Olb2004a

Olb2004a
Antarctic Science 16 (4): 439–470 (2004) © Antarctic Science Ltd Printed in the UK
DOI: 10.1017/S0954102004002251
The dynamical balance, transport and circulation of the Antarctic
Circumpolar Current
DIRK OLBERS1*, DANIEL BOROWSKI1, CHRISTOPH VÖLKER1 and JÖRG-OLAF WOLFF2
1
Alfred Wegener Institute for Polar and Marine Research, D-27515 Bremerhaven, Germany
2
ICBM, University of Oldenburg, D-26111 Oldenburg, Germany
*corresponding author: [email protected]
Abstract: The physical elements of the circulation of the Antarctic Circumpolar Current (ACC) are
reviewed. A picture of the circulation is sketched by means of recent observations from the WOCE decade.
We present and discuss the role of forcing functions (wind stress, surface buoyancy flux) in the dynamical
balance of the flow and in the meridional circulation and study their relation to the ACC transport. The
physics of form stress at tilted isopycnals and at the ocean bottom are elucidated as central mechanisms in
the momentum balance. We explain the failure of the Sverdrup balance in the ACC circulation and highlight
the role of geostrophic contours in the balance of vorticity. Emphasis is on the interrelation of the zonal
momentum balance and the meridional circulation, the importance of diapycnal mixing and eddy processes.
Finally, new model concepts are described: a model of the ACC transport dependence on wind stress and
buoyancy flux, based on linear wave theory; and a model of the meridional overturning and the mean density
structure of the Southern Ocean, based on zonally averaged dynamics and thermodynamics with eddy
parametrization.
Received 22 December 2003, accepted 1 September 2004
Key words: ACC, Southern Ocean, form stress, meridional overturning, transient and standing eddies
1. Introduction
Roughly 75% of the world ocean volume has temperatures
below 4°C, connected (at polar latitudes) with only 2% of
the ocean surface. Palaeoceanographic data have revealed
that this was not always the case. Before Drake Passage
opened due to continental drift about 30 Myr ago the
climate of the ocean was considerably warmer. In the course
of the establishment of the Southern Ocean in its present
shape the difference between surface and bottom
temperatures in equatorial regions changed from about 7°C
to its present value of about 26°C (e.g. Berger 1981). The
polar climate of the Southern Hemisphere became
increasingly colder by the growth of glacial ice on the
Antarctic continent and the gradual development of the sea
ice cover around it, leading to the formation of deep cold
water masses propagating as Antarctic Bottom Water
(AABW) to the adjacent northern ocean basins. The
opening of Drake Passage also established the strongest and
longest current system in the world ocean, the Antarctic
Circumpolar Current (ACC), with a volume transport of
roughly 130–140 Sv (1 Sv ≅ 106 m3 s-1), extending around
the globe with a length of roughly 24 000 km. As the most
important link between the ocean basins of the Atlantic,
Pacific and Indian Oceans, the ACC serves as a conduit of
all active and passive oceanic tracers which affect Earth’s
climate, notably heat and salt which strongly influence the
oceanic mass stratification, circulation and consequently the
ocean heat transport, and the greenhouse gas carbon
dioxide. But in contrast to this strong zonal exchange
Fig. 1. Schematic map of major currents in the southern
hemisphere oceans south of 20°S. Depths shallower than
3500 m are shaded. The two major cores of the Antarctic
Circumpolar Current (ACC) encircling Antarctica are shown:
the sub-Antarctic Front and Polar Front. F = front, C = current
and G = gyre. From Rintoul et al. (2001).
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Fig. 2. Magnitude of the mean sea surface temperature gradient, from 44 months of observations by the Along-Track Scanning Radiometer
on the ERS-1 satellite. Superposed are positions of (from north to south) the Subtropical Front, sub-Antarctic Front, Polar Front, South
ACC Front, and southern boundary of the ACC, taken from Orsi et al. (1995). From Hughes & Ash (2001).
brought about by the deep reaching and strong zonal current
these same characteristics of the ACC act to limit
meridional exchange and tend to isolate the ocean to the
south from heat and substance sources in the rest of the
world ocean.
The ACC system is sketched in Fig. 1 by its major fronts.
These are traced by the regionally (and temporally, see
Section 3) highly variable surface temperature gradient
displayed in Fig. 2, which shows that the ACC is a
fragmented system of more or less intense jet streams. The
thermal fronts have a close correspondence in density and
extend to depth, in most places to the bottom (see Fig. 3),
but can also be correlated with surface elevations as
detected in satellite altimetry data (e.g. Gille 1994). The
ACC resides mainly in the two circumpolar fronts, the sub-
Antarctic Front and the Polar Front, which, due to regional
and temporal variability, appear as multiple branches in the
hydrographic section of Fig. 3.
From Fig. 3 it becomes apparent that watermass
properties do penetrate across the ACC and, in fact, there is
a prominent meridional circulation associated with the
predominantly zonal ACC. It was described as early as 1933
by Sverdrup (see also Sverdrup et al. 1942) and has lately
been interpreted as the Southern Ocean part of the ‘global
conveyor belt’ circulation (Gordon 1986, Broecker 1991,
Schmitz 1995). The ‘sliced cake’ view of the Southern
Ocean watermasses and their propagation shown in Fig. 4
(Gordon 1999) presents this overturning circulation by the
distribution of salinity and temperature of Antarctic
Intermediate Water (AAIW), Circumpolar Deep Water
Fig. 3. Properties versus pressure along the WOCE SR3 repeat section between Australia and Antarctica (≈ l40°E). Left: potential
temperature (°C; contour interval is 1° for Θ > 3°C, and 0.2° for Θ < 2.6°). Right: salinity (on the practical salinity scale, contour interval
is 0.1 for solid contours, and 0.02 for dashed contours). From Rintoul et al. (2001).
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
441
Fig. 4. A ‘cake’ with large slice removed view of the Southern Ocean. Isotherms of the annual average sea surface temperature (SST °C) are
shown on the plane of the sea surface. The core of the eastward flowing ACC and associated polar front occurs near the 4°C isotherm. The
right plane of the slice shows salinity (S). These data are derived from the oceanographic observations along the Greenwich Meridian
shown on the floor of the figure (dots, from Fahrbach et al. 1994). Deep relatively saline water, > 34.7 (Circumpolar Deep Water CDW,
arrow) spreads poleward and upwells towards the sea surface. It is balanced by a northward flow of lower salinity waters, < 34.4 near
1000 m (Antarctic Intermediate Water AAIW, arrow) and by sinking of slightly lower salinity water along the continental slope of
Antarctica (arrow). This process (salty water in, fresher water out) removes the slight excess of regional precipitation from the Southern
Ocean. Along the left plane is temperature (T°C) based on data collected at the same points as used for the salinity section. Shallowing of
the isotherms is evident as the deep water rises up towards the sea surface. There it is cooled and sinks flooding the bottom layers with
waters of less than 0°C. This cold bottom water spreads well into the global oceans (Antarctic Bottom Water AABW). Along the outer
edge of the figure, latitude 35°S, is salinity (S). The low salinity water (AAIW) is shown as the less than 34.45 band near 1000 m. More
saline deep water is seen spreading southward near the 4000 m depth. From Gordon (1999).
Fig. 5. Left: section integrated (south to north) baroclinic transport relative to the deepest common level for SR1 (Drake Passage) and SR3
(Australia to Antarctica) from various hydrographic section along SR1 and SR3 between 1991 and 2001. The location and names for the
fronts are according to Orsi et al. (1995). Right: transport in neutral density classes. For SR3 the data of the various surveys are shown as
bins, for SR1 only the mean transport profile of the cruises is given (full line). From Cunningham et al. (2003) and Rintoul & Sokolov
(2001).
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D. OLBERS et al.
(CDW) and AABW. The figure also depicts the circulation
and nicely illustrates one of the central questions of
Southern Ocean science: how do water properties (heat,
salt, nutrients and other chemical substances) cross the
strong deep reaching ACC? We will address this question of
mass and property balances in Section 5, which is closely
connected to the question of the dynamical balance of the
ACC (treated in Section 3): what are the forces driving the
zonal current, which act as brake and what physical
mechanisms are responsible for the deep reaching current
profile?
It should be borne in mind that the answer to these
questions will not necessarily give insight into the problem
of the relation of the magnitude of the zonal transport of
water by the ACC to the flux of (zonal) momentum through
the system (as the balanced transfer of money to and from a
bank account does not determine the account balance). We
discuss the dependence of transport on forcing functions
and present a new simple linear transport model (Section 4).
In this paper we review some of the concepts and theories
which are currently discussed for the circulation of the
Antarctic Current system. It expands and complements
other recent reviews of the ACC system (e.g. Olbers 1998,
Rintoul et al. 2001) and summaries of the ACC dynamics
contained in original research articles (e.g. Gnanadesikan &
Hallberg 2000, Tansley & Marshall 2001). But we should
point out that this review does not cover all the current
research on the ACC, e.g. we do not address regional
properties of the ACC system and its temporal variability;
we not report on teleconnections and links of the Southern
Ocean with the global ocean circulation and the possible
dependence of the stratification and transport of the ACC on
remote conditions and mechanisms.
2. The zonal transport
The meridional momentum balance of the ACC is basically
geostrophic, i.e. the zonal current velocity (at each
geopotential level) is related to the meridional pressure
gradient, resulting from a dip of about 1.5 m (from north to
south) of the sea surface across the current system, and the
gradient of density in the fronts as, for example, can be
inferred from temperature and salinity in the SR3 section in
Fig. 3. The surface pressure gradient yields an overall
eastward surface velocity and the mass stratification yields
a positive shear (ug)z = (g/f)ρy of the geostrophic part of the
current1, the velocity thus diminishes with depth but
generally not as strongly as to imply a reversal of the flow.
The above ‘thermal wind relation’ is utilized to infer from
hydrographic section data the ‘baroclinic’ transport (normal
to the section and referred to a common depth) or the DCL
transport (referred to the bottom depth or deepest common
1
We work with the Boussinesq approximation. Density, pressure and
stresses are divided by a constant reference density.
level (DCL) for station pairs) of the ACC. Various attempts
have been made to determine the absolute or net transport
by taking reference velocities from moorings or LADCP
(Lowered Acoustic Doppler Current Profiler) or by
levelling bottom pressure gauges (see the recent discussion
of Cunningham et al. 2003). Prior to WOCE most efforts
were made in the International Southern Ocean Studies
(ISOS) experiment at Drake Passage (Whitworth 1983,
Whitworth & Peterson 1985). More recently, transport
estimations have been made at Drake Passage (WOCE SR1)
and the section between Australia and Antarctica (WOCE
SR3, see Fig. 3) at 140°E, where multiple surveys have been
made during WOCE.
The average DCL transport of SR1 for six hydrographic
sections across Drake Passage (see Fig. 5) is 136.7 ± 7.8 Sv,
with about equal contributions from the Polar Front (57.5 ±
5.7 Sv) and the sub-Antarctic Front (53 ± 10 Sv)
(Cunningham et al. 2003). The analysis of ISOS and WOCE
data (spanning 25 years) gave no indication of significant
trends or unsteadiness. Following Rintoul & Sokolov
(2001) the mean transport south of Australia at SR3 is 147 ±
10 Sv (relative to a ‘best guess’ reference level: at the
bottom except near the Antarctic margin, where a shallower
level is used consistent with westward flow over the
continental slope and rise). It is about 13 Sv larger than the
ISOS estimate of absolute transport through Drake Passage
and about 10 Sv larger than the SR1 DCL transport (see
Fig. 5). The transport south of Australia must be larger than
that at Drake Passage to balance the Indonesian
throughflow, which is believed to be of order 10 Sv.
However, given the remaining uncertainty in the barotropic
flow at both locations, the agreement is likely to be
fortuitous. Variability of transport at SR3 has been detected
in a six year record of repeat hydrographic section (Rintoul
et al. 2002). It is fairly small (1–3 Sv). Figure 5 shows also
the contribution of transport in the main classes of
watermasses. In both sections the CDW range carries most
of the zonal transport and no systematic temporal change of
the relative contribution could be detected.
Monitoring the transport through Drake Passage is by
now a standard diagnostic of numerical global ocean
models. The resulting values are spread over a large range
from well under 100 Sv to well over 200 Sv. The reasons for
this diversity are not fully understood. But wind forcing and
thermohaline processes (Cai & Baines 1996, Gent et al.
Table I. Volume transport of the ACC, diagnosed from some eddy
permitting and eddy resolving ocean-only models.
Model
FRAM
SC 92
SC 92
POP 11
Resolution
ACC transport Reference
1/4° x 1/2°, 32 levels
1/2°, 20 levels
1/4°, 20 levels
roughly 10 km in
Drake Passage, 20 levels
OCCAM
1/4°, 36 levels
195 Sv
180 Sv
163 Sv
136 Sv
FRAM Group 1991
Semtner & Chervin 1992
Semtner & Chervin 1992
Maltrud et al. 1998
152 Sv
Fox et al. 2000
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
2001), model resolution and the representation of
topography (Best et al. 1999), as well as the parametrization
of subgrid scale tracer fluxes (e.g. Danabasoglu &
McWilliams 1995, and Fritzsch et al. 2000) are known to be
important factors. Most eddy resolving models (or
‘permitting’ since the achieved resolution does not resolve
all of the relevant eddy scales) yield transport values closer
to but mostly above the observations (see Table I).
Integrating the thermal wind balance (ug)z = (g/f)ρy twice
vertically we get
0
∂h g ∂ 0
gh
U g = ∫ u g dz = hu g ( z = − h) − ρ ( z = − h) −
zρdz
−h
∂y f ∂y ∫− h
f
(1)
for the geostrophic transport (per unit length along the
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section). The geostrophic velocity at the bottom can be
expressed by the gradient of pressure taken at the bottom,
ug(z = -h) = - (py)-h/f, which can be combined with the
second term to yield
U g = −h
∂pb ∂χ
−
∂y ∂y
(2)
In this relation the geostrophic transport is expressed by the
gradients of bottom pressure pb = p(z = -h) and the density
0
moment
χ = g ∫ zρdz
which is the total baroclinic potential
−h
energy (referred to z = 0). But Ug is not the total transport,
which is the integral of absolute velocity from top to
bottom. The total volume transport through a section also
contains the Ekman transports (due to wind stress and
Fig. 6. Streamfunction ψ [upper panel] and reconstruction of the streamfunction with (4) [middle panel]. The data are from a global OGCM
with variable resolution (1° x 2° at Drake Passage latitudes, 25 levels, MOM code). Geostrophic contours f/h are shown in the lower panel.
There are two prominent regions where geostrophic contours are blocked by continents: the region between Australia and Antarctica, and
Drake Passage between South America and Antarctica. Here, the ACC must cross geostrophic contours. Although the geostrophic
contours are not blocked there, the ACC also crosses the geostrophic contours at the East Pacific Rise and several other locations. From
Borowski et al. (2002).
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D. OLBERS et al.
frictional bottom stress) normal to the respective section
and other contributions induced by nonlinearities and lateral
friction. In large-scale currents the latter two are usually
small and for north–south sections the Ekman contributions
can be neglected (for predominantly zonal winds). But
turning from the section coordinate y to the general case we
add to the geostrophic transport the Ekman parts and get the
expression for the total transport U in vector form,
fk × U = − f∇ψ = − h(∇p ) − h − ghρ ( z = − h)∇h − ∇χ + τ 0 − τ b
= − h ∇ pb − ∇ χ + τ 0 − τ b
(3)
The total transport conserves mass and is thus representable
by a streamfunction ψ. With the approximation made so far
(neglecting lateral stresses and nonlinearities) the above
equation is the vertically integrated balance of momentum.
The relative importance of the different pressure gradient
contributions in Eq. (1) or (2) to geostrophic transport has
been addressed by Borowski et al. (2002). They argue on
the basis of the balances of barotropic momentum and
vorticity that the deep transport h(Lp)-h in Eq. (3) across
geostrophic contours f/h should be small if these are
blocked by continents (as in Drake Passage and other places
in the path of the ACC, see lowest panel of Fig. 6). Then,
neglecting the variation of f on the lhs of Eq. (3) and the
deep and the Ekman transports on the rhs and integrating
along a section of constant bathymetry h = const, we find
that the transport normal to such a contour is related to the
difference of baroclinic potential energy between the ends,
i.e. f0∆ψ ~ ∆χ, a relation which we encounter again in
Section 3.5. In models with simplified geometry such
conditions can easily be established. In a series of
experiments with zonal channel geometry (see Fig. 20), but
also in global coarse resolution OGCMs, Borowski et al.
(2002) could grossly verify the relation
∂ψ ∂χ
∂h ∂χ
=
+ ghρ ( z = − h)
≈
∂y ∂y
∂y ∂y
(4)
between the meridional gradients of streamfunction and
potential energy. Figure 6 compares the transport pattern
(upper panel) and its reconstruction via Eq. (4) (middle
panel) in a global coarse resolution OGCM. While there are
clear differences in the closed basins of the major oceans,
the overall agreement of the streamfunction and its
reconstruction is rather good within the ACC region (ψ and
its reconstructed values from the gradient of potential
energy coincide within 10%). By and large the contribution
from the bottom pressure gradient to the transports is thus
small.
circulation has already been mentioned. But the zonality
also acts as a brake. In the basins which are zonally blocked
by continents there is a meridional exchange of heat
accomplished by the time mean gyre current systems. There
is no such mean transport of heat across the latitudes of the
ACC (DeSzoeke & Levine 1981). Instead, the loss of heat
from the ocean in the area south of the ACC must be carried
across the current by smaller-scale and/or time varying
features in the current field, usually summarized as the
meso-scale eddy field. Transient eddies with scales tens to a
few hundred kilometres (much larger than the baroclinic
Rossby radius which is of order 10 km in the Southern
Ocean, Houry et al. 1987) are a very prominent feature
along the path of the ACC. There are also small stationary
features, sometimes attached to outstanding topographic
peculiarities, which scale in the category mesoscale eddies.
With exception of the western boundary currents in the
subtropical gyres the variance of transient features in the
ACC dominates the global distribution of variability of
surface displacement obtained from satellite altimetry,
particular in areas of shallow rough topographic obstacles
(e.g. Gille et al. 2000) and meridional excursions in the path
of the flow. Estimates of the meridional eddy heat flux from
a number of moored instruments confirmed the southward
transfer with sufficient magnitude to close the overall heat
budget (see Fig. 7). Recently Gille (2003) confirmed this
a.
b.
f
3. What is so special about the dynamics of the ACC?
The zonal periodicity of the Southern Ocean, creating a
circumpolar pathway of watermasses to circle the globe and
allowing the ACC to play a major part in the conveyor belt
Fig. 7. Profiles of cross-stream eddy heat flux observed in the
AUSSAF experiment. a. four AUSSAF moorings, symbols
indicate statistical significance (triangle 95%, circle 90%),
b. representative profiles from all available SAF sites: AUSSAF
(circles), south-east of New Zealand (SENZ: Bryden & Heath
1985, triangles), northern Drake Passage (NDP: Nowlin et al.
1985, squares). From Phillips & Rintoul (2000). [geog:
northward velocity is used, shear: velocity normal to mean
current shear is used; the correlations are either determined from
all frequencies or from a bandpass covering the eddy
frequencies].
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
445
Fig. 8. The sea surface height in the Southern Ocean of a
simulation with the POP ocean model. Contour interval is 0.1 m.
The contours between -1.0 m and 0 m are as solid lines. The plot
(above) shows the mean latitude associated with the SSH
contours. These latitudes are used in Fig. 9 as lateral coordinate.
enormous extrapolation from point observations to the
circumpolar area by determining the eddy heat flux from a
combination of climatological hydrographic data and the
ALACE (Autonomous LAgrangian Circulation Explorer)
float trajectories tracing out the Southern Ocean in the last
decade.
3.1. Transient and standing eddies
In the description of the global atmospheric circulation it is
custom to reduce the information contained in observations
by considering zonally averaged time mean fields and
deviations from it (see e.g. Peixoto & Oort 1992), e.g. the
meridional velocity v is split into its zonal-plus-time mean
v and deviation v* so that v = v + v*. The mean
meridional heat flux (divided by ρcp) is then
vT = v T + v *T * which identifies a flux achieved by the
mean fields and a flux induced carried by the covariance of
the deviation fields (the ‘eddies’). Clearly, the combined
zonal-plus-time mean of v* vanishes, i.e. ν * = 0, but the
Fig. 9. Comparison of zonally averaged fields [upper panels] and mean fields constructed from an ACC path following averaging procedure
[lower panels]. The left panel displays the mean current, the middle and right panels display the transient and standing components of the
potential density flux (note that contour intervals are different). The heavy lines in the upper panels mark the latitudes of Drake Passage.
The path following average fields is displayed with respect to the mean latitudes of the corresponding SSH contours, as given in the plot in
Fig. 8. Units: velocity in ms-1, density fluxes in m2 s-1.
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D. OLBERS et al.
time mean alone does not vanish. Denoting the temporal
average by cornered brackets we get a separation into a time
mean (‘standing eddy’) component and the time varying
(‘transient eddy’) component, v* = +v*, + v’, and likewise
for the ‘eddy’ heat flux, v * T * = v * T * + v' T ' with the
contributions from standing and transient eddies.
Motivated by the zonal unboundedness of the ACC this
separation of fields and covariances has been applied to data
(from models since synoptic maps of ACC properties do not
exist) for the belt of latitudes passing Drake Passage (e.g.
Killworth & Nanneh 1994, Stevens & Ivchenko 1997,
Olbers & Ivchenko 2001). We elucidate the typical results
of zonal averaging using results from the global ocean POP
model (Parallel Ocean Program, see Maltrud et al. 1998)
which marginally resolves the transient eddy field with a
resolution of roughly 6.5 km in polar latitudes. The time
mean sea surface topography of the simulation is shown in
Fig. 8, revealing a quite realistic ACC (compare to Fig. 2) as
a collection of strong, regionally bounded jets which break
up at topographic features and in summary pass Drake
Passage but do not at all follow the corresponding belt of
latitudes. Consequently, in the zonally averaged picture
(Fig. 9 upper panels) many details of the ACC current
system are lost. The averaged state in the Drake Passage
band of latitudes is picked from the stronger features at the
southern rim of the ACC in that latitude interval and thus
misses most of its circumpolar structure. Moreover, the
average does not at all represent the local structure of the
current in Drake Passage. The transport of the ACC through
Drake Passage is 130 Sv in this POP simulation, which is
very close to observations. In contrast, the transport of the
zonally averaged current is only about 50 Sv.
Most of the ACC actually finds its representation in the
averaged picture at latitudes north of the Drake Passage belt
(see for example the zonally averaged zonal current in
Fig. 9). As a consequence we have standing eddy
contributions which are strong compared to the transient
eddy contributions. This is exemplified in Fig. 9 by the eddy
density flux v * ρ * and +v'ρ',, respectively. In the Drake
Passage belt these fluxes are of comparable size and
northward - at blocked latitudes - the flux of the standing
eddies overwhelms the flux of the transient eddies by an
order of magnitude.
The lower suite of panels in Fig. 9 displays the same
fields using an alternative average which is oriented along
the contours of the time mean sea surface height (SSH; the
POP code has a free surface implemented): we show the
mean tangential velocity and the component of the density
flux by standing and transient eddies which is normal to the
SSH contours (‘standing’ now refers to the deviations from
the convoluted path). This path following average clearly
captures more of the properties in the ACC region than a
zonal average. A similar streamwise average analysis is
presented in Ivchenko et al. (1998) and Best et al. (1999).
The mean tangential velocity collects all jets into a strong
current - surprisingly with one single core. It is eastward
everywhere and centred at the height contour -0.5 m (mean
latitude of -49°) with the highest speeds at the surface of
about 0.2 m s-1 which is two times the maximum of the
zonally averaged zonal velocity. The eddy fluxes, shown in
the middle and left panels, demonstrate that time mean and
transient field are separated to a large extent: the standing
eddy component is still non-zero (because the flow slightly
turns with depth) but is clearly much diminished compared
with the zonal mean and negligible compared to the
transient component of the path following mean.
In summary, we conclude that the zonal average does not
separate the time mean and the transient motion in a simple
way. Zonal mean, standing eddy and transient eddy
components arise and the standing eddy component is a
major player. Dynamically it belongs to the time mean flow
but it overrides the transient component. When analyzing
dynamical balances in the latitudinal-longitudinal
coordinate system standing and transient components have
their physical meaning (e.g. in the balance of zonal
momentum which will be discussed in many places in this
paper). But building models of a mean circulation in a
zonally average framework is inherently hampered by an
inadequate treatment of the standing eddy component.
Because it is intractable to parameterizations it is generally
neglected but is larger than the transient component for
which reasonable parameterizations are known (Johnson &
Bryden 1989).
The path following (or convoluted) average produces a
much clearer separation of the flow into time mean and
transient components. In this framework the coordinate
system is attached to the specific flow (the model SSH
contours in the above example). Analysing balances or
setting up models in the convoluted average frame is
conceptually simpler because standing eddies can be
neglected but the fields (velocities, fluxes etc) are oriented
at the convoluted coordinates. For instance, in a convoluted
average analysis we would consider the balance of the
along-stream component of momentum with the alongstream component of wind stress entering, rather than the
balance of zonal momentum. In the course of the paper we
will frequently have recourse to these different averaging
concepts.
3.2. Interfacial and bottom form stress
Eddies not only carry heat in the mean but also establish a
transfer of momentum as well. While lateral eddy
momentum fluxes turned out to be rather small (compared
to the wind-stress) and indifferent in sign (Morrow et al.
1994, Phillips & Rintoul 2000, Hughes & Ash 2001) the
ACC is the outstanding example in the ocean for diapycnal
transport of momentum by eddies. Since the momentum
imparted to the ocean surface layer by the strong zonal
winds in the Southern Ocean cannot be balanced in the
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
Fig. 10. Schematic demonstrating the interfacial form stress for an
isopycnal interface in the water (shown is the zonal depth).
There is higher pressure at the depth of the density surface
where it is rising to the east compared with where it is deepening
to the east. This results in an eastward pressure force (interfacial
form stress) on the water below. This is related to the fact that
the northward flow occurs where the vertical thickness of water
above the density surface is small, and southward flow where
the thickness is large, so there is a net southward mass flux at
lighter densities due to the geostrophic flow. The same kind of
pressure force acting on the sloping bottom topography leads to
the bottom form stress. Redrawn from Rintoul et al. (2001).
Drake Passage belt by large scale zonal pressure gradients a consequence of the lack of zonal boundaries - and because
lateral Reynolds stresses are too small for significant
transport away from the ACC towards boundaries, a
downward transfer of momentum is the only mechanism to
prevent indefinite acceleration of the zonal flow in the
surface layer. The diapycnal momentum transport cannot be
achieved by small-scale three-dimensional turbulence (it
would require viscosities far too large); it must be done by
the eddies. Depending on the framework - layer or level
coordinates - different kinds of eddy terms arise in the
dynamical balances. We first present the dynamical
balances for a layer framework, and later describe the
corresponding physics in level coordinates (at the end of
Section 3.3 and in Section 4.1).
The most important mechanism of momentum transfer in
layer coordinates is the eddy interfacial form stress (IFS). It
operates everywhere in the ocean where eddies are present
and deform isopycnals but the unique sign and magnitude in
the vast circumpolar area is truly outstanding. IFS transfers
horizontal momentum across inclined (by eddies)
isopycnals by fluctuations (by eddies) in the zonal pressure
gradient. Imagine two interfaces (isopycnals) z = -d1(x) and
z = -d2(x) along any circumpolar path with coordinate x
along it and integrate the (negative) pressure gradient -px
between the interfaces and around the path,
−∫∫
− d1 ( x ) ∂p
−d2 ( x)
∂x
∂d 
∂d

dzdx = − ∫  p( x, z = −d1 ( x)) 1 − p( x, z = − d 2 ( x)) 2  dx
∂x 
∂x

∂

 ∂
p( x, z = − d 2 ( x))  dx
= ∫ d1 p( x, z = − d1 ( x)) − d 2
x
x
∂
∂


(5)
to get its contribution to the rate of change of x-momentum
447
in the corresponding volume. The pressure is taken at the
isopycnal depth and its gradient appearing in the second
formulation thus acts across the inclined isopycnal.
Obviously, to get a non-zero dp x (the overbar indicates the
path and time mean) the pressure must vary at the isopycnal
depth in a way that an out-of-phase part with respect to the
depth variations is present (see Fig. 10). Evidently, the strip
of ocean gains x-momentum by the amount d1 p1x from the
fluid above z = -d1(x) and loses d 2 p2 x to the fluid below
z = -d2(x). Thus, for infinitesimally distant isopycnals the
vertical divergence of the interfacial form stress IFS
= pd x = − dp x enters the momentum balance. The mean
depth d is not relevant, only the eddy component
d * = d − d contributes (also for pressure) so that
IFS = − d * p *x . The starred quantities contain the signal
from the time-mean ‘standing’ eddies and the signal from
the time varying transient eddies and the IFS may be
separated accordingly.
Equating the zonal pressure gradient with the northward
geostrophic velocity, fv*g = p*x, and the layer depth
fluctuation with (potential) temperature anomaly,
d * = θ * / θ z , we find that the IFS relates to the meridional
eddy heat flux, − d * p * x = ( f / θ z )ν *gθ *. A poleward eddy
flux of heat is just a downward transport of zonal
momentum by IFS in the water column. These processes are
strictly coupled. The transient eddies which carry the
poleward heat flux shown in Fig. 7 thus establish a
downward transport of momentum.
In summary, though horizontal pressure gradients can
only establish a transfer of horizontal momentum in
horizontal direction they do transport horizontal momentum
across tilted surfaces from one piece of ocean to another. A
layer bounded by tilted isopycnals is thus forced by stresses
(IFS) at the bounding top and bottom surfaces (in the same
way as the Ekman layer is driven by frictional stresses at top
and bottom).
Deriving the relation Eq. (5) it was assumed that the
isopycnal strip does not run into the bottom nor touches the
sea surface. If this situation occurs additional pressure terms
arise from the bounding outcrops. These terms present a
flux of horizontal momentum through these boundaries into
the strip. For the bottom contact the corresponding flux is
part of the bottom form stress (see below).
Notice that the same mathematical operations used to
derive Eq. (5) apply if the interface is solid as, for example,
the ocean bottom at z = -h(x). The bottom form stress BFS =
− hpbx = hx pb operates here to transfer zonal momentum out
of the fluid to the solid earth (since h is constant in time only
the time mean bottom pressure is relevant). BFS works
everywhere in the ocean where the submarine ocean bed is
inclined but to be of significance the gradient of the bottom
pressure (the normal geostrophic velocity) must be
correlated to the ocean depth variations, or vice versa: the
bottom pressure must be out of phase with the depth along
the respective circumpolar path, e.g. there must be high
448
D. OLBERS et al.
Fig. 11. Zonal section of the observed potential density at 60°S, taken from the gridded data of the WOCE Hydrographic Special Analysis
Center (SAC, see http://www.dkrz.de/~u241046/SACserver/SACHome.htni). The section is viewed from the south.
pressure at rising topography and low pressure at the
opposite falling slope to the east to let eastward momentum
leak out to the earth. A depth-pressure correlation can in fact
be seen in circumpolar hydrographic sections passing
through Drake Passage around Antarctica, as shown in
Fig. 11. From the density ρ we can infer the baroclinic bottom
pressure p clin = g 0 ρdz
∫z
contained in the mass stratification.
It is obvious in the section that there is more lighter water to
the west of the submarine ridges than to the east.
Surprisingly the BFS derived from such a pattern
accelerates the eastward current, acting thus in cooperation
with the eastward wind stress - a feature of the ACC dynamics
which will be reconsidered in the course of this paper.
3.3. The dynamical balance of the zonal flow
The IFS and BFS contributions to the physics of zonal
currents can be elucidated by a simple conceptual model.
Consider a strip of ocean from Antarctica to the northern
rim of the ACC and split the water column into three layers
(which may be stratified), separated by interfaces which
ideally are isopycnals (see Figs 11 & 12). The upper layer
from the sea surface z = η0 = ζ to some isopycnal at depth
z = -η1 and includes the Ekman layer, the intermediate layer
with base at z = -η2 lies above the highest topography in the
Drake Passage belt (the range of latitudes which run
through Drake Passage), and the lower layer reaches from
z = -η2 to the ocean bottom at z = -η3= -h. We apply a time
and zonal average to the balance equations of zonal
momentum in the three layers and use Eq. (5) to get
− f V 1 = −η1* p1*x + τ 0 − τ 1 − R1
− f V 2 = η1* p1*x − η 2* p2* x + τ 1 − τ 2 − R2
*
− f V 3 = η 2* p2* x − hpbx
+ τ 2 − τ b − R3
Fig. 12. A schematic view of the meridional overturning
circulation in the layers of the Southern Ocean watermasses as
an enlarged view of the cut in the ‘cake’ of Fig. 4. An upper cell
formed primarily by northward Ekman transport beneath the
strong westerly winds and southward transport in the Upper
Circumpolar Deep Water (UCDW) layer is assumed
‘unblocked’ by topography in the three-layer model of Section
3.3. The lower cell driven primarily by formation of dense
AABW near the Antarctic continent and inflowing NADW is
assumed ‘blocked’ beneath the depth of the highest topography.
From Speer et al. (2000).
(6)
where the depth and zonally integrated northward volume
flux in each layer are denoted by V i , i = 1, 2, 3.
Furthermore, pi is the pressure at the respective layer
depths, p3 = pb the bottom pressure and the overbar denotes
time and ACC path following mean. Note that the surface
term ζp0 x drops out in the first equation because the
surface pressure is p0 = gζ. As before, the star denotes the
deviation from this average, τ0 is the wind stress, τi the
frictional stresses at interfaces, τ3 = τb the frictional bottom
stress, and Ri = ∂ui*vi* / ∂y the divergence of the appropriate
lateral Reynolds stress. We will assume that the interfacial
friction stresses τ1, τ2 and the Ri can be neglected (which is
confirmed by measurements, e.g. Phillips & Rintoul 2000,
and eddy resolving models). The meridional circulation is
defined by transports between isopycnals and is thus of
Lagrangian quality. The wind-driven component
-τ0/f (the Ekman transport) in the top layer, a similar
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
frictional transport τb/f in the bottom layer, and a
geostrophic component in the bottom layer, associated with
the bottom form stress − hpbx , also appear if the flow is
averaged between geopotential (constant depth) levels (see
end of this section). These Eulerian quantities form the
Deacon cell of the Southern Ocean meridional overturning
(see Döös & Webb 1994 and Fig. 24).
Since Σi V i = 0 by mass balance of the ocean part to the
south (neglecting the very small effect of precipitation and
evaporation on mass balance) the overall balance of zonal
momentum is between the applied wind stress, the bottom
form stress and the frictional stress on the bottom,
τ 0 − hpbx − τ b•0
(7)
The frictional stress τb and the here neglected Reynolds
stresses are generally small in the ACC. Munk & Palmen
(1951) were the first to discuss this balance of momentum
for the ACC (but surprisingly, much of the research on the
ACC after Munk & Palmen’s article had forgotten the
importance of the bottom stress and tried frictional
balances, e.g. Hidaka & Tsuchiya (1953), Gill (1968)).
Hence, the momentum put into the ACC by wind stress is
transferred to the solid earth by bottom form stress. The
transfer is at the same latitude because the divergence of the
Reynolds stress is small. This balance has been confirmed
in most numerical models which include submarine
topographic barriers in the zonal flow and have a realistic
(small) magnitude of the Reynolds stress divergence. If the
ocean bottom is flat (in models the bottom can be made flat)
either bottom friction may get importance and/or the
neglected Reynolds terms could come into play. Eddy
effects seem to be unimportant in the vertically integrated
449
balance but it is worth mentioning that most coarse OGCMs
do not confirm Eq. (7), see Cai & Baines (1996). The reason
is that such models use very large lateral viscosities so that
the parameterized Reynolds stresses become large, even
though the simulated current is broad and smooth.
Figure 13 exemplifies the total zonal momentum balance
with results from the eddy resolving POP model (Parallel
Ocean Program, Maltrud et al. 1998). It is instructive to
write the pressure p as sum of the baroclinic (density related)
part p clin = g 0 ρdz
∫z
and the barotropic (surface related) part
gζ. While the total bottom form stress clearly takes out the
momentum put in the ocean by wind stress we have seen
clin
above in Fig. 11 that the baroclinic part h( pb ) x does not
have the corresponding sign: according to the phase shift of
density with respect to the submarine topography the
baroclinic bottom form stress should accelerate the
eastward current. Indeed, this has been found in the analysis
of the eddy permitting model FRAM (Fine Resolution
Antarctic Model, Fram Group 1991). The right hand panel
of Fig. 13 displays the balance with the pressure terms
h( pbclin ) x and g hζ x separated. Individually they are much
larger than the zonal wind stress, by about an order of
magnitude, but of opposite sign and thus they nearly cancel.
This feature in the dynamical balance of the ACC will be
further discussed in Section 4.2. A summary of the balance
of zonal momentum in the ACC is displayed in Fig. 14.
A global perspective of the zonal balance is presented in
the experiments which Bryan (1997) has performed with a
non-eddy-resolving OGCM. In his findings the balance
between zonal wind stress and bottom form stress prevails
everywhere in the world ocean and likewise, we find in all
Fig. 13. Left: Vertically integrated balance of total momentum from the POP model in the Southern Ocean, indicating a balance between
wind-stress [solid line] and negative bottom form stress [dashed] at each latitude (small deviations are due to Reynolds stress effects; the
terms are normalized by f0 = 1.25 x 10-4s-1, units are Sv). From Olbers (2005). Right: vertically integrated balance of total momentum from
the FRAM model in the Drake Passage belt: 1 is baroclinic form stress, 2 barotropic form stress, 3 zonal wind-stress (units Nm-2). From
Stevens & Ivchenko (1997).
450
D. OLBERS et al.
Fig. 14. Sketch of the zonal balance of momentum for the ACC:
the flow establishes a high of barotropic (surface) pressure and a
low of baroclinic (hydrostatic) pressure upstream of a zonal
ridge and a corresponding low/high downstream. The associated
barotropic and baroclinic bottom form stresses almost balance,
their residual counteracts the wind stress. The wind drives a
northward Ekman transport in the surface layer (⊗), and
corresponding to the bottom form stress there is southward
geostrophic return flow in the valleys between the ridges which
partly block the zonal path (u). The system is viewed from
Antarctica.
experiments (run with different wind climatologies) the
approximate cancellation of the barotropic and baroclinic
form stresses which individually are an order of magnitude
larger than the wind stress (see fig. 11 of Bryan 1997). It is
noteworthy that the signature of the momentum balance
found here for the ACC - with driving by the baroclinic
form stress and braking by the barotropic - is only found in
Bryan’s results poleward of the subtropical gyres.
If, in addition to the assumptions of small Ri and τ1 , τ2
and τb, the flow conserves potential density then there
cannot be transport across isopycnals and the meridional
transport in each layer must vanish, V i = 0, by mass
conservation. We find that the interfacial form stress η i* pix*
is vertically constant and equal to the wind stress τ0 and to
the bottom form stress,
ηi* pix* •τ 0 • hpbx
(8)
Then, in each layer, the meridional mass fluxes induced by
wind stress and pressure gradients are compensated (in
models with a flat bottom, the bottom form stress must be
replaced by the frictional bottom stress in the above
relation). This scenario of ‘constant vertical momentum
flux’ is realized in quasigeostrophic layer models (Wolff
et al. 1991, Marshall et al. 1993, Olbers et al. 2000, Olbers
2005) which are by construction adiabatic.
The real ocean is diabatic, i.e. there is mixing across
isopycnals by small-scale turbulence and air-sea fluxes, but
it is still in debate if it occurs predominantly between the
outcropping isopycnals in the surface layer or in the interior
as well (see Section 5). The meridional overturning
transports Vi at a certain latitude circle can be non-zero
only if there is exchange of mass between the layers south
of the respective latitude - implying conversion of
watermasses south of the ACC. In fact, by mass
conservation, the Vi equal the net exchange with the
neighbouring layers over the area south of the respective
latitude. At the same time, the overturning transports imply
a Coriolis force in the individual isopycnal layers which is
in balance with the vertical divergence of the interfacial
form stress. The divergence of the heat flux due to transient
eddies can clearly be deduced from Fig. 7 (we have shown
that roughly − η ' p ' x ≈ − fv' g θ ' / θ z ). Eddy effects at the
respective latitude and diabatic interior effects of smallscale turbulence occurring to the south must thus adjust
according to mass and momentum requirements of the zonal
current and the meridional overturning. The isopycnal
analysis of the zonal momentum balance in the FRAM
model by Killworth & Nanneh (1994) can be taken to
exemplify the importance of diabatic processes and the
inapplicability of the ‘constant vertical momentum flux’
scenario: there is a net meridional circulation at all depths in
balance with a divergent IFS and wind stress (the latter
influences also deeper isopycnal layers which outcrop at
some longitude along the circumpolar path).
We have so far discussed the momentum balance in a
Lagrangian framework by using isopycnal layers. But many
dynamical concepts and the numerical models are written in
an Eulerian framework where geopotential (depth) horizons
are the vertical coordinates. Interfacial form stress is then
invisible, and apart from Reynolds stress terms the zonally
averaged balance of zonal momentum
∂
∂{τ } ∂
− {u * v*} − {u * w*} + ∑ δ p b
∂z
∂y
∂z
ridges
(9)
has no obvious signature of eddy effects at all. Here, τ is
stress (vertical transport of zonal momentum by turbulent
motions) in the interior. The sum of the bottom pressure
differences δ p b is extended over all submarine ridges
interrupting the integration path at depth z (continents are
included). Each ridge or continent contributes to the
difference between the values on the eastern side and the
western side, i.e. δpb = p(xE, y, z = -h) - p(xW, y, z = -h). The
curly bracket operator denotes zonal integration on level
surfaces and a* = a - {a} /L is the deviation (again standing
plus transient eddy component), L is the path length and the
overbar is now only the time mean. From mass balance it
may be shown that the vector {v} = -∂φ/∂z, {w} = ∂φ/∂y has
a streamfunction φ (the meridional Eulerian streamfunction) despite possible interruptions of the zonal path by
submarine topography. Equation (9) may then be integrated
vertically from the surface to some depth z to yield the
balance for the depth interval from the surface to the level z,
− f {v} =
– fφ = {τo} – {τ} – R + F
(10)
where R collects the integrated Reynolds stress divergence
and F is the bottom form stress cumulated at the level z from
the bottom pressure term,
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
F ( z) = ∫
0
z
∑ δ p dz = ∑ ∫
b
ridges
ridges
xE ( z )
xW ( z )
pb
∂h
dx
∂x
(11)
Since φ (z = -h) = 0 the balance Eq. (7) is recovered if R
neglected as before, with F(z = -h) as the total form stress.
However, instead of the interfacial form stress balance we
are now facing in the interior a balance between the
integrated Coriolis force and frictional, Reynolds and
bottom pressure stresses. The balance is described in
Stevens & Ivchenko (1997) for FRAM and repeated for
many other eddy resolving models (see the summary in
Olbers 1998). For POP we show the terms of the vertically
integrated balance Eq. (10) in Fig. 15. It is obvious which
terms are the main players in the different depth ranges. In
the top layer these are Coriolis force and wind stress. At
intermediate depths where topography is not yet
intersecting there is little change in φ with depth, i.e. small
meridional transport and thus small Coriolis forces balanced
by small Reynolds and frictional effects2. And in the deep
blocked layers the balance between Coriolis force and
bottom form stress can be seen. The total balance of zonal
momentum in this POP experiment is shown in Fig. 13.
Where are the eddy effects in this framework? They are
hidden in the Eulerian Coriolis force, as will be discussed in
Section 5 where we proceed with the Eulerian framework
and include the still missing connection to thermohaline
forcing and turbulent mixing.
3.4. Failure of Sverdrup balance
An outstanding feature of Southern Ocean dynamics is the
failure of one of the cornerstones of theoretical
oceanography – the Sverdrup balance βψx = curl τ0. It
relates the northward transport V = ψx (the meridional
velocity vertically integrated from the bottom to the
surface) to the local curl of the wind stress vector τ0. Here, β
= df/dy is the meridional gradient of the Coriolis parameter
f. Closure of the circulation occurs by a western boundary
current to satisfy mass conservation. Apparently, in the
range of latitudes of Drake Passage the Sverdrup balance
must fail: ∫ Vdx must be zero to ensure mass conservation
of the piece of ocean to the south, yet the wind stress curl
will not integrate to zero in general. Contrary to the
circulation in an ocean basin, we cannot overcome this
problem by some kind of boundary current returning the
mass flux.
Nevertheless we are aware of many attempts to generate
a Sverdrupian solution for the ACC. Notable is Stommel’s
approach (Stommel 1957) where the Antarctic Pensinsula is
expanded to the north to block the Drake Passage latitude
band and allow only for a northward passage. Similar
barotropic theories have been presented more recently by
Webb (1993), Ishida (1994), and Hughes (2002). While we
may classify these studies as theoretical test cases, the
approach of Baker (1982) is more intriguing: in an attempt
to estimate the ACC transport from wind data Sverdrup’s
balance is integrated along 55°S, i.e. just north of Drake
Passage in a possibly ‘Sverdrupian regime’, starting at the
west coast of South America and extended to the east flank
of the ACC system, leaving out the part where it shoots
northward after leaving Drake Passage (see Figs 1 & 2).
Because of mass conservation the ACC transport running
through a piece of the section that must be equal to the
(negative) integrated curl of the wind stress of the
remaining part - the ACC transport could then be explained
entirely in terms of a certain property of the Southern Ocean
wind system. In fact, for particular wind stress climatology
data Baker found reasonably good agreement with the
observed ACC transport.
The fallacy in Baker’s approach is not in the particular
choice of the integration path, it is that the Sverdrup balance
is not applicable to most of the Southern Ocean (and
possibly most of the world ocean, see Hughes & De Cuevas
2001). It neglects the interaction of the circulation with the
topography, which could be suspected to be important from
the penetration of the ACC to great depth, unlike currents in
basin gyres. In ocean basins the deep pressure gradients are
shut off during spin-up of the circulation by westward
propagating baroclinic Rossby waves of successively
increasing vertical mode number (Anderson & Gill 1975).
In the Southern Ocean the strong and deep reaching
eastward current hinders even the fastest (first baroclinic)
mode from westward propagation (see e.g. Hughes et al.
1998). The establishment of deep pressure gradients not
only makes the work of the bottom pressure on topography,
the BFS, effective, it also modifies the Sverdrup theory. The
Sverdrup balance derives from the planetary vorticity
conservation, βv = fwz + curl τz, which states that a piece of
the water column which is affected by friction (τ is the
frictional stress appearing locally in the water column) or is
stretched vertically must experience an appropriate
advection of planetary vorticity (βv = vdf/dy). The Sverdrup
balance results by vertical integration of the vorticity
balance under the assumption of vanishing vertical motion
at great depth so that there is no stretching of the total water
column and no friction at depth. In the presence of
submarine topography and deep pressure gradients this is
not valid: geostrophic flow across topography induces a
vertical motion, wg(z = -h) = -ug(z = -h) · Lh = - (1/f) J(pb,
h), and stretching4. Hence, with the rigid lid assumption at
the surface, w(z = 0) = 0, we get
∂ψ
= curl (τ 0 − τ b ) + J ( pb , h)
∂x
(12)
The frictional stress τb of the flow on the bottom is
β
2
Locally the Coriolis forces are large. They generate the pressure gradients
which are needed to establish the interfacial form stress.
451
4
We introduce here the Jacobian operator, J(a, b) = axby - aybx.
452
D. OLBERS et al.
Fig. 15. Terms of the vertically integrated momentum balance (10) for the POP model as function of depth and latitude (note: the values at
depth z reflect the contribution to the balance of zonal momentum in the depth interval from the surface to the level z): Coriolis force fφ
[upper left panel], wind stress {τ0} [upper right panel], Reynolds stress R [lower left panel], bottom form stress F [lower right panel].
Units: all quantities are normalized by a mean Coriolis frequency; Contour interval is 5 Sv except for Reynolds stress where it is 1 Sv.
From Olbers & Ivchenko (2001).
generally small but the so called bottom pressure (or
topographic) torque J(pb, h) (Holland 1973) can locally be
very large, even overwhelming the torque by the wind stress
by an order of magnitude or more. This is demonstrated in
Fig. 16 showing the streamfunction and the bottom pressure
torque in a simulation with the global eddy permitting
OCCAM model (Ocean Circulation and Climate Advanced
Modeling Project, see Coward (1996) for details). Clearly,
northward excursions of the current are correlated with
positive bottom torques and southward with negative, as
suggested by the barotropic vorticity balance Eq. (12). We
should mention that this view applies on scales of a few
degrees which are clearly larger than those of individual
eddies. On smaller scales the neglected advection of relative
vorticity comes into play and the dominant balance is
between the bottom torque and nonlinear advection terms
(Wells & De Cuevas 1995). In any case the simple Sverdrup
theory does not apply.
Notice that the zonally averaged balance of barotropic
vorticity is consistent with the balance of total momentum
Eq. (7): integrating Eq. (12) around a latitude circle yields
the meridional divergence of Eq. (7). Though there is local
compensation of the β-term and the bottom form stress as
indicated in Fig. 16, the wind curl and the bottom torque
balance in the zonal mean.
We should like to point out that Eq. (12) is merely a
balance that the circulation has to satisfy (possibly
augmented by the so far neglected terms such as lateral
friction, see below). It is not sufficient to determine the
streamfunction because the bottom pressure and frictional
torques are not prescribed functions like the wind stress curl
but rather must be determined from a complete solution.
How this can be achieved is the subject of the next section.
3.5. The geostrophic contours
In the β-term of Eq. (12) the transport V = ψx appears which
is normal to latitude circles. From a mathematical point of
view, latitude circles are the characteristics of the
differential equation. Some of the problems discussed above
arise from the periodicity of these characteristics in the
latitude belt of Drake Passage. There is another vorticity
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
453
Fig. 16. Barotropic streamfunction from the OCCAM model (contour interval 15 Sv) and bottom pressure torque (shading) in units of
10-6 N m-3. Both quantities have been smoothed by 4.25° of longitude and 3.25° of latitude. Note that northward flows are associated with
positive torques, southward flows with negative torques. For comparison, typical wind stress curl in this region gives a torque of
10-7 N m-3. From Rintoul et al. (2001).
statement about the circumpolar flow with other
characteristics, namely the geostrophic contours f/h, which
in the majority do not close in themselves around
Antarctica: they either run into continents or they close
around topographic plateaus in the Southern Ocean (like
Kerguelen) or run around the immediate rim of Antarctica
(see Figs 6 & 18). The role of f/h contours as characteristics
becomes evident if we divide the momentum balance (3) by
the depth h before taking the curl. We arrive at
1
f
 1
 f
= J ψ ,  = J  χ ,  + curl (τ 0 − τ b + F )
h
h
 h
 h
(13)
which identifies the agents which force transport across the
geostrophic contours f/h =const. These are the baroclinic
torque (or the JEBAR term, Joint Effect of Baroclinicity
And Relief, Sarkysian & Ivanov 1971), and the curl of the
depth averaged frictional stresses applied at the top and the
bottom. We have augmented the barotropic vorticity
balance by the term F, indicating the effect of hitherto
neglected lateral Reynolds stresses in the vertically
integrated momentum balance Eq. (3). Both the lateral
transport F and the bottom frictional stress τb depend on the
actual state of the flow. To make the balance more specific
for later use we put in linear bottom friction and lateral
U ⋅∇
diffusion of the depth integrated velocity, thus τb = εU, F =
AhL2U so that
τ
1
F
1
curl b = ε ∇ ⋅ ∇ψ
curl = Ah ∇ ⋅ ∇ 2 ∇ψ
h
h
h
h
(14)
The advantage of Eq. (13) over Eq. (12) is obvious in
homogeneous ocean: for constant density the JEBAR term
drops from Eq. (13) whereas the latter would still contain
the bottom torque of the barotropic (surface) pressure
contained in pb.
With wind stress and potential energy prescribed Eq. (13)
is able to predict the streamfunction if suitable boundary
conditions are set. Besides conditions required by the lateral
friction term (usually no-slip condition for U on the coasts)
we have to satisfy mass conservation, which requires ψ =
constant on coasts, with different constants on the different
islands because these values determine the transports
between them. One constant may be set to zero without
restriction (e.g. ψ = 0 on the American continent), the other
constants must be predicted, which states the need for
additional equations. These follow from the requirement
that a solution of Eq. (13) must allow the calculation of the
pressure field pb from the momentum balance Eq. (3) (with
F included; pb is calculated by path integration from one
454
a.
D. OLBERS et al.
b.
c.
d.
Fig. 17. Experiments with the BARBI model forced by wind stress from NCEP. a. streamfunction for a homogeneous flat bottom ocean,
b. streamfunction for a homogeneous ocean, using realistic topography, c. streamfunction for a simulation including baroclinicity,
d. baroclinic potential energy for case c. [CI for streamfunction: 10 Sv for values lower than -5 Sv, 5 between -5 and 25, 10 between 25
and 200 and 100 for values greater than 200 Sv, CI for potential energy: 1000 m2 s-2]. From Olbers & Eden (2003).
coastal point where its value may be set arbitrarily). In a
multi-connected domain, with islands present, the
uniqueness of pb is guaranteed by the integrability
conditions
1
∫island ds ⋅∇pb = ∫island ds ⋅ h [f ∇ψ − ∇χ + τ 0 − τ b + F ]= 0
(15)
around each island (e.g. Antarctica) on an arbitrary path.
With n islands (or continents) there are thus n - 1 such
conditions which render the reconstruction of pb pathindependent.
The barotropic vorticity Eq. (13), together with the
constraint Eq. (15), is evidently a central tool for the
determination of ocean transports. The potential energy χ
has to come from the baroclinic equations of heat and salt
conservation where ψ couples in via advection. Numerical
ocean models using the rigid-lid approximation actually
determine the depth-integrated velocity vector from a
vorticity equation setup as Eqs (13) & (15).
The importance of the geostrophic contours and of
JEBAR were demonstrated in a series of early numerical
experiments with the GFDL model of the world ocean
circulation. Bryan & Cox (1972) presented the circulation
for a homogeneous fluid (constant density, thus having zero
JEBAR) in an ocean with continents but of constant depth.
Cox (1975) extended the studies to the cases of variable
topography (with blocked f/h contours but still zero
JEBAR) and also to a topographic ocean with baroclinicity,
hence nonzero JEBAR. Due to limited computer resources,
the last experiment was largely a diagnostic simulation, i.e.
the thermohaline fields do not deviate much from the initial
state taken from observations. But full prognostic
experiments have since then been repeated many times (e.g.
Han 1984a, 1984b, Cai & Baines 1996) with very similar
results. New simulations of these cases are discussed in
Olbers & Eden (2003) and depicted in Fig. 17. They were
obtained with the BARBI reduced physics model which
uses (13) and a balance equation for χ,
∂χ
χ 1
1


+ hU ⋅ ∇ 2 − N 2 ∇ ⋅  u'+ h 2U  = K h∇ 2 χ
∂t
h
2
3


1 2
1
fk × u' = h ∇χ − h 2τ
3
3
(16)
in which the time rate of change, the barotropic transport of
χ, the vertical advection (the divergence term, see below) of
a background stratification (represented by the BruntVaisala frequency, N 2 = − gd ρ / dz ) and diffusion is
included. The potential energy χ is then associated with the
deviation of density from the mean density ρ which leaves
JEBAR unchanged. The term proportional to N2 represents
the generation of baroclinic potential energy by lifting or
lowering the background isopycnals, i.e. it derives
from ∂w ρ ( z ) / ∂z in the density balance. The vertical
pumping is done by a contribution from the barotropic
velocity U/h and a vertical moment of the baroclinic
2
velocity, u' = ∫ z (u − U / h)dz for which a separate equation
is derived from the momentum balance (for details see
Olbers & Eden 2003). In essence the coupled system Eqs
(13) & (16) is representing a wave system consisting of
planetary-topographic Rossby waves forced by wind stress.
The flat bottom, homogeneous ocean has an ACC
transport of a couple of hundreds of Sverdrups (more than
600 Sv in Bryan & Cox 1972, 700 Sv in BARBI). The
homogeneous ocean with topography has very low ACC
transport (22 Sv in Cox 1975, 35 Sv in BARBI), and the
third experiment, now considering baroclinic conditions in
a topographic ocean, generally gets a realistic transport for
the ACC (187 Sv in Cox 1975, 130 Sv in BARBI). How can
we explain this behaviour?
The flat bottom case has an almost zonal ACC driven by
the zonal wind. Since bottom form stress and bottom torque
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
cannot operate, friction is the only momentum sink and with
the diffusive parametrization of lateral eddy induced
transports of momentum by a diffusivity Ah, as in Eq. (14),
the zonal transport is proportional to Y 3τ0/Ah where Y is the
width of the current. We are facing ‘Hidaka’s dilemma’
(Hidaka & Tsuchiya 1953, see Wolff et al. 1991): either we
implement a reasonably sized diffusivity and then get an
unrealistically large transport or we must use an
unrealistically large eddy viscosity to get a reasonable size
of the ACC transport.
The topographic homogeneous case has a transport that is
far too low. The system now establishes a bottom form
stress and a bottom pressure torque (but not JEBAR) from
the surface pressure being out-of-phase with the submarine
barriers of the flow (see Fig. 14). The current is mostly
along the geostrophic contours which are north of the Drake
Passage belt. Apparently, the frictional torques in Eq. (13)
are too weak to push mass across these contours in a
significant amount.
There is an interesting lesson to learn from the
momentum balance Eq. (3): the component oriented along
f/h = const is given by
∂ψ h ∂pb 1
=
+ (τ 0 − τ b + F )⋅ ×
∂s
f ∂s
f
(17)
where s is the path length coordinate along geostrophic
contours and × is the tangential unit vector. If the contours
are blocked by continents (as in the Pacific sector, see Figs 6
& 18) the transport between them is ∆ψ • (h/f)∆pb + VEk
where VEk is the Ekman transport across the contour and ∆pb
the pressure difference between the continents (the other
friction terms are small in this regime). We may interpret
this latter term as a net geostrophic transport sustained by
the pressures on the coasts. On the other hand, with small or
zero ∆ψ we see that the net wind stress along f/h-contours is
taken up by a pressure difference on the continents - just as
in the Sverdrup circulation regime in a flat bottom basin
bounded by continents. For the f/h-contours closed on the
rim of Antarctica we have a Hidaka-type transport regime
where wind stress is balanced by friction. The transport,
however, is small because the wind is much weaker (and
actually westward, see Fig. 19) and the width of this region
small. The closed f/h-regimes on the Mid-Atlantic Ridge
and around Kerguelen are governed by friction as well as
balancing a small net Ekman transport into these regions.
The final case which considers topography and
baroclinicity gets a reasonably sized ACC transport for
which clearly JEBAR is responsible. It is an order of
magnitude larger than the wind curl but that property alone
would not explain why this new forcing should not be
blocked by the f/h-contours as the wind curl is blocked. In
fact, the JEBAR field has a very particular spatial structure:
highs and lows are placed right along the undulating path of
the geostrophic contours (see Fig. 18) to help the current to
circumvent the f/h-constraint. We may state this property in
455
a different framework: from Fig. 17 c & d it appears that ψ
and χ are highly correlated, suggesting a functional relation
χ = C(ψ) established by the dynamics. This relation is
plotted in the last panel of Fig. 18. A reasonable fit is
suggested from Eq. (4), thus roughly χ = f0ψ + const, and
this casts the vorticity balance (13) into
β ∂ψ ~
1
− curl (τ 0 − τ b + F )
h ∂x
h
(18)
The topographic β-term and JEBAR in Eq. (13) combine to
achieve new (unblocked) characteristics which actually are
those of the flat bottom problem.
4. The dependence of transport on forcing
What are the mechanisms and forcing functions that
determine the transport of the zonal flow? The
considerations of momentum and vorticity balances and the
associated fluxes through the circulation system, outlined in
the previous sections, do not answer this question. Indeed,
one cannot expect a prediction of the ACC transport from
just one or two integral balances. They indicate, however,
that wind forcing and vertical momentum flux by transient
and standing eddies and waves are important but also the
processes which set up the surface and interior pressure
field, which is not really helpful because it covers nearly all
possible mechanisms. The baroclinic pressure aspect brings
the local surface fluxes of heat and freshwater into focus
(see Fig. 19). Combined as the surface density flux they
determine the density field in concert with advection and
diapycnal mixing - and then it might correctly be suspected
that side issues such as mixing by small-scale turbulence
and remotely forced agents, such as the import of NADW
into the Southern Ocean, could have an influence on the
ACC transport. A complete theory capable of predicting the
absolute transport of the ACC is thus a formidable
challenge. For quantitative answers a full model including
external forces by the wind stress and the surface fluxes of
density (or buoyancy) as well as the advection of mass
(volume) and density must be solved, which points towards
studies with numerical OGCMs. Though quite a suite of
carefully designed numerical experiments exists (some have
been mentioned in the previous sections and more will be
discussed below) their contribution towards an
understanding of the shaping of ACC transport is limited.
Many of the studies have the flavour of an engineering task:
changing parameters and/or forcing and monitoring the
results. These provide qualitative answers but a deeper
insight into the dynamics of the ACC can be obtained by
cheaper methods which may reveal mechanisms in trade for
completeness.
We have mentioned above the Hidaka regime, describing
an entirely frictional zonal current in a flat bottom ocean
with a simple transport formula. Another simple concept
follows from the momentum balance in adiabatic conditions
456
D. OLBERS et al.
Fig. 18. Upper panels: curl of wind stress curl (τ0/h) [left] and
JEBAR [right] with f/h-contours overlaid (notice the different
scale of the two plots, the colourbar for the wind curl runs from
-0.03 to 0.03, for JEBAR from -0.1 to 0.1, units are 10-11 s-2).
Lower panel: correlation between ψ and χ. All for the Southern
Ocean sector of the BARBI experiment c) from Fig. 17.
Fig. 19. Climatology data of zonal wind stress [upper left, unit Nm-2], surface heat flux [upper right, unit Wm-2], and freshwater flux [lower
left, unit 10-8 m s-1] and density flux [lower right, unit 10-6 kg m-2 s-2] (from SOC Climatology, Josey et al. 1998). The wind stress is above
0.1 Nm-2 over most of the ACC region and largest in the Indian Ocean. Values for the air-sea heat exchange are controversial. In the SOC
climatology the Atlantic-Indian Antarctic waters gain heat at a rate of roughly 25 Wm-2 but values may be biased. The Southern Ocean
gains freshwater from the atmosphere and continental runoff at a rate of roughly 10-5 kg m-2 s-1, net estimates for the Antarctic region south
of 60°S lie around 0.25 Sv. The flux of density is dominated by the freshwater gain: the Antarctic region looses density at the surface, thus
gains buoyancy (see also Fig. 23) where surface waters are transported to the north. Closer to the coast, where winds are westward and the
Ekman transport is southward a loss of buoyancy is expected (see also Fig. 12). The thick line in all figures is the zero contour.
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
which results in Eq. (8). Instead of emphasizing the lateral
eddy flux of momentum as in the previous flat bottom
model it is based on the ‘constant vertical flux scenario’
(appropriate to an eddy-active ocean with no diapycnal
mixing). Replacing the IFS by the lateral eddy heat or
density flux as indicated in Section 3.2 we get the JohnsonBryden relation (Johnson & Bryden 1989), f v' ρ ' / ρ z = τ 0 ,
here written for density and transient eddies (denoted by a
dash). The standing eddy component is neglected (which is
a severe assumption because it exceeds the transient
component in realistic conditions) or - if the mean is
interpreted as average along the current path - τ0 is not the
zonal wind stress but rather the path following component
(so we use dashes in the following arguments). According to
the formula the ‘northward’ eddy density flux v' ρ ' in the
circumpolar belt of the ACC, suitably normalized, is of the
size of the ‘zonal’ wind stress τ0 (so there would no
meridional overturning circulation, which may rightly be
questioned). In a first step Johnson & Bryden parameterize
the transient lateral eddy flux by a down-gradient form,
v' ρ ' = − K ρ y , and find that the wind stress and the eddy
diffusivity constrain the slope of the isopycnals,
s = − ρ y / ρ z = τ 0 / ( fK ) . Such a relation is roughly
consistent with the observed slopes in the ACC belt4 if the
eddy diffusivity is of order K ~ 103 m2 s-1 (take s = 10-3,τ0 =
10-4 m2 s-2). Johnson & Bryden (1989) proceed replacing the
lateral density gradient using the thermal wind relation,
fu z = g ρ y , and find
f2
uz = τ 0
N2
(19)
where the vertical density gradient is replaced by the
2
squared Brunt-Vaisala frequency N = − g ρ z . Apparently,
K(f/N)2 defines an equivalent diffusivity for the vertical
momentum transfer which is achieved by lateral density
diffusion (see e.g. Rhines & Young 1982, Olbers et al.
1985). We see here the same equivalence between vertical
momentum transfer and horizontal heat transfer by eddies
as in Section 3.2. In a second step Johnson & Bryden used
Green’s form (Green 1970, Stone 1972) of the diffusivity K
= αR 2 /T. It is obtained for a baroclinically unstable zonal
current, where R is a measure of the eddy transfer scale, and
T = N/|fuz| is the Eady time of growth of the unstable eddies
(Eady 1949). The constant α measures the level of
correlation between v’ and ρ’ in the density flux (α = 0.015
± 0.005 according to Visbeck et al. 1997). Johnson &
Bryden’s result for the ACC transport is obtained by relating
the turbulence scale R to the baroclinic Rossby radius λ =
Nh/(|f|π). For R = π2λ we get their estimate of the shear
K
1
2
 τ
N (z )
u z =  2 0 2 ⋅
f 
 π αh
(20)
which yields by integration the transport relative to the
4
In detail there are substantial and dynamically relevant deviations from
this adiabatic model, which we reconsider in Section 5.
457
bottom. The shear and thus also the transport is proportional
to the square root of the wind stress. With some reasonable
values of parameters a transport of about hundred Sv
relative to the bottom can be obtained.
The Johnson-Bryden model has been much discussed as a
theory of the ACC transport. Attempts to verify the squareroot relation with numerical models are plentiful (e.g.
Gnanadesikan & Hallberg 2000 with coarse-resolution
models with simple geometry, Gent et al. 2001 for coarseresolution global models, Tansley & Marshall 2001 for twolayer channel models) but generally without success. This is
not surprising in view of the many assumptions put together
in this model. First, there is the assumption of the adiabatic
state of the flow (zero diapycnal mixing) which is violated
in the real ocean but also in numerical models operating on
z-levels (isopycnic models may be adjusted close to an
adiabatic state). Second, it is not clear whether the Eady
model and other details of baroclinic instability theory are
appropriate in the ACC eddy field. Certainly some of these
features are generated by baroclinic instability but they are
not in the initial growth state but rather in some state of
equilibration. Furthermore, the above parameterizations are
not implemented in most coarse OGCMs. Even eddy fluxes
deduced from eddy resolving models show a quite poor
agreement with eddy flux parameterizations (Bryan et al.
1999 and Olbers & Ivchenko 2001 for the POP model).
Finally, we might also expect that the stratification, entering
in the Johnson-Bryden concept only in a prescribed N(z),
would at least partly be set by the action of the wind and the
overturning circulation, to point here again at the missing
thermohaline forcing. This is convincingly demonstrated by
Gnanadesikan & Hallberg (2000) with simple models in
which the buoyancy forcing has a direct feedback on the
density structure (the tilt of the interface in a two-layer
ocean). The wind stress and buoyancy feedback interplay in
a complex way via the balance of northward Ekman
transport and the upwelling through the thermocline to
produce a meridional pressure gradient across the
unblocked latitudes which is in balance with the baroclinic
part of the ACC transport. The net transport is then clearly
not uniquely determined by the wind stress.
The above concepts miss the influence of topography. The
total momentum balance Eq. (7) contains the part of the
bottom pressure which is out of phase with variations of the
topography along the zonal path of integration. Some
insight into the mechanism which the flow uses to generate
bottom form stress has been gained from heavily truncated
images of the full dynamics, so called low-order models
where the flow fields are represented by very few spectral
components (hopefully those of dynamical relevance, see
Olbers 2001). In the barotropic Charney-DeVore model
(Charney & DeVore 1979, see also Olbers & Völker 1996,
Völker 1999) the topography is taken sinusoidal in the zonal
direction: if topography is sine, form stress is cosine. The
model resolves the zonal current u and the sine and cosine
458
D. OLBERS et al.
components of pressure. The latter are established by a
standing barotropic Rossby wave which is generated by the
mean flow u going over the topography. At the upstream
side of the hills the fluid must be lifted up thus making high
pressure, at the downstream side a pressure low follows.
The naturally westward propagating wave becomes
stationary by eastward advection in the zonal current and
friction: it is locked in resonance with the mean flow and
produces a form stress which becomes a nonlinear
functional of the zonal velocity,
( fδ ) 2
1
BFS [u ]= − εuh 2
2
ε + k 2 (u − cR ) 2
(21)
Here k is the zonal wavenumber of the topography, δ the
ratio of height of the hills above the ocean floor to the mean
depth, cR = β/k2 the speed of barotropic Rossby waves and ε
a parameter of linear bottom friction, τb = -εuh. The total
momentum balance (7), written now as τ0 - ε uh + BFS[u] =
0, then determines the zonal transport uh. Three equilibria
are found if τb/(hε) is well above cR, two are stable
circulation regimes. For the two solutions in the resonant
range (u close to cR) the friction in the momentum balance is
negligible, these solutions are balanced by form stress. The
off-resonant solution is controlled by friction. It is
remarkable that friction is essential in all cases to shift the
pressure field out-of-phase with respect to the topography.
Charney & DeVore (1979) have developed this model for
atmospheric flow regimes. In the ocean the resonant
solutions do not exist - flow speeds are much less than
speeds of barotropic Rossby waves - and reasonable values
for the wind stress and the bottom friction allow only for the
frictionally controlled solution
uh =
τ0 /ε
1 + δ 2 (ak )2 / 2
1
BFS [u ]= − εuhδ 2 (ak )2
2
(22)
where a = |f/β| is the earth radius times tangent of latitude.
The transport in this barotropic model decays away from the
frictional solution uh = τ0/ε (with hundreds Sv transport)
with increasing height of the topography. The drag of the
form stress increases quadratically with the height of the
topography and attains higher values than friction for
moderately sized submarine ridges. In a baroclinic
extension the Charney-DeVore resonance can operate in
realistic ACC conditions. This will be discussed in Section
4.2.
We should mention that the above model works on an
infinite β-plane but also in a zonal channel - a set-up which
might be more appropriate for the ACC. However, here the
f/h contours become blocked at some critical topography
height (δ > δc = 2Y/(π|a|), Y = channel width) and then the
flow regime is not well represented by a few low-order
modes (see Olbers et al. 1992). In fact, analytical and
numerical solutions (Krupitsky & Cane 1994, Wang &
Huang 1995) of barotropic channel flow reveal a current
regime which is entirely unrealistic for application to the
ACC: the flow is in narrow frictional boundary layer
currents at the walls, switching side from south to north in a
narrow internal layer along a connecting f/h-contour. In this
blocked regime the flow is weak due to substantial drag by
form stress and the transport is independent of friction.
In the rest of this section we extend the Johnson-Bryden
concept to include the missing thermohaline forcing, and
present a linear wave barotropic-baroclinic theory of the
establishment of the bottom form stress.
4.1. Extended Johnson-Bryden type models
We proceed with the zonally averaged Eulerian model of
Section 3.3 to extend the Johnson-Bryden concept towards
the missing thermohaline component. The vertical eddyinduced flux of momentum enters via the TEM theory
(Transformed Eulerian Mean, see Andrew et al. 1987,
McIntosh & McDougall 1996) in which Eq. (10) is
augmented by a correspondingly averaged balance of
potential density ρ. Assuming stationary conditions we start
with the density balance (ρu)x + L · ρv = -Ix - L · J where (u,
v) is the three-dimensional velocity, (I, J) is the small-scale
turbulent flux of ρ, and L the (y, z) derivative. We separate
density and velocity into a zonal mean part and deviation,
e.g. ρ = {ρ} /L + ρ*. As in Section 3.3 the curly bracket
operator denotes zonal integration on level surfaces and L is
the path length. The balance of mean density B = {ρ} /L is
then obtained by zonal integration of the density balance
and expressed by
∂
∂
J (φ , B ) = − {v * ρ *} − {w * ρ *} − ∇ ⋅ J
∂y
∂z
(23)
where φ is the Eulerian overturning streamfunction, {v} =
-∂φ/∂z, {w} = ∂φ/∂y. The eddy density flux is treated as
follows: the flux vector is split in the components oriented
at the isopycnal, ({v*ρ*}, {w*ρ*}) = -φed(-Bz, By) - Kdia(By,
Bz), which introduces a diapycnal eddy-induced diffusivity5
Kdia and an eddy-induced streamfunction φed, given by
φed =
{v * ρ *} Bz − {w * ρ *} B y
∇B
2
K dia = −
{v * ρ *} B y + {w * ρ *} Bz
∇B
2
(24)
This allows Eq. (23) to be rewritten as
J(φres, B) = L · Kdia LB – L · J
(25)
The mean density is advected by a combination of the
Eulerian current and the eddies with φres = φ + φed which is
called residual streamfunction. No approximation has been
made yet, Eqs (23) & (25) are identical. According to the
common belief, however, the diapycnal flux of density by
eddies is small and so we neglect the first term on the rhs of
Eq. (25). Eliminating then the Eulerian streamfunction from
5
Since the curly bracket is zonal integration rather than average the
diffusivity gets the dimension m3s 1.
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
the momentum balance we get the TEM model
fφed – fφres = {τ0} – {τ} – R + F
J(φres, B) = –L · J
(26)
The close correspondence of this momentum balance and
the isopycnal form (6) becomes obvious when we use Kdia ≡
0 to write the eddy streamfunction as φed = - {v*ρ*} /Bz
which is the ‘heat flux equivalent’ of the interfacial form
stress, as outlined in Section 3.3. Consequently we may
view Eq. (26) as extension of the Johnson-Bryden model
Eq. (8). With the residual streamfunction it includes the
thermohaline part, it also includes Reynolds and bottom
form stresses. We shall use the model Eq. (26) below in
Section 5 to deduce the density field and the residual
streamfunction (the overturning circulation) from the
forcing of the system by wind and surface buoyancy flux.
Here we attempt to infer from Eq. (26) the magnitude of the
zonal transport.
We neglect subgrid and Reynolds stresses (which are
small) and the standing eddy term (which is small if the
mean is ACC path following, as discussed before) and use a
downgradient parametrization of the transient eddy flux,
{v'ρ'} = - KLBy, by which φed = -LKBy /Bz = LKs below the
mixed layer. The residual streamfunction is inferred from
(26), and φres, taken just below the surface mixed layer,
relates to the surface density flux ,0 by φresBy = ,0. The
two balances in (26) then lead to the relation
τ
,
Ks − 0 = O
f LB y
(27)
just below the mixed layer base. A similar relation is used
by Speer et al. (2000) to examine transformation of
watermasses around Antarctica. Assuming that mixing by
turbulence is small in the interior the slope at the mixed
layer base is related by Eq. (26) to that at greater depth and
Ks - τ0/f = const on an isopycnal (see also equation Eq. (43)
below). Thus, the relation Eq. (27) holds in the interior as
well but the terms on the lhs are taken at the respective
latitude and depth and the rhs at the corresponding
isopycnal outcrop to the south. If the meridional gradient of
density in the surface layer is known the complete interior
density field can in fact be determined from τ0 and ,0.
Marshall & Radko (2003) use Eq. (27) in this ‘diagnostic’
mode (By at the surface is given from observations) to infer
the structure of the overturning circulation in the Southern
Ocean.
Another diagnostic form of Eq. (27) is achieved by the
assumption that the vertical gradient Bz, or the BruntVaisala frequency N2 = -gBz, is known from observations.
Replacing then the meridional gradient by the vertical
current shear, fuz = gBy, we recover the Johnson-Bryden
model in an extended form,
g, O
f2
K 2 u z2 = u zτ 0 −
L
N
(28)
It now includes thermohaline forcing. If the wind stress
459
term dominates over the surface density flux term6 the
relation Eq. (19) is recovered. Under weak wind stress
conditions we find a buoyancy-driven zonal flow (if the
surface density flux ,0 is negative, as actually found over
most of the ACC, see Figs 19 & 23). Implementing the
Green-Stone parametrization into the first case yields the
above discussed square-root dependence of transport on the
wind stress, and the second case leads to a cubed-root
dependence of transport on the buoyancy forcing -g,0.
We would like to clarify that the above analysis is not a
complete transport theory. Neither By in the surface layer
nor the profile of N2 can be regarded as universally given
parameters, as they clearly will depend on the forcing τ0 and
,0. Equation (27), though valid over a large depth range if
internal turbulence can be neglected, is not sufficient to
completely determine the density field or the current profile
from the forcing functions and universal parameters. An
additional relation is needed, for example an equation which
determines the meridional profile of the slope s or of the
residual streamfunction φres beneath the mixed layer from
the forcing. This means that somewhere in the overturning
circulation diffusion and mixing must come into play mathematically speaking a non-local problem has to be
solved (the complete problem is elliptical, see Section 5).
The shortcut described by Karsten et al. (2002) and Karsten
& Marshall (2002) where φres is simply set equal to the
Eulerian transport τ0/f (or a specified fraction of it) lacks
physical grounds. A similar restriction is found in Bryden &
Cunningham (2003): in their considerations the residual
transport is entirely ignored.
4.2. The shaping of bottom form stress
With the presentation of the barotropic Charney-DeVore
model we have highlighted the role of long Rossby waves
(with wavelength of the underlying topography) in shaping
the bottom form stress. We have pointed out that the
resonant behavior in the model cannot occur in a barotropic
ocean because barotropic waves are too fast. But we can
have such a mechanism in a baroclinic set-up: baroclinic
oceanic Rossby waves have a speed comparable to zonal
velocities in the ACC. In a baroclinic version of the
Charney-DeVore model with ACC conditions Olbers &
Völker (1996) and Völker (1999) show that baroclinic
waves are generated in resonance with the topography and
become stationary when the barotropic current speed equals
the baroclinic Rossby wave speed. The transport decreases
strongly with increasing topography height, starting from a
frictionally controlled state at low heights, followed by a
transition to a complex resonant regime with multiple
equilibria at intermediate heights, and further to a state
controlled by barotropic and baroclinic bottom form stress
With ,0 = 5 x l0-9 m2 s-3, Y = 2000 km, ∆B = 5 x 10-3 ms-2 strong wind
stress is τ0 >> 2 x 10-4 m2 s-2.
6
460
D. OLBERS et al.
Fig. 20. Model results from the primitive equation channel model for configuration SIN1 [left panels] and SIN2 [right panels] (for
description see text). The three columns show experiments with horizontal eddy viscosity 104m2s-1. The top row shows the barotropic
streamfunction ψ , the second row shows the baroclinic potential energy χ divided by a mean Coriolis frequency f0, the third row shows a
zonal section of the potential temperature in the center of the channel. From Borowski (2003).
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
461
Fig. 21. Terms of the zonal momentum
balance in the channel experiments SIN1
and SIN2 of Borowski (2003), as a function
of the horizontal eddy viscosity [dotted:
friction, dash-dotted: wind stress, dashed:
baroclinic form stress, full: barotropic form
stress].
at high topography. Within the limits of such a simple model
this latter regime would be appropriate to the ACC. Though
the momentum balance Eq. (7) seems to operate here
without friction, it is important that the phase shifts of the
topographically induced pressure gradients with respect to
the topographic undulations are proportional to the
coefficients of bottom and interfacial friction of the model,
in close correspondence to the barotropic model as given by
Eqs (21) & (22). The baroclinic topographic resonance
theory determines the transport in adiabatic models in a
manner similar to the barotropic Charney-DeVore
mechanism: the bottom form stress is a complicated
resonance function of the barotropic and baroclinic
velocities and the transport follows from Eq. (7) and a
corresponding balance for the baroclinic momentum. The
structural properties of this low-order model are preserved
when the degrees of freedom are increased from the
simplest nontrivial model with 11 modes to a number
representing a moderately resolved coarse model (with 75
modes). We return to such a low-order model in more detail
below.
There have been numerous numerical studies with a
realistic ACC configuration with coarse resolution models
(Olbers & Wübber 1991, Cai & Baines 1996, Gnanadesikan
& Hallberg 2000, Gent et al. 2001, Borowski 2003) which
investigated the dependence of transport on the buoyancy
forcing at the surface, however, without revealing a clear
concept of how the transport depends on the forcing
functions. Distinguishing between the pressure forces
generated by the topographic resonance mechanism or by a
thermohaline forcing with zonal variations is non-trivial. It
must be kept in mind that a forcing by a restoring term, say
γ(Tobs(y) - T) for temperature, will always give rise to a nonzonal surface flux if the flow and thus T are non-zonal.
Borowski (2003) presents a large number of channel
experiments with a numerical primitive equation model
(MOM with 1° meridional x 2° zonal grid and 16 levels)
studying the sensitivity to forcing and frictional parameters.
The forcing fields (wind stress and Tobs) are strictly zonal
but restoring is used for temperature. Figure 20 displays the
streamfunction, potential energy and a zonal section of
temperature through the middle of the channel for two
experiments which differ only in the restoring surface
temperature (10°C gradient across the channel for
experiments SIN1, 20°C for SIN2). However, the
momentum balance is drastically different (see Fig. 21): in
SIN1 the baroclinic form stress drives the eastward flow
(clearly visible in Fig. 20: as in Fig. 11 the water on the
western slope is lighter compared to the eastern slope), in
SIN2 it decelerates (here the lighter water is on the eastern
slope), in both cases the baroclinic form stress is in
opposition to the barotropic form stress. It seems that the
stronger gradient of surface restoring temperature puts the
system into a different regime by implementing a stronger
non-zonal thermal forcing.
Analytical theories of the ACC which include
baroclinicity are mostly done in the quasi-geostrophic
framework and use a modal truncation: the layer
streamfunctions are represented by a set of structure
functions for the (x, y)-dependence so that the dynamical
equations are written only as ordinary differential equations
for the time dependence of amplitudes. If the system is
truncated to a few modes, a so called ‘low-order model’ is
obtained and steady states can be investigated analytically
using the mathematics of dynamical system theory or by use
of numerical bifurcation tools (Dijkstra 2000). The
Charney-DeVore-type models discussed above are of such
kind. Here, we extend the model of Völker (1999) by adding
thermohaline forcing and implementing lateral diffusion of
momentum (more details are given in Borowski 2003). A
two-layer zonal channel with quasigeostrophic dynamics is
considered, with a sine-shaped topography in the zonal
direction. The two streamfunctions are expressed by
suitably chosen sinusoidal structure functions. After some
eliminations the resulting system is written in terms of six
amplitudes representing the barotropic transport T (upper
layer plus lower layer transport), the baroclinic transport S
(upper layer minus lower layer transport), and respective
barotropic and baroclinic sine and cosine components, Ts, Ss
and Tc, Sc (more accurately, the T and S quantities are based
on barotropic and baroclinic velocities). The latter generate
the barotropic and baroclinic form stress, respectively. It is
clear that with such an enormous reduction, transient eddies
are not present. Their effect on the downward transfer of
momentum - the interfacial form stress IFS - is
parameterized by friction acting on the interface of the two
462
D. OLBERS et al.
layers with a coefficient κ. The physical mechanism of this
friction is completely equivalent to a diffusion of layer
thickness (see Gent & McWilliams 1990, Gent et al. 1995).
In addition we have lateral diffusion of momentum with a
viscosity ε. We have scaled these variables7 and the
dimensionless form of the set of differential equations
becomes
∂T
= W0 + δ (Tc − hS c )− εT
∂t
(29)
∂S λ2W0 λ2
Q
=
−
δ (Tc − hS c )+ l[Tc S s − Ts S c ]+ 0 − (κ + λ2ε ) S
∂t
1− h
1− h
h
∂Tc
= β Ts − 2δ (T − hS )− m[TTs + h(1 − h) SS s ]− εTc
∂t
(30)
(31)
2
∂S c
2λ
= βλ2 S s +
δ (T − hS )− n[TS s − STs ]− (κ + λ2ε ) S c
∂t
1− h
(32)
∂Ts
= − β Tc + m[TTc + h(1 − h) SS c ]− εTs
∂t
(33)
∂S s
= − βλ2 S c + n[TS c − STc ]− (κ + λ2ε ) S s
∂t
(34)
The forcing appears in W0 (wind stress) and Q0 (external
buoyancy flux). Terms derived from nonlinear advection are
found in the cornered brackets (R,m,n are numerical
coupling coefficients depending only on the channel
dimensions, see footnote). The respective term in the zonal
baroclinic balance Eq. (30) is the interfacial form stress
(IFS) induced by the standing eddies. The κ-term in that
equation is the corresponding transient eddy interfacial
form stress. The baroclinicity enters via the internal Rossby
radius λ, and the topography height is δ. Lateral friction
operates in the barotropic T-equations and interfacial
friction in addition in the baroclinic S-equations. The W0term in the baroclinic Eq. (30) arises from the Ekman
pumping acting on the background stratification. The sine
and cosine Eqs (31) to (34) describe planetary-topographic
Rossby waves with the same zonal wave number as the
topography; β is a scaled planetary coefficient df/dy. There
are terms arising from nonlinearities (advection) and
diffusion. Note that Eq. (29) is the balance of vertically
integrated momentum, and is congruent to Eqs (7) & (10).
7
The scaling and coefficients are as follows: All transport variables are
scaled by π2 /(|f0|Y2) to yield dimensionless T, S, Tc,…. Parameters are b
= Y/(πL), ε = π2Ah/(b|f0|Y2),κ = π2K/(b|f0|Y2), λ = πR/Y, β = 2Ydf/dy/|f0|
where L is the zonal length and Y the width of the channel. R = √(g' / f02 )
H1H2/H is the internal Rossby radius of a 2-layer fluid with densities ρ1,
ρ2, mean layer thicknesses H1,H2,H = H1 + H2 and reduced gravity g' =
g (1 - ρ1/ρ2), furthermore h = H1/H is a thickness ratio. Coupling
coefficients are R = 3π2b2/8, m = 64π2/3, n = 16/3. The scaled depth is 1 +
(δ/π) sin(2πx/L), thus δ/π is the relative height of the topography. The
forcing amplitudes are W0 = (3/16)π2τ0/(bHYf02), Q0 = (3/32)π3
B0/(b|f0|3Y2) in terms of the zonal wind stress τ = τ0 sin2(πy/Y) and the
surface buoyancy flux B = B0 cos(πy/Y) sin(πy/Y). Here,τ and B are
dimensioned m2 s-2 and m2 s-3, respectively.)
We can identify the barotropic and baroclinic contributions
to the bottom form stress.
Considering the steady state of the above model we
recover the most important physical mechanisms which we
have outlined in this article.
The barotropic form stress drag: a barotropic state is
obtained for λ = Q0 = 0. Then all S-fields are identically
zero and a barotropic solution as described by Eqs (21) &
(22) emerges in which the transport decays quadratically
with increasing topography height due to the drag of the
barotropic form stress.
Johnson-Bryden dynamics: if lateral friction is small and
the influence of topography and standing eddy IFS are
ignored we regain from Eq. (30) the Johnson-Bryden state
Eq. (28),
κ
W
Q
S= 0+ 2 0
λ2
h λ (1 − h)
(35)
in which the transport contained in the shear current is
governed by wind stress and buoyancy flux.
The compensation of barotropic and baroclinic form
stresses: for strong stratification (large λ) or vanishing
nonlinearities (cornered bracket terms neglected, i.e. no
advection) we learn from the zonal mean baroclinic balance
Eq. (30) that if the form stress terms δTc and δhSc are
individually increasing with topography height they must
compensate since the remaining terms in the balance do not
increase. The reason for the compensation of the barotropic
and baroclinic form stresses, discussed in detail in Section
3.3, is thus found in the compensation of the vertical lifting
or lowering of the background stratification (given by λ2)
by the pumping induced by the Ekman velocity and the
barotropic and baroclinic vertical velocities of the flow (the
first and second terms on the rhs of Eq. (30)), the latter two
being generated by the flow passing across the topography
(they are proportional to δ). It is not clear yet, however,
which of the two form stresses drives and which brakes the
zonal flow and under what circumstance they would
increase with height of the barriers.
Breaking of ‘Hidaka’s dilemma’ by stratification and
eddies: adding Eqs (29) & (30) in a suitable way to
eliminate the form stress parts we find the balance of upper
layer transport T + (1 - h) S,
W0 Q0
1− h
+
− ε (T + (1 − h )S ) = 2 (κS + l[Tc S s − Ts S c ])
h λ2
λ
(36)
and eliminating the wind forcing between Eqs (29) & (30),
we get the balance
δ
hQ0
−
+
(T − hS )− ε (T − hS )= − h2 (κ S + l[Tc S s − Ts Sc ]
(1 − h )λ2 1 − h c c
λ
(37)
for the lower layer transport T - hS. On the rhs we find the
IFS due to transient and standing eddies.
Suppose for the moment that the IFS are small or the
463
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
Rossby radius is large (strong stratification). We then arrive
at the physically intuitive statement that the transport in the
surface layer, T + (1 - h)S, is decoupled from the
topography: it is given by the wind stress and buoyancy flux
acting in the above combination against lateral friction. The
surface layer transport is then in a ‘Hidaka’-type state
(inversely proportional to the eddy viscosity ε and thus large
for reasonably sized ε). The transport of the lower layer, T hS, is governed by the magnitude of the bottom form stress
and external heating Q0. This Hidaka dilemma must be
resolved by weak stratification and large IFS from transient
or/and standing eddies. Notice that the crucial term [TcSs TsSc] of the standing eddies is nullified for the flat bottom
wave solution of the Eqs (31) to (34). It is, however,
supported by linear topographic waves. The wave equations
yield after some manipulations
Tc S s − Ts S c = −4(δ / β ) 2
(T − hS ) 2
βλ2 ((ε / β ) 2 + 1)((ϑ / β ) 2 + 1)
(38)
κ
The standing wave IFS is - in this approximation - negative
if the topography is undulated and the bottom flow nonvanishing, it must transfer eastward momentum downward.
It is worth noticing that it depends marginally on the
viscosity ε but is entirely supported by the interfacial
friction κ.
The form stress terms δTc and δhSc arise by the response of
the barotropic and baroclinic wave system, described by the
four Eqs (31) to (34), to the zonal flow crossing the
topography (there is no direct forcing because the external
heating function was assumed strictly zonal). A reasonable
solution of the model should yield transports in the range
β » T, S / βλ2, meaning that the flow velocities are much
less than the speeds of barotropic Rossby waves but
supercritical with respect to the baroclinic waves, as in ACC
conditions. Then the nonlinearities in Eqs (31) & (33) are
small and these equations represent a linear barotropic
planetary-topographic wave. The barotropic form stress δTc
can thus be explained by long linear Rossby waves
generated by the deep current, T - hS in the above model,
crossing the large ridges blocking the circumpolar path of
the ACC. In contrast, the baroclinic form stress δhSc might
be governed by a nonlinear response, according to T, S /
βλ2 in (32) and (34). We discuss the consequences further
below.
Still it is worth considering the linearized version of the
above model.
4.3. A linear transport model
We thus neglect the standing eddy IFS in Eq. (30) and the
advection terms in the wave equations. The form stress
terms become
2 ε (T − hS )
2 ~ ϑ (T − hS )
δhS c = 2(δ / β ) h
δTc = −2(δ / β )
(ε / β )2 + 1
(ϑ / β )2 + 1
(39)
~
with ϑ = κ / λ2 + ε , h = h / (1 − h ) . The barotropic form stress
is thus supported by lateral friction, the baroclinic by
interfacial friction and stratification. However, both extract
eastward momentum from the flow if the deep transport is
eastward: the linear wave model thus does not imply the
afore mentioned form stress compensation effect. The total
transport T and the shear transport S are readily evaluated as
lengthy expressions of δ, ε, κ and λ,
~
T = AW0 + B (W0 + h Q0 / λ2
~
hS = C W0 + D (W0 + h Q0 / λ2 )
(40)
where the direct wind effect and the baroclinic forcing have
been separated. The factors attached to the forcing
amplitudes are due to friction and form stress by
topographically induced Rossby waves. These transport
response functions are
ϑ + 2(δ / β )2 h~η 
A


 
2


(
)
B
2
δ
/
β
η
1
 


=
C 
~
2~
2

  εϑ + 2(δ / β ) η ϑ + h ε  2(δ / β ) h η
 D


2
 
 ε + 2(δ / β ) η  (41)
~
2
2
with η = ε / 1 + (ε / β ) + h ϑ / 1 + (ϑ / β ) . There are three
sources of transport according to the relation Eq. (40): T and
hS are generated by direct wind forcing, indirect wind
forcing via Ekman pumping on the mean stratification, and
buoyancy flux. For strictly barotropic conditions where λ 6 0
we get ϑ 6 ∞ and the Charney-DeVore result Eq. (22) is
(
(
)
)
(
)
Fig. 22. Left: height dependence of the response
functions A [full], B [dashed], C [dashdotted] and
D [dotted] of the linear transport model. Middle:
the contributions to the total transport T of direct
wind forcing [full], Ekman pumping [dashed] and
heating [dashdotted]. Right: same for shear
transport hS. Parameters are: W0 corresponds to
0.1 Nm-2, Q0 to 10 Wm-2 and λ to a Rossby radius
of 10 km, furthermore h = 0.25, Ah = 104 m2 s-1, K
= 500 m2s-1.
464
D. OLBERS et al.
δhS c =
Fig. 23. The forcing data obtained by an ACC-path following
average from the NCEP analysis (wind stress: full, units 10-4
m2 s-2, density flux: dashed, units 10-6 kgm-2 s-1). From Olbers &
Visbeck (2004).
regained.
The response functions A, B, C and D are displayed in
Fig. 22 as height dependence for typical parameters. All
functions flatten out to a plateau at large topography heights
but have different critical heights above which this happens.
The critical height of the barotropic response is clearly δ ~ β
(see also the Charney-DeVore model Eq. (22)). The
baroclinic response scale depends very much on the sizes of
ε, κ and λ. The figure also elucidates the shares of the
different transport sources for typical W0 and Q0 in the total
and the shear transports. It becomes clear that only below
quite moderate heights of the topography we find the direct
wind effect dominating. At larger heights and for the present
parameters the Ekman pumping acting on the stratification
is the most important driving agent of transport. Only if
~
h Q0 / λ2 becomes of the order of W0 the buoyancy forcing
is getting effective for the transport.
Finally we consider a nonlinear extension of the above
model. We want to improve on the baroclinic bottom form
stress and thus still neglect the standing eddy IFS. Including
now all terms in the baroclinic wave Eqs (32) & (34) the
baroclinic bottom form stress becomes
ϑ
nhS 

2~
 2(δ / β ) h (T − hS )+ (δ / β )Tc 2 
ελ 
(ϑ / β )2 + 1 − (nT / βλ2 )2 
(42)
and it is evident that supercritical conditions, T > βλ2, can
indeed lead to a negative baroclinic bottom form stress
which then drives the current eastward (if the bottom flow
T - hS is eastward and the last term on the rhs is small actually requiring a large viscosity ε as indicated by the
numerical experiments in Fig. 21). We conclude that the
supercriticality of the ACC with respect to baroclinic
Rossby waves is essential for the system to achieve the
observed balance of zonal momentum in which wind and
baroclinic bottom form stress drive and barotropic form
stress decelerates the current.
5. The meridional overturning
Much of the recent perception of the ACC circulation is
centered on the meridional overturning and ventilation of
water masses in the Southern Ocean. The classical view
(Sverdrup et al. 1942) of water mass storage and spreading
is sketched in Fig. 4 (Gordon 1999) and repeated in Fig. 12
where the role of eddies in the unblocked part of the
watercolumn is highlighted. We have pointed out at various
places in this paper that eddies and turbulent mixing might
accomplish a major task in shaping and balancing the
overturning circulation and it remains to put up a simple
strawman model to demonstrate how it might work.
We follow the concept presented in Olbers & Visbeck
(2004) which extends the work of Marshall & Radko (2003)
to a predictive model of the overturning. We assume that all
mixing and watermass formation processes take place in an
upper layer of the ocean - basically a turbulent layer where
Fig. 24. Left: sketch of the structure function T(z). The depth z = -d marks the base of the upper frictional boundary layer; z = -h is the
bottom (maximum depth along circumpolar path), and below z = -D the watercolumn is blocked by topography. Middle: the Euler
streamfunction for the average wind stress from Fig. 23 and the structure function T(z). Right: the averaged potential density [white
contours] on top of the salinity (ACC-path following average, computed from the SAC climatology). According to the eddy-adiabatic
model presented here, the isopycnals coincide with the residual streamfunction pattern. From Olbers & Visbeck (2004).
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
Fig. 25. Solution for parameter values Ks = 1000 m2s-1 at the
southern boundary, linearly increasing by a factor 5 towards the
northern side, αd = 8.44 x 10-7 ms-1, αa = 3 x 10-6 ms-1, d = 100
m, a = 500 m. The upper two panels display the resulting density
fields Bd and Ba [both full] and the middle panels show the
slopes [both full]. They are compared to the SAC climatology
~
[dotted]. The deep B density is dashed [upper right panel]. The
lower two panels show the streamfunctions and pumping
velocities at the shear layer base z = -a: Ekman streamfunction
[dashed], eddy streamfunction [dashdotted], residual
streamfunction [full], eddy streamfunction from SAC
climatology [dotted], same for the pumping velocities. From
Olbers & Visbeck (2004).
Ekman transport and pumping is established by the wind
and buoyancy is imprinted on the surface waters by heat and
freshwater flux from the overlying atmosphere - while the
ocean interior is void of turbulence but eddies are present
that transport and mix substances along isopycnals. We
465
refer to this concept as ‘adiabatic eddy regime’ and see it as
an extreme scenario. The real ocean might have substantial
mixing by turbulence in the interior as well (see Heywood
et al. 2002, Naveira Garabato et al. 2004). Eddies might
contribute by a diapycnal flux to watermass formation, too.
However, the above simplified view allows for an analytical
treatment. We choose set-up where the equations are
averaged along a mean ACC path. We thus neglect the
standing eddy component and use the TEM model derived
in Section 4.1 with a downgradient parameterization of the
meridional density flux by transient eddies, {v'ρ'} = - KLBy.
This casts the interior eddy streamfunction into φed = Ks
where s = -By/Bz is the isopycnal slope. We neglect the eddy
Reynolds stress in (26) and assume that the frictional stress
τ acts as a body force which is nonzero only in the surface
Ekman layer. Since wind stress and bottom form stress must
balance, as discussed in Section 3.3, the amplitude of the
form stress is that of the wind stress. Consequently, the
Eulerian streamfunction is set by the wind stress and,
abbreviating the Ekman transport by M = -{τo}/(Lf), we
finally get φ = MT(z) and φres = MT(z) + Ks where T(z) is a
structure function as sketched in Fig. 24.
With no turbulence in the interior we have J ≡ 0 and the
density budget of (26) becomes J(φres, B) = 0, saying that
residual streamlines and isopycnals coincide. This can be
written as a differential equation for the slope,
∂
(Ks + M )+ s ∂ Ks = 0
∂y
∂z
(43)
The isopycnals are the characteristics of this equation.
Initial data of s are required on some non-isopycnal curve,
in our case this will be the depth level z = - a below which
turbulence stops acting. Because s is infinite in the mixed
layer where B = Bd(y) we assume between the mixed layer
Fig. 26. Left panel: the density field for the upper layer simulation of Fig. 25. The eddy diffusivity is Ki = 500 m2s-1 at the southern boundary,
linearly increasing by a factor 5 towards the northern side and vertically constant. For the upper 500 m the solutions Ba and Ba have been
converted to contours, below the characteristics are coloured according to the corresponding density. At black characteristics upward and
downward motion changes direction. Right panel: the resulting density values on the northern boundary at -40°S [black and grey curve;
grey is upwelling] in comparison to the SAC densities [data from 42°S to 38°S; dashed profiles]. From Olbers & Visbeck (2004).
466
D. OLBERS et al.
Fig. 27. A sketch of the ACC system showing the zonal flow and the meridional overturning circulation and watermasses. Antarctica is at the
left side. The east-west section displays the isopycnal and sea surface tilts in relation to submarine ridges, which are necessary to support
the bottom form stress signatures discussed in the text. The curly arrows at the surface indicate the buoyancy flux, the arrows attached to
the isopycnals represent turbulent mixing. Sinking of Antarctic Intermediate Water (AAIW) is not shown (see Fig. 24). Redrawn using a
figure from Speer et al. (2000).
base z = - d and the depth z = - a an intermediate layer with
finite slopes. As in the mixed layer we implement a
prescribed structure of the density field in the ‘slope layer’
- d ≤ z ≤ - a,
B (y, z ) = Bd (y )
z+d
z+a
− Ba (y )
a−d
a−d
(44)
The slope at z = - a is then sa = (a - d)( ∂Ba/∂y)/(Ba - Bd).
It remains thus to determine the upper layer densities
Bd(y) and Ba(y). In this depth range the complete nonadiabatic density balance Eq. (23) must be applied. We
insert the density structure (mixed layer and slope layer)
and deduce a coupled advective-diffusive set of balance
equations,
∂ 
∂B
 Km d
∂y 
∂y

∂B
 = −, O + (M + Ed ) d − α d (Ba − Bd )
∂y

∂  ∂Ba  ∂  ∂Bd 
∂M
 Ka
 +  Kd
=−
(Ba − Bd )+ Ea ∂Ba − Ed ∂Bd +
∂y 
∂y  ∂y 
∂y 
∂y
∂y
∂y
~
+ α d (Ba − Bd )+ α a Ba − B
(
)
(45)
Advection by the eddy streamfunction enters as Ed = φed
(z = - d) = K(z = - d)sd and Ea = φed(z = - a) = K(z = - a)sa.
Apart from advection eddies also lead to diffusion of
density: the diffusivities on the lhs are vertical integrals of
K(y,z). The equations describe our basic perception of the
processes that happen in the upper layer. The mixed layer
density is forced by the surface density flux ,0 from
exchange by heat and freshwater with the atmosphere (we
use the forcing data shown in Fig. 23). There is meridional
advection by the Eulerian Ekman currents and by eddies,
there is vertical pumping by these agents (eddy pumping
would appear if the advection terms are split into complete
divergences and vertical advection), with mixing by
turbulence to sustain the vertically mixed state Bd(y), and
entrainment of density at the mixed layer base,
parameterized by a mixing coefficient αd. Corresponding
~
terms arise in the slope layer balance. The density B that is
entrained into the slope layer from below is the density at
depth ranges with inward flow at the northern boundary of
the model domain, e.g. NADW at 40°S. This water proceeds
up the isopycnals to the slope layer base without changing
its density. The coupling to the interior ocean occurs via the
predicted slopes at the base of the slope layer.
The performance of the model is exemplified in Figs 25 &
26, using reasonable parameter values (see figure caption).
It should, however, be borne in mind that the solution
structure might change substantially if another choice is
made, in particular in the interior where the slopes are
inversely proportional to the value of the eddy diffusivity K
and changes at some level influence via the characteristics
the entire deeper structure. For the simulation shown in
Fig. 26 we have used the most simple form oriented at the
diffusivity estimates discussed in Olbers & Visbeck (2004):
CIRCULATION OF ANTARCTIC CIRCUMPOLAR CURRENT
K(y, z) is vertically constant and a linear function of y with
significant increase (by a factor of 5) towards the north.
This particular solution yields an up-down-up-down pattern
of pumping (by eddies and Ekman) at the slope layer base
(see the solid curve in the lower right panel of Fig. 25)
which in this structure may be associated with upwelling
NADW, downwelling AAIW and upwelling Subantarctic
mode water (the latitudes and densities are roughly
consistent with this interpretation). Propagation of the upper
layer solution via the characteristic Eq. (43) into the interior
yields the associated isopycnals (and residual
streamfunction) which also coincide well with the observed
interior density structure (see Fig. 24 and the left panel of
Fig. 26). We would like to point out that the eddy field has a
dominant share in this simulation: without eddies the
streamfunction and densities would mirror the Deacon cell
depicted in Fig. 24. This is also evident from the
streamfunctions and associated pumping velocities of the
solution shown in the lower panels of Fig. 25.
6. Conclusions
There are five threads running through this review of the
ACC system: submarine topography, standing and transient
eddies, long barotropic and baroclinic Rossby waves,
turbulent mixing and surface flux of momentum and
buoyancy. They are woven into the physics governing this
extraordinary current, attempting to contribute to an answer
of the most interesting and most urgent questions about the
ACC: what is the balance of the zonal momentum; what
mechanism and forcing functions determine the transport;
how do watermasses and the substances they carry penetrate
the strong and deep-reaching zonal flow? We base our
discussion on the research on these topics which has
accumulated in the last decade and partly answer these
questions. Our tools are observations, theory and models
coming from a variety of instrumental techniques, field
expeditions and modelling concepts. Fig. 27 gives a sketch
of the physics in the ACC system showing its most
important ingredients.
The flow achieves a balance of its zonal momentum in
which the input of eastward momentum from the wind
stress acting all the way around Antarctica is transferred by
standing and transient eddies through the water column to
the bottom where the bottom pressure field adjusts such that
bottom form stress acts as a sink. Clearly, if adjustment is
not present, the imbalance will accelerate the current,
inducing changes of the pressure field until a balance has
been reached. Separating this form stress into the one
arising from the surface pressure and the one due to the
internal mass stratification it is seen that these individual
components overwhelm the wind stress by an order of
magnitude, the barotropic form stress is retarding the
eastward current and the baroclinic is accelerating it (see
east-west interface of Fig. 27). Both, however, must
467
compensate to a high degree, a constraint that arises from
the balance of the zonal mean isopycnal thickness structure
where vertical pumping by the Ekman, barotropic and
baroclinic velocities of the current acting on the mean
stratification are dominating and must cancel each other.
The large scale vertical flow comes about by the zonal flow
passing across the large-scale topographic barriers, mainly
the midocean ridges along the path of the ACC in the
Southern Ocean. We have found evidence that the driving of
the eastward current by the baroclinic bottom form stress
needs a supercriticallity of the flow with respect to the
associated long planetary Rossby waves. They must be
advected eastward by the current to become locked with the
required phase shift to the zonal topographic undulations.
The downward transport of zonal momentum is
established by the mechanism of interfacial form stress by
which pressure forces act across the isopycnal interfaces
which are tilted by the action of transient and standing
eddies. The form stress is associated with a meridional eddy
density (or heat) flux, and a downward flux of eastward
momentum supports a poleward heat flux, in agreement
with observations and eddy-resolving models. The vertical
divergence of interfacial form stress drives the meridional
overturning circulation. In isopycnal ranges which are
zonally unblocked by submarine topography the divergence
of the interfacial form stress is the dominant driving force of
the meridional flow - it is eddy-driven as indicated by the
wavy arrows at intermediate depths on the frontside of
Fig. 27. Below in the blocked range of isopycnals we have
geostrophic meridional flow in the valleys between the
submarine ridges, these are supported by the pressure
gradients associated with the bottom form stress. And above
the eddy-driven regime we find the northward Ekman flow
which is driven by the eastward zonal wind. Clearly, the
balance of momentum and the forces driving the meridional
overturning circulation correspond to each other (in a
mathematical frame they are described by the same
equations).
The isopycnals in the Southern Ocean connect the deep
ocean to the north of the ACC to the surface areas to the
south, the ACC being attached to stronger tilts correlated
over the depth. We have turned existing concepts on
processes shaping the isopycnal stack in the Southern
Ocean into a prognostic theory. Also for these processes we
refer to Fig. 27. The Eulerian mean flow and the eddies
combine to transport density (heat and substances) to and
from the upper ocean layer where mixing by small-scale
turbulence and exchange of heat and freshwater with the
atmosphere must occur. Two assumptions imply that the
mean transport in the interior - below the mixed layer - by
the mean flow and eddies is entirely along the isopycnals.
These are: eddies do not carry stuff across isopycnals, only
along them; diapycnal mixing by small-scale turbulence is
absent below the mixed layer. Then streamlines of transport
by mean flow and eddies, representing the residual
468
D. OLBERS et al.
circulation, coincide with isopycnals. The concept allows
the complicated mathematics of an advection-diffusionmixing regime to be broken into manageable parts - mixed
layer physics with Ekman and eddy advection and a diabatic
interior - which may be solved by simpler means. We are
able to predict the density field - the decrease of surface
density with increasing latitude and the shape of the
downward sloping isopycnals - from the wind field and
buoyancy flux through the ocean surface. The theory
applies to an ACC path following average and thus is rather
qualitative but it demonstrates the overwhelming
importance of the transient eddy field in shaping the density
field in the Southern Ocean.
The last concern of our review is the issue of transport of
the ACC. Over many decades the current was considered
basically wind-driven and its transport attributed to the
direct action of the wind over the Southern Ocean as the
most prominent driving agent, though early experiments
with global OGCMs clearly showed the importance of
baroclinicity on the ACC transport. A more detailed
transport theory is recapitulated below in two steps.
If a mean stratification is considered as a given
horizontally uniform background (a mean Brunt-Vaisala
frequency profile or an isopycnal layer stack) the effect of
the wind stress is not limited to a direct driving of currents
by friction but there is in addition the Ekman pumping
acting to deform the stratification. Together with a
prescribed input of buoyancy (say, cooling in the south and
heating in the north) this Ekman pumping sets up a
baroclinic pressure force which gives rise to a baroclinic
component of the current. Feedback by bottom form stress
(the Rossby wave connection mentioned above) directly
influences the zonal transport. In summary, the ACC
transport has a direct contribution from the wind stress, an
‘indirect’ contribution from the Ekman pumping on the
mean stratification, and a contribution from the prescribed
surface buoyancy flux. The latter two contributions only
appear if the topography is undulated (if bottom form stress
may act). Our simple linear transport model elucidates these
mechanisms and clearly shows that for sufficiently high
topography amplitudes the indirect wind forcing and the
external heating may be dominant over the direct wind
effect.
In a second stage we realize that the stratification is not
given but depends on many processes, among them also
features of the local wind stress and surface buoyancy flux
not considered so far. Non-zonal pattern in these fluxes (e.g.
more cooling in the Atlantic sector of the Southern Ocean
than elsewhere) may generate zonal pressure forces in the
bottom form stress and influence the transport. There is also
the possibility of remote control via the NADW/conveyor
belt connection. There are other second order effects such as
the influence of the winds on turbulence and mixing and
regional differences in the transient eddy energy. We did not
attempt to quantify these effects on the ACC transport.
The concepts and models discussed in this review would
greatly benefit from extended knowledge in some specific
areas. From the experimental and field work community we
need information about the level and regional distribution of
diapycnal mixing in the area south of the ACC, in particular
where in the water column it occurs predominantly.
Furthermore we would profit from better knowledge about
the parametrization of isopycnal eddy fluxes which
certainly needs a combination of theoretical work and eddyresolving modelling efforts in idealized and realistic
configurations. Finally, we must admit that many of the
considerations in this review are based on relatively simple
models and model diagnostics. We certainly would profit
from improved diagnostic utilization of realistic eddyresolving models toward an understanding of mechanisms
outlined in this article, e.g. the establishment of form stress
in the interior and at the bottom by waves and eddies.
Acknowledgements
We appreciate numerous discussions with Carsten Eden and
Sergey Danilov and are grateful for the extensive and
helpful comments of three referees. We greatly benefited
from Stephen Rintoul’s comments on the manuscript.
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