scheme of Large et al. (1994). Underneath the surface mixed layer, convective adjustment is used to remove gravitational instabilities.
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The model is configured over 6808 in latitude with 28 horizontal spacing in longitude and latitude and 22 levels in the vertical. Free-slip bottom, and nonslip lateral wall boundary conditions are used. Horizontal (not isopycnal) mixing of momentum and tracers are parameterized as Laplacian, with the viscosity n h of 5 3 10 4 m 2 s 21 and diffusivity k h of 10 3 m 2 s 21 . The background vertical viscosity and diffusivity are set to be n y 5 10 23 m 2 s 21 and k y 5 10 25 m 2 s 21 . Near the sea surface, the vertical eddy viscosity and diffusivity calculated by the KPP scheme can be an order of magnitude or more higher than the background values. To allow a time step of 1 h, an implicit scheme is used for the vertical mixing. The control run is initialized with the temperature and salinity (T–S) climatological conditions of Levitus and Boyer (1994) and Levitus et al. (1994) and is driven by twice-daily wind stress and daily heat and virtual salt fluxes from the NCEP–NCAR reanalysis. The surface heat and salt fluxes are modified by restoring the sea surface temperature (SST) and salinity (SSS) to the monthly climatological conditions. The constrained run differs from the control run in its initial temperature and salinity fields and the surface forcing fields during the entire 9-yr period. No restoring of SST and SSS is included, that is, the solution is entirely driven by the estimated surface fluxes. The initial T–S fields and surface forcing are obtained through the data assimilation procedure by constraining the model solution with the observed data. Constraining data include both mean and time-varying altimetry from the Ocean Topography Experiment (TOPEX)/Poseidon and the European Remote Sensing Satellites (ERS)-1/2, the Levitus and Boyer (1994) and Levitus et al. (1994) monthly hydrographic climatological conditions over the full water column, monthly mean surface temperature fields, as well as wind stress and heat and freshwater fluxes from the NCAR–NCEP reanalysis. Details on the optimization, especially on the relative weights specified for each of those datasets, are provided by Stammer et al. (2002). The model simulations cover 9 yr from 1992 to 2000. The global kinetic energy diagnostic indicates that the model solutions are quasi stationary after the initial adjustment in about one year. All analyses shown below are based on time-mean fields averaged over the last 7 yr, that is, 1994–2000. 3. Analysis procedure After averaging between two arbitrary depth levels, z1 and z 2 , the horizontal momentum equation governing the ocean circulation can be written as
E E
z1
Ut 1
[= · (uu) 1 (wu) z ] dz 1 f k 3 U
z2
z1
52
z2
=p/r o dz 1 t (z1 ) 2 t (z 2 )
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E
z1
= · (n h=u) dz.
(1)
z2
Here u 5 (u, y) is the horizontal velocity vector, w is vertical velocity component, U 5 (U, V) 5 # zz12 u dz, p is the pressure, r o is a constant reference value for density r, t 5 (t x , t y ) is the turbulent stress (scaled by r o ) between vertical layers of the fluid, f is the Coriolis parameter, and ¹ is the horizontal gradient operator. The hydrostatic pressure p is calculated as
E
h
p(z) 5
gr dz9,
(2) 4. Depth-integrated circulation
z
where g is the acceleration due to gravity and h is the free surface elevation above z 5 0. The vorticity balance of the vertically integrated flow field follows by taking the curl of (1),
[
b V 5 curl 2U t 2
E
z1
z2
By using the kinematic boundary conditions at the surface (z1 5 h) and bottom [z 5 2H(x, y)], the vorticity balance of the depth-integrated circulation can be derived from (3) as
b V 5 curl(t s ) 1 BPT 1 LFT 1 ADV,
]
=p/r o dz 1 t (z1 ) 2 t (z 2 )
(4)
where t is the surface wind stress. The turbulent stress at z 5 2H is usually small in comparison with t s , and it vanishes in our cases because the model uses a freeslip bottom boundary condition. The time derivative term in (3) is assumed to be negligible after averaging the model field over seven years. The term BPT stands for the bottom pressure torque, which is defined as s
1 f (w 2 u · =z1 )(z1 ) 2 f (w 2 u · =z 2 )(z 2 ) 1 LFT 1 ADV,
Lesser Antilles. Both features are not directly related to the wind stress but must be a feature of the flow field at that depth level. At 2200 m (Fig. 2b) the large-scale structures in w still show some resemblance with those at the shallower depth, especially in the North Pacific. However, the field now looks generally much more complex and shows many features that reflect the topography, in particular over the Southern Ocean. The role played by these complicated, highly nonuniform vertical flows will be a major concern in the following vorticity analysis.
(3)
where b 5 ] y f, and LFT and ADV stand for the curls of the lateral friction and the advection terms, respectively. The terms multiplied by f are introduced through the vertical integral of the continuity equation. We note that (1) and (3) apply without any restriction to the circulation integrated between any two surfaces in the vertical direction. In particular, z1 and z 2 can vary in the horizontal. In the following sections we will first examine the vorticity balance for the depth-integrated flow and then extend the analysis to the flow integrated between different depth levels. To justify the subdivision of layers, we first show in Fig. 1 the time-mean meridional and vertical velocity components along a zonal section at 248N. In the interior basins of the Pacific and Atlantic Oceans, the minima in the meridional and vertical flows are found between 1000- and 2000-m depth. At greater depths, the horizontal and vertical flow structures are significantly more complicated, especially near topography. Based on Fig. 1, we define the base of the upper ocean to be at 985-m depth, the intermediate layer to reach from 985- to 2200-m depth, and the deep ocean to cover the depth range below 2200 m. The horizontal distributions of the time-mean vertical velocity, at the depths of 985 and 2200 m, are shown in Fig. 2. At 985 m, the large-scale pattern of w reflects primarily the wind stress curl. In the North Atlantic, upwelling occurs along the northern flank of the Gulf Stream and over the entire subpolar gyre. Downwelling occurs over the entire North Atlantic subtropical gyre south of the zero wind stress curl line. Maximum downwelling at this depth is found east of the Mid-Atlantic Ridge south of the North Atlantic Current and along the
1
E
h
BPT 5 curl 2
2
=p/r o dz
2H
(5)
and is mathematically equivalent to BPT 5 J(p b /r o , H).
(6)
Here J(A, B) 5 A x B y 2 A y B x is the Jacobian operator and p b is the bottom pressure. To be consistent with the numerics of the z-level models, BPT is calculated using (5) by first computing =p in each level and then taking the vertical integration. Here =p is set to zero whenever a grid for pressure gradient encounters topography. Figure 3 displays the spatial distribution of the first two terms in (4). Their difference, bV 2 curl(t s ), and the remaining three terms in (4) are shown in Fig. 4. Note that the term ADV is inferred from the residual of the balance instead of being calculated directly. Here and in the following, a moving 68 longitude 3 68 latitude spatial smoothing has been applied, which greatly reduces grid-scale noises in the BPT, LFT, and residual fields. On large scales we observe a correspondence between bV and the curl of the wind stress in the interior basins, as one would expect for a wind-driven gyre circulation in the absence of topography. However, there are clear deviations from this correspondence on smaller scales, especially near western boundary currents (WBCs), in the subpolar North Atlantic, and over the Southern Ocean, where large values of BPT, LFT, and ADV can be found. Moreover, the curl of the wind stress is neg-
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FIG. 1. Seven-year averaged (a) meridional flow and (b) vertical flow at 248N from the constrained run. In (a), values between 20.3 and 0.3 cm s 21 are shown by contour lines spaced by 0.05 cm s 21 ; values larger than 0.3 and smaller than 20.3 cm s 21 are shown by dark and gray shadings, respectively. In (b), values between 25 3 10 27 and 5 3 10 27 m s 21 are shown by contour lines spaced by 10 27 m s 21 ; values larger than 5 3 10 27 and smaller than 25 3 10 27 m s 21 are shown by dark and gray shadings, respectively. In both (a) and (b), positive, zero, and negative values are distinguished using thinner solid, thick solid, and dashed lines.
ative over the Antarctic Circumpolar Current (ACC), whereas bV has both positive and negative signs. In the interior basins, the deviation of bV from curl(t s ) near topographic features is primarily balanced by the BPT term. According to (6), BPT vanishes if one of the following conditions is met: 1) H or p b is spatially uniform or 2) the contours of H are parallel to those of p b . An analysis of the model results shows that the variation of p b is closely related to the sea surface elevation h in areas with low stratification. In stratified areas, p b is related to h 2 h s , where h s is the change in sea level due to the steric effect. Hence, large values of BPT correspond to enhanced barotropic flows. By assuming the near-bottom flow to be in geostrophic balance and by invoking the kinematic condition at the bottom, one can derive (away from the equator) the relation
BPT 5 2 fw b 5 f u b · =H,
(7)
where w b and u b are the vertical and horizontal velocities at the bottom. Our analysis shows that (7) generally holds in the interior basins but breaks down near steep topography, a common feature of z-level models (Hughes 1992). According to (7), BPT vanishes where the bottom flow aligns with the isobath, and large values of BPT indicate strong deep flow impinging onto topographic features. Further, BPT can be decomposed into BPT 5 Hby b 2 H 2u b · =( f /H),
(8)
that is, a part associated with the advection of planetary vorticity by the near-bottom flow and a part associated with the near-bottom flow crossing f /H contours. In areas where baroclinicity is weak and LFT/ADV are not
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FIG. 2. Seven-year average of vertical velocity at depths of (a) 985 and (b) 2200 m, from the constrained run. Values between 20.5 3 10 26 and 0.5 3 10 26 m s 21 are shown by contour lines spaced by 0.1 3 10 26 m s 21 . Positive, zero, and negative values are distinguished using thinner solid, thick solid, and dashed lines. Strong upwelling and downwelling with magnitudes greater than 0.5 3 10 26 m s 21 are shown by dark and gray shadings, respectively.
important, the remaining balance in the vorticity budget is between H 2 u b · =( f /H) and the wind stress curl. Such a simple balance has been discussed earlier (Willebrand et al. 1980). In our model solutions, however, the baroclinicity, friction, and nonlinear advection terms are sufficiently large to cause departure from this balance. The influence of topography, nonlinear advection, and friction in the depth-integrated vorticity balance has been analyzed previously based on eddy-permitting simulations (e.g., Bryan et al. 1995; Hughes and de Cuevas 2001). Our coarse-resolution solution generally agrees with these previous analyses. In particular, the spatial distributions and magnitudes of the terms presented in Figs. 3 and 4 bear great similarity to those shown by Hughes and de Cuevas (2001) from an eddy-permitting global ocean simulation. (We note for a quantitative comparison purpose that 10 210 m s 22 in Figs. 3 and 4
corresponds to 10 27 N m 23 in Fig. 4 of Hughes and de Cuevas.) Note that Hughes and de Cuevas (2001) used a horizontal resolution of 0.258 longitude 3 0.258 latitude, and they applied a moving 28 longitude 3 28 latitude spatial smoothing to reduce noise. For both the coarse- and high-resolution simulations, the spatial smoothing helps to reduce the grid-size noises, which appear to be unavoidable in z-level ocean models. It is encouraging that the role played by BPT in our coarseresolution simulations is similar to that in eddy-permitting simulations, at least on the scales considered here. However, a noticeable difference between coarseresolution and eddy-permitting simulations can be found in the relative importance of LFT and ADV. In eddypermitting simulations ADV has larger magnitudes than LFT, whereas in coarse resolutions LFT is more significant. This difference is to be expected because
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FIG. 3. Vorticity balance of the depth-integrated circulation from the constrained run: (a) bV and (b) curl(t s ). Values between 210 210 and 10 210 m s 22 are shown by contour lines spaced by 0.2 3 10 210 m s 22 . Positive, zero, and negative values are distinguished using thinner solid, thick solid, and dashed lines. Values larger than 10 210 and smaller than 210 210 m s 22 are shown by dark and gray shadings, respectively.
coarse-resolution models lack eddy motions and are excessively viscous. Does the constrained model show different dynamics than the unconstrained run? An analysis of the control run shows that the terms in (4) are all similar in their spatial distributions to those shown in Figs. 3 and 4. However, all five terms are quantitatively different between the two runs, implying somewhat different balances. Figure 5 shows the differences in bV, curl(t s ), and BPT between the constrained and control runs. The difference in bV is statistically significant in the WBCs and at high latitudes. At these locations, both the differences in curl(t s ) and in BPT contribute to the difference in bV. At higher latitudes, the difference in BPT is more significant than the difference in curl(t s ). Note that the differences in these vorticity terms correspond to significant changes in the depth-integrated circulation between the two solutions. For example, in the con-
strained run the volume transport in the Gulf Stream region increases by 5 Sv (1 Sv [ 10 6 m 3 s 21 ), while in the ACC it decreases by 15 Sv. The difference in curl(t s ) shows large-scale zonal bands that have no correspondence in the differences in bV and BPT. These zonal bands are particularly significant in the interior basins where differences in bV and BPT are small. As shown in Fig. 3, the zonal bands are evident in the distribution of curl(t s ) as obtained through the state estimation procedure. 5. Zonal bandedness It is important to explore the cause and realism of the zonal bands in curl(t s ). The curl of the unadjusted NCEP–NCAR reanalysis winds, used as forcing in the control run, already shows some zonal bands. Their amplitude is enhanced through data assimilation. The bands
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FIG. 5. Difference between the constrained and control runs in (a) bV for the depth-integrated flow, (b) BPT, and (c) curl(t s ). All terms are in units of 10 210 m s 22 .
FIG. 4. Vorticity balance of the depth-integrated circulation from the constrained run: (a) bV 2 curl(t s ), (b) BPT, (c) LFT, and (d) ADV (see text), corresponding to (4). All terms are in units of 10 210 m s 22 .
may be induced through information in the datasets used as constraints, but it is not immediately obvious through which one. Major datasets being assimilated include altimetry and hydrographic climatological data. The time-mean model sea surface height (SSH) is constrained by the time-mean sea surface topography determined by TOPEX/Poseidon (T/P) altimeter after
subtracting the ‘‘EGM96’’ estimate of geoid (Lemoine et al. 1997). Wunsch and Stammer (2003) discussed the difference between the model-estimated SSH and the time mean SSH from T/P. The difference shows clear structures of residual geoid error, implying that the estimated SSH field has an improved skill above the T/P data and the unconstrained model but that some geoid error could have leaked into the time-mean model SSH field as well. See Wunsch and Stammer (2003) for a detailed discussion. The monthly T–S climatological data were compiled by objective analysis of hydrographic observations (Levitus and Boyer 1994; Levitus et al. 1994). From the hydrographic data alone, the steric height h s can be determined. Figure 6a shows the meridional gradients of the time-mean T/P SSH, ]h/]y, and the steric height ]h s /]y at 1998E across the Pacific. It is notable that both gradient terms are consistent with each other on large meridional scales but are different on small scales. Both terms show some similarities with the zonal variation
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lation. The adjustment is similar in pattern and amplitude to the difference between the scatterometer-based winds from ERS-1/2 and NCEP–NCAR reanalysis winds (Fig. 7b), indicating a large degree of skill, at least in low latitudes. [See Stammer et al. (2003b) for more detailed discussion.] 6. The upper ocean At the base of the upper ocean, the turbulent stress t can be neglected in comparison with the surface wind stress. Excluding shelf areas shallower than 985 m and neglecting the time derivatives, the vorticity budget for the upper ocean becomes
bV 5 curl(t s ) 2 fwub 1 LFT 1 ADV, FIG. 6. (a) The meridional gradients of the T/P time-mean SSH ( h) minus the geoid estimate and the steric height (h s ) computed from the Levitus hydrographic climatological conditions, both at 1998E. (b) The meridional variation of curl(t s ) from the constrained run at 1998E.
of curl(t s ) from the constrained run (Fig. 6b), for example, around 108 and 258N, but do not allow an immediate conclusion as to from where the banded structures might arise. Could unrealistic model physics, for example, the lack of eddy activity and excessive viscosity, cause the enhancement in zonal wind stress pattern? In the eddypermitting simulation of Hughes and de Cuevas (2001), the zonal bands can be found in ADV and LFT terms. Apparently, such bands do not show up in bV or in curl(t s ) of their fields. It is not clear if the zonal bands in ADV balance those in LFT in their analysis. In a recent study, Kessler et al. (2003) showed further evidence of zonal bandedness in ADV and LFT in model simulations. By using enhanced meridional resolution near the equator, they are able to show that the zonal bands in ADV and LFT do not cancel out exactly in equatorial Pacific, thus leading to zonal bands in the velocity field. They also showed zonal bandedness in wind stress curl from the scatterometer-based wind product, which also leads to zonal bands in the flow field. It is reasonable to argue that the use of higher resolution and smaller viscosity enables better simulation of zonal bands in ocean currents than is possible in our 28 model solution. It is possible, then, that the enhancement of zonal bands acts in our wind stress to compensate the lack of resolution and the associated excessive lateral friction. At this point it is important to ask whether the adjusted wind stress show any improvement over the NCEP–NCAR reanalysis winds, despite possible artifacts of model and data errors. Figure 7a shows the magnitudes of the adjustment of curl(t s ) obtained relative to NCEP–NCAR reanalysis through data assimi-
(9)
where wub denotes the vertical velocity at the base of the upper ocean. In the classic Sverdrup relation, (9) reduces to the balance of the first two terms, that is,
bV 5 curl(t s ).
(10)
The appealing aspect of this relation is twofold: 1) it relates the meridional volume transport in the interior ocean directly to the surface wind and 2) it allows mapping of a streamfunction for the upper-ocean gyre circulation because it demands wub 5 0 [see Wunsch (1996) for a detailed discussion]. Figure 8a shows the difference between the first two terms in (9), bV 2 curl(t s ), from the constrained run. If the Sverdrup balance were to hold, this difference would be negligible. As compared with curl(t s ) (Fig. 3b), the amplitudes of bV 2 curl(t s ) are small only in the subtropical regions outside the boundary current regions, especially from about 308S to the equator in the Pacific. A general departure from Sverdrup balance is clearly visible over many other parts of the global ocean, especially in the WBCs and at high latitudes. Similar deviations from the Sverdrup relation are also found in the control run (Fig. 8b), although with smaller amplitude. Figure 9 shows the distribution of 2 fwub and the sum of LFT and ADV [plotted as the difference of the first three terms in (9)] from the constrained run. At higher latitudes, the departure of bV from curl(t s ) (Fig. 8a) is primarily balanced by 2 fwub on the large scale. However, near WBCs the sum of LFT and ADV also becomes important in closing the vorticity budget. The streamfunction can be computed by integrating V or curl(t s ) zonally, starting from the eastern boundary while excluding the WBCs. Figure 10 shows the zonal integrals of V, V 2 curl(t s )/b, and (LFT 1 ADV)/b. It is clear that the Sverdrup relation does not hold over the subpolar gyre regions where the departure of V from curl(t s )/b can be explained to a large extent by the zonal integral of 2 fwub /b. In the interior subtropical areas where the Sverdrup relation is generally expected to apply, the model results show large discrepancies. The peak values of the differences reach 65 Sv in the in-
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FIG. 7. Differences in wind stress curls: (top) adjusted wind (ECCO) minus NCEP– NCAR reanalysis fields and (bottom) ERS-1/2 wind stress minus NCEP. The fields are averaged from 1992 through 1997. Units are 10 210 m s 22 .
terior subtropical Atlantic 615 Sv in the North Pacific. These large discrepancies are confined to the abovementioned zonal bands, which results from the wind stress curl and are balanced by the zonal integral of (LFT 1 ADV)/b. In fact, the zonal bands correlate well with the large lateral shear in the gyre circulation. Analogous computations from the control run lead to similar, although somewhat weaker, results (peak values reach 610 Sv in subtropical Pacific; cf. Fig. 8b). 7. The intermediate layer The intermediate layer is bounded by two vertical levels taken here to be at 985- and 2200-m depth, respectively. If the turbulent stress t can be neglected, the vorticity equation can be written as
bV 5 f (wmt 2 wmb ) 1 LFT 1 ADV,
(11)
where wmt and wmb denote the vertical velocity at the top and bottom of the intermediate layer, respectively. The bV term in the intermediate layer is negligible at low latitudes away from the western boundaries (Fig. 11a). In the interior basins, bV is clearly balanced by f (wmt 2 wmb ) (Fig. 11b), the vortex stretching/compression due to the divergence of vertical flow. The imbalance between bV and f (wmt 2 wmb ) occurs nears the WBCs and at high latitudes, indicating that LFT 1 ADV is important there. Figure 12a shows a vector plot of the volume transport integrated from 985- to 2200-m depths calculated by the constrained run. The middepth flow is strongest in the ACC. In the Atlantic Ocean, there is a strong southward-going WBC; it likewise shows up in the Indian Ocean as a quasi-continuous band along the western boundary. In contrast, in the Pacific poleward mo-
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FIG. 8. Vorticity balance in the upper ocean from surface to 985-m depth: bV 2 curl(t s ) from (a) the constrained run and (b) the control run. The setup of contours and shadings is the same as Fig. 4.
tion appears in both hemispheres. The intermediate flow in the interior Pacific is weak and is confined to a series of zonal bands. The control run shows flow patterns that are similar to those of the constrained run and results in a distribution of the four terms in (12) similar to that found from the constrained run. The existence of zonal flows does not conflict with the vertically integrated vorticity balance (11). However, (11) cannot explain why the flow tends to align in zonal directions. With regard to the momentum balance (1), a plausible explanation is the role played by turbulence stress t, which represents the diffusion of momentum between vertical layers of the fluid. Nakano and Suginohara (2002) examined the roles played by vertical and horizonal momentum diffusion in model simulations using 18 horizontal resolution. The zonal flows in the intermediate layer in their simulations can be traced back to the gyre circulation in the upper ocean driven by wind, which is also primarily in the zonal direction. They found that the strength and distribution of the
zonal flows are sensitive to the vertical and horizonal viscosity parameters. 8. The deep ocean By using the kinematic boundary condition at the bottom, the vorticity equation for the deep layer can be written as
bV 5 fwdt 1 BPT 1 LFT 1 ADV,
(12)
where wdt is the vertical velocity at the top of the deep layer. A classical description of the abyssal circulation is given by the Stommel–Arons theory (Stommel 1958; Stommel and Arons 1960a,b), which is based on vorticity arguments and assumes a flat ocean bottom and uniform upwelling. Downwelling is assumed to happen at two isolated locations, one in the subpolar North Atlantic and one in the Weddell Sea. Friction was limited in their theory to the WBC; hence the vorticity balance in the interior basin results in
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FIG. 9. The distribution of (a) 2 fwub and (b) LFT 1 ADV from the constrained run. The setup of contours and shadings is the same as in Fig. 4.
bV 5 fwdt .
(13)
Figure 13 shows that the bV term is largely balanced by fwdt 1 BPT except near the deep WBC in the Atlantic. In the interior basins, the magnitude of fwdt is small as compared with BPT (Fig. 4b). In the Southern Ocean, there are large cancellations between BPT and fwdt . The LFT and ADV terms (not shown) are most important in the deep WBCs, in particular in the Atlantic. Although the Stommel–Arons theory is attractively simple, it does not hold in an ocean with complex bottom topography: our analysis above shows that 1) in the presence of topography, BPT makes an important contribution to the vorticity budget and 2) the vertical velocities at depth are far from uniform. Both findings are in clear violation of the assumptions that went into the Stommel–Arons theory. By using (8), the vorticity equation (13) (with LFT and ADV being neglected) can be rewritten as
b(V 2 Hy b ) 5 fwdt 2 H 2u b · =( f /H).
(14)
In regions where baroclinicity is weak, the balance is between the two terms at the right-hand side of (15). Such a balance implies that the mass exchange with the upper ocean drives horizonal flow crossing local f /H contours, instead of only meridional flow. Figure 12b shows the volume transport of the constrained model flow field below 2200-m depth. The deep flow differs substantially from the prediction of the Stommel–Arons theory. In the interior basins, the deep flow aligns zonally on many occasions. In the eastern North Pacific, the flow is basically equatorward instead of poleward as predicted by the theory. Some of these deep flow pathways are consistent with observations. For example, in the South Atlantic near 208S, there are two branches of northward flow; both join the deep WBC just south of the equator. Near 258S, an offshore branch out of the deep WBC flows zonally across the Brazil Basin and reaches the mid-Atlantic Ridge. This pattern matches
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FIG. 10. Zonal integration from the eastern boundaries of the Pacific and Atlantic Oceans of the following quantities: (a) meridional transport V, (b) V 2 curl(t s )/b, and (c) 2 fwub /b. Here V is the volume transport from surface to 985-m depth and wub is the vertical velocity at 985-m depth. The results are obtained from the constrained run. In (a), the contour interval is 5 Sv. In (b) and (c), the contour interval is 2 Sv for values between 210 and 10 Sv. Values larger than 10 Sv and smaller than 210 Sv are shown by dark and gray shadings, respectively. In all panels, positive, zero, and negative values are distinguished using thinner solid, thick solid, and dashed lines.
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FIG. 11. Vorticity balance in the intermediate layer from 985 to 2200 m: (a) bV and (b) f (wmt 2 wmb ). Values between 20.5 3 10 210 and 0.5 3 10 210 m s 22 are shown by contour lines spaced by 0.1 3 10 210 m s 22 . Positive, zero, and negative values are distinguished using thinner solid, thick solid, and dashed lines. Values larger than 0.5 3 10 210 and smaller than 20.5 3 10 210 m s 22 are shown by dark and gray shadings, respectively.
nicely with the measurements made during the Deep Basin Experiment (see Hogg 2001). On the other hand, the direction and location of deep WBCs in the model solution correspond well with the map shown by Stommel (1958). The only exception is in the North Pacific, where Stommel showed a southward flow in the northern part of the western boundary while our solution shows a continued northward flow. In the Atlantic, the direction and strength of the deep WBC may be entirely determined by downwelling at high latitude. The model results show that friction is important in the Atlantic deep WBC, consistent with the assumption in Stommel–Arons theory. As was shown in Fig. 13, the Pacific deep WBC is nearly determined by fwdt 1 BPT; hence the role played by topography is more important than that played by friction. In the interior ocean,
no indication can be found that the Stommel–Arons theory holds, not even in our coarse-resolution results. Figure 14 shows the differences between the constrained and control runs in terms of fwdt and BPT. The magnitudes of the differences are significant; in fact they are comparable in amplitude to those of fwdt and BPT over large areas. As a result, the difference in bV is also significant, both in the interior basins and in the deep WBCs. Those changes in the vorticity balance are brought about by differences in the initial temperature and salinity conditions and in the time-varying forcing fields. To identify the relative importance of changes in both, we computed the above terms from an additional test run, which was forced by the optimized surface forcing, but which was initialized in the same way as the control run. With the exception of the WBC in the Atlantic
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FIG. 12. Volume transport (a) in the intermediate layer from 985- to 2200-m depths and (b) in the deep ocean from 2200 m to the seafloor. Both panels show the solution from the constrained run. Transports less than 2 m 2 s 21 are not shown. Transports from 2 to 5 m 2 s 21 are shown by blue arrows, 5 to 10 m 2 s 21 by red arrows, and greater than 10 m 2 s 21 by green arrows. Isobaths of 2000, 3000, and 4000 m are shown by black curves.
Ocean, the results are much closer to the control run in terms of bV, fwdt , and BPT, underlining the significant impact of the deep density field on the abyssal flow field and the associated vorticity balances. Away from the deep WBC in the Atlantic, changes in initial conditions cause the majority of the difference between the constrained and control runs in the deep ocean. Changes in
surface forcing mainly cause changes in the deep WBC in the Atlantic during the 9-yr period of model simulations. Because only WBCs adjust quickly to variations in surface forcing, we can conclude that changes in wind stress variations on seasonal, interannual, and possibly decadal timescales do not play a crucial role on those timescales for the vorticity balance in the deep ocean.
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FIG. 14. Difference between the constrained run and the control run in terms of the vorticity balance in the deep ocean from 2200 m to the seafloor: (a) D( fwdt ) and (b) DBPT (10 210 m s 22 ).
FIG. 13. Vorticity balance in the deep ocean from 2200 m to the seafloor: (a) bV, (b) fwdt , and (c) fwdt 1 BPT (10 210 m s 22 ).
9. Discussion In the past, considerable efforts went into seeking observational evidence of a Sverdrup relation in the ocean. The common approach was to calculate the meridional flow from cross-basin hydrographic sections and to compare the result with the climatological curl of wind stress. Roemmich and Wunsch (1985) provided computations along two zonal sections at 248 and at 368N, with the upper ocean bounded by the density surface of s u 5 27.4 (nominal depth 900 m). They found an approximate Sverdrup balance in the eastern part of the Atlantic along 248N over a distance of 4000 km. However, a significant departure was reported in the western basin at the same latitude (west of 568W) and along the entire section at 368N. Circulation in the western basin near 248N is sometimes described as the ‘‘Antilles Current,’’ and high-order dynamics is speculated to operate there (Schmitz et al. 1992). In the Pacific Ocean, Hautala et al. (1994) showed that the Sverdrup relation generally holds at 248N almost independent of
the choice of the base of the upper ocean from s u 5 26.3 to s u 5 27.5. The most significant deviation (about 7 Sv) was found over a range of 208 longitude east of 1508W, but it was not clear whether this deviation represented a transient or permanent feature of the circulation. Despite the fact that observational studies support the Sverdrup balance to hold in the subtropical interior basins, Wunsch and Roemmich (1985) cautioned about the validity of these results. One of the problems in calculating the geostrophic flow from hydrographic data is the assumption of a level of no motion, which is commonly chosen as 1000 m in observational studies. The authors presented a consistency check of this assumption by computing vertical profiles of y and w from data collected in 1981 and during the International Geophysical Year (IGY) of the 1950s. After y was calculated by vertically integrating the thermal wind relation under the assumption of y 5 0 at 1000-m depth, w was computed by vertically integrating the geostrophic vorticity relation
by 5 fw z ,
(15)
with fw 5 curl(t s ) at surface. They found that the nearbottom y and w have substantial magnitudes and do not agree with the bottom kinematic condition. Moreover, the near-bottom velocities from the 1981 data and IGY data have opposite signs, although both data agree with the Sverdrup relation in the upper ocean. They concluded that, although the assumption of y 5 0 near 1000 m leads to agreement with the Sverdrup relation in the
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upper ocean, it is inconsistent when the entire water column is considered. Our model results show considerable deviation from the Sverdrup relation, but this discrepancy cannot be explained in terms of a linear geostrophic vorticity relation because of nonlinear advection and friction. Figure 15 shows vertical profiles of y and w averaged over 108 in longitude in the eastern Atlantic centered at 248N, 318W. Near the surface, the amplitudes in y and w from the constrained and control runs are essentially consistent with those from Wunsch and Roemmich (1985). However, large differences exist below about 1000 m: Here our two solutions show considerably smaller amplitudes in y and w, and the near-bottom y and w values are now consistent with the bottom kinematic boundary condition. For a further comparison, Fig. 16 compares vertical profiles of by and fw z averaged over the same area as in Fig. 15. The figure suggests that (15) holds fairly accurately at middepth in both the constrained and unconstrained runs, roughly between 500- and 2500-m depths. The slight deviation at depth below 3000 m may be due to the increase of lateral friction near topography. Obvious deviations are observed in the upper 500 m, where the increased lateral friction and nonlinear advection must play a role. The deviation from geostrophic vorticity relation in the upper 500 m is the cause of the deviation from Sverdrup relation in the upper ocean. 10. Summary The vorticity analysis presented in the previous sections serves two purposes: 1) to examine the dynamics that set the time-mean circulation in coarse-resolution models and 2) to understand differences between constrained and unconstrained simulations. The conclusions from this analysis can be summarized as follows. 1) The large-scale structure and magnitude of the bottom pressure torque in coarse-resolution simulations are similar to the BPT in the eddy-permitting simulation Hughes and de Cuevas (2001). In the interior basins, BPT acts to balance the difference between wind stress curl and bV for the depth-integrated flow and is a controlling factor of the interior abyssal flow. In the western boundary currents and at high latitudes, BPT has large magnitudes and is at least as important as the sum of lateral friction and nonlinear advection in driving the depth-integrated circulation there. 2) The geostrophic vorticity relations hold in the interior basins at intermediate and deep layers but break down in the upper layer. Thus, in the interior intermediate layer, bV is mainly balanced by vorticity stretching/compression associated with the divergence of vertical flow. In the deep layer, the vorticity stretching/compression is due to the combined
3)
4)
5)
6)
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effect of BPT and the mass exchange with the layer above. The curls of lateral friction and nonlinear advection have the largest magnitudes in WBCs and at high latitudes. As compared with eddy-permitting models, coarse-resolution simulations use larger lateral friction and hence obtain larger LFT and smaller ADV. In the interior basins, LFT and ADV are confined to zonal bands in the upper ocean. In the interior basins of the upper ocean, the model simulations show significant deviation from the Sverdrup balance. In the subpolar gyre regions, the departure can be explained to a large extent by the vorticity stretching/compression associated with the vertical velocity at the base of the upper layer. In the subtropical gyre, the deviation is confined to zonal bands that are associated with the curls of lateral friction and nonlinear advection. The constrained simulation obtains adjustments to surface forcing from NCEP–NCAR reanalysis. The adjustment shows similar structure and magnitudes with low-latitude differences between scatterometerbased wind and reanalysis wind. However, at midand high latitudes, enhancement of zonal bands in the adjusted wind stress curl may include compensations for model deficiencies. The differences between the constrained and unconstrained simulations are significant in vorticity terms. The adjustment to Levitus hydrographic climatological conditions as the model’s initial condition causes the most significant changes in BPT, which is the main reason for changes in abyssal flow. Changes in the depth-integrated circulation are due to changes in both BPT and wind stress. The magnitudes of the changes are substantial for all the major ocean currents.
Our analysis also points to needs for further improvement in ocean state estimation. First of all, errors in the constrained data require further assessment and need to be reflected in assigning weights in the optimization procedure. Next, the model needs to be further improved. The lack of spatial resolution and the use of large viscosity are the most obvious model deficiencies. Although not assessed in the above analysis, the short integration duration and crude representation of topography may also contribute to the deficiency of the present simulations. One indication of model deficiency is the fact that the bottom pressure torque in these simulations is a very noisy quantity. It remains to be examined whether the noise can be reduced in long-term simulations after the deep density field is fully adjusted to the flow field and whether the balances can be improved with improved representation of topography, for example, by using a lopped (or partial) cell formulation (Adcroft et al. 1997). This analysis is intended to examine mainly the largescale structure of the time-mean circulation. The sub-
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FIG. 15. Profiles of (a) meridional and (b) vertical flow in the eastern basin of the Atlantic at 248N. The profiles are averaged over 108 in longitude with the center at 318W. The thick solid curves are from the constrained run, and the dashed curves are from the control run.
division of layers according to fixed height levels is natural for a z-level model; however, it is not ideal from the perspective of ocean dynamics. In the interior oceans, the vertical flow is mainly due to diapycnal mixing. It would be more natural to integrate the horizontal flow between isopycnal surfaces and hence to
examine the budget of potential vorticity. For the deep layer, such analysis will better reveal the different roles played by topography and diapycnal mixing in driving the abyssal flow, and may point out more deficiencies of the model in mixing parameterization. The subdivision of vertical layers is also not ideal for
FIG. 16. Profiles of by and fwz in the eastern basin of the Atlantic at 248N, for the (a) constrained and (b) control runs. The profiles are averaged over 108 in longitude with the center at 318W.
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regional interests. In a recent study, Gent et al. (2001) discussed momentum and vorticity balances at three different depth levels in the Southern Ocean. Their objective was to understand what determines the strength of the ACC. They defined the upper ocean as the Ekman layer, the intermediate layer as between 200 and 2000 m, and the deep layer as below 2000 m. They showed that, over the ACC, the geostrophic vorticity relation does not hold in the whole intermediate layer. Lateral friction and nonlinear advection are important in the vorticity balance and in setting the strength of the ACC. Acknowledgments. We thank Carl Wunsch and Rolf Ka¨se for stimulating discussions and Richard Greatbatch and Dan Wright for comments on an earlier draft. Two anonymous reviewers provided insightful comments that greatly improved the manuscript. Kyozo Uejoshi helped with the preparation of Fig. 7. Computational support was provided by the National Center for Atmospheric Research and through a National Resource Allocation Committee (NRAC) grant from the National Partnership for Computational Infrastructure (NPACI). Reanalysis surface forcing fields from NCEP–NCAR are obtained through NCAR. This work was supported in part through ONR (NOPP) ECCO Grant N00014-991-1049, through NASA Grant NAG5-7857, and through a contract with the Jet Propulsion Laboratory (1205624). This article is a contribution of the Consortium for Estimating the Circulation and Climate of the Ocean funded by the National Oceanographic Partnership Program. REFERENCES Adcroft, A., C. Hill, and J. Marshall, 1997: Representation of topography by shaved cells in a height coordinate ocean model. Mon. Wea. Rev., 125, 2293–2315. Arakawa, A., and V. R. Lamb, 1977: Computational design of the basic dynamical processes of the UCLA general circulation model. Methods Comput. Phys., 17, 174–267. Bryan, F. O., C. W. Boning, and W. R. Holland, 1995: On the midlatitude circulation in a high-resolution model of the North Atlantic. J. Phys. Oceanogr., 25, 289–305. Gent, P. R., W. G. Large, and F. O. Bryan, 2001: What sets the mean transport through the Drake Passage? J. Geophys. Res., 106, 2693–2712. Hautala, S. L., D. H. Roemmich, and W. J. Schmitz Jr., 1994: Is the North Pacific in Sverdrup balance along 248N? J. Geophys. Res., 99, 16 041–16 052. Hogg, N. G., 2001: Quantification of the deep circulation. Ocean Circulation and Climate, Observing and Modelling the Global Ocean, G. Siedler, J. Church, and J. Gould, Eds., Academic Press, 259–270. Hughes, C. W., 1992: The effect of topography on ocean flow. Ph.D. thesis, Oxford University, 141 pp.
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——, and B. A. de Cuevas, 2001: Why western boundary currents in realistic oceans are inviscid: A link between form stress and bottom pressure torques. J. Phys. Oceanogr., 31, 2871–2885. Kessler, W. S., G. C. Johnson, and D. W. Moore, 2003: Sverdrup and nonlinear dynamics of the Pacific Equatorial Currents. J. Phys. Oceanogr., 33, 994–1008. Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with nonlocal boundary layer parameterization. Rev. Geophys., 32, 363–403. Lemoine, F., and Coauthors, 1997: The development of the NASA GSFC and NIMA joint geopotential model. Proceedings of the International Symposium on Gravity, Springer-Verlag, 746 pp. Levitus, S., and T. P. Boyer, 1994: Temperature. Vol. 4, World Ocean Atlas 1994, NOAA atlas NESDIS 4, 117 pp. ——, R. Burgett, and T. P. Boyer, 1994: Salinity. Vol. 3, World Ocean Atlas 1994, NOAA atlas NESDIS 3, 99 pp. Marotzke, J., R. Giering, Q. K. Zhang, D. Stammer, C. Hill, and T. Lee, 1999: Construction of the adjoint MIT ocean circulation model and application to Atlantic heat transport sensitivity. J. Geophys. Res., 104, 29 529–29 548. Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997a: A finite-volume, incompressible Navier–Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 5753–5766. ——, C. Hill, L. Perelman, and A. Adcroft, 1997b: Hydrostatic, quasihydrostatic and non-hydrostatic ocean modeling. J. Geophys. Res., 102, 5733–5752. Nakano, H., and N. Suginohara, 2002: A series of middepth zonal flows in the Pacific driven by winds. J. Phys. Oceanogr., 32, 161–176. Roemmich, D., and C. Wunsch, 1985: Two transatlantic sections: Meridional circulation and heat flux in the subtropical North Atlantic Ocean. Deep-Sea Res., 32, 619–664. Schmitz, W. J., Jr., J. D. Thompson, and J. R. Luyten, 1992: The Sverdrup circulation for the Atlantic along 248N? J. Geophys. Res., 97, 7251–7256. Stammer, D., and Coauthors, 2002: The global ocean circulation during 1992–1997, estimated from ocean observations and a general circulation model. J. Geophys. Res., 3118, doi:10.1029/2001JC000888. ——, and Coauthors, 2003a: Volume, heat and freshwater transports of the global ocean circulation 1993–2000, estimated from a general circulation model constrained by WOCE data. J. Geophys. Res., 3007, doi:10.1029/2001JC000937. ——, K. Ueyoshi, A. Ko¨hl, W. B. Large, S. Josey, and C. Wunsch, 2003b: Estimating air–sea flux estimates through global ocean data assimilation. J. Geophys. Res., in press. Stommel, H., 1958: The abyssal circulation. Letter to the editors. Deep-Sea Res., 5, 80–82. ——, and A. B. Arons, 1960a: On the abyssal circulation of the World Ocean. I. Stationary planetary flow patterns on a sphere. DeepSea Res., 6, 140–154. ——, and ——,1960b: On the abyssal circulation of the World Ocean. II. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep-Sea Res., 6, 217–233. Willebrand, J., S. G. H. Philander, and R. C. Pacanowski, 1980: The oceanic response to large-scale atmospheric disturbances. J. Phys. Oceanogr., 10, 411–429. Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cambridge University Press, 437 pp. ——, and D. Roemmich, 1985: Is the North Atlantic in Sverdrup balance? J. Phys. Oceanogr., 15, 1876–1880. ——, and D. Stammer, 2003: Global ocean data assimilation and geoid measurements. Space Sci. Rev., 108, 147–162.
