1. Introduction
Antarctic ice shelf and land ice masses are declining in response to climate change (e.g., Cook et al. 2005; Rignot et al. 2014; Joughin et al. 2014; Rignot et al. 2019; Joughin et al. 2021) with implications for global climate (Bronselaer et al. 2018) and sea level rise (DeConto and Pollard 2016; Pan et al. 2021). A major contributor is the transport of warm circumpolar deep water (CDW) onto the Antarctic continental shelf producing basal melting of adjacent ice shelves (Jacobs et al. 1992; Rignot and Jacobs 2002; Pritchard et al. 2012). This results in a thinning and retreat of ice shelves exposed to the warm water, which reduces their buttressing effect and accelerates the mass release of marine-terminating glaciers into the ocean (DeConto and Pollard 2016; Paolo et al. 2015).
In the Weddell Sea, the onshore transport of warm deep water (WDW), a derivative of CDW formed through mixing with colder and fresher water within the Weddell Gyre (Vernet et al. 2019), is concentrated at locations where dense water spills over the continental shelf and is topographically steered down the continental slope (Morrison et al. 2020). Indeed, observations within the Filchner Trough, a major pathway for the export of dense water from the Weddell Sea continental shelf, show a coherence between the downslope transport of dense waters and onshore WDW transport (Darelius et al. 2023).
On the Weddell Sea continental shelf, winter surface cooling and salt rejection during sea ice formation transforms cold and fresh Antarctic surface water (AASW) into denser high-salinity shelf water (HSSW), some of which then circulates through the Filchner and Ronne ice shelf cavities (Gordon et al. 2001; Nicholls et al. 2001, 2009; Hattermann et al. 2012; Janout et al. 2021). HSSW induces basal melting at the ice shelf–ocean interface where it is transformed into ice shelf water (ISW) (Jenkins and Doake 1991; Jacobs et al. 1992; Orsi et al. 1999; Foldvik et al. 2004). The dense water subsequently propagates down the continental slope into the abyssal ocean while entraining WDW (Orsi et al. 1999; Gordon et al. 2001; Nicholls et al. 2009). The resulting Weddell Sea bottom water (WSBW) forms the densest and most oxygenated contribution to the Antarctic Bottom Water (AABW), which flows northward as the lower limb of the meridional overturning circulation (MOC) (Fahrbach et al. 1995; Gordon et al. 2001; Orsi and Whitworth 2005).
Together with Ekman convergence and downwelling in response to alongshore winds, the dense water export sets up a characteristic V-shaped isopycnal structure of the Antarctic Slope Front (ASF) (Jacobs 1991; Gill 1973). The ASF separates the continental shelf from WDW, and its offshore flank is associated with the Antarctic Slope Current (ASC) flowing westward along the continental shelf break (Thompson et al. 2018).
The downslope flow of dense water creates an isopycnal connection between the continental slope and shelf so that no work against buoyancy forces is required to move a water parcel onto the shelf (e.g., Nicholls et al. 2009, their Fig. 8). There is, however, a gradient in potential vorticity (PV) resulting from the decreasing thickness of isopycnal layers toward the shelf break (Thompson et al. 2014), forming a dynamical barrier. The descent of dense water generates mesoscale eddies that transfer westward momentum upward. The resulting momentum convergence in the WDW layer then balances the topographic vorticity gradient and allows the onshore flow of WDW (Stewart and Thompson 2016).
Other drivers of shoreward WDW transport include residual tidal flow (Wang et al. 2013), interactions of the ASC with submarine troughs and Rossby wave propagation therein (St-Laurent et al. 2013), bottom boundary layer transport (Wåhlin et al. 2012), and wind forcing (Hellmer et al. 2012; Darelius et al. 2016; Daae et al. 2017; Ryan et al. 2017).
Capturing eddy-driven exchanges across the ASF is challenging for numerical ocean models because the small deformation radius at high latitudes can only be resolved at fine horizontal resolutions. To resolve the first baroclinic radius of deformation on a continental shelf and slope at a latitude of 65°S, an ocean model requires a grid resolution of approximately 1 km (Hallberg 2013), much higher than currently feasible in global climate models. Idealized numerical experiments representing the Antarctic continental slope and shelf confirm that a horizontal resolution on the order of
When eddies are not resolved, a parameterization of their effects on the model solution is required. For this purpose, a combination of the Gent and McWilliams (GM, Gent and McWilliams 1990) and the Redi (Redi 1982) scheme is commonly used. The GM scheme reduces isopycnal slopes by means of an advective tracer flux where the advective velocity, often labeled bolus velocity, is a function of the slope of the local isentropic surface. The Redi scheme in turn imposes a downgradient diffusion of tracers along neutral surfaces, representing isopycnal diffusion of mesoscale eddies (Redi 1982). Both schemes require setting a transfer coefficient, the thickness or GM diffusivity κGM, and the isopycnal or Redi diffusivity κRedi.
Initially often set constant, it is clear that the GM and Redi diffusivities should vary in space and time. Several schemes to compute a spatially varying GM coefficient have been proposed based on mixing length theory, in which the diffusivity is related to the product of an eddy length scale and velocity (e.g., Green 1970; Stone 1972; Visbeck et al. 1997; Cessi 2008; Eden and Greatbatch 2008; Fox-Kemper and Ferrari 2008; Jansen et al. 2015) or based on properties of the eddy stress tensor (Marshall et al. 2012). In a subclass of schemes, the GM diffusivity is related to the subgrid eddy energy (e.g., Cessi 2008; Eden and Greatbatch 2008; Marshall et al. 2012; Jansen et al. 2015).
Frameworks for spatially varying estimates of κGM are usually developed for the case of a flat bottom. Sloping bathymetry, however, influences baroclinic instability depending on the ratio between topographic and isopycnal slope δ = stopo/siso (Blumsack and Gierasch 1972; Mechoso 1980; Isachsen 2011; Brink and Cherian 2013). For δ < 0, the bottom slope has a stabilizing effect so that growth rates and length scales reduce with |δ|. When isopycnals moderately slope in the same direction as the bathymetry (0 < δ < 1), the bottom slope acts to destabilize the flow with maximum growth rates obtained for δ = 0.5. Finally, in the case of topographic slopes steeper than the slope of the isopycnals (δ > 1), the growth of instability is entirely suppressed.
The mean topographic slope adjacent to the dense shelves is approximately 0.02 in the Weddell Sea and reaches 0.07 in the Cape Darnley region and Adelie Coast (Mensah et al. 2021). Within the ASF, isopycnal slopes tilt in both the same and opposite direction compared to the continental slope, resulting in values of |δ| of up to 15 (Stewart and Thompson 2013; Le Paih et al. 2020). In a process model of the ASF and ASC, Stewart and Thompson (2013) infer reduced diffusivities over the continental slope where δ < 0. Scalings that diagnose the eddy diffusivity from the output of process model simulations of continental slopes perform better when they incorporate information about the topographic slope for both δ < 0 and δ > 0 (Wei and Wang 2021; Wei et al. 2022). Only recently, modifications to make the GM/Redi scheme slope-aware have been implemented in numerical ocean models (Wei et al. 2024; Nummelin and Isachsen 2024). Nevertheless, these implementations remain to be tested in more detail, in particular, in the context of downslope flows of dense water and onshore flow of WDW at middepth.
In this work, we apply the GM/Redi parameterization to a numerical ocean model representing the ASF and address the following questions:
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Does the GM/Redi parameterization for mesoscale eddies reproduce eddy-driven shoreward heat flux associated with the presence of WDW?
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What is the effect of the GM/Redi parameterization on the simulated hydrographic fields?
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What are suitable choices for the diffusivities within the GM/Redi scheme to represent the exchange of heat across the continental slope?
For this purpose, we use an idealized model of the Weddell Sea continental slope and shelf and compare high- and coarse-resolution simulations with and without the GM/Redi parameterization. The model setup and parameterization are described in section 2, and the performance of the GM/Redi scheme using different diffusivity estimates is evaluated in section 3, followed by a discussion and conclusions in section 4.
2. Model setup and analysis
For this work, an idealized model of the Weddell Sea continental slope and shelf is set up. The configuration closely resembles the one described in Stewart and Thompson (2016), for which we will only give a brief description and refer the reader to the original publication for more details. As a reference, we run the model at high-resolution resolving the first baroclinic radius of deformation and then compare the outcome to a coarse-resolution simulation in which the Rossby radius is not resolved. Subsequently, we add the GM/Redi parameterization at coarse resolution and investigate its influence on cross-slope heat fluxes and the hydrographic mean state.
a. Reference simulations
All experiments are performed using the hydrostatic version of the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al. 1997; MITgcm Group 2023). The domain has a horizontal extent of 400 km × 450 km, featuring periodic boundaries in the x direction and closed boundaries in the y direction. The bathymetry of the Weddell Sea continental slope is represented through an idealized, meridionally homogeneous slope connecting a 500-m-deep shelf section to the ocean bottom at 3000-m depth (Fig. 1). At the surface, the model is forced by a time-invariant zonal wind stress profile τx with a maximum stress of τmax = −0.075 N m−2 representing westward wind. Over the first 50 km of the shelf, salt is injected at the surface at a rate of ssurf = 2.5 mg m−2 s−1 to produce dense water. To maintain realistic surface water conditions, a two-equation thermodynamic sea ice model employing a constant turbulent heat exchange velocity of 6 × 10−5 m s−1 is used (Schmidt et al. 2004). Here, surface heat and salt fluxes representing freezing and melting are determined from surface temperature and salinity. Within a 50-km-wide sponge layer at the open ocean boundary, velocities are restored to zero and temperature and salinity are restored to the initial profiles with time scales of 27 and 54 days, respectively. For the experiments, we select a nonlinear equation of state of McDougall et al. (2003) and a third-order direct space-time advection scheme with flux-limiting. The nonlocal K-profile parameterization (KPP; Large et al. 1994) represents vertical mixing in the surface boundary layer and the ocean interior. At the bottom, momentum is extracted by bottom drag parameterized using a linear bottom drag coefficient of rb = 10−3 m s−1. Here, the absence of along-slope topographic variations and the associated topographic form drag requires setting an untypically large bottom drag coefficient to simulate ASC velocities in the range of observed values. The model is run on an f plane with β = 0 since the vorticity gradient resulting from the sloping topography is 100 times larger than the change in planetary vorticity. All simulations are initialized from rest using profiles of potential temperature θ and salinity S representative of the western Weddell Sea (Thompson and Heywood 2008). The model is then integrated with a horizontal grid spacing of 10 km for 40 years after which mean kinetic and potential energies have stabilized and no drift in the domain-averaged temperature and salinity is observed. This coarse resolution ensures that eddies are mostly unresolved over the continental slope while the slope is still represented by a reasonable number of 15 grid points. To obtain the high-resolution reference simulation, the output fields are interpolated to a horizontal resolution of 2 km after which the model is run to equilibrium again. This procedure is then repeated for a horizontal resolution of 1 km. Further refinements in resolution did not produce major changes to the model solution, and therefore, the simulation with a resolution of 1 km will serve as our reference. The numerical parameters of the reference simulation are summarized in Table 1.
(a) Input profiles for surface salt flux ssurf and zonal wind stress τx, (b) topographic slope and along-slope and time-averaged potential temperature θ at 1-km resolution (color shading) and neutral density (contours), and initial and restoring profiles of (c) potential temperature and (d) salinity S.
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
Parameter choices for the high-resolution reference simulation. Parameters for the coarse-resolution runs are given in parentheses if different from the high-resolution case.
b. Gent–McWilliams/Redi parameterization
c. Simulation analysis
3. Results
a. Model solutions at high and coarse resolution
We start by discussing the differences in the model solutions at horizontal resolutions of 1 and 10 km, which a suitable parameterization has to overcome. We note here that running the model at a resolution of 1 km increases the computational cost by a factor of 600 compared to the resolution of 10 km.
At the surface, a fresh and cold layer of water is maintained by interactions with the simplified thermodynamic sea ice model (Figs. 2a,b). The westward wind stress leads to shoreward Ekman transport resulting in the depression of the isopycnals where the surface water converges over the shelf break. The salt input over the shelf produces dense water flowing down the continental slope in the form of a gravity current. The warm and salty water in between is connected to the continental shelf through sloping isopycnals resulting from both Ekman pumping and dense water export. With the strong idealization of the model setup in mind, we will refer to these waters as AASW, WDW, and WSBW, separated by the neutral density surfaces of 28.10 and 28.45 kg m−3. For a detailed discussion of the dynamical processes in the high-resolution setup, the reader is referred to Stewart and Thompson (2016).
(left) Along-slope and time-averaged potential temperature and (right) salinity for horizontal resolutions of (a),(b) 1 and (c),(d) 10 km without the GM/Redi scheme and (e),(f) with the GM/Redi scheme setting
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
At a resolution of 10 km, the isopycnal slopes are steeper as they cannot be relaxed as effectively in the absence of small-scale eddies (Figs. 2c,d). Consequently, the surface water is displaced further downward and pushes the WDW further offshore. As a result, both the shelf and the gravity current on the continental slope are colder. On the shelf, the isopycnals are now particularly steep and the salt input cannot be distributed as effectively in the horizontal. Close to the shelf break, interactions with the downward-displaced fresh surface water lead to an even fresher gravity current.
At such coarse resolution, the along-slope averaged EKE is orders of magnitude smaller compared to the high-resolution reference simulation (Figs. 3a,b). Similarly, the eddy component of the heat flux strongly reduces over the slope and shelf (Figs. 3c,d). At both resolutions, the eddy heat flux is balanced by a mean offshore heat flux over the shelf and slope. On the open ocean side, the net shoreward heat flux is balanced by the thermodynamic sea ice model and the restoring layer (not shown). The salt fluxes are dominated by the mean component over the slope and open ocean, moving the salt injected over the shelf offshore, whereas the eddy component of the salt flux is generally small (Figs. 3e,f). Since the model sustains no meridional mean flow over the shelf, all meridional fluxes are counted as eddy fluxes here [Eq. (9)], explaining how salt fluxes on the shelf can still be resolved at coarse resolution. In the following, we focus our discussion on the eddy heat fluxes.
(a),(b) Along-slope and time-averaged EKE (shading and contours), (c),(d) onshore heat, and (e),(f) salt fluxes for horizontal resolutions of (left) 1 and (right) 10 km.
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
b. Estimating the GM diffusivity
(a) Mean and (b) eddy contributions to the residual overturning streamfunction (color shading and contours) and (c) estimate of the GM diffusivity according to Eq. (21). Contour lines in (c) show the neutral density (same levels as in Fig. 2).
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
Vertically averaged diffusivity
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
We note that over the open ocean section, the slope-aware GM scheme following Visbeck et al. (1997) produces a lower GM diffusivity compared to
Methods to diagnose κRedi (e.g., Bachman et al. 2015) require the determination of the elements of the eddy diffusivity tensor from model integrations with multiple passive tracers. Since our focus is on the GM scheme, this approach is beyond the scope of the paper and we therefore do not attempt to diagnose κRedi. For all simulations, we choose a spatially uniform isopycnal diffusivity of κRedi = 15 m s−2 approximately matching the GM diffusivity over the slope. Spatially varying Redi coefficients (e.g., Ferrari and Nikurashin 2010; Wei and Wang 2021) did not yield significant improvements over our choice of constant κRedi (not shown). We proceed by first evaluating the performance of the parameterization using
c. Using the diagnosed GM diffusivity to parameterize shoreward heat fluxes
With the GM/Redi parameterization, isopycnal slopes relax, particularly at the AASW–WDW interface (Figs. 2e,f). The V-shaped isopycnals move upward, lifting the layer of warm and salty WDW by around 200 m. WDW is found further onshore where it can reach the shelf break. This also affects the deep water exported within the gravity current, which becomes slightly warmer with GM/Redi. Over the continental shelf, the flattened isopycnals reduce the accumulation of salt and thus the salinity error locally. Nevertheless, the overturning cell of the bolus streamfunction is shallower than the eddy overturning in the high-resolution run (Figs. 4b and 6b). Consequently, the exchange with the fresh surface water does not extend to the bottom of the shelf so that the gravity current is slightly too salty.
(a) Onshore heat fluxes decomposed into the contributions of the GM scheme, the Redi scheme, and the resolved eddies for a horizontal resolution of 10 km using GM/Redi with the diagnosed vertically averaged
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
In total, the domain-integrated RMSD computed between the coarse resolution and the coarse-grained high-resolution fields reduces by 58.7% for temperature and 44.6% for salinity with the GM/Redi scheme. We conclude that the eddy parameterization generally improves the hydrographic structure in this application although some differences persist. In particular, the gravity current on the continental slope remains too broad, whereas it is strongly confined to the slope at high resolution. This is a well-known phenomenon in z-coordinate ocean models where the downslope transport of dense water is subject to excessive entrainment unless Δx < Δz/α (Winton et al. 1998). Considering a vertical grid spacing of Δz = 75 m at the center of the slope and a topographic slope of stopo = 0.02, the “slope-resolving” horizontal resolution Δz/stopo = 3.75 km is only reached in the high-resolution reference simulation. Therefore, we cannot expect the eddy parameterization to resolve this issue.
In the simulation with the GM/Redi parameterization, the shoreward heat flux, consisting of contributions from the eddy parameterization and from resolved eddies, is considerably larger compared to the case without GM/Redi over most of the domain (Fig. 6a). Mainly, the GM scheme produces a strong heat flux over the central continental slope, which is very similar to the high-resolution simulation. This is consistent with the bolus streamfunction ψbolus, which generally compares favorably to the computed eddy streamfunction ψeddy (Fig. 6b). Here, the positive vertical gradient of ψbolus corresponds to a shoreward bolus velocity in the WDW layer according to Eq. (2). Approaching the shelf break, the vertical gradient of the bolus streamfunction becomes small, and consequently, the advection of heat produced by the GM scheme is weak. The heat flux from the GM scheme thus becomes smaller than in the eddy-resolving simulation. We note that even at coarse resolution, there is a contribution of resolved eddies at the lower continental slope where the GM coefficient is low. In this part of the domain, the Rossby radius of deformation is about 30 km so that some eddies are resolved. In the open ocean section, the GM coefficient is high, resulting in the damping of resolved eddies and the associated heat flux. The damping of resolved eddies could have been avoided by choosing an even coarser resolution, which would however have resulted in fewer grid points over the slope leading to an even less realistic representation of the gravity current. Since we also expect the sponge layer to influence the open ocean side, we focus our discussion on the continental slope. Furthermore, we discuss the implications of the interaction of GM and resolved eddies in section 4.
On the upper slope, the Redi scheme takes over and captures some of the shoreward heat flux across the shelf break. Moving onshore along an isopycnal, the potential temperature decreases by more than 1°C within a short distance over the shelf break. The strong isopycnal temperature gradient then drives an onshore heat flux in the Redi scheme [cf. Eq. (4)], whereas the bolus streamfunction no longer sustains a shoreward heat flux in the GM scheme. Still, the heat flux produced by the Redi scheme is about 50% smaller than in the high-resolution reference. Some improvements to the heat fluxes over the shelf may be achieved by locally setting a higher κRedi. A detailed investigation of how to set κRedi is an important task for future work, especially for the modeling of ocean–ice shelf interactions, which requires the correct amount of heat to be transported onto the shelf.
d. Parameterized slope-aware GM coefficients
With a properly designed diffusivity, an idealized model of the Weddell Sea continental slope with the GM scheme shows improved cross-slope heat fluxes and hydrographic mean state. An appropriate diffusivity informed by a high-resolution reference simulation, however, is usually not available beforehand. Instead, a modeler usually chooses a constant value for the GM diffusivity or employs a flow-dependent scheme (e.g., Visbeck et al. 1997). Neither solution takes into account the suppressive effect of the continental slope as shown in Fig. 5.
We now contrast the results obtained with and without the slope-aware versions of the GM scheme [Eqs. (24) and (26)]. With a high diffusivity appropriate for shelf or open ocean, the onshore heat fluxes are strongly overestimated (Fig. 7, dash–dotted lines). Here, WDW can directly access the continental shelf and erode the V-shaped isopycnal structure of the ASF, once the suppressive influence of the topographic slope is neglected (Fig. 8). Choosing a diffusivity appropriate only for the continental slope instead, the onshore heat flux is underestimated at the transition from the slope to the open ocean (Fig. 7, dotted lines). Moreover, the isopycnal slopes over the continental shelf become too steep, which again leads to the accumulation of salt similar to the coarse-resolution simulation without the GM/Redi parameterization (not shown). Also, the low diffusivity choice is less realistic since a diffusivity suitable for the open ocean would most likely be given preference in a larger model domain.
Along-slope and time-averaged onshore eddy heat flux at 1-km resolution and onshore heat flux at 10-km resolution using the GM/Redi scheme with (a)
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
Along-slope and time-averaged potential temperature at horizontal resolution (a) 1 km and at 10 km using GM/Redi with (b)
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
The slope-aware version of the GM scheme yields both reasonable heat fluxes across the continental slope and some improvements to the isopycnal slopes on the shelf. Furthermore, the heat fluxes do not depend very much on the choice of the slope parameter ϵc or ϵVb97 (Fig. 7, gray envelope). The slope-aware modification to the GM scheme thus seems to perform fairly robustly in the given application.
At the open ocean side of the domain, the slope-aware modification to the constant diffusivity yields a better heat flux estimate compared to the slope-aware Visbeck scheme. We note again that some eddies are resolved here and that the boundary restoring influences this part of the domain. We therefore refrain from further interpreting these differences.
In summary, the GM/Redi scheme improves the coarse-resolution simulation in every aspect that we have investigated (Fig. 9). In particular, the largest improvements are observed for the mean hydrographic fields and cross-slope heat fluxes where the RMSD to the high-resolution reference simulation reduce by half compared to the simulation without GM/Redi. While the effect on the total velocity of the ASC is small, the baroclinic component also improves considerably as the isopycnal slopes are relaxed by the parameterization. The mean fields over the continental shelf also improve even though the differences to the high-resolution reference simulation are larger than the ones computed over the whole domain (Fig. 10). Potentially, further improvements can be achieved by choosing κRedi more carefully or by introducing along-slope topographic variations that allow a topographically steered onshore flow of WDW at the top of the continental slope (St-Laurent et al. 2013). Making the GM coefficient depend on the topographic slope reduces the differences to the high-resolution reference simulation as much as using a diagnosed GM diffusivity. Over the shelf break, the slope-aware GM version produces higher coefficients and moves the WDW further onshore compared to the run with the diagnosed diffusivity (Figs. 5 and 8). This further reduces the difference to the high-resolution fields in an integral measure (Figs. 9a,b) but introduces a warm and salty bias over the shelf break. We conclude that a carefully chosen, small GM diffusivity over the continental slope is essential to simulating correct cross-slope heat fluxes. Employing a high diffusivity value that is derived from an open ocean simulation everywhere in the domain will lead to excessively large cross-slope heat fluxes. Only the slope-aware versions of the GM scheme produce a diffusivity that matches the diagnosed diffusivity over the continental slope, shelf, and open ocean parts of the model domain and allows more realistic simulations.
Volume weighted RMSD of (a) potential temperature, (b) salinity, and (c) barotropic, baroclinic, and total along-slope velocities ubt, ubc, and utot between the coarse-grained high-resolution simulation and the coarse-resolution simulation with different κGM. The bars show the relative RMSD compared to the simulation without the GM/Redi scheme. (d) Integrated cross-slope heat fluxes (sum of contributions from GM/Redi scheme and resolved eddies) relative to the integrated cross-slope eddy heat flux of the high-resolution simulation. All integrals are computed for the complete model domain excluding the sponge layer.
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
As in Fig. 9, but computed over the continental shelf only (0–125 km).
Citation: Journal of Physical Oceanography 54, 8; 10.1175/JPO-D-23-0215.1
4. Summary and discussion
In this work, we assess the effect of the GM/Redi parameterization for mesoscale eddies in an idealized model of the Weddell Sea continental shelf and slope. We find that with the GM/Redi scheme, WDW is generally moved toward the continental shelf, and the heat flux is better simulated compared to the case with no GM/Redi. Here, the GM scheme transfers WDW across the central continental slope, whereas the Redi scheme generates a diffusive heat flux across the continental shelf break. As the main result, a successful simulation with the GM/Redi parameterization crucially depends on a choice of the GM diffusivity that reflects the suppressive effect of the continental slope, where in this application, the diffusivity is reduced by an order of magnitude. Schemes designed for the open ocean that diagnose κGM only from the resolved flow—represented here by the Visbeck et al. (1997) scheme—cannot capture this behavior and instead yield a fairly constant thickness diffusivity. Neglecting the attenuation of the eddy diffusivity over the continental slope here results in a strong overestimation of onshore WDW transport or in a misrepresentation of shelf and open ocean hydrographic mean states.
Our experiments clearly illustrate the advantage of slope-aware eddy parameterizations, in which the GM diffusivity depends on the topographic slope, for simulating exchanges across the ASF. In idealized simulations, the diagnostic scaling of cross-slope eddy buoyancy fluxes improves when a dependency on the continental slope is introduced directly through the slope parameter δ for retrograde fronts (Wang and Stewart 2020) or through the slope Burger number for prograde fronts (Wei et al. 2022). Sterl et al. (2024) find an analytical expression for the suppression of eddy mixing across sloping topography in a barotropic framework, providing physical context for the proposed empirical functions. The empirical and analytical suppression functions remain to be tested in detail in numerical ocean models, where isopycnals slope both in the same and in the opposite direction compared to the topography. First implementations of slope-aware GM schemes in idealized channel models and a global ocean model show an improved performance compared to the traditional GM scheme (Wei et al. 2024; Nummelin and Isachsen 2024). In this work, we tested a topographic scaling in coarse-resolution simulations in a case with particularly complex hydrographic conditions where δ switches sign in the vertical. We chose to modify the GM scheme based on the slope Burger number producing similar damping as previously reported (Brink 2012; Brink and Cherian 2013; Brink 2016; Wei et al. 2022). In a vertical average, empirical scalings constructed from the slope parameter δ (e.g., Stewart and Thompson 2013; Wang and Stewart 2020) yielded similar results.
So far, we have exclusively focused on vertically averaged GM and Redi diffusivities. It is established, however, that eddy mixing is not vertically uniform in the ocean and that vertical variations should eventually be considered in eddy parameterizations (e.g., Ferreira et al. 2005; Abernathey et al. 2011; Griesel et al. 2014; Bachman et al. 2015; Poulsen et al. 2019). Although the vertical structure of eddy mixing over continental slopes has been studied (Isachsen 2011; Wang and Stewart 2020), no scaling from which vertically varying GM and Redi coefficients can be constructed has been proposed yet.
Overall, our findings motivate a larger-scale testing of slope-aware parameterizations including other sections of the ASF. A good starting point could be to modify diagnostic schemes that already include the aspects of the dynamic flow, where the computation of κGM can be easily adjusted. Testing the scheme by Visbeck et al. (1997) revealed some weaknesses unrelated to topographic suppression effects, motivating a detailed evaluation of the performance of the scheme over continental slopes. More complicated schemes that integrate a prognostic subgrid eddy energy equation (Eden and Greatbatch 2008; Marshall et al. 2012; Mak et al. 2018) potentially produce more accurate GM coefficients but may require more substantial modifications when integrating the effect of sloping topography.
As computing power increases, global ocean models will (at least partially) resolve mesoscale eddies in the open ocean while smaller eddies on the slope remain unresolved. Various techniques have been proposed to limit the damping effect of GM onto the resolved eddies, including scaling κGM by the first baroclinic deformation radius and the horizontal grid spacing (Hallberg 2013) or a splitting approach where GM only acts on the large-scale field (Mak et al. 2023).
In our configuration, the Redi scheme produces an onshore diffusive heat flux. The choice of κRedi, however, is the result of tuning and not backed by dynamical considerations. A κRedi that is a function of the topographic slope may enhance the performance of the Redi scheme over continental slopes (Wei and Wang 2021). We conclude that the behavior of the Redi scheme and its interaction with the GM scheme in the context of the ASF raises questions to be answered in future work.
The idealized model setup carries some limitations. First of all, we do not consider topographic variations in the along-slope direction that can influence both the intensity and distribution of cross-slope buoyancy fluxes. Around the Antarctic continental margin, dense water export and associated eddy-driven shoreward heat fluxes concentrate in bathymetric depressions (e.g., Orsi and Wiederwohl 2009; Williams et al. 2010; Stewart et al. 2018; Morrison et al. 2020; Stewart 2021). Additionally, along-slope topographic features act as drivers of buoyancy transfers across continental slopes through the generation of standing eddies (e.g., Abernathey and Cessi 2014; St-Laurent et al. 2013; Bai et al. 2021; Si et al. 2022). Even when along-slope topographic variations are present, we may still expect the presented topographic scaling to lead to improvements since transient eddy fluxes have been shown to dominate over standing eddy fluxes across slope currents such as the ASC (Wei et al. 2022; Si et al. 2022). So far, slope-aware diagnostic scalings of eddy buoyancy fluxes across idealized slope fronts have been tuned over smooth topography. Still, they outperform schemes without a slope dependency when applied to cases in which topography varies along the slope (Wang and Stewart 2020; Wei et al. 2022). Furthermore, the idealized model neglects the variability in the wind forcing and associated impacts on the outflow of dense water from the ice shelf cavities in the Weddell Sea (Wang et al. 2012; Daae et al. 2018), the local modification of sea ice growth in polynyas (Wang et al. 2021), and the inflow of warm water into the cavities through the modification of coastal currents (Hellmer et al. 2012; Darelius et al. 2016). Moreover, we do not account for the effect of tides, which contribute to setting up the structure of the ASF through tidal rectification (Flexas et al. 2015), shape heat fluxes across the ASF (Stewart et al. 2018; Stewart 2021; Si et al. 2022, 2023), and drive an onshore residual flow of CDW (Wang et al. 2013). While considering the thermodynamic effects of sea ice, we also do not account for the influence of sea ice dynamics on the transfer of momentum between the atmosphere and ocean (Si et al. 2022).
The central role of the Weddell Sea in producing bottom water and thereby shaping the global ocean circulation requires an accurate estimation of heat transports across the Weddell Sea continental slope. Our application and improvement of existing parameterizations represent a promising step toward improving the simulation of these heat transports at non-eddy-resolving and eddy-permitting resolutions. In future work, it should be extended to other Antarctic marginal seas, where eddies mediate exchanges across the ASF (Stewart et al. 2018; Si et al. 2023). Furthermore, reducing the degree of idealization by adding an ice shelf cavity would allow tracking the influence of the improved parameterization on the melting of ice shelves and the sources of dense water. Finally, our results encourage the integration of slope-aware eddy parameterization into regional and global ocean models.
Acknowledgments.
This paper is a contribution to the project T3 (Energy Transfers in Gravity Currents) of the Collaborative Research Centre TRR 181 “Energy Transfers in Atmosphere and Ocean” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Project 274762653). We thank Andrew Stewart for providing configuration files for the MITgcm reference simulation. We also thank Sergey Danilov for helpful discussions and two anonymous reviewers for their constructive comments.
Data availability statement.
The MITgcm code can be accessed at https://github.com/MITgcm, and documentation is provided at https://mitgcm.readthedocs.io/en/latest. Modifications to the model code required to reproduce the simulations are available at https://github.com/nicolasdettling/weddell_gm.git. Input files and namelists to rerun all experiments are stored at https://doi.org/10.5281/zenodo.10033249.
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