Plain Language Summary
Deep ocean is a significant source of heat and salt for the Antarctic coasts. Thus, its behavior is relevant for a wide range of climate sciences, such as Antarctic glacial melt, sea ice variability, and global ocean overturning. The warm deep water is presumably transported to the Antarctic coasts by ocean eddies, but observational evidence has not been provided to date. By synthesizing ocean measurements and satellite altimetry, we estimate the eddy-driven onshore transport of the deep water over the Antarctic margin. It is shown that shoreward heat transport by eddies is generally balanced with heat gain by solar heating and heat loss expected from surface freezing and glacial melt. This result indicates that eddy transport plays a fundamental role in the coastal heat supply. The thickness of the deep water primarily controls the ability to mix along density surfaces that bridge the open ocean to the continental shelves; more homogeneous thickness allows for easier mixing. Our results facilitate the possibility of predicting the diffusion rate of eddies using the thickness of the deep water. This idea is helpful in simulating deep-water transport in global climate models, where sub-grid and unresolved effects of eddies need to be prescribed.
1 Introduction
Throughout the Antarctic Circumpolar Current (ACC), mesoscale eddies transport water masses across time-mean streamlines, comprising the adiabatic pathway of the global meridional overturning circulation (Marshall and Radko, 2003; Cessi, 2019). Isopycnal eddy diffusion is fundamental for the poleward heat flux across the ACC because bottom-enhanced diapycnal mixing (Kunze et al., 2006) and surface water transformation (Abernathey et al., 2016) are unlikely to penetrate the intermediate and deep layers. Recent observations have indicated that mesoscale eddies play a key role in bridging the Antarctic meridional overturning from deep ocean basins to continental shelves (e.g., Mckee et al., 2019; Yamazaki et al., 2021), but its conditioning from the ACC to the Antarctic Slope Current (ASC) largely remains unknown from an observational standpoint (Thompson et al., 2018).
Circumpolar Deep Water (CDW), the primary source of heat and salt for the Antarctic coasts, is transported across the ASC predominantly by mesoscale eddies in the absence of large-scale zonal pressure gradients (Stewart and Thompson, 2013). In reality, the pressure gradient associated with topographic features generates standing eddies and meanders, facilitating meridional water exchange (e.g., Hogg and Blundell, 2006). Topography-controlled geostrophic flows can transport CDW poleward in the continental margin (Morrison et al., 2020; Hirano et al., 2021). However, the steep barotropic potential vorticity (PV) gradient in the upper continental slope (shallower than ~3,000 m isobaths) is unlikely to allow the presence of cross-slope mean flow, and thus eddy diffusion and tidal mixing might be essential to determine the onshore CDW flux near the shelf break (Stewart et al., 2018; Yamazaki et al., 2020). To the south of the ACC, the spatial variability of eddy diffusion has yet to be reported except for Foppert et al. (2019; FRE19), and its controlling factor remains unclear. This study aims to: (1) provide an observational estimate of isopycnal diffusion in the Antarctic margin, which is portrayed as a poleward extension of the estimation by Naveira Garabato et al. (2011; GFP11), and (2) quantify the isopycnal CDW flux by eddy diffusion towards the continental shelves.
The horizontal circulation in the Antarctic margin is shaped by the subpolar gyres between the ACC and ASC (e.g., Park and Gamberoni, 1995). The Weddell and Ross gyres are wide enough to isolate cold shelf water from warm CDW; in contrast, the proximity of the ACC to the continent creates an eastward slope current in the eastern Pacific sector (Spence et al., 2017; Thompson et al., 2020) and standing eddies in the Indian sector (Mizobata et al., 2020; Yamazaki et al., 2020), resulting in a relatively warmer coastal condition than in the other sectors (e.g., Stokes et al., 2022). The present study targets the Indian sector corresponding to the East Antarctic margin (30–160°E; Fig. 1), where a sufficient amount of in-situ data exists for the estimation. In this region, the eddy conditions have recently been explored (FRE19; Stewart et al., 2018), wherein lateral tidal mixing is known to be weaker than that in the rest of the Antarctic margin (Padman et al., 2018), and the frontal structure of the ASC is relatively prominent (Pauthenet et al., 2021). The onshore CDW flux towards East Antarctica is collecting attention as a cause of glacial melting (e.g., van Wijk et al., 2022).
The remainder of this paper is set out as follows: Section 2 reviews the theoretical background for observation-based eddy diffusivity calculations and introduces the concept of the mixing length framework. Section 3 describes data and methods used for the diffusivity calculations, and Section 4 presents the quantification of the mixing length, eddy diffusivity, and eddy fluxes together with their spatial variations. Lastly, Section 5 assesses the validity of the presented results and discusses controlling factors for eddy diffusion. Section 6 is the conclusion.
2 Theoretical background
The mixing length framework serves as the basis for the analysis. Conventional ways to estimate oceanic eddy diffusion are hydrographic variability (Armi and Stommel, 1983; GFP11) and altimetric eddy scaling (Klocker and Abernathey, 2014; Bates et al., 2014). In both methods, the diffusivity k is obtained via the mixing length formulation (Taylor, 1921):
\begin{equation} k\ =\ \Gamma U_{\text{eddy}}L_{\text{mix}}\ \ \ \ \ \ \ \ \ \ (1),\nonumber \\ \end{equation}
where \(\Gamma\) is mixing efficiency, \(U_{\text{eddy}}\) is characteristic eddy velocity measured by the standard deviation of downgradient velocity \(\sigma(v)\), and \(L_{\text{mix}}\) is mixing length scale. This formulation relies on two major assumptions: (i) tracer fluctuations are generated by the local stirring of the large-scale tracer gradient, with weak advection of tracer variance from upstream regions; and (ii) the tracer gradient varies slowly over the distance \(L_{\text{mix}}\). The validity of these assumptions is confirmed a posteriori .
The mixing length framework has been widely applied in the closure of geostrophic turbulence because it can link the eddy tracer transport to the downgradient flux in Eulerian form. Diffusivity k of generalized tracer \(\varphi\) (which approximately follows PV contours) due to isopycnal stirring is parameterized as
\begin{equation} \overset{\overline{}}{v^{\prime}\varphi^{\prime}}\ =\ -k\frac{\partial\varphi}{\partial y}\text{\ \ \ \ \ \ \ \ \ \ }\left(2\right),\nonumber \\ \end{equation}
where the overbars indicate the temporal average in the isopycnal layer, the primes indicate deviations from the averages, and the tracer gradient is assumed to be meridional. \(v\) is meridional velocity, so the tracer flux is the covariance between tracer anomaly and cross-frontal velocity. Here, it is also assumed that the tracer\(\varphi\) mixes purely along the isopycnal. This assumption is aligned with the conditions where the mixing process is statistically stable, adiabatic, and solely caused by linear waves (e.g., Vallis, 2017). We may choose any tracer for \(\varphi\) if the tracer concentration represents the PV field to the extent of interest, where its diffusion satisfies a requirement for the GM flux mimicking baroclinic instability (Gent and McWilliams, 1990). Then, the scalar coefficient of downgradient flux can express the skew component of the diffusivity tensor. In this framework, GFP11 and FRE19 derived the eddy diffusivity using hydrographic variability of the isopycnal temperature and spiciness (as \(\varphi\)) and inferred the volume transport as a result of the thickness diffusion.
One possible choice for \(L_{\text{mix}}\) is the characteristic eddy scale determined by an altimetric velocity field (Klocker and Abernathey, 2014; Bates et al., 2014); this approach requires the eddy’s deformation radius to be resolved by altimetry. Another choice is an empirical method using hydrographic data. Emulating the arguments of Armi and Stommel (1983), GFP11 estimated \(L_{\text{mix}}\) in the Southern Ocean from hydrographic variability as follows:
\begin{equation} \ L_{\text{mix}}=\frac{\sigma(\varphi)}{|\nabla\varphi|}\text{\ \ \ \ \ \ \ \ \ \ }\left(3\right)\nonumber \\ \end{equation}
Although GFP11 used potential temperature for the isopycnal tracer\(\varphi\), other tracer variables (e.g., isopycnal spiciness and layer thickness) are possible. From equations (1)–(3), the mixing efficiency is calculated as:
\begin{equation} \Gamma=\frac{\overset{\overline{}}{v^{\prime}\varphi^{\prime}}}{\sigma\left(v\right)\ \sigma(\varphi)}\text{\ \ \ \ \ \ \ \ \ \ }\left(4\right)\nonumber \\ \end{equation}
This expression is identical to the correlation coefficient between\(v\) and \(\varphi,\) but a wide range of estimates exists for\(\Gamma\) (ranging 0.01–0.4; Holloway and Kristmannsson, 1984; Visbeck et al., 1997; Karsten and Marshall, 2002) possibly depending on the variety of definitions. The only observational estimate is\(\Gamma=0.16\) by Wunsch (1999), which GFP11 suggested could be used to illustrate absolute diffusivity.
The hydrographic estimate of eddy diffusivity by GFP11 is generally consistent with independent estimations via Lagrangian tracer dispersion numerically advected with altimetric velocity (Marshall, 2006; Sallée et al., 2011), falling between 500 and 3,000 m2s˗1 within the ACC core and 2,000–3,500 m2 s˗1 in the jet’s equatorward flank. The resulting map of diffusivity can be explained by the suppression theory deduced from weakly nonlinear wave­–mean flow interaction (Ferrari and Nikurashin, 2010), interpreted as that jet-induced advection reduces the duration of isopycnal mixing for the same water mass, leading to suppression of mixing length. The suppression of eddy stirring ceases in “leaky jets” associated with non-parallel shear flows and meanderings steered by the topography (GFP11; Sallée et al., 2011; Tamsitt et al., 2018). Klocker and Abernathey (2014) conducted numerical simulations to test the quantitativeness of the mixing length framework. They remarked that diffusivity could equivalently be estimated in a hypothetical unsuppressed mixing regime by either the eddy scale/tracer-based mixing length formulations if choosing 𝛤=0.15 for the tracer-based mixing length, supporting the estimate by Wunsch (1999). These studies rationalize using the hydrographic variability method to quantify eddy diffusion.
3. Data and methods
3.1 Satellite altimetry for \(\mathbf{U}_{\mathbf{\text{eddy}}}\)
An observational estimate of characteristic eddy velocity\(U_{\text{eddy}}\) is obtained by altimetric velocity. The advent of synthetic aperture interferometric radar altimeter enabled to measure sea ice freeboard remotely, and its application to dynamic ocean topography has recently been developed (Armitage et al., 2018; Dotto et al., 2018; Mizobata et al., 2020). The present study adopts the monthly reconstructed 0.2° grid dynamic ocean topography from 2011 to 2020 by Mizobata et al. (2020) to derive the climatological geostrophic velocities (Fig. 2). This dataset has an advantage over the product by Armitage et al. (2018) as its empirical orthogonal function filtering can remove spurious stripe patterns.
\(U_{\text{eddy}}\) is calculated as the standard deviation of the altimetric flow speed (Fig. 2). Its reliability is underpinned by the mooring measurements at 113°E (Peña-Molino et al., 2016), which mark standard deviations of 0.04–0.06 m s­-1 in zonal and meridional directions at the CDW layer (~500 dbar) over the continental slope of 500–4,000 m. The typical value of\(U_{\text{eddy}}\) is somewhat larger than the choice of FRE19 (0.017 m s­-1), as they adopted the temporal mean velocity from the same mooring data. In principle, \(U_{\text{eddy}}\) is the standard deviation of the cross-frontal velocity. However, in contrast to the ACC’s mainstream, the flow field in the Antarctic margin is quiescent, and the mean flow directions are unclear (upper panel of Fig. 2). To bypass this problem, we simply define \(U_{\text{eddy}}\) as the root-mean-squared velocity, accounting for its good agreement with the direct flow measurement (Peña-Molino et al., 2016). The vertical variations of the eddy velocity are not considered since the dynamic topography does not monotonically descend poleward in the subpolar zone (e.g., Yamazaki et al., 2020), and thus the gravest empirical mode technique performed by GFP11 is not applicable. Nevertheless,\(U_{\text{eddy}}\) adopted for CDW is deemed acceptable because the vertical attenuation caused by the geostrophic shear is considerably small by the quasi-barotropic flow structure (Meijers et al., 2010; Peña-Molino et al., 2016; Mizobata et al., 2020) and the CDW isopycnals shoaling to 200–400 dbar (Yamazaki et al., 2020).
3.2 CTD profiles for \(\mathbf{L}_{\mathbf{\text{mix}}}\)
The mixing length \(L_{\text{mix}}\) is calculated from the hydrographic variability using equation (3). Historical CTD profiles from the World Ocean Database (https://www.ncei.noaa.gov/), Argo Global Data Assembly Center (Argo, 2000), and Marine Mammals Exploring the Oceans Pole to Pole archive (https://www.meop.net/; Treasure et al., 2018) are compiled. Data for December–March and 1990 onwards are extracted and filtered to remove poorly flagged data and fragmented profiles. Thereafter, 1-dbar Akima interpolation is performed for the CTD profiles. Surface data averaged within the neutral densities (Jacket and McDougall, 1997) are then constructed (Fig. 3), corresponding to CDW (defined as 28.0–28.1 kg m-3) and Antarctic Surface Water (ASW; defined as 27.9–28.0 kg m-3). The bounding densities are selected based on the overturning streamfunction in the Indo-Pacific sector, whereas these definitions can change in the Atlantic sector (Lumpkin and Speer, 2007). Fig. 3 indicates that, in contrast to the meridional spiciness gradient of ASW being stronger than CDW, the thickness gradient of CDW is generally stronger than ASW. We compare results from the CDW and ASW layers to check the layer dependency of mixing length and the quantitativeness of estimation.
Previous studies have adopted potential temperature and spiciness as the isopycnal tracer \(\varphi\) (GFP11; FRE19). However, the diffusion of these tracers does not necessarily reproduce the PV diffusion in the real ocean, and it is unclear whether these tracers yield diffusivity\(k\) that is conformant to the eddy volume flux and the downgradient (i.e., Fickian) diffusion. Since the isopycnal thickness is a possible candidate for the PV-conservative tracer (e.g., Vallis, 2017), the present study compares the thickness-based \(L_{\text{mix}}\) estimates with the spiciness-based estimates. This comparison can demonstrate how estimated \(L_{\text{mix}}\) changes depending on the tracer variables and the validity of the previous estimates (GFP11; FRE19). Conservative Temperature, Absolute Salinity, and spiciness (at 0 dbar) are calculated using the Gibbs Sea Water Oceanographic Toolbox (http://www.teos-10.org/), and the layer thickness is derived from the pressure difference between the upper and lower isopycnal surfaces of each water mass (e.g., 28.0 and 28.1 kg m-3 for CDW).
Mapping isopycnal climatology onto 0.2° grids is performed with the radius basis function interpolation (Yamazaki et al., 2020), which can reproduce the best representative surface of noisy data nonparametrically in the least-squares sense. Grid data with no less than 10 points within a 75 km data radius are adopted, selecting grids with data from multiple years. Although the data coverage is partially reduced at 30–60°E, a sufficient amount of data is found within the region of interest, particularly over the continental slope of 1,000–3,000 m. The correspondence observed among the 3,000 m isobath, CDW spiciness of 0.15 kg m-3, and CDW thickness of 300 m (Fig. 3) guarantees the reliability of the interpolated field. After calculating deviations of surface data from the interpolated climatological field, root-mean-squared tracer variations\(\sigma(\varphi)\) are derived in each grid from the deviation data within the 75 km radius. This procedure minimalizes artifacts in\(\sigma(\varphi)\) due to the spatial variation of the tracer field within the data radius. Although the choice of the radius size is a trade-off between the data amount and the resolution, the data criterion (10 points within the 75 km radius) is determined by the characteristic scale of the continental slope topography and the degree to which the mixing length is not dependent on data amount. Our choice is comparable to the discussion by GFP11 that “about 5–10 stations per 100 km” is a reasonable baseline required for the \(L_{\text{mix}}\) calculation to capture the basic distribution patterns.
3.3 Validation of mixing efficiency \(\mathbf{\Gamma}\)
One of the largest uncertainties of diffusivity \(k\) lies within the mixing efficiency \(\Gamma\). Based on equations (1) and (3), FRE19 indicated the along-slope variability of eddy condition in the East Antarctic margin via mapping standard deviation of isopycnal spiciness, while their formulation does not include \(\Gamma\) and spatially variable \(U_{\text{eddy}}\), leaving some ambiguities for the absolute value of \(k\). For a trial, we directly calculate \(\Gamma\) from the correlation coefficient between \(v\) and \(\varphi\), using a 17-month mooring record (for 2010–2011) across the ASC in 113°E over the slope of 500–4,000 m (reported by Peña-Molino et al., 2016). Vertical/meridional linear gridding (by 50 dbar for 200–1,500 dbar and by 0.1° for 65.5–61.5°S) is performed for hourly meridional velocity and temperature profiles. Their correlation coefficient for 12 months (8,761 steps) is calculated for each grid, assuming that the temperature variation is aligned with the PV change and the temperature gradient is directed northward on average. Although the temperature is used for\(\varphi\), the temperature variation is almost proportional to the spiciness variation within the layers of interest.
From the histogram of \(\Gamma\), the estimated mean value is 0.12 for down-gradient cases and 0.10 for all cases (Fig. 4). If the mean eddy velocity steadily directs the downgradient of the mean temperature, the upgradient cases may be irrelevant to the climatological eddy condition. Wunsch (1999) derived \(\Gamma=0.16\) from a global inventory of mooring records, broadly consistent with our estimates but larger by 30–40%. We must admit that 12 months is too short to determine eddy statistics with certainty (additional low-pass filtering may effectively cut off uninterested short-term variations at the cost of underestimation), and the cross-slope section cannot represent the diverse flow regimes of the Southern Ocean. Nevertheless, we adopt the mixing efficiency \(\Gamma=0.16\) by Wunsch (1999) based on the general agreement with the local estimate. The validity of choice is further discussed in Section 5.1.
4. Result
4.1 Mixing length
The standard deviation and normed gradient of the isopycnal tracer\(\varphi\) of each water mass are shown in Figs. 5 and 6 for the spiciness and layer thickness, respectively. The large gradient of spiciness is found near the ACC’s southern boundary (SB; defined as the southernmost extent of 1.5 °C isotherms) in ASW (27.9–28.0 kg m-3), while in CDW (28.0–28.1 kg m-3), the gradient is largest over the upper continental slope to the south of SB (Fig. 5; middle panels). The large standard deviation (top panels) broadly corresponds to its steep gradient. Relative to the spiciness, the thickness gradient is somewhat homogeneous, and the coherence between the standard deviation and gradient is less noticeable (Fig. 6). As in the spiciness, the steep thickness gradient of CDW is found in the proximity of the SB, indicating a poleward volume flux facilitated by the thickness diffusion.
The bottom panels in Figs. 5 and 6 are the mixing length\(L_{\text{mix}}\) derived from equation (3) for each tracer. The spatial distributions of the spiciness/thickness-based\(L_{\text{mix}}\) are analogous regarding their small values near the SB. These estimates are quantitatively consistent with the previous estimate by GFP11, where \(L_{\text{mix}}\) can exceed 150 km in the unsuppressed part of the ACC. Although the spiciness/thickness-based diagnostics are highly dependent on the choice of isopycnal layer, the two \(L_{\text{mix}}\) estimates for CDW and ASW exhibit the highest value of ~150 km in the ACC domain and its suppression towards the continental slope. These results suggest the robustness of the estimated \(L_{\text{mix}}\). The spatial variation of\(L_{\text{mix}}\) is generally consistent with the jet-induced suppression theory (Ferrari and Nikurashin, 2010) as discussed in the following, while near-boundary turbulent suppression or “law of the wall” likely becomes more influential over the Antarctic margin than in the ACC domain, analogously to what GFP11 speculated.
The dependence of \(L_{\text{mix}}\) on the flow regime is shown in Fig. 7. The estimates of \(L_{\text{mix}}\) are averaged in bins of the mean flow speed and are individually shown for the ACC frontal zones following categorization by Orsi et al. (1995) (Figs. 1 and 2). Here, the frontal zones refer to the dynamic topography data of Mizobata et al. (2020); the subpolar zone (south of SACCF-S to the ASF): < –1.85 m, the southern zone (from the SACCF-S to SACCF-N): –1.85 ~ –1.6 m, and the antarctic zone (from SACCF-N to PF): –1.6 ~ –1.0 m. Readers are advised to compare Fig. 7 with the result by GFP11 (their Fig. 10), which puts emphasis on the more energetic part of ACC to the north. In the Antarctic and southern zones, \(L_{\text{mix}}\) tends to decrease from 70–90 to 30–60 km as the flow speed increases to 0.5 m s-1, indicating suppressed mixing due to wave–mean flow interaction. In the antarctic zone, \(L_{\text{mix}}\) partly increases with the mean flow exceeding 0.5 m s-1, corresponding to leaky jets in the lee of topographic features (GFP11; Sallée et al., 2011; Tamsitt et al., 2018) such as the Kerguelen Plateau (~80°E) and the Southeast Indian Ridge (~150°E; see Fig. 2). On the other hand,\(L_{\text{mix}}\) is not so dependent on flow speed in the subpolar zone as in the ACC, ranging from 20 to 60 km. These results suggest that the jet-induced mixing suppression previously documented in the northern part of the ACC is less effective poleward. We posit that the mixing suppression in the subpolar zone is likely associated with the near-boundary turbulent suppression by the continental slope topography and the flow regime more quiescent than the ACC domain. In Fig. 7, differences between CDW and ASW are unclear, accounting for the different data coverages of ASW and CDW (Fig. 3). Meanwhile, the thickness-based \(L_{\text{mix}}\) for ASW in the subpolar zone is exceptionally large for strong flows with relatively large standard errors, possibly due to the less distinct thickness gradient than spiciness in ASW (Figs. 5 and 6). Notably, while the choice of the isopycnal layer and tracer \(\varphi\) occasionally affects the outcome, the hydrographic variability method yields\(L_{\text{mix}}\) consistent with previous estimates (GFP11; FRE19).
To examine whether the tracer variance or inverse tracer gradient is more important in setting the \(L_{\text{mix}}\) variation, a histogram of \(L_{\text{mix}}\) is plotted on\(\sigma(\varphi)-1/|\nabla\varphi|\) space (Fig. 8), in which the isolines of \(L_{\text{mix}}=\) 20, 100 km are shown by white contours. The poleward suppression of \(L_{\text{mix}}\) is observed by comparing the positions of the largest population and center of mass among the frontal zones. In all presented layers and methods, modes and averages of \(L_{\text{mix}}\) are aligned with the \(1/|\nabla\varphi|\) axis in the antarctic zone, and they migrate towards the \(\sigma(\varphi)\)axis across the diagonal line as moving poleward. A key conclusion from this plot is that the inversed tracer gradient \(1/|\nabla\varphi|\)becomes more influential poleward to the spatial variation of\(L_{\text{mix}}\) than \(\sigma(\varphi)\) does (i.e., the variation of\(L_{\text{mix}}\) in the cross-isoline direction is hardly explained by\(\sigma(\varphi)\) in the subpolar zone in contrast to the Antarctic and southern zones). This control might be because the poleward PV gradient becomes steeper (equivalently, the width of baroclinic zone becomes narrower) to the south, plausibly due to the continental slope topography. The topographic control of \(L_{\text{mix}}\ \)implicates a possibility of parameterizing the eddy diffusivity using prescribed topographic information in an ocean model, as recently explored by idealized numerical simulations (Stewart and Thompson, 2016). We anticipate that, in the subpolar zone, \(L_{\text{mix}}\) and \(k\) can be predicted by the topographic gradient, and this idea will be assessed in the next section.
4.2 Isopycnal diffusivity
Based on the conformance of the results in the ACC domain with those of previous studies, the diffusive parameters in the Antarctic margin are further investigated. The along-slope variation of the thickness-based mixing length for CDW is represented in Fig. 9 (middle panel), which is generally controlled by the thickness gradient rather than the thickness variability (top panel). Using the mixing length formulation of equation (1), the isopycnal diffusivity \(k\) is calculated by multiplying the mixing efficiency Γ, eddy velocity \(U_{\text{eddy}}\), and mixing length \(L_{\text{mix}}\) (bottom panel). The isopycnal diffusivity for the two tracer variables is mapped in Fig. 10. Here, the climatological flow direction is represented by the overlain mean dynamic topography and contours that are characteristic of the subpolar circulation (˗1.97 and ˗1.85 m) highlighted in blue. Fig. 10 suggests that the isopycnal diffusivity \(k\) typically ranges from 100 to 500 m2 s-1 in the subpolar zone for both tracer variables. Its variation within the subpolar zone is attributable to the spatial variation of \(L_{\text{mix}}\) (Figs. 5 and 6) rather than \(U_{\text{eddy}}\) (Fig. 2) as observed in Fig. 9. To scrutinize the spatial variability, regional maps of the thickness-based diffusivity along with the isopycnal CDW temperature are presented in Fig. 11. Diffusivity is typically higher where CDW intrudes onshore at 70°, 90°, 110°, and 120°E (Yamazaki et al., 2020) as seen in Fig. 9. The CDW intrusions generally corresponds to the smaller thickness gradient, resulting in the larger \(L_{\text{mix}}\) and higher \(k\) at the location (Fig. 11). The enhanced diffusivity is also observed at 140°E (Fig. 10), where the intervals between the ACC and ASC become narrow and clockwise subgyres are meridionally squeezed.
The spatial variation of \(k\) results from those of \(L_{\text{mix}}\)and \(U_{\text{eddy}}\), and its functional dependency (i.e., control of\(k\) by \(L_{\text{mix}}\) and \(U_{\text{eddy}}\)) varies in space. A histogram of \(k\) in\(\ U_{\text{eddy}}\ -\ L_{\text{mix}}\)coordinates is generated for each layer and method (Fig. 12), analogously to Fig. 8. In any of the frontal zones, neither\(L_{\text{mix}}\) nor \(U_{\text{eddy}}\) is a dominant controlling factor since both the population and center of mass are located close to the diagonal line. Nevertheless, we may state that \(k\) is more dependent on \(L_{\text{mix}}\) than \(U_{\text{eddy}}\) in the subpolar zone comparing to the southern or antarctic zones. The result supports the aforementioned idea that the spatial scale of tracer gradient can parametrize eddy diffusivity in the Antarctic margin. This idea is further tested by Fig. 13, in which \(k\), \(L_{\text{mix}}\), and inversed topographic gradient within the subpolar zone are regressed onto the inversed tracer gradient. The results show significant correlations (p-value < 0.01) of \(k\) and \(L_{\text{mix}}\)with \(1/|\nabla\varphi|\) (0.62 and 0.78 for spiciness; 0.65 and 0.82 for thickness, respectively). In contrast, the correlation with the topographic gradient is insignificant for both tracers, implying that additional information is required to determine the climatological tracer gradient. Although the controlling factors for the tracer gradient field remain unknown, the derived correlations are still encouraging as it suggests a possibility that the eddy diffusion can be estimated by only providing the isopycnal tracer gradient.
The correlation of the thickness-based estimation with diffusivity is slightly more significant than the spiciness-based result. The higher correlation of thickness implies that the thickness gradient better represents the PV gradient, whereas the difference in the correlation coefficients can fall within the uncertainty. Provided that the ambient PV field is well approximated by the isopycnal layer thickness within the subpolar zone, where the flow condition is relatively quiescent and the relative vorticity tends to become small, the isopycnal thickness can be regarded as the PV-conservative variable. The mixing length framework principally requires that the diffusion of the tracer is Fickian, and this likely holds in case the tracer is PV-conservative (e.g., Vallis, 2017). In addition, the thickness diffusion is likely more representative of the eddy volume transport (eddy advection) than the spiciness diffusion. Predicated on these conditions, we estimate diffusive fluxes by applying the thickness-based diffusivity.
4.3 Eddy fluxes
Assuming that the isopycnal thickness diffuses downgradient in a GM-flux manner, we can estimate the diffusive volume flux of CDW (Fig. 14). Bolus transport \(\psi\) is calculated as
\begin{equation} \psi\ =\ -k_{H}\nabla\text{H\ \ \ \ \ \ \ \ \ \ }\left(5\right),\nonumber \\ \end{equation}
where \(H\) and \(k_{H}\) are the isopycnal layer thickness and thickness-based diffusivity, respectively. \(\psi\) is equivalent to the layer-integrated bolus velocity (in m2s-1), and its horizontal integration gives a unit of transport. The zonal eddy transport (middle panel) is typically eastward in the lee of topography and westward in the other area, reflecting the directions of the streamline and the thickness gradient steered by the topography. The eddy volume transport generally directs shoreward in the subpolar zone, as shown by the transport vector (top panel) and its meridional component (bottom panel). We can observe the poleward CDW transport continuously extending from the eastern flank of the Kerguelen Plateau, where isopycnal eddies are favorably generated, facilitating the CDW flux towards the Antarctic margin. Along-slope variation of the meridional eddy transport is not so pronounced as \(k\) (Fig. 10), and the most significant poleward CDW transport is obtained around 140°E. This result is related to the fact that the magnitude of eddy transport is \(|\psi|=\ \Gamma U_{\text{eddy}}\sigma(H)\) by equation (3) and is not proportional to the inversed thickness gradient (whether the CDW flux becomes uniquely proportional to \(U_{\text{eddy}}\) is unclear even in zonally-symmetric configuration due to possible variability of mixing efficiency; e.g., Stewart and Thompson, 2016). Partially northward eddy transport along the continental slope (e.g., around 70°E) likely reflects the multiple-cored ASC over the gentle continental slope, which has emerged in previous literature (Meijers et al., 2010; Stewart and Thompson, 2016).
The meridional component of \(\psi\) is zonally integrated to derive the cross-slope fluxes over the 1,000–3,000 m isobaths (Fig. 15). Strictly, this is not the cross-isobath transport, whereas the method does not lose its fidelity regarding the averaging depth range within which the direction of the slope gradient greatly varies. Standard errors associated with the cross-slope variation are shaded. The gross onshore CDW transport is 0.39/0.12 Sv (= m3s-1) in the eastern/western Indian sectors (divided by the Princess Elizabeth Trough ~90°E), respectively. Using the along-slope average of the sensible heat of CDW, these transports are translated to the onshore heat fluxes of 3.6/1.2 TW, where the heat flux change due to the along-slope temperature variation safely falls within the error range. The interbasin contrast in the thermal forcing, wherein a more significant amount of heat is transported onshore in the eastern sector than in the western sector, seems consistent with the coastal regimes, represented by warm Totten Ice Shelf and cold Amery Ice Shelf (Stokes et al., 2022).
Offshore transport of ASW (defined as 27.9–28.0 kg m-3) to the west of 130°E is 0.15 Sv, locally compensating ~40 % of the onshore volume flux by CDW. In contrast, ASW eastward of 130°E is transported to the pole, and its contribution to the onshore heat flux (~0.4 TW) might not be negligible. As discussed in Section 5.3, these estimates are quantitatively consistent with the coastal heat sink due to sea ice formation and glacial melting.
5. Discussion
5.1 Diffusivity estimation
The present study is fundamentally based on the assumption that the mixing length framework is valid to the extent of our interest. One of the necessary conditions for the formulation (see Section 2) is a scale separation between \(L_{\text{mix}}\) and the spatial variation of\(\nabla\varphi\). We estimated the typical value of \(L_{\text{mix}}\)to be 20–60 km in the subpolar zone (Fig. 7). \(\nabla\varphi\) likely varies in the cross-slope direction by a scale comparable to or larger than the slope width (~100 km for the 1,000–3,000 m interval), so it is possible to regard this condition as holding in the Antarctic margin. The other necessary condition for \(L_{\text{mix}}\)estimation is that tracer fluctuations must reflect local eddy stirring rather than tracer anomalies advected from upstream. This condition also holds in the Antarctic margin, given its weaker nonlinearity than the ACC’s mainstream (Fig. 3). Another caveat for the method is that the estimates of tracer variance are obtained from anomalies relative to the climatological mean field spreading within a data radius (75 km in this study), whereas the formulation of the mixing length employs a temporal average over fluctuations at a spatially fixed point. Although this issue is inevitable when deriving diagnostics from in-situ data, an appropriate choice may be obtained by testing the result’s sensitivity to the data radius (Section 3.2).
No significant difference is found between the thickness-based and spiciness-based \(L_{\text{mix}}\) (Figs. 5 and 6). To our knowledge, the present study is the first example to demonstrate that the two choices of tracer yield very similar \(L_{\text{mix}}\) estimates based on hydrographic variability. This result supports the validity of previous estimates in which isopycnal tracers physically independent of PV are used (GFP11; FRE19; Armi and Stommel, 1983). Meanwhile, a small but noticeable difference between the spiciness/thickness-based estimations is obtained; e.g., the large thickness-based (spiciness-based) \(k\) in 70°E (110°E) seems weak by the counterpart method (Fig. 10). The flow speed dependency of \(L_{\text{mix}}\) can also vary by choice of tracer (Fig. 7). These generally pertain to the regional difference in the tracer gradient, as the large diffusivities tend to result from the weak tracer gradient. The quiescent flow regime in the subpolar region indicates that the isopycnal thickness may be deemed more PV-conservative than in the ACC. Although the spiciness variation can also be used for the mixing length estimation, the PV-conservative nature of isopycnal thickness and the thickness-based\(L_{\text{mix}}\) rationalize the calculation of thickness-diffusive transport.
The estimated diffusivity of 100–500 m2s-1 is significantly smaller than the along-slope estimation of 950 ± 400 m2 s-1 by FRE19. However, their estimation was for the relative diffusivity in that they implicitly assumed the mixing efficiency \(\Gamma\) to be unity (far exceeding its range; 0.01–0.4) and hence seems incompatible with our estimation. If \(\Gamma=0.16\ \)by Wunsch (1999), also supported by Klocker and Abernathey (2014), is consistently applied for their values, the isopycnal diffusivity of 90–220 m2s-1 is obtained, somewhat smaller \(k\) than our estimates. Further, our estimation is consistent with previous studies in the ACC’s mainstream, typically ranging 500–2000 m2 s−1 (Marshall et al., 2006) and 1500–3000 m2 s−1 (Sallée et al., 2011) with a poleward decrease. Although the uncertainty of \(k\) is inaccessible and the mixing length estimated from the hydrographic variability does not necessarily reflect diffusion solely due to eddies by principle, the accordance with the estimations derived from the altimetric speed supports its plausibility.
One of the most uncertain parts is the spatial variability of mixing efficiency. Visbeck et al. (1997) argue that the eddy transfer coefficient, which determines the proportionality of diffusivity to the horizontal/vertical stratification and the width of the baroclinic zone, is a universal constant (equal to 0.015) regardless of the flow regime. Mixing efficiency is different from this coefficient by its formulation, but they are possibly associated with each other. The mooring data (Section 3.3) and the argument by Klocker and Abernathey (2014) generally support the local validity of \(\Gamma=0.16\) by Wunsch (1999). However, its global applicability is a future task and requires the utility of numerical models. The spatial variation of \(\Gamma\) can alter the correspondence between the enhanced diffusivity and the CDW intrusions (Fig. 11), while the along-slope analysis leastwise suggests that the mixing length becomes larger where CDW intrudes shoreward (Fig. 9). Furthermore, it is presumable that the effect of its spatial variation is negligible when considering a basin-wide transport as performed in Fig. 15.
5.2 Variability in the Antarctic margin
\(L_{\text{mix}}\) and \(k\) tend to be larger where the onshore CDW intrusion occurs (Figs. 9 and 11). The scale of this correspondence is observed in a somewhat larger scale than the data radius = 75 km, which provides the highest limit of the resolving scale. This result supports that the cross-slope eddy flux is fundamental for transporting CDW onto the shelves. Since the intrusion sites are strongly associated with the topography-controlled subgyres (Yamazaki et al., 2020), the structure of the subpolar gyre and bathymetric feature are expected to be essential for determining the eddy field. The locations of CDW intrusion in Fig. 9 and the major contribution of eddy fluxes are reproduced in an eddy-resolving simulation (Stewart et al., 2018). The isopycnal fields further indicate that the CDW thickness gradient tends to be gentle where the intrusion occurs (Figs. 9 and 11). This situation likely pertains to the tracer gradient control on \(L_{\text{mix}}\) and \(k\)(Figs. 8 and 12). In the ACC domain, however, \(k\) is more dependent on\(U_{\text{eddy}}\) and \(\sigma(\varphi)\) than in the subpolar region (Figs. 8 and 12), and thus \(L_{\text{mix}}\) is more related to the flow speed via the jet-induced suppression mechanism (Fig. 7). These results highlight the transition of the Southern Ocean eddy regime towards the Antarctic margin.
The correspondence between the CDW intrusion and gentle thickness gradient further suggests that the baroclinic structure of the ASC behaves as a barrier to the onshore CDW intrusion, as hypothesized by modeling studies (Thompson et al., 2018). The theoretical prediction of jet-induced mixing suppression (Ferrari and Nikurashin 2010) might be consistent with the unintrusive situation associated with a steep thickness gradient associated with the ASC; however, we could not obtain\(L_{\text{mix}}\) dependency on flow speed (Fig. 7). Although this is perhaps because of the coarse resolution of the altimetric velocity data (1/4°) with respect to the scale of ASC, currents in the subpolar zone are weaker than the ACC (Fig. 2), and the ASC is not as distinguished as the ACC jets (Peña-Molino et al., 2016), implying that the jet-induced suppression is not necessarily the primary factor for the shoreward eddy diffusion.
The dynamic driver governing the isopycnal thickness field remains unknown, yet we can posit that topographic steering plays an indispensable role. To investigate meridional overturning circulation across the ASC jets in a zonally symmetric configuration, Stewart and Thompson (2016) demonstrated that \(L_{\text{mix}}\) scaled by slope width accurately predicts the simulated onshore flux of CDW (R2 = 0.89). In contrast, our estimated\(L_{\text{mix}}\) is significantly correlated with the thickness gradient but not with the topographic gradient (Fig. 13). This result might be due to the indispensable role of the surface layer and distribution of ASW in controlling the layer thickness.
5.3 Eddy flux and heat budgets
The shoreward eddy heat flux (Fig. 15) is quantitatively consistent with the previously reported coastal budgets; integrating within the eastern Indian sector (90–160°E), the annual cumulative sea ice production is 520 ± 75 km3 (Tamura et al., 2016; accounting for Shackleton, Vincennes, Dalton, Dibble, and Mertz Polynyas), translated to heat loss of 4.2–5.6 TW. On the other hand, the integrated ice shelf basal melt rate is 198 ± 39 Gt yr-1 (Rignot et al., 2013; accounting for Mertz, Dibble, Holmes, Moscow Univ., Totten, Vincennes, Conger, Tracy, and Shackleton Ice Shelves), translated to 1.7–2.5 TW. Therefore, the diffusive CDW heat flux of 3.2–3.9 TW (within the 28.0–28.1 kg m-3 neutral density) compensates for nearly half of the sum of cryospheric heat sinks by sea ice formation and ice shelf melt, and thus is a major source of heat for the Antarctic coasts. Missing source of heat (~3 TW) and offshore heat advection is likely balanced by the annual cumulative solar heating (~5 TW within 100 km from the coastline of 90–160°E; Tamura et al., 2011) and the partial onshore intrusion of ASW (to the east of 130°E). The volume imbalance between CDW and ASW may imply the local exporting flux of Antarctic Bottom Water, while an effect of convergent flux due to the along-slope mean flow cannot be distinguished in our analysis.
Results by Stewart and Thompson (2016) indicate a possibility of underestimating the onshore heat flux derived from the mixing length formulation when estimation is solely based on the thickness-diffusive CDW flux, or “eddy advection,” because the isopycnal “eddy stirring” can also contribute to the heat flux without transporting water volume, especially near the shelf break. The coastal heat budget closure by the eddy volume flux pertains to the situation that the eddy stirring and tidal mixing are not dominant over the targeted slope (1000–3000 m). This is consistent with a realistic simulation (Stewart et al., 2018), in which most of the heat flux explained by eddy advection over the isobaths subsequently reaches the Antarctic coast beyond the shelf break. The poleward CDW transport by the cross-slope geostrophic current was measured seaward of the 3000 m isobath (Mizobata et al., 2020), and this might be generally confined to the lower continental slope, regarding the numerical model (Stewart et al., 2018) and the trajectories of profiling floats (Yamazaki et al., 2020).
6. Summary and outlook
To describe the CDW diffusion towards the Antarctic margin, the present study conducted an extensive analysis of hydrographic measurements and satellite altimetry data. The mixing length formulation served as the primary basis for analysis. The spiciness/thickness-based estimations yielded similar results, which validated the mixing length estimates previously made using hydrographic variability. The same analysis was applied for ASW, and its mixing length close to CDW was obtained. Over the ACC domain (antarctic and southern zones), concurrences with previous studies on the mixing suppression theory and its exception in the lee of the topography (leaky jets) were found. However, the mixing length dependency on the mean flow was not found in the subpolar zone, reflecting a quiescent flow regime in the Antarctic margin. Mixing length tends to be larger where the CDW intrusion occurs. This observation is attributable to the thickness control on the diffusivity, in which the gentle thickness gradient allows for the ease of isopycnal mixing. Volume transport was estimated in a GM-flux manner, and the thickness-diffusive onshore heat flux over the continental slope was quantitatively consistent with cryospheric heat sinks (i.e., sea ice formation and ice shelf basal melt). Closure of the coastal heat budget was nearly achieved among the CDW eddies over the 1,000–3,000 m isobaths (+3.2‒3.9 TW), solar heating (+5 TW), sea ice formation (\(-\)4.2‒5.6 TW), and basal melt of ice shelves (\(-\)1.7‒2.5 TW) for 90–160°E. This result suggests that the isopycnal eddy advection is substantial for the cross-slope CDW intrusion.
As a concluding remark, we underscore that the isopycnal thickness field is essential for determining the eddy fluxes in the Antarctic margin. Our findings allow predicting the eddy diffusivity by solely providing the layer thickness. This idea might be valuable for simulating CDW transport in climate models, where subgrid effects of eddy fluxes need to be parameterized. The mesoscale process is crucial for the multidecadal variability of onshore CDW flux (e.g., Yamazaki et al., 2021), and thus its appropriate expression in climate models is fundamental for projecting the Antarctic glacial melt with accuracy and confidence.