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\begin{document}

\title{Diffusion of Circumpolar Deep Water towards Antarctica}



\author[1]{Kaihe Yamazaki}%
\author[2]{Shigeru Aoki}%
\author[3]{Kohei Mizobata}%
\affil[1]{National Institute of Polar Research}%
\affil[2]{Hokkaido University}%
\affil[3]{Tokyo University of Marine Science and Technology}%


\vspace{-1em}



  
  \date{May 18, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Warm, salty Circumpolar Deep Water (CDW) is recognized as the primary
driver for Antarctic glacial melt, but the mechanism by which it reaches
the continental shelves remains highly uncertain from an observational
standpoint. With the scarcity of eddy flux estimation in the Antarctic
margin, we quantify the isopycnal diffusivity of CDW using hydrographic
variability and satellite altimetry under the mixing length framework.
For comparison, the spiciness and thickness are used as isopycnal
tracers, and the two tracers yield qualitatively similar estimates. Over
the Antarctic Circumpolar Current (ACC), spatial variation of mixing
length is generally aligned with the jet-induced mixing suppression
theory, including its exception in the lee of the topography. In
contrast, the mixing length does not depend on the mean flow in the
subpolar zone, likely reflecting the relatively quiescent flow regime.
The estimated isopycnal diffusivity ranges from 100 to 500 m2 s-1 south
of the ACC. The eddy diffusivity tends to be enhanced where the gradient
of isopycnal thickness becomes small and CDW intrudes onshore. The
cross-slope eddy CDW flux is estimated, and the associated onshore heat
flux across is calculated as \textasciitilde{}3.6 TW in the eastern
Indian sector. The eddy heat flux and coastal solar heating are
generally balanced with cryospheric heat sinks including glacial melting
and surface freezing, suggesting that the eddy advection is substantial
for the onshore CDW flux. The thickness field is essential for
determining mixing length and eddy fluxes in the subpolar zone, whereas
the situation does not hold for the ACC domain.%
\end{abstract}%



\sloppy


Kaihe Yamazaki\textsuperscript{1,2,3}, Shigeru Aoki\textsuperscript{1},
Kohei Mizobata\textsuperscript{4}

\textsuperscript{1} Institute of Low Temperature Science, Hokkaido
University, Hokkaido, Japan

\textsuperscript{2} Graduate School of Environmental Science, Hokkaido
University, Hokkaido, Japan

\textsuperscript{3} National Institute of Polar Research, Tokyo, Japan

\textsuperscript{4} Tokyo University of Marine Science and Technology,
Tokyo, Japan

Corresponding author: Kaihe Yamazaki (kaiheyamazaki@gmail.com)

Key Points

\begin{itemize}
\tightlist
\item
  Eddy diffusivity in the subpolar Southern Ocean is estimated to be
  100--500 m\textsuperscript{2} s\textsuperscript{-1} based on
  hydrographic variability and satellite altimetry.
\item
  Eddy heat flux towards the Wilkes Land is \textasciitilde{}3.6 TW,
  nearly balancing with surface freezing, glacial melt, and solar
  heating.
\item
  The thickness gradient controls mixing length and eddy diffusivity in
  the subpolar zone.
\end{itemize}

Abstract

Warm, salty Circumpolar Deep Water (CDW) is recognized as the primary
driver for Antarctic glacial melt, but the mechanism by which it reaches
the continental shelves remains highly uncertain from an observational
standpoint. With the scarcity of eddy flux estimation in the Antarctic
margin, we quantify the isopycnal diffusivity of CDW using hydrographic
variability and satellite altimetry under the mixing length framework.
For comparison, the spiciness and thickness are used as isopycnal
tracers, and the two tracers yield qualitatively similar estimates. Over
the Antarctic Circumpolar Current (ACC), spatial variation of mixing
length is generally aligned with the jet-induced mixing suppression
theory, including its exception in the lee of the topography. In
contrast, the mixing length does not depend on the mean flow in the
subpolar zone, likely reflecting the relatively quiescent flow regime.
The estimated isopycnal diffusivity ranges from 100 to 500
m\textsuperscript{2} s\textsuperscript{-1} south of the ACC. The eddy
diffusivity tends to be enhanced where the gradient of isopycnal
thickness becomes small and CDW intrudes onshore. The cross-slope eddy
CDW flux is estimated, and the associated onshore heat flux across is
calculated as \textasciitilde{}3.6 TW in the eastern Indian sector. The
eddy heat flux and coastal solar heating are generally balanced with
cryospheric heat sinks including glacial melting and surface freezing,
suggesting that the eddy advection is substantial for the onshore CDW
flux. The thickness field is essential for determining mixing length and
eddy fluxes in the subpolar zone, whereas the situation does not hold
for the ACC domain.

\textbf{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
\textasciitilde{}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\selectlanguage{ngerman}°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 \emph{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 \emph{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\selectlanguage{ngerman} 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; Sall\selectlanguage{ngerman}ée et
al., 2011), falling between 500 and 3,000
m\textsuperscript{2}s\selectlanguage{english}\textsuperscript{-1} within the ACC core and
2,000--3,500 m\textsuperscript{2} s\textsuperscript{-1}\selectlanguage{english} 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; Sall\selectlanguage{ngerman}ée 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 \selectlanguage{english}?=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 0.2deg 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 113degE (Pena-Molino et al., 2016), which mark
standard deviations of 0.04--0.06 m s\textsuperscript{-1} in zonal and
meridional directions at the CDW layer (\textasciitilde{}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\textsuperscript{-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 (Pena-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; Pena-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\textsuperscript{-3}) and Antarctic Surface
Water (ASW; defined as 27.9--28.0 kg m\textsuperscript{-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\textsuperscript{-3} for CDW).

Mapping isopycnal climatology onto 0.2deg 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--60degE, 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\textsuperscript{-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 113degE over the slope of
500--4,000 m (reported by Pena-Molino et al., 2016). Vertical/meridional
linear gridding (by 50 dbar for 200--1,500 dbar and by 0.1deg for
65.5--61.5degS) 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 degC isotherms) in ASW (27.9--28.0 kg
m\textsuperscript{-3}), while in CDW (28.0--28.1 kg
m\textsuperscript{-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
\textasciitilde{}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): \textless{}
--1.85 m, the southern zone (from the SACCF-S to SACCF-N): --1.85
\textasciitilde{} --1.6 m, and the antarctic zone (from SACCF-N to PF):
--1.6 \textasciitilde{} --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\textsuperscript{-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\textsuperscript{-1}, corresponding to leaky jets in the lee of
topographic features (GFP11; Sallee et al., 2011; Tamsitt et al., 2018)
such as the Kerguelen Plateau (\textasciitilde{}80degE) and the Southeast
Indian Ridge (\textasciitilde{}150degE; 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 \selectlanguage{greek}Γ, \selectlanguage{english}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 (-\selectlanguage{english}1.97 and -\selectlanguage{english}1.85 m)
highlighted in blue. Fig. 10 suggests that the isopycnal diffusivity
\(k\) typically ranges from 100 to 500
m\textsuperscript{2} s\textsuperscript{-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\selectlanguage{ngerman}°, 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 \textless{}
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 m\textsuperscript{2}s\textsuperscript{-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\selectlanguage{ngerman}°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 (=
m\textsuperscript{3}s\textsuperscript{-1}) in the eastern/western Indian
sectors (divided by the Princess Elizabeth Trough
\textasciitilde{}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\textsuperscript{-3}) to the west of 130°E is 0.15 Sv, locally
compensating \textasciitilde{}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 (\textasciitilde{}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 (\textasciitilde{}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
m\textsuperscript{2}s\textsuperscript{-1} is significantly smaller than
the along-slope estimation of 950 ± 400 m\textsuperscript{2}
s\textsuperscript{-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
m\textsuperscript{2}s\textsuperscript{-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 m\textsuperscript{2} s\selectlanguage{english}\textsuperscript{-1} (Marshall
et al., 2006) and 1500--3000 m\textsuperscript{2} s\selectlanguage{english}\textsuperscript{-1}
(Sallee 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 (1/4deg) 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 (Pena-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
(R\textsuperscript{2} = 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--160degE), the annual cumulative sea ice production is
520 +- 75 km\textsuperscript{3} (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\textsuperscript{-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\textsuperscript{-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 (\textasciitilde{}3 TW) and
offshore heat advection is likely balanced by the annual cumulative
solar heating (\textasciitilde{}5 TW within 100 km from the coastline of
90--160degE; Tamura et al., 2011) and the partial onshore intrusion of ASW
(to the east of 130degE). 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\selectlanguage{english}-3.9 TW), solar heating (+5 TW), sea ice formation
(\(-\)4.2\selectlanguage{english}-5.6 TW), and basal melt of ice shelves
(\(-\)1.7\selectlanguage{english}-2.5 TW) for 90--160degE. 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.

\textbf{Acknowledgments}

We are grateful to the Argo Program and the MEOP project for significant
deployment efforts in the Southern Ocean and successive QC processes. B.
Pena-Molino provided the I9S mooring records. This study was financially
supported by JSPS KAKENHI (grant number 21H04918, 22J00870). The authors
declare no competing interest.

\textbf{Open Research}

All hydrographic data are available to the public, as detailed in
Section 3.2. Dataset including dynamic ocean topography data (Mizobata
et al., 2020) and Python codes used for figure creation can be
downloaded from https://doi.org/10.5281/zenodo.6463138.

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\textbf{Figures and Tables}

\textbf{Figure 1} . Circulation and topography in the East Antarctic
margin. The 3,000 m isobath is highlighted in white. Polar Front (PF;
green), Northern/southern branches of Southern ACC Front (SACCF-N/S;
yellow/magenta), and subpolar gyres (blue) are derived from dynamic
ocean topography (Section 3.1), and Southern Boundary (SB; red) and
Antarctic Slope Front (ASF, by 0degC at 400 dbar; cyan) are reproduced
from temperature field of a climatological dataset by Shimada et al.
(2017). The SACCF-S corresponds to the southernmost eastward jet of ACC
(\textasciitilde{}4,000 m isobath), whereas the SB is located along the
center of subpolar recirculating gyres (4,000--3,000 m) about zero lines
of zonal velocity.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Figure 2} . Altimetric flow speed averaged for 2011--2020. The
upper panel is the temporal-mean geostrophic velocity, and the lower
panel is the eddy velocity as root-mean-squared speed. Frontal positions
are drawn as in Fig. 1.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Figure 3} . Distribution of hydrographic data and isopycnal
watermass properties of ASW and CDW. Data density per 75 km data radius
(top), Conservative Temperature (middle), and isopycnal pressure
(bottom) are presented for ASW (left) and CDW (right). Areas with less
than 10 data points within the 75 km radius are masked in gray. White
contours in middle/bottom panels denote isopycnal spiciness/thickness
(by 0.05 kg m\textsuperscript{-3}/100 m intervals), respectively, and
the thick contour is --0.15 kg m\textsuperscript{-3}/300 m for each
panel. SB (red) and isobaths with 1,000 m intervals (black) are also
shown.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Figure 4} . Histogram of mixing efficiency (or correlation
coefficient between meridional velocity and temperature) from the
cross-slope mooring section in 113\selectlanguage{ngerman}°E. ``Down-gradient'' denotes the
correlation by southward velocity (i.e., opposite to the temperature
gradient), and ``up-gradient'' means the correlation by northward
velocity. Since the downgradient flux must direct southward (shoreward),
``up-gradient'' possibly reflects transient events irrelevant to the
climatological eddy condition. Hence, in addition to the whole mean
(black line) and the previous estimate (dashed line; 0.16), the mean
value only for the ``down-gradient'' is also presented (gray line).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Figure 5} . Mixing length calculation using isopycnal spiciness.
Spiciness variability (top), normed spiciness gradient (middle), and
mixing length (bottom) are presented for ASW (left) and CDW (right).
White contours in top/middle panels denote isopycnal spiciness as of
Fig. 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Figure 6} . Mixing length calculation using isopycnal thickness.
Thickness variability (top), normed thickness gradient (middle), and
mixing length (bottom) are presented for ASW (left) and CDW (right).
White contours in top/middle panels denote isopycnal thickness as of
Fig. 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Figure 7} . Mixing length dependency on the mean flow. Upper and
lower panels are based on spiciness and thickness, and left/right panels
are for ASW/CDW, respectively. The results are separately shown for the
three frontal zones: subpolar (south of SACCF-S), southern (from SACCF-S
to SACCF-N), and antarctic (from SACCF-N to PF) zones. Standard errors
due to the spatial variation are shaded.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Figure 8} . Two-dimensional histogram of mixing length as a
function of tracer variability (horizontal) and inversed tracer gradient
(vertical), indicating relative dependency of mixing length on the two
variables. Rows correspond to the methods (spiciness and thickness) and
layers (ASW and CDW), whereas columns correspond to the three frontal
zones. Color shade is normalized to unity, and yellower indicates a
larger data population. The axes are also normalized to illustrate their
functional dependency. White cross denotes the averaged value of tracer
variability and inversed tracer gradient, and white contours are the
mixing length of 20 and 100 km. The diagonal dotted line indicates where
controls by the two variables become comparable.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\textbf{Figure 9} . Along-slope variability of the diagnostics averaged
within 1\selectlanguage{ngerman}° bin in longitude over the 1,000--3,000 m isobaths. (top)
Inversed gradient and standard deviation of the CDW thickness. (middle)
thickness-based diffusivity and eddy velocity. (bottom) Isopycnal
diffusivity for CDW. Red shades are the locations of CDW intrusion
described in Yamazaki et al. (2020), where ``Wx'' is warm signals in the
Australian-Antarctic Basin, ``PB'' is for the Prydz Bay, and ``LHB'' is
for the Lützow-Holm Bay.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

\textbf{Figure 10} . Spatial distribution of isopycnal diffusivity for
CDW. Upper and lower panels are based on spiciness and thickness,
respectively. In addition to SB (red) and isobaths, dynamic topography
is overlaid by 2 cm intervals (white contours). Characteristic contours
of dynamic topography are highlighted in blue (thick: subpolar gyre as
--1.97 m, thin: SACCF-S as --1.85 m).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

\textbf{Figure 11.} Eddy diffusion and CDW intrusion focusing on three
key regions (Prydz Bay, Princess Elizabeth Trough, and off Wilkes Land).
(top) Thickness-based diffusivity for CDW (Same as the lower panel of
Fig. 10). (middle) The inversed thickness gradient. (bottom) The
isopycnal temperature (same as Fig. 3). Vectors annotate where the CDW
intrusion corresponds to the enhanced diffusivity.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}

\textbf{Figure 12} . Histogram of isopycnal diffusivity on phase diagram
as a function of eddy velocity (horizontal) and mixing length
(vertical), analogously to Fig. 8. White contours denote the isopleths
of 100 and 500 m\textsuperscript{2} s\textsuperscript{-1}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}

\textbf{Figure 13} . Tracer gradient control of CDW diffusion. Isopycnal
diffusivity (top), mixing length (middle), and topographic gradient
(bottom) over the 1,000--3,000 m isobath are regressed onto the inversed
tracer gradient as spiciness (left) and thickness (right). The
background color is the normalized histogram. Horizontal and vertical
axes are normalized by the averages.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}

\end{center}
\end{figure}

\textbf{Figure 14} . Spatial distribution of eddy CDW fluxes. The top,
middle, and bottom panels are transport vectors, zonal and meridional
transport, respectively. SB, isobaths, and dynamic topography are
denoted as in Fig. 10. For illustrative purposes, colormap for the
meridional transport is flipped.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

\textbf{Figure 15} . Cross-slope volume/heat transport. Meridional eddy
transports for ASW and CDW (averaged in 1\selectlanguage{ngerman}° bin over the 1,000--3,000 m
isobaths) are zonally integrated. Standard errors due to the cross-slope
variation are shown by shade. Volume transport can precisely be
translated to heat flux using the mean temperature of CDW and ASW as
indicated by ticks to the left. The 90°E meridian for CDW corresponds to
the interbasin boundary between eastern/western Indian sectors, while
130°E is the transitional longitude for the ASW transport direction.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig2/fig2}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig4/fig4}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig3/fig3}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/fig1/fig1}

\end{center}
\end{figure}

\selectlanguage{english}
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\end{document}

