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.