The Antarctic Circumpolar Current in three dimensions

43 Pages·2004·1.61 MB·English

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The Antarctic Circumpolar Current in three
dimensions
Timour Radko and John Marshall
Department of Earth, Atmospheric and Planetary Sciences,
Massachusetts Institute of Technology.
[email protected]; [email protected].
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Abstract. A simple theory is developed for the large scale three-dimensional
structure of the Antarctic Circumpolar Current and the upper cell of its overturning
circulation. The model is based on a perturbation expansion about the zonal-average
residual mean model developed in Marshall and Radko (2003). The problem is solved
using the method of characteristics for idealized patterns of wind and buoyancy forcing
constructed from observations. The equilibrium solutions found represent a balance
between the Eulerian meridional overturning, eddy-induced circulation and downstream
advection by the mean flow. Depth and strati(cid:12)cation of the model thermocline increases
signi(cid:12)cantly in the Atlantic-Indian sector where the mean wind-stress is large. Residual
circulation in the model is characterized by intensi(cid:12)cation of the overturning circulation
in the Atlantic-Indian sector and reduction in strength in the Paci(cid:12)c region. Predicted
three-dimensional patterns of strati(cid:12)cation and residual circulation in the interior of the
ACC are compared with observations.
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1. Introduction
The absence of land barriers in the Antarctic Circumpolar Current (ACC) results in
distinct dynamical features that have no direct counterpart in the theory of midlatitude
ocean gyres. Sverdrup dynamics, the cornerstone of subtropical thermocline theory
(Rhines and Young, 1982, Lyuten et al., 1983), do not apply here. Both wind and
buoyancy forcing play a role, as do geostrophic eddies which appear to be crucial in
determining the strati(cid:12)cation and transport of the ACC (see the review by Rintoul et
al. 2001). The di(cid:14)culty of incorporating eddy transfer led to a rather slow development
of conceptual models of the ACC (see, e.g. Johnson and Bryden, 1989; Marshall et
al., 1993). Recently, however, residual-mean theories have been applied { see Karsten
et al. (2002), Marshall and Radko (2003, MR hereafter) { which, we believe, capture
the essence of the zonally averaged circulation and strati(cid:12)cation of the Southern
Ocean and fully embrace the central role of eddies. It is assumed that the Eulerian
meridional circulation driven by the westerly winds (the Deacon cell), tending to
overturn isopycnals, is largely balanced by the geostrophic eddies which act in the
opposite sense. The Transformed Eulerian Mean formalism (Andrews and McIntyre,
1976) is used to represent the combined e(cid:11)ect of eddy-induced and mean flow advection.
While the simpli(cid:12)ed two-dimensional zonal-average view attempts to explain the
integral characteristics of the ACC, such as the overall strength of the overturning
circulation and eastward baroclinic transport, zonal averaging masks important
three-dimensional e(cid:11)ects. For example, inspection of the pattern of surface heat flux in
the Southern Ocean (Fig. 1) reveals its strikingly nonuniform distribution along the path
of the ACC, with air-sea heat fluxes warming the ocean at rates reaching (cid:24) 60W/m2
into the ocean in the Atlantic and Indian ocean and becoming smaller and changing
sign in the Paci(cid:12)c region. The downstream variation in the surface forcing may be a
consequence of the asymmetry in the trajectory of the ACC; the current is partially
steered by bottom topography (Marshall, 1995) and therefore does not exactly follow
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latitude circles. As a result, the surface heat flux in to the ocean increases (relative to
the streamline average) where the ACC meanders equatorwards in to warmer regions
downstream of Drake Passage (see Fig. 1) and reduces when the ACC drifts polewards
in the Paci(cid:12)c sector. Wind stress (Fig. 2a) also varies along the path of the ACC being
signi(cid:12)cantly larger in the Atlantic-Indian sectors. In this paper we will argue that the
spatially nonuniform forcing leads to downstream variation of the thermocline depth
and strati(cid:12)cation of the ACC (see observations of Sun and Watts, 2002), and modulates
the strength of the upper cell of the meridional overturning circulation. Figure 1.
This study attempts to explain the three-dimensional time mean structure of
the ACC and the associated pattern of the overturning circulation by extending the
zonal average model of MR. Exploiting the asymptotic limit in which the downstream
variation in buoyancy is assumed to be weak relative to its variation in the meridional
plane, the problem is reduced to a system of equations which can be readily solved
using the method of characteristics. The theory is analogous to linear models of forced
stationary waves in the earth’s troposphere (see the review by Held, 1983).
The paper is set out as follows. After presenting key observations of the ACC in
section 2, we develop a theoretical framework (Sec. 3) by introducing the quasi-zonal
jet approximation. Residual-mean theory is used to incorporate the e(cid:11)ects of mesoscale
eddies. Simpli(cid:12)cations introduced by the analytical model are then used to determine
the interior structure of the ACC for various surface and boundary conditions. In
Sec. 4 we briefly consider the simplest \diagnostic" model which involves prescribing
the idealized distribution of the surface buoyancy (b ), buoyancy flux (B) and the wind
m
stress (τ) and computing the resulting interior (cid:12)elds. In Sec. 5 we study a di(cid:11)erent, and
perhaps more physical, \prognostic" model in which the surface buoyancy distribution
and fluxes are considered unknown and computed as a part of the problem. Model
solutions are compared with the oceanographic observations. We summarize and
conclude in Sec. 6.
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2. Observational background
Figs. 2-4 present key observations of surface properties and forcing of the Antarctic
Circumpolar Current (to be used in our theoretical model). The pattern of zonal wind
stress (τ) is shown in Fig. 2a, the surface buoyancy distribution (b ) in Fig. 3a, and
m
the air-sea buoyancy flux (B) in Fig. 4a.1 To analyze the variation of these (cid:12)elds
along the path of the ACC, it is convenient to reference our along-stream coordinate
to mean surface geostrophic contours; these are indicated by the heavy solid lines in
Figs. 2a,3a,4a (deduced from satellite altimetry), which mark the boundaries and the
axis of the ACC. The width of the region bounded by these streamfunction contours is
1000-2000km (depending on the longitude). Projection of the zonal wind stress, surface
buoyancy, and buoyancy flux on to this coordinate system is shown in Figs. 2b,3b,4b
respectively, where the ordinate is now a geostrophic streamline rather than latitude. Figure 2.
Several comments on the structure of the forcing (cid:12)elds are in order. Downstream (x)
variation in (τ,B,b ) contains a large signal in the fundamental (in x) harmonic whose
m
wavelength equals the zonal extent of the ACC. The two lowest Fourier components in
x { the along stream average (n=0 mode) mode and the fundamental harmonic (n=1
mode) { capture much of the large scale spatial variability of the ACC. Downstream
variation in the wind stress (Fig. 2b) is rather moderate, about 20% of the mean. Winds
over the ACC intensify in the Indian and Western Paci(cid:12)c sectors and weaken over
Eastern Paci(cid:12)c and Western Atlantic. Surface buoyancy (Fig. 3b) has a similar pattern,
characterized by weak downstream variation, whereas the air-sea flux (see Fig. 4b) is
highly inhomogeneous. Buoyancy flux over the ACC is mostly positive (i.e. into the
ocean) but changes sign in the eastern Paci(cid:12)c (120(cid:14)W −60(cid:14)W). Figure 3.
1It should be noted that there is a signi(cid:12)cant uncertainty in the observations, partic-
ularly with regard to the air-sea fluxes; the estimates of the surface buoyancy flux from
various datasets (e.g. NCEP, COADS, SOC) di(cid:11)er by as much as 50%.
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The following discussion will focus on the dynamics of the long wave components
and therefore we also show (in panels 2c,3c,4c) the (cid:12)ltered data for (τ,B,b ) consisting
m
of the two lowest Fourier components in x: the streamline average mode and the
fundamental harmonic. Fourier analysis indicates that the large-scale variation in
surface buoyancy and air-sea buoyancy flux are very much in phase with each other but
shifted somewhat to the west relative to the phase of the wind stress. Our objective is
to study how the surface forcing (cid:12)elds drive and interact with large scale motions in the
interior of the ACC and thereby shape the three-dimensional patterns of buoyancy and
meridional overturning in the Southern Ocean. Figure 4.
3. Formulation
3.1. Elements of residual-mean theory
Our starting point is the three-dimensional time mean equations of motion. We
ignore the inertial terms in the time mean momentum equations, and write them as
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>
>
<−fv = −∂P + ∂τx
∂x ∂z
(1)
>
>
:fu = −∂P + ∂τy,
∂y ∂z
where (u,v) are the horizontal components of the Eulerian mean velocity v, (τ ,τ )
x y
are the wind stress components which we assume to be signi(cid:12)cant only in a surface
boundary layer, and P is the dynamical pressure. The time-mean buoyancy equation is
∂B
v(cid:1)rb = −r(cid:1)(v0b0)+ , (2)
∂z
where b is the time mean buoyancy and primes denote the perturbations from this
mean due to transient eddies; B represents the vertical buoyancy flux due to small-scale
processes and air-sea fluxes. In application to the ACC we interpret x as the along
stream coordinate; y is the coordinate normal to the stream, referenced to a mean
surface geostrophic contour.
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Andrews and McIntyre (1976) showed that it is possible to incorporate the eddy-flux
terms in the Eulerian equations by introducing ‘residual velocities’ given by:
(cid:3)
v = v+v , (3)
res
where v(cid:3) = (u(cid:3),v(cid:3),w(cid:3)) is the eddy induced velocity of residual mean theory, which is
assumed to be non-divergent:
r(cid:1)v(cid:3) = r(cid:1)v = 0. (4)
We also require that the vertical velocity (w(cid:3)) is zero at the surface (z = 0), and
therefore integration of (4) from surface to depth z results in w(cid:3) = ∂ Ψ(cid:3) + ∂ Ψ(cid:3), where
∂x u ∂y v
R R
Ψ(cid:3) = − zu(cid:3)dz,Ψ(cid:3) = − zv(cid:3)dz are components of a vector streamfunction. The eddy
u 0 v 0
velocities can be compactly written thus:
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>
>
<(u(cid:3),v(cid:3)) = − ∂ Ψ(cid:3),
∂z
(5)
>
>
:w(cid:3) = r (cid:1)Ψ(cid:3)
h
where Ψ(cid:3) = (Ψ(cid:3),Ψ(cid:3)).
u v
In Appendix A we relate the vector streamfunction Ψ(cid:3) to the eddy buoyancy fluxes
(v0b0). The resulting model provides a physical basis for a parameterization scheme
in which the eddy streamfunction (below the vertically homogeneous mixed layer) is
determined by the local isopycnal slope as follows:
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>
>
<Ψ(cid:3) = k s js j, u(cid:3) = −k ∂ (s js j),
u 0 x x 0∂z x x
(6)
>
>
:Ψ(cid:3) = k s js j, v(cid:3) = −k ∂ (s js j).
v 0 y y 0∂z y y
where k is a constant which sets the magnitude of the eddy transfer process. This
0
parameterization is a direct extension of a two-dimensional closure introduced in MR
and supported by the laboratory experiments in Cenedese et al. (2004).
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The buoyancy equation written in terms of the residual velocities (see Appendix A)
reduces to
∂B~
v (cid:1)rb = , (7)
res
∂z
where B~ = B +B(cid:3) includes the explicit buoyancy forcing (B) and a contribution from
the diabatic component of eddy fluxes (B(cid:3)). Here, however, we assume that eddies are
adiabatic2 and therefore B(cid:3) = 0.
3.2. Quasi-zonal jet approximation
Our study is focused on a regime in which departure of the solution from its along
stream average is asymptotically small. As shown below, this approximation makes
it possible to theoretically analyze three-dimensional e(cid:11)ects, while retaining a direct
connection with the two-dimensional ACC model in MR.
To develop a linear theory for the quasi-zonal current, we search for a solution by
expanding in a parameter
hb i
1
(cid:15) = << 1,
hb i
0
which measures the variation in buoyancy along the streamlines of the ACC relative to
the cross-stream variation. Inspection of the surface buoyancy (cid:12)eld in Fig. 3c indicates
that (cid:15) (cid:24) 0.1, which justi(cid:12)es the asymptotic expansion in (cid:15). Next, we separate the mean
variables into the dominant two dimensional part [a function of (y,z) only] and a weak
2This assumption is violated in the near surface regions, where the presence of the
surface tends to suppress the vertical component of the eddy fluxes, which become di-
rected across the isopycnals. However, the estimates in MR suggest that in the near
surface mixed layer the explicit buoyancy forcing B signi(cid:12)cantly exceeds B(cid:3) and there-
fore diabatic component of the eddy fluxes can be neglected in this simple model of the
ACC.
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((cid:24) (cid:15)) three dimensional component. The linear analysis is focused on the dynamics of
the fundamental harmonic in x:
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>
>
>>v = v (y,z)+v (x,y,z)+O((cid:15)2), v = Re[v^ (y,z)exp(ikx)]
> 0 1 1 1
>
>
>
>
>
>>v(cid:3) = v(cid:3)(y,z)+v(cid:3)(x,y,z)+O((cid:15)2), v(cid:3) = Re[v^(cid:3)(y,z)exp(ikx)]
> 0 1 1 1
>
>
>
>
<b = b (y,z)+b (x,y,z)+O((cid:15)2), b = Re[^b (y,z)exp(ikx)]
0 1 1 1
(8)
>
>
>>P = P (y,z)+P (x,y,z)+O((cid:15)2), P = Re[P^ (y,z)exp(ikx)]
> 0 1 1 1
>
>
>
>
>
>>>B = B0(y,z)+B1(x,y,z)+O((cid:15)2), B1 = Re[B^1(y,z)exp(ikx)]
>
>
>
>
:τ = τ (y)+τ (x,y)+O((cid:15)2), τ = Re[τ^ (y)exp(ikx)]
x 0x 1x 1 1x
where k is a wavenumber. B is the buoyancy forcing in Eq.(7), and τ is the zonal
x
component of the wind stress. The subscript (0,1...) in (8) pertains to (zero,(cid:12)rst....)
order quantities in (cid:15).
To illustrate the expansion procedure, we write the buoyancy equation in (7) as
follows:
∂B ∂B
(v +v +...)(cid:1)r(b +b +...)+(v(cid:3) +v(cid:3) +...)(cid:1)r(b +b +...) = 0 + 1 +... .
0 1 0 1 0 1 0 1
∂z ∂z
The leading (zero order) balance in (cid:15) is:
∂B
v (cid:1)rb = 0, (9)
0 res 0
∂z
whereas the (cid:12)rst order balance requires that
∂B
v (cid:1)rb +v (cid:1)rb = 1 (10)
1 res 0 0 res 1
∂z
The momentum equations (1), continuity equation (4), and the eddy closure (6) are
similarly expanded in powers of (cid:15). Not surprisingly, at the zero order in (cid:15) we recover
a set of equations identical to those used in the two-dimensional model (MR), which
is briefly reviewed below. The (cid:12)rst order correction will give us information about
longitudinal variations in the stream, the information we seek.
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3.3. Zero order solution: the ACC in two dimensions
The zero order component of the (y,z)-velocities can be expressed in terms of the
scalar streamfunctions (time-mean and eddy induced) as follows:
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>
><v(cid:3) = −∂Ψ∗0 w(cid:3) = ∂Ψ∗0,
0 ∂z 0 ∂y
>
>
:v = −∂Ψ0 w = ∂Ψ0,
0 0
∂z ∂y
and the residual circulation is described by the streamfunction Ψ (y,z) = Ψ +Ψ(cid:3).
0 res 0 0
The eddy parameterization (6) at the leading order reduces to
∂Ψ(cid:3) ∂ b
Ψ(cid:3) = k js js = −k s2, v(cid:3) = − 0 = k s2, s = − 0y, (11)
0 0 0 0 0 0 0 0 0 0
∂z ∂z b
0z
and the zero-order y-momentum equation is
∂τ
−fv = 0. (12)
0
∂z
The problem is solved separately in a thin, vertically homogeneous mixed layer
(−h < z < 0) and in the strati(cid:12)ed interior (z < −h ). For the interior it is assumed
m m
that eddies are adiabatic, and the forcing, mechanical and thermodynamical (B,τ),
vanishes. Hence, the buoyancy equation (9) reduces to
J(Ψ ,b) = 0, (13)
0 res
where J(A,B) (cid:17) A B −A B . Eq.(13) implies that the residual streamfunction and
y z z y
buoyancy are functionally related: Ψ = Ψ (b ).
0 res 0 res 0
Integrating the buoyancy equation (2) over the depth of the vertically homogeneous
mixed layer (−h < z < 0), for the (two-dimensional) zero order component we obtain:
m
∂b
Ψ j 0 m = B , (14)
0 res z=−hm ∂y 0
where b is the surface buoyancy. Finally, integrating the momentum equation (12)
0 m
vertically from the surface to depth z (noting that Ψ = 0 at the surface and τ is zero
0