A survey of theoretical models of the Antarctic circumpolar current.

N PS ARCHIVE
1969
CALLAHAN, J.

A SURVEY OF THEORETICAL MODELS OF THE
ANTARCTIC CIRCUMPOLAR CURRENT

by

Jeffrey E. Callahan

' •

':N'?':

■i

A Survey of Theoretical Models

of

the Antarctic Circumpolar Current

by

Jeffrey E. Callahan

//

An essay submitted to The Johns
Hopkins University in conformity
with the requirements for the de-
gree of Master of Arts.

DUDLEY KNOX LIBRARY
NAVAL POSTGRADUATE SCHOOL
MONTEREY, CA 93943-5101

Baltimore, Maryland
1969

^^POSTGRADUATES
MONTEREY, CA 93943-Stfti

Abstract

Five theoretical studies of the Antarctic Circumpolar
Current are critically reviewed. The structures of the
models, including significant assumptions and approximations,
are discussed. Theoretical results are compared with
observed features of the Circumpolar Current. Progress
in the effort to understand the dynamics of the Current is
summarized, and suggestions are made for future work
related to this problem.

to-t

Acknowledgments

The author wishes to acknowledge the assistance of the
following people: Professor R. B. Montgomery, for many-
useful suggestions regarding the form and content of the
essay; Mr. Richard Linfield and his staff, for preparing
the figures; and Mrs. Emma Hammond, for preparing the
manuscript.

This essay was written while the author was studying
under the Burke Scholar program of the United States Navy.

Table of Contents

Page

Introduction 1

Descriptive Features of

the Circumpolar Current . 2

Channel Flow Models 11

Asymmetric Models 20

Discussion of Model Characteristics 27

Concluding Remarks 32

Appendix 35

Bibliography 37

Vita 40

List of Illustrations and Tables

Page

Figure 1. Zone of maximum westerly winds over the

Southern Ocean . 4

Figure 2. Computed mass transport of the Circumpolar

Current relative to the 3000-decibar surface . . 5

Figure 3. Geostrophic velocities relative to 4000

decibars in a section from Cape Leeuwin,

Australia, to the Antarctic continent 7

Figure 4. Schematic representation of circulation in a
meridional section across the Southern
Ocean . 10

Figure 5. Hypothetical circulation patterns based on

assumption of Sverdrup-like solution 21

Figure 6. Model geometry and some numerical solutions
of Gill's (1968) model of the Circumpolar
Current 25

Table 1. Numerical results from Hidaka & Tsuchiya

(1953) 19

Table 2. Summary of model characteristics 28

Introduction

The Antarctic Circumpolar Current, the only ocean current
which circles the earth, is the principal agent of water exchange
among the world's oceans. Calculations of mass transport
through Drake Passage indicate that the Circumpolar Current is
also the strongest of the world's ocean currents. In spite of
these unique and interesting features, the Circumpolar Current
has received relatively little attention from oceanographers,
either in theoretical models or in field studies. As a consequence,
the dynamics of the Current are not yet clearly understood, and
its role in the general circulation of the ocean has not been
properly evaluated.

This essay is a survey of theoretical papers dealing with the
Circumpolar Current. Contributions by Munk & Palmen (1951),
Hidaka & Tsuchiya (1953), Stommel (1957; 1962), and Gill (1968)
are critically reviewed. Throughout the study the emphasis is on
the physical structure of models. Are the models realistic
analytical representations of natural conditions? Are the
assumptions and approximations reasonable? How meaningful
are the results ?.

Two reasons may be given for conducting this survey. One
is to find out how much has been learned about the dynamics of the
Circumpolar Current: what dynamical features have been revealed
by past models ? The complementary purpose is to identify
unanswered questions concerning the Current, that is, to suggest
directions for future work.

Descriptive Features of the Circumpolar Current

In order to provide a basis for evaluating the models to be
studied, a summary of major features of the Southern Ocean and
of the Circumpolar Current will first be given. Much of the
information contained in this section has been taken from the
descriptive accounts by Deacon (1937a; 1963} and Sverdrup et al
(1942, Chapter XV).

Southern Ocean is the name given to the great body of water
which surrounds the Antarctic continent. To the north it merges
with the Atlantic, Pacific, and Indian oceans. The absence of
natural boundaries makes it difficult to delineate the northern
limit; 40 °S may be used as an arbitrary boundary.

With the exception of regions where major submarine ridges
are found, average depth is about 4000 m. The greatest
depression is the South Sandwich Trench with a maximum sounding
of roughly 8300 m.

During part of the year a large portion of the Southern Ocean
is covered with ice. In October, at the end of the austral winter,
pack ice extends to 55 °S - 57°S everywhere except in the Pacific,
where it extends only to 63 °S. By the end of summer (March) the
edge of the pack has retreated almost to the Antarctic coast
(Mackintosh and Herdman, 1940).

Knowledge of the wind field over the Southern Ocean is
somewhat limited. Except in the Drake Passage area, almost all
data come from ships. Ship data are variable in quality and
uneven in time/space distribution. The available data indicate
that the mean wind field is characterized by strong westerlies
between about 40°S and 60 °S, with weaker and more variable

easterlies south of that latitude (von Arx, 1957; Vowinckel,
1957). The westerlies do exhibit polar asymmetry with respect
to speed, as is shown in figure 1.

This essay is concerned primarily with the segment of the
Southern Ocean known as the Circumpolar Current. Here, too,
a certain amount of arbitrariness is required to define the
subject. The interior boundary falls at about 60 °S in the Atlantic
and Indian sectors, somewhat further south in the Pacific. A
countercurrent, driven by prevailing easterlies, flows between
the Circumpolar Current and the Antarctic continent.

Surface current charts depict a general eastward motion,
known as the West Wind Drift, from 60 °S to roughly 40 °S.
However, the true Circumpolar Current covers only part of this
zone. Approximate limits are shown in figure 2, in which
transport relative to the 3000-decibar level is plotted. The
center of the Current, say the transport line marked "2", is
found at 50 °S in the Atlantic and Indian Ocean sectors, but it
swings south to 60°S in the Pacific. Large-scale meanders may
be observed in several places.

Several oceanographic expeditions, notably the Discovery
Investigations (Great Britain), Ob cruises (U.S.S.R. }, and
Eltanin cruises (U.S.A. ), have made observations in the
Southern Ocean. Climatic conditions make work in this area
difficult, and data is sparse over large sections of the Ocean.
As a result, water motions within the Circumpolar Current can
be described only in broad outline.

The mean velocity field at the naviface has been deduced
in large part from ship drift reports. Taljaard (1957) plotted

ANTARCTIC CONV.-MACKINTOSH,

1946.
• An t.Conv. from BT'fi, 1947-48,

1955-57.
A Ant. Conv. from Whalers.

Figure 1. Zone of maximum westerly winds over the
Southern Ocean, computed from sea-level barometric
pressure data. (From Wexler, 1959. )

60'W

I2tf£

Figure 2. Computed mass transport of the Circumpolar
Current relative to the 3000-decibar surface. Transport
between two lines is about 20 x 10 12 g sec"*. Light
shading covers areas with depth less than 3000 m. (From
Sverdrup et al, 1942, p. 615.)

surface currents around Antarctica using data from several
sources, including the British Admiralty "Antarctic Pilot" and
U. S. Naval Hydrographic Office "Sailing Directions for
Antarctica". If the area near Drake Passage (where currents
are sometimes greater than 1 knot) is excluded, the drift is
generally toward the east at 0. 2 to 0. 6 knots (12 to 36 cm sec )..
Deacon (1963) states that the average surface current in the
West Wind Drift is 8 miles per day (15-17 cm sec ). This
figure is based on drift bottle measurements.

Geostrophic computations based on hydrographic data
provide an indication of subsurface velocities in the Current. An
example for a section between Antarctica and Australia is shown
in figure 3. Sections such as these give the impression that the
Circumpolar Current is significantly deeper than most wind-
driven currents. It must be emphasized that geostrophic profiles
depend heavily on choice of reference level, and any uncertainty
associated with this choice introduces uncertainty in the results.

The importance of the reference level chosen for
geostrophic computations is illustrated in a summary of Drake
Passage mass transport calculations published by Gordon (1967).
Gordon includes the work of seven previous authors and his own

transport figures, too. The calculated transports range from

12 -1

85 to 218 x 10 g sec , discounting Ostapoff's extremely low

values. The lower values are those computed using the 3000-

decibar surface as a reference level, while the higher values are

generally those based on Defant's method for determining the

level of no motion.

12 - 1

Gordon himself estimates the transport at 218 x 10 g sec

using a set of seven Ob stations evenly spaced across the Passage.

878

879

880

831

882

883

084

885

886

887

'20

U

2io

14 9^

cm/scc

35°

45'

55'

65*

_i

SOUTH LATITUDE

Figure 3. Horizontal geostrophic velocity, relative to
4000 decibars, normal to a section from Cape Leeuwin,
Australia, to the Antarctic continent. (From Sverdrup
et al, 1942, p. 614.)

-

8

He obtains a reference level using the "equivalent-baratropic
assumption, " namely, that the mean density between the
reference level and the naviface is constant. Starting from an
assumed level of no motion in the southern end of Drake Passage,
he finds that the reference level slopes down sharply toward the
north. In fact, the reference level lies below the ocean floor in
the northern two thirds of the Passage.

If Ostapoff's calculations are excluded, only two of the

12 -1

remaining nine listed by Gordon fall below 100 x 10 g sec

Thus, although there is a considerable spread in computed mass

transport values, it is probable that the transport of the

12 -1

Circumpolar Current is greater than 100 x 10 g sec

Geostrophic mass transport calculations made for several

sections across the Current indicate a large longitudinal

variation in transport. This variation is evident in figure 2 and

also in the following transport estimates by Kort (1962):

12 - 1

Section Transport (10 g sec }

Drake Passage 150

Antarctica - South Africa 190

Antarctica - Tasmania 180

While zonal flow is the most conspicuous mean water motion
(at least at the naviface), it is not the only significant one taking
place. Analysis of water types and the distributions of
temperature, salinity, and oxygen content indicate the presence
of a well developed meridional circulation. Estimates of
meridional transport are even rougher than those of zonal

9

transport, but they show that the former may be significant, at
least in certain parts of the Southern Ocean. In the Atlantic, for

example, Sverdrup et al (1942, p 629) estimate from Meteor

12 "-1

data that 35 x 10 g sec flows south across 30°S, balanced by

an equal flow to the north. Note that meridional transport of
this order would be required to account for the longitudinal
variations in zonal transport found by Kort.

Deacon's (1937a) interpretation of the meridional circulation
is given in figure 4. South of the 50th parallel it is basically a
three-layer system. Between about 200 m and 1500 m there is a
southward flowing deep current of relatively warm (y2C), saline
(^ 34. 7%o) water. The major source of this deep water is
thought to be the area southeast of Greenland, where surface
water cools during winter, sinks, and spreads south. The supply
of deep water is augmented by mixing with intermediate and
bottom water. Deep water is also called circumpolar water
because of its uniform distribution around the continent (Sverdrup
et al, 1942, p 607).

Wedging beneath the deep layer is Antarctic bottom water,
which is nearly as saline as, and 1 C to 2 C colder than, deep
water. Bottom water is probably formed from a mixture of deep
water and extremely cold surface water sliding down the Antarctic
continental shelf, mostly in the Weddell Sea (Fofonoff, 1956).
From there bottom water spreads north along the western trough
of the Atlantic Ocean, where it can be detected well into the North
Atlantic, and around the Antarctic continent. The rate of
formation of bottom water is probably greatest during winter.

The upper few hundred meters of the Antarctic zone consist
of cold, poorly saline surface water. This layer is influenced

10

30*

40c

50°
Antarctic

South

1000 m

Subtropical water

Antarctic
intermediate current

Antarctic

■<

2000 m

3000 m

1000 m

Subtropical
convergence convergence
1 Sub-Antarctic zone

y j/ ,Aiixed- ^"surface current

water region

Warm
deep current

/

Antarctic
bottom current

/

Figure 4. Schematic representation of circulation in a
meridional section across the Southern Ocean. (From
Deacon, 1963. )

11

strongly by seasonal fluctuations in air temperature and ice
conditions. It is uniform in winter, but patches of water with
anomalous properties are found there in summer.

Before concluding this section, mention will be made of
the prominent feature known as the Antarctic polar front or
Antarctic convergence. The front has been observed in all
sectors of the Southern Ocean. At the naviface, the polar front
is manifested by a relatively sharp north-south gradient in water
temperature, occasionally as much as 2 C in five miles. A few
authors, for example, Wexler (1959) have attempted to identify
the front on the basis of subsurface features in the water column,
but these methods have not met with general acceptance.
According to Mackintosh (1946) the polar front is found near
50°S in the Atlantic and Indian sectors, while in the Pacific it
lies closer to 60°S. Mackintosh found that the mean monthly
position of the front varies only slightly with season. Somewhat
larger variations with time scales of several days do occur.

Channel Flow Models

Among the earliest theoretical discussions of the Antarctic
Circumpolar Current are those of Munk & Palmen (1951) and
Hidaka & Tsuchiya (1953). These papers differ little in their
fundamental concept of the current as an axi- symmetric channel
flow. Wind stress is balanced primarily by frictional forces.
It is found that this simple balance will not yield realistic transport
figures unless unusually large values are taken for the eddy
viscosity coefficients.

Munk & Palmen start with a simple analytical model. The
equations are written in cylindrical coordinates (see Appendix).

12

It is assumed that the current is a steady, purely zonal flow of
uniform depth H. Nonlinear terms are neglected. Under these
restrictions only the tangential component of the horizontal
equations of motion remains,

If (1) is integrated from the bottom to the naviface and if
bottom friction is neglected, then

Ak (tf m - f.) + r = °,

where M = net zonal mass transport (per unit width)
o

5 i fu. Az

-H

and t = zonal wind stress.

Because of the axial symmetry,

tf- + &(ri)-

Thus,

or

(2) r + Ak Jr ^

Boundary conditions are M = 0 (i) at the coast of Antarctica
(f-Yi) and (ii) at some other latitude circle to the north (Y"=VT). The
appropriate solution of (2) is

3Ah

W[ ~- —

13
Integrating M from To to Yt gives the net zonal transport,

l&Aw

I .~l

(«•-«'- ^, AiJ

In order to make a quantitative estimate of the transport Munk &

-2 8 2-1

Palmen set t = 2 dynes cm , A =10 cm sec , and place the

boundaries at latitudes 70 °S and 45 °S. The computed transport
is over 10 g sec . If the boundaries are moved to 65 °S and

55 °S, the approximate limits of the Drake Passage, computed

,«15 -1

transport is still more than 10 g sec

It was shown earlier that the geostrophic transport through

,-x , « . , t„14 -I 14 -1

Drake Passage is between 10 g sec and 2 x 10 g sec

Thus the theoretical transport is at least an order of magnitude
greater than that computed from hydrographic data. It is
unlikely that the computed value is in error by an order of
magnitude. Munk & Palmen offer two ways to bring the theoretical
result into line with the observed.

The first and most direct way is to increase the eddy co-
efficient A^ by one or more orders of magnitude. However, other

studies of large-scale oceanic flows indicate that A ranges

7 8 h

between 10 and 10 . There is no reason to suppose that it should

be considerably larger in this region.

Instead, Munk & Palmen favor introducing bottom stress.
This source of retarding action was explicitly neglected in the
original model.

Bottom stress could manifest itself as "skin friction. "
Sverdrup et al (1942, p 479) give the following formula, based on
Prandtl's mixing length theory, for the mean horizontal velocity

14
within the turbulent boundary layer over a rough bottom:

u.k = 2.5 y^T ^{J-irJ

The roughness length £0 is given as 2 cm from measurements made

by Revelle and Fleming in San Diego Harbor*. If the wind stress

-2
of 2 dyne cm were balanced entirely by bottom friction, u

would be 14 cm sec only 1 m above the sea floor. Since this

is approximately equal to the maximum surface velocities of the

Circumpolar Current, Munk & Palmen consider skin friction an

unlikely retarding mechanism.

Alternatively, bottom stress might be caused by the so-called

mountain effect. This term refers to the retarding effect of a

pressure drop across a submerged barrier, e. g. , a submarine

ridge, imbedded in the flow. Assume there is a pressure

difference ATw across each ridge over which the current flows.

Then the average bottom stress is

= -A- Affc A ti ,

where C = distance around a latitude circle

and An = sum of heights of ridges.

The Circumpolar Current crosses four major ridge systems
in its path around the Antarctic continent. East of Drake Passage
lies the Scotia Ridge, including the South Sandwich and South
Orkney island groups, having a height of 4 km; at 75 °E is the
Kerguelen-Gaussberg Ridge, h = 3 km; at 165 °E is the Macquarie
Ridge, h = 2 km; and at 150 °W is the Pacific -Antarctic ridge,

*One must wonder if a roughness parameter measured at the
bottom of San Diego Harbor is appropriate to the bottom of the
Southern Ocean.

15

h = 1 km. The combined height of the ridges is 10 km; at 60°S

2

_2

C = 18, 000 km. Setting ?b = 2 dyne cm" , Munk & Palmen find

A lb = 4000 dyne cm , which is equivalent to 4 dynamic
centimeters. Deacon (1937b) calculated cross-current dynamic
height differences on the order of 1 m at the surface of the
Southern Ocean. Pressure gradients over ridges amounting to
just a few percent of those found at the surface would be sufficient
to balance the wind stress.

Use of bottom stress, whatever the mechanism, requires
that the Current penetrate to the sea floor over at least part of its
path. What reason is there to expect the Circumpolar Current to
be unusually deep? Munk & Palmen reject the possibility that
momentum is transmitted from the surface to extreme depths by

vertical turbulent exchange. Such a mechanism would imply a

4 2 -1

vertical eddy coefficient greater than 10 cm sec , which is

one to two orders of magnitude larger than commonly used values.

A different mechanism is revealed by considering the balance
of angular momentum in the current. Under steady conditions the
absolute angular momentum of the Current about the earth's axis
is constant, and any changes in angular momentum caused by
stresses or transport phenomena must cancel.

Somewhat artificially Munk & Palmen divide the meridional
circulation into two layers. They suppose that angular momentum
is exported in the top layer while it is imported to the Current in
the bottom layer. Next they hypothesize that all the angular
momentum produced by the wind torque over the Current is
advected across the northern boundary of the Current in the upper
layer. If the northward mass flux is Q then

16
70° S

q(Rcos«s*).a = j tUtf Rcos (j>)(rcos (?) R A<$> ,

where -U. = earth's angular velocity
R = earth's radius

and Y = latitude.

12 -1

The flux Q is found to be about 30x10 g sec , roughly the

amount of water transported north in the Peru and Benguela
currents.

Water to replace that lost in the upper layer comes from
three sources: precipitation, runoff from the Antarctic continent,
and southward transport of deep water. Precise data measuring
the first two sources are not available, but rough estimates show
that they provide at most a few percent of the required amount.
Virtually all the water lost in the upper layer is replaced by water
from the lower layer.

Southward flowing deep water comes from a region of
higher absolute angular momentum than water in the Circumpolar
Current. In order to preserve that angular momentum it must
develop an eastward drift as it moves toward the pole. The
momentum excess of deep water relative to "local" water is
balanced by the loss of angular momentum associated with the
mountain effect. Thus the total angular momentum of the Current
is conserved and, at the same time, the entire water column
acquires an eastward drift. *

*This argument has a certain qualitative appeal, but it also has at
least one serious quantitative defect. The eastward velocity which
a particle of water must acquire in order to compensate for the
loss of angular momentum as it moves south is much larger than
the maximum velocities in the Current. For example, a particle
moved without friction from 45 °S to 46 °S would develop an eastward
velocity of 800 cm sec relative to the earth.

17

Like Munk & Palmen, Hidaka & Tsuchiya (1953) model the
Circumpolar Current as a steady flow between solid boundaries
at 45 °S and 70 °S. Density is assumed constant, and nonlinear
terms are neglected. Hidaka & Tsuchiya write their equations
in terms of the horizontal velocity components instead of mass
transport.

With the x, y, z-axes positive eastward, northward, and
vertically downward, the horizontal equations of motion are

Av t^ t Ah (7^ + ry*) + **" = j y5
Av ^ * AfcClx* + ayv "*+UL s f ^y '

where u, v are the x, y-components of velocity
and f = Coriolis parameter = 2 il sin latitude.

It is postulated that a uniform wind is blowing in the x-
direction only. Therefore u, v, p, and the surface elevation >
are independent of x. Using the hydrostatic equation and
letting W = u + iv, the two equations above may be written as

(3) n" hlX

Continuity is

s O

^v "^ w

ay a*

where w = vertical velocity component.

Boundary conditions are

<[ at z = H (the bottom)

W = 0

''[ at y = ±{(45°S, 70°S)

and ny r — + *» - 0 at z = 0 (naviface).

o i

Hidaka & Tsuchiya assume a Fourier series solution of W,

(4) W(y^) = h ^v co* — Fh— '

2 C , v (25-1 ) TT X

where u>* Cy) r J \ W ( X) OS J H ~ ** •

By substituting (4) in (3) and expressing (&>\>f) % ^(.y) and Z v.y)
also as Fourier series, it is found that

T wrrU + y) Cis-t) TT£

Wty#ri =2^ DMS sm — ^J- cos —jTi '

The quantity D is a complex coefficient whose value depends on
ms

A , A . and tm , the Fourier coefficients of tCyJ.
v h

Having derived expressions for the velocity field and surface
elevation, Hidaka & Tsuchiya compute u, v, ^ , and the total mass

transport T. Water depth is set equal to 4 km, and wind stress is

-2

set equal to 2 dyne cm . The unknowns are evaluated for two

r a ,„8 2 -1 10 2 -1 T , ,

values of A, : 10 cm sec and 10 cm sec . In both cases

3 2-1
A = 2 x 10 cm sec . Results are given in table 1.

19

Table 1

Numerical results from Hidaka & Tsuchiya (1953)

Ah

[cm sec )

u
max

(cm sec )

V

max
(cm sec )

T
(g sec )

£ 5 across
Current

Cm)

io8

100

3

15
8. 1 x 10

25

io10

14

2

14
9.3 x 10

3

10 2 -1

Values computed with A = 10 cm sec are more

8 2 -1

realistic than those with A = 10 cm sec . Transport and

surface slope are rather large, but the maximum velocities are

comparable to observed values. Vertical plots of the velocity

components reveal that the meridional component has a nonzero

value only in the upper few hundred meters. The zonal

component stays near its maximum value from the naviface

almost to the bottom. At the naviface the velocity has a

parabolic distribution between the latitudinal boundaries, with

a maximum at 58 °S.

Hidaka & Tsuchiya conclude that the gross features of the

Circumpolar Current can be explained without including

submarine topography effects if unusually high levels of lateral

turbulence exist in the Southern Ocean.

20

Asymmetric Models

Henry Stommel was the first to suggest that the Circumpolar
Current can not be treated analytically as a zonal, axisymmetric
flow. * Stommel (1957) observes, " . . . if one plots the minimum
depth for each complete latitude circle, it is found that the
latitude circles that pass through Drake Passage are blocked by
the island arc somewhat to the east ... . It is seen that
nowhere in the Antarctic Water-ring is there a latitude with a
deeper threshold than 1000 m. The Antarctic Circumpolar Current
therefore cannot be purely zonal. "

Stommel concludes that, because of the partial barrier
across the Passage, the Southern Ocean is more nearly an
enclosed basin than a uniform channel. With this hypothesis it is
reasonable to treat the Current in the manner of Sverdrup {1947).
That is, over most of its extent only wind stress, Coriolis, and
pressure-gradient terms play an important part in the equations
of motion. Near Drake Passage, Stommel expects boundary
currents similar to those found along meridional boundaries in
other ocean current systems, and he predicts that any instability
or higher-order processes associated with the Current occur here.

In this paper Stommel does not develop an analytical model
to test his notions about the dynamics of the Current. He only
presents a qualitative interpretation of how such a current might
evolve. Several schematic representations taken from his paper
are shown in figure 5.

Sverdrup et al (1942) did note that the Current "... is locally
deflected from its course, partly by the distribution of land and
sea and partly by the submarine topography. " (p 615)

21

\ ig. (a). The schematic Southern Ocean.
Antarctica is the solid black circle. The
meridional barrier extending northward from
Antarctica is represented by the solid heavy
black vertical line. The schematic wind system
(purely zonal) is depicted by the heavy arrows
on the lower left. The concentric circles arc
latitude circles. Latitudes of Ekman conver-
gence and sinking at the surface are indicated
by minus signs, latitudes of Ekman divergence
and upwelling are indicated by plus signs. The
direction of the required meridional geostro-
phic flow is indicated by light radial arrows.

(b). Transport lines of the solution
for the model depicted in Fig. (a). The
western boundary currents are to be interpreted
schematically.

Fig. (c). Modification of the transport field
produced by introduction of other meridional
barriers corresponding to Africa, Australia, and
New Zealand, and by breaking the American-
Antarctic barrier so as to admit a very con-
stricted Davis Straits.

Fig. (d). Hypothetical form of the solution
that results from rupturing the American-
Antarctic barrier in such a way as to permit
water to flow through, but to obstruct all
latitude circles.

Figure 5. Hypothetical circulation patterns based on
assumption of Sverdrup-like solution. (From Stommel,
1957.)

22

One difficult aspect of Stommel's approach is visualizing the
role of Drake Passage in the flow regime. Stommel suggests that
higher-order processes take place there, but he does not indicate
what these might be. He regards the island arc to the east as a
partial barrier to zonal flow, but the analytical representation of
the barrier is not readily apparent.

In an attempt to gain some insight into the problem without
directly confronting these analytical obstacles Stommel (1962)
proposes a laboratory model of the Antarctic Circumpolar
Current. The model consists of a rotating cylinder with a single
radial (meridional) barrier. The meridional barrier is so
constructed that it can be changed from a solid boundary to a
porous boundary of variable flow resistance R. Poleward
convergent flow is superposed on solid-body rotation by a source-
sink combination which has the net effect of a distributed sink over
the water surface. The distributed sink models the divergent
Ekman drift caused by westerly winds over the Southern Ocean.

Recalling the results of earlier experiments with rotating
cylindrical models (Stommel etal, 1958; Faller, I960), Stommel
predicts the effect of the barrier on the flow regime. When the
barrier is solid (R = oQ), poleward geostrophic flow dominates.
With R < oO a zonal component is introduced in the geostrophic flow,
giving it a spiral form, and non-geostrophic radial flow develops
in the bottom Ekman layer. At R = 0 the geostrophic flow is
purely zonal. Its magnitude is just great enough to drive the
radial Ekman flow required by continuity.

As an analogue to the Circumpolar Current, Stommel
envisions the following configuration*.

23

a

(After Stommel, 1962)

The black portions of the wall are solid, the gap is porous.
In this case flow is poleward everywhere except in the narrow
ring which passes through the porous gap. In the gap itself flow
is zonal and nongeostrophic; in the remainder of the ring flow
is quasi-zonal and geostrophic. Several isobars have been
drawn, and regions of relative high and low pressure are marked.
The arrows indicate direction of flow.

This model is much too simple to duplicate actual conditions
in the Circumpolar Current. The purpose is to study the
influence of the Passage on the Current. Unfortunately Stommel
has yet to perform the experiment (Stommel, personal
communication), so his idea remains untested.

Stommel's discussions of the Circumpolar Current have
stimulated further theoretical work. Two models based on his

24

suggestion of a Sverdrup-like solution are those by Wyrtki (I960)
and Gill (1968). The basic equations used in both studies are
almost identical. Only Gill's model will be discussed in detail,
because he deals directly with the effect of Drake Passage on the
primary longitudinal flow. Wyrtki devotes most of his paper to
the transverse water motions and their influence on the
Antarctic convergence.

The analysis is carried out in rectangular coordinates,
shown in figure 6 (upper); the model is also shown in polar
projection. Line y = 0 is the coast of Antarctica. South America
is represented by the solid boundary x = 0, L. The gap from
y = B to y = 0 is Drake Passage.

The dynamical equations are patterned after those of
Stommel (1948); pressure gradient, Coriolis, and vertical
friction are the dominant forces. Bottom friction is assumed to
be a linear function of velocity, * and the P -plane approximation
is made.

The equations are integrated over depth, which is assumed
uniform. The assumption of incompressible flow allows Gill to
introduce a transport function H* , yielding the governing
equation

S Ox + *yy) ■» *x = Yx - \y ,

By including friction as a retarding force Gill and Wyrtki deviate
slightly from the Sverdrup solution. Physically, it is reasonable
to expect that friction acts on the current at least near the
Antarctic coast. Strictly speaking this is lateral rather than
vertical friction, but the friction law used in these models is so
general that it can be considered to represent either case.

25

y-D

y = B

y = 0

x=\L

x=\L

V

A

^" = ^tot

(<z) e = 045 fj{=522 (16xl08m3/s) (jb) e=l-25 j^tot//=2-22 (67x10s m3/s)

{C^iit

(c) e = 3-33 ^tot//=l-08(3-2xl08m3/s) (d) e=l-25 ^tot//=l-05 (3\Lxl08m3/s)

Figure 6. (Upper) The model geometry in Mercator and
polar projection.

(Lower) Some numerical solutions showing the
dependence on the friction parameter £ ( = SL./B), and on
the wind stress distribution X(y). For cases (a), (b), and
(c), X = 6/iT sin (TT/4 + TTy/6; for case (d), X = 5/lT sinTTy/5.
The maximum wind stress is further north in case (d).
The contour interval forY is 1/4^.* . The equivalent
dimensional mass transport is given in parentheses.
(From Gill, 1968. )

26

where X, Y = components of dimensionless wind stress

O = dimensionless friction parameter

and subscripts denote differentiation.

An analytical solution (valid for small b ) and a series of
numerical solutions are given. Plots of numerical results for
various combinations of 6 and wind stress illustrate the effect of
these parameters on the solution. Two wind stress functions are
used. In both cases the meridional component Y is set equal to
zero, and the zonal component is assumed to be a function of y
only.

The theoretical transport lines, figure 6 (lower), resemble
figure 2 in several respects. After passing through Drake
Passage the model current swerves sharply to the north. In the
western (Atlantic -Indian) portion of the basin the Current is
broad and remains displaced to the north, but as it moves into
the Pacific the transport lines converge and shift to the south.
Within the Passage the transport lines are crowded in the northern

part, where velocities are known to be highest. Total transport in

14 -1 15 -1

the plots shown ranges from 3x10 g sec to 1. 6 x 10 g sec

depending on the combination of friction parameter and wind

stress. However, the general shape of the Current is similar

in every plot regardless of the choice of parameters. This would

seem to indicate the overriding influence of the boundaries in

determining the form of the flow.

Gill finds that the Current consists of two strongly coupled

components: a zonal part which accounts for less than half the

total transport and, to the north, an asymmetric part which makes

up the rest. He shows that the asymmetry of the Current is due

27

more to the effect of the Passage* than to longitudinal variations
in the wind field.

Discussion of Model Characteristics

Several analytical models of the Circumpolar Current have
been presented above. In this section certain aspects of these
models will be examined more closely to bring out similarities
and differences in approach. The structures of the models will be
compared with observed features of the current. Table 2
summarizes important characteristics of the models.

First, three assumptions which are shared by all the models
(and are common in other studies of ocean currents) will be
discussed: (i) unsteady and (ii) nonlinear terms are neglected in
the dynamical equations, and (iii) depth is taken to be constant.

The lack of data with which to evaluate the relative
importance of time -dependent terms forces a long-term average
approach in the models. The neglect of unsteady terms in the
equations of motion is consistent with this approach.

The relative importance of nonlinear accelerations in the
equations of motion may be estimated with the Rossby number,

R = U/fLi, where U is a characteristic velocity, L is a

o '

characteristic length, and f is the Coriolis parameter. With

U = 20 cm sec , L = 1000 km = 10 cm, and f = 10 sec , it

_2
is found that R < 10 . Neglect of nonlinear terms is a
o 5

It is noteworthy that the bending of transport lines is achieved
over a flat bottom. Stommel (1957) regards the barrier formed by
the Scotia Ridge as essential, but Gill shows that the constricting
effect of the Passage is enough to distort the flow. Perhaps the
large-scale meanders in the Indian and Pacific sectors are related
in a similar fashion to the influence of the African and Australian -
New Zealand land masses.

Table 2
Summary of model characteristics

28

Net

Transport
(g sec )

vD
O

in
o

in
o

13

o 5

^* o

Id
U

in
o

i
o

M

2 °
A tf)

Solid Wall
at 45°S

Solid Wall
at 45°S

Solid Wall at

Unspecified

Latitude

id

-* ii

o
co

c

0

• r-4

o

id
u

■*-»
id

Lateral
Vertical

nt

o

>

id

o

>

Wind
Stress
Function

zonal,

*• = Const

■4->
W

C
O

■3 °
a H

0 XJ

Northward Ekman
Drift Simulated by
Distributed Sink
at Surface

C II

Zonal
Pressure
Grad.

0

c

O

c

to
1)

Zonal

Cbriolis

Force

0

c

to

CD

Homo-
geneous
Water

o

c

10

(A
<D
>>

o
c

1 s

rH ■•->

o >»

o w
O

id

o

'u

C
*>>

U

c

nj
o

"id

u

13

a
u

u
a

00

c
Id

o

0)

u

o

2

Munk &
Palmen

(1951)

Hidaka &;
"Tsuchiya

(1953)

Stommel
(1962)

Gill
(1968)

0)

iS
oo

oo

4)

2

c
o

+■»
<d

0

*J

CO

<d

0)

o

T3

<d

c
o

id

oo

2

Uh

o

s

29

reasonable approximation in these models of the large-scale flow.

It is more difficult to make a priori judgments as to the
effect of bottom topography on the Circumpolar Current, for, with
the exception of the area near the Scotia Ridge, it is not generally-
known whether the Current is deep enough to feel the bottom. If
it isn't, bottom topography may be ignored. But if the flow is deep
enough to feel the bottom, large-scale topographic features may be
important. Sverdrup (1941) showed theoretically that the bending
of streamlines over the Scotia Ridge may be caused by the effect
of the Ridge on the Current. Similar distortions over the other
submarine ridges are evident in figure 2. For the present the
constant depth assumption must be regarded as being of question-
able validity.

The models also display important differences in approach.
These will be discussed under the headings Geometry and
Boundary Conditions, and Driving and Retarding Mechanisms.

1. Geometry and Boundary Conditions

Defining model boundaries is the fundamental problem here.
The southern limit is obvious --Antarctica- -and the appropriate
condition is that velocity vanishes at the boundary. However, the
northern limit and boundary condition are not as clear-cut.

In the papers by Munk & Palmen and Hidaka & Tsuchiya the
Current is constrained to flow between solid walls, i. e. , in a
channel of uniform cross section. The northern boundary
condition is then u = 0 along some latitude circle. It is evident
from surface current charts (e. g. , Sverdrup et al, 1942, Chart
VII; Dietrich, 1963, Chart 5) that there is no zone of intense
velocity shear surrounding Antarctica, as would be expected in
the presence of a solid boundary. Such a boundary would prohibit

30

exchange between the Current and the oceans which border it to
the north. In the ocean, however, a significant meridional
exchange does occur. This exchange probably influences the
dynamics of the zonal current, and it should not be arbitrarily
eliminated.

Gill is more careful about choosing the northern boundary
condition. He regards the northern limit of the Current as the
position where it matches the Sverdrup-like circulation which
develops in lower latitudes. In other words, the meridional
extent of the Current is determined by the wind field. The
boundary at y = D may result in the formation of boundary
currents which modify the Sverdrup regime. However, it is
possible, through a judicious choice of parameters, to ensure
that the dependence of the solution on D is weak.

Wyrtki also assumes that the northern boundary of the
Current is determined by the anticyclonic gyre which exists at
lower latitudes. He does not investigate the influence of different
wind field distributions on this boundary as Gill does.

Stommel, Wyrtki, and Gill include a partial meridional
barrier in their models. This has a profound effect. The most
obvious result is that it eliminates the axial symmetry implicit
in the channel configurations. It also has dynamical effects
which are discussed below.

2. Driving and Retarding Mechanisms

Wind is the driving agent for the models studied in this

paper. Munk & Palmen and Hidaka & Tsuchiya assume a constant

-2

eastward wind stress of 2 dyne cm over the entire zone from

45 °S to 70 °S. Like the solid wall at 45 °S, this is a highly
artificial representation of natural conditions.

31

Gill and Stommel (1957) use zonal wind stress functions
which vary in the north- south direction but not in the east-west

direction. Over the region of the Circumpolar Current winds

-2

are westerly. Gill has a maximum stress of about 1. 5 dyne cm

Wyrtki includes both longitudinal and latitudinal variations
in the zonal wind stress. In figure 1 it is seen that the zone of
maximum westerlies shifts toward the south in the Pacific. It is
not clear whether this shift in the wind is an important factor in
the asymmetry of the Current, since Gill obtains an asymmetric
Current without allowing for longitudinal wind dependence.

Friction is included as a retarding mechanism in all
models. Hidaka & Tsuchiya use vertical and lateral friction
terms, but they indicate that lateral friction dominates. Munk &
Palmen use only lateral friction. Wyrtki and Gill employ a
simple form of bottom friction. Gill shows that the equations
can also be written using lateral friction instead of vertical
friction. The two models are similar in form and results.

The meridional barrier in Stommel' s, Wyrtki' s, and
Gill's models introduces another retarding mechanism not
found in the channel-flow models --a zonal pressure gradient.
Its significance can be seen in the fact that the eddy coefficients

required to give observed transport in Gill's model are

32-1 82-1

A = 10 cm sec or A, = 10 cm sec , while the co-
v h

efficients required for the channel models are one to two orders
of magnitude larger.

32

Concluding; Remarks

In retrospect, Stommel's (1957) discussion of the Antarctic
Circumpolar Current stands out as a turning point in the effort
to explain theoretically the dynamics of the Current. He was
the first to note the implications of boundary asymmetry with
respect to dynamical structure. Polar asymmetry is a
fundamental characteristic of the Current. Therefore, it is not
surprising that inconsistencies appear in the simple channel-
flow models of Munk & Palmen and Hidaka & Tsuchiya.

Recent papers by Wyrtki and Gill carry forward Stommel's
suggestion. Polar symmetry is abandoned, and a partial
meridional barrier is introduced. Boundary conditions and the
wind stress distribution are more realistic. These models
predict the transport and general form of the Current much better
than the earlier zonal models. It is concluded that lateral
boundary geometry, in particular the Drake Passage constriction,
and latitudinal variations in wind stress are key elements in the
dynamics of the Circumpolar Current.

Several of the large-scale features of the Circumpolar
Current have been explained, but a number of interesting questions
concerning its dynamics remain unanswered. Some of these are
of general oceanographic interest and will not be discussed here.
Four questions, however, are of particular significance to this
subject:

1. Is there any reason to expect the Circumpolar
Current to extend to much greater depths than do wind-
driven currents in lower latitudes ?

In several of the papers reviewed above reference

33

is made to the extreme depth of the Current. Observational
support for this statement is rather scanty, consisting
principally of geostrophic velocity sections such as figure 2.
Nor has the matter received adequate theoretical attention.
Physically, three factors might be expected to cause
an unusually deep flow: (i) the strength of the west winds,
(ii) the great fetch over which they blow in the Southern
Ocean, and (iii) the relatively homogeneous structure of
Antarctic waters compared with tropical and subtropical
waters.

2. If the Current is very deep, is it retarded
significantly by bottom topography?

This question is aimed at investigating further the
mountain effect discussed by Munk & Palmen. Although
their analysis is too superficial to be conclusive, the
possibility of zonal pressure gradients caused by submarine
ridges should not be ignored. The retarding effect of
underwater barriers could be studied in a laboratory model
similar to the one proposed by Stommel (1962).

3. What is the nature of the meridional circulation,
and what role does it play in the zonal flow?

Meridional flow must have both wind-driven (Ekman)
and thermohaline components. The former has been
studied theoretically by Sverdrup (1933) and Wyrtki (I960).
Deacon (1937a) has discussed the thermohaline circulation.
These papers are concerned primarily with the relationship
between transverse water motions and the Antarctic

34

Meridional flow may play a larger part in the
overall dynamics of the Circumpolar Current. The quasi-
steady transverse circulation may act as a momentum-
transfer agent in the manner proposed by Munk & Palmen.
Barcilon (1966; 1967) suggests that the thermohaline
meridional circulation driven by runoff from Antarctica
sets up a westward countercurrent which opposes the
Circumpolar Current.

To properly evaluate the role of the meridional
circulation it will be necessary to determine the size of
the meridional transport more accurately than has been
done in the past. Conventional geostrophic methods are
of little help. A new approach will be required, perhaps
use of the heat balance equation for the southern hemisphere,

4. What is the effect on the Current of the annual
pack ice cycle ?

It was noted earlier that during winter, pack ice
covers a large portion of the Southern Ocean. The ice
cover may enhance or reduce the transfer of energy from
the wind to the water, depending on such factors as the
roughness of the ice surface and internal friction within
the pack. It is interesting to observe that Gordon (1967)
finds a slight decrease in transport through Drake
Passage in winter.

35

Appendix

Equations of motion in cylindrical coordinates for an
incompressible fluid relative to a frame rotating with clockwise
angular velocity £l (adapted from Batchelor, 1967).

(tangential)

tH- + U-VUL + \L£ + 2 XIV

at r

+ Aw[vU-£, + £^] + Av g-

(radial)

at

- it - 2iltL

r

-J <3

-JL df

? 3~r

Ak[vlv - ?*

r

i it] A Pv

36

Appendix (cont'd),
(vertical)

|^ + y-yuj = ^Fi + Ah 7h w + Av ^* - <j

A A

where \J = U.J+ITtf' + UJjju.v, w = velocity-

components in vT £ - directions and

A a • J

J (T, 3 = unit vectors

XL = magnitude of earth's rotation

& - pressure

Q - density

Oi = gravity

Ak = kinematic coefficient of lateral eddy
viscosity

Av = kinematic coefficient of vertical eddy
viscosity

i
X7\n = horizontal Laplace operator

7 Tr v.r Sv- J + v1- a^

and V . *£ + fr£ +I|

37

Bibliography

Barcilon, V. (1966) On the influence of the peripheral Antarctic
water discharge on the dynamics of the Circumpolar
Current. Jour. Marine Res. 24 (3) 269-275.

(1967) Further investigation of the influence of the

peripheral Antarctic water discharge on the Circumpolar
Current. Jour. Marine Res. 25 (1) 1-9.

Batchelor, G. K. (1967) An Introduction to Fluid Dynamics.
London: Cambridge University Press. 615 pp.

Deacon, G. E. R. (1937a) The hydrology of the Southern Ocean.
"Discovery" Report 15: 1-124.

(1937b) Note on the dynamics of the Southern Ocean.

"Discovery" Report 15: 125-152.

- -- (1963) The Southern Ocean in The Sea, Vol 2 (M. N.

Hill, Ed.). New York: Interscience Publishers, Inc.,
pp. 281-296.

Dietrich, G. (1963) General Oceanography. New York:
Interscience Publishers. 588 pp.

Faller, A. (I960) Further examples of stationary planetary

flow patterns in bounded basins. Tellus 12 (2) 159-171.

Fofonoff, N. P. (1956) Some properties of sea water influencing
the formation of Antarctic Bottom Water. Deep -Sea Res. 4:
32-35.

Gill, A. E. (1968) A linear model of the Antarctic Circumpolar
Current. Jour. Fluid Mech. 3_2 (3) 465-488.

Gordon, A. L. (1967) Geostrophic transport through the Drake
Passage. Science _156_ (3783) 1732-1734.

Hidaka, K. and M. Tsuchiya (1953) On the Antarctic Circumpolar
Current. Jour. Marine Res. 12 (2) 214-222.

Kort, V. G. (1962) The Antarctic Ocean. Sci. Am. 207 (3)
113-128.

38

Bibliography (cont'd. )

Mackintosh, N. A. (1946) The Antarctic convergence and the
distribution of surface temperatures in Antarctic waters.
"Discovery" Report 23: 177-212.

and H. F. P. Herdman (1940) Distribution of the pack-
ice in the Southern Ocean. "Discovery" Report 19: 285-296.

Munk, W. H. and E. Palmen (1951) Note on the dynamics of the
Antarctic Circumpolar Current. Tellus 3(1) 53-55.

Stommel, H. (1948) The westward intensification of wind-driven
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(1957) A survey of ocean current theory. Deep Sea

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(1962) An analogy to the Antarctic Circumpolar Current.

Jour. Marine Res. 20(1): 92-96.

1 a. B. Arons, and A. Faller (1958) Some examples of

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Sverdrup, H. V. (1933) On vertical circulation in the ocean due
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the Antarctic Circumpolar Current. "Discovery" Report 7_:
139-170.

(1941) The influence of bottom topography on ocean

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(1947) Wind-driven currents in a baroclinic ocean; with

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, M. W. Johnson, and R. H. Fleming (1942) The Oceans:

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39

Bibliography (cont'd. )

Taljaard, J. J. (1957) Geographical and hydrological features
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von Arx, W. (1957) An experimental approach to problems
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Vowinckle, E. (1957) Climate of the Antarctic Ocean in

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Wexler, H. (1959) The Antarctic convergence-or divergence?
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13.(4) 153-174.

40

Vita

Jeffrey Edwin Callahan was born in Cambridge,
Massachusetts on 24 September 1943. He entered the
United States Naval Academy in June 1961. Upon
graduating in June 1965 he was awarded a Bachelor
of Science degree and commissioned as an Ensign,
United States Navy. His first duty was aboard the
U. S. S. Willis A. Lee (DL-4) in the Atlantic Fleet.
The author began his graduate studies at The Johns
Hopkins University in June 1967.

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