column is inappropriate as well. The influence of
stratification is visible in other forms of the vorticity balance
(kٌ⌿) и ٌ ;
U
g
и ٌH:kи ٌ(/( 0 H))
(4.6.21)
or
H
2
u b иٌ ;U
g
иٌf:f kи ٌ(/ 0 f):fw E (4.6.22)
obtained from (4.6.1) using kٌ⌿:Hu b ;
U
g
9k(/f) (ignoring lateral stresses and bottom
friction for simplicity). Here, U
g is the baroclinic
(thermal wind) transport relative to the bottom.
Obviously, in the case of weak stratification when
the baroclinic transport term in the above balances
could be ignored, the transport streamfunction
and the bottom velocity both would follow f/H
contours where the corresponding stress curls are
weak. If, in addition, the variation of the planetary
vorticity f along the path of the flow is small the
bathymetry contours act as characteristics (the
topographic T: f⌬H/(H⌬L) is in fact generally
larger than the planetary ).
A more detailed consideration of stratification
effects in models would obviously be required to
distinguish between these different possibilities. An
intelligent shortcut has been pursued by Marshall
(1995a,b) using the homogeneous potential vorticity model of Marshall et al. (1993b). With a functional dependence f z :Q(), the density field
and the baroclinic transport U
g is determined by a
boundary value, say (x, z:0): s (x). Also the
bottom density b is determined by s . Furthermore, the bottom is a material surface and – since
Montgomery potential, M:p;gz, and density
are conserved along the three-dimensional flow in
adiabatic conditions – we have a functional dependence M b :M b ( b ). Assuming no friction in the
abyss, the bottom velocity is geostrophic, i.e.
fu b :k(ٌM b ;gHٌ b ). Inserting these relations
into (4.6.22) we find (after some manipulation)
A(kٌ s )иٌ ;B(kٌ s ) и ٌf: Q s w E
(4.6.23)
with
A:H
2
(H9H ref )Q b ,
B:͵
0
9H
Qz dz
(4.6.24)
This vorticity equation determines the surface density from the Ekman pumping w E , provided the
functional relations H ref ( b ):(1/g)ѨM b /Ѩ b and
Q() are specified. The extreme cases with their
characteristics f/H:constant and f:constant
appear again in (4.6.23) – corresponding to a
homogeneous ocean with large A (because here
H ref ӷ H) and motionless abyss with large B
(because here Q ӷ Q b ) . For constant H ref and linear
Q():a;b (as found by Marshall et al., 1993b)
the characteristic problem (4.6.23) is linear and
characteristics are easily computed. Marshall
(1995b) presents a solution for the transport
streamfunction and the interior circulation for
an unforced case (w E :0), prescribing a surface
density section between Antarctica and Australia
(Fig. 4.6.11). The solution transports 160 Sv, and
streamlines are largely zonal and quite indifferent
to topography in large portions of the domain.
Nevertheless, topography clearly steers the current over the major topographic features such as
the Atlantic Ridges, the South East Indian and
Macquarie Ridges, the Pacific–Antarctic Ridge and
the East Pacific Rise. These patterns are enhanced
for larger values of the reference depth H ref , which
leads to an increase of the deep currents. Outstanding in the solution is an exaggerated northward
deflection over the Kerguelen Plateau, whereas the
observed equatorward displacement of the current
behind Drake Passage and the gradual poleward
migration in the rest of the Southern Ocean is not
reproduced, presumably due to neglect of the wind
forcing.
4.6.4 Water mass formation and
conversion
Isopycnals shoal steeply to the south across the
Southern Ocean, reflecting the baroclinicity of the
ACC (Fig. 4.6.3). Water spanning a wide range of
density is thus directly forced by exchange of
momentum, heat and fresh water with the overlying atmosphere and sea ice. The air–sea–ice interactions strongly modify the physical and chemical
properties of outcropping layers and transform
water from one density class to another. The water
masses formed in this way ventilate a substantial fraction of the world ocean volume and are
a key link in the global overturning circulation
(Section 4.6.5).
f
2
ᎏ
g
f
ᎏ
H
f
ᎏ
H
f
ᎏ
H
2
f
ᎏ
H
4.6 The Antarctic Circumpolar Current System
291
Rintoul, Hughes and Olbers
stratification is visible in other forms of the vorticity balance
(kٌ⌿) и ٌ ;
U
g
и ٌH:kи ٌ(/( 0 H))
(4.6.21)
or
H
2
u b иٌ ;U
g
иٌf:f kи ٌ(/ 0 f):fw E (4.6.22)
obtained from (4.6.1) using kٌ⌿:Hu b ;
U
g
9k(/f) (ignoring lateral stresses and bottom
friction for simplicity). Here, U
g is the baroclinic
(thermal wind) transport relative to the bottom.
Obviously, in the case of weak stratification when
the baroclinic transport term in the above balances
could be ignored, the transport streamfunction
and the bottom velocity both would follow f/H
contours where the corresponding stress curls are
weak. If, in addition, the variation of the planetary
vorticity f along the path of the flow is small the
bathymetry contours act as characteristics (the
topographic T: f⌬H/(H⌬L) is in fact generally
larger than the planetary ).
A more detailed consideration of stratification
effects in models would obviously be required to
distinguish between these different possibilities. An
intelligent shortcut has been pursued by Marshall
(1995a,b) using the homogeneous potential vorticity model of Marshall et al. (1993b). With a functional dependence f z :Q(), the density field
and the baroclinic transport U
g is determined by a
boundary value, say (x, z:0): s (x). Also the
bottom density b is determined by s . Furthermore, the bottom is a material surface and – since
Montgomery potential, M:p;gz, and density
are conserved along the three-dimensional flow in
adiabatic conditions – we have a functional dependence M b :M b ( b ). Assuming no friction in the
abyss, the bottom velocity is geostrophic, i.e.
fu b :k(ٌM b ;gHٌ b ). Inserting these relations
into (4.6.22) we find (after some manipulation)
A(kٌ s )иٌ ;B(kٌ s ) и ٌf: Q s w E
(4.6.23)
with
A:H
2
(H9H ref )Q b ,
B:͵
0
9H
Qz dz
(4.6.24)
This vorticity equation determines the surface density from the Ekman pumping w E , provided the
functional relations H ref ( b ):(1/g)ѨM b /Ѩ b and
Q() are specified. The extreme cases with their
characteristics f/H:constant and f:constant
appear again in (4.6.23) – corresponding to a
homogeneous ocean with large A (because here
H ref ӷ H) and motionless abyss with large B
(because here Q ӷ Q b ) . For constant H ref and linear
Q():a;b (as found by Marshall et al., 1993b)
the characteristic problem (4.6.23) is linear and
characteristics are easily computed. Marshall
(1995b) presents a solution for the transport
streamfunction and the interior circulation for
an unforced case (w E :0), prescribing a surface
density section between Antarctica and Australia
(Fig. 4.6.11). The solution transports 160 Sv, and
streamlines are largely zonal and quite indifferent
to topography in large portions of the domain.
Nevertheless, topography clearly steers the current over the major topographic features such as
the Atlantic Ridges, the South East Indian and
Macquarie Ridges, the Pacific–Antarctic Ridge and
the East Pacific Rise. These patterns are enhanced
for larger values of the reference depth H ref , which
leads to an increase of the deep currents. Outstanding in the solution is an exaggerated northward
deflection over the Kerguelen Plateau, whereas the
observed equatorward displacement of the current
behind Drake Passage and the gradual poleward
migration in the rest of the Southern Ocean is not
reproduced, presumably due to neglect of the wind
forcing.
4.6.4 Water mass formation and
conversion
Isopycnals shoal steeply to the south across the
Southern Ocean, reflecting the baroclinicity of the
ACC (Fig. 4.6.3). Water spanning a wide range of
density is thus directly forced by exchange of
momentum, heat and fresh water with the overlying atmosphere and sea ice. The air–sea–ice interactions strongly modify the physical and chemical
properties of outcropping layers and transform
water from one density class to another. The water
masses formed in this way ventilate a substantial fraction of the world ocean volume and are
a key link in the global overturning circulation
(Section 4.6.5).
f
2
ᎏ
g
f
ᎏ
H
f
ᎏ
H
f
ᎏ
H
2
f
ᎏ
H
4.6 The Antarctic Circumpolar Current System
291
Rintoul, Hughes and Olbers