Deep-Sea Research, 1971, Vol. 18, pp. 685 to 721. Pergamon Press. Printed in Great Britain.
Effects of geometry on the circulation
of a three-dimensional southern-hemisphere ocean model
A. E. GILL* a n d K. BRYAN~"
(Received 11 August 1970; in revisedform 11 Jamtary 1971; accepted 12 January 1971)
Abstract--A series of numerical experiments on the circulation of the Antarctic region have been
carried out in order to study the effect of baroclinicity and geometry on the circumpolar current
and the Antarctic Convergence. Comparisons are made between model oceans which do and do not
have barrier free zones ('Drake Passage'). Limitations of computer storage require the parameters
used to be somewhat different than those applicable to the ocean. However, certain conclusions can
be drawn. The existence of the barrier free zone is offered as an explanation of the formation of
Antarctic Intermediate Water in latitudes a little north of the latitude of Cape Horn. It is found
that when the model 'Drake Passage' is as deep as the rest of the model ocean (which is of uniform
depth), the circumpolar current is predominantly wind-driven. When the model 'Drake Passage'
is only half as deep, the transport of the current is increased nearly three times and the predominant
driving mechanism appears to be 'thermal'.
1. INTRODUCTION
WHAT w o u l d h a p p e n to the w o r l d ' s ocean circulation if the D r a k e Passage, between
S o u t h A m e r i c a a n d A n t a r c t i c a , were closed? Certainly there w o u l d no longer be a
c i r c u m p o l a r current, b u t o t h e r features o f the circulation would be affected too.
W o u l d there still be downwelling at an A n t a r c t i c convergence to f o r m A n t a r c t i c
I n t e r m e d i a t e W a t e r ? W o u l d there still be upwelling o f w a r m deep water further
south to bring nutrients from lower latitudes near the surface? I f so, the w o r l d ' s
climate w o u l d be greatly effected as w o u l d the fish a n d plant life o f the sea at high
southern latitudes.
In this paper, we r e p o r t a series o f numerical experiments designed to show the
effect on the circulation o f changes o f g e o m e t r y akin to closing o r o p e n i n g D r a k e
Passage. T h e m o t i v a t i o n for the study is not, o f course, connected with any engineering
project, b u t is merely to p r o v i d e a greater u n d e r s t a n d i n g o f the d y n a m i c s o f the
S o u t h e r n Ocean. The numerical experiments s u p p o r t the hypothesis that the existence
o f D r a k e Passage or, rather, o f a zone o f latitudes across which there is no continental barrier, has a p r o f o u n d effect on the meridional circulation as well as on the
h o r i z o n t a l circulation.
The reason for expecting the presence o f a barrier-free zone to effect the meridional circulation is as follows. W h e n barriers exist, the Coriolis force, per unit area,
associated with the n o r t h w a r d (or southward) m o t i o n integrated along a line o f c o n s t a n t
latitude a n d c o n s t a n t depth, can be b a l a n c e d g e o s t r o p h i c a l l y by a pressure difference
*Department of Applied Mathematics and Theoretical Physics, Cambridge.
1"Geophysical Fluid Dynamics Laboratory, N.O.A.A., Princeton University, Princeton,
New Jersey 08540.
685
,0 ~
ANTarCTIC
(b)
|
sul~ ~XOelC~L
4----
s
~J
/
~T ,~CT C
4
|
ANTAIC~IC
Fig. 1. (a) Horizontal circulation in the Antarctic region after KORT (1962). Contours of transport
stream function are shown in units of 106 m3sec -~, and the volume flux between adjacent stream
lines is 60 × 106 m3sec-k (b) The meridional circulation of water in the South Atlantic Ocean
(from D~ACON, 193"/, Fig, 1).
(a)
90 ° EAST
90 °
~
J
Z
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model
687
between the eastern and western boundaries (at the relevant latitude and depth). If
the line of constant latitude and constant depth intersects no boundaries, there can
be no such pressure difference and hence no such geostrophic motion. In particular,
the general southward movement of the upper layers of the ocean north of the
Antarctic Convergence would tend to be halted by this effect around the latitude
of Cape Horn, resulting in sinking of this water mass. On the other hand, motion
in the surface Ekman layer, being non-geostrophic, is not effected, so Antarctic
st~rface water can move northward across the barrier free zone in such a layer, and so
contribute to the formation of Intermediate Water.
The model oceans, used for the numerical experiments, are enclosed in basins of
three different geometries, as depicted in Fig. 2. The barrier-free zone in one case
extends over the whole depth of the ocean, and in another case extends over only
half the depth. The circulation is driven by an imposed wind stress and an imposed
surface temperature distribution. The differences between the five different cases are
summarised in Table 2 below. The finite-difference formulation is similar to that of
BRYAN and Cox (1967) and B~VAN and Cox (1968), hereinafter called studies A and
B respectively. For each computation the final state is represented by about I0,000
numbers and over 200 hr of computer time are required. Because the available
information in each computed solution is so great, and the investment of computer
time involved in obtaining this information is so much, it is proposed that magnetic
tapes containing the information be made available on request. (A charge will be
made to cover cost of tapes, postage, etc.). Prof. H. Stommel has suggested that this
will launch us into the post-Gutenberg era of publication!
The following four sections are mainly about method. Some readers may prefer
to skip these at a first reading and turn directly to the results sections which begin
with §6.
2.
EQUATIONS
OF
THE MODEL
The numerical model consists of a set of finite-difference equations and boundary
conditions. These equations may be derived in two ways. One way is to divide the
domain into a finite number of cells. Then for each cell one can consider the balance
of mass, momentum and heat, and write appropriate finite difference equations to
express these balances. In the limit as the cell size tends to zero these equations
should become the partial differential equations expressing balances of mass,
momentum and heat in the continuum. The second method is to begin with the
continuum equations and to write finite-difference equations which 'approximate'
them; that is, to write finite-difference equations which become the continuum
equations in the limit as the grid spacing tends to zero. Finite-difference equations
obtained by the second method may not be capable of interpretation as balances
for individual cells. The difference between solutions of finite difference equations
obtained by the two different methods is probably unimportant if the smallest scale
features of the corresponding solution of the continuum equations are 'resolved',
but may be vital in cases such as ours where such resolution is not attempted. For
instance, the upper layer of the model ocean, which represents the 'mixed layer'
is thicker than the Ekman layer, but the correct expressions for the Ekman flux (and
therefore Ekman divergence) are obtained if the cell philosophy is used. This is because
the expression for the Ekman flux comes from a balance of forces between that of
688
A . E . GILL and K. BRYAN
\Z
I
- 1 / g
L_
!
-----------_
I/
I
+
r
--~
J
L
I r----(
- - - .... ~ - . _
,(~,~
I/-"
Fig. 2. The three different geometries used in the computations (a) closed basin, (b) deep gap,
(c) shallow gap. A symmetry condition is applied at the equator (~0 ~ 0) in each case. In (b), the
grid is also shown.
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 689
the wind on the surface and the Coriolis force associated with the Ekman flux. Both
these forces are properly represented in the finite differences scheme [see appendix
to BRYAN and Cox (1968, part I)].
For purposes of presentation, it is convenient to proceed as in the second method ;
that is, to first write down the continuum equations and then derive the finite-difference
'equivalent'. However, it is to be emphasized that the cell philosophy was very much
in mind when the finite-difference equations were formulated, so that the finitedifference equations can be regarded as expressing balances of mass, momentum
and heat for individual cells. The scheme was also designed so that the energy equation,
which can be derived from the others, was satisfied for individual cells (see BRYAN
(1969) for further discussion of the above points). In the absence of dissipative
processes, the finite-difference scheme is such that mass, momentum, heat and energy
are conserved.
The continuum equations and their nondimensional forms are precisely the same
as in study B, so we will do little more than quote the equations in nondimensional
form. The finite-difference formulation is different from study B and will be presented
in the next section.
There are five basic physical approximations made. These are: (i) The Boussinesq
approximation, (ii) the hydrostatic approximation, (iii) the neglect of Coriolis forces
associated with the vertical component of velocity, (iv) the use of an eddy viscosity
and an eddy diffusivity to model effects of motions on a scale too small to be resolved
by the numerical grid, (v) the representation of convective overturning effects by a
process which vertically mixes any part of a fluid column which is tending to become
statically unstable. Eight dimensional parameters may be defined. These are f~,
the planetary rotation rate, a, the planetary radius, H, the ocean depth, A M, the
lateral eddy viscosity, An, the lateral eddy diffusivity, ~, the vertical eddy diffusivity
(and viscosity), r*/Po, a surface stress/density scale and c~90", a scale of surface
buoyancy per unit mass where c~ is the expansion coefficient, g, the gravitational
acceleration and 0", the surface temperature scale. From these we define three derived
scales based on the thermocline equations, which we expect to govern the system
away from the boundaries. There is no explicit calculation of the salinity field, except
as a diagnostic calculation in case 15-I. Since salinity differences make significant
contributions to buoyancy differences, it is best to compare the 'temperature'
fields of the model with observed fields of potential density that is, to regard 9 as
an 'apparent temperature' (FOFONOFF, 1962, p. 368).
Initially, we define a velocity scale V* and depth scale d which correspond to
motion forced by temperature contrasts. V* and d are related to other parameters by
2f~ V* /d - ~,qO*/a
(2.1)
which comes from the equation used to compute relative geostrophic currents from
the density field, and known as the 'thermal wind' equation by meteorologists. A
second formula is
V*/a
=
K/d 2
(2.2)
which comes from the heat equation (vertical diffusion only). Another horizontal
690
A.E. GILL and K. BRYAN
velocity scale V** m a y be defined which corresponds to forcing by the E k m a n
divergence. V** is given by
V**d = z*/2f2po
(2.3)
which comes from the Sverdrup relation.
The 8 dimensional parameters only involve the dimensions o f length and time,
so 6 independent nondimensional parameters may be defined. These are the Rossby
number
Ro = V*/2f~a
(2.4)
= (o~gO*)213~c1/3/(2~'))5/3a 4/3
the Reynolds n u m b e r
R e = V*a/A,,
(2.5)
Pd = V * a / A ,
(2.6)
the Pdcl6t n u m b e r
the ratio o f ocean depth to thermocline depth
(2.7)
h = H / d = H (ce~tO*/2fbc)l//3aZ/3
the aspect ratio a/H, and the velocity scales ratio
(2.8)
? = V * * / V * = z*/p(TgO*a)l/3(2f~tc) 2/3.
The last definition is different from that used in papers A and B. We note that in
many papers on ocean circulation an E k m a n n u m b e r E is defined rather than a
Reynolds number. E is given by
(2.9)
E = R o R e - 1 = AM/2f~a 2.
Note that in the present study E is based on lateral friction and horizontal scale.
Table 1. Scales used to define nondimensional variables.
Quantity
Horizontal velocity components
Vertical velocity components
Vertical coordinate
Time
Pressure/density
Temperature
Stress]density
Transport stream function
Nondimensional
variables
Scale
st, v
V*
w
V*d/a
z
d
t
a/V*
2~) V*a
~b
T*/po
V*ad
P
~
r6,-ra
0*
Value in
present study
0"17 msec -1
1-16 × 10-Smsec -1
430 m
3-7 X 107S~ 1"2 yr
16 m2sec- 2
15°C
1-16 × 10-3m2sec -2
470 × 106masec- 1
The scales used to define the nondimensional variables are given in Table 1. If
95 represents latitude and 2 longitude, the equations are as follows.
(wcos95)z + u~ + (vcos95)¢ = 0
(2.10)
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 691
Ro(u, + D u - tan ¢uv) - 7zz= = v sin ¢ - see ck Pa + EFX
(2.11)
Ro(vt + D v + tan ¢ uu) - ~¢'~ = - u sin ¢ - Pee + EF4'
(2.12)
0 = P,
(2.13)
0 t + DO = Q~ + P d - X A O
where
(2.14)
DO - see ~{(ug)~ + (v cosCg)~ + (w cos ~g),}
(2.15)
A# = sec ¢{(sec ff g)aa + (cos ff g+),}
(2.16)
F a = Au + (I - tan 2 ~b)u -- 2 tan ff sec ¢va
(2.17)
F + = z~v + (1 - tan 2 ~b)v + 2 tan ¢ see ~bua.
(2.18)
The function, Q, represents a vertical flux of heat due to either of two processes.
The first is vertical diffusion which occurs whenever the fluid is statically stable,
that is,
Q = 0 z when 0~ > 0.
(2.19a)
This transfer of heat is necessarily downwards. The second process, which we call the
convective process, occurs where the fluid is tending to become statically unstable.
In these regions it is assumed that vertical mixing occurs to just the extent required
to keep the fluid in these regions neutrally stable. Such a heat transfer is necessarily
upwards, that is,
Q < 0, 0 z = O.
(2.19b)
The way this process is achieved numerically is given in the next section.
The stress components g(~a,T¢) are equated to Ro(u,,vz) at interior points and
given the values indicated below at the surface and ocean bottom.
For purposes of defining boundary conditions, the geometry of the model, shown
in Fig. 2, is defined as follows. The southern boundary is placed at ¢ = - ~ a ,
while the meridional boundaries are located at 2 = 0,21. The nondimensional
depth of the basin is 9.3 units, where the scale depth is defined by (2.7). The gap in
the meridional walls, when it exists, extends from latitude -@1 to latitude - ~ 2
and from the surface to either z = - 4 . 0 or the bottom z = - 9 . 3 .
The boundary conditions are
(i) no slip and no normal heat flux across solid vertical boundaries, that is
u = o = 9, = 0,
(2.20)
where 0, is the normal derivative of 0.
(ii) symmetry about the equator; that is
u¢=v
= 0~=0at¢
=0
(iii) periodicity through the gap; that is
for points (~,z) in the gap, the values ofu,v,O at 2 = 0 and 2 = 21 are the same.
(iv) given conditions at the surface
za
W
..~ T ¢
g : ( ¢ ) t at z -- 0
(2.21)
(2.22)
(2.23)
692
A.E. GILL and K. BRYAN
(v) E k m a n stress condition at the bottom. F o r the southern hemisphere this is
~(2/Ro)½"r ~ = v + u,
?(2/Ro)½~
W=
= v -
u,
Q=0.
The important conditions are the ones at the surface as these drive the model.
C o m p u t a t i o n s were carried out for five different cases, which have been numbered
14-1 to 18-1. We will use the same numbers here for consistency with the laboratory
classification. In all cases, the dimensions o f the basin are given by
2t = (I)3 = 1.222 radians (70 °)
and in all cases the applied wind stress was the same. The function G2 is shown in
Fig. 3 and is based on an analytical formula which roughly approximates the zonally
averaged eastward c o m p o n e n t o f the stress as computed by Hellerman (1967). The
scale z* o f stress is chosen so that the m a x i m u m value o f IG2'(~) [ is unity. The
OO
--I 0
-,GL5 I
0,0
+0.5
+1.0
,J
10°
20*
40+
",.. t
+'
)
+~].
70'
T
+"z"Y
,.................. ] .............
Woll
Fig. 3. Given conditions at the surface. G2 is the eastward component of the surface wind stress,
scaled so that the maximum gradient of G2 is unity. G1 is the temperature in the surface layer, scaled
to have unit range. The linear distribution (solid line) was prescribed in cases 14-I, 15-I and 18-I,
and the curved distribution (broken line) in case 16-I.
northerly c o m p o n e n t o f the stress was set at zero. The variations between models
consisted o f variations in the temperature forcing function G1 (if) and in the geometry.
The five cases are summarised in Table 2. The gap, when it exists, is b o u n d e d by
latitudes qbj = 1-016 radians (58-2 °) and qb2 = 1.104 radians (63.2°), and the sill,
when it exists, is at a depth given by
k o=
5½, i.e. z =
-3.96.
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 693
Table 2. Summary of the five different cases for which computations were carried out.
The different geometries are shown in Fig. 2 and the surface temperature distributions
in Fig. 3.
14-1
15-I
16-1
17-I
18-1
Closed basin
Deep gap
Deep gap
Deep gap
Shallow gap
Linear surface temperature distribution
Linear surface temperature distribution
Curved surface temperature distribution
Uniform temperature (barotropic case)
Linear surface temperature distribution
In the three cases, 14-1, 15-I and 18-I, designed to show the effects of geometry, the
surface temperature was required to be a linear function of latitude (solid line in
Fig. 3), that is
Gl(q~) = 1 + ,;/,/03.
In case 16-I, the distribution was chosen to model the observed surface density distribution as found, for instance, by LYNN and REID (1968, Fig. 4). The curve used is
shown as a broken line in Fig. 3.
3.
NUMERICAL METHOD
a. Basis of the method
The basic method is the same as in B, but the finite-difference formulation is
different, following a more general scheme outlined by BRYAN (1969). Essentially,
the method consists of integrating the equations forward in time starting from an
initial state of rest and of stable horizontally-uniform stratification. Pressure is
eliminated from the momentum equations to give vorticity equations. Because of
the hydrostatic assumption, the vertical component of the vorticity equation need
only be considered in its vertically integrated form. In this form it may be regarded
as an equation for the transport stream function, if, which, by continuity, can be
defined such that
f OR dz = --$¢, f 0 v cos $ dz = ~ .
-h
-h
(3.1)
The equivalent equation in barotropic models is very familiar. At each time step an updated value of A~k is calculated. ~b is then calculated by successive over-relaxation,
using the value at the previous time step as a first guess. The calculation is carried
forward until it is close to a steady state or appears to oscillate about an approximately
steady state. In the latter case the fields reproduced in the figures represent averages
over a suitable interval of time.
The vorticity equations used for the calculation are as follows. Firstly, the horizontal components of the vorticity equation are merely the z-derivatives of (2.1 l) and
(2.12), with (2.13) being used to substitute for P~. These may be written
(Rou r - s i n C v ) : = -sec¢oq~ + G~
(3.2)
(Ro vt + sin q~u)z = --~4, + G~
(3.3)
694
A.E. GILLand K. BRYAN
where
G ~ = - R o ( D u - tan q~uv) + ~
+ EF ~
and
G ~ = - R o ( D v + tan ~buu) + VT~ + EF ¢).
(3.4)
(3.5)
The vertical integral of the vertical component of the vorticity equation (obtained by
eliminating P between 2.11 and 2.12) is
0
Ro(t+~a=sec¢,f
where
[G~-(G
./
~ COS ,k),~]dz,
(3.6)
-h
( = see2 ~ ~ a,l + sec ~ (cos ~ ¢,~),/, = A¢.
b. The arrangement of grid points
The grid points are in a rectangular array in (~,2,z) space with varying intervals
designed to give extra resolution near boundaries and, in particular, near the edges
of the gap. Grid points are defined by numbers i,j,k which can take on integer values
or differ from integer values by t. In the latter case, we say they have 'half-integer'
values. Time points are defined by integers I. A one-one correspondence between
values of i,j,k,l and values of 2,~,z,t is established by first defining three sets of
intervals Ai, Aj, A k in the three coordinates 2,~,z respectively, and defining a timeinterval At. Then 'half-integer points' are defined by
i
~-i+1/2 = 2 AI,,
i'=l
j
~j+1/2 = - tfi3 "~ 2 Aj,,
j'=l
k
Zk+l/2 = -- ~ A k,
k'=l
(3.7)
the sums being over integer values. Integer points are defined by
2i = !(2i+1/2 + ~i-1/2) i = I, 2 . . .
(3.8)
etc., and time points by t t = lAt. Additional sets of intervals are defined by
Ai+I/2 = 1(Ai -I- A i + I )
i = l, 2 . . .
etc. In the horizontal, values of u,v, are stored at half-integer values of i,j, and values
of,9 are stored at integer values. In the vertical, u,v, and S are stored at integer values
of k. The spacing in the horizontal was based on an analytical formula designed to
meet the following requirements:
(i) smooth changes in spacing so higher order differences are well behaved,
(ii) constant spacing, b, a long way from boundaries,
(iii) fractional rate of change of spacing bounded above.
The formula was also designed to make the rate of change of spacing small near the
boundary. The formula, as applied near the western boundary was of the form
Ai
0.25 cosh 0.8(i - 1)
-b- = l + 0.25 cosh 0-8(i - 1) "
(3.9)
The position of the points used are shown in Table 3 to the nearest tenth of a degree.
There are four half-integer and five integer points within the gap.
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 695
Table 3. Position of grid points in the horizontal to the nearest tenth of a degree of
longitude or latitude.
(i + ½ = 1, 19): O, 1"2, 2'9, 5'6, 9-3, 13.8, 18.8, 24"1, 29"5,
35"0, 40-5, 45"9, 51"2, 56"2, 60'7, 64-4, 67-1, 68'8, 70"0.
(j + ½ = 1, 29): 70-0, 68'9, 67"6, 65'7, 64-2, 63"2, 62"2, 61'2, 60'2,
59"2, 58'2, 57-2, 55"8, 53'5, 50"4, 46"6, 42"4, 38"0,
33"4, 28"8, 24"3, 19'7, 15"3, 11"1, 7"3, 4"2, 1'9, 0-5, --0"5.
Table 4. Grid intervals in the vertical and initial distribution of temperature with depth.
Layer k -Thickness (m)
Ak (nondimensional)
Initial temp. 0
1
60
0-14
1-16
2
120
0"28
0-94
3
240
0"56
0'61
4
480
1"12
0'37
5
800
1"86
0"14
6
1000
2.33
0'04
7
900
2"09
0'01
8
400
0"93
0"01
It is convenient now to define the finite difference operators
~ag = ( g i + l i 2 - g l - i i 2 ) / A +
(3.10)
g2 = ½(gi+l/2 -'F ffi-1/2)
(3.11)
which may be defined at integer and half-integer points. Similar operators can be
defined for the coordinates ~b, z and t. A calculus of these operators is easily worked
out.
The finite difference forms of the equations will now be given, with only brief
comments about the reason for the choice of scheme. For further details, the reader
is referred to BRYAN (1969). First of all, the finite difference form of (3.1) is
8
8
Z ukA~ = -<$<+ffz, E v,,,Ak = 6,tff'/,,
k=l
(3.12)
k=l
the stream function, ~k, being stored at integer points. Some care is now required in
writing the finite difference form of the continuity equation (2.10). In order for w to
vanish at both k = ½ and k = 8½, the vertical integral of the finite difference form of
(2.10) must be consistent with (3.12). Because temperature and velocity are stored
at different points, different cells are used for temperature and momentum equations,
and so different forms of (2.10) are required. Consider first the temperature equation.
Since 0 is stored at integer values of i,j,k, the value of w is required at integer values
of i,j and half-integer value of k. The form of (2.10) used is
cos¢6=w+
\A s]+
vA,
+1--~ cos q$)
= 0.
(3.13)
This is consistent with (3.12) since
8 u A~~
~,~, ~
Ak = _,$+(~,~<t.).
(3.14)
696
A.E. GILLand K. BRYAN
The advection term in the temperature equation (2.14) must be written so that it
vanishes when v~ is constant. The form used is
6"fit= - s e c ~b 6juAj~
0 ~Aj
I\
/ - s e c q~ 6~ (-~i~\
A, cos q~ ~ )
-6z(w'O z) + ~z~z9 + Pd-l{sec2 q~6~6;.0 + sec ~b6~(cos q~fi~0)}. (3.15)
The advection terms are calculated at time step l and diffusion term of time step l - 1.
The convective mixing process works as follows. At the end of each time step, predicted values of 0 z+ ~ are tested for static stability (9k-1 < Ok). If this criterion is not
satisfied for part of the column, the heat of the offending cells is mixed uniformly
over these cells. The test for static stability is repeated and, if necessary, further mixing
takes place. Eventually the static stability criterion is satisfied, and the values of 9 ~+1
obtained at this stage are the ones used for further time-stepping.
The finite-difference form of (3.2) is
6z[Ro 6tfd - sin q5 (-~vt+l + :~Vl-l)] = ~zG~ -- sec ~b 6a0 z~.
(3.16)
The implicit treatment of the v terms on the left-hand side is used so that inertial
waves need not be resolved in time and will be damped out (KURmARA, 1965). In
the expression for G~ (see 3.4), ~,z~ is written ~6~z ~,
where
G2(,k), k = ½
za = ~ - l Ro~zu, 1½ < k < 7½
~,-1 (½Ro)1/2(u8 + Vs), k = 8½.
(3.17)
The finite-difference form for F ~ is
F ~ = sec 2 ~b6zf~u + sec ~b6~ (cos q~64,u)
+(1 - tan z ~b)u - 2 tan q5 sec q5 6~fia,
(3.18)
this being calculated at the time step before the previous one in order to give computational stability. The finite difference form for the term Du in (3.4) requires w at
half-integer values of i,j, and k. This value will be called w' and the finite-difference
form for (2.10) used to calculate w' is
6~w' + see 4~ ~ { a ~ + h - ~ ( i ~
~ -
~)}
+ sec ~b 6o{v cos ~ - h-16~(~ 4'° - ~b)} = 0.
(3.19)
This is consistent with (3.12) since
8
y {a~ ~ + h-~6~(~ ~ - ~,)}zx~ = - 6 , ¢
k=l
The form used for Du ensures that Du vanishes when u is constant, the form being
Du = sec q~ 6a{tiz[az + h-'f,(ff~z _ O)]}
+ s e e ~ ~4~ {~14'[v Cos (~4~ _ h - 16z(ffq~, _ I]/)]}
+ 6z(w'ft').
(3.20)
Effects of geometry on the circulation of a three-dimensional southernohemisphereocean model 697
This completes the finite-difference formulation of (3.2). That of (3.3) is similar.
The remaining equation is the finite difference form of (3.6). This is written
8
Ro ~,~' = - ~ i ~
+ sec ~ ~ [~(G~) ~ - ~,(G ~ cos ~)~]Ak
(3.21)
k=l
with
= s e c 2 t~ ~,t~;I// + sec ff ~ (cos ~b ~ , ) .
(3.22)
is calculated from ( by successive over-relaxation.
When the gap is present, another condition (KAMENKOVlCH, 1961) is required to
make the determination of @ unique. The finite-difference form of this condition
(see BRYAN, 1969) used was
18
8
{Ro 6t~4,~tx + ~ G'~'kAk)i+~Ai+½ = O,
i=1
(3.23)
k=l
the condition being applied a t j = 7½.
The finite-difference tbrm of the conditions applied at the vertical boundaries
are the obvious ones. For instance, on the southern b o u n d a r y j = ½,
u = v = fi¢oq = 0.
(3.24)
The conditions at the top and bottom surfaces for the continuity and momentum
equations have already been given. For the heat equation (3.15) the conditions are
0 = Gt(~b) a t k = 1
6zO = 0
at k = 8½.
(3.25)
Note that the temperature in the top layer is prescribed, modelling the homogeneous
surface layer of the ocean. This means that (3.15) is only applied for k = 2 to k = 8.
For the diagnostic calculation of salinity in case 18-1, the velocity field was fixed
at the value obtained from the primary calculation, and (3.15) was used with salinity
s replacing oa. The boundary conditions were the same as those for oa, that is, s was
prescribed as a function of latitude at k = 1 and the normal derivative of s was
made zero at solid boundaries. At the end of each time step, convective mixing
(preserving salt) was carried out wherever convective mixing of ~ had been occurring
at the end of the primary calculation. The integration was continued until the field of
s was close to a steady state.
4 . C H O I C E OF PARAMETERS
One of the conditions that dictates the choice of parameters is that of sufficient
resolution of the western boundary layer. Two possible widths of this layer are
suggested in the literature, namely
Lr
= E 1/3 = ( A M / 2 f l a ) 1/3
and
Lr
= R o 1/2 = ( V * / 2 f l a ) 1/2
= (~g0,)1/3 ~cl/6(2f0- 5/6 a - 2/3
(4.1)
698
A.E. GILL and K. BRYAN
corresponding to boundary layers associated with friction and inertia effects respectively. In this model, bottom friction is too weak to be relevant. A further possible
scale L u = P d - s is derived in B. Now because of storage difficulties, the parameters
have to be chosen to give widths somewhat larger than those applying to the ocean
in practice. In this study both the Ekman number and Rossby number are made
large, keeping Lg and L e approximately equal. The larger values could be associated
either with excessively large values of friction or with an artificially reduced rotation
rate. The parameters used in the present study are shown in Table 5. There is some
advantage in thinking of these values as corresponding to an earth with a slower
rotation rate. F o r instance, if the value of f~ is taken to be one tenth o f that of the
earth, the nondimensional numbers chosen correspond to the scales shown in table
1, together with the values 2f~ = 1.47 × I0 - s s -s, A,, = 4.9 x 104m 2 s - s , At1 =
1.2 × 1 0 4 m 2 s - s and Z = 5 x 10 -a m 2 s -1.
In addition to the width scales associated with the western boundary current,
there are width scales associated with the circumpolar current. The scales for a lateral
friction model have been discussed by GILL (1968, p. 485) and arise from a balance
Table 5. Parameters o f the computation compared with those used by BRYAN and Cox
(1967) and BRYAN and Cox 0968).
Present study
Ro
Re
E
Pd
y
h
Lt
LI,
Lu
As
2
X 10 - 3
22
9 x 10-5
90
1.1
9"3
0"044
0"044
0'011
0"021
Study A
Study B
3 x 10-5
8
4 x 10-6
32
1
13
0"006
0"019
0"031
2 x 10-4
150
1"5 x 10-6
150
1
10
0"014
0'011
0"007
0"005
As represents the spacing A~ (i= 1) at the western wall.
o f the fl-term, ~k~ and the friction term EA2~k in the equation for the transport stream
function. In the present notation, the appropriate boundary layer has average width
E s/4 reducing to values of L r = E 1/3 in the neighborhood of the gap. In the present
study E s/4 = 0"1 so that this layer is satisfactorily resolved.
To make any quantitative comparison with observations, it would be necessary
to extrapolate results to more realistic (that is, lower) values of E and Ro. Because each
calculation involves so much computer time, however, it has not been possible even
to examine trends resulting from changes in E and Ro. Therefore, in making comparison with observation general patterns have been compared rather than values.
Since terms involving E and Ro are only important in boundary-layers if E and Ro
are small, it is hoped that patterns will not change very much as E and Ro are varied.
(The resolution problem is discussed by GILL, 1971.)
5. TIME-DEPENDENTBEHAVIOUR OF THE MODEL TIME STEP
Time step
There are various criteria for choosing a time step which is small enough for a
stable numerical integration. The Courant-Friedrichs-Lewy condition (discussed,
for instance, in RICr~TMF.YER, 1957) for a stable advection scheme requires a time step
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model
699
less than the grid-spacing divided by velocity. Inspection of the final solution gives
a minimum value of this ratio of about 3 × 10 -3, although smaller values might be
required to handle transients. The condition for a stable diffusion scheme is that the
time step be less than
1
F r o m Tables 3 and 4, the smallest values of A i cos ~b, Aj and A k correspond to cells
in the gap near the surface, the values being A~ cos ~b= 0"010 (rad), Aj = 0.107 (rad)
and Ak = 0.14. From Table 5, E/Ro = 0.045. Substitution in the above formula gives
a time step of
1
(450 + 150 + 50) -1 = 4 × 10 -4
Although such formulae give a guide, the final choice of time step for a complicated
calculation must depend, to some extent, on trial and error. In the present calculation,
the time step used was 104 sec = 3 × 10 -4 nondimensionai units.
Adjustment time
The slowest adjustment process which determines the time for the model to reach
a quasi-equilibrium is the temperature diffusion process. By definition the time for
diffusion over the thermocline depth is one unit of time (3700 time steps). However,
the time to diffuse to deeper layers goes up as the square of the depth, so that the deeper
layers are much slower in coming to adjustment. The computations were run for
about 24 time units. Table 6 shows how the temperature at levels k = 3 (z = - 0 - 7 )
and at k = 6 (z = - 5 . 1 ) varies with time for cases 14-I and 16-1. Also shown is
the variation of kinetic energy per unit mass and transport through the passage.
In case 15-1 a steady state was not reached but an oscillation of period about 4.2
units of time was found. Another oscillation of about one-third of this period is also
shown in Fig. 4, which displays curves of the kinetic energy per unit mass and total
transport through the passage as functions of time. Also indicated is the period over
which averages were taken to give the quantities presented later as the 'steady-state'
solution.
Table 6. Values of certain quantities as functions of time.
Case 14-1
t
0
2.5
5"0
7'5
10"0
12'5
15"0
17"5
20-0
22"5
Meantemponj=19and
k = 3
k = 6
0'605
0"460
0"459
0"458
0"458
0'458
0-458
0'458
0"458
0"458
0'044
0"047
0'051
0'055
0'058
0"060
0-062
0"063
0-064
0"065
Case 16-I
,~Ieankinetic M e a n t e m p o n j = 1 9 a n d
energy per
k = 3
k = 6
unit ma~s
0
0"43
0'44
0"47
0'45
0-46
0"46
0'47
0"47
0"47
0'605
-0'349
0-348
0'348
0"348
0'347
0"347
0"347
0"347
0"044
-0'078
0'084
0'090
0'094
0'097
0"100
0' 102
0'105
M e a n k i n e t i c Transport
energy per through the
unit mass
passage
0
0"34
0"36
0"38
0"39
0.41
0"42
0'42
0'43
0"43
0
0"44
0'46
0'47
0-48
0-48
0"49
0"49
0-49
0"49
700
A.E. GILL and K. BRYAN
Machine requirements
The calculations were carried out on a Univac 1108 computer with a core storage of
64,000 words of approximately 8 decimal places. The basic operating speed of the
machine is 15 /~sec for multiplication and 10 /zsec for addition and subtraction.
The present program required 29,000 words of core to store the variables, u, v, and 9,
at two time levels. A single run extending over 24 units of nondimensional time
required over 88,000 time steps in the numerical integration. Each time step took
approximately 8 sec, giving a total computation time of about 200 hr for each case.
6. THE VELOCITY
FIELDS
The purpose of this section and the next is to present the velocity and temperature
fields obtained from the computations. There will be some discussion of the way these
fields compare with observed fields, but the main discussion will be about two most
important distinctive features of the circulation in the Southern hemisphere, namely
1.0
0,
:~
- -
,\
e
//'
'_,,
"J
". . . . . .
%;
Transport Through the
~).4
Gop
0,2
_
18
'
'"
:
19
'
~
_
20
'
A v e r a g i n g ]nlervo|
.1,
,'2
_
/,
_
3.7
/,
2'~
/,
Fig. 4. Variations of kinetic energy (solid line) and transport through the gap (broken lines) in
cases 15-I and 16-I. The fields presented in the following figures are averages over the interval shown.
the circumpolar current and the formation of Antarctic Intermediate Water. This
discussion follows in sections 8 and 9.
The method of presentation of the velocity field is as follows. Firstly, pictures of
the horizontal transport, meridional transport and zonal transport are shown. These
show the mass transport in each of the three co-ordinate planes and so may be regarded
as depicting certain average properties of the velocity field. Secondly, details of the
velocity field are presented for one case (18-I) by presenting contours of pressure and
of the vertical velocity component on a few horizontal planes. Thirdly, some typical
particle trajectories are shown for case 15-I.
Mass transport stream functions
Figures 5, 6, and 7 show contours of the three different mass transport stream
functions 0, ~9("), and O(z) that may be defined after integrating the continuity equation
:.
5/~-
,o,
(a)
" " 2
~
o.
,E
o"
1
,~.
o.f
63.2"-
30'
.
.....
:.ON, , i ;
.
~
lO"
~,,:
2O'
/
.
~o,
.
(d)
3O
4c'
LON'S,TUDE
~oo
P
b~)"
.
6O"
I
70*
(b)
.
.
LONGITUDE
'
.
(e)
LONa,TUOE
.
.
.2
a,o
......
(C)
~,o,
2
4,o.' ~,o.
L O r , / G ' T t j fgE
6,o.
.
Fig, 5. Horizontal circulation in the model ocean for cases (a) 14-1 (closed basin), (b) 15-1 (deep gap,
linear surface-temperature distribution), (c) 16-I (deep gap, curved surface temperature distribution),
(d) 17-I (deep gap barotropic case), and (e) 18-I (shallow gap, linear surface-temperature distribution).
Contours of the mass transport stream function ~b, are shown in non-dimensional units. ¢ is zero
on boundaries north of the gap, and has a value ~btot on boundaries south of the gap, where
Ctot = (b) 0"64, (c) 0"48, (d) 0"42 and (e) 1-78. ~btot is the volume flux through the gap.
2 0 - ~
lo'-
,
70.
O
.-...1
O
O
O~
B
,.a
g
R
'O
R
~r
8
O
A.
O
O
q
8
ell
702
A.E. GmL and K. BRYAN
(2.10) over the ranges of the three co-ordinates z, 2 and q~ respectively. The function
is defined by (3.1) and represents horizontal mass transport. The function O(m) is
defined by
21
21
j vcos~d2
= -O= ("),
j w c o s q ~ d 2 = ~k~(m,,
0
(6.1)
0
and represents meridional mass transport. The third function, ~('), is defined by
0
0
j'_ w c o s ~ d ~ b = -ifia (=},
6.(z)
_t- ud~b
-@3
(6.2)
-@3
and represents zonal mass transport. The figures are drawn directly over the output
from the line printer, the contours being based on those obtained by linear interpolation between grid points.
(a) The pattern of horizontal mass transport (Fig. 5) obtained in the barotropic
LATITUDE
10" 20*
!~
30 °
40 °
LATITUDE
50 °
::)*
o8:
g
0*
I
10" 20*
I
I
30 °
,
,
3.0'
40 °
I
50*
I
i
7(3 °
i
_.
i
-Z
5.1
,
8.8
,
'o'
i
,
3.0
-Z
5.1
8.8
9.3
(c)
(d)
Fig. 6. Meridional circulation in the model ocean for cases (a) 14-1 (closed basin), (b) 15-I (deep
gap, linear surface-temperature distribution), (c) 16-1 (deep gap, curved surface-temperature distribution), and (d) 18-1 (shallow gap, linear surface-temperature distribution). Contours of the mass
transport stream function, ff('~),are shown in non-dimensional units. ~b(m) is zero on the boundaries.
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 703
case (17-I) is very similar to the pattern one would expect from linear theory (c.f.
MuNK, 1950; GILL, 1968). The main features are the subtropical ~ r e , the circumpolar
current and the clockwise gyre south of the circumpolar current. The last two features
may be compared with their counterparts in the observed circulation as found by
KORT (1962, Fig. 1(a)). In the deep-gap baroclinic cases, the currents are strengthened
and distorted somewhat, but the basic features are the same. The most spectacular
effect is the increase in the strength of the circumpolar current in the shallow-gap case.
This is due to a thermal effect which will be discussed in §9.
Some quantitative conclusions can be drawn from the barotropic case, for which
non-linear effects are unimportant. Analysis on the lines of GILL (1968) indicates
that Ifltot is given approximately by an expression of the form
~ttot
= ~((I)2 -
~l)3f(q)
where
E
(E21) 1/4
r/-
(6.3)
qb2 - @1
The result of case 17-I gives
f(1) = 0.02.
(6.4)
In dimensional terms, (6.3) becomes
"c*l3
~tJl/tot =--'~
J(?])
(AaZ21/2f~) 1/4
where
~/ =
(6.5)
l
and l = a ((I)2 - ~1) is the width of the gap. Because of the strong dependence on A,
the formula (6.5) cannot be used to predict Ll/tot, but one can ask instead what value
of A gives the observed value of trfltot.9 It turns out that a value of A of 2.5 × 104 m2s -1
gives, by (6.5), a value of r/ near unity and also, by (6.4), a value of q~tot about the
observed value (2.3 x 10 s m3s -1, from REID and NOWLIN, 1971). Since, however,
this study shows considerable sensitivity of Wtot to the geometry of the Drake Passage
region and also to baroclinic effects, one cannot make any strong conclusions from
this result.
(b) The patterns ofmeridional circulation (Fig. 6) show the very pronounced effect
of the gap. In the deep-gap cases, the gap tends to prevent meridional transport
across it, so the single gyre of the closed gap case is split into two. In the shallow-gap
case there is more transport across the gap but the pattern is still very different from
that for the closed-basin case and there are still two gyres. The reason for the strong
effect of the gap on the meridional circulation may be seen if the eastward momentum
equation (2.11) is integrated with respect to 2 across the basin. In the closed-gap case,
the leading terms are, in most places, the geostrophic terms, that is, the approximate
balance is
sin
¢
v cos q5 sin ~ d2 = P(21) - P(0)
(6.6)
0
This equation indicates how the gap can have a profound effect on the meridional
motion, for, in the gap, P(21) = P(0), and so the right-hand-side of (6.6) vanishes.
This implies that unless other terms become important, Cztin) will be greatly reduced
in the gap. The reduction in the deep-gap cases is quite dramatic. The pictures of the
meridional motion obtained in the computations may be compared with the picture
704
A.E. GILL and K. BRYAN
deduced from observations by DEACON (1937, Fig. 1(b)). Although there are considerable differences, the following points of agreement m a y be noted: (i) downwelling at the
southern b o u n d a r y (ii) upwelling further north, and then (iii) a zone of downwelling
north o f the gap. The latter zone of downwelling represents the formation of Antarctic
Intermediate W a t e r and this feature will be discussed in §8. In the numerical models
this downwelling is much deeper than in reality.
(c) The zonal circulation shown in Fig. 7, is closely linked to the north-south
temperature gradients by the relation
u~ sin q~ = - 0 +
(6.7)
which is valid if the flow is geostrophic. This requires a circulation of the sign shown
in the figures. In the closed-basin case, the circulation must be completed by upwelling
and downwelling near the boundaries. In the other cases the transport through the
gap in the circumpolar current is clearly indicated by the fact that ~(~) is not zero
on the lower boundary.
LONGITUDE
c, 0 °
O.O'~
.
1.0"
1.5 4111 !
9.3
20 °
30 °
~-
40"
LONGITUDE
50 °
60'
70 °
"
.--,----1",'11
-'--"--'--
~
0 °
"L.
10 °
20 °
I
.2 4. I
I/,-"
I
30 °
I
,---,___.-12~
# ....
"
70 °
"-'hi
~
I
IffP,
60 °
I
tbj
O O~
+o
50 °
I
.
Io)
f/h-'-C'%Z~o~
,
40 °
I
J~
-
I
4
I
I
I
I
~";o,,
:~:
,71
,/+ -u
-Z
ii
(c)
(d)
Fig. 7. Zonal circulation in the model ocean for cases (a) 14-1 (closed-basin), (b) 15-1 (deep gap,
linear surface-temperature distribution), (c) 16-I (deep gap, curved surface-temperature distribution),
and (d) 18-I (shallow gap, linear surface-temperature distribution). Contours of the mass transport
stream function, ~b~z), are shown in non-dimensional units. The position of the sill is indicated in
(d). ~b~') is zero on the upper surface, and has the value ~htot on the lower boundary (see Fig. 5).
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 705
Details of velocity field
Figure 8 shows contours of pressure and contours of the vertical velocity component at four different levels. The motion was found to be closely geostrophic except
at grid points next to boundary points and near the equator. Hence the horizontal
velocity components are given, to a first approximation, by
v s i n ~ = secffP~,
usin~b = - P ~
(6.8)
and so the horizontal velocity is approximately parallel to the pressure contours
except very close to the boundaries. Near the surface, the motion in the Ekman
layer must be added to the geostrophic motion. The pressure calculated is actually
an average over neighbouring grid points, so appears to be discontinuous at the north
and south edges of the gap.
It is interesting to compare the horizontal velocity field at each level with the
vertically integrated field, shown in Fig. 5(e), to see if the vertically integrated motion
is similar to the motion at any particular level. One can see, for instance, the circumpolar current and the sub-tropical gyre in the upper sections, but the deep currents
(Fig. 8(g)) tend to be in the opposite direction. It follows that the main contributions
to the vertically integrated field come from the upper layers. Of the four horizontal
sections shown in Fig. 8, the one that looks most like the vertically integrated field is
the one for mid-depths (Fig. 8(e)). This section even shows features like the tropical
gyre and the clockwise "Weddell Sea" gyre south of the circumpolar current. The
uppermost section does not show these features.
The horizontal currents shown in Fig. 8 can be compared with those deduced
indirectly from observations by studies, for instance, of properties on isentropic
surfaces (TAFT, 1963 ; REID, 1965) or through calculations of pressure relative to some
arbitrary reference level. The two ideas are combined in some studies and REID (1965)
has published maps of 'acceleration potential' relative to some reference level, on
surfaces of constant 'thermostatic anomaly'. For a precise definition of these
quantities, the reader is referred to the original paper. It is sufficient for our purposes
to regard these as contours on a surface of variable depth, of a quantity which, if
substituted for P in (6.8), gives horizontal velocity components. The southern hemisphere portion of these maps is reproduced in Fig. 9. The following points of similarity
with Figs. 8(a), (c) and (e) may be noted: (i) the eastward circumpolar current (ii) the
anti-clockwise sub-tropical gyre and (iii) the southward shift with increasing depth
of the centres of the sub-tropical gyres.
The difference in pressure between the eastern and western boundaries may be
estimated from Fig. 8. This difference is closely related to the meridional circulation
through the relation (6.6) which is approximately valid where this motion is geostrophic. Taking, in particular, the surface values, these predict that the sea-level on the
Pacific coast of southern South America is higher than the sea level on the Atlantic
coast. This agrees with deductions REID (1961) has made from observational data.
The sign of the pressure difference corresponds, by (6.6), to general southward motion
near the surface (but below the Ekman layer), in agreement with the results shown
in Fig. 6(d). At Cape Horn, the difference in sea level vanishes so that if (6.6) were
strictly true, the southward motion would cease altogether. Figure 6(d) shows that
although the southward motion does not vanish at Cape Horn, it is considerably
706
A.E. GILLand K. BRYAN
weaker there than further north. As a consequence downwelling is induced to the
north of the latitude of Cape Horn. This downwelling region found in the model
appears to correspond to the zone of downwelling where Antarctic Intermediate
Water is observed to be formed.
The vertical velocity component varies very rapidly, so only a limited amount of
information can be shown in the figures. These show the sign of the vertical velocity
and the regions of very strong vertical motion.
Figure 8(b) shows the vertical velocity just below the first layer and is therefore
just that required to compensate the divergence in the first layer (principally the
o o.~:
L O N G I T U DE
?o
~o~ ~o~ ~oo ~oo ~oo ioo 0o
~o~-I\~',,,?"
/\\,w
~
,~.
~.~o~_~
o °
~oJ\'[
=
/
~
-~'.o~ I •
\~.o~
'-.'
ooo
Jl I
~o$~,,,, .--:- ........
IX":,.,,;-
L O N G I T U DE
~oo ~o~ ~oo ~oo'~oo
,(°)
I
,
\
I
•
I
I
J
(b) I
l ..... .--!::--:_~_J ~
Ic)
Fig. 8 (a-d).
(d)
i
~o~
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 7 0 7
E k m a n d i v e r g e n c e ) . L a r g e r v e l o c i t i e s are f o u n d at m i d - l e v e l s . T h e m o s t i n t e n s e
v e r t i c a l m o t i o n is f o u n d in t h e cells a d j a c e n t to the v e r t i c a l b o u n d a r i e s , a n d m u c h
o f t h e u p w e l l i n g a n d d o w n w e l l i n g is c o n c e n t r a t e d in n a r r o w b a n d s ( o n e cell t h i c k )
a d j a c e n t t o t h e s e b o u n d a r i e s . F i g u r e 7 s h o w s this f a c t v e r y clearly. F i g u r e 10 s h o w s
h o w t h e p r e s e n c e o f the g a p c a u s e s l a r g e c h a n g e s in t h e field o f v e r t i c a l m o t i o n .
LONGITUDE
O°
0o . ~
1o°"
10 °
I
20 °
I
",
30 ~
I
40 °
I
, .............
%"',
"--
i
LONGITUDE
50 °
I
""
60 °
I
..7.'-
.....
3 0 °-
4
204
I
I
I/
......
20 °
I
30 °
I
40 °
I
i
50 °
~
60 °
I
70 °
I
I
I
I
'
I
'
.
~
L3.1
3125~
1
-'-..
I/
I
I
.J
<
(f)
/
I
I
I
c
%F
"-
II
LJ I
~1 40
I
10 °
I
,J
(e)
0o=. I
)°
. I
--
70°i
]0°
70 °
I
.
2~
7O o / H ~ ' - ~
"
"
~
~
-
(h)
Fig. 8. Contours of pressure, P, and vertical velocity, w, at different levels for case 18-1. (a) P at
k = 1 (z = --0'07). (b) w at k = 1½ (z = --0.14). (c) P at k = 3 (z = --0"7). (d) w at k -- 3½
(z = --1.0). (e) P at k = 5 (z -- --3"0). (f) w at k = 5½ (z = --4.0). (g) P at k = 7 (z = - 7 . 3 ) .
(h) w at k = 7½ (z = - 8 - 4 ) . Pressure is shown relative to a value of 3.13 ( = 5 0 m2s -z) at q~ = 0,
A = ~t.
708
A . E . GILL and K. BRYAN
Strong vertical motion is produced in and near the gap. Particularly intense downwelling is found in the eastern side of the basin just north of the gap, and intense
upwelling is found on the western side just south of the gap. These intense motions
do not, however, have dramatic effects on the temperature field.
Trajectories
Some particle trajectories were computed on the following lines. The basin was
divided into cells according to the grid system and in each cell the velocity was
assumed to be uniform. If the particle reached a boundary where zero normal
velocity was prescribed, or if it reached a cell boundary such that normal velocities
on either side were opposed, the particle was moved back normal to the boundary
one-fifth of a cell width and the computation continued. Positions and times of cellboundary crossings were recorded and used to draw paths. Two portions of trajectories for case 15-I are shown in Fig. 11.
Particles often get caught in a ' l o o p ' and tend to go over similar paths again
and again. However, on each circuit the path is somewhat different and eventually
the particle 'escapes' from the loop and moves into a different region. For instance.
Equator
(a)
Equator
(b)
Fig. 9. Acceleration potential (from whose gradient geostrophic velocities may be calculated) on
surfaces where the thermosteric anomaly is (a) 1.25 and (b) 0"80 litres per ton (from REID, 1965).
The depth of the constant-anomaly surfaces are shown in Rr,m (1965) and vary from zero to 700 m
in case (a) and zero to 1200 m in case (b). The reference level for calculating the potential is 1000
decibars in case (a) and 2000 db in case (b).
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 709
L'
~------~
L
O,
.E
o,
.~G
.
i °z~,.
o
Q~
' ~-~
~.o
-
~~2~_~
;~e'~
o
o . =_
o
o
"~ (B FI.L LLV-[
Effects o f g e o m e t r y o n the c i r c u l a t i o n o f a t h r e e - d i m e n s i o n a l s o u t h e r n - h e m i s p h e r e o c e a n m o d e l 7 1 1
I C) lq ([; f f U D [ [
o~t '°
~,':
i"
I ON(, [~JDE
~V" 4i°' ~i'~ ~,°° '~'
~i~o .,.,
.~;
, 'di,iTI L)E
4~.~ ~i'" ':i°'
7O °
1
"
,:°
~
: 3" ~
~
'
~
,,"
'
,,"
]¢ '
'
d
i,)"
\
/![,o,
\
; '-"
Y.
f
o
'°:I
, ,.Ao,
,
\\
/"
\........ ~'fl
7O"
0
o
(4
(el
(f)
Fig. 12. H o r i z o n t a l t e m p e r a t u r e sections. (a) at k = 5 (z = - - 3 ' 0 ) for case 14-I (closed basin).
(b) at k = 5 (z = --3-0) for case 15-1 (deep gap, linear s u r f a c e - t e m p e r a t u r e distribution). (c) at
k = 5 ( z = - - 3 . 0 ) for case 16-1 (deep gap, s u r f a c e - t e m p e r a t u r e distribution). (d), (e), (f) at k = 3
( z = --0"7), k = 5 ( z = --3"0), a n d k = 7 (z = - 7 . 3 ) for case 18-I ( s h a l l o w gap, linear s u r f a c e t e m p e r a t u r e distribution). T h e slightly negative values in ( f ) are d u e to s m a l l t r u n c a t i o n errors.
effect of temperature differences, yet the temperature itself is determined as a balance
of advection (by the velocity field with which it is so closely related) and mixing
processes. Both aspects of this interplay can be detected in the horizontal section of
temperature shown in Fig. 12. On one hand, isotherms give the direction of the
vertical gradient of horizontal velocity. On the other hand, the effect of vertical
advection is particularly noticeable in upwelling zones near the western boundary
and (at depth) in the tropics, where lower temperatures are found.
Figure 13 shows meridional sections of zonal mean temperatures. One interesting
feature is that the average temperature for the whole basin is higher when the gap is
present than it is when the gap is not present. This is probably due to the existence
of the circumpolar current when the gap is present leading to higher values of uz and
therefore of 0~. Since O is zero on the southern boundary, this implies higher average
temperatures when the gap is present. This is in agreement with observations for it is
well-known (see, for instance, R~ID, 1961) that mean north-south density gradients
are larger in the southern hemisphere than they are in the northern hemisphere.
712
A.E.
LATITUDE
o°"
•
7
;o. 2o°
3o ° 4o"
I
so"
,'¢-S~-'~
07 1-
" " J
/
G I L L a n d K . BRYAN
LATITUDE
,
,
m
•
/
/
70* 0 °
.~.
10 ° 2 0 ° 3 0 ~ 4 0 °
i
i
,
I
.~,'
--'-
--7
L
70 <
4..---/y f l /
J
•- - - - - - ' ~
• ~
.
5,1
50 °
/IC
~-J
7
.,4
I
I
I
7,3
8,8
9.3
0.000,28"
0,7
(o)
j '
I
!
(b)
I
q.,A
'I
I I
II
1•5
3.0
II
J
II
--Z
5.1
I1
7.3
i I
J i
8.8
9•3
II
!
I
I
;
(c)
(d)
Fig. 13. Meridional sections of zonal mean temperatures for cases (a) 14-I ('closed basin), (b) 15-1
(deep gap, linear surface-temperature distribution, (c) 16-I (deep gap, curved surface-temperature
distribution) and (d) 18-1 (shallow gap, linear surface-temperature distribution)•
In the model, the only contributions to buoyancy come from temperature, which
should therefore be regarded as an 'apparent temperature' (FoFoNOEF, 1962, p. 368)
and compared with quantities whose horizontal gradients give reasonable approximations to horizontal gradients of buoyancy forces. Thus the fields shown in Figs. 12
and 13 can usefully be compared with maps of potential specific volume anomaly
(e.g. TAFr, 1963) or of at (e.g. DEFANT, 1961, Plate 7) or of potential density (LYNN and
REID, 1968, Figs. 4 and 10). The following points of similarity between computed
and observed fields may be noted: (i) small horizontal gradients at the lower levels,
(ii) steeply sloping contours in the south, (iii) density minima at mid-latitudes at
mid-levels and (iv) a tendency for isolines in horizontal sections to run south of east
and north of west, often showing abrupt changes near the boundaries•
8. A N T A R C T I C
INTERMEDIATE
WATER
FORMATION
The striking effect, shown in Fig. 6, of the existence of the gap on the meridional
circulation has already been noted. The reason for the effect has also been given,
Effect s of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 713
a n d will be highlighted further in this section. In brief, there is a d y n a m i c c o n s t r a i n t
which prevents a flux o f water across the latitudes o f the gap. T h u s the water which,
further n o r t h , is flowing p o l e w a r d in the u p p e r layers, is forced to sink as it a p p r o a c h e s
the latitude o f C a p e H o r n . W e suggest t h a t the sinking p r o d u c e d in this m a n n e r is
responsible for A n t a r c t i c I n t e r m e d i a t e W a t e r f o r m a t i o n , a n d t h a t the downflow
region n o r t h o f the gap in Figs. 6(b), (c) a n d (d) should be identified with the d o w n flow region near the A n t a r c t i c Convergence shown in Fig. l(b).
Tile d y n a m i c a l reasons for the downflow in the models, will n o w be discussed
further. T h e n the solutions will be e x a m i n e d to see i f there are any o t h e r features
which m a y be identified with the A n t a r c t i c Convergence o r the f o r m a t i o n o f A n t a r c t i c
I n t e r m e d i a t e Water. Several a u t h o r s , e.g. MtrNv, a n d PALM~N (1951) a n d STOMMEL
(1957), have p o i n t e d o u t the difference in d y n a m i c s between a region where continental
barriers cross each parallel, a n d a region where they d o not. W h e r e barriers exist, as
in most o f the ocean, pressure differences between the eastern a n d western b o u n d a r i e s
are possible, a n d hence the t y p e o f b a l a n c e inherent in S v e r d r u p ' s relation. W h e r e
barriers do n o t exist, as in the latitudes o f D r a k e Passage, there can be no such
pressure difference, a n d the d y n a m i c a l balances are quite different. O b v i o u s l y there
must be s o m e transition zone in which the d y n a m i c a l balances change f r o m one t y p e
F
/
:~ "f
/
r I
'~'i
\\
\
/
,,
->,,__
X
i
e GAP~,
/
\
o~i
\\
,L
,
°3F
f'~
'~,,,
,;~
/
",,
?
I
I
5jO°s
i
' [i-i
50*s
70Os
/~-"
// \
-Oi
I
-0'2
/z
/
I
70°s
\
,,/
\
\
\
-03
- C 5~
i
-
\
i
Cb)
(a)
.~/~,/
~
--- • 70Os
(c)
Fig. 14. (a) Comparison of the five terms in equation (8.1), the integrated east-west momentum
equation, for the upper half (-- 3"96 < z < 0) of the ocean:
pressure term, - - - - Coriolis term,
lateral friction term,
vertical friction term, - - - non-linear term. (b) The same for the
lower half (--h < z < --3.96). (c) Comparison of terms in (8.1) for the complete ocean depth
( - h ~ z < 0) - - pressure term, - - - - surface stress term, - - - lateral friction term, - - - - bottom
friction term, - - - non-linear term.
-
-
-
714
A.E. GILLand K. BRYAN
to another, the width of the zone depending on the importance of various non-linear
or frictional effects. It seems natural to look for features of the ocean which are
circumpolar in extent and are so placed geographically that they can be identified
with such a transition in dynamical structure. The Antarctic Convergence and the
associated region of formation of Antarctic Intermediate Water are such features,
and we suggest they are associated with the change in dynamical structure.
One function of the models has been to show what features may appear when such
a transition zone is present. It is possible to go further than this by looking at the
balances of terms in the equations (for one case, 15-I). The steady state solution in
this case was used to calculate integrals of different terms in the east-west component
(2.11) of the momentum equation. The equation was integrated with respect to
longitude from the eastern to the western boundary, and with respect to depth
between fixed levels. The division into terms is shown by the square brackets in the
integral of (2.11) below:
21 z2
21
[- Ro f ( (Du 0
tan q~ uv)cos ~bd2dz] + [7 ((z~(z2) - z~(zl))cos ~b d2]
zz
0
•~1 Z2
Z2
+ [f fvsinqbcos¢d2dz]+[- f (P(2,)-P(O))dz]
0
•1
At
z~
+ w f f
0
(8.1)
Zl
cos
d dzl = O.
zl
The five terms are called respectively the non-linear term, the vertical friction term,
the Coriolis term, the pressure term and the lateral friction term. Figure 14(a) shows
the five terms for the upper half of the water column (Z 2 = 0 , Z 1 = - - 3 " 9 6 ) and
Fig. 14 (b) the five terms for the lower half(z2 = - 3.96, z~ = - h ) . Figures 14(a) and
(b) show the main effect of the presence of the gap. Except near the gap and the
south wall, there is an appropriate geostrophic balance (6.6) between the pressure
term and the Coriolis term. In the gap, the pressure term vanishes identically and
Figs. 14(a), (b) show that the Coriolis term becomes much smaller there. It does
not vanish, as (6.6) would require, but is balanced instead by other terms, principally
that representing lateral friction. The reduction in Coriolis force in the upper half
implies a reduction in flux, which, by continuity, requires downwelling.
Away from the gap, the most significant non-geostrophic term is the vertical
friction term for the upper half (see Fig. 14(a)). This term represents the effect of the
surface stress, which produces a (non-geostrophic) Ekman flux in the uppermost
layer.
Figure 14(c) shows the terms when the integral is over the complete depth (z~ = - h,
z2 = 0), and so is the sum of the terms shown in Figs. 14(a), (b). The vertical
friction term has been split into its contributions from the surface stress and the bottom
stress. The Coriolis term vanishes identically as there is no net mass flux north or
south. Well to the north of the gap, one would expect a balance between the pressure
term and the surface stress term. Figure 14(c) indicates that this is true for the most
northerly point shown, but the lateral friction term becomes relatively more important as the gap is approached. The pressure term vanishes identically in the gap,
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 715
where the two most important terms are the surface stress term and the lateral friction
term. The non-linear term is almost as big as the stress term in the middle of the gap.
Figure 14 has shown the dynamical reasons for the most dramatic features of
Fig. 6, which are associated with the vanishing of the pressure term in (8.1). However
it should be remembered that Fig. 6 shows only the )~-integrated motion. Is the
downwelling north of the gap found at all longitudes, or is it found only on average?
Figures 8 and 10 show there is in fact both upwelling and downwelling (at level
k = 5½ say) just north of the gap, except in case 16-I. Before discussing case 16-1,
it is pertinent to ask if the observed features of the ocean temperature and salinity
fields are possibly consistent with a velocity pattern like that shown in Fig. 8. The
question is whether the temperature and salinity sections at a particular longitude
are mainly locally determined or depend more on average conditions around the
latitude circle. Because there is a strong eastward motion, to carry information from
one longitude to another, it seems possible that the latter alternative could be true,
in which case the observed fields need not imply downwelling at all longitudes. A
calculation of salinity distribution discussed later in this section seems to bear this out.
Case 16-1 is rather exceptional, in that downwelling is found at all longitudes
just north of the gap (Fig. 10(c)). Since this case was chosen especially with the region
around the Antarctic Convergence in mind, this is an appropriate time to discuss it.
The reason for doing an extra case at all is due to the rather inflexible surface boundary
condition used in the model, the surface temperature being prescribed. (This was done
because numerical convergence is much slower when a surface heat flux is prescribed.)
If a sharp jump in temperature like that observed at the Convergence is prescribed
one could not be sure if features found in the solution were due to this imposed
surface condition or to other factors. For this reason, the surface temperature in
most cases was prescribed as a linear function of latitude. Case 16-I was computed
LATITUDE
O°
I
•
/
10 ° 20 °
1
30 °
1
I
35.0 J
40 °
I
70 °
50 °
1
I
r
34.6//
.01 f" 4.4J
5.1 -I
7.3 t
34
34.6
8.8
9.3
Fig. 15. Meridional section of zonal mean salinity for case 18-1.
"s~ssoJo Aq u ~ o q s
~ae slu.lod p!.t8 jo stroll!sod i~luoz.t.toq ~q& "uo.tl0~s ~pn:t!SUOl-p!tu e (q) ~epunoq u . t ~ l s ~ ~ql ol
~SOlO (e) lu~uodtllOO /~I!3OlOA pae,~lseo ~ql jo sanoluo3 ~u!aoqs suo!13as leuo!p!aotu o,~& "91 "~.t~I
I
\
k
I
\
i'],,
\
\
I
/t
I
0
L~
\
)
2 f --
,
~--'
I \
]
-,
('
"~'~-\
_
\
) -j-I,I \
)
.
I
'I
Js
\ "l
1
/ \ k'
i
~"
(~)
l q
~S
I
• t~ "
I
I
ok\f
2'
/
/
R
i
?
I
(/
i
i
'it
"--
~h
5
%
ji
:00_
o ~,
NVAUU. "M P ~ T n D '~ "V
91L
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 717
to see if the solution is sensitive to the surface temperature condition. The surface
temperature distribution chosen (see Fig. 3) was based on some observed meridional
distributions. Now since in the model, the only contributions to buoyancy forces
come from the temperature field, ' t e m p e r a t u r e ' in the model should be regarded
rather as an ' a p p a r e n t temperature', so the surface temperature in the model
should be compared with observed distributions of density (a0. Although there is
usually a sharp change in temperature at the Convergence, there is not usually a
sharp change in o-t. Sometimes, as in the sections of DEACON (1937, Figs. 12, 13) and
BURLING (1961, Fig. 10), there is a local m a x b n u m of o-t at the Convergence and this
feature has been included in the surface distribution for case 16-I, although the
maximum is by no means invariably found at the Convergence. The effect of" this
maximum density in the models is to produce convection and downwelling immediately below it, and so give the effect shown in Fig. 10(c). Thus case 16-I shows
that the solution is sensitive to the surface condition in this one respect, although
otherwise the solution is n o t too sensitive to the surface condition.
Thus the models leave the question of the detailed velocity distribution around
the Convergence open. They suggest that it would be preferable to have a much freer
surface condition that that used. The ideal condition would be one allowing both
atmosphere and ocean to adjust freely, as in the model of MANABE and BRYAN (1969).
So far the discussion has only referred to the behaviour relatively near the surface.
Further down, the models are in conflict with observation since Fig. l(b) shows
that the Intermediate Water, once formed, seems to level out at about l km, whereas in the model (Fig. 6) it appears to descend right to the bottom. Now, in practice,
the movements of Intermediate Water are traced by its low salinity. In order to see
how good a tracer salinity is in the model, a further computation was done for case
18-I. Using the steady state velocity field found for case 18-I, the distribution of
salinity established by this velocity field was calculated, the surface salinity being
given a prescribed zonal distribution and no flux being allowed across solid boundaries.
The zonally-averaged distribution so obtained is shown in Fig. 15 (which incidentally
shows the prescribed surface distribution). The tongue of fresh water is seen to extend
to the bottom as one would expect from the velocity field shown in Fig. 6. The conclusion is that salinity is an adequate tracer of the Intermediate Water. Incidentally,
the calculations seem to show that, in the model, the salinity distribution depends
more on the zonally-averaged conditions than on local conditions. Otherwise the
salinity field (which does not vary greatly with longitude) would not fit the zonallyaveraged velocity field (shown in Fig. 6) so well.
It seems appropriate here to discuss another method used in oceanography to
trace water masses, namely isentropic analysis. This assumes that diffusion effects
are so slight that potential density is conserved over large distances in the ocean.
Applied to the present model, the idea would be that motion tends to follow lines
of constant 9. In particular, water formed just north of the gap would, in cases 15-I
and 18-I, tend to follow the 0.2 isotherm. Comparison of Figs. 6 and 13 shows that,
on average at least, the motion does n o t follow the isotherms, but is usually closer
to the vertical than the slope of the isotherms would suggest. This is necessary in fact.
to allow a balance between vertical advection and vertical diffusion in the deep subtropics as assumed, for instance, by MurqK (1966). The conclusion is that isentropic
718
A.E. GILLand K. BRYAN
analysis is not consistent with the idea of a balance between vertical diffusion and
vertical advection in the deep ocean.
9.
THE
CIRCUMPOLAR
CURRENT
The main features of the circumpolar current are reproduced in the computations.
It is a major feature of the circulation, it is eastward moving, it decreases in strength
with depth, and it moves to the north after passing through the gap. Two sections
across the current are shown in Fig. 16. It will be noticed (Fig. 8(g) and Fig. 16) that
there is also a westward countercurrent near the bottom near the southern boundary.
This westward bottom current exists near the southern boundary whether the gap
is present or not. Such a current apparently exists in reality, but is deflected northwards by submarine ridges (GORDON, 1966; HEEZEN, THARP and HOLLISTER,
1968).
The most striking effect of geometry on the circumpolar current is (Fig. 5) the
dramatic increase in transport obtained when the bottom half of the gap is closed.
Comparison of cases 15-I and 18-I shows that the transport increases three-fold
when the area of the gap is reducedby closing the bottom half. Most of this section is
devoted to a further investigation of this effect. The method used is to investigate the
balance of eastward forces on a circumpolar strip of ocean bounded by the two
parallels of latitude which define tile gap, by the ocean floor and by the ocean surface.
The forces involved can be found by integrating the east-west component (2.11)of
the momentum equation over the strip, or, equivalently by integrating equation (8.1)
with respect to q~ from - ( 1 ) 2 to -qb I (and with zl = - h , z 2 ~ 0 ) , The resulting
equation is
-01
21
0
f [-Rof -~ (Du-02
0
-01
+
2~
j_
-@2
";~l
-
4,,14+
-02
0
-h
-0~
cos
0
-O l
-~2
t a n ~ u v ) cos~bd2dz]dq~
-h
0
-@t
-02
31
0
0
-h
The terms in this equation are shown in Table 7 for the case where the strip extends
over the whole depth (Zl = - h , z: = 0). The first term is referred to as the rate of
influx of momentum over the sides, although there is a very small additional contribution from the term involving the product tan c>uv.The remaining terms in (9.1)
are respectively the wind force on the surface of the strip, the frictional force on the
bottom of the strip, the force due to pressure differences between east and west
(non-zero in case 18-I only) and the sum of the frictional forces on the north and
south sides of the strip.
As did MUNK and PALMI~N (1951), we ask what forces on the strip balance the
driving force of the wind. In linear barotropic models (GILL, 1968), the opposing
force is frictional. MUNK and PALMI~N (1951), suggested that pressure differences
across submarine ridges might provide an important opposing force. Table 7 shows
Effects of geometry on the circulation of a three-dimensional southern-hemisphere ocean model 719
the balances o f forces on the strip obtained for cases 15-I and 18-I. In the deep-gap
case (15-I), the result is not too surprising except perhaps for the fact that the combined effect o f non-linear terms and the b o t t o m friction terms is 9 0 ~ o f the wind
force and in the same direction as the wind force. These forces are balanced by
horizontal friction forces. Incidentally, the force due to b o t t o m friction is in the
same direction as the wind force because o f the counter currents near the bottom.
However, case 18-I shows a dramatic change. As MUNK and PALMI~N (1951)
suggested, the force due to pressure differences is significant. The surprising thing,
however, is that this force is in the same direction as the wind force and four times
as large. This seems to indicate that the pressure difference does not develop as a
response to the wind, but is the result o f independent forces, presumably thermal.
The way such a pressure difference could be set tip can be seen qualitatively from
Table 7.
Eastward components of the Jorces on the zonal strip which passes through
the gap. The nondimensional unit is 2f~p V* a2d.
Force on surface due to wind
Rate of influx of eastward momentum
from north and south
Frictional force on bottom
Frictional force on northern and
southern sides
Force due to east-west pressure
differences below the sill
15-1
÷ 0.0040
18-I
+0.0040
F0'0019
-F0"0016
-0'0023
-0.0004
--0.0075
--0"0170
--
-~ 0'0157
Fig. 8(g). This shows that the cold water that sinks to the b o t t o m on the southern
b o u n d a r y forms a westward moving b o t t o m current which carries cold water to the
western b o u n d a r y where it is deflected northward past the ' r i d g e ' which blocks the
b o t t o m half of the gap. As Fig. 12(f) shows, this water is colder than the water on the
other side o f the ridge, that is, on the eastern b o u n d a r y of the basin. N o w at the top
of the ridge, (z = - 3 - 9 6 ) the pressure is the same on each side o f the ridge (which
has no width), so at greater depths the pressure is higher on the side o f the ridge on
which the water is denser (by the hydrostatic relation). This gives a pressure difference
o f the sign shown in Table 7. It does not oppose the wind force, but acts in the same
direction, and therefore helps to drive the circumpolar current. The dramatic increase
in the transport in case 18-I is due to this effect.
The question now raised is whether the real ocean behaves in this way. The b o t t o m
circulation is something like that shown in Fig. 8(g) and the water is certainly denser
on the Atlantic side o f Drake Passage than on the Pacific side (see e.g. LYNN and
REID, 1968, Fig. 9). If the pressure could be taken as constant at 2500 m (the top o f
the ridge) from the South Sandwich Trench to the western end of Drake Passage,
GORDON'S (1967, Plate 10) map of the dynamic t o p o g r a p h y o f the 2500 db surface
relative to the 4000 db surface shows a difference in dynamic height across the ridge o f
0.2 dynamic metres, (2 x 103 newton m-2). If the average pressure difference is half this,
the corresponding force over a cross section o f 1500 m x 5"5 x 105 m is 8 x 1011 newton.
N o w c o m p a r e this with the wind force on the surface of the strip. According to
HELLERMAN (1967), the mean zonal wind stress at 60 ° S is 0.08 n e w t o n m -2, giving
a net wind force on the strip of about 0 . 0 8 x 2 x 1 0 7 x 5 . 5 x 10 s = 9 x 10 ~1 newton.
720
A.E. GILL and K. BRYAN
Thus if the pressure were constant at 2500 m, the pressure force across the ridge
would be the same as the wind force on the surface of the strip and in the same
direction. However, the ridge is so broad (2500 kin) that there is no justification
for assuming the pressure is constant at 2500 m so the above calculation may well be
meaningless. The only way to resolve the question of whether there is a significant
pressure force across the ridge is to find some means of calculating absolute dynamic
topography oil an east-west transect of the ridge (cf. REID and NOWLIN'S (1971)
calculation of absolute dynamic topography on a north-south section across Drake
Passage).
10. CONCLUSIONS
The model studies show that simple geometrical differences can make fundamental
differences to the circulation. The presence of a gap like Drake Passage led to sinking
north of the gap and upwelling south of the gap, the sinking region corresponding
to the region where Antarctic Intermediate Water is formed. The transport of the
circumpolar current was found to depend on whether the gap was as deep as the rest
of the ocean or only half as deep, the transport being larger in the latter case. Tile
increase in transport was due to a pressure difference across the barrier in the bottom
half of the gap, the pressure difference being due to colder water being carried past
the east side of the barrier. In the real ocean, colder water is found on the Atlantic
side of Drake Passage than on the Pacific side, so there may be a similar pressure
difference across the barrier that exists there. One cannot calculate this difference,
however, without a knowledge of absolute dynamic topography on an east-west
section from the South Sandwich Trench to the western end of Drake Passage.
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