382 V
OLUME
29JOURNAL OF PHYSICAL OCEANOGRAPHY
q 1999 American Meteorological Society
Behavior of Double-Hemisphere Thermohaline Flows in a Single Basin
B
ARRY
A. K
LINGER
Oceanographic Center, Nova Southeastern University, Dania Beach, Florida
J
OCHEM
M
AROTZKE
Center for Global Change Science, Massachusetts Institute of Technology, Cambridge, Massachusetts
(Manuscript received 26 May 1997, in final form 12 March 1998)
ABSTRACT
A coarse resolution, three-dimensional numerical model is used to study how external parameters control the
existence and strength of equatorially asymmetric thermohaline overturning in a large-scale, rotating ocean basin.
Initially, the meridional surface density gradient is directly set to be larger in a ‘‘dominant’’ hemisphere than
in a ‘‘subordinate’’ hemisphere. The two-hemisphere system has a broader thermocline and weaker upwelling
than the same model with the dominant hemisphere only. This behavior is in accord with classical scaling
arguments, providing that the continuity equation is employed, rather than the linear vorticity equation.
The dominant overturning cell, analogous to North Atlantic Deep Water formation, is primarily controlled by
the surface density contrast in the dominant hemisphere, which in turn is largely set by temperature. Consequently,
in experiments with mixed boundary conditions, the dominant cell strength is relatively insensitive to the
magnitude Q
S
of the salinity forcing. However, Q
S
strongly influences subordinate hemisphere properties, in-
cluding the volume transport of a shallow overturning cell and the meridional extent of a tongue of low-salinity
intermediate water reminiscent of Antarctic Intermediate Water.
The minimum Q
S
is identified for which the steady, asymmetric flow is stable; below this value, a steady,
equatorially symmetric, temperature-dominated overturning occurs. For high salt flux, the asymmetric circulation
becomes oscillatory and eventually gives way to an unsteady, symmetric, salt-dominated overturning. For given
boundary conditions, it is possible to have at least three different asymmetric states, with significantly different
large-scale properties. An expression for the meridional salt transport allows one to roughly predict the surface
salinity and density profile and stability of the asymmetric state as a function of Q
S
and other external parameters.
1. Introduction
Though the earth’s global thermohaline circulation is
dominated by temperature (the deepest water is gener-
ally the coldest), salinity variations play a crucial role
in determining the location of deep-water formation.
The surface temperature distribution is roughly sym-
metric about the equator, but surface salinity has notable
north–south asymmetries [e.g., cf. Figs. 8.8 and 8.10 in
Peixoto and Oort (1992)], with winter northern North
Atlantic water roughly 0.5 psu saltier than australwinter
Weddell Sea water (Levitus and Boyer 1994). As a re-
sult, deep-water formation is equatorially asymmetrical
within the World Ocean, with substantial transports of
North Atlantic Deep Water crossing the equator and
flowing into the Southern Hemisphere [Speer and Mc-
Cartney (1991); Warren (1981) for a review of earlier
Corresponding author address: Dr. Barry A. Klinger, Oceano-
graphic Center, Nova Southeastern University, 8000 North Ocean
Drive, Dania Beach, FL 33004.
E-mail: klinger@ocean.nova.edu
work]. This asymmetry is interesting for purely ocean-
ographic reasons as well as for its influence on climate.
For example, the heat transport in the South Atlantic is
equatorward rather than poleward (e.g., Macdonald
1993).
As Rooth (1982) and Bryan (1986) have shown, using
a box model and a general circulation model, respec-
tively, such an asymmetry in salinity and meridional
overturning can occur even when the boundary condi-
tions are equatorially symmetric. Surface boundary con-
ditions form a relatively tight constraint on surface tem-
perature but can be thought of as equivalent to speci-
fying a flux of salinity rather than the salinity itself.
Because of these ‘‘mixed boundary conditions,’’ more
than one circulation state can exist given such a set of
boundary conditions (Stommel 1961). In a single ocean
basin spanning the equator, it is possible to have either
equatorially symmetric temperature-dominated (T-dom)
sinking near the poles, equatorially symmetric salinity-
dominated (S-dom) sinking near the equator, or equa-
torially asymmetric deep-water formation at one pole
only (Welander 1986).
M
ARCH
1999 383KLINGER AND MAROTZKE
How overturning strength depends on external pa-
rameters has received relatively thorough scrutiny in
single-basin systems driven by temperature-only bound-
ary conditions (Bryan 1986; Colin de Verdiere 1988;
Winton 1996; Marotzke 1997) or freshwater-only
boundary conditions (Huang and Chou 1994). The two-
hemisphere, mixed-boundary-condition case has been
explored with two-dimensional models by Marotzke et
al. (1988), Thual and McWilliams (1992), Quon and
Ghil (1992, 1995), Cessi and Young (1992), Schmidt
and Mysak (1996), Vellinga (1996), and Dijkstra and
Molemaker (1997). When salinity forcing is sufficiently
stronger than temperature forcing, only the S-dom state
exists, and when temperature forcing is sufficiently
stronger than salinity forcing, only the T-dom state ex-
ists. It is only when both temperature and salinity forc-
ing are in some sense at intermediate strengths that the
equatorially symmetric states and the asymmetric state
can all exist [see Thual and McWilliams (1992) Fig. 3].
These two-dimensional studies did not relate the flow
strength and regime boundaries to parameters that could
be readily applied to three-dimensional basins in which
basin width, planetary rotation, and planetary radius
play a role. There has been no three-dimensional study
of this system in which forcing parameters were sys-
tematically varied, though Weaver and Sarachik (1991)
used such a model to study the time evolution of tran-
sitions from state to state.
The thermohaline circulation is a global system in
which the flow in each basin is influenced by that in
the other basins (Warren 1983; Gordon 1986; Rintoul
1991; Marotzke and Willebrand 1991; Stocker and
Wright 1991; England 1993; Macdonald and Wunsch
1996). This system has an extremely complicated de-
pendence on a multiplicity of parameters, including ba-
sin geometry (Hughes and Weaver 1994), atmospheric
freshwater transports between basins (Marotzke and
Willebrand 1991; Stocker et al. 1992), and the coupling
to the atmosphere (e.g., Mikolajewicz and Maier-Reimer
1994; Rahmstorf and Willebrand 1995; Weber 1998).
Therefore, it is useful to understand the less compli-
cated, single-basin cell as a stepping stone to under-
standing the more complete system.
The basic question of this paper is: How do the ex-
ternal parameters in a two-hemisphere, mixed-bound-
ary-condition system determine the temperature, salin-
ity, and overturning in the equatorially asymmetric
state? We subdivide this into two smaller problems. In
the asymmetric state, one hemisphere will possess the
densest surface water in the basin. Peterson (1979) and
Cox (1989) have shown that this water must dominate
the bottom of the entire basin (when the equation of
state is linear), but the resulting state has not been care-
fully studied. Thus, given such an asymmetric surface
density distribution, what is the resulting overturning
strength and pycnocline structure? This will be exam-
ined in section 3, which discusses experiments in which
the surface density is restored to a reference profile.
Then the full question can be answered if, for a given
mixed boundary condition, we can predict the resulting
surface density. We consider this question via mixed-
boundary-condition experiments in section 4.
Hughes and Weaver (1994) showed in an idealized
global GCM under mixed boundary conditions that At-
lantic overturning strength is linearly related to the bas-
inwide meridional gradient in zonally and vertically av-
eraged steric height (i.e., a double vertical integral of
zonally averaged density). Rahmstorf (1996) found in
his global GCM coupled to a diffusive atmospheric en-
ergy balance model that Atlantic overturning is pro-
portional to the zonally averaged middepth density dif-
ference between northern and southern Atlantic bound-
aries. In contrast to these studies, we want to relate the
strength of the thermohaline circulation directly to ex-
ternal parameters; to clarify the relation we use simpler
geometry and forcing. Wang et al. (1999a,b) used an
idealized, global, hybrid coupled GCM and Scott et al.
(1999) uncoupled and coupled box models to study in-
terhemispheric thermohaline flow and its interaction
with the atmosphere, in particular the effects of equa-
torially asymmetric atmospheric water vapor transports.
Here, we leave out asymmetries in forcing (except in
restoring-boundary-condition experiments) and detailed
considerations of ocean–atmosphere interactions, to fo-
cus on the rotating fluid dynamics of the interhemi-
spheric thermohaline circulation. The impact of wind,
nonlinearities in the equation of state, and north–south
asymmetries in the forcing are all important factors that
deserve careful treatment, and we defer these to a later
paper. For simplicity we use a relatively idealized sys-
tem here.
2. Numerical model
All experiments are conducted with MOM-2, the
Modular Ocean Model version of the GFDL Model (Pa-
canowski 1996; Cox 1984), a B-grid (Arakawa and
Lamb 1977) finite-difference discretization of the prim-
itive equations that computes solutions by stepping for-
ward in time. The domain is a sector of a sphere with
zonal and meridional boundaries and a flat bottom. De-
fault parameters are shown in Table 1; experiments with
any of these parameters changed are noted in the text
in section 4. The model is run at coarse, uniform hor-
izontal resolution with vertical grid spacing increasing
from 50 m at the surface to 500 m at the bottom. All
advective terms are retained in the temperature, salinity
and momentum equations, and density
r
is related to
temperature T and salinity S by
r
5
r
0
1
b
S 2
a
T, (1)
where
a
and
b
are the thermal and haline expansion
coefficients, respectively. The values chosen correspond
to a linearization of the equation of state at surface
pressure and a temperature of about 138C (see Table 1).
Though Gargett and Holloway (1992) argue that dif-
384 V
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29JOURNAL OF PHYSICAL OCEANOGRAPHY
T
ABLE
1. Summary of numerical experiments.
Parameter Symbol Value
Basin width, length
Basin depth
Vert. diffusivity momentum,
tracers
Hor. diffusivity momentum,
tracers
Isopycnal diffusivity
Expansion coefficients
T restoring timescale
Long, lat grid spacing
Number of levels
Momentum time step
Tracer time step
L,
f
p
H
(n
V
,
k
V
)
(n
H
,
k
H
)
k
r
a
,
b
t
dl, d
f
N
z
dt
M
dt
T
608,648
4500 m
(1, 0.5) 3 10
24
m
2
s
21
(250, 1.0) 3 10
3
m
2
s
21
2000 m
2
s
21
0.2 kg m
23
C
21
,
0.8 kg m
23
psu
21
30 days
3.758, 4.08
15
3600 s
5d
fusivities of T and S should be different, the parame-
terization of Zhang et al. (1998) shows this to be a
relatively small effect; here we use the same diffusivity
for both fields (see Table 1).
All walls and the bottom are insulating in both T and
S. Temperature is forced by restoring the surface layer
to a zonally uniform reference profile. The reference
profile as a function of latitude
f
in either hemisphere
is given by
1
T 5 T 1DT[cos(
pf
/
f
) 2 1], (2)
ep
2
where 2
f
p
#
f
#
f
p
, T
e
is the restoring surface tem-
perature at the equator, and DT is the restoring surface
temperature difference between the equator and the po-
lar boundary (in restoring boundary condition experi-
ments, this is DT
N
in the northern hemisphere and DT
S
in the southern hemisphere). Thus as we move north-
ward from the southern boundary, the restoring tem-
perature smoothly increases from a low of T
e
2DT
S
to
a high of T
e
and then decreases back to a low of T
e
2
DT
N
.
In the restoring-boundary-condition runs (section 3),
salinity is constant and hence no salinity forcing is nec-
essary. In the mixed-boundary-condition runs (section
4), the reference temperature is symmetric about the
equator (DT
N
5DT
S
). The salinity is driven by setting
a zonally uniform surface salinity flux to represent the
effects of freshwater fluxes produced by evaporation,
precipitation, and runoff. The dependence on latitude
for forcing strength Q
S
is given by
q
S
5 Q
S
cos(
pf
/
f
p
)/cos(
f
). (3)
This form represents the generally negative evaporation
minus precipitation (E 2 P) at high latitudes and pos-
itive E 2 P at low latitudes found on the real earth.
The real earth also has a narrow region of negative E
2 P near the equator, which we ignore here under the
assumption that the resulting shallow equatorial salinity
minimum has a relatively small effect on the large-scale
thermohaline circulation. The cosine factor in the de-
nominator is used so that the zonal integral of q
S
would
be a simple cosine in latitude, with no net salt added to
or removed from the basin.
Experiments are started with initial conditions of ei-
ther a resting, isothermal, isohaline (35 psu) ocean, or
a previous experiment. Each run is integrated until a
nearly steady state is reached, unless noted otherwise.
The usual criteria for steady state being reached is that
the drift in meridional volume transport exponentially
decreases over the last several centuries of the integra-
tion and is no more than 0.0025 Sv per century (Sv [
10
6
m
3
s
21
) at the end. Spot checks on selected runs
show that, when these criteria are met, other measures
such as temperature and salinity are also nearly steady.
In order to satisfy the Courant–Friedrichs–Levy crite-
rion at the lowest computational cost, a shorter time step
is used for the momentum equations than the tracer
equations (Bryan 1984). Each experiment generally
takes from 1000 to 5000 tracer years to reach steady
state, which takes on the order of 1 day of cpu time on
a DEC AlphaStation 250 4/266 workstation.
Most of the experiments are conducted using hori-
zontal diffusion of properties to parameterize mixing
induced by geostrophic eddies. Some runs with restoring
boundary conditions are also repeated with the ‘‘Gent–
McWilliams’’ parameterization, which supplements iso-
pycnal diffusion of T and S with additional advection
by a tracer velocity representing the untilting of iso-
pycnals by baroclinic instability (Gent and McWilliams
1990; Gent et al. 1995). This parameterization has a
stronger dynamical justification than horizontal diffu-
sion, and allows numerical models to better represent
the relatively thin thermocline and small deep-water for-
mation regions of the real ocean and to eliminate spu-
rious diapycnal diffusion in regions of strong horizontal
gradients such as western boundary currents (Veronis
1975; Bo¨ning et al. 1995; Danabasoglu et al. 1994). The
Gent–McWilliams runs are conducted with a flux-cor-
rected transport scheme added to MOM-2 by Weaver
and Eby (1997). Flux-corrected transport is a finite dif-
ference scheme that attempts to retain the accuracy of
centered-difference schemes while also suppressing
nonphysical overshoots of temperature and salinity (a
property shared by the less accurate upstream differ-
encing scheme).
3. Restoring boundary conditions
a. Experiments
We conduct two-hemisphere experiments, which are
forced only by restoring to a temperature profile that is
asymmetric about the equator. We wish to understand
how the maximum meridional overturning volume
transport and the pycnocline structure of a double-hemi-
sphere overturning cell differ from the case of a single-
hemisphere system. As section 4 will show, the surface
density given by (2), with different values of DT in the
M
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1999 385KLINGER AND MAROTZKE
F
IG
. 1. Meridional overturning streamfunction (contours) and zonal average temperature (shading) for restoring BC
experiments, (a) 1H and (b) 2H, DT
N
5 0. Overturning contours are 2 Sv apart; isotherms are at 0.05, 0.1, 0.2, 0.4, and
0.8 of DT
S
5 308C. In this and all subsequent plots of overturning streamfunction, dashed, solid, and dotted contours
represent negative (southern sinking), positive (northern sinking), and zero values, respectively.
northern and southern hemispheres, is a reasonable ap-
proximation to the shape of the surface density profiles
produced by mixed-boundary-condition experiments.
In all the experiments, the southern hemisphere is
made the ‘‘dominant’’ hemisphere by giving it the dens-
est surface water. Thus the dominant deep water, anal-
ogous to North Atlantic Deep Water in the Atlantic
Ocean, is formed in the southern hemisphere in our
experiments. We chose the southern hemisphere because
it happened to be the dominant hemisphere in the mixed-
condition experiments (section 4); since our experi-
ments have completely equatorially symmetric geom-
etry (unlike the real ocean), the choice of dominant
hemisphere is arbitrary. The weaker and deeper flow of
Antarctic Bottom Water does not appear in these ex-
periments because the equation of state is linear and
there is no circumpolar channel corresponding to the
Southern Ocean.
The relationship between the behavior in one-hemi-
sphere (‘‘1H’’) and two-hemisphere (‘‘2H’’) basins is
demonstrated by runs with DT
S
set to 308,68, and 18C.
In each of these experiments, we set DT
N
5 0, thus
exploring the case of most extreme equatorial asym-
metry first. For each 2H experiment, a 1H experiment
is run with forcing identical to the dominant hemisphere
of the corresponding 2H run. Outside of section 3b, ‘‘1H
experiment’’ refers to an experiment with equatorial
symmetry rather than an actual wall at the equator.
These two variations are nearly identical (11.86 Sv over-
turning with the wall and 11.84 Sv with symmetry).
In another series of experiments, DT
S
is fixed while
DT
N
is varied. We are especially interested in the situ-
ation in which DT
N
is almost as large as DT
S
. In this
case, the degree of asymmetry between the hemispheres
is small, yet the circulation must be qualitatively dif-
ferent from a symmetric experiment because deep water
is required to spread from the dominant hemisphere to
fill the deepest region of the other ‘‘subordinate’’ hemi-
sphere. In this series of experiments, DT
S
5 308C, while
DT
P
[DT
N
2DT
S
is set to 158,68,38, 1.58, 0.68, and
08C.
b. Dependence on dominant hemisphere temperature
gradient
The zonally integrated thermohaline circulation is
characterized by small, relatively intense downwelling
regions associated with deep convection and large areas
of weak, diffusively driven upwelling (Fig. 1). In runs
with DT
N
5 0, the presence of the nonconvecting hemi-
sphere means that a strong thermocline covers about
two times the area that it does in a single-hemisphere
run.
The classical scaling for upwelling velocity W, ther-
mocline depth D, and horizontal velocity V in large-
scale, buoyancy-driven circulation (Bryan and Cox
1967; Bryan 1987; Colin de Verdiere 1988) is based on
the vertical advective–diffusive balance,
wb
z
5
k
b
zz
, (4)
and thermal wind,
f
y
z
52b
x
, (5)
where f is the Coriolis parameter,
y
and w are merid-
ional and vertical velocities, and b 52g
r
/
r
0
is the
buoyancy (g is the gravitational acceleration, 9.8 m
2
s
21
). These equations yield the scale relations
W 5
k
/D, (6a)
fV/D 5Db/M, (6b)
where Db is the imposed meridional surface buoyancy
range and M is the basin zonal length scale. One might
ask whether the zonal buoyancy difference in (5) must
386 V
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29JOURNAL OF PHYSICAL OCEANOGRAPHY
scale like Db, but Marotzke (1997) demonstrates that
this is actually a reasonable assumption, and we show
below that it holds fairly well in our numerical exper-
iments. Another weakness of the classical scaling em-
ployed here (and indeed of the vertical mixing param-
eterization in the model) is that vertical or diapycnal
mixing in the ocean is known not to be uniform but
concentrated near the margins (e.g., Munk 1966;
Wunsch 1970; Armi 1978; Ledwell and Bratkovich
1995; Toole et al. 1997). Again, Marotzke (1997) has
shown that some scaling and numerical results are rea-
sonably insensitive to assumptions about how localized
the mixing is.
One more scale relation must be included in order to
close the system and find W, D, and V. Often this is
done using the linear vorticity relation (see Bryan 1987;
Colin de Verdiere 1988),
b
0
y
5 fw
z
, (7)
where
b
0
is the meridional gradient of f. However, this
equation does not apply to the western boundary current,
an important contributor to the zonal average of
y
. When
k
or Db is varied, this inapplicability does not matter,
because the western boundary current strength has the
same sensitivity to
k
and Db as the interior flow. How-
ever, the relationship between the western boundary and
the interior currents changes when the geometry of the
flow changes. Thinking of the deep flow as a homo-
geneous layer driven by a point source and a distributed
sink (Stommel et al. 1958; Stommel and Arons 1960),
we see that, if the sink area is changed but upwelling
speed w remains the same, the interior flow is unaffected
but the western boundary current must change to satisfy
continuity. In order to compare 2H flows to 1H, it is
therefore more appropriate to use the continuity equa-
tion (see Marotzke 1997; Winton 1996).
Assuming that the volume transport into the region
of deep-water formation equals the upwelling over al-
most the entire basin, we have
MDV 5 MLW, (8)
where L is the meridional length of the basin. Equations
(6) and (8) yield the scale relations
1/3
k
LM f
D 5 , (9a)
12
Db
1/3
2
Db
k
W 5 , (9b)
12
fLM
1/3
2
Db
k
L
V 5 , (9c)
22
12
Mf
1/3
22 2
Db
k
LM
F5MDV 5 , (9d)
12
f
where F is the meridional overturning streamfunction.
If linear vorticity (7) were used instead of continuity
(8), then (9) would be the same except that L would be
replaced with the radius of the earth, R. In one hemi-
sphere, L ø R, and the two assumptions about the ver-
tical velocity scale would yield the same result. In two
hemispheres with one of them nonconvecting, however,
L ø 2R, and thus the scaling containing the continuity
equation predicts that the two-hemisphere case will have
a broader thermocline and weaker vertical velocity,
whereas scaling containing the linear vorticity equation
predicts that the thermocline and vertical velocity will
be the same in the two cases.
We test the above scaling relationships with 2H (DT
N
5 0) and 1H experiments. In the 1H experiments, D }
Db
21/3
and w }Db
1/3
, as expected. Here, D is taken to
be the integral length scale of the zonal average tem-
perature at the equator,
00
D 5 zT dz T dz . (10)
EE
12@12
zz
00
In each run, water with temperature in the bottom 1%
of the temperature range for the water column is con-
sidered ‘‘subthermocline’’ and is excluded from the cal-
culation. Here W is found by taking the maximum zonal
average vertical velocity at each latitude and averaging
this from latitudes 28 to 308S, which is much of the
region dominated by upwelling but excludes recircu-
lation close to the convection region. The total volume
transport of the meridional overturning cell is also
roughly proportional to Db
1/3
, though this scaling law
is closer to Db
1/2
for small Db because the upwelling
area decreases somewhat for smaller Db.
The maximum buoyancy difference between the east-
ern and western boundaries is approximately b
E
2 b
W
5 0.25Db for all 1H and 2H runs, confirming the the-
oretical result of Marotzke (1997) and the hypothesis
that the zonal buoyancy difference scales like Db. The
zonal buoyancy difference has a weak dependence on
L (2H runs have about a 20% smaller proportionality
constant). All these minor factors can be ignored in (9).
The factors of L
1/3
in (9) imply that, since 2
1/3
5 1.3,
each 2H run should have a 30% broader thermocline
and 30% weaker upwelling than the corresponding 1H
run. We measure the thermocline depth at the equator
as before, and average the maximum zonal average w
from 308Sto628N. The numerical experiments display
the relationship between 2H and 1H D and W predicted
by the scaling.
The combination of a somewhat weaker W and a larg-
er area of upwelling cause the total overturning volume
transport of the 2H experiments to be somewhat less
than double that of the 1H experiments: the scaling pre-
dicts a factor of 2
2/3
5 1.6, whereas actual values are
slightly higher (Table 2). Roughly equal amounts of
upwelling occur in both hemispheres. For DT
S
of 308C
and 68C, the convecting hemisphere has somewhat
greater upwelling than the nonconvecting hemisphere.
The western boundary current is stronger in this hemi-