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The Ocean Circulation in Thermohaline1
Coordinates2
Jan D. Zika,∗3
Matthew H. England and Willem P. Sijp
The Climate Change Research Centre, The University of New South Wales, Australia
4
For the Journal of Physical Oceanography5
August 3, 20116
∗Corresponding author address: The Climate Change Research Centre, University of New South Wales,
2037, NSW, Australia.
E-mail: [email protected]
1
ABSTRACT7
The thermohaline streamfunction is presented. The thermohaline streamfunction is the in-8
tegral of transport in temperature - salinity space and represents the net pathway of oceanic9
water parcels in that space. The thermohaline streamfunction is proposed as a diagnostic to10
understand the global oceanic circulation and its role in the global movement of heat and11
freshwater. The coordinate system used is fully Lagrangian. As such, physical pathways12
and ventilation timescales are naturally diagnosed, as are the roles of the mean flow and13
turbulent fluctuations. As potential density is a function of temperature and salinity, the14
framework is naturally isopycnal and is ideal for the diagnosis of water-mass transformations15
and advective diapycnal heat and freshwater transports. Crucially, the thermohaline stream-16
function is computationally and practically trivial to implement as a diagnostic for ocean17
models. Here, the thermohaline streamfunction is computed using the output of an equili-18
brated intermediate complexity climate model. It describes a global cell, a warm tropical19
cell and a bottom water cell. The streamfunction computed from eddy-induced advection is20
equivalent in magnitude to that from the total advection, demonstrating the zero order im-21
portance of parameterised eddy fluxes in oceanic heat and freshwater transports. The global22
cell, being clockwise in thermohaline space, tends to advect both heat and salt towards23
denser (poleward) water-masses in symmetry with the atmosphere’s poleward transport of24
moisture. A re-projection of the global cell, from thermohaline to geographical coordinates25
reveals a thermohaline mean circulation reminiscent of the schematised ‘global conveyor’.26
1
1. Introduction27
It has long been recognised that the ocean displays variability across a large spectrum of28
spatial scales, from global scale oceanic gyres and circumpolar currents to millimetre scale29
turbulent motions, and on a myriad of temporal scales from millennial overturning to waves30
with micro second periods (Davis et al. 1981). All temporal and spatial scales of motion31
contribute to the global ocean circulation and its role in the climate system (Munk and32
Wunsch 1998). Despite this, descriptive oceanographers have identified specific pathways of33
net oceanic motion from observed tracers fields (Wust 1935; Deacon 1937; Gordon 1986).34
These pathways have been schematised in a variety of ways, the ‘global conveyor’ of Broecker35
(1991) being the most universally recognised (see Richardson 2008, for a review of such36
schematics).37
Over the past half century, numerical models have moved to finer and finer resolution38
and have progressively resolved an increasing number of scales. A key challenge remains39
however: distilling the plethora of motions of a numerical model into simplified indices and40
diagrams with minimal loss of information. It is pertinent, also, to test integral metrics of41
the ocean circulation and quantify how they relate to the climate system.42
A common way of understanding the global circulation is to average oceanic velocities43
at constant latitude and depth and look at the circulation in a meridional-vertical coordi-44
nate. The circulation revealed is known as the Meridional Overturning Circulation (MOC,45
Kuhlbrodt et al. 2007). The MOC describes net vertical and meridional motion. it is often46
used to infer how the ocean distributes properties around the globe by transposing the diag-47
nosed MOC onto zonally averaged quantities. This approach is somewhat valid in the North48
2
Atlantic where a strong mean overturning exists and properties are reasonably homogeneous49
at constant depth and latitude. In other regions however, overturning cells in the depth-50
latitude plane do not correspond to property transports at all. In the Southern Ocean, for51
example, a vigorous ‘Deacon Cell’ exists, apparently driving 30-40 Sv of light surface water52
down to the depth of Drake Passage at around 2000m (Manabe et al. 1990). However, when53
the circulation is averaged in density-latitude space, correlations between the meridional54
velocity and the thickness of density layers counter the Deacon Cell (Doos and Webb 1994).55
Thus, by averaging the flow in density, rather than depth space, a complimentary view, more56
consistent with the distribution of tracers, is revealed.57
In the example of the Southern Ocean, the vertical coordinate, density, is used to capture58
meridional exchanges of water-masses. Nycander et al. (2007) and Nurser and Lee (2004)59
propose diagnosing the ocean circulation in density - depth coordinates to capture the vertical60
exchanges of water-masses. Using this approach, Zika (2011) demonstrate that the processes61
which contribute to meridional exchanges of water-masses in the Southern Ocean can be62
very different to those which contribute to vertical exchanges. Boccaletti et al. (2005) and63
Ferrari and Ferreira (2011) have analysed the meridional circulation in temperature-latitude64
coordinates and reveal a different circulation to that in density-latitude coordinates. O65
Observed tracer distributions reveal oceanic exchanges of both heat and salt which are66
not only meridional, but also zonal, between ocean basins, and vertical, between the ocean67
surface and the interior (e.g. Broecker 1991; Gordon 1991; Schmitz 1996). We seek a68
coordinate system which captures this net global circulation.69
Principally, the MOC is considered important in the climate system because of its role70
in transporting heat, freshwater, nutrients and carbon. By transporting heat and freshwater71
3
around the globe the ocean influences global temperatures and plays a key role in the hydro-72
logical cycle. The water-masses through which the global-conveyor is thought to travel are73
each defined by unique temperature and salinity properties. Here we investigate the ocean74
circulation in temperature-salinity (thermohaline) coordinates. By doing so, we are able to75
determine whether a ‘global conveyor’ of the scale previously proposed based on observed76
tracer fields (Gordon 1986; Broecker 1991) exists in a climate model.77
Zika and McDougall (2008); Zika et al. (2009b) use a conservative temperature-neutral78
density coordinate (locally equivalent to a rotated conservative temperature-salinity coordi-79
nate) to understand the balance of advection and mixing in the ocean interior from obser-80
vations. They establish the fundamental link between flow in this coordinate and diabatic81
processes, and have extended this to a general inverse method (Zika et al. 2009a, 2010).82
Some previous studies have analysed numerical model output in a thermohaline coordinate83
(Cuny et al. 2002; Marsh et al. 2005). Indeed, Blanke et al. (2006) compute a streamfunction84
in thermohaline space for a regional model of the North Atlantic using Lagrangian particle85
tracking. Here we compute a streamfunction in thermohaline coordinates from the output86
of a global ocean model using a straight-forward methodology. We then discuss the meaning87
and implications for flow in such a coordinate system.88
Section 2 of this article defines the thermohaline streamfunction (THS) and describes how89
it can be easily diagnosed from a finite volume ocean model. Section 3 describes the THS90
as diagnosed from a climate model. Section 4 discusses the Eulerian mean and fluctuating91
contributions to the THS and in Section 5 the calculation of advective diapycnal heat and92
salt fluxes is discussed. Section 6 describes the diagnosis of a thermohaline transit time while93
Section 7 shows a diagnosis of the THS for different basins and the use of the THS to infer94
4
water-mass pathways. We summarise the main conclusions in Section 8.95
2. The thermohaline streamfunction96
Here we review some standard streamfunction diagnostics used in the literature, then97
describe the computation of a thermohaline streamfunction.98
a. Barotropic and Meridional Overturning Streamfunctions99
Take a flow in the 3 coordinates: longitude, x, latitude, y and height, z bounded by100
[x1, x2], [y1, y2] and [−H(x, y), η(x, y)] respectively with velocity u = [u, v, w]. If we have101
∇ · u = 0; (1)
we can define a streamfunction, ψxy, such that102
∂ψxy∂y
= −∫ η
−Hu dz ;
∂ψxy∂x
=
∫ η
−Hv dz. (2)
So long as v = 0 at y = y1 (in the case of the global ocean, y1 would be a latitude circle103
wholly inside the Antarctic continent), the streamfunction is diagnosed using104
ψxy(x, y) = −∫ y
y1
∫ η
−Hu(x, y′, z) dz dy′. (3)
ψxy is commonly known as the barotropic streamfunction. The barotropic streamfunction105
for an intermediate complexity climate model (to be discussed in detail in Section 3) is plotted106
in Fig. 1. The barotropic streamfunction reveals the wind driven basin scale gyres, their107
boundary currents, and the pathway and strength of the Antarctic Circumpolar Current.108
5
Alternatively, integrating at constant depth and latitude, one can derive a meridional109
overturning streamfunction ψzy such that110
∂ψzy∂y
= −∫ x2
x1
w dx ;∂ψzy∂z
=
∫ x2
x1
v dx (4)
and again, so long as w = 0 at z = −H (e.g. below the deepest topography), ψzy can be111
diagnosed as:112
ψzy(y, z) = −∫ z
−H
∫ x2
x1
v(x, y, z′) dx dz′. (5)
The meridional overturning streamfunction (MOC) for an intermediate complexity climate113
model is plotted in Fig. 2. The MOC reveals meridional cells linking the equatorial and114
poleward waters and the surface and deep waters. As discussed in Section 1, the MOC115
does not accurately represent the meridional exchanges of waters of different densities and116
temperatures. In order to diagnose such exchanges, and infer the processes which contribute117
to those exchanges, it is illuminating to compute a streamfunction as a function of latitude118
and a second, time evolving tracer. Following Ferrari and Ferreira (2011), we define a119
meridional streamfunction ψCy, for an arbitrary tracer C, such that120
ψCy =
∫ ∫C′≤C
v dx dz (6)
where∫ ∫
C′≤C dxdz is the area over a surface of constant latitude, where C ′ ≤ C. The121
streamfunction ψCy can be used to diagnose the advective meridional transport of C and to122
understand which mechanism give rise to that transport (Ferrari and Ferreira 2011).123
Equation (7) applied to an instantaneous velocity field gives an instantaneous streamfunc-124
tion. In this case the streamfunction represents not only the rate at which water parcels move125
from one concentration, C, to another, but also the rate at which iso-surfaces of constant126
6
C move in space. Averaging over some period ∆t we may diagnose a mean streamfunction127
ΨCy such that128
ΨCy =1
∆t
∫ t+∆t
t
∫ ∫C′≤C
v dx dz dt (7)
Here and throughout, time mean streamfunctions will be defined using a capital Ψ. Eule-129
rian mean and fluctuating components will later be distinguished using Ψ and Ψ′ respec-130
tively. As (Ferrari and Ferreira 2011) point out, if C is steady over the period ∆t, (i.e.131 ∫ t+∆t
t(dC/dt)dt = 0), then ΨCy represents the flow of water from one C value to another.132
Whether or not the flow is steady, ΨCy, can still be diagnosed, in the unsteady case it repre-133
sents both the transformation of water parcels and the movement of the tracer (Nurser and134
Marsh 1998).135
Here we compute the Ψyσ2 , that is, a streamfunction in potential density - latitude co-136
ordinates (σ2 being density referenced to 2000m depth; Fig. 2 b). The density-latitude137
streamfunction gives a complementary view of the circulation to the depth-latitude stream-138
function (Fig. 2a). In particular the Deacon Cell is reduced to 3-4 Sv in the density-latitude139
case. The parameterised eddy-induced velocity and zonal asymmetries allow a circulation to140
exist in the Southern Ocean in latitude-depth space without large exchanges of mass (den-141
sity) across latitude circles. In addition, the bottom water cell around Antarctica and the142
deep ’abyssal cell’ below 2000m depth are here linked together in one bottom water cell at143
densities greater than 37 kg m−3.144
What is not clear from the depth-latitude, nor the density latitude streamfunction, is145
the route taken by dense waters after they have sunk in formation regions such as the North146
Atlantic. Both diagnostics show a sinking around 60◦N. Where these waters eventually upwell147
7
and which geographical regions they transit through (either isopycnally or diapycnally) is148
unclear.149
b. A streamfunction in two general coordinates150
We seek a streamfunction which is not solely meridional, but can also encapsulate vertical151
and zonal motions as well. As such, we compute a streamfunction ψC1C2 in terms of two152
tracers, C1 and C2. Hence153
ψC1C2 =
∫C′
1≤C1|C2
u · nC2 dA (8)
where nC1 is the direction normal to a C2 iso-surface and∫C′
1≤C1|C2dA is the area over the154
C2 iso-surface where C ′1 ≤ C1. Again, the instantaneous ψC1C2 represents both the trans-155
formation of water parcels from different tracer concentrations and the adiabatic movement156
of tracer iso-surfaces (see Griffies 2007, pages 138-141 for a discussion of dia-surface flow).157
The mean streamfunction in the C1 − C2 coordinate is then158
ΨC1C2 =1
∆t
∫ t+∆t
t
∫C′
1≤C1|C2
u · nC1 dA dt (9)
In this study we will only consider cases where tracers are statistically steady. Interpretations159
of unsteady streamfunctions are left to future work.160
Note that a streamfunction ΨC1C2 could be formulated for iso-surfaces of constant C1161
where ΨC1C2 = −ΨC2C1 . Replacing C2 with the meridional coordinate y, (9) is equivalent to162
(7) as u ·ny = v. Indeed, one could also replace C1 with the vertical coordinate z and recover163
the mean of the latitude-depth overturning Ψyz (5; i.e. the MOC). Hence (9) represents a164
general equation for a streamfunction in two coordinates, whether in standard geographical165
8
coordinates or coordinates defined by time variable tracers.166
For the remainder of this study we will consider oceanic flow in thermohaline coordinates.167
That is, where our two tracer variables are potential temperature, θ, and salinity, S. The168
mean thermohaline streamfunction is given by169
ΨθS =1
∆t
∫ t+∆t
t
∫θ′≤θ|S
u · n dA dt. (10)
In practice (9) could be applied to any three-dimensional, incompressible flow. We use θ170
and S as these are the variables which describe heat and salt content respectively in our171
model. In calculations based on more precise models or observations, the appropriate heat172
variable is conservative temperature (Θ; McDougall 2003) and the appropriate salt variable173
is absolute salinity (SA; McDougall et al. 2010).174
If the flow and tracer distributions are steady, ΨθS(S, θ) represents the transformation of175
water-masses from different temperature and salinity values; i.e. the warming and freshening176
of water parcels as they move through the ocean. This conversion need not be diapycnal177
as water parcels can have compensating changes in heat and salt and remain at the same178
isopycnal.179
Schematically represented in Fig. 3 are four streamfunctions: two in fixed geographical180
coordinates, x−y and z−y, one with a Lagrangian vertical coordinate, θ−y and finally the181
θ − S streamfunction in purely Lagrangian co-ordinates. The flow in Fig. 3 is reminiscent182
of both a mid-laltitude gyre and diffusively upwelled thermohaline conveyor (Stommel and183
Arons 1960). A gyre circulation moves warm salty waters poleward (Fig. 3 a). The warm184
salty water then cools becoming cold and salty. The surface waters then freshen with some185
sinking into the deep ocean (Fig. 3 a). The surface gyre circulation warms and evaporates as186
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it moves southward becoming warm and salty again. The deep portion warms and freshens187
via interior mixing as it upwells (Fig. 3 b). Averaging at constant latitude we see that this188
circulation transports warm water northward and cold water southward (Fig. 3 c). Averaging189
in temperature salinity space we see that, in this special case, both the gyre circulation and190
the diffusively upwelled overturning involve a circulation from warm and fresh to warm and191
salty, then to cold and salty, to cold and fresh and back to warm and fresh.192
Ocean models do not describe a continuous flow field and temperature and salinity dis-193
tributions but rather, a discretised approximation of it. The most common discretisation is194
one where the equations of motion and property conservation are solved on a grid defined195
by finite volumes. For each volume, a property concentration is assigned, as are transports196
across grid volume interfaces.197
Here we describe how ΨθS is calculated from a finite volume ocean model. Consider a198
model with 1 : N grid box interfaces and 1 : M discrete time steps. Volume fluxes, Uij,199
across all grid box interfaces, i, at time steps, j, are determined from the interface velocity200
and the interface area (e.g. U = [u∆y∆z; v∆x∆z;w∆x∆y] for longitudinal, latitudinal and201
vertical grid spacing ∆x, ∆y and ∆z respectively); and the respective temperatures at the202
interfaces θij and the salinities of the adjoining grid boxes S+ij and S−ij are stored in computer203
memory. Then, over all time steps the thermohaline streamfunction is computed using204
10
ΨθS(S, θ) =1
M
M∑j=1
N∑j=1
δθij δSij Uij,
δθij =
1 if θij <= θ,
0 otherwise.
(11)
δSij =
1 if S+
ij > S and S−ij < S,
−1 if S+ij < S and S−ij > S,
0 otherwise.
Above, the delta function δθ eliminates from the calculation all interfaces where θi > θ205
and δS eliminates fluxes which do not cross the S iso-surface and also sets the sign of the206
flux depending on whether the flow is up gradient or down gradient.207
Fig. 4 shows how a thermohaline streamfunction, ΨθS, is calculated from a discretised208
field. In Fig. 4 , the grid boxes where δSδθ =1 or -1 are those along the light blue line in209
panel c) between S ′ < S and S ′ > S with θ′ < θ. Computing ΨθS for a velocity field such as210
the output of an ocean model, we effectively integrate over all water-masses of like properties211
in time and space.212
3. The thermohaline streamfunction of a climate model213
We diagnose the ocean component of the final 10 years of a 3000 year simulation of214
the University of Victoria Climate Model (1.8◦ latitude by 3.6◦ longitude grid spacing, 19215
levels, 2D energy balance atmosphere) as used by (Sijp et al. 2006, , specifically their GM216
case). The ocean model is MOM2 (GFDL MOM Version 2.2, Pacanowski 1995). The217
11
vertical mixing coefficient increases with depth, taking a value of 0.6 x 10−4 m2 s−1 at218
the surface and increasing to 1.6 x 10−4 m2 s−1 at the bottom. The model employs the219
eddy-induced advection paramaterization of Gent and McWilliams (1990) with a constant220
diffusion coefficient of 1000 m2 s−1. Tracers are diffused in the isopycnal direction with a221
constant coefficient of 2000 m2 s−1.222
The ocean model displays a plausible global θ − S distribution and MOC and by the223
year 3000 the ocean model has reached a stable equilibrium. The. This model is typical of224
those used to understand different climate states and the stability of the global overturning225
circulation. As per all such ocean climate models, the model MOC displays several isolated226
cells including a shallow clockwise (north) and an anti-clockwise (south) cell in the tropics,227
a Northern Hemisphere deep cell, a bottom water cell towards Antarctica, a Deacon Cell228
between 60◦S and 40◦S and finally an abyssal cell below 2000m depth.229
A scatter plot of ocean θ−S with a colour code corresponding to ocean basin and latitude230
allows one to relate points in θ − S space to particular geographical regions (Fig. 5). The231
warmest and saltiest waters are found in the North Atlantic (red), as are colder salty waters232
reaching below 5◦C (red). The coldest and freshest waters are found in the Arctic (red), cold233
and modestly fresh waters are found in the Southern Ocean (yellow) and fresh but generally234
slightly warmer waters in the North Pacific (dark blue). The warmest waters are found in235
the equatorial Indo-Pacific Oceans (cyan). Only along a locus of water-mass classes, narrow236
in the saline coordinate, do waters from the three major ocean basins exist together.237
Using monthly velocity, potential temperature and salinity fields from the ocean model,238
we compute the mean thermohaline streamfunction (Fig. 6). It reveals, as its dominant fea-239
ture, a large15 Sv cell. This global cell reaches from the warm salty waters found only in the240
12
Atlantic, through to the cold and fresh waters found largely in the Southern Ocean and then241
to the warm fresh waters found only in the Indo-Pacific basin. Two other counterclockwise242
cells appear in the THS. These are the very warm tropical cell (θ > 25◦ C) with a transport243
of around 8 Sv and the very cold (θ < 2◦ C) bottom water cell, also with a transport of244
around 8 Sv. Within the global cell lies a warm and a cold subcell, separated around the245
10◦ C isotherm each with an additional transport of 5–10 Sv.246
North Atlantic Deep Water (NADW) is characterised in this model as having a potential247
temperature colder than 5◦ C and a salinity of around 35 g / kg. Only 4 – 6 Sv achieves such248
temperatures and salinities. NADW, at its coldest, is close to 2◦ C with a salinity of around249
34.5 g / kg. Once formed, the NADW warms, presumably through entrainment and interior250
mixing. The NADW branch then joins the remaining streamlines heading towards colder251
fresher regions of the ocean. In Section 6 we decompose this circulation by ocean basin.252
A circulation of around 6 Sv links water-masses found uniquely in the Atlantic, Southern253
and Indo-Pacific (compare Fig. 5 and Fig. 6) The many vertical and horizontal circulations,254
once integrated in θ − S space, give rise to one interconnected flow. In effect, this is a255
quantification of the ‘global conveyor’ proposed by Broecker (1991).256
4. Eddy-induced and seasonal transports257
In the ocean model discussed here, as in virtually all contemporary ocean models which258
do not explicitly resolve eddy fluxes, tracers are advected by both an Eulerian mean velocity259
13
(uEM) and an eddy-induced velocity (uGM) such that260
u = uEM + uGM . (12)
The eddy-induced velocity is prescribed using the scheme of Gent and McWilliams (1990).261
In the MOC, (uGM) is significant mainly in the Southern Ocean and North Atlantic where262
isopycnal slopes are very large. When computing the THS of Fig. 6, we used both the263
eddy-induced and Eulerian velocities and averaged these for each monthly field. We can264
now assess the influence of the Eulerian flow and the eddy-induced flow and the fluctuations265
actually resolved by the model by making the following decomposition:266
ΨEM
θS (S, θ) =
∫θ′≤θ|S
uEM · n dA dt,
ΨGM
θS (S, θ) =
∫θ′≤θ|S
uGM · n dA dt (13)
where the overline represents an Eulerian average (i.e. ξ = 1∆t
∫ t+∆t
tξdt for some variable ξ).267
At its peak, the Eulerian mean contribution to the thermohaline streamfunction, ΨEM
θS ,268
is 39 Sv (Fig. 7 b). The global cell of the Eulerian plus eddy-induced THS is 15 Sv and269
peaks at 25 Sv in its sub-cells (Fig. 6). The Eulerian mean THS is compensated by an270
anti-clockwise circulation due to the eddy-induced velocity of around 5 - 10 Sv (Fig. 7b).271
This compensation occurs across a significant fraction of the water-mass classes covered by272
the global THS cell (-1◦C > θ > 15◦C and 34 g / kg > S > 35 g / kg). We find that, while273
uGM is small across most latitudes, it is large where water flows from one water-mass to274
another, particularly in the cooler fresher water-masses associated with the Southern Ocean.275
As above, the eddy-induced velocity is introduced to account for the effect of transient276
eddies not resolved on the 1.8◦ by 3.6◦ grid. The model does however, resolve fluctuations on277
14
seasonal and inter-annual timescales and hence we do expect a difference between ΨθS and278
ΨθS = ΨEM
θS + ΨGM
θS ; the difference being the role played by resolved temporal fluctuations279
(Ψ′θS, Fig. 7d). The major difference between the two is in the tropical waters (θ > 20◦C)280
and in the region linking the warm salty portion and the cold fresh portion of the global281
cell (θ ≈ 15◦C and S ≈ 35g/kg). The ocean model used here is known to display a robust282
seasonal cycle but small interannual fluctuations. It is thus likely that that in this model,283
and perhaps in the real ocean, seasonal fluctuations that contribute around 5 Sv of exchange284
between warm salty and cold fresh waters.285
5. Diapycnal fluxes and the symmetric diapycnal flux286
theorem287
By way of introduction to the idea of diagnosing the advective heat transport from a288
streamfunction, we again revisit Ferrari and Ferreira (2011) where their latitude-θ stream-289
function Ψy−θ can be used to diagnose the advective meridional heat transport (MHT) using290
MHT (y, θ) =
∫ ∞−∞
ρ0 cp Ψyθ dθ (14)
where ρ0 is a reference density and cp is the heat capacity of sea water. Equation (14)291
represents the flux of heat in the meridional direction due to advection, at the latitude y. In292
the case of the thermohaline streamfunction, ΨθS can be used to determine a heat function293
describing the flux of heat across isohalines and a salt function describing the flux of salt294
across isotherms.295
As potential density is a function of temperature and salinity, a heat function and salt296
15
function can also be defined, describing the flux of heat and salt across isopycnals. The total297
heat and salt transports across an isopycnal are given by298
FHeat(σ) = ρ0 cp
∫ ∞−∞
∂ΨθS
∂θθ dθ|σ + ρ0 cp
∫ ∞−∞
∂ΨθS
∂Sθ dS|σ, (15)
FSalt(σ) =
∫ ∞−∞
∂ΨθS
∂θS dθ|σ +
∫ ∞−∞
∂ΨθS
∂SS dS|σ. (16)
The salt transport can be converted into the more tangible ‘freshwater transport’, FFW ,299
in units of Sverdrups simply using FFW = −FSalt/S. Here we use S = 35 g / kg. The300
diapycnal heat and freshwater transports across potential density surfaces (σ0) from the301
ocean model simulation are shown in Fig. 8.302
For most isopycnal ranges, the ocean transports heat towards denser waters and freshwa-303
ter towards lighter waters. This is consistent with a balance between the ocean circulation304
and the atmosphere; the subtropical waters being associated with a heat flux into the ocean305
and a freshwater flux out of the ocean. The denser polar waters are associated with a heat306
flux out of the ocean and a freshwater flux in. Greatbatch and Zhai (1990) used observed307
meridional heat flux estimates and the ocean hydrography to estimate a global heat-function308
in terms of a diffusivity for temperature. Our analysis suggests that accurate estimates of309
atmosphere-ocean heat and freshwater fluxes could be used to estimate the thermohaline310
streamfunction in a similar way. This would effectively be a combination of the surface311
water-mass analysis method of Walin (1982) extended to 2 tracers and the tracer-contour312
framework of Zika et al. (2010) used to relate interior diabatic pathways to mixing.313
The sign of the diapycnal heat and salt flux is evident geometrically from the thermo-314
haline streamfunction itself. Clockwise cells advect heat and salt toward denser waters and315
16
anticlockwise cells advect heat and salt toward lighter waters (Fig. 9). This symmetry316
between downward heat and freshwater fluxes can be formally stated in the following.317
For most θ, S and pressure ranges of the ocean, density increases with increasing salinity318
and reducing temperature, i.e.319
dσ = −α ∂θ + β ∂S (17)
where α is a positive thermal expansion coefficient and β is a positive haline contraction320
coefficient. Here we will make the sole assumption that α/β is constant for each σ. Applying321
the chain rule to (15) we obtain322
FHeat(σ) = ρ0 cp [ΨθSθ]θ=∞θ=−∞ − ρ0 cp
∫ ∞−∞
ΨθS dθ|σ
+ρ0 cp [ΨθSS]S=∞S=−∞ − ρ0 cp
∫ ∞−∞
ΨθS∂θ
∂SdS|σ. (18)
As ΨθS = 0 outside the maximum and minimum salinities and temperatures323
FHeat(σ) = −ρ0 cp
∫ ∞−∞
ΨθS dθ|σ − ρ0 cp
∫ ∞−∞
ΨθS∂θ
∂SdS|σ. (19)
and likewise for the salt flux324
FSalt(σ) = −∫ ∞−∞
ΨθS dS|σ −∫ ∞−∞
ΨθS∂S
∂θdθ|σ. (20)
Now using (17; where dσ = 0) we find325
FHeat(σ) = −ρ0 cp
∫ ∞−∞
ΨθS dθ|σ − ρ0 cp
∫ ∞−∞
ΨθSβ
αdS|σ. (21)
FSalt(σ) = −∫ ∞−∞
ΨθS dS|σ −∫ ∞−∞
ΨθSα
βdθ|σ. (22)
and hence given we are assuming α/β is constant for constant σ the ratio of diapycnal326
advective heat flux to salt flux is simply327
FHeat(σ)
FSalt(σ)= ρ0 cp
β
α. (23)
17
The significance of (23), hereafter the diapycnal flux symmetry theorem, is that where328
there is a net advective heat flux across isopycnals by the statistically steady flow, salinity329
must be advected also. Hence freshwater must be advected in the opposite direction. So330
where the ocean advects heat toward denser water-masses, it must advect freshwater towards331
lighter water-masses.332
In an ongoing study, we are investigating whether the diapycnal flux symmetry theorem333
has broader implications for global heat and freshwater transports. It may be the case that334
this theorem places constraints on the relationship between atmospheric moisture transport335
and ocean heat transport.336
6. Thermohaline transit time337
Given the thermohaline streamfunction and estimates of the mean gradients of temper-338
ature and salinity, it is possible to gain a measure of the time taken for a water parcel to339
make a full circuit in θ − S space. We name this measure the thermohaline transit time,340
TθS(Ψ). As we discuss in the next Section, individual water parcels, even in the absence of341
dispersion, do not complete entire ‘loops’ in θ−S space, as with any three dimensional flow.342
However, the concept of transit time is still insightful as it gives a measure of the minimum343
time for a water parcel to be advected in this fully Lagrangian co-ordinate system.344
For a fluid parcel following streamline in θ − S space, the time increment, dt|Ψ, for a345
given θ and S increments ∂θ|Ψ and ∂S|Ψ is346
dt|Ψ =Dt
Dθ∂θ|Ψ +
Dt
DS∂S|Ψ. (24)
18
That is, the time taken for a a fluid parcel to get between two points along a streamline347
separated by ∂θ|Ψ and ∂S|Ψ, depends on both the rate of change of temperature following the348
the fluid parcel Dθ/Dt, and the rate of change of salinity following the fluid parcel DS/Dt.349
If θ and S are statistically steady, in order for a fluid parcels to change its temperature and350
salinity, it must be advected to somewhere with a different temperature and salinity such351
that352
Dθ
Dt= u · ∇θ ;
DS
Dt= u · ∇S. (25)
By computing ΨθS from (10) we have naturally computed an average of both components of353
the velocity needed in (25) as (10) can be written354
∆Ψ|S =1
∆t
∫ t+∆t
t
∫ θ+∆θ
θ
u · ∇θ|∇θ| dAS; (26)
∆Ψ|θ =1
∆t
∫ t+∆t
t
∫ S+∆S
S
u · ∇S|∇S| dAθ. (27)
Assuming the absolute gradients |∇θ| and |∇S| do not vary greatly in time or across the355
areas dAS and dAθ respectively, we can make the following approximation:356
∆Ψ|S∆AS
=⟨
u·∇θ|∇θ|
⟩≈ < u · ∇θ >
< |∇θ| > (28)
∆Ψ|θ∆Aθ
=⟨
u·∇S|∇S|
⟩≈ < u · ∇S >
< |∇S| > . (29)
Above, the angular brackets (< >) represent the average in time and over an isotherm357
between S and S+ ∆S and the average over an isohaline between θ and θ+ ∆θ respectively.358
From the time mean θ and S fields of the model, we can compute < |∇θ| > and < |∇S| >359
simply by averaging the gradients across the areas dAS and dAΘ. Thus from (25) we have360
Dθ
Dt≈ ∆Ψ|S
∆AS< |∇θ| >;
DS
Dt≈ ∆Ψ|θ
∆Aθ< |∇S| > . (30)
19
Substituting into (24) then yields361
dt|ψ ≈1
∆Ψ|S∆AS
< |∇θ| >∂θ|Ψ +
1∆Ψ|θ∆Aθ
< |∇S| >∂S|Ψ. (31)
One can then estimate the time rate of change following the ΨθS streamline by integrating362
(31) such that the total thermohaline transit time, TθS(Ψ), is363
TθS(Ψ) =
∮dt|Ψ
≈∮
1∆Ψ|S∆AS
< |∇θ| >∂θ|Ψ +
∮1
∆Ψ|θ∆Aθ
< |∇S| >∂S|Ψ. (32)
We calculate TθS by following the longest -4 Sv streamline in θ− S space (Fig. 10). The364
full circuit takes 1,350 years using (32). The cumulative time taken along the contour shows365
how the fluid moves from the warmest freshest waters around 25◦ C and 34.4 g / kg to the366
cool high saline waters at 10◦ C and 35.5 g / kg in a relatively short time-scale of 100 years367
or so. This short time-scale is likely to be because of the strong, near surface wind driven368
circulation allowing rapid transitions between different water-masses. The transit from this369
water-mass (NADW) to the cold fresh waters around 0◦ C and 34.5 g / kg, a relative short370
distance in θ − S space, takes the considerable majority of the remaining 1250 year transit.371
These dense water-masses are found only in the deep ocean where advection is weak and372
temperature and salinity gradients are large meaning water-mass transitions occur gradually.373
The thermohaline transit time of THS contours are shown in Fig. 11. In each case we374
take the longest streamline for a given transport value. The tropical cell, defined as anti-375
clockwise cells warmer than 10oC, has the shortest transit times of order 20 years. For the376
global cell and it’s subcells, the transit times range from hundreds of years and asymptote377
20
to around 1500 years for the -2 Sv contour. We do not discern between subcells in the378
warmer or colder portions of the global cell. The longest bottom water cell transit times are379
equivalent to global cell at order 1500 years. Increased resolution of the THS in θ− S space380
may change the bottom water transit times estimated here.381
7. Basin specific flow and water-mass pathways382
Here we address whether streamlines in thermohaline space correspond to pathways of383
individual water parcels or whether this circulation is simply the accumulation of many,384
geographically independent cells. To answer this question we start by investigating the385
nature of the THS in separate geographical regions where such a separation is possible.386
The North Atlantic is generally saltier than the Indian and Pacific basins. However the387
two share waters of like θ−S properties. Looking at the maximum temperatures and salinities388
along the sections which separate the Atlantic from the Indo-Pacific, we can determine θ−S389
ranges which are either ‘wholly’ within the Atlantic or ‘wholly’ within the Indo-Pacific, and390
of course, not in the Southern Ocean. That is, although like θ − S properties may exist in391
both basins, they are not geographically linked at any time. Any volume transport from392
Atlantic water to Indo-Pacific water must occur through some other intermediary θ − S393
range.394
The θ − S ranges we choose for this strictly ’non Southern Oceanwater are S > 35g/kg,395
S > 34.8g/kg and θ > 21.5oC or S < 34.8g/kg and θ > 16oC (i.e. only those regions in θ−S396
space which have green or red contours in Fig. 12) . Any water in that range is either in the397
Atlantic, or in the Indo-Pacific, but not in the Southern Ocean. Hence, a streamfunction can398
21
be defined independently by integrating form θ =∞ back to the edge of this domain in θ−S399
space. The result and the geographical domain of the two water-masses is shown in Fig. 12.400
The streamlines for the Indo-Pacific are shown in green and the Atlantic in red. Clearly the401
wide, anvil shape of the global THS is the result of a combination of a fresher Indo-Pacific402
branch and a saltier Atlantic branch. The warmest saltiest waters only exist in the Atlantic403
and the coldest freshest waters only exist in the the Indo-Pacific. The Indo-Pacific takes404
freshwater from the deep or southern waters and returns them as saltier waters while the405
Atlantic takes intermediate salinity waters and returns them as high salinity waters.406
The same decomposition as is done for the warmest waters by ocean basin can be done407
for the coldest freshest waters by hemisphere. Cold fresh water exists in both the Southern408
Ocean and the Arctic/North Pacific. However, no cold fresh water exists around the Equator.409
Hence we can partition the two. Specifically, we compute a streamfunction only for waters410
with θ < 12oC and S < 34.5g/kg for each hemisphere. The resulting Southern Ocean411
streamfunction is almost the same as that derived from the sum of both the Northern412
and Southern Hemispheres, demonstrating the dominant role of the Southern Ocean in the413
circulation in this θ − S range.414
An interpretation of the global water-mass pathway thus emerges of water moving from415
the Atlantic to the Southern Ocean and into the Pacific. We now attempt to track this416
pathway more precisely by following a streamline from one basin to the next. We take the417
THS plotted in Fig. 12 and follow the θ − S values corresponding to the longest -4 Sv418
streamline. We start with the freshest point in the Atlantic on this contour. This contour is419
tracked until the edge of the Atlantic water-mass is reached at S = 35g/kg. At this point the420
Atlantic streamfunction matches the global one. Continuing along the global streamline to421
22
S = 34.5g/kg the global THS matches almost perfectly with the Southern Ocean and again422
to θ = 12◦C where it matches with the Pacific. Although one could continue indefinitely, we423
stop where this streamline reaches θ = 21.5oC and S > 34.5g/kg. By allowing streamlines424
to transit from one basin to the next, the circulation is no longer closed and no one cell425
exists in a unique sense.426
In order to determine the possible geographical route of water parcels along this ther-427
mohaline pathway we colour water-masses in the ocean according to their location in θ − S428
space. For each point along the contour, a colour is chosen corresponding to the ‘distance’429
along the contour; distance being in θ − S space, not geographical space. The colour is430
then used to label the actual points in the ocean which have that temperature and salinity.431
The resulting analysis is shown in Fig. 13. Following the pathway as it moves from the432
surface Atlantic to the North Atlantic, then down into the deep ocean, and eventually up433
into the Southern Ocean and out into the North Pacific, reveals a pathway reminiscent of434
the schematic diagrams of the thermohaline circulation, first depicted by Gordon (1986).435
An important difference between the conveyor of Gordon (1986) and Broecker (1991)436
and that revealed by Fig. 13 is that the model diagnosed THS transits through the surface437
waters of the Southern Ocean with significant transformation of θ − S properties. This438
is despite the meridional overturning in density space showing North Atlantic Deep Water439
upwelling diapycnally at mid-latitudes (Fig. 2). This suggests that waters are exchanged440
meridionally at constant density, but enter and exit with different temperatures and salinities441
(i.e. different spiciness).442
23
8. Summary and conclusions443
The circulation of a global ocean model has been presented in temperature salinity coor-444
dinates for the first time. The resulting thermohaline streamfunction, the THS, distils the445
ocean circulation into three distinct cells: a tropical warm cell, a global cell and an Antarctic446
Bottom Water cell. Many of the streamlines (around 4-5 Sv) of the global cell link the cold447
and salty waters of the North Atlantic to the warm fresh waters of the Indo-Pacific Ocean,448
providing a quantification of the models ‘global conveyor’. In addition, a number of novel449
advantages of computing a thermohaline streamfunction are revealed.450
First, the THS can be used to diagnose the role of resolved and parameterised transient451
fluxes in water-mass transformation zones. Parameterised transient eddy fluxes are found452
to be leading order across the majority of water-mass classes. Averaging in thermohaline453
space provides a unique Lagrangian framework in which the role of diabatic processes and454
parameterised fluxes are highlighted.455
Second, once the circulation is averaged in thermohaline coordinates, diapycnal heat and456
salt transports are easily quantified. Clockwise cells in thermohaline space flux heat and salt457
towards denser waters and anti-clockwise cells flux heat and salt towards lighter waters. This458
symmetry between diapycnal heat and salt fluxes is formalised with the symmetric diapycnal459
flux theorem (23). In our model, heat and salt are transported from light waters to dense460
waters by the clockwise global cell. Heat and salt are transported from light waters associated461
with sub-tropical latitudes to denser waters associated with polar regions. This poleward462
salt transport is in symmetry with the atmosphere’s poleward transport of freshwater at463
subtropical to subpolar latitudes.464
24
A measure of the transit time of the global conveyor, the thermohaline transit time, is465
easily quantified from the THS. The thermohaline transit time in our model is 1350 years for466
typical streamlines associated with the the conveyor. Such a simple diagnostic may prove467
useful in comparing the ventilation time of various models, especially in cases where the468
incorporation of more involved tracer diagnostics is prohibitive.469
Separating the THS by region, the Southern Ocean is found to dominate the coolest and470
freshest water-masses. The THS trajectory from cool fresh waters to cool and salty waters is471
the sum of both a fresher tropical/subtropical branch in the Indian and Pacific Oceans and472
a saltier tropical to subpolar branch in the Atlantic Ocean. The sum of these three branches473
reveals a globally interconnected circulation.474
A major difference between the global conveyor of this model and the schematic view of475
Gordon (1986) and Broecker (1991) is that most of the diagnosed THS (-5 to -15 Sv; narrow476
core of the THS in Fig. 6) occurs at the surface rather than in deep waters. In addition477
almost all streamlines (0 to -15 Sv) transit through the Southern Ocean. This is despite478
the majority of diapycnal upwelling in the model occurring in the abyssal ocean (Fig. 2).479
This implies that even a density space overturning, driven by interior diapycnal mixing, may480
still transit through the Southern Ocean in order for global balances of heat and salt to be481
maintained.482
9. Acknowledgements483
This work was supported by the Australian Research Council. We thank Steven Griffies484
for providing helpful comments on a draft of this manuscript.485
25
486
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30
List of Figures568
1 Mean barotropic streamfunction (Ψxy; Sv) from the ocean component of the569
final 10 year average of a 3000 year run of a climate model (described in570
Section 3). The 20 Sv (blue) and -20 Sv (red) contours are shown. Ψxy571
is computed by integrating northward from the Antarctic, the value at the572
African continent is then subtracted. 35573
2 Top: Mean depth-latitude streamfunction or MOC (Ψzy; Sv) from from ocean574
component of a climate model (described in Section 3). Bottom: Density-575
latitude streamfunction (Sv). The 4 Sv (red) and -4 Sv (blue) streamlines are576
shown with solid contours. 36577
3 Schematic describing four streamfunction diagnostics: a) barotropic stream-578
funciton showing horizontal motion, b) meridional overturning streamfunction579
showing vertical motion, c) temperature-latitude streamfunction showing net580
effect of horizontal motion in transporting heat meridionally and d) thermo-581
haline streamfunction showing net effect of all motion in transporting water582
from regions of different temperatures and salinities. Overlaid in a), b) and583
d) are schematic isolines of constant potential temperature and salinity (at584
the surface in a) and zonally averaged in b). 37585
31
4 Schematic describing the way the thermohaline streamfunction is computed586
from the output of a finite volume ocean model. Model grid boxes are defined587
on some native coordinate (e.g. x and y, or y and z) by their salinity (a)588
and potential temperature (b). Where potential temperature θ′ < θ all fluxes589
are summed across grid box interfaces from salinity S ′ < S to S ′ > S in the590
native coordinate (c). Summing these fluxes, a thermohaline streamfunction591
at (θ, S) is determined and represented in thermohaline coordinates (d). 38592
5 Scatter plot of potential temperature and salinity from the mean state of the593
ocean model. Each grid box is represented by one dot in θ − S space. The594
colour of each dot corresponds to its latitude and basin using the colour code595
shown in the insert. 39596
6 The thermohaline streamfunction derived from 10 years of monthly, u, v ,597
w, θ and S fields of the ocean model. Positive values are anti-clockwise and598
negative values are clockwise. The 4 Sv streamline is shown in red and the -4599
Sv streamline in blue. 40600
7 Top left: THS from the sum of the time mean Euleran velocity and the time601
mean eddy-induced velocity (uEM + uGM). Top right: THS from the time602
mean Euleran velocity. Bottom left: THS from the time mean eddy-induced603
velocity. Bottom right: Difference between total THS and THS from the604
sum of the time mean Eulerian and eddy-induced velocities. That is, the605
contribution to the THS due to resolved fluctuations such as the seasonal cycle. 41606
32
8 a) Thermohaline Streamfunction (THS, ΨθS) as in Fig 6 but with contours607
of surface referenced potential density (σ0) overlaid. b) Diapycnal heat and608
c) diapycnal freshwater flux from the ocean model computed directly from609
the THS. The freshwater flux is simply computed as Fsalt/35g/kg. The total610
(solid), Eulerian (dashed) and eddy-induced (dotted) components are shown.611
Consistent with the symmetric diapycnal flux theorem (23), a positive heat612
flux corresponds to a negative freshwater flux throughout. 42613
9 Schematic describing the geometric relationship between the THS and diapy-614
cnal heat and salt fluxes. If water is advected toward denser waters on the615
warmer-saltier parts of an isopycnal surface (σ; left) this appears as a clock-616
wise circulation in θ − S space (right). As the same trajectory has different617
temperatures as it crosses the same isopycnal surface it represents a net pos-618
itive diapycnal transport of heat and salt. The circulation shown in physical619
space (left) can be both a horizontal (e.g. gyre in x-y space) circulation where620
this diapycnal transport is balanced by surface heating, cooling and evapo-621
ration minus precipitation and/or a vertical overturning (in z-y space) where622
the transport is balanced by interior mixing or nonlinear processes. 43623
10 The accumulated thermohaline transit of the -4Sv streamline. The colour of624
each dot represents the relative time taken along the streamline estimated625
using only the thermohaline streamfunction itself and the mean θ−S gradients. 44626
33
11 The thermohaline transit time of the longest THS streamlines in θ − S space627
(i.e the longest streamlines in Fig. 6 for each transport). Shown are transit628
times for the global cell (green), the bottom water cell (temperatures below629
10oC), and the tropical cell contours (longest positive streamlines with tem-630
peratures above 10oC). The Antarctic Bottom Water transit times may be631
underestimated due to interpolation onto a coarse θ − S grid. 45632
12 The thermohaline streamfunction (THS) for various oceanic regions. Shown633
are: THS for water-masses found only in the Indian and Pacific basins (green;634
green volume in insert), water-masses found only in the Atlantic (red) and635
finally water-masses found only in the Southern Ocean (blue). 46636
13 A thermohaline pathway projected into physical space. The -4 Sv THS con-637
tour is followed from the Atlantic water-masses (red in Fig. 12) to the South-638
ern Ocean (blue in Fig. 12) and then finally to the Indo-Pacific water-masses639
(bottom right panel). At points equally spaced in θ − S space along the640
contour, a colour is chosen to represent water-masses with those water-mass641
properties (bottom right panel). All grid boxes in the corresponding region642
with potential temperatures and salinities within 0.1◦C and 0.05 g / kg re-643
spectively of that point in θ − S space are then assigned the colour of that644
point. Top panel shows all grid boxes above 300m. Middle panel: all grid645
boxes below 300m. Bottom left panel: only Altantic grid boxes. Bottom646
middle panel: Indo-Pacifc grid boxes. 47647
34
130120110100908070605040302010
0102030405060708090100110120130
Longitude
Latit
ude
Barotropic Streamfunction (Sv)
50 100 150 200 250 300 35080
60
40
20
0
20
40
60
80
Fig. 1. Mean barotropic streamfunction (Ψxy; Sv) from the ocean component of the final10 year average of a 3000 year run of a climate model (described in Section 3). The 20 Sv(blue) and -20 Sv (red) contours are shown. Ψxy is computed by integrating northward fromthe Antarctic, the value at the African continent is then subtracted.
35
Dep
th (m
)
Depth Latitude Streamfunction (Sv)
80 60 40 20 0 20 40 60 80
0
1000
2000
3000
4000
5000
6000
252321191715131197531
135791113151719212325
Latitude
Pote
ntia
l Den
sity
(2 ,
kg m
3 )
Density Latitude Streamfunction (Sv)
80 60 40 20 0 20 40 60 80
29
30
31
32
33
34
35
36
37
Fig. 2. Top: Mean depth-latitude streamfunction or MOC (Ψzy; Sv) from from ocean com-ponent of a climate model (described in Section 3). Bottom: Density-latitude streamfunction(Sv). The 4 Sv (red) and -4 Sv (blue) streamlines are shown with solid contours.
36
Fig. 3. Schematic describing four streamfunction diagnostics: a) barotropic streamfunci-ton showing horizontal motion, b) meridional overturning streamfunction showing verticalmotion, c) temperature-latitude streamfunction showing net effect of horizontal motion intransporting heat meridionally and d) thermohaline streamfunction showing net effect of allmotion in transporting water from regions of different temperatures and salinities. Overlaidin a), b) and d) are schematic isolines of constant potential temperature and salinity (at thesurface in a) and zonally averaged in b).
37
Fig. 4. Schematic describing the way the thermohaline streamfunction is computed fromthe output of a finite volume ocean model. Model grid boxes are defined on some nativecoordinate (e.g. x and y, or y and z) by their salinity (a) and potential temperature (b).Where potential temperature θ′ < θ all fluxes are summed across grid box interfaces fromsalinity S ′ < S to S ′ > S in the native coordinate (c). Summing these fluxes, a thermohalinestreamfunction at (θ, S) is determined and represented in thermohaline coordinates (d).
38
Fig. 5. Scatter plot of potential temperature and salinity from the mean state of the oceanmodel. Each grid box is represented by one dot in θ − S space. The colour of each dotcorresponds to its latitude and basin using the colour code shown in the insert.
39
252321191715131197531
135791113151719212325
Salinity (g/kg)
Pote
ntia
l Tem
pera
ture
(o C)
10 year mean Thermohaline Streamfunction (Sv)
33.5 34 34.5 35 35.5 36 36.5
0
5
10
15
20
25
30
Fig. 6. The thermohaline streamfunction derived from 10 years of monthly, u, v , w, θand S fields of the ocean model. Positive values are anti-clockwise and negative values areclockwise. The 4 Sv streamline is shown in red and the -4 Sv streamline in blue.
40
15
15
5
5
5
5
55
5
5 55
5
Pote
ntia
l Tem
pera
ture
(o C)
THS from u (!! S, Sv)
15
15
5
5
5
5
55
5
5 55
533.5 34 34.5 35 35.5 36 36.5
0
5
10
15
20
25
30
25
15
1515
5
55
5
5
5
5
5 5
5
THS from uEM (!EM
! S , Sv)
25
15
1515
5
55
5
5
5
5
5 5
533.5 34 34.5 35 35.5 36 36.5
0
5
10
15
20
25
30
5
5
5
Salinity (g / kg)
Pote
ntia
l Tem
pera
ture
(o C)
THS from uGM (!GM
!S , Sv)
5
5
5
33.5 34 34.5 35 35.5 36 36.5
0
5
10
15
20
25
30
39373533312927252321191715131197531
13579111315171921232527293133353739
5
55
5
5
5
55
Salinity (g / kg)
THS from u ! u (!!! S, Sv)
5
55
5
5
5
55
33.5 34 34.5 35 35.5 36 36.5
0
5
10
15
20
25
30
Fig. 7. Top left: THS from the sum of the time mean Euleran velocity and the time meaneddy-induced velocity (uEM + uGM). Top right: THS from the time mean Euleran velocity.Bottom left: THS from the time mean eddy-induced velocity. Bottom right: Differencebetween total THS and THS from the sum of the time mean Eulerian and eddy-inducedvelocities. That is, the contribution to the THS due to resolved fluctuations such as theseasonal cycle.
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Fig. 8. a) Thermohaline Streamfunction (THS, ΨθS) as in Fig 6 but with contours of surfacereferenced potential density (σ0) overlaid. b) Diapycnal heat and c) diapycnal freshwaterflux from the ocean model computed directly from the THS. The freshwater flux is simplycomputed as Fsalt/35g/kg. The total (solid), Eulerian (dashed) and eddy-induced (dot-ted) components are shown. Consistent with the symmetric diapycnal flux theorem (23), apositive heat flux corresponds to a negative freshwater flux throughout.
42
Fig. 9. Schematic describing the geometric relationship between the THS and diapycnalheat and salt fluxes. If water is advected toward denser waters on the warmer-saltier partsof an isopycnal surface (σ; left) this appears as a clockwise circulation in θ−S space (right).As the same trajectory has different temperatures as it crosses the same isopycnal surfaceit represents a net positive diapycnal transport of heat and salt. The circulation shown inphysical space (left) can be both a horizontal (e.g. gyre in x-y space) circulation where thisdiapycnal transport is balanced by surface heating, cooling and evaporation minus precipita-tion and/or a vertical overturning (in z-y space) where the transport is balanced by interiormixing or nonlinear processes.
43
33.5 34 34.5 35 35.5 365
0
5
10
15
20
25
30 Cumulative Thermohaline Transit Time (years)
Salinity (g/kg)
Pote
ntia
l Tem
pera
ture
(o C)
200
400
600
800
1000
1200
Fig. 10. The accumulated thermohaline transit of the -4Sv streamline. The colour ofeach dot represents the relative time taken along the streamline estimated using only thethermohaline streamfunction itself and the mean θ − S gradients.
44
25 20 15 10 5 0 5 10 15 200
500
1000
1500Thermohal ine Transi t Time (T! S)
Thermohaline Streamline (Sv)
Year
s
Global CellBottom Water Cell
0 5 10 15 200
10
20
30
Thermohaline Streamline (Sv)Ye
ars
Tropical Cell
Fig. 11. The thermohaline transit time of the longest THS streamlines in θ − S space (i.ethe longest streamlines in Fig. 6 for each transport). Shown are transit times for the globalcell (green), the bottom water cell (temperatures below 10oC), and the tropical cell contours(longest positive streamlines with temperatures above 10oC). The Antarctic Bottom Watertransit times may be underestimated due to interpolation onto a coarse θ − S grid.
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Fig. 12. The thermohaline streamfunction (THS) for various oceanic regions. Shown are:THS for water-masses found only in the Indian and Pacific basins (green; green volume ininsert), water-masses found only in the Atlantic (red) and finally water-masses found onlyin the Southern Ocean (blue).
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Fig. 13. A thermohaline pathway projected into physical space. The -4 Sv THS contouris followed from the Atlantic water-masses (red in Fig. 12) to the Southern Ocean (blue inFig. 12) and then finally to the Indo-Pacific water-masses (bottom right panel). At pointsequally spaced in θ−S space along the contour, a colour is chosen to represent water-masseswith those water-mass properties (bottom right panel). All grid boxes in the correspondingregion with potential temperatures and salinities within 0.1◦C and 0.05 g / kg respectivelyof that point in θ − S space are then assigned the colour of that point. Top panel shows allgrid boxes above 300m. Middle panel: all grid boxes below 300m. Bottom left panel: onlyAltantic grid boxes. Bottom middle panel: Indo-Pacifc grid boxes.
47