Journal of Marine Research, 69, 167–189, 2011
Spin-up and adjustment of the Antarctic Circumpolar
Current and global pycnocline
by Lesley C. Allison1 , Helen L. Johnson2 and David P. Marshall3
ABSTRACT
A theory is presented for the adjustment of the Antarctic Circumpolar Current (ACC) and global
pycnocline to a sudden and sustained change in wind forcing. The adjustment timescale is controlled
by the mesoscale eddy diffusivity across the ACC, the mean width of the ACC, the surface area of the
ocean basins to the north, and deep water formation in the North Atlantic. In particular, northern sinking
may have the potential to shorten the timescale and reduce its sensitivity to Southern Ocean eddies, but
the relative importance of northern sinking and Southern Ocean eddies cannot be determined precisely,
largely due to limitations in the parameterization of northern sinking. Although it is clear that the main
processes that control the adjustment timescale are those which counteract the deepening of the global
pycnocline, the theory also suggests that the timescale can be subtly modified by wind forcing over the
ACC and global diapycnal mixing. Results from calculations with a reduced-gravity model compare
well with the theory. The multidecadal-centennial adjustment timescale implies that long observational
time series will be required to detect dynamic change in the ACC due to anthropogenic forcing. The
potential role of Southern Ocean mesoscale eddy activity in determining both the equilibrium state of
the ACC and the timescale over which it adjusts suggests that the response to anthropogenic forcing
may be rather different in coupled ocean-atmosphere climate models that parameterize and resolve
mesoscale eddies.
1. Introduction
The Antarctic Circumpolar Current (ACC) plays a pivotal role in the global ocean and
climate system. The ACC is associated with steeply tilted isopycnals, exposing abyssal water
masses to the surface on the southern flank of the current; interactions with the overlying
atmosphere and sea ice lead to the formation of water masses that fill a substantial part of
the global ocean volume. Water masses formed on the northern flank of the ACC sequester
large amounts of heat from the atmosphere (Gille, 2002), with the potential to influence
the global response to anthropogenic warming. The outcropping isopycnals in the Southern
Ocean also provide an important pathway for the uptake of anthropogenic carbon dioxide
1. NCAS-Climate, Department of Meteorology, University of Reading, Earley Gate, Reading, RG6 6BB,
United Kingdom. email: [email protected]
2. Department of Earth Sciences, University of Oxford, South Parks Road, Oxford, OX1 3AN, United Kingdom.
3. Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU,
United Kingdom.
167
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(CO2 ) into the ocean interior (Sabine et al., 1999; Sabine et al., 2004; Caldeira and Duffy,
2000). The climate system is therefore sensitive to changes in ocean circulation at high
southern latitudes, with the potential for feedback mechanisms altering the rate at which
CO2 is removed from the atmosphere (Mignone et al., 2006).
The ACC is driven by a complex nonlinear combination of forcing mechanisms, and
obtaining a clear theory for its transport remains a significant challenge (see e.g., Rintoul,
Hughes, and Olbers, 2001). The westerly winds play an important role, accelerating the
current by transferring momentum to the barotropic (depth-independent) component of the
flow. As well as this, the winds also drive a northward Ekman transport at the surface, which
is an important part of the meridional circulation in the Southern Ocean. The vertical Ekman
pumping associated with divergence to the south of the ACC and convergence to the north
impacts upon the stratification and causes isopycnals to tilt upward toward the Antarctic
continent. Through thermal wind balance, this effect on the stratification drives an eastward
baroclinic transport, and is an indirect mechanism by which the wind drives the ACC (e.g.,
Gnanadesikan and Hallberg, 2000).
Since the baroclinic component of the ACC is associated with meridional density gradients across the Southern Ocean, then in addition to the indirect effect of the wind, any
process that affects the meridional slope of isopycnal surfaces and the depth of the pycnocline to the north can play a role in controlling the strength of the ACC. The transport
can therefore be influenced by processes such as surface heat and freshwater forcing
(Saenko et al., 2002; Hogg, 2010), mesoscale eddy processes (Karsten et al., 2002),
diapycnal mixing (Gnanadesikan and Hallberg, 2000; Munday et al., 2011), the formation rate of Antarctic Bottom Water (AABW) (Cai and Baines, 1996; Gent et al., 2001),
and the rate at which North Atlantic Deep Water (NADW) is formed and imported into the
Southern Ocean (McDermott, 1996; Gnanadesikan and Hallberg, 2000; Fučkar and Vallis,
2007).
In recent years significant progress has been made toward understanding how the transport
of the ACC varies on timescales ranging from subseasonal to interannual. The subseasonal
variability is predominantly barotropic and observable using bottom pressure recorders
(Whitworth, 1983; Meredith et al., 1996). Hughes et al. (1999) showed that transport variability on timescales between 10 and 220 days is dominated by a barotropic mode that
follows contours of f/H (where f is the Coriolis parameter and H is the ocean depth),
and that this variability is linked to variations in the wind stress which alter the meridional
sea level slope. Observational evidence for this barotropic response to high frequency wind
variability was provided by Gille et al. (2001).
Baroclinic processes become important for ACC variability on timescales longer than
around a year. Using a coarse resolution coupled model, Hall and Visbeck (2002) showed
that interannual wind stress variability associated with the Southern Annular Mode (SAM)
affects the baroclinic ACC transport via the effect of Ekman pumping on the isopycnal slope.
Meredith et al. (2004) provided observational evidence for this link between interannual
ACC transport variability and the SAM.
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Allison et al.: Spin-up & adjustment of ACC
169
Quantitative theories for the adjustment of the ACC to a change in wind stress have
been put forward. Using three years of bottom pressure measurements, Wearn and Baker
(1980) suggested a simple relaxation model for the fast barotropic adjustment mode. Their
model is based on an increase in dissipation which eventually balances the increased ACC
momentum associated with strengthened winds and predicts an adjustment timescale of
around a week. Olbers and Lettmann (2007) extended the Wearn and Baker (1980) model
by introducing a mechanism for baroclinic adjustment. They linked the ACC transport to
bottom form stress and the baroclinic potential energy stored in the density field to produce
a linear model that has both barotropic and baroclinic components. Their model has two
adjustment modes, and yields a barotropic timescale of a few days and a baroclinic timescale
of 2–3 years.
While much has been learned about the short-term adjustment of the Southern Ocean
to local changes in wind forcing, far less is known about how the ACC may respond
to changes in wind forcing on timescales longer than interannual. This is an important
question for understanding the response of the global ocean to anthropogenic forcing, in
terms of both change in the meridional circulation across the ACC, which affects the oceanic
uptake of anthropogenic carbon (e.g., Le Quéré et al., 2007; Lovenduski and Ito, 2009), and
change in the stratification to the north of the ACC (e.g., Böning et al., 2008), which is an
important control on global ocean heat content. Furthermore, understanding the timescale
of ACC equilibration is central to understanding the phase-lag between the temperature
changes seen in Greenland and Antarctic ice cores during glacial cycles (Blunier et al.,
1998; Broecker, 1998; Stocker, 1998; Barker et al., 2009), as well as for determining the
time over which ocean models must be integrated to reach equilibrium.
Theoretical understanding for how the ACC may adjust to a change in forcing is based
on local Southern Ocean dynamics. Indeed, many of the numerical modeling studies that
examine the ACC’s transient adjustment to wind stress changes use model domains of
limited latitudinal extent. In these channel model studies, there is the assumption that at
equilibrium, the northward Ekman transport is balanced by a southward eddy return flow,
as in Johnson and Bryden (1989). However, many studies (e.g., Gnanadesikan and Hallberg, 2000; Fučkar and Vallis, 2007) have demonstrated that the strength of the ACC at
equilibrium is influenced by processes that occur outside the Southern Ocean, so it is reasonable to expect that the global ocean may have some role to play in the adjustment of the
ACC.
After an initial geostrophic adjustment of the local stratification to a sudden change
in southern hemisphere wind forcing (see e.g., Cushman-Roisin, 1994; Gill, 1982), or
through the local baroclinic adjustment mechanism discussed by Olbers and Lettmann
(2007), it is likely that a response would be felt elsewhere in the global ocean through oceanic
teleconnections. A change in the density field (such as the deepening of isopycnal surfaces
to the north of the ACC predicted by Hall and Visbeck (2002) in response to strengthening
winds) would induce a combination of wave mechanisms that can communicate the signal
to remote locations across the globe. McDermott (1996) and Gnanadesikan and Hallberg
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(2000) show that the response of the density field to changes in Southern Ocean wind stress
is global in extent. Using an interhemispheric eddy-resolving ocean model, Wolfe and Cessi
(2010) demonstrate that the density structure within the circumpolar current determines the
mid-depth stratification in the entire domain. This result is supported by the recent work
of Radko and Kamenkovich (2011), which highlights the central role that Southern Ocean
processes play in setting the global stratification in a two-and-a-half-layer analytical model.
Kamenkovich and Radko (2011) also demonstrate that the stratification just north of the
Southern Ocean is important for controlling the stratification of the Atlantic and its steadystate meridional overturning. In this paper we use the link between the baroclinic ACC
and the global stratification to examine their transient response to a change in forcing.
Similar global balances connect the ACC, global pycnocline and Atlantic overturning in
the theoretical models of Samelson (2004, 2009).
We develop a simple time-dependent model for the adjustment of the ACC and global
pycnocline to changes in local and remote forcing. We find that both the equilibrium ACC
and its adjustment timescale are controlled by the mesoscale eddy diffusivity across the
ACC, the area of the global ocean to the north, and deep water formation in the North
Atlantic. The theory suggests that the timescale may be subtly modified by the relative
strength of the wind forcing over the ACC and global diapycnal mixing, but this is a
relatively minor effect. The e-folding adjustment timescale is found to be multidecadal to
centennial.
While preparing this manuscript, we have become aware of two related studies by Samelson (2011) and Jones et al. (2011) that reach broadly similar conclusions, albeit using
analytical models that have a much simpler representation of the ACC and a less detailed
analysis of the adjustment timescales.
The details of the analytical model are presented in Section 2, and its time-dependent
solution is given in Section 3. In Section 4 we validate the theoretical estimates against
numerical calculations performed with a reduced-gravity model in an interhemispheric
basin with a circumpolar channel. Section 5 considers the influence that northern deep water
formation may have on the adjustment process, and a concluding discussion is presented in
Section 6.
2. Theoretical model for ACC adjustment
In this section, we formulate a simple model of the global pycnocline and baroclinic
ACC in terms of a surface “pycnocline” layer overlying a deep abyssal layer. The model
represents an extension to that presented by Gnanadesikan (1999) to include a more consistent formulation of Southern Ocean wind forcing and eddies (Allison et al., 2010). The
model is sketched schematically in Figure 1. The global pycnocline depth in a steady state
is set by processes that act as sources and sinks of water to the surface layer. The depth of
the pycnocline is increased by the equatorward Ekman transport out of the Southern Ocean
(TEk ) and diapycnal upwelling over the basins to the north of the ACC (TU ), and decreased
2011]
Allison et al.: Spin-up & adjustment of ACC
171
Figure 1. Schematic of the simple analytical model of Gnanadesikan (1999).
by the poleward eddy bolus transport in the Southern Ocean (TEddies ) and northern deep
water formation (TN ):
TS + TU − TN = 0,
(1)
where TS = TEk − TEddies is the residual Southern Ocean transport.
As discussed by Gnanadesikan and Hallberg (2000), a deepening of the global pycnocline
leads to an increase in baroclinic ACC transport through an increased meridional density
gradient across the current. Assuming that the ACC is in geostrophic balance, and that the
pycnocline outcrops to the south of the ACC, the baroclinic ACC transport can be estimated
from the pycnocline depth, h, via
TACC ≈
g # h2
,
2|fS |
(2)
where g # is the reduced gravity and |fS | is the magnitude of the Coriolis parameter at the
latitude of the ACC (assumed constant across the current). This relationship motivates the
use of a simple reduced-gravity framework as an idealized approach to explore the response
of the baroclinic ACC and global stratification to altered forcing. The ACC transport can
be approximated via a single variable, h, the interfacial depth between the upper and lower
layers of differing density.
To explore the adjustment of the pycnocline to a change in forcing, the rate of change
of volume in the surface layer can be equated to the combined transports from each of
the density class transformation processes. We arrive at a time-dependent extension of the
Gnanadesikan (1999) model:
!!
∂
(3)
h dA = TS + TU − TN ,
∂t
(c.f. Jones et al., 2011) where A is the surface area of the basins to the north of the ACC.
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In this section, we first motivate the representation of the pycnocline depth through a
single global variable, then we review the rationale for the form of the wind forcing and
eddies over the Southern Ocean.
a. Pycnocline adjustment north of the ACC
A change in Southern Ocean wind stress will modify the density structure on the northern
side of the ACC. This density signal will be transmitted by westward-propagating Rossby
waves to the eastern coast of South America, and would then propagate equatorward via fast
boundary waves (e.g., Kawase, 1987; McDermott, 1996; Johnson and Marshall, 2002, 2004;
Ivchenko et al., 2006). Equatorial Kelvin waves communicate density anomalies eastward
along the equator, and upon reaching the eastern side of the basin, boundary waves carry the
signal poleward in both hemispheres, with baroclinic Rossby waves spreading the signal
westward into the basin interior. Through these wave mechanisms, a change in Southern
Ocean wind stress can be rapidly communicated throughout the global ocean. Here we
model the oceans north of the ACC through a single basin. Allison (2009) extends this
analysis to two basins (representing the Atlantic and Pacific) to include the seiching mode
between the basins (c.f. Cessi et al., 2004) but finds that this has little impact on the overall
adjustment.
Following Johnson and Marshall (2002), the dynamics in the ocean interior, assuming
small Rossby number, are described by the Rossby wave equation:
∂h
c(φ) ∂h
κv
−
= .
∂t
R cos φ ∂λ
h
(4)
Here φ is latitude, λ is longitude, R is the radius of the Earth, t is time and c(φ) = βg # h0 /f 2
is the Rossby wave speed, where g # is the reduced gravity, h0 is a reference depth for the
pycnocline, f is the Coriolis parameter and β its meridional gradient. Included in the Rossby
wave equation is the contribution from diapycnal mixing, with diffusivity κv , but not wind
forcing (the latter is easily incorporated but does not change the results significantly).
Integrating Eq. 4 over the basin, but excluding the western boundary layer, we find:
∂
∂t
!!
interior
h dA = TU +
!φN
c(he − hb )R dφ,
(5)
φS
where he is the layer thickness on the eastern boundary and hb is the layer thickness just
outside the western boundary layer. As in Johnson and Marshall (2002), we further assume
that the boundary wave dynamics prevent significant pressure gradients along the eastern
boundary; i.e., that he is independent of latitude at any instant. The surface layer thickness on
the western side of the basin, just outside any western boundary current/viscous boundary
layer, is therefore set by the layer thickness at the eastern boundary, at a time-lag given
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Allison et al.: Spin-up & adjustment of ACC
173
by the Rossby wave speed and the basin width, L = L(φ), and by the diapycnal mixing
experienced during the passage of the Rossby wave across the basin:
hb (t, φ) ≈ he (t − L/c, φ) +
κv L
.
he (t) c
(6)
Here we assume that he varies sufficiently slowly that the contribution from diapycnal
mixing can be approximated using he (t).
Since the adjustment timescale is likely to be much longer than the maximum Rossby
wave crossing time, we can now use a Taylor expansion to write
hb (t, φ) ≈ he (t) −
∂he L L κv
+
,
∂t c
c he (t)
and hence, integrating over latitude, we obtain
!!
∂
∂he
A,
h dA =
∂t
∂t
(7)
(8)
interior
where A is the area of the basin. This result is not unexpected: it simply states that the Rossby
waves have sufficient time to radiate he anomalies into the basin near-instantaneously,
relative to the overall adjustment, and hence the thickness throughout the interior can be
approximated by the thickness on the eastern boundary.
Finally, if the volume of the western boundary layer is small, then the volume over the
entire basin (Eq. 3) is roughly equivalent to the volume over the interior (Eq. 8):
∂he
A ≈ TS + TU − TN ,
∂t
(9)
where
TU ≈
κv A
.
he
(10)
b. Southern Ocean transport
We now consider the processes that set TS . As discussed and illustrated in Allison et al.
(2010) for the equilibrium ACC, to obtain good agreement between the theory and more
detailed numerical calculations, we must distinguish between the fraction of the wind stress
which drives the ACC and the fraction which drives a gyre circulation in the basin. The
northern limit of this region in which momentum is imparted to the ACC can be defined by
the northernmost geostrophic streamline of the ACC; i.e., where h = he , the equilibrium
layer thickness on the eastern boundary of the basin. Here we expand on the dynamical
reasons for evaluating the Southern Ocean transports within the region of circumpolar
streamlines.
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[69, 2-3
When the Rossby number is small, the large-scale approximation to the momentum
equation over the Southern Ocean can be written
f k × u + g # ∇h =
τs
.
ρ0 h
(11)
The transport velocity is therefore
hu =
1
k×∇
f
"
g # h2
2
#
−k×
τs
,
ρ0 f
(12)
and, using the parameterization of Gent and McWilliams (1990) for the eddy-induced transport velocity,
hu∗ = −κGM ∇h,
(13)
where κGM is the eddy diffusivity, the net transport velocity is
" # 2#
1
gh
τs
∗
−k×
− κGM ∇h.
h(u + u ) = k × ∇
f
2
ρ0 f
(14)
The transport across a closed layer thickness contour, TS = TEk − TEddies , can be obtained
by integrating the cross-contour component of the forcing along its length. The integration
can be simplified by assuming the contours are nearly zonal:
$
$ (x)
$
τ
∂h
TS ≈ h(v + v∗ )dx ≈ −
dx − κGM dx.
(15)
ρ0 f
∂y
We now integrate Eq. 15 across the circumpolar streamlines and divide by the zonal mean
meridional width of this integration region, Ly to give the final result:
TS ≈ −
1
Ly
!!
circumpolar
streamlines
τ(x)
1
dA −
ρ0 f
Ly
$
κGM he dx,
(16)
where he is the pycnocline depth north of the ACC (approximately equal to the eastern
boundary layer thickness, as discussed above) and we have assumed h ≈ 0 south of the
ACC. Although in this study we shall treat Ly as a time-invariant quantity, its magnitude
is determined by the details of the circulation. This goes some way towards introducing a
dynamical representation of Ly as suggested by Levermann and Fürst (2010).
c. Northern sinking
There are several different approaches for scaling the northern sinking term, TN , and for
a summary we refer the reader to Fürst and Levermann (2011). Most approaches agree that
TN should scale with the square of some vertical length scale (either the pycnocline depth
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Allison et al.: Spin-up & adjustment of ACC
175
or the level of no motion in the overturning) and linearly with a meridional (or vertical)
density difference. A simple scaling can be written
TN ≈
γg # h2
,
fN
(17)
where fN is the Coriolis parameter at the latitude of the sinking region and γ is a constant
with details dependent on the scaling method. The assumption that the density gradient is
constant means that this scaling does not account for the thermohaline feedbacks that are
important for controlling the behavior of the overturning circulation (e.g., Johnson et al.,
2007; Fürst and Levermann, 2011). This inadequacy, and the increased complexity resulting
from the inclusion of a quadratic expression for TN in the volume budget, motivates us to
first consider the adjustment in the limit of no northern sinking. We shall proceed by setting
TN = 0, but note that the effect of the northern sinking can be included by linearizing Eq. 17
and incorporating the coefficients into the other terms. We defer further discussion of the
effects of the northern sinking term to Section 5.
3. Analytical time-dependent solution
To simplify the following analysis, the coefficients associated with the scalings for TEddies
and TU are combined to form one coefficient for each term:
G=
κGM Lx
Ly
(18)
is the constant associated with the Gent and McWilliams (1990) eddy scaling, where κGM
is the eddy diffusivity, Lx is the zonal extent of the ACC and Ly is its meridional extent
as discussed in the previous section (when multiplied by h this is equivalent to the second
term on the right hand side of Eq. 16), and
D = κv A
(19)
is the constant associated with the diapycnal upwelling in the basin, where κv is the diapycnal
diffusivity and A is the basin area.
With no northern sinking term (TN = 0), the volume budget for the surface layer is now
written
∂h
D
A = TEk − Gh + ,
∂t
h
and the equilibrium solution for the pycnocline depth is:
%
2
TEk + TEk
+ 4GD
heq =
.
2G
(20)
(21)
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[69, 2-3
Journal of Marine Research
Figure 2. Adjustment timescale (years) from the linear theory, evaluated using Eq. 23. (a) Dependence
on Southern Ocean wind stress, τACC , and eddy diffusivity coefficient, κGM , with the basin area and
diapycnal diffusivity held constant (at 2 × 1014 m2 and 10−5 m2 s−1 respectively). (b) Dependence
on diapycnal diffusivity, κv , and basin area, with the wind stress and eddy diffusivity held constant
(at 0.16 N m−2 and 2000 m2 s−1 respectively). In each case, Lx = 2 × 107 m and Ly = 2 × 106 m.
The time-dependent volume budget (Eq. 20) can be integrated to give an implicit solution for
the evolution of the pycnocline depth. It is assumed that the adjustment takes place following
a step change in one (or more) of the volume budget transport terms. All coefficients (TEk ,
G and D) are assumed constant in the integration. The solution is
−Gh2 + TEk h + D = C
&
G(h − heq )
G(h + heq ) − TEk
"
'
TEk
TEk −2G heq
#
e−
2Gt
A
,
(22)
where C is a constant set by the initial conditions. For a small perturbation ∆h about the
equilibrium state, Eq. 22 can be linearized to give:
 






1
 2G

∆h ≈ ∆h0 exp − 1 −
t
,
(23)
%


A 


1 + 1 + 4GD
2
TEk
where ∆h0 = ∆h(t = 0). Thus the adjustment timescale is controlled mainly by the
mesoscale eddy transport across the ACC (set by G), and the area of the ocean basins to the
north of the ACC. The adjustment timescale also depends weakly on the Southern Ocean
Ekman transport and global diapycnal diffusivity, through the size of the nondimensional
2
parameter 4GD/TEk
. This allows the parameter in brackets in Eq. 23 to vary between 0.5
and 1, altering the timescale by up to a factor of two. The (e-folding) adjustment timescale
is plotted as a function of wind stress, eddy diffusivity, diapycnal diffusivity and basin area
in Figure 2. The adjustment timescale is typically of order 500 years.
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Allison et al.: Spin-up & adjustment of ACC
177
4. Numerical model
We now present a series of numerical calculations with a reduced-gravity model to test
the analytical theory presented in Sections 2 and 3, for the regime without northern sinking.
a. Model formulation
The model consists of a dynamically-active surface layer overlying a motionless abyss.
The model equations are:
∂u
τs
+ u · ∇u + f k × u + g # ∇h = Ah ∇ 2 u +
,
∂t
ρ0 h
∂h
κv
+ ∇ · (hu) = ∇ · (κGM ∇h) + ,
∂t
h
(24)
(25)
where we include the Gent and McWilliams (1990) eddy parameterization and the same
parameterization of diapycnal mixing as in Gnanadesikan (1999). The remaining symbols
are the horizontal velocity u, layer thickness h, Coriolis parameter f , wind stress τs , lateral
viscosity Ah , mesoscale eddy diffusivity κGM and diapycnal diffusivity κv . The equations
are discretized on a C-grid (as described in Johnson and Marshall, 2002) with a lateral
grid spacing of 1◦ . Time-stepping is with a leap-frog scheme. The model does not include
a representation of deep water formation in the northern high latitudes; the experiments
presented here are designed to test the analytical theory in the regime where there is no
northern sinking.
The domain extends from 74◦ S to 65◦ N in latitude and is 97◦ wide in longitude
(A ≈ 1014 m2 , Lx ≈ 5 × 106 m), with a circumpolar channel extending from the southern
boundary to 56◦ S. Lateral boundary conditions are no normal flow and no slip. The calculations are initialized with a uniform layer thickness of 10 m and no flow, and the model is
integrated for 5000 years with applied wind stress and diapycnal mixing. The wind stress
is zonal and restricted to a narrow latitude band in the southern hemisphere:
"
# "
#
∆φ
∆φ
(x)
2 π(φ − φ0 )
τ = τ0 cos
φ0 −
≤ φ ≤ φ0 +
,
(26)
∆φ
2
2
where φ is latitude and the maximum wind stress is located at φ0 . The wind stress profile in
the control experiment is shown in Figure 3a. A very basic representation of high-latitude
buoyancy loss is incorporated by strongly relaxing the layer thickness (over 11 days) to
its initial value of 10 m at the southern margin of the domain. This acts to keep the layer
interface close to the surface at the southern boundary. Reference model parameters are:
κGM = 2000 m2 s−1 , κv = 10−5 m2 s−1 , Ah = 16 × 103 m2 s−1 , g # = 0.02 m s−2 .
b. Numerical model results
The wind stress and equilibrium layer thickness from the control simulation are shown
in Figure 3a, with wind forcing confined to the circumpolar channel (τ0 = 0.16 N m−2 ,
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Journal of Marine Research
[69, 2-3
Figure 3. (a) Wind stress (N m−2 ) and surface layer thickness (m) at the end of the 5000-year control
integration. The contour spacing for the layer thickness field is 100 m, and the 100 m and 800 m
contours are labeled. (b) Time series of the basin-averaged (north of 30◦ S) layer thickness (m; solid
line, left axis) and circumpolar transport (Sv; dashed line, right axis).
φ0 = 62.5◦ S, ∆φ = 14◦ ). The equilibrium circumpolar current is aligned with the latitude
of the wind stress and is associated with a strong meridional gradient in layer thickness.
North of 40◦ S, the layer thickness varies by less than 4 m, consistent with the rapid (relative
to the overall adjustment) erosion of pressure gradients by eastern boundary waves and
westward propagating long Rossby waves. Figure 3b shows the evolution of the basinaveraged surface layer thickness in the control simulation, and the circumpolar transport
through the model’s representation of Drake Passage. The initial change in circumpolar
2011]
Allison et al.: Spin-up & adjustment of ACC
179
Table 1. Summary of numerical model experiments. The maximum wind stress in the westerly jet
(τ0 ), the latitude of the wind jet axis (φ0 ), the eddy diffusivity (κGM ) and the diapycnal diffusivity
(κv ) are varied.
τ0 (N m−2 )
φ0
κGM (m2 s−1 )
κv (m2 s−1 )
Control
0.16
62.5◦ S
2000
10−5
Varying τ0
0.208
0.112
0
Varying φ0
Varying κGM
Varying κv
55.5◦ S
48.5◦ S
1000
3000
10−4
transport is approximately linear with time. This can be anticipated from the analytical
theory, since diapycnal mixing dominates the adjustment for small h causing the pycnocline
to grow as a diffusive boundary layer at a rate proportional to (κv t)1/2 . The adjustment to
equilibrium occurs over roughly 2000 years.
To test the sensitivity of the solution to the various parameters involved, a further eight
model experiments are performed with varying wind stress magnitude τ0 , wind stress latitude φ0 , eddy diffusivity κGM and diapycnal diffusivity κv . Each experiment is integrated
for at least 4000 years, and they are summarized in Table 1.
c. Comparison between model and theory
To calculate the theoretical layer thickness evolution that corresponds to each model
simulation, Eqs. 21 and 22 are evaluated using the parameter values from the model. The
Southern Ocean transport terms are evaluated using the method outlined in Section 2b (Eq.
16); TEk is integrated over the region of circumpolar streamlines taken from the end of each
numerical simulation (the shape of this region varies little during the adjustment), and Ly
(used in the evaluation of both Southern Ocean terms) is the mean meridional width of the
integration region.
Figure 4 shows the evolution of surface layer thickness in the numerical model averaged over the basin (north of 30◦ S) as a function of time for the experiments in which
we vary: (a) the wind stress latitude; (b) the wind stress magnitude; (c) the mesoscale
eddy diffusivity; and (d) the diapycnal mixing coefficient. Also shown are the theoretical
estimates from Eq. 22. There is generally excellent agreement between the results of the
numerical model (symbols) and theoretical estimates (lines), both for the magnitude of the
equilibrium solution and the adjustment timescale. The one exception is for high levels of
diapycnal mixing (dashed curve and triangles in Fig. 4d) where the theory underestimates
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Figure 4. Adjustment of the surface layer thickness, h, as a function of (a) wind stress latitude, φ0 ; (b)
wind stress magnitude, τ0 ; (c) mesoscale eddy diffusivity, κGM ; (d) diapycnal mixing coefficient,
κv . The symbols show the results from the reduced-gravity numerical model, and lines are the
predictions from the analytical theory (Eq. 22). In panel (a), the grey lines show the theoretical
spin-up estimates in which TEk is calculated using the peak wind stress and both TEk and TEddies
are evaluated using the characteristic width (Ly = 106 m) in Eq. 22 instead of values integrated
over the circumpolar streamlines. In all the subfigures, the control experiment is shown with filled
circle symbols, and the corresponding theoretical estimate is shown with a solid line.
the equilibrium layer thickness. The reason for this discrepancy is not clear, but may be
related to the simple way in which diapycnal mixing is parameterized in the model: the
upwelling is inversely proportional to the depth of the layer interface, which is very shallow
near the southern boundary. The large values of diapycnal mixing which arise here are rather
unphysical and not included in the theory, but only significantly influence the model result
when the diapycnal diffusivity is set to a very high background value.
Figure 4b shows that in both the theoretical estimate and the numerical model, a reasonably deep pycnocline is obtained (deeper than 550 m) without explicit wind forcing.
In this simple setup, this corresponds to a circumpolar transport of around 25 Sv. In this
regime, diapycnal mixing is the sole process that maintains a deep pycnocline, through the
downward mixing of heat. This supports the idea that diapycnal mixing in the basins to the
north of the Southern Ocean may play a role in setting the baroclinic transport of the ACC
(Munday et al., 2011).
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Allison et al.: Spin-up & adjustment of ACC
181
Crucially, the results in Figure 4c show that the adjustment timescale is sensitive to the
Southern Ocean eddy diffusivity: the timescale is shorter when the eddy diffusivity is large,
consistent with the theoretical prediction from Eq. 23; see Figure 2. The theoretical solution
also suggested a subtle influence of the wind stress magnitude and diapycnal diffusivity on
the timescale, through the size of the nondimensional parameter in Eq. 23. This effect can
be seen in Figure 4b (the adjustment under zero wind stress has a shorter timescale than
the adjustment under finite wind stress) and Figure 4d (the timescale is slightly shorter for
larger diapycnal diffusivity, although this effect is less obvious).
Also shown in Figure 4a (grey lines) is the theoretical prediction using the peak wind
stress, τ0 , and the characteristic width of the wind forcing, Ly = 106 m, rather than values
obtained by integrating over the circumpolar streamlines following Allison et al. (2010), as
discussed in Section 2b. Not only is the variation of the equilibrium pycnocline depth with
wind stress latitude reversed, but the adjustment timescale is also significantly underestimated when the integrated values are not used.
5. Impact of northern deep water formation
In the previous sections, we have explored the theoretical solution and numerical results
for the limit without northern sinking (TN = 0). We now consider the effect that including
the northern sinking might have on the solution.
The commonly used scaling for TN (Eq. 17) has a squared dependence on h, which if
included in the time-dependent volume budget would make an analytical solution difficult
to achieve. However, for small perturbations from equilibrium, a linear approximation to
TN can be obtained through a truncated Taylor expansion:
5
∂TN 55
(27)
TN (h) ≈ Nc + h
∂h 5h=heq
where
5
∂TN 55
Nc = TN (heq ) − heq
∂h 5h=heq
is a constant. These coefficients can be combined with those for TEddies and TEk , so that the
surface layer volume budget can be written:
6
7
5
∂TN 55
D
∂h
h+ .
A = (TEk − Nc ) − G +
(28)
∂t
∂h 5h=heq
h
The derivative in Eq. 27 should strictly include thermohaline feedbacks associated with a
change in the strength of the overturning circulation. However, the commonly used scaling
for the northern sinking transport (Eq. 17) relies on the assumption that the meridional
(or zonal) density difference in the upper ocean (and assuming the pycnocline outcrops in
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[69, 2-3
the sinking region, then equivalently g # ) is constant and independent of TN . This relates
to the fact that the scaling only accounts for mechanical forcing; it cannot support the
thermohaline feedbacks that are fundamental to the dynamics of the overturning circulation
(see Johnson et al., 2007; Fürst and Levermann, 2011). Griesel and Morales Maqueda
(2006) and Levermann and Fürst (2010) showed that variations in the meridional density
difference are important for determining the magnitude of TN , and that the density difference
can vary independently of the pycnocline depth, suggesting that g # should not be treated as
constant. Furthermore, de Boer et al. (2010) demonstrated that the northern sinking need
not be positively correlated with the upper ocean meridional density gradient at all, and
suggest that the positive linear relationship seen in most model simulations arises from a
thermohaline feedback of the circulation.
A second assumption of the scaling in Eq. 17 is that the depth of the level of no motion in
the overturning is the same as the depth of the pycnocline. Although Fürst and Levermann
(2011) point out that these two depth scales are linked, de Boer et al. (2010) question the
validity of this assumption by showing that the two depth scales can respond in opposite
ways to changes in forcing, and the level of no motion has a better association with the
overturning transport than the pycnocline depth. In the experiments of Griesel and Morales
Maqueda (2006), changes in the strength of the overturning were not accompanied by
significant changes in the pycnocline depth.
These concerns aside, Wolfe and Cessi (2010) highlight the connection with the circumpolar channel by demonstrating that the stratification at the northern edge of the channel
provides an appropriate vertical length scale for h in the TN scaling (i.e., that the depth of the
pycnocline at the northern edge of the channel sets the depth of the pycnocline throughout
the domain, including the latitude of the maximum overturning).
The version of the TN scaling that is arguably most applicable to this study is that of
Johnson and Marshall (2002), since their method is based on a model of reduced gravity.
Using this approach, the northern sinking term scales via:
TN ≈
g # h2
.
2fN
(29)
A comparison of Eqs. 2 and 29 reveals the implication that the baroclinic ACC transport
and the transport of the Atlantic meridional overturning circulation (AMOC) are approximately equal (since |fS | ≈ fN ). The fact that the observed baroclinic ACC transport (e.g.,
Cunningham et al., 2003) is many times larger than the observed AMOC transport (e.g.,
Cunningham et al., 2007) suggests a problem with this common scaling. Also, Fučkar and
Vallis (2007) have shown that TN and TACC do not vary together. In their experiments, a
decrease in TN leads to a deepened pycnocline and a concomitant increase in TACC .
Despite these inadequacies, we may still make some progress by accepting the available
scaling for TN (Eq. 29), and assuming that g # is independent of h. Substitution into Eq. 28
leads to
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Allison et al.: Spin-up & adjustment of ACC
183
7 "
6
#
g # h2eq
g # heq
∂h
D
A = TEk +
− G+
h+ ,
∂t
2fN
fN
h
(30)
where the equilibrium pycnocline depth is now given by the solution to the cubic volume
budget:
−
g # h3eq
2fN
− Gh2eq + TEk heq + D = 0,
(31)
which for an appropriate range of parameter space has one positive real root. The analytical
solution to Eq. 30 is equivalent to Eq. 22 with the substitutions TEk → TEk + (g # h2eq /2fN )
and G → G + (g # heq /fN ). It is worth noting that including the northern sinking term leads
to a damping effect on the steady state response to a change in forcing (the northern sinking
increases in response to a strengthening of the Southern Ocean winds, which suppresses
the effect of the forcing), as also found by Radko and Kamenkovich (2011).
Figure 5 shows how the adjustment timescale and equilibrium pycnocline depth vary
as a function of wind stress, eddy diffusivity, diapycnal diffusivity and basin area, now
that a representation of the northern sinking has been included. The timescale can be compared with Figure 2 for the case without TN . Including northern sinking reduces the adjustment timescale from a few hundred years to slightly less than a century. The response
of the timescale to variations in key parameters is also altered, particularly the sensitivity to the Southern Ocean wind stress and eddy diffusivity (compare Figure 5a with
Figure 2a). In both cases (with and without TN ) the timescale is inversely related to the
eddy diffusivity. However, the timescale is less sensitive to the eddy diffusivity when TN is
included.
The quadratic dependence of the northern sinking term on the pycnocline depth means
that, in setting the adjustment timescale, TN dominates over TEddies (which in this framework
scales linearly with the pycnocline depth). However, the relative importance of northern
sinking and Southern Ocean eddy activity is far from clear, since observations and eddyresolving model experiments show that the eddy field itself exhibits nonlinear behavior
(Meredith and Hogg, 2006; Hogg et al., 2008). This issue is partially explored by Jones
et al. (2011) who consider the impact of different power laws for the variation of TEddies with
h on the e-folding adjustment timescale. Despite the caveats associated with the scalings
in the Gnanadesikan (1999) framework, it is clear that both the Southern Ocean eddies and
the northern sinking are important for setting the adjustment timescale.
6. Discussion
This study has considered the long-term transient response of the baroclinic ACC and
global stratification to a change in forcing (such as a step change in Southern Ocean wind
stress). We have built upon the analytical model of Gnanadesikan (1999) by introducing
time-dependence to the volume budget for the surface layer above the global pycnocline, and
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[69, 2-3
Figure 5. Adjustment timescale (years) from the linear theory evaluated using the solution to Eq. 30
(i.e., the timescale in Eq. 23 but with reinterpreted coefficients) which includes the effect of northern
sinking. (a) Dependence on Southern Ocean wind stress, τACC , and eddy diffusivity coefficient,
κGM , with the basin area and diapycnal diffusivity held constant (at 2 × 1014 m2 and 10−5 m2 s−1 ,
respectively). (b) Dependence on diapycnal diffusivity, κv , and basin area, with the wind stress
and eddy diffusivity held constant (at 0.16 N m−2 and 2000 m2 s−1 , respectively). In each case,
Lx = 2 × 107 m, Ly = 2 × 106 m and g # = 0.01 m s−2 . The central panels (c,d) show how
heq varies with the parameters when TN is included. The lower panels (e,f) show how TN varies
throughout the parameter space.
2011]
Allison et al.: Spin-up & adjustment of ACC
185
including a more consistent measure of the Southern Ocean transport terms. In this model,
the pycnocline depth (and baroclinic ACC) is set by Southern Ocean Ekman transport and
basin-wide diapycnal upwelling, which deepen the surface layer, and by Southern Ocean
eddies and northern sinking, which counteract the deepening. We first explored the analytical
solution for adjustment in the case without northern sinking (noting that its effect can be
incorporated into the solution by reinterpreting the coefficients for the other terms), and then
presented results from simple numerical simulations with a reduced-gravity ocean model.
Finally, we considered the effect of including the northern sinking term.
Our results suggest that the e-folding adjustment timescale for the global pycnocline and
baroclinic ACC is multidecadal to centennial, with the adjustment taking several centuries
to fully complete. We find that without an interactive representation of NADW formation,
the long-term adjustment of the ACC is acutely sensitive to the mesoscale eddy diffusivity
over the Southern Ocean (with stronger eddy activity leading to a faster adjustment). When
a representation of deep water formation in the North Atlantic is included, the adjustment
timescale becomes shorter and its sensitivity to Southern Ocean eddy activity is significantly
reduced. In both cases, the area of the global ocean is also important for setting the timescale
(the timescale is shorter in a smaller domain). The theory also predicts that the Southern
Ocean wind stress and global diapycnal mixing may subtly modify the timescale (and the
numerical results are suggestive of this) but their effect is substantially weaker than that
of the other processes. In this model, it is clear that while the winds and mixing cause the
global stratification to deepen, it is the processes which counteract the deepening that set
the timescale for adjustment.
The relative importance of Southern Ocean eddies and northern sinking for setting the
timescale is not clear, because the available scalings for both processes are somewhat inadequate. As discussed in the previous section, the simple dependence of TN on the square of
the pycnocline depth cannot account for thermohaline feedbacks associated with changes
in the overturning circulation (Johnson et al., 2007; Fürst and Levermann, 2011). The linear
scaling for the Southern Ocean eddy transport is also considered to be a severe simplification; eddy-resolving calculations suggest that the ACC can exhibit threshold behavior, with
eddy activity increasing markedly in response to changes in wind forcing (Tansley and Marshall, 2001; Hallberg and Gnanadesikan, 2006), as originally anticipated by Straub (1993)
(but also see Nadeau and Straub (2009), for an alternative explanation). Observations and
eddy-resolving numerical model experiments indicate that eddy energy increases a year
or two after an increase in wind forcing (Meredith and Hogg, 2006; Hogg et al., 2008),
partially compensating the increased Ekman transport. Such threshold behavior is not captured in models that parameterize eddies. The extent to which threshold behavior occurs on
longer, baroclinic timescales remains unknown and awaits eddy-resolving models that can
be integrated to equilibrium (Treguier et al., 2010). The outcome that the adjustment can
be sensitive to the representation of mesoscale eddy activity is disturbing since Southern
Ocean eddy processes remain poorly understood and poorly represented in state-of-the-art
climate models.
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[69, 2-3
Irrespective of any eddy threshold behavior, the global pycnocline deepens only very
gradually in response to increased wind forcing, due to the large surface area of the basins
north of the ACC. Thus anthropogenic forcing is unlikely to have led to appreciable change
in the mean baroclinic transport of the ACC to date, consistent with observations (Böning
et al., 2008). However, note that the pycnocline deepens faster in a channel model of
limited latitudinal extent, suggesting that results from channel models should be treated
with caution.
Despite the caveats associated with the representation of northern sinking in our analytical
model, the results suggest that this process has the potential to play a central role in setting
the long-term adjustment timescale of the baroclinic ACC and the global stratification, and
to control the sensitivity of the timescale to Southern Ocean eddy activity. Further work is
needed to better understand the relationship between the global pycnocline depth and the
rate of deep water formation in the North Atlantic, to properly understand its influence on
the adjustment.
A corollary of our study is that (depending on the influence of northern sinking) Southern
Ocean eddy fluxes may be a rate-limiting process that sets the spin-up timescale for non-eddy
resolving coupled ocean climate models. This raises the intriguing possibility that a future
generation of eddy-resolving ocean models may equilibrate faster due to nonlinear eddy
threshold behavior. More pressingly, this emphasizes the importance of better resolving
ocean eddies in model projections of anthropogenic climate change.
Acknowledgments. This study is funded by the UK Natural Environment Research Council. HLJ is
supported by a Royal Society University Research Fellowship. DPM is supported by the James Martin
21st Century School, University of Oxford. We thank Andy Hogg and one anonymous reviewer for
their helpful feedback.
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Received: 6 April, 2011; revised: 27 June, 2011.