Plain Language Summary
CDW is a significant source of heat and salt for the Antarctic coasts,
so its behavior is topical for a wide range of climate sciences:
especially in the contexts of Antarctic glacial melting, sea ice
variability, and global ocean overturning. Numerical simulations have
previously suggested that CDW is transported onshore by advection of
ocean eddies, but there has been no observational basis. Synthesizing
in-situ pressure/temperature/salinity measurements and satellite
altimetry, we provide a rigorous estimate of the onshore CDW transport.
Shoreward heat flux by CDW eddies is generally balanced with heat loss
expected by surface freezing and glacial melt, indicating that the eddy
transport plays a fundamental role in the coastal heat supply. The
gradient of CDW thickness primarily controls the ability to mix, i.e.,
spatially homogeneous thickness allows for ease of mixing. Our results
facilitate a possibility to predict the eddy diffusivity solely from the
layer thickness. This idea is valuable for simulating CDW transport in
global climate models, where subgrid, unresolved effects of eddies need
to be parameterized.
1 Introduction
Over the extent of the Antarctic Circumpolar Current (ACC), mesoscale
eddies transport water masses across the streamlines, building up 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 in the Southern Ocean because the
bottom enhanced diapycnal mixing (Kunze et al., 2006) and the surface
water transformation (Abernathey et al., 2016) unlikely penetrates the
intermediate and deep layers in the interior. Recent observations have
indicated that mesoscale eddy plays a key role in bridging the Antarctic
meridional overturning from deep ocean basins to the continental shelves
(Thompson et al., 2014; Mckee et al., 2019), while the eddy condition
from the ACC to the shoreward Antarctic Slope Current (ASC; Thompson et
al., 2018) remains unconstrained.
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 gradient
(Stewart and Thompson, 2013). In reality, pressure gradient associated
with topographic features generates standing eddies and meanders,
facilitating the 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). Meanwhile, the steep barotropic potential vorticity (PV)
gradient on the upper continental slope (inshore of
~3,000 m isobaths) unlikely allows for the presence of
the cross-slope mean flow, and hence eddy diffusion and/or tidal mixing
might be essential for the onshore CDW flux near the shelf break
(Stewart et al., 2018; Yamazaki et al., 2020). To the south of the ACC,
the spatial distribution of eddy diffusion is yet to be described except
few analyses (Foppert et al., 2019; FRE19 henceforth). From an
observational standpoint, this study quests (1) to delineate the
controlling factor of eddy diffusion in the Antarctic margin (portrayed
as a poleward extension of Naveira Garabato et al., 2011; GFP11
henceforth) and (2) to quantify the isopycnal CDW flux by eddy diffusion
towards the continental shelves.
The horizontal circulation in the Antarctic margin is shaped by the
eastward ACC and the westward ASC, and in the transition zone between
them exist clockwise subpolar gyres (e.g., Park and Gamberoni, 1995).
The Weddell and Ross gyres are wide enough to isolate cold shelves from
warm CDW, whereas ACC’s proximity to the continent creates the 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 the
relatively warmer coastal conditions than the other sectors (Jenkins et
al., 2016; Silvano et al., 2016). A lack of knowledge on the subpolar
ocean circulation motivates further reanalysis of in-situ observations.
Although FRE19 inferred the along-slope variability of eddy transport
over the continental slope using seal-mounted
conductivity-temperature-depth (CTD) data, the correspondence between
the flow regime and eddy diffusion remains unclear. The present study
approaches this question regarding the importance of eddy flux for the
multidecadal change in the Antarctic thermal conditioning (Yamazaki et
al., 2021). The East Antarctic margin (30–160°E; Fig. 1) is mainly
targeted, where a sufficient amount of in-situ data exists thanks to
ceaseless efforts of deploying profiling float and biologging. In this
region, the eddy condition has recently been explored (FRE19; Stewart et
al., 2018, 2019), lateral tidal mixing is weaker than the rest of the
Antarctic margin (Beckman and Pereira, 2003), the frontal structure of
ASC is relatively prominent (Pauthenet et al., 2021), and the onshore
CDW flux collects attention for the future climate projection (Yamazaki
et al., 2021; Hirano et al., 2021).
This paper is set out as follows. In Section 2, we review the
theoretical background for observation-based eddy diffusivity
calculation and introduce the concept of mixing length framework. In
Section 3, we describe data and methods used for diffusivity
calculation. In Section 4, mixing length, eddy diffusivity, and eddy
fluxes are quantified, and their spatial variations are delineated with
respect to the topographic structure in the continental margin. In
Section 5, validity of presented results is assessed, and controlling
factors of eddy diffusion are discussed. We conclude in Section 6.
2 Theoretical background
This section briefly reviews arguments of the mixing length framework,
which provides the basis for our analysis. The observational estimates
of oceanic eddy diffusion may branch in four ways: hydrographic
variability (Armi and Stommel, 1983; GFP11), altimetric eddy scaling
(Klocker and Abernathey, 2014; Bates et al., 2014), tracer patch
deformation (Marshall et al., 2006), and dispersion time scale (LaCasce
and Bower, 2000; Sallée et al., 2011). For the first two methods, the
diffusivity k is derived 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 (sometimes referred to as eddy
transfer coefficient), \(U_{\text{eddy}}\) is characteristic eddy
velocity measured by the standard deviation of downgradient velocity\(\sigma(v)\), and \(L_{\text{mix}}\) is mixing length scale. This
formulation rests on two major assumptions (quoted from GFP11): (i)
tracer fluctuations are generated by local stirring of the large-scale
tracer gradient, with the advection of tracer variance from upstream
regions being weak, and (ii) the tracer gradient varies slowly over the
distance \(L_{\text{mix}}\).
The mixing length framework has widely been applied for the closure of
geostrophic turbulence since it can link the eddy tracer transport to a
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 temporal average in the isopycnal layer, the
primes indicate deviations from those averages, and the tracer gradient
is assumed to be meridional. \(v\) is meridional velocity so that the
tracer flux is the covariance between tracer anomaly and cross-frontal
velocity. Here, it is assumed a priori that tracer \(\varphi\)mixes purely along isopycnals. This assumption is equivalent to
conditions that the mixing process is statistically steady, adiabatic,
and solely caused by linear waves (e.g., Vallis, 2017). We may choose
any passive tracer for \(\varphi\) if the tracer concentration
represents the PV field, where its diffusion satisfies a requirement for
the GM flux (Gent and McWilliams, 1990) mimicking baroclinic instability
and the scalar coefficient of downgradient PV flux can express the skew
component of diffusivity tensor.
One possible choice for \(L_{\text{mix}}\) is characteristic eddy scale
(Klocker and Abernathey, 2014; Bates et al., 2014), which can be
determined by the altimetric velocity field. Another possible choice is
a rather empirical way of using hydrographic data. Emulating the
arguments of Armi and Stommel (1983), GFP11 derived \(L_{\text{mix}}\)in the Southern Ocean from hydrographic variability, i.e.,
\begin{equation}
\ L_{\text{mix}}=\frac{\sigma(\varphi)}{|\nabla\varphi|}\text{\ \ \ \ \ \ \ \ \ \ }\left(3\right).\nonumber \\
\end{equation}Although they used isopycnal temperature for the conservative tracer\(\varphi\), other candidates exist for the tracer variable (e.g.,
isopycnal spiciness and layer thickness). By the equations (1)–(3), the
mixing efficiency follows as
\begin{equation}
\Gamma=\frac{\overset{\overline{}}{v^{\prime}\varphi^{\prime}}}{\sigma\left(v\right)\ \sigma(\varphi)}\text{\ \ \ \ \ \ \ \ \ \ }\left(4\right),\nonumber \\
\end{equation}that is identical to the correlation coefficient between \(v\) and\(\varphi\). There are a wide range of estimates for \(\Gamma\)(0.01–0.4; Holloway and Kristmannsson, 1984; Visbeck et al., 1997;
Karsten and Marshall, 2002) likely depending on the variety of\(L_{\text{mix}}\) definitions. GFP11 noted that the only observational
estimate \(\Gamma=0.16\) provided by Wunsch (1999) might be used for
illustrating absolute values of k .
The hydrographic estimate of eddy diffusivity by GFP11 is broadly
consistent with a more direct estimation via Lagrangian tracer
dispersion numerically advected with altimetric velocity (Marshall,
2006; Sallée et al., 2011), generally falling into 500–3000
m2 s−1 within the ACC core and
2000–3500 m2 s−1 in its 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 eddy’s continuous action for the same
water mass and suppresses mixing length. The suppression of eddy
stirring ceases in “leaky jets,” likely 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 of Wunsch (1999). These studies rationalize using the
hydrographic variability method: the equations (1) and (3), and thus we
apply them for quantifying eddy diffusion.
3. Data and methods
3.1 Satellite altimetry for \(\mathbf{U}_{\mathbf{\text{eddy}}}\)
An observational estimate of characteristic eddy velocity\(U_{\text{eddy}}\) can be given by altimetric velocity in the open
ocean, while the satellite altimetry has previously been unavailable in
the seasonal ice zone. Later, 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; Mizobata et al., 2020). The
present study adopts the monthly-reconstructed 0.2° grid dynamic ocean
topography during 2011–2020 by Mizobata et al. (2020) to derive
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
altimetric flow speed (lower panel of Fig. 2). Its reliability is
underpinned by the mooring measurements at 113°E (Pena-Molino et al.,
2016), marking standard deviations of 0.04–0.06 m
s-1 in zonal and meridional directions at the CDW
depth (~500 dbar). 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 standard
deviation of the cross-frontal velocity. However, in contrast to the
ACC’s mainstream, the flow field in the Antarctic margin is stagnant,
and the mean flow directions are ambiguous (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).
Vertical variations of eddy velocity are neglected in this study. GFP11
treated this issue by applying the gravest empirical mode analysis to
derive geostrophic shear. The gravest empirical mode scheme is very
effective in the ACC domain, while it cannot be applied for the
Antarctic margin as the dynamic topography does not descend poleward
monotonically. Nevertheless, we consider that \(U_{\text{eddy}}\)adopted for CDW is acceptable because the vertical attenuation due to
geostrophic shear is considerably small in the Antarctic margin by the
quasi-barotropic flow structure (Pena-Molino et al., 2016; Mizobata et
al., 2020; Yamazaki et al., 2020).
3.2 CTD profiles for \(\mathbf{L}_{\mathbf{\text{mix}}}\)
Mixing length \(L_{\text{mix}}\) is calculated from hydrographic
variability by the equation (3). We assemble historical CTD profiles
from World Ocean Database (https://www.ncei.noaa.gov/; for
shipboard CTD), 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). Data are extracted
for December–March and 1990 onwards. After removing bad flagged data
and fragmented profiles, 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 figures
indicate that, in contrast to isopycnal temperature gradient of ASW
stronger than CDW, isopycnal thickness gradient of CDW is generally
stronger than ASW. Our focus is CDW, while a comparison to the ASW
layer, with a larger number of data than the CDW layer, facilitates to
check the layer dependency and the quantitativeness of \(k\).
The previous studies adopted potential temperature and spiciness as the
isopycnal tracer \(\varphi\) (GFP11; FRE19). However, it is unclear if
these tracers yield diffusivity \(k\) conforming to volume transport
expected by the downgradient PV diffusion (e.g., Marshall and Radko,
2003). Since the layer thickness is a possible candidate for the PV
conservative variable (e.g., Vallis, 2017), the present study adopts
both spiciness and layer thickness as the isopycnal tracer \(\varphi\),
and the diffusivities derived from the two variables are compared.
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 watermass.
Mapping surface data onto 0.2° grids is performed with the radius basis
function interpolation (Yamazaki et al., 2020), which reproduces the
best representative surface of noisy data nonparametrically in the
least-squares sense. Grid data with less than 10 points inside a 75 km
data radius are masked (gray area). Although the data coverage
particularly reduces in 30–60°E, a sufficient number of data exist
within the region of our interest (e.g., the continental slope of
1,000–3,000 m isobaths). Correspondence among the 3,000 m isobath,
–0.15 kg m-3 CDW spiciness, and 300 m CDW thickness
(Fig. 3 right panels) guarantees fidelity of the interpolation. After
calculating deviations of surface data from the gridded 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. The choice
of the radius size is a trade-off between the data amount and the
resolution, while 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 \(k\) rests within the
mixing efficiency \(\Gamma\). Based on the 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 did 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 across the ASC (in 113°E for 2010–2011; Pena-Molino et
al., 2016). Vertical/meridional linear gridding (by 50 dbar for
200–1500 dbar and by 0.1 degrees for 65.5–61.5°S) is performed for
hourly meridional velocity and temperature profiles to yield 1040 grids
in total. During the 12 months (8761 steps), their correlation
coefficient is calculated for each grid, assuming that the temperature
variation is approximately coherent with the PV change and its gradient
directs northward on average.
From the histogram of \(\Gamma\), the mean value is estimated as 0.12
for down-gradient cases and 0.10 for all cases (Fig. 4). The up-gradient
cases are possibly irrelevant to the climatological eddy condition since
the downgradient velocity must direct southward by the mean temperature
field (Fig. 3). 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, but such
filtering possibly leads to underestimation). Based on the general
agreement of the local value with the global estimate, the present study
adopts the mixing efficiency \(\Gamma=0.16\) by Wunsch (1999)
consistently with previous investigations (GFP11; Klocker and
Abernathey, 2014). The validity of our choice is further discussed in
Section 5.1.
4. Result
4.1 Mixing length
Standard deviation and normed gradient of isopycnal tracer \(\varphi\)for spiciness and layer thickness of each watermass are shown in Figs. 5
and 6, respectively. The large gradient of spiciness is concentrated
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), it emerges over the upper
continental slope to the south (Fig. 5; top panels). The large standard
deviation of spiciness broadly corresponds to its steep gradient.
Relative to the spiciness gradient, the thickness gradient is likely
homogeneous, and the coherence between the standard deviation and the
gradient is less noticeable (Fig. 6). As in the spiciness, the sharp
thickness gradient of CDW is found in the proximity of the SB,
indicating a poleward volume flux represented by thickness diffusion
(Yamazaki et al., 2020). Rounded patchy patterns appearing in the
thickness-based diagnostics are likely associated with the distribution
of standing eddies, while those signals are not visible in the
spiciness-based values.
The lowermost panels in Figs. 5 and 6 present the mixing length\(L_{\text{mix}}\) derived from the equation (3). The patchy patterns in
the thickness-based diagnostics do not emerge for \(L_{\text{mix}}\).
The spatial distributions of the spiciness/thickness-based\(L_{\text{mix}}\) are analoguous in terms of their meridional
variations. 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. Even though 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 near the SB. These results suggest the quantitative
robustness of the \(L_{\text{mix}}\) estimates. 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.
The dependency of \(L_{\text{mix}}\) on the flow regime is detailed in
Fig. 7. Estimates of \(L_{\text{mix}}\) are averaged in bins of mean
flow speed and individually shown for the ACC frontal zones categorized
by Orsi et al. (1995; see Figs. 1 and 2). The frontal zones refer to the
dynamic topography data of Mizobata et al. (2020); the Subpolar Zone
(south of SACCF-S): < –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 GRP11 (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 from zero 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 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 dependent of flow speed in the Subpolar Zone,
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 associated with the near-boundary turbulent suppression
by the continental slope topography. We also confirmed that discussion
for the \(L_{\text{mix}}\) dependency on the flow speed unchanged in
case the inversion of suppression factor (Ferrari and Nikurashin, 2010)
is taken as the horizontal axis. In Fig. 7, inter-layer dependencies 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. Its difference from the
spiciness-based estimates is possibly due to the less distinctive
gradient of thickness than spiciness in ASW (Figs. 5 and 6). It should
be noted that the hydrographic variability method can yield\(L_{\text{mix}}\) and isopycnal diffusivity \(k\) quantitatively
consistent with the previous estimates, while the choice of the
isopycnal tracer \(\varphi\) occasionally affects the outcome and thus
requires some rationale (as considered in the next section).
To monitor the transition of \(L_{\text{mix}}\)’s controlling factor
towards the Antarctic margin, a histogram of \(L_{\text{mix}}\) is
plotted on \(\sigma(\varphi)\)–\(1/|\nabla\varphi|\) space (Fig. 8), in
which all coordinates are normalized by their averages, the isolines of\(L_{\text{mix}}=\) 20, 100 km are shown by white contours, and the
averaged diagnostics for each layer/method are marked by plus. The
poleward suppression of \(L_{\text{mix}}\) is readily observed by
comparing the positions of population and plus 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. Significant learning
drawn 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 is 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 \(L_{\text{mix}}\) signifies a
possibility to parameterize 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 general agreement with the previous studies in the ACC
domain, the diffusive parameters in the Antarctic margin are
investigated more closely. Using the mixing length formulation of the
equation (1), the isopycnal diffusivity \(k\) is calculated as the
product of mixing efficiency Γ, eddy velocity\(U_{\text{eddy}}\), and mixing length \(L_{\text{mix}}\). Fig. 9
provides diffusivity maps for CDW diagnosed by spiciness and thickness,
focusing on the Subpolar Zone. The climatological flow direction is
represented by the mean dynamic topography overlaid, and contours
characteristic to the subpolar circulation (–1.97 and –1.85 m) are
highlighted in blue. The isopycnal diffusivity \(k\) typically ranges
100–500 m2 s-1 in the Subpolar Zone
for both tracer variables, and \(k\) likely becomes small near the SB,
which shapes the transition zone from ACC to ASC. The spatial variation
of \(k\) within the Subpolar Zone seems attributable to the spatial
variation of \(L_{\text{mix}}\) (Figs. 5 and 6) rather than\(U_{\text{eddy}}\) (Fig. 2) and thus to the PV gradient change (as seen
in Fig. 8). To visualize the along-slope variability of \(k\), local
maps of thickness-based diffusivity are shown in Fig. 10 with the
isopycnal CDW temperature. Importantly, diffusivity is likely higher
where the onshore CDW intrusion occurs: 70°, 90°, 110°, and 120°E (these
intrusion pathways are documented in Yamazaki et al., 2020).
Additionally, enhanced diffusivity is observed in 140°E (Fig. 9), where
intervals between ACC and ASC become narrow and clockwise subgyres are
meridionally squeezed. The mechanism for this nontrivial correspondence
between the eddy diffusion and the onshore CDW intrusion will be
argumented in Section 5.4.
The spatial variation of \(k\) results from those of \(L_{\text{mix}}\)and \(U_{\text{eddy}}\), and its functional dependency varies in space.
Analogously to Fig. 8, a histogram of \(k\)in\(\ U_{\text{eddy}}\)–\(L_{\text{mix}}\) coordinates is plotted for
each layer and method (Fig. 11). In any frontal zone, neither of\(L_{\text{mix}}\) and \(U_{\text{eddy}}\) is a dominant controlling
factor as the population and the center of mass are located close to the
diagonal line. Still, we may state that \(k\) is more dependent on\(L_{\text{mix}}\) than \(U_{\text{eddy}}\) in the Subpolar Zone,
contrasting to the Southern and Antarctic Zones. The result supports the
aforementioned idea that the spatial scale of tracer gradient can
parametrize eddy diffusivity in the Antarctic margin via mixing length
formulation. This idea is further tested by Fig. 12, in which \(k\),\(L_{\text{mix}}\), and the inversed topographic gradient within the
Subpolar Zone are regressed onto the inversed tracer gradient\(1/|\nabla\varphi|\), coordinated with nondimensionalized axes, and the
scatters are colored by the altimetric mean velocity. Not surprisingly,
significant correlations of \(k\) and \(L_{\text{mix}}\) with\(1/|\nabla\varphi|\) are obtained (0.73 and 0.89 for spiciness; 0.81
and 0.92 for thickness, respectively). On the other hand, the
correlation between the topographic and tracer gradients is
insignificant for both tracers, implying that additional information is
required to derive the climatological tracer gradient from the
topographic data. Despite that controlling factors for the tracer
gradient field remain veiled, the present result is encouraging since it
allows us to predict eddy diffusion adequately if only we somehow
determine the gradient of isopycnal tracers.
Compared to the spiciness-based estimation, the correlation of the
thickness-based estimation with diffusivity is more statistically
significant. The higher correlation of thickness implies that the
thickness gradient better represents the PV gradient and the width of
the baroclinic zone than the spiciness gradient does. This result seems
quite reasonable provided that the ambient PV field is well approximated
by the isopycnal layer thickness within the Subpolar Zone, where the
flow condition is stagnant, and the relative vorticity likely becomes
small. Predicated on these facts, we proceed to estimate the diffusive
transport applying the thickness-based diffusivity to the isopycnal
thickness field.
4.3 Volume and heat transport
Assuming that the isopycnal thickness simply diffuses downgradient in a
GM-flux manner, we can estimate diffusive volume flux of CDW (Fig. 13).
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 the
thickness-based diffusivity, respectively. This is equivalent to the
layer-integrated bolus velocity (in m2s-1), and its horizontal integration gives a unit of
transport. The zonal eddy transport likely directs downstream in the lee
of topography and upstream in the other area (middle panel), indicative
of the internal form stress balance within the ACC (Marshall et al.,
2017). As a result of the thickness gradient, the volume transport
generally directs shoreward in the Subpolar Zone, as represented by the
transport vector direction and its meridional component (lower panel).
We can observe the poleward CDW transport continuously extending from
the eastern flank of the Kerguelen Plateau, where isopycnal eddies are
favorably generated, to the continental margin. Along-slope variation of
the meridional eddy transport is not so pronounced as \(k\) (Fig. 9),
and the most significant poleward CDW transport is obtained around
140°E. This is because the magnitude of transport is\(|\psi|=\ \Gamma U_{\text{eddy}}\sigma(H)\) by the equation (3) and
is not proportional to the inversed thickness gradient (whether CDW flux
becomes uniquely proportional to \(U_{\text{eddy}}\) is unclear even in
zonally-symmetric configuration regarding 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 of volume and heat (Fig. 14; over the 1000–3000 m
isobaths). Standard errors associated with the cross-slope variation are
shaded, within which heat flux change due to the along-slope temperature
variation safely falls. The gross onshore CDW transport is 0.39/0.12 Sv
(= m3 s-1) in the eastern/western
Indian sectors (divided by the Princess Elizabeth Trough
~90°E), respectively, translated to the onshore heat
fluxes of 3.6/1.2 TW. The interbasin contrast in thermal forcing seems
consistent with the stratification regimes inshore, represented by warm
Totten Ice Shelf and cold Amery Ice Shelf (Silvano et al., 2016).
Offshore transport of ASW to the west of 130°E is 0.15 Sv, balancing
with ~40 % onshore volume flux by CDW. On the contrary,
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 5.2, 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
likely holds in the Antarctic margin, given the weaker nonlinearity than
the ACC’s mainstream (Fig. 3).
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. This
infers quantitativeness of a series of previous estimates, in which
isopycnal tracers not necessarily dependent on PV have been adopted
(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. The flow dependency
of \(L_{\text{mix}}\) also likely varies by choice of tracer (Fig. 7).
These subtle contrasts generally pertain to the local difference in the
tracer gradient, as the large diffusivities likely result from the weak
tracer gradient. We found that the thickness gradient better represents
the variations of \(L_{\text{mix}}\) and \(k\) than the spiciness
gradient (Fig. 12) attributable to the PV-conservative nature of
isopycnal thickness. The thickness-based \(L_{\text{mix}}\) and \(k\)rationalize the calculation of thickness-diffusive transport, accounting
for the residual overturning theory (Marshall and Radko, 2003).
Although the estimated diffusivity of 100–500 m2s-1 is significantly smaller than the along-slope
estimation of 950 ± 400 m2 s-1presented by FRE19 (based on spiciness variability), their estimate
implicitly assumed the mixing efficiency \(\Gamma\) to be unity (far
exceeding its previous estimates; 0.01–0.4) and hence seems
incompatible as an absolute diffusivity estimation. In case\(\Gamma=0.16\) by Wunsch (1999) is consistently applied for their
values, the isopycnal diffusivity of 90–220 m2s-1 is obtained from their result, rather smaller\(k\) than our estimate. Further, our estimation is quite consistent
with previous studies in the ACC’s mainstream, typically ranging for
500–2000 m2 s−1 (Marshall et al.,
2006) and 1500–3000 m2 s−1 (Sallée
et al., 2011) with a poleward decrease.
To investigate the meridional overturning circulation across the ASC
jets in zonally symmetric configuration, Stewart and Thompson (2016)
conducted idealized numerical experiments. They demonstrated that\(L_{\text{mix}}\) scaled by the slope width accurately predicts the
simulated onshore flux of CDW (R2 = 0.89). However, we
found that \(L_{\text{mix}}\) is significantly correlated with the
thickness gradient but not with the topographic gradient (Fig. 12). This
dissociation with the topographic slope scale may be interpreted because
of thickness control by the surface layer, expected from shelf water
export in the clockwise subgyres (Yamazaki et al., 2020). We assume that
the thickness field itself is strongly connected to the zonally
asymmetric structures of circulation and topography (see Section 5.4).
The inaccessible but most uncertain part of our estimate is the spatial
variability of mixing efficiency. Visbeck et al. (1997) argumented that
eddy transfer coefficient, which determines the proportionality of
diffusivity to the horizontal/vertical stratification and the width of
baroclinic zone, is a universal constant (equal to 0.015) regardless of
flow regime. Mixing efficiency is different from this coefficient by its
formulation, but they are possibly associated with each other. Validity
of \(\Gamma=0.16\) is dependent on, let alone mooring data analyzed in
Section 3.3, discussion by Klocker and Abernathey (2014) that\(\Gamma=0.15\) is suitable for the tracer-based mixing length
calculation to be consistent with diffusivity by altimetric eddy scale.
Examination for its universality is a future task and requires a utility
of numerical models. Although the spatial variation of \(\Gamma\) can
alter the correspondence between the enhanced diffusivity and the CDW
intrusions (Fig. 10), the presented result leastwise suggests that
mixing length is large where CDW intrudes shoreward. Furthermore, it is
presumable that its spatial variation is negligible when considering a
basin-wide transport as in the next section.
5.2 Coastal transport and heat budgets
The estimated onshore heat/volume flux (Fig. 14) is quantitatively
consistent with the previously reported coastal budgets. As for
integration within the eastern Indian sector (90–160°E), the
annually-cumulative sea ice production is 520 ± 75 km3(Tamura et al., 2016; a sum of Shackleton, Vincennes, Dalton, Dibble,
and Mertz Polynyas), being translated to heat loss of 4.2–5.6 TW. The
integrated ice shelf basal melt rate is 198 ± 39 Gt
yr-1 (Rignot et al., 2013; a sum of Mertz, Dibble,
Holmes, Moscow Univ., Totten, Vincennes, Conger, Tracy, and Shackleton
Ice Shelves), being translated to 1.7–2.5 TW. Therefore, the CDW heat
flux of 3.2–3.9 TW (within the 28.0–28.1 kg m-3neutral density) compensates for nearly half of the cryospheric heat
sink 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 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; Fig. 14).
As connectivity of the on-shelf current over the Antarctic coastline is
likely weak in the Indian sector (Dawson et al., 2021), the volume
imbalance between CDW and ASW implies the local exporting volume of
Antarctic Bottom Water. In this sense, the partial intrusion of ASW to
the east (Fig. 14) is likely consistent with the intensive bottom water
formation in the Adelie/Mertz region (Williams et al., 2010), which
might be ~0.3 Sv on the annual mean basis (from a
numerical simulation by Kusahara et al., 2017). To the west of 130°E,
the ASW export only balances with ~40% of the CDW
influx, so that the remaining volume (~0.2 Sv) may be
attributable to the bottom water export in Vincennes Bay (Kitade et al.,
2014), Cape Darnley (Ohshima et al., 2013), and the rest of minor
formation sites. The CDW volume compensation for the bottom water export
can be numerically simulated over the circumpolar domain (Morrison et
al., 2020). This study provides the first observational implication for
the phenomenon with the quantitative estimation of the coastal
heat/volume budgets.
Results by Stewart and Thompson (2016) indicate a possibility to
underestimate the onshore heat flux derived from the mixing length
formulation solely based on the thickness-diffusive CDW flux (likely
corresponding to “eddy advection”), as the isopycnal “eddy stirring”
can also contribute to the heat flux without transporting water volume,
especially near the shelf break. The remarkable heat budget closure
pertains to the situation that, compared to the eddy advection, the eddy
stirring and tidal mixing are not dominant over the targeted slope
(1000–3000 m; Fig. 14), as indicated by a realistic simulation (Stewart
et al., 2018), and most of the heat flux explained by eddy advection
over the isobaths subsequently reaches the Antarctic coast beyond the
shelf break. On the other hand, the poleward CDW transport by the
cross-slope geostrophic current (measured in seaward of the 3000 m
isobath; Mizobata et al., 2020) might be confined to the lower
continental slope, consistent with the numerical model (Stewart et al.,
2018) and the weak shoreward advection of profiling floats (Yamazaki et
al., 2020).
5.3 Diapycnal fluxes
The divergence of \(\psi\) is also calculated to evaluate the diapycnal
flux in the Antarctic margin (Fig. 15, top panel). It can be decomposed
into the thickness squeezing term and the symmetric diffusion term:
\(\nabla\cdot\psi\ =\ -\nabla k_{H}\cdot\nabla H-\ k_{H}\nabla^{2}H\ \ \ \ \ \ \ \ \ \ (6)\),
and both are explicitly computable (Fig. 15; middle and bottom panels,
respectively). Since \(\nabla k_{H}\) likely reflects the spatial
variation of \(k_{H}\) at the upper surface (28.00 kg
m-3) rather than a tranquil deeper layer, divergent
(convergent) thickness squeezing \(-\nabla k_{H}\cdot\nabla H\) can
be interpreted as upward (downward) diapycnal flux through the upper
surface (left panel of Fig. 15). Likewise, since \(\nabla H\) likely
reflects its variation at the lower surface (28.10 kg
m-3) rather than undulation of shallower isopycnals,
divergent (convergent) symmetric diffusion \(-\ k_{H}\nabla^{2}H\) can
be interpreted as downward (upward) diapycnal flux through the lower
surface. These ideas are translated to the diapycnal velocity over the
Subpolar Zone; the net convergence of 1.0 ± 11 µm s-1for the CDW density (28.00–28.10 kg m-3) is
decomposed into upward diapycnal fluxes of 1.8 ± 16 µm
s-1 (at the upper surface) and 2.8 ± 21 µm
s-1 (at the lower surface). Even though the spatial
variability is quite large, these averaged values are very comparable to
Ekman upwelling of ~2 µm s-1 typical
in the Antarctic margin (Liang et al., 2017). This agreement might
further underpin the quantitativeness of our estimation.
The net upward diapycnal flux due to the symmetric diffusion term\(-\ k_{H}\nabla^{2}H\) may be a manifestation of the convex curvature
of the lower isopycnal (Fig. 15, right panel). On the other hand, the
net upward diapycnal flux by the thickness squeezing term\(-\nabla k_{H}\cdot\nabla H\) can be interpreted due to the seaward
gradient of \(k_{H}\). It is attributable to the gradual inclination of
upper isopycnal from the SB to the continental shelf, since \(k_{H}\) is
highly correlated with the magnitude of thickness gradient (Fig. 12).
This situation is checked by the fact that\(-\nabla k_{H}\cdot\nabla H\) tends to be positive to the south of
SB (Fig. 15, middle panel). These arguments imply that, even though the
isopycnal gradient is well correlated with that of topography (Yamazaki
et al., 2020), the spatial distribution of CDW thickness is not simply
determined by the structure of topography but also by the interface
between CDW and ASW, so their discordance encountered in Fig. 12 appears
to be reasonable.
The divergence of isopycnal eddy advection indicates the net upward
diapycnal fluxes through the upper and lower surfaces of the CDW layer
(Fig. 15, right panel). The net diapycnal upwelling seems consistent
with the kinematic analysis of the layer thickness, in which both
thickness squeezing and symmetric diffusion terms are controlled by
thickness Laplacian as the isopycnal diffusivity \(k\) is highly
dependent on the inversed thickness gradient (Fig. 12). These diapycnal
fluxes are likely significant for modifying CDW along the isopycnal
pathway over the continental slope, controlling the property of modified
CDW inshore. Furthermore, the local diapycnal upwelling can explain why
isopycnal/temperature surfaces tend to be shallow where the CDW
intrusion occurs (Yamazaki et al., 2020; see their Figs. 8 and 10). Its
vertical position relative to the topography is critical to whether the
CDW isopycnal is bridged to the continental shelf. On the other hand,
offshore advection of ASW might also play a crucial role in the vertical
adjustment of CDW, and thus the effect of diapycnal flux needs to be
further evidenced. Even though a basin-scale upwelling is naturally
expected from the divergent wind stress in the Subpolar Zone, the
presented result is valuable as an observational estimate of the
climatological diapycnal flux, possibly demonstrating its spatial
variation associated with the circulation and topography.
5.4 Circulation and eddy fluxes
We found that \(L_{\text{mix}}\) and \(k\) are likely large where the
onshore CDW intrusion is localized (Fig. 10), indicating that the
onshore CDW intrusion is achieved by the cross-slope eddy advection.
Upon this result, we can speculate how the CDW intrusion is established.
First, the recirculating gyres steered by the barotropic PV and
horizontal shear between the ACC and ASC determine the location of
shoreward intrusion. Offshore CDW then approaches the continental slope
advected by a quasi-barotropic flow branched from the ACC. Due to the
ambient PV constraint, the mean flow cannot reach the upper slope
(inshore of ~3000 m); instead, this encroachment
steepens the gradient of CDW isopycnal. The steepened isopycnal locally
causes baroclinic instability, and, subsequently, the generated eddies
facilitate the onshore eddy advection.
The explanation in the previous paragraph should be rationalized along
with two facts: (i) the magnitude of \(\psi\) (i.e.,\(\Gamma U_{\text{eddy}}\sigma(H)\)) is independent of \(\nabla H\) and
(ii) \(L_{\text{mix}}\) and \(k\) are “inversely” proportional to\(\nabla H\) by the equations (1) and (3). The latter suggests that the
inversed thickness gradient is associated with isopycnal tracer
fluctuations and \(U_{\text{eddy}}\) (in short, “ability to mix”)
rather than the geostrophic stability. This infers that a steep
thickness gradient is associated with the strong mean flow and likely
prevents cross-frontal eddy transport, as schematized in Fig. 16. On the
other hand, the former infers that the spatial variation of eddy
transport \(\psi\) reflects that of tracer fluctuations and\(U_{\text{eddy}}\) rather than the \(\nabla H\) field. Since\(U_{\text{eddy}}\) likely has a minor effect on the spatial variation
(accounting for Fig. 11), we may interpret the large shoreward eddy
transport in the intrusion sites as manifesting a large \(\sigma(H)\)and \(L_{\text{mix}}\), as suggested by FRE19. The baroclinic eddy
generation accompanied by the cross-slope CDW flux is expected to occur
intermittently, with a gentle thickness gradient on average. In
contrast, the sharp thickness gradient is likely associated with a
baroclinically stable part of the ASC, hence unintrusive (Fig. 16). This
situation may be noticed by comparing the CDW thickness and isotherm
(Yamazaki et al., 2020; their Fig. 10), where thicker CDW and its
smaller gradient can be observed in the intrusion sites. The situation
illustrated in Fig. 16 implicates that the ASC behaves as a barrier to
the onshore CDW intrusion. The dynamical driver governing the thickness
field remains unknown, yet we posit that ASW’s property, as well as
topographic steering, plays an indispensable role.
6. Conclusion
To investigate the controlling factor of onshore CDW intrusion across
the Antarctic continental slope, the present study conducted an
extensive analysis of hydrographic measurements and the satellite
altimetry data taking advantage of the mixing length formulation. The
spiciness/thickness-based estimations yielded qualitatively similar
results, supporting the fidelity of the mixing length estimates
previously made using hydrographic variability. The same analysis is
applied for ASW, and its mixing length close to CDW was obtained. Over
the ACC domain (Antarctic and Southern Zones), a general agreement with
the mixing suppression theory and its exception in the lee of the
topography is found, as previously reported (GFP11). In contrast, no
mixing length’s dependency on mean flow is obtained in the Subpolar
Zone, reflecting a stagnant flow regime in the Antarctic margin. Eddy
diffusion is likely enhanced where the CDW intrusion is localized by the
recirculating gyres, which are steered by the barotropic PV (i.e.,
topography). This correspondence is primarily attributable to the
spatial variation of diffusivity controlled by the isopycnal thickness
gradient, and the gentle thickness gradient allows for ease of isopycnal
mixing. Volume transport is estimated in a GM-flux manner, and it is
shown that thickness-diffusive onshore heat flux over the continental
slope is quantitatively consistent with cryospheric heat sinks (sea ice
formation and ice shelf basal melt), suggesting that the isopycnal eddy
advection is the main factor of the onshore CDW intrusion. Upward
diapycnal fluxes across the CDW isopycnals are indicated by kinematic
analysis of eddy flux divergence, in which thickness squeezing and
symmetric diffusion terms cause upward fluxes in the upper and lower
surfaces, respectively. The estimated diapycnal flux is broadly
consistent with upwelling predicted by cyclonic wind stress, further
supporting our quantification. Predicated on these findings, the
mechanism of eddy flux localization is speculated, and the controlling
factors of the onshore CDW intrusion are illustrated in Fig. 16. Our
findings may break new ground on the Southern Ocean dynamics, in which a
connection between the meridional overturning circulation and the
coastal buoyancy budget has been hypothesized (e.g., Snow et al., 2016;
Morrison et al., 2020).
As a concluding remark, we underscore that the isopycnal thickness field
is essential for determining the eddy fluxes in the Antarctic margin.
The presented results facilitate a possibility to predict the eddy
diffusivity by solely determining the layer thickness. This idea might
be valuable for simulating CDW transport in global climate models, where
subgrid effects of eddy fluxes need to be parameterized. Detailed
reproduction of eddy flux is substantial for the multidecadal
variability of onshore CDW flux (Yamazaki et al., 2021) and is
inevitable for climate projection with higher credibility.