Eddy-Resolving Model Estimate of the Cabbeling Effect on the Water Mass Transformation in the Southern Ocean

L. Shogo Urakawa Atmosphere and Ocean Research Institute, University of Tokyo, Chiba, Japan

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Hiroyasu Hasumi Atmosphere and Ocean Research Institute, University of Tokyo, Chiba, Japan

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Abstract

Cabbeling effect on the water mass transformation in the Southern Ocean is investigated with the use of an eddy-resolving Southern Ocean model. A significant amount of water is densified by cabbeling: water mass transformation rates are about 4 Sv (1 Sv ≡ 106 m3 s−1) for transformation from surface/thermocline water to Subantarctic Mode Water (SAMW), about 7 Sv for transformation from SAMW to Antarctic Intermediate Water (AAIW), and about 5 Sv for transformation from AAIW to Upper Circumpolar Deep Water. These diapycnal volume transports occur around the Antarctic Circumpolar Current (ACC), where mesoscale eddies are active. The water mass transformation by cabbeling in this study is also characterized by a large amount of densification of Lower Circumpolar Deep Water (LCDW) into Antarctic Bottom Water (AABW) (about 9 Sv). Large diapycnal velocity is found not only along the ACC but also along the coast of Antarctica at the boundary between LCDW and AABW. It is found that about 3 Sv of LCDW is densified into AABW by cabbeling on the continental slopes of Antarctica in this study. This densification is not small compared with observational and numerical estimates on the AABW formation rate, which ranges from 10 to 20 Sv.

Corresponding author address: L. Shogo Urakawa, Atmosphere and Ocean Research Institute, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan. E-mail: surakawa@aori.u-tokyo.ac.jp

Abstract

Cabbeling effect on the water mass transformation in the Southern Ocean is investigated with the use of an eddy-resolving Southern Ocean model. A significant amount of water is densified by cabbeling: water mass transformation rates are about 4 Sv (1 Sv ≡ 106 m3 s−1) for transformation from surface/thermocline water to Subantarctic Mode Water (SAMW), about 7 Sv for transformation from SAMW to Antarctic Intermediate Water (AAIW), and about 5 Sv for transformation from AAIW to Upper Circumpolar Deep Water. These diapycnal volume transports occur around the Antarctic Circumpolar Current (ACC), where mesoscale eddies are active. The water mass transformation by cabbeling in this study is also characterized by a large amount of densification of Lower Circumpolar Deep Water (LCDW) into Antarctic Bottom Water (AABW) (about 9 Sv). Large diapycnal velocity is found not only along the ACC but also along the coast of Antarctica at the boundary between LCDW and AABW. It is found that about 3 Sv of LCDW is densified into AABW by cabbeling on the continental slopes of Antarctica in this study. This densification is not small compared with observational and numerical estimates on the AABW formation rate, which ranges from 10 to 20 Sv.

Corresponding author address: L. Shogo Urakawa, Atmosphere and Ocean Research Institute, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan. E-mail: surakawa@aori.u-tokyo.ac.jp

1. Introduction

The global thermohaline circulation (THC) is composed of localized deep-water formation and its upwelling. It carries an enormous amount of heat, which amounts to about 2 PW (Trenberth and Caron 2001). Because of this large heat transport, this circulation is considered to have a great effect on climate. Therefore, much attention has long been paid to the way it is driven or sustained. In the present climate, deep water is formed in the northern end of the North Atlantic and around Antarctica (Schmitz 1995). On the other hand, the Indian and Pacific Oceans are characterized by upwelling of such deep-water masses. The Southern Ocean connects these formation and upwelling areas of the deep water and plays an important role in the global THC.

The meridional overturning circulation in the Southern Ocean is considered to be composed of northward flows in the surface and bottom layers and a southward flow in the deep layer (e.g., Speer et al. 2000; Ganachaud and Wunsch 2000; Sloyan and Rintoul 2001b). Strong westerly winds there lead to a significant amount of northward Ekman transport near the sea surface. Around Antarctica, dense shelf water (DSW) is formed on the continental shelves. It descends the continental slope with increasing its volume by entraining overlying Circumpolar Deep Water (CDW) and is transformed to Antarctic Bottom Water (AABW). This bottom water flows northward near the bottom and enters the Atlantic and the Indo-Pacific. There must be a large southward volume transport in the deep layer of the Southern Ocean to compensate these equatorward volume transports. For example, Ganachaud and Wunsch (2000) use an inverse model and show that deep water flows into the Southern Ocean at the rate of 29 Sv (1 Sv ≡ 106 m3 s−1) around 30°S. Sloyan and Rintoul (2001b) also use an inverse model and find that about 52 Sv of upper deep water flows into the Southern Ocean around 30°S. Iudicone et al. (2008a) carry out a numerical experiment with an ocean general circulation model (OGCM) and show that Upper Circumpolar Deep Water (UCDW) is transported into the Southern Ocean at the rate of 16 Sv at 30°S. Previous works have tried to quantify the meridional overturning circulation in the Southern Ocean, but their estimates widely range, as described above. The definition of deep water differs among these previous studies. It partially accounts for the large discrepancy among these different estimates. However, there are large differences between meridional transports of water mass classified in accordance with the same definition. For example, Sloyan and Rintoul (2001b) show that 21 Sv of Lower CDW (LCDW) is transported northward, whereas 1 Sv of LCDW is transported into the Southern Ocean in the study of Iudicone et al. (2008a). Overall, a consensus has not yet been reached on the strength of the meridional circulation in the Southern Ocean.

These meridional volume transports in the surface, deep, and bottom layers indicate that there is large water mass transformation in the Southern Ocean. Deep water, which flows into the Southern Ocean, should be modified into lighter mode and intermediate waters or heavier deep and bottom waters to close the volume budget. Such transformation requires buoyancy gain or loss, where the air–sea buoyancy flux and diapycnal mixing have been considered to play a dominant role. Therefore, there have been many studies which focus on the water mass transformation due to these processes (e.g., Speer et al. 2000; Sloyan and Rintoul 2001b,a; Iudicone et al. 2008a). On the other hand, it is known that nonlinearity of the equation of state (EOS) produces large diapycnal velocity (Klocker and McDougall 2010), which coincides with a local water mass transformation rate per unit area. Our study focuses on cabbeling, which is the most important process that results from the nonlinearity of EOS (McDougall 1987).

When two water masses with different temperature and salinity are mixed, the density of water becomes higher than the mean density of precursor water masses. The importance of this cabbeling effect on water mass transformation in the Southern Ocean has been pointed out by several preceding studies. Hirst and McDougall (1998) use a low-resolution OGCM under a realistic configuration. Following McDougall (1987), they decompose water mass transformation into contributions of vertical diffusion, cabbeling, and the effect of slope limitation of the isopycnal diffusion. They show that the vertical diffusion plays a dominant role in the interior domain, which is not directly influenced by the sea surface buoyancy flux. The second largest contribution is from cabbeling. Its large transformation is confined near the surface. Marsh (2000) investigates water mass transformation due to cabbeling with the use of an isopycnic-coordinate OGCM. An intermediate water is transformed to a higher-density water at a rate of up to 7 Sv in the Southern Ocean. Iudicone et al. (2008a,b) also quantitatively investigate the water mass transformation rate in the Southern Ocean, using a depth-coordinate OGCM coupled with a sea ice model, and show importance of water mass transformation due to the cabbeling effect in the Southern Ocean.

Cabbeling is also considered to be important in the energetics of the global THC. Densification due to cabbeling is a result of volume contraction. It makes the center of gravity of the water column fall and leads to a gravitational potential energy (GPE) loss. Gnanadesikan et al. (2005) conduct a numerical experiment with an OGCM under a realistic configuration and investigate the GPE budget of the global THC. They show that the convergence/divergence of horizontal buoyancy transport leads to a large GPE loss and conclude that this GPE sink results from the cabbeling effect, although their analysis does not explicitly extract the cabbeling effect. Urakawa and Hasumi (2009) also investigate the GPE budget with an OGCM and find that cabbeling is a major GPE sink in the GPE budget of the THC. They separately calculate the GPE budgets in the Atlantic, the Indo-Pacific, and the Southern Ocean and show that most of GPE consumption by cabbeling occurs in the Southern Ocean. However, their model adopts a horizontal diffusion scheme rather than an isopycnal diffusion scheme. It might lead to overestimation of the cabbeling effect. Urakawa and Hasumi (2010) carry out a numerical experiment with an OGCM that adopts parameterizations for mesoscale eddies (isopycnal tracer diffusion and layer thickness diffusion). Although their estimate of the GPE loss due to cabbeling is smaller than the estimate of Urakawa and Hasumi (2009), the GPE consumption by cabbeling is significantly large.

These preceding studies are based on non-eddy-resolving models. Mesoscale eddies play an important role in the water mass transformation. Strong stirring by mesoscale eddies favors cabbeling in the Southern Ocean. Therefore, eddy diffusivity is an important matter in estimating the cabbeling effect from non-eddy-resolving models. However, most of previous studies adopt a constant isopycnal diffusion coefficient. This assumes that mesoscale eddies are ubiquitous in the ocean and diffuse water masses with the same strength all over the World Ocean. Such constant eddy diffusivity is questionable, because mesoscale eddy activity differs according to location. To gain a comprehensive insight into roles of cabbeling in the global THC, its effect on the water mass transformation in the Southern Ocean should be quantitatively investigated with the use of a high-resolution model.

This paper is composed of the following sections: Section 2 explains the experimental design of this study. A method for diagnosing the water mass transformation rate is briefly introduced in section 3. In section 4, reproducibility of the high-resolution model is checked and the water mass transformation is quantitatively investigated. Section 5 gives the conclusions and discussion.

2. Experimental design

The model employed for this study is the Center for Climate System Research (CCSR) Ocean Component Model (COCO), version 4 (Hasumi 2006). This model is based on the primitive equations under the hydrostatic and Boussinesq approximations with explicit free surface and is formulated on a general orthogonal curvilinear horizontal coordinate and the geopotential height vertical coordinate. The model domain is the Southern Ocean from 75° to 20°S. The northern and southern limits of the model domain are solid boundaries. A realistic bathymetry based on the General Bathymetry Chart of the Oceans (GEBCO; IOC et al. 2003) is used, and the coastline around Antarctica is modified with the use of Moderate Resolution Imaging Spectroradiometer (MODIS) Mosaic of Antarctica (Haran et al. 2005). Ice shelves and grounded icebergs in this model are treated as land. Horizontal resolution is ⅛° in longitude and in latitude. The model has 81 levels in the vertical direction, and the grid spacing is 50 m for the upper 2000 m and linearly increased to 100 m toward the bottom.

An eddy-resolving model tends to simulate large horizontal velocities compared with non-eddy-resolving models. A previous modeling study indicates that numerical diffusion associated with advection by such large velocities could inhibit proper evaluation of the role of eddies in the Southern Ocean (Lee et al. 2002). The second-order moment advection scheme developed by Prather (1986) with the flux limiter of Morales Masqueda and Holloway (2006), which is known to cause much less numerical diffusion than conventionally used ones, is employed for tracer advection in this model. A biharmonic horizontal diffusion scheme is employed and its coefficient is 1 × 1010 m4 s−1. The range of horizontal biharmonic diffusion coefficient adopted by previous modeling with a similar horizontal resolution is very wide. Discussion on our choice of this coefficient is given later. Vertical diffusion coefficient is 5 × 10−5 m2 s−1. For momentum equations, an enstrophy-conserving scheme (Ishizaki and Motoi 1999) and a biharmonic friction with Smagorinsky-like viscosity (Griffies and Hallberg 2000) are employed.

The model is spun up for 1 yr from a state of rest. During this spin-up period, potential temperature and salinity are restored to World Ocean Circulation Experiment (WOCE) global hydrographic climatology (WGHC; Gouretski and Koltermann 2004) from top to bottom with a restoring time of 30 days. Then, the model is forced for 10 yr by monthly climatological average of Röske’s (2001) wind forcing. Sea surface temperature and salinity are restored to the monthly climatological values of World Ocean Atlas 2001 (WOA01; Stephens et al. 2002; Boyer et al. 2002) with a restoring time of 50 days. At the northern boundary, potential temperature and salinity from 23° to 20°S are restored to WGHC from top to bottom with a restoring time of 30 days. Dense shelf water on the Antarctic continental shelf (shallower than 1000-m depth) is prescribed by restoring potential temperature and salinity to the monthly climatological average of a high-resolution East Antarctic model (EAA; Kusahara et al. 2010, 2011). A restoring time is 30 days, except for 5 days at the southern boundary of the model domain. A set of 3-day interval snapshots for the last 1 yr is analyzed.

A steady state is not achieved in our model because of the short integration time, and our model state gradually drifts away from the initial state. However, it is not feasible to run such an eddy-resolving model for hundreds or thousands of years under the presently available numerical resources. The drift indicates shortcomings of our model and/or forcing. A steady state achieved in such a model will be unrealistic. Our main focus is the cabbeling effect on the water mass transformation in the Southern Ocean. To gain the realistic estimate of that, it is essential to represent realistic distributions of potential temperature and salinity. From this point of view, the short integration time is advantageous to us because it leads to less deviation of our model from initial potential temperature and salinity distributions.

3. Methodology for the water mass transformation analysis

Advection–diffusion equations for potential temperature and salinity are described as
e1
e2
where (ϕ) is the convergence/divergence term of diffusion flux including surface flux for an arbitrary tracer ϕ. The tendency equation of potential density σ can be derived as
e3
where α* and β* are defined as α* ≡ ∂σ/∂θ and β* ≡ ∂σ/∂S. The right-hand side of this equation can be decomposed into the diffusion term of potential density and the cabbeling term,
e4
This diffusion term can furthermore be decomposed into the surface buoyancy flux term and the internal mixing term,
e5
On the other hand, the material derivative of potential density can be described as
e6
where ωσ is the diapycnal velocity across an arbitrary isopycnal surface Sσ and σ is the gradient of potential density in the direction normal to Sσ. Note that (negative) ωσ denotes the velocity toward higher (lower) density. The total volume flux Ωσ, which corresponds to the water mass transformation rate, across the surface Sσ can be calculated by integrating ωσ over Sσ,
e7
The right-hand side of this equation can be transformed with the use of a generalized form of Leibnitz theorem as (Iudicone et al. 2008b)
e8
where Vσ is the volume bounded by Sσ and the ocean bottom (Fig. 1). The three terms in the final expression are contributions of the surface buoyancy flux, the internal mixing, and the cabbeling,
e9
e10
e11
respectively. A water mass transformation rate due to each process can be calculated with this method. Horizontal and vertical components of contributions of the internal mixing and the cabbeling are separately calculated and discussed (horizontal and vertical components are denoted by subscripts h and υ, respectively; i.e., D = Dh + Dυ). The net water transformation rate Ωσ is composed of effects of body forcing (restore at the northern boundary and on the Antarctic Continental shelves), convective adjustment, and cabbeling due to numerical diffusion in addition to these five terms (F, Dh, Dυ, Ch, and Cυ). The contribution of convective adjustment is small and not discussed in this study. The effect of cabbeling induced by the numerical diffusion is diagnosed with the method described in the appendix.
Fig. 1.
Fig. 1.

A schematic for water mass transformation.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

From the viewpoint of volume conservation, the net water mass transformation rate Ωσ satisfies the following equation (Nurser et al. 1999):
e12
where ψ denotes the volume export from the layer bounded by Sσ and the bottom at the northern boundary of volume integration (Fig. 1). It is assumed that there is no water exchange at the surface (which is the case in this model) and the southern boundary. This relationship shows that the net water mass transformation rate Ωσ coincides with the meridional streamfunction calculated in the σ coordinate with the reversed sign in a steady state (Marsh et al. 2000). However, a steady state is not achieved in this study because of the short integration time. Care must be taken when we compare the meridional streamfunction and the water mass transformation rate.

4. Results

a. Reproducibility of the model

Annual-mean zonal averages of potential temperature and salinity in the model are compared with those of WGHC in Fig. 2. In this figure, layer thickness-weighted averages are first calculated along σ2 isopycnal surfaces and then remapped to the depth coordinate with the use of the mean depths of the isopycnal surfaces. Because the height of isothermal and isohaline surfaces widely changes with space, zonal averages at a constant height lead to spurious mixing of water masses. Characteristics of water masses are represented better in the isopycnal layer thickness-weighted averages. Cold water covers the sea surface at high latitudes in the observation, whereas sea surface water is relatively warm in the model. At middepth, slopes of the isothermal surfaces from 1° to 3°C are a little smaller and the water mass around 60°S is more fresh than that in the observation. The signal of Antarctic Intermediate Water (AAIW; characterized by the salinity minimum) is being weakened in the model. Near the bottom, the northward intrusion of a cold water (θ < −0.8) is weak in the model. However, the overall distributions of potential temperature and salinity are realistically kept in this model.

Fig. 2.
Fig. 2.

Zonal average of (a),(c) potential temperature and (b),(d) salinity of the (left) WGHC and (right) model. Note that contour intervals are uneven.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

The meridional overturning circulation in the model is shown in Fig. 3. The zonal integration is performed along isopycnal surfaces. There are three major cells (Döös and Webb 1994): a subtropical cell (σ2 < 35), a subpolar cell (35 < σ2 < 36.7), and a deep cell (36.7 < σ2). The subpolar cell is associated with the North Atlantic Deep Water (NADW) formation and the deep cell is associated with the AABW formation. The subtropical and subpolar cells in this study are comparable to those of a previous eddy-permitting model (Lee and Coward 2003) or an eddy-resolving model (Lee et al. 2007). On the other hand, the deep cell is larger than that of these preceding studies. Lee and Coward (2003) use an ocean-only model, where DSW is not well reproduced. This is the reason why the deep cell is not properly represented in their model (Lee et al. 2007). The model used by Lee et al. (2007) includes a sea ice component. The deep cell in their study carries about 20 Sv of bottom water. The transport by the deep cell in this study exceeds that in Lee et al. (2007) by about 8 Sv. Although a sea ice model is not incorporated into the present model, we try to reproduce the dense water around Antarctica by the restore to the high-resolution East Antarctic model result (Kusahara et al. 2010), where sea ice production and dense shelf water formation in coastal polynyas are realistically reproduced. Therefore, we believe that such a strong deep cell is possible in the real ocean.

Fig. 3.
Fig. 3.

Meridional overturning circulation in the Southern Ocean (a) in σ2 coordinates and (b) in depth coordinates. The streamfunction of (b) is calculated by remapping that of (a) to depth coordinates with the use of the mean depths of isopycnal surfaces. The units are Sv, and the contour interval is 4 Sv.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

Figure 4a shows root-mean square of the sea surface height anomaly (SSHA) calculated by using the Ocean Topography Experiment (TOPEX)/Poseidon satellite altimeter data of the year 1993. High SSHA variability indicates strong eddy activity. Such large variance is found to the east of Australia, in the Brazil–Malvinas confluence and in the Agulhas retroflection region. These areas are well known for energetic activity of mesoscale eddies. Our model well captures the distribution of such SSHA variance (Fig. 4b), although its amplitude is slightly larger than the satellite data. Such a tendency is also found in previous studies (Tanaka and Hasumi 2008b; Masumoto et al. 2004). Tanaka and Hasumi (2008b) suspect that this is due to sparseness of the satellite altimeter data. Pascual et al. (2006) show that merging several satellite data leads to larger SSHA variance and improving the representation of the mesoscale variability in the global ocean.

Fig. 4.
Fig. 4.

The root-mean-square of the SSHA (a) for TOPEX/Poseidon (in the year 1993) and (b) for the model. The units are centimeters. Areas below 3 cm or with no data are stripped.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

b. Water mass transformation due to processes other than cabbeling

Water is classified into the following water masses in this study: surface/thermocline water (STW; σ2 < 34.70), Subantarctic Mode Water (SAMW; 34.70 < σ2 < 35.75), Antarctic Intermediate Water (35.75 < σ2 < 36.30), Upper Circumpolar Deep Water (36.30 < σ2 < 36.86), Lower Circumpolar Deep Water (LCDW; 36.86 < σ2 < 36.98), and Antarctic Bottom Water (σ2 > 36.98). Figure 5 shows water mass transformation rates (the volume flux across isopycnal surfaces) in the Southern Ocean.

Fig. 5.
Fig. 5.

(a) Water mass transformation rates in the Southern Ocean (30°–75°S). The dashed and dotted lines denote contribution of sea surface buoyancy flux and DSW formation by restore to the EAA of Kusahara et al. (2010), respectively. The solid line represents the sum of these. A positive value indicates densification of a water mass. (b) Water mass transformation due to horizontal diffusion (solid line) and vertical diffusion (dotted line). The cabbeling effects are not included there. (c) Water mass transformation rates by cabbeling due to horizontal and numerical diffusion (thick line), vertical diffusion (dotted line), and numerical diffusion (thin solid line). (d) As in (c), but the thin line represents cabbeling due to horizontal and numerical diffusion at high latitudes (60°–75°S). (e) Net transformation rate (dotted line). The solid line denotes the volume change rate of the domain between each isopycnal surface and the bottom. (f) The thick line denotes the residual of two in (e). The thin line represents the meridional streamfunction at 30°S with the sign reversed (as in Fig. 3a). Note that dense water classes are magnified in this figure.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

Previous works point out that the sea surface buoyancy flux plays a dominant role in the water mass transformation (Iudicone et al. 2008a,b). The present study also shows dominance of the surface buoyancy flux (Fig. 5a). The surface buoyancy flux strongly modifies SAMW, AAIW, and UCDW toward the lighter water masses in our study. Its maximum impact is found for AAIW, which amounts to over 20 Sv. Densification by the surface buoyancy flux is found for STW and LCDW. These results are consistent with those of preceding studies (Iudicone et al. 2008a,b). Iudicone et al. (2008a,b) show that AABW is densified by the surface buoyancy flux, whereas it acts to make AABW lighter in the present study. The model used by Iudicone et al. (2008a,b) is coupled to a sea ice model, where DSW is produced on the Antarctic continental shelves. Such a dense water formation is brought about by restoring potential temperature and salinity on the Antarctic continental shelf to the results of a high-resolution East Antarctic model in our model. Over 5 Sv of water mass transformation rate is found for UCDW, LCDW, and AABW in the present study.

About 7 Sv of LCDW and about 10 Sv of AABW are modified by vertical diffusion into UCDW and LCDW, respectively (Fig. 5b). About 5 Sv of AAIW is transformed into SAMW in our study. This is not consistent with the result of previous studies (Iudicone et al. 2008a,b), which show strong densification of SAMW into AAIW. Although the contribution of vertical diffusion in the present study does not contain the cabbeling effect and that in the previous works does, the cabbeling due to vertical diffusion in the present study is relatively small (Fig. 5c) and does not explain this discrepancy. It is not clear what causes this difference. Most of the strong modification of Iudicone et al. (2008a,b) occurs in the mixed layer. Near the sea surface, vertical resolution of their model is finer than our model. On the other hand, stratification near the surface seems to be stronger than the climatological data in their model (Fig. 1 of Iudicone et al. 2008b). This leads to large vertical mixing.

Our model explicitly resolves mesoscale eddies, which tend to stir water masses in an isopycnal direction. Horizontal biharmonic diffusion induces large water mass transformation in the present study (Fig. 5b), however, which indicates that stirring by mesoscale eddies leads to strong diapycnal diffusion. It is remarkable that an enormous amount of dense water is modified into lighter water by horizontal diffusion1 (about −50 Sv is found in the AABW class). Discussion on this point is given later.

c. Cabbeling effect on the water mass transformation

There are three types of cabbeling in our model. Two of those are cabbeling associated with horizontal biharmonic diffusion (horizontal cabbeling) and vertical diffusion (vertical cabbeling), and the other is cabbeling due to numerical diffusion (numerical cabbeling) induced by errors in the advection scheme. On the other hand, cabbeling is defined in a limited sense as a process associated with isopycnal diffusion in preceding studies (e.g., McDougall 1984). Hirst and McDougall (1998) and Iudicone et al. (2008a) exclude a contribution of vertical diffusion from their calculations of the cabbeling effect. It is better to separately discuss numerical cabbeling by its horizontal and vertical components. However, we cannot do so with the method proposed in the appendix. The pattern of numerical cabbeling is much closer to that of horizontal cabbeling rather than vertical cabbeling (Fig. 5c). In addition, the effect of vertical cabbeling is smaller than that of horizontal cabbeling. Therefore, we assume here that numerical cabbeling occurs horizontally rather than vertically and refer to the sum of horizontal and numerical cabbeling simply as cabbeling hereafter.

In our model, SAMW and AAIW undergo large densification by cabbeling with a peak at the boundary between SAMW and AAIW. SAMW is modified into AAIW at a rate of about 7 Sv, and about 5 Sv of AAIW is transformed into UCDW. Sloyan and Rintoul (2001b) estimate the water mass transformation from AAIW to UCDW by interior mixing at 31 Sv with the use of an inverse model of the Southern Ocean. Densification at the rate of 5 Sv accounts for about 16% of this estimate. Such strong modification of AAIW is also found in Marsh (2000) (about 4 Sv) and Iudicone et al. (2008a,b) (over 10 Sv).

There is another positive peak of densification due to cabbeling at the boundary between LCDW and AABW, and about 9 Sv of LCDW is modified into AABW. Such a strong densification of LCDW is also found in Marsh (2000) (over 3 Sv). Iudicone et al. (2008a,b) show strong modification of LCDW, about 6 Sv, into AABW. They also show that water mass transformation induced by the isopycnal diffusion is negative for AABW. They discuss that thermobaricity is important at this high-density class. However, the transformation rate with the use of surface-referenced thermal expansion and saline contraction coefficients, which do not include the thermobaric effect, also shows negative value in the AABW class of their study. The effect of the isopycnal diffusion in their study basically coincides with the cabbeling effect. In areas where slopes of isopycnal surfaces are steep such as in the surface mixed layer, however, the isopycnal diffusion parameterization acts as diapycnal diffusion due to the slope limitation. It is speculated that their negative value in the AABW class might be due to the slope limitation of the isopycnal diffusion parameterization. Our study has advantages in the estimation of the cabbeling effect over these previous works. Our model resolves mesoscale eddies and is free from uncertainty of mesoscale eddy diffusivity. To calculate the realistic estimate of its effect, it is important to reproduce realistic activity of mesoscale eddies and temperature and salinity distributions. We confirm that temperature and salinity distributions are realistically maintained in our model and variance of SSHA, which is an indicator of activity of mesoscale eddies, is quite similar to that calculated with the satellite altimeter data.

Diapycnal velocity (velocity across an isopycnal surface) can be calculated by applying Eq. (7) to each column of the model and dividing the transport by the area of the grid box as is done by Iudicone et al. (2008b). Its distributions at the boundaries for STW/SAMW, SAMW/AAIW, AAIW/UCDW, and LCDW/AABW are shown in Figs. 6a–d, respectively. About 4 Sv of STW is transformed into SAMW in this model (Fig. 5c), and this transformation is located north of the Subtropical Front (Fig. 6a). Its diapycnal velocity is especially strong around the Brazil–Malvinas confluence zone and the Agulhas retroflection region, where mesoscale eddies are active (Fig. 4b). Densification of SAMW into AAIW and of AAIW into UCDW by cabbeling is widely distributed along the Antarctic Circumpolar Current (ACC; Figs. 6b,c). Diapycnal volume transports of SAMW to AAIW are large to the north of the Subantarctic Front, where SAMW is often found (e.g., Sallée et al. 2006). Diapycnal velocity at this boundary is equally strong among the Atlantic, Indian, and Pacific sectors of the Southern Ocean. Antarctic Intermediate Water is modified into UCDW around the Polar Front. Although large diapycnal velocity on this isopycnal surface is also widely found over the Southern Ocean, densification is especially large in the Pacific sector. On the other hand, cabbeling induces the large diapycnal velocity along not only the ACC but also the coast of Antarctica at the boundary of LCDW and AABW (Fig. 6d). This densification at high latitudes occurs on the continental slope (at around 1200-m depth; Fig. 6e). It is consistent with the common understanding of the AABW formation (Orsi et al. 1999): DSW is created on the Antarctic continental shelves and descends on the continental slopes with increasing its volume by entraining the overlying CDW. Cabbeling contributes to raising density during this mixing process. Although its effect is thought important for the AABW formation (Foster 1972), it has not been quantitatively investigated. The present study shows that its impact reaches over 3 Sv (thin line in Fig. 5d). The estimation of the AABW formation rate from observation and models ranges from 10 to 20 Sv (Orsi et al. 1999; Hellmer and Beckmann 2001; Naveira Garabato et al. 2002; Kusahara et al. 2010). The contribution of cabbeling is considerably large, compared to these estimates.

Fig. 6.
Fig. 6.

Diapycnal velocity induced by cabbeling on the isopycnal surfaces of (a) 34.65, (b) 35.75, (c) 36.35, and (d) 36.985 σ2. A positive value indicates densification of a water mass. The units are 10−6 m s−1. Blue lines in (a)–(c) denote the Polar Front, the Subantarctic Front, and the Subtropical Front from the south, respectively (Orsi et al. 1995). (e) As in (d), but East Antarctica is magnified here. Contours denote the depth of the isopycnal surface (contour interval is 300 m).

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

A net water mass transformation rate (dotted line in Fig. 5e) is calculated by summing contributions of sea surface buoyancy flux, body forcing, diffusion, and cabbeling (including vertical cabbeling). In a steady state, the net water mass transformation rate coincides with the meridional streamfunction at 30°S. If this model were in a steady state, this water mass transformation rate would indicate that about 35 Sv of AABW is transported into the Southern Ocean. Of course, this is not the case (Fig. 3), and it indicates that the volume inflation term in Eq. (12) (the first term in the right-hand side) has a significant contribution to the net water mass transformation rate. This effect is calculated from annual changes of depths of isopycnal surfaces. It basically shows a negative value, which means that the depth of the isopycnal surface deepens and the volume of water below this isopycnal surface decreases during this period for analysis (Fig. 5e). It also has a large negative peak around 37 σ2, which is mainly due to the horizontal diffusion (Fig. 5b). This is evidence that our model is drifting away from the initial potential temperature and salinity fields. Subtracting the volume inflation effect from the net water mass transformation, we can calculate the “true” water mass transformation (thick line in Fig. 5f). The thin line in Fig. 5f shows the sign reversed meridional streamfunction calculated from meridional velocities (the same as Fig. 3a) at 30°S. These two lines well agree with each other. Slight differences would be due to the effect of numerical diffusion of density, which cannot be diagnosed with the method proposed in the appendix. Compared with these meridional streamfunctions, the effect of cabbeling (Figs. 5c,d) is significantly large in the deep-water mass transformation in the Southern Ocean and has a big impact on the global THC.

5. Conclusions and discussion

In this study, the cabbeling effect on the water mass transformation in the Southern Ocean is quantitatively investigated with the use of a high-resolution model. The water mass transformation rate by cabbeling has double peaks: one at the boundary between SAMW and AAIW and the other between LCDW and AABW. About 7 Sv of SAMW is estimated to be transformed into AAIW. Large diapycnal velocity, corresponding to this transformation, is found to the north of the Subantarctic Front and widely distributed over the Southern Ocean. Densification of LCDW into AABW amounts to about 9 Sv, which accounts for one-third of the diapycnal volume transport from LCDW to AABW in the Southern Ocean estimated by Sloyan and Rintoul (2001b). Over 3 Sv of that occurs to the south of 60°S. There is large diapycnal velocity not only along the ACC but also on the continental slopes of Antarctica. The amount of this densification is remarkably large, compared to the estimates of AABW formation rate, which ranges from 10 to 20 Sv (e.g., Orsi et al. 1999).

An important question is left to be answered: whether the large diapycnal volume flux by the horizontal diffusion (Fig. 5b) is true or not. As already noted, deep stratification drifts away from reality in our model. A significant amount of dense water is modified into the lighter water by the horizontal diffusion. If such modification is true, a large amount of dense water has to be formed in the Southern Ocean to maintain the realistic stratification. To check this out, we calculate a water mass transformation rate by surface forcings [Eq. (9)] in the East Antarctic model of Kusahara et al. (2010), which well reproduces dense water formation around Antarctica. Monthly-mean data from 1979 to 2008 are used to calculate the climatological water mass transformation rate. It shows strong densification at the rate of about 30 Sv around the LCDW and AABW classes (Fig. 7a), which is significantly larger than that of our Southern Ocean model (solid line in Fig. 5a). It implies that the dense water formation is still insufficient in our model. Figure 7a also shows the sign reversed net transformation rate without contributions of sea surface buoyancy flux and DSW formation by restore on the Antarctic continental shelves in our model. Its sharp peak found in the dense water classes originates from the effect of horizontal diffusion. Most part of that is smaller than dense water formation by surface forcing in the East Antarctic model. If such strong dense water formation is realized in our model, the intense drift of deep stratification might significantly diminish in the model.

Fig. 7.
Fig. 7.

As in Fig. 5, but for water mass transformation rates due to sea surface buoyancy fluxes (a) of the EAA of Kusahara et al. (2010) and (b) associated with the sea ice growth/melt (TONH11). The thin solid line in (a) denotes the sign reversed net transformation rate without contributions of sea surface buoyancy flux and DSW formation by restore on the Antarctic continental shelves in the Southern Ocean model. Temperature and salinity data of WOA05, ECCO2, and the EAA are used for calculations of the dotted, thick solid, and thin solid lines in (b), respectively. The calculation area ranges from 90° to 30°S for the solid lines in (a) and from 80° to 50°S for the dotted line in (a) and for the lines in (b). Note that dense water classes are magnified in this figure.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

Then, another question emerges: Is such large dense water formation found in the East Antarctic model realistic? Tamura et al. (2008) calculate the sea ice formation rate in the Antarctic coastal polynyas from the satellite data and heat budget calculation. Tamura et al. (2011, hereafter TONH11) extend their study and establish a dataset of surface heat/salt fluxes associated with sea ice growth/melt in the Southern Ocean (80°–50°S; http://wwwod.lowtem.hokudai.ac.jp/polar-seaflux/). Using that, we can estimate the water mass transformation rate by surface forcing from observational data. Ocean temperature and salinity data are required for this estimation. We utilize three different datasets: World Ocean Atlas 2005 (WOA05; Locarnini et al. 2006; Antonov et al. 2006), Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2; http://ecco2.jpl.nasa.gov/data1/cube/cube92), and the EAA. These three estimates are named TONH-WOA05 (dotted line), TONH-ECCO2 (thick solid line), and TONH-EAA (thin solid line), respectively (Fig. 7). Accordingly, the estimate shown in previous paragraph is named EAA-EAA, which is recalculated in the area from 80° to 50°S (Fig. 7a). Note that monthly means of heat/salt fluxes and temperature and salinity profiles from 1992 to 2007 are used for calculation.

Positive peaks of TONH-WOA05 and TONH-EAA are as large as that of EAA-EAA but found in the UCDW class. Transformation rates at 37.04 σ2, where the dense water is significantly modified into the lighter water in our model, are about 5 Sv for TONH-WOA05 and 13 Sv for TONH-EAA. TONH-ECCO2 takes its positive peak around 37.04 σ2, but its value is a little small (about 20 Sv). Quite small transformation of TONH-WOA05 in the AABW class might result from the low horizontal resolution of WOA05, which leads to poor representation of the coastal polynyas, and sparse observational data during winter. Although TONH-ECCO2 and TONH-EAA show relatively large densification of AABW, those are smaller than EAA-EAA by 10 and 17 Sv, respectively.

The heat/salt fluxes of TONH11 do not include all the components of heat exchange between ocean and atmosphere or sea ice. Figure 8 shows cumulative heat/salt input into the ocean starting from 80°S in the East Antarctic model (solid lines) and the observational dataset of TONH11 (dashed lines). The ocean is cooled by about 0.4 PW from 80° to 60°S in the East Antarctic model, whereas the ocean is cooled by 0.16 PW in the dataset of TONH11. On the other hand, about 1.1 Gt s−1 of salt is taken from the ocean from 80° to 60°S in the East Antarctic model, whereas about 5.3 Gt s−1 of salt is released into the ocean in the dataset of TONH11. This is because the effects of evaporation, precipitation, and runoff are not incorporated in the salt flux of the dataset of TONH11. These differences between heat and salt fluxes lead to the differences between the estimates of water mass transformation rate by surface forcings. Temperature and salinity distributions are also important for the calculation of transformation rate by surface forcings as well as the sea surface heat/salt fluxes. Although the sea ice formation rate reproduced in the East Antarctic model agrees well with that of Tamura et al. (2008) (Kusahara et al. 2010), these two do not exactly coincide with each other. Such a discrepancy could be problematic in the estimation of transformation rate such as the thin solid line in Fig. 7b. The sea surface buoyancy flux and ocean density field are considered to be highly correlated with each other. For example, cold and dense (warm and light) water tends to be found in strongly cooled (warmed) areas. Inconsistency between the sea surface buoyancy flux and ocean density field might lead to an error in estimation of water mass transformation rate. To quantitatively investigate the water mass transformation rate by surface forcings, it is better to use consistent datasets for sea surface fluxes and ocean temperature and salinity fields. Further studies are needed on this problem. Although we cannot confirm that strong densification in the high-density class is true in the real ocean, we could say that such densification is possible in the real ocean.

Fig. 8.
Fig. 8.

Cumulative (a) heat and (b) salt input into the ocean starting from 80°S. Solid lines are for the EAA of Kusahara et al. (2010), and dashed lines are for the data of TONH11. Both are the averages from 1992 to 2007.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

Even supposing that the water mass transformation rate by surface forcings in the East Antarctic model (Fig. 5a) is true, modification of high-density water into lighter water by horizontal diffusion is still larger than that. This might result from the horizontal diffusion coefficient adopted in our Southern Ocean model. Although this model resolves mesoscale eddies, it still has uncertainty in submesoscale diffusivity. The range of horizontal biharmonic diffusion coefficients is very wide among models of preceding studies that have a similar horizontal resolution. For example, it is 9 × 109 m4 s−1 in Masumoto et al. (2004) and 5 × 107 m4 s−1 in Tanaka and Hasumi (2008a,b). No explicit horizontal diffusion of tracers is used in Lee et al. (2007). The biharmonic diffusion coefficient used in our model is slightly larger than that of Masumoto et al. (2004). There is a possibility that our choice for the horizontal diffusivity is too large to adequately represent the water mass transformation rate due to submesoscale processes. We need to investigate sensitivity of the cabbeling effect to horizontal diffusivity.

However, a sensitivity study of the whole Southern Ocean demands enormous computational resources, and it is beyond the scope of this study. We have conducted a sensitivity study with an idealized model and a high-resolution experiment of the Southern Ocean under the same configuration as this study except for low horizontal biharmonic diffusivity (5 × 107 m4 s−1). Here, we show a brief conclusion of these experiments. Using low horizontal biharmonic diffusion coefficient results in significant reduction of the model’s drift (Fig. 9a). Although it still shows large values in the dense water class, most of this drift can be explained by shortage of dense water formation in our model. By contrast, the cabbeling effect, which is the main target of the current study, is relatively insensitive to the explicit diffusivity (Fig. 9b). Over 7 Sv of LCDW is modified into AABW by cabbeling there. The conclusion of this study quantitatively changes little in this experiment with low diffusivity. Readers might think that much lower horizontal diffusivity would lead to reduction of the drift and the cabbeling effect. The sensitivity study with idealized model shows that this is not the case: the magnitude of the cabbeling effect does not depend on a horizontal biharmonic diffusion coefficient lower than 5 × 107 m4 s−1. This indicates that the estimate of the cabbeling effect does not vary at all as long as the Southern Ocean stratification is well reproduced as in the experiment with low diffusivity shown above.

Fig. 9.
Fig. 9.

As in Fig. 5, but for (a) the volume change of the domain between each isopycnal surface and the bottom and (b) water mass transformation rate due to cabbeling. Thick (thin) lines denote the results of the experiment with high (low) horizontal diffusivity. Note that dense water classes are magnified in this figure.

Citation: Journal of Physical Oceanography 42, 8; 10.1175/JPO-D-11-0173.1

Readers might also think that estimates in the experiment with low diffusivity are more plausible than that in the experiment with high diffusivity. However, another problem is found there: explicit diffusion hardly works with such low diffusivity. Cabbeling due to numerical diffusion dominates in the water mass transformation due to cabbeling shown in Fig. 9b there. We find that numerical diffusion of second-order moment scheme imitates the explicit diffusion in the experiment with low diffusivity and seems to be physically meaningful. Further discussion is required before reaching this conclusion, and it will be reported elsewhere shortly. On the other hand, explicit diffusion dominates in the experiment with high diffusivity. As already noted, the cabbeling effect varies little between these two experiments. We believe that the experiment with high diffusivity well reproduces at least the cabbeling effect in the real ocean.

Acknowledgments

The authors thank A. Oka, Y. Komuro, H. Tatebe, M. Kurogi, M. Watanabe, K. Kusahara, Y. Sasajima, Y. Matsumura, T. Kawasaki, and Y. Hiraike for their helpful comments and discussions. The authors are also grateful to Eric Firing and two anonymous reviewers for their comments. This study is supported by Japan Science and Technology Agency/Core Research for Evolutional Science and Technology; Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists; and Initiative on Promotion of Supercomputing for Young Researchers, Supercomputing Division, Information Technology Center, University of Tokyo. Numerical calculations were done by HA8000 at Information Technology Center, University of Tokyo. Figures in this paper are produced by Grid Analysis and Display System and gnuplot.

APPENDIX

Cabbeling Associated with Numerical Diffusion

Advection in a model contains the effect of numerical diffusion. It does not match the true advection effect. The advection–diffusion equations for potential temperature and salinity in a model are described as
ea1
ea2
where (ϕ) is the advection term for an arbitrary tracer ϕ in a model. This term can be decomposed into the effect of true advection and numerical diffusion as
ea3
The contribution of the numerical diffusion to the tendency of potential density can be described as
ea4
The last two terms in the second line can be decomposed into the density diffusion term and the cabbeling term as in the case of the physical diffusion,
ea5
where num is the cabbeling effect induced by numerical diffusion. Here, we describe the sum of true advection and numerical diffusion terms of density as follows:
ea6
The advection–diffusion equation for potential density can be rewritten as
ea7
Therefore, the cabbeling effect associated with the numerical diffusion can be diagnosed by
ea8
Advection terms of potential temperature and salinity [(θ), (S)] coincide with those solved in a model. By integrating this term as in Eq. (11) and taking its derivative with respect to potential density as in Eq. (8), we can diagnose the cabbeling effect on the water mass transformation.

It has not been clarified the way to evaluate the advection term of density ′(σ). Equation (A6) well resembles Eq. (A3). Therefore, one possible way is to impose the same advection operator as potential temperature and salinity [(θ) and (S)]. However, it is not plausible because the numerical diffusion term of density represents the diffusive component of the right-hand side of Eq. (A5) rather than the numerical diffusion of density due to advection scheme num(σ). In other words, num(σ) does not necessarily coincide with . Magnitude of numerical diffusion depends on the advected tracer field. The term num(σ) depends on density field, whereas depends on fields of potential temperature and salinity. This could lead to a difference between these two terms. We find that numerical cabbeling diagnosed with this method shows negative water mass transformation, which means modification to lighter waters, all over the Southern Ocean. It means that numerical diffusion often brings about up-gradient fluxes of potential temperature and salinity. This is unlikely because the second-order moment scheme hardly causes such an antidiffusive process (Morales Masqueda and Holloway 2006). Another simple way, which we adopt in the current study, is to calculate advected potential density by substituting advected potential temperature and salinity into the equation of state. Unphysical cabbeling is seldom found in the numerical cabbeling effect diagnosed with this method. Therefore, we assume that this method adequately evaluates the advection term ′(σ).

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1

Biharmonic diffusion is not diffusion in a strict sense, because it could cause an up-gradient flux of a tracer. However, it is shown afterward that cabbeling induced by biharmonic horizontal diffusion results in water mass densification in every density class and almost all over the Southern Ocean. This means that antidiffusive tracer fluxes are scarcely found in our model. Therefore, we use the term “horizontal diffusion” here and hereafter in the current study.

Save
  • Antonov, J. I., R. A. Locarnini, T. P. Boyer, A. V. Mishonov, and H. E. Garcia, 2006: Salinity. Vol. 2, World Ocean Atlas 2005, NOAA Atlas NESDIS 62, 182 pp.

  • Boyer, T. P., C. Stephens, J. I. Antonov, M. E. Conkright, R. A. Locarnini, T. D. O’Brien, and H. E. Garcia, 2002: Salinity. Vol. 2, World Ocean Atlas 2001, NOAA Atlas NESDIS 50, 165 pp.

  • Döös, K., and D. J. Webb, 1994: The Deacon cell and the other meridional cells of the Southern Ocean. J. Phys. Oceanogr., 24, 429442.

    • Search Google Scholar
    • Export Citation
  • Foster, T. D., 1972: An analysis of the cabbeling instability in sea water. J. Phys. Oceanogr., 2, 294301.

  • Ganachaud, A., and C. Wunsch, 2000: Improved estimates of global ocean circulation, heat transport and mixing from hydrographic data. Nature, 408, 453457.

    • Search Google Scholar
    • Export Citation
  • Gnanadesikan, A., R. D. Slater, P. S. Swathi, and G. K. Vallis, 2005: The energetics of ocean heat transport. J. Climate, 18, 26042616.

    • Search Google Scholar
    • Export Citation
  • Gouretski, V. V., and K. P. Koltermann, 2004: WOCE global hydrographic climatology. Bundesamt für Seeschifffahrt und Hydrographie Tech. Rep. 35/2004, 54 pp.

  • Griffies, S. M., and R. W. Hallberg, 2000: Biharmonic friction with a Smagorinsky-like viscosity for use in large-scale eddy-permitting ocean models. Mon. Wea. Rev., 128, 29352946.

    • Search Google Scholar
    • Export Citation
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  • Fig. 1.

    A schematic for water mass transformation.

  • Fig. 2.

    Zonal average of (a),(c) potential temperature and (b),(d) salinity of the (left) WGHC and (right) model. Note that contour intervals are uneven.

  • Fig. 3.

    Meridional overturning circulation in the Southern Ocean (a) in σ2 coordinates and (b) in depth coordinates. The streamfunction of (b) is calculated by remapping that of (a) to depth coordinates with the use of the mean depths of isopycnal surfaces. The units are Sv, and the contour interval is 4 Sv.

  • Fig. 4.

    The root-mean-square of the SSHA (a) for TOPEX/Poseidon (in the year 1993) and (b) for the model. The units are centimeters. Areas below 3 cm or with no data are stripped.

  • Fig. 5.

    (a) Water mass transformation rates in the Southern Ocean (30°–75°S). The dashed and dotted lines denote contribution of sea surface buoyancy flux and DSW formation by restore to the EAA of Kusahara et al. (2010), respectively. The solid line represents the sum of these. A positive value indicates densification of a water mass. (b) Water mass transformation due to horizontal diffusion (solid line) and vertical diffusion (dotted line). The cabbeling effects are not included there. (c) Water mass transformation rates by cabbeling due to horizontal and numerical diffusion (thick line), vertical diffusion (dotted line), and numerical diffusion (thin solid line). (d) As in (c), but the thin line represents cabbeling due to horizontal and numerical diffusion at high latitudes (60°–75°S). (e) Net transformation rate (dotted line). The solid line denotes the volume change rate of the domain between each isopycnal surface and the bottom. (f) The thick line denotes the residual of two in (e). The thin line represents the meridional streamfunction at 30°S with the sign reversed (as in Fig. 3a). Note that dense water classes are magnified in this figure.

  • Fig. 6.

    Diapycnal velocity induced by cabbeling on the isopycnal surfaces of (a) 34.65, (b) 35.75, (c) 36.35, and (d) 36.985 σ2. A positive value indicates densification of a water mass. The units are 10−6 m s−1. Blue lines in (a)–(c) denote the Polar Front, the Subantarctic Front, and the Subtropical Front from the south, respectively (Orsi et al. 1995). (e) As in (d), but East Antarctica is magnified here. Contours denote the depth of the isopycnal surface (contour interval is 300 m).

  • Fig. 7.

    As in Fig. 5, but for water mass transformation rates due to sea surface buoyancy fluxes (a) of the EAA of Kusahara et al. (2010) and (b) associated with the sea ice growth/melt (TONH11). The thin solid line in (a) denotes the sign reversed net transformation rate without contributions of sea surface buoyancy flux and DSW formation by restore on the Antarctic continental shelves in the Southern Ocean model. Temperature and salinity data of WOA05, ECCO2, and the EAA are used for calculations of the dotted, thick solid, and thin solid lines in (b), respectively. The calculation area ranges from 90° to 30°S for the solid lines in (a) and from 80° to 50°S for the dotted line in (a) and for the lines in (b). Note that dense water classes are magnified in this figure.

  • Fig. 8.

    Cumulative (a) heat and (b) salt input into the ocean starting from 80°S. Solid lines are for the EAA of Kusahara et al. (2010), and dashed lines are for the data of TONH11. Both are the averages from 1992 to 2007.

  • Fig. 9.

    As in Fig. 5, but for (a) the volume change of the domain between each isopycnal surface and the bottom and (b) water mass transformation rate due to cabbeling. Thick (thin) lines denote the results of the experiment with high (low) horizontal diffusivity. Note that dense water classes are magnified in this figure.

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