Lecture 23. Hydrological cycle and haline circulation
5/9/2006 12:07 PM
1. Dynamics role of hydrological cycle in climate
Poleward heat flux in the climate system has been one of the main focuses of climate study.
A study by Trenberth and Caron (2001) showed that the atmospheric component of the poleward
heat flux dominates the total poleward heat flux at high latitudes. As stated by the authors in the
abstract, "At 35o latitude, at which the peak total poleward transport in each hemisphere occurs, the
atmosphere transport accounts for 78% of the total in the Northern Hemisphere and 92% in the
Southern Hemisphere. In general, a much greater portion of the required poleward transport is
contributed by the atmosphere than the ocean, as compared with previous estimates." The oceans
seem unimportant for high-latitude climate, in terms of carrying the heat flux much needed for
warming up high latitudes. Is that true? In particular, the Southern Hemisphere is covered primarily
by the oceans, especially between 35o S and 70o S . Therefore, it seems incomprehensible how can
the oceans play such a minor role in poleward heat flux.
A. Definition of poleward heat flux
Oceanic heat flux has been discussed in many papers and textbooks. As both instruments
and numerical models improved, poleward heat fluxes are better diagnosed. For the most update
information the readers are referred to the review by Bryden and Imawaki (2001).
The main point is that although heat flux data may be further improved, there seems a
fundamental problem in the definition of poleward heat fluxes that gives rise to misconception and
similar mistakes have been made by many people. From thermodynamics, it is well known that
heat flux cannot be well defined for a system that has net mass gain (or loss) through the lateral
boundary of the system. For example, Trenberth and Caron (2001) made a point that the poleward
heat fluxes in the South Pacific and Indian are not well-defined because the existence of the
Indonesian Throughflow.
The same caution should apply to both the oceans and atmosphere because both the ocean
and atmosphere are not closed systems in terms of mass -- there is water exchange between them.
In fact, the hydrological cycle in forms of evaporation and precipitation is an essential ingredient
for the heat flux in atmosphere -- the latent heat flux. The mass flux associated with evaporation
and precipitation has been traditionally ignored in heat flux calculation by oceanographers. This
mass flux is not accounted for in many previous calculations because the follows.
First, such a small mass flux is rather hard to be identified from the classical dynamic
calculation based on either a reference level or other means. Second, most oceanic general
circulation models have been based on the Boussinesq Approximations, thus, the dynamic effects
associated with evaporation and precipitation is simulated in terms of the virtual salt flux condition
on the upper surface; while the mass flux associated with evaporation and precipitation is totally
ignored. As the oceanic general circulation models improved, the new generation of models is
formulated under the natural boundary condition (Huang, 1993). As a result, the mass flux
associated with evaporation and precipitation is exactly accounted for. A more accurate definition
of heat flux is, thus, necessary. A simple definition of heat flux is defined in the Appendix, which
can be used for an oceanic section with a net mass flux exchange.
Strictly speaking, the poleward heat flux in the climate system should be defined as three
terms, as shown in Fig. 1:
1) Sensible heat flux in the oceans;
2) Sensible heat flux in the atmosphere;
3) Latent heat flux in the atmosphere-ocean coupled system.
1
Over the latitudinal range from the equator to pole there is a continuous exchange of heat
between these three loops.
Fig. 1. Sketch of the atmosphere-ocean coupled climate system.
As an example, we calculate the freshwater flux through the air-sea interface. We use the
evaporation and precipitation rate over the world oceans by Da Silva et al. (1994), Fig. 2. This
dataset has taken into consideration of river run-off, so the globally integrated evaporation and
precipitation rates are balanced. Near the equator precipitation dominates, especially between 0o N
to 10o N . However, evaporation dominates over the subtropics and thus produces a net water vapor
flux that is transported by the atmosphere to high latitudes. At high latitudes (roughly 40o off the
equator) precipitation dominates.
2
Fig. 2. Water vapor source and sink due to evaporation and precipitation, integrated zonally, from
Da Silva et al. (1994)
However, this dataset seems to give rise a poleward freshwater flux that is too large in the
southern hemisphere. As an alternative we use the water vapor flux calculated from atmospheric
circulation models, Gaffen et al. (1997). The freshwater flux reported in their study is based on
average of 25 atmospheric general circulation models. This poleward water vapor flux M w is the
major mechanism of poleward heat flux in the climate system, Fig. 3. The poleward heat flux
associated with the water vapor cycle is closely related with the latent heat content of water vapor
H f = ( Lh − hw ) M w
(1)
where Lh = 2500 J / g is the latent heat content for water vapor, hw is the enthalpy of the return
water in the ocean, which is much smaller than L Lh and neglected in the calculation. The heat flux
associated with the water vapor flux should be attributed to the ocean-atmosphere-land coupled
system.
However, the mass flux associated with the water vapor flux is two orders of magnitude
smaller than other mass fluxes in the climate system (Table 1). This is one of the reasons why this
mass flux has been overlooked in many papers discussing climate. On the other hand, this
seemingly tiny mass flux is associated with a large heat flux. One kilogram of water vapor can
deliver 2.5 × 106J of heat. When one kilogram of water is cooled down by 10oC, the heat released is
4.18 × 104J. Thus, water vapor is 60 times more efficient than the water in transporting heat.
Here we use Sverdrup as a unit for mass flux. Sverdrup has been widely used as a volume
flux unit in oceanography, and 1Sv=106m3s-1. Since sea water density is very close to 103kg/m3, as
a mass flux unit we have 1Sv=109kg/s. As shown in Table 1, Sverdrup is a very convenient unit for
climate system because mass fluxes in both the oceans and atmosphere have the same order of
magnitude, although the density of water is one thousand times larger than air (Held, personal
communication, 2001). The mass flux in the Jet Stream is inferred from the angular momentum.
According to Peixoto and Oort (1992), the maximum angular momentum in the North Hemisphere
is about 9.6 × 1025kgm2s-1. Assuming the mean latitude is at 30oN, this gives a rough estimate of
500 Sv for the mass flux associated with the westerlies in the Northern Hemisphere.
Current
Systems
Mass flux
(Sv)
Gulf Stream Kuroshio ACC
150
130
134
Meri. Cell in
atmosphere
200
Jet Steam
~500
Poleward
water vapor
1
Table 1. Mass transport maxima in the climate system. The source of the values: Gulf Stream,
Hogg and Johns (1995); Kuroshio, Wijffels et al. (1998); ACC (Antarctic Circumpolar Current),
Rintoul et al. (1998); meridional overturning rate and the Jet Stream in the atmosphere, Peixoto and
Oort (1992); poleward water vapor flux, Gaffen et al. (1997).
3
Fig. 3. Northward water vapor flux (Sv) and associated latent heat flux (PW) due to evaporation
and precipitation, as calculated from atmospheric general circulation models.
In the Northern the water vapor flux due to oceanic process peaks at 40o N , with a volume
flux of 0.84Sv and the corresponding poleward heat flux of 2.1PW. In the Southern Hemisphere,
the poleward water vapor flux reaches the maximum at 40o S , with a volume flux of 1.05Sv and
the corresponding poleward heat of 2.64PW. It is clear that the latent heat flux associated with the
evaporation and precipitation over the world oceans consists of a substantial portion of the
poleward heat flux in the current climate system. From the hydrological cycle alone, we can
estimate the poleward heat flux associated with latent heat, and this is a major component of the
poleward heat flux, e.g., Schmitt (1995). Such a large poleward heat flux is, of course, intimately
connected to the general circulation in the oceans. In the events of climate change, surface
temperature and salinity change will certainly affect the latent heat flux, and thus the entire climate
system.
After introducing the poleward heat flux associated with water vapor cycle, the poleward
heat fluxes in the current climate system are shown in Fig. 4. The sum of these three fluxes is the
total poleward heat flux diagnosed from the satellite measurement. The oceanic sensible heat flux is
the oceanic heat flux from the NCEP data (Trenberth and Caron, 2001). Although the atmospheric
sensible heat flux is still the largest component in the both hemisphere, it is not much larger than
the other components.
4
Fig. 4. Northward heat flux in PW; RT indicates the heat flux required for radiation balance as
diagnosed from satellite data, AT indicates the atmospheric sensible heat flux, OT indicates the
oceanic sensible heat flux, and LT indicates the latent heat flux associated with the hydrological
cycle in the atmosphere-ocean coupled system.
B. Heat flux divergence
Caution must be taken in the interpretation of the poleward heat flux. Heat flux itself may
not be very important -- what really matters is the divergence of the heat flux. For example, at
65o S atmospheric sensible heat flux is stronger than the latent heat flux. However, a close
examination reveals that the latent heat flux divergence is much stronger than that of the
atmospheric sensible heat flux, Fig. 5. Similarly, in the Northern Hemisphere oceanic sensible heat
flux divergence is the dominating source of heat south of 40o N , north of this latitude the
divergence of latent heat flux becomes more important, and north of 50o N the atmospheric
sensible heat flux dominates.
5
Fig. 5. Heat flux divergence. Atmos indicates the atmospheric sensible heat flux divergence, Ocean
indicates the oceanic sensible heat flux divergence, and Water Vapor indicates the latent heat flux
divergence associated with the hydrological cycle in the atmosphere-ocean coupled system.
In both hemispheres, poleward heat flux is carried by three components and it works like a
relay team. In subtropics oceanic sensible heat flux is the dominating contributor to the poleward
heat flux divergence, and in mid latitudes the latent heat flux divergence is the dominating
contributor. Finally, in high latitudes the atmospheric sensible heat flux divergence dominates, Fig.
5. The situation in the Southern Hemisphere is an excellent example demonstrating the roles of
these three components. The most noticeable term is the latent heat flux divergence that dominates
over mid latitudes. The atmospheric sensible heat flux divergence plays major role south of 70o S ,
where the Southern Ocean meets the continent.
The heat flux balance for a given water column is
Short wave radiation + divergence of oceanic sensible heat flux =
latent heat + long waves radiation + sensible heat
It is important to note that divergence of the oceanic sensible heat flux is a relatively small term in
the oceanic heat flux balance at any given location, Fig. 6. In comparison, latent heat released to the
atmosphere is much larger than the divergence of the oceanic sensible heat flux at any given
location. The latent heat flux released to the atmosphere is, thus, a continuous source of heat for the
atmospheric circulation. Although the atmosphere may play an important role in sending heat to
higher latitudes, without the continuous heat support from the oceans, the warm climate we enjoyed
now may not be realized at all.
6
Fig. 6. Heat fluxes through from the ocean to the atmosphere, integrated zonally, and the
divergence of the oceanic sensible heat flux, labeled as Ocean Current.
C. The hydrological cycle
In the climate system, the solar radiation is primarily absorbed by the ocean first. The
atmosphere is primarily heated through evaporation and latent heat release in the atmosphere, plus
the so-called greenhouse gases absorption of heat in the atmosphere itself. As shown in Fig. 4, the
amount of latent heat flux from the ocean to the atmosphere is huge. The total amount of
evaporation is 12.45Sv, so the latent heat associated with evaporation is about 31.1 PW. As the
oceanic circulation changes, the latent heat flux will be substantially modified; thus, we should
study how this is going to happen.
In previous studies, the important role of the atmosphere-ocean hydrological cycle
associated with evaporation and precipitation has been overlook in oceanographic literature. In
oceanic heat flux calculations, the poleward heat flux associated with the hydrological cycle has
been neglected, and it was incorrectly attributed to the atmosphere alone. Thus, what most
oceanographers calculated in the past should be called oceanic sensible heat flux.
On the other hand, meteorologists have claimed the poleward heat flux associated with the
hydrological cycle as entirely due to the atmospheric processes, and it has very little connection
with the ocean circulation. Such an entire separation of heat fluxes is inconsistent with the physics.
Without the oceanic currents to bring heat northward, there would be no high surface temperature
in the Gulf Stream (or Kuroshio) region. As a result, there would be no strong latent heat release in
these areas.
It is well known that the latent heat flux is the dominating form of poleward heat flux in the
atmospheric circulation; however, this component has long been thought of primarily an
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atmospheric process. The reason of such a misconception is, at least partially, due to the seemingly
small amount of mass associated with this water vapor flux. The global integrated meridional water
vapor flux is on the order of 109 kg / s , which is much smaller than the mass flux of other
components of the oceanic general circulation and the atmospheric circulation. Thus, the return
branch of the global hydrological cycle, which takes place in the oceans, has been overlook. This
misconception is widely spread in many documents, papers, and books about climate.
First, although the main branch of the global hydrological cycle is through the oceans, many
existing scientific programs related to hydrological cycle completely ignore the oceanic component
of the cycle.
Second, the thermohaline circulation in many existing climate models is not realistically
simulated. For example, the oceanic component in many climate models is represented in terms of
the so-called swap ocean, i.e., a very shallow layer of water (50 m or slightly deeper) with no
current.
In such models the atmosphere takes up water vapor at low latitudes and transports it to
high latitudes. After releasing the latent heat from the water vapor, the water has been discarded
from the model, and not too many people thought about dynamic consequence of this seemingly
small amount of water flux. Obviously, an ocean without current would be unable to transport the
fresh water from high latitudes, where precipitation prevails, back to lower latitudes, where the
evaporation prevails. Thus, without the oceanic currents the subtropical ocean will eventually dry
up, with all water piles up at high latitudes -- an equilibrium state impossible to reach. It is very
unlikely that when a quasi-steady state is reached such models can reproduce the strong
thermohaline circulation in the oceans. Thus, the sea surface conditions, such as surface
temperature, surface salinity, and air-sea heat fluxes, from such a model may be quite different
from the current climate.
Many of the existing climate models have an oceanic component based on the so-called
Boussinesq approximations. In such models, the ocean is treated as an incompressible fluid
environment, and the salinity balance in the model is controlled by the so-called virtual salt flux
condition at the sea surface, or the traditional technique of requiring the surface salinity relaxation
to the climatological mean surface salinity. Changes in the water vapor cycle, thus, in such models
have no dynamic consequence at all.
As the oceanic general circulation models improved, the dynamic role of freshwater will be
simulated more realistically. Among many other features, evaporation and precipitation can drive
the so-called Goldsbrough--Stommel circulation (Huang and Schmitt, 1993) that is a barotropic
circulation with a horizontal circulation on the order of a Sverdrup. This circulation is much smaller
than the wind-driven or thermohaline circulation, so that it has been neglected from most
discussion about ocean circulation and climate. However, the seemingly small freshwater flux
associated with evaporation and precipitation is responsible for driving the haline circulation in the
oceans. In fact, evaporation and precipitation alone (without wind stress and heating) can drive a
three-dimensional baroclinic circulation, two orders of magnitudes stronger than that of the
barotropic Goldsbrough--Stommel circulation, i.e., the same order of strength as the wind-driven
circulation or the thermally-driven circulation (Huang, 1993).
Thus, evaporation and precipitation is one of the major forces in driving the oceanic general
circulation. As the climate system shifts to a new state, the freshwater flux through the oceanatmosphere system will change in response. As a result, the thermohaline circulation will be in a
different state, and the sea surface temperature distribution will be different from the present-day
distribution. Consequently, the surface air-sea heat fluxes shown in Fig. 6 can be dramatically
8
different. A comprehensive understanding will require much effort toward the ocean-atmosphere
coupled system.
D. The implication of a swap ocean of no current:
The role of the oceans in the ocean-atmosphere coupled system is illustrated in Fig. 7. In the
current climate, there are three components in this climate heat conveyor, and they work together
like a relay team. Although the atmosphere is responsible for sending heat to high latitudes, it might
be a misleading statement in saying the atmosphere heat transport alone is responsible for the high
latitudes poleward heat flux.
If there were no water vapor cycle that is closely related to the thermohaline circulation in
the oceans, there would be no strong latent heat flux in the climate system. Thus, the whole Earth
would be cooled down substantially. Since there is no evaporation and precipitation, there will be
no strong salinity gradient in the oceans. Due to less heat flux transport from low latitudes,
temperature at high latitudes should be lower than at current climate condition. Due to the strongerthan-current meridional temperature difference and almost zero salinity difference, the oceanic
sensible heat flux would be intensified, which can partially compensate the loss of poleward heat
flux due to lack of the hydrological cycle, as shown in experiment B.
In some climate models, the oceanic component has been treated as a swap, i.e., without the
current. Although it is clear that such a model would not allow heat flux transport associated with
current, a far more strong constraint on the climate system has not been clearly spelled out. A close
examination reveals the following constraints for such a model:
1) No current nor meso-scale eddies, so there is no horizontal heat flux -- molecular heat diffusion
is negligible.
2) No current to transport freshwater horizontally. As a result:
i) Evaporation and precipitation must be locally balanced.
ii) No latent heat flux in the atmosphere.
Consequences:
Although the ocean can still play the role of absolving short wave radiation from the sun
and heat up the atmosphere through latent heat release associated with evaporation, long wave
radiation and sensible heat flux, there is no oceanic sensible heat flux nor latent heat flux in the
atmosphere, and the only mechanism to transport heat is the dry atmosphere circulation.
9
Fig. 7. Sketches for (a) climate model with current and (b) climate model with a swap ocean.
Summary
(1) It is more appropriate to separate the poleward heat flux into three components: the
atmospheric sensible heat flux, the oceanic sensible heat flux, and the atmosphere-ocean coupled
latent heat flux.
(2) It is important to examine the divergence of these fluxes and their interaction.
(3) It is very important to explore how changes in the oceanic circulation affect the surface
conditions, such as surface temperature and the air-sea heat fluxes, and the global climate
conditions on the Earth.
(4) Hydrological cycle plays important role in setting up the sea surface temperature and salinity;
thus, the so-called swamp ocean model cannot simulate the oceanic role in climate correctly.
I hope this note can promote people's new interest in study the ocean-atmosphere coupled
system from a slightly different angle.
2. Haline Circulation induced by Evaporation and Precipitation
Historically, evaporation/precipitation was the first mechanism explored as the driving force
for the oceanic general circulation. However, the haline circulation in the ocean has been
overlooked for many decades. In fact, in most previous oceanic circulation models the haline
circulation was driven by either salt relaxation condition or the virtual salt flux condition; thus, the
real physics of haline circulation remained obscure for long time. Even before 1990, there was only
a very few papers published, in which the oceanic circulation was driven by evaporation and
precipitation. It is important to note that surface freshwater flux does not provide the mechanic
10
energy required for sustaining the circulation against friction; thus, evaporation and precipitation
along cannot drive haline circulation energetically. In the following discussion we keep use this
terminology for historical reason, and sometime we will use the term induce instead.
A. Classical circulations driven by evaporation and precipitation
(1) Hough (1897) solution
Evaporation and precipitation can drive oceanic circulation, and it was first explored by
Hough in his tidal paper published in 1897. For simplicity, it was assumed precipitation over
Northern hemisphere and evaporation over the southern hemisphere. (Hough originally assumed
the E-P pattern to be a second Legendre polynomial P2 ( x ) = ( 3 x 2 − 1) / 2 , i.e., precipitation over
high latitudes and evaporation over low latitudes.) Hough did not know how to parameterize
friction, so there was no friction in his model. In fact, his solution may be valid only for the initial
stage of the problem, when the solution can be treated as infinitesimal, so that the nonlinear effect
can be neglected. As time goes on, the free surface elevation and the zonal velocity would be
unbounded, so the solution is not valid.
Hough did not include any figures in his paper; his solution became much clearer after it
was illustrated graphically by Stommel (1957), Fig. 8.
Fig. 8. Two successive stages of the Hough-type circulation pattern, driven by precipitation
distributed over the northern hemisphere (P) and evaporation distributed over the Southern
hemisphere. The hovering arrows indicate the distribution of precipitation-evaporation. The arrows
drawn on the surface of the spheres are velocity components. The zonal currents grow with time.
The central solid portion of the earth is shown as shaded in cut-away mid-section. (Hough, 1897;
Stommel, 1957)
(2) Goldsbrough (1933) solution
Goldsbrough discussed the case of the steady flow driven by evaporation and evaporation.
By choosing the E-P pattern in a special form, he was able to construct a steady solution without
involving the friction terms. Most importantly, he first derived the vorticity constraint for largescale circulation
11
β hv = f ( E − P )
(2)
This is now called the Goldsbrough relation, according to this relation evaporation and
precipitation work as "pumping" in driving large-scale oceanic circulation.
Since precipitation drives equatorward flow, to close the circulation evaporation is needed
at the same latitudes to bring the water back poleward. Thus, the trick in finding the Goldsbrough
circulation is to assume a pattern of E-P that is zonally alternating so that the zonal integration of EP is zero along any latitude circle.
To illustrate the solution, we choose the following evaporation and precipitation pattern for
a basin 60o by 30o (Fig. 9):
⎧1
⎛π x ⎞
cos ⎜
0≤ x≤Δ
⎟,
⎪
⎡ π ( y − yn ) ⎤
⎝ 2Δ ⎠
⎪Δ
(3)
E − P = ( e − p ) sin ⎢
⎥, e − p = ⎨
⎛
⎞
π
x
1
⎣ ( yn − y s ) ⎦
⎪
cos ⎜
⎟, Δ ≤ x ≤ 1
⎪⎩ 1 − Δ
⎝ 2(1 − Δ ) ⎠
where the unit is m/yr, and x is the non-dimensional coordinate for each zonal circle, Δ = 0.06 .
Therefore, the zonal sum of evaporation and precipitation is zero, as proposed by Goldsbrough.
This freshwater flux induces a circulation that is closed within the basin, without requesting the
help of other dynamic processes. It is also important to note that barotropic circulation induced by
evaporation and precipitation is on the order of 1Sv, which is too small compared with the
circulation observed in the oceans; thus, this circulation cannot be used to explain the much
stronger circulation observed in the oceans.
45
4
−0.4
−0.6
0.40.3
0.5
Latitude
−1
−0.4
8
6
Latitude
0.1
0.2
1
30
4
.2
−0
25
−0.4
2
20
0
10
0
20
30
40
Longitude
−0.6
0.5
0.4
30
−0.8
0.2
35
0.3
−0.6
0
35
6
0.
0.4
40
0.1
3
−0.8
2
40
0.2
0.1
−0
.2
45
b) Streamfunciton (Sv)
0.
50
1
a) Evaporatin and precipitation (m/yr)
0
0
−0.2
50
0.3
0.2
25
0.1
0.1
−0.2
0
50
60
20
0
10
20
30
40
Longitude
50
60
Fig. 9. Goldsbrough's solution for a closed basin, with precipitation and evaporation balanced for
each latitudinal circle.
(3) Stommel (1957, 1984) solution
Goldsbrough's main contribution is to derive the vorticity constraint for the ocean interior.
He did not know how to treat the friction; thus, in order to find a steady solution he used a
distribution of evaporation minus precipitation that is quite unrealistic. The general case of arbitrary
evaporation minus precipitation was solved by Stommel (1957). Stommel pointed out that, in
principle, circulation driven by evaporation minus precipitation in the ocean interior can be closed
by western boundary currents. Therefore, the fresh-water-driven circulation should be called the
12
Goldsbrough-Stommel circulation. Fig. 10 shows Stommel's generalization of Goldsbrough's for a
single hemisphere basin.
Fig. 10. Left: The Goldsbrough gyre driven by evaporation-precipitation and presented by him in
1933 as a model of the North Atlantic; right panel: Effects of using a more realistic distribution of
evaporation-precipitation, and including western boundary currents. (Stommel, 1985)
D. Goldsbrough-Stommel circulation in the world oceans.
Using the existing data sets of evaporation minus precipitation, Huang and Schmitt (1992)
calculated the Goldsbrough-Stommel circulation in the world oceans. The solution is based on
zonally uniform evaporation and precipitation. The western boundary currents that play the role of
closing the meridional mass flux at each latitude in each basin is then attached to the interior
solution.
13
Fig. 11. The Goldsbrough-Stommel circulation of the world oceans. Each arrow indicates the
horizontal mass flux integrated over a 5o × 5o box, in Sv. Along the western boundary of each
basin, there is a curve indicting the northward mass flux (in 106 m3 / s within the western boundary,
which is required to close the circulation. In this calculation, the inter-basin transports are neglected
(Huang and Schmitt, 1993).
B. Is there something missed?
The circulation discussed above is only the barotropic mass flux, which is tiny compared
with the wind-driven circulation. There is no salinity involved in the dynamic analysis, were the
salinity included the entire picture would be totally different.
A. Fresh water flux induced circulation in a salty estuary
The circulation in an estuary depends on many factors, such as the amount of river run-off,
the mean salinity, and the tidal mixing. If there were no vertical mixing, the fresh water from river
run-off would overflow on top of the salty water in the estuary, and this is the only circulation
pattern -- the salty water below would be stagnant, if we neglect the tidal flow, Fig. 1.5a. However,
with the tidal mixing, a small amount of river run-off can induce a huge return flow in an estuary,
Fig. 1.5b. In our discussion we will assume that there is always energy source available for mixing,
such as the barotropic and internal tides, internal waves, and other sources; however, the exact
nature of the energy source is not our concern here.
14
Fig. 12. Sketch of models with freshwater-driven circulation in an estuary and an open ocean. a)
and c) Model without vertical mixing; b) and d) Model with vertical mixing (Huang, 1993).
C. The North Atlantic as an estuary
Similar to the case discussed above, the North Atlantic can be treated as a huge estuary,
with precipitation at high latitudes that is exactly balanced by precipitation at low latitudes. First,
let us assume there is absolutely no mixing. At time t=0 , rain starts to come into the subpolar basin,
and evaporation starts at low latitudes. Precipitation at high latitudes builds up the free surface, and
water starts to flow toward low latitudes (rotation would modifies the path). At low latitudes, in the
beginning evaporation would make some water saltier and sinks to depth, and this would give rise
to motion in the salty water. However, as fresh water arrives the low latitude and gradually covers
up the entire upper surface of the basin, evaporation can only affect the plain water, but not the
salty water. As the residual motion in the deep water gradually loses their kinetic energy, the only
motion will be the equatorward flow of plain water on top of the stagnant deep and salty water, Fig.
12c.
Second, if there is vertical mixing, there will be a very strong return flow induced by
vertical mixing, Fig. 12d. The overturning rate is
S
R= 0 FF
(4)
ΔS
Because ΔS is much smaller than S0 , a small amount of precipitation can drive a strong
meridional circulation.
3. Upper boundary conditions for the salinity balance
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A. The rigid-lid approximation
The physical boundary condition on the free surface is
Dη
w=
+ ( e − p ) , at z = η
(5)
Dt
D ∂
∂
∂
where w is the vertical velocity,
= +u +v
is the total derivative, η is the free surface
Dt ∂t
∂x
∂y
elevation, e-p is the evaporation minus precipitation. For the motions in the ocean interior, the
following scaling can be used
(6)
( x, y ) = L( x ', y '), (u, v) = U (u ', v ')
fLU
t = Tt ', η =
η'
(7)
g
w = δ Uw ', e − p = λδ U ( e − p ) '
(8)
where δ = H / L 1 is the aspect ration, and λ > 0.01 1 . An assumption has been made that the
time scale is determined by the horizontal advection. Substituting these relations into the boundary
condition and dropping the primes, one obtains
G
∂η
(9)
w = εT F
+ ε Fu ⋅∇η + λ (e − p), at z = ε Fη
∂t
U
1
f 2 L2
where ε T =
1 is the Kible number, ε =
1 is the Rossby number, and F =
> O(1) .
fL
fT
gH
If the advection time scale L/U is chosen as the time scale, the first two terms are of the same
magnitude. For motions with horizontal scale of 1000 km, the non-dimensional number ε F is
about 10−4 , which is much smaller than λ . Thus, the upper boundary condition can be linearized as
w = e − p, at z = 0
(10)
The rigid-lid approximation is not accurate for motions with time scale shorter than decadal,
motions on the planetary scale, or motions in the shallow seas. For example, for motions of
planetary scale, L = 107 m , for wave motion on seasonal time scales or shorter, T ≤ 107 s , so
εT > 0.02 . Thus, the vertical velocity due to wave motions of the free surface elevation has the
same magnitude as that of the evaporation minus precipitation. Under such circumstance, the rigidlid approximation, in which the free surface motion is neglected, is inaccurate. For the large scale
circulation on climate time scale the rigid-lid approximation is valid.
However, in the past the rigid-lid approximation has been equal to set w=0 on the upper
surface. Since no water can go through this so-called rigid lid, to simulate the salinity change a
virtual salt flux across the air-sea interface is introduced. This virtual salt flux formulation has
become a "real" flux simply because it has been used almost in every oceanographic books and
papers. Some time people forget that this salt flux is not real.
B. Three types of boundary conditions
In this section the focus is on different types of boundary conditions for the haline
circulation in the x-z plane. For convenience, Cartesian coordinates will be used in this section.
The salinity balance in a surface box, Fig. 13, can be written as
∂S 1 ⎡
1
κ
κ
+
−
−
(11)
+
( uS ) − ( uS ) ⎤⎦ + ⎡⎣( wS )+ − ( wS ) ⎤⎦ = H S x+ − S x− + V S z+ − S z−
⎣
∂t Δx
Δz
Δx
Δz
(
)
(
)
16
where ( uS ) and ( uS ) are the salt flux advected across the right and left boundaries, S x+ and S x−
+
−
are the horizontal salinity gradient at the right and left boundaries; similarly, ( wS ) and
+
+
z
( wS )
−
are
−
z
the salt flux advected through the upper and low boundaries, and S and S are the vertical
salinity gradient at the upper and lower boundary; S f = κ V S z+ is the salt flux cross the air-sea
interface. In addition, vertical velocity is prescribed on the upper surface
w = w+ , at z = 0
(12)
1) Relaxation Condition
+
(a) w = 0 , so ( wS ) + vanishes.
(13)
This boundary condition is usually called the rigid-lid condition. As will be discussed
above, the rigid-lid approximation does not necessarily require the vertical velocity to be zero at the
upper surface.
(b) S f = Γ( S * − S ) ,
(14)
where S * is specified according to climatological mean surface salinity.
2) Mixed Boundary Condition
+
+
(a) w = 0 , so ( wS ) vanishes.
(15a)
(b) S f = ( E − P) S ,
(15b)
where E and P are the evaporation and precipitation, S is the salinity in the box. This is the virtual
salt flux required for the salinity balance.
3) Natural Boundary Condition
+
(a) w = E − P ,
(16a)
and this is the fresh water flux controlling the salinity balance.
+
(16b)
(b) ( wS ) − κ V S z+ = 0 .
Note that within the water column there is always turbulent salt flux κ V S z and advective
salt flux wS. However, in the air there is no salt, so both these terms are identically zero, although
w is not zero. At the air-sea interface the turbulent flux exactly cancels the advective flux so that
there is absolutely no salt flux crossing the air-sea interface, as required by the physics. Thus, we
may call this turbulent flux S f = κ v S z an anti-advective salt flux. As seen from discussion above,
S f = ( E − P ) S at the air-sea interface; however, this flux is identically zero above the water
because there is no salt advection and no need for the anti-advective salt flux.
17
Fig. 13. A finite difference grid box in the x-z plane (Huang, 1993).
Accordingly, the salinity balance for a surface box is reduced to
∂S 1 ⎡
1
κ
κ
+
−
−
(17)
uS ) − ( uS ) ⎤ − ( wS ) = H S x+ − S x− − V S z−
+
(
⎦ Δz
∂t Δx ⎣
Δx
Δz
If the mixing terms are negligible, (17) is further reduced to
DS
=0
(18)
Dt
This equation means that the total salt is conserved. The local salinity change is due to fresh water
dilution/concentration, and there is no need for the virtual salt flux.
In many previous mixed layer models and primitive equation models for the oceanic general
circulation the so-called ``rigid-lid'' approximation has been interpolated as equivalent to setting the
vertical velocity equal to zero at the sea surface. Without a vertical flux of plain water through the
upper surface, the salinity change due to evaporation and precipitation has to be simulated by the
virtual salt flux as discussed above. Therefore, the so-called anti-advective salt flux has been
substituted as a forcing to simulate the salt balance. However, the essential assumption of the rigid
lid approximation is to move the upper boundary from the free surface z = η to the rigid lid z=0,
and it is not inconsistent with adding a small vertical velocity at the air-sea interface. As long as we
are only concern about large-scale geostrophic motions of climate time scale, the rigid-lid
approximation is valid.
In fact, the rigid-lid approximation has been used in many models with non-zero vertical
velocity at z=0. For example, in many quasi-geostrophic models the dynamic effect of wind stress
curl is simulated by imposing the equivalent Ekman pumping velocity at the upper surface. In
general, the Ekman pumping velocity is about 30 times larger than the vertical velocity associated
with precipitation and evaporation. Since all these quasi-geostrophic numerical models have been
used extensively without any trouble associated with the non-zero vertical velocity on the upper
surface, this should be true for the primitive equation models too. For motions of planetary scale or
(
)
18
motions associated with seasonal time scale (or shorter), the rigid-lid assumption may not be valid,
and a free surface should be included in the calculation.
C. Remark:
The natural boundary condition discussed above is based on the rigid lid approximation,
which was used extensively in the past. However, in the new generation of numerical models, the
free surface is a preferred choice. The suitable boundary condition for salinity balance is a simple
mathematical statement that (P-E) is a source of freshwater at the upper surface of the ocean, with
no salinity flux through the air-sea interface. It is to note that the upper surface of the ocean is
neither an Eulerian surface nor a Lagrangian surface because freshwater moves across this surface,
as discussed in Lecture 4.
In a pressure-sigma coordinate, the vertical coordinate is defined as σ = ( p − pt ) / pbt ,
pbt = pb − pt , where pb and pt are pressure on the bottom and the sea surface, which can change
with time. The upper boundary condition is
pbtσ = − g ρ f ( e − p )
(19)
where ρ f is density of freshwater through the air-sea interface. This boundary condition means
that due to precipitation minus evaporation the pressure in each grid point in the water column is
increased. The corresponding salinity condition at the upper surface is that the next salt flux due to
salt advection and vertical salt diffusion exactly cancel each other, i.e.,
S f = S f ,adv + S f ,diffu = 0, at the surface
(20)
In the traditional z-coordinate, the vertical velocity at the sea surface satisfies
∂η
(21)
+ u ⋅ ∇η + ( e − p ) ρ f / ρ s
∂t
where ρ s is the sea surface density. Vertical integration of this equation leads to a prognostic
equation for the free surface
⎛∂ η
⎞
∂η
∂ η
= − ⎜ ∫ udz + ∫ vdz ⎟ + ( p − e ) + RT
(22)
∂t
∂y − H
⎝ ∂x − H
⎠
where RT means rest of terms associated with thermohaline processes in the water column. Thus,
precipitation tends to increase the sea level. However, the local sea level is also closely related to
the horizontal convergence/divergence of the vertically integrated velocity field. For the details of
sea surface elevation change induced by precipitation/evaporation, the reader is referred to Huang
and Jin (2002).
w=
D. The Pitfalls of the Relaxation and Mixed Conditions
These three boundary conditions are quite different in their physical meaning and numerical
stability. Since the relaxation condition is a negative feedback, the solutions are mostly stable.
Although the ability to match the observed surface salinity may be an advantage for
simulating the present climate, it may not be suitable for simulating the oceanic circulation for
general situations. For example, the surface salinity for the world oceans in the remote past and in
the future is unknown, so the relaxation condition cannot apply. Similarly, using the salinity
relaxation condition for oceanic forecasting is questionable. Neither the reference salinity
distribution S * nor the suitable relaxation constant is clear.
19
There are essential difficulties associated with the first and second types of boundary
conditions. First, to balance the salt in the oceans a huge virtual salt flux through the air-sea
interface and the atmosphere is required for taking up the salt from the subpolar basin, transporting
it equatorward, and dumping it there, Fig. 3.2. The virtual salt flux required in the North Atlantic
can be estimated as
35
(23)
∑ ( E − P ) S > 109 kg / s × 1000 > 3.5 ×107 kg / s
It is obvious that such a huge virtual salt flux is unphysical and should be avoided.
Second, using the ( E − P ) S to force a model can cause the total salt in a basin to increase
unboundedly. The argument is very simple indeed. At the present day, the global integration of
precipitation minus evaporation should be zero:
∫ ∫ ( E − P ) dA = 0
(24)
However, the global integration of the virtual salt flux is larger than zero:
∫ ∫ ( E − P ) SdA > 0
(25)
This is due to a positive correlation between evaporation and high salinity. For example,
evaporation is very strong in the subtropical North Atlantic that gives rise to a salinity of more than
37 psu; meanwhile excess precipitation in the subpolar Pacific gives rise to a surface salinity lower
than 32 psu. Thus, to avoid the salinity explosion one must use the virtual salt flux defined by
S f = (E − P) S s
where S s is the mean sea surface salinity averaged over the whole domain of the model. Notice
that the original definition of virtual salt flux includes a positive (negative) feedback wherever
evaporation is stronger (weaker) than precipitation. However, in this new form the virtual salt flux
has to be entirely fixed and it has nothing to do with the local salinity. Apparently, the feedback due
to the local salinity is all gone. Even if the virtual salt flux can be accepted as a parameterization,
the physics associated with the virtual salt flux is distorted. In addition, this constraint can
introduce large errors wherever the local salinity is much different from the basin or global mean.
For example, surface salinity in the North Pacific can be lower than 33psu, and the surface salinity
in the North Atlantic can be higher than 37psu. Suppose S s is defined as 35psu for a world ocean
circulation mode, a systematic bias of 10% will be introduced through the upper boundary
condition for the salinity.
Remark: Although the virtual salt flux has been used in general circulation models only
within the past few years, it has been used in many books and papers involving air-sea interaction
for a long time. For example a virtual salt flux, ( P0 − E0 ) S0 , appeared in the salinity equation in the
classic paper by Niiler and Kraus (1977). As discussed above, this virtual salt flux is unphysical. In
addition, the global salinity conservation requires replacement of the local S0 with the global mean
S s ; thus, errors are introduced through use of the virtual salt flux.
The natural boundary conditions for the salinity balance apply to all other cases involving
fresh water flux cross the air-sea interface, including mixed layer models, ice-ocean coupling
models, and atmosphere-ocean coupling models.
E. A virtual natural boundary condition
20
Evaporation and precipitation data is hard to collect, and there is no reliable historical data.
As a compromise, one can use the sea surface salinity as forcing field to reconstruct the salinity
balance in the past, using the virtual natural boundary condition as following.
First, the equivalent evaporation and precipitation field can be inferred from the historical
sea surface salinity data from the model as
E − P = Γ( S * − S ) / S s ,
where S * is the observed salinity at a given time in the past, and S s is the global mean sea surface
salinity. Second this equivalent evaporation and precipitation field is used the freshwater flux
through the air-sea interface in the model.
4. Thermohaline Circulation under Freshwater Forcing Alone
The example shown here is for a 4o × 4o resolution model ocean of 60o × 60o basin, driven
by a "linear" profile of E-P, shown in Fig. 14, with w0 = 1m / year . Note that the model's only
forcing is the fresh water flux through the upper surface. The total meridional mass flux (indicated
by the dashed line) associated with E-P is about 0.3 Sv, this is really very small, compared with the
wind-driven circulation.
Fig. 14. Meridional distribution of mass fluxes: E-P in the evaporation minus precipitation (in
10−5 cm / s ), M int erior is the poleward mass flux in the interior, M wbc is the mass flux of the western
boundary current, M total is the total poleward mass flux; all these mass fluxes are in Sverdrups.
The model reaches a state of periodic oscillation with a regular period of 18 years, not
shown here. Fig. 5.2 shows the barotropic streamfunction (upper panel) and the total barotropic
meridional mass flux within each 4o × 4o grid-box (lower panel). These two figures show the
southward flow driven by precipitation in the subpolar basin, the poleward flow driven by
evaporation in the subtropical basin, and the western boundary currents required for closing the
21
circulation, as suggested by Stommel.
Fig. 15. (a) Barotropic streamfunction and (b) integrated meridional mass flux through each box, in
Sv.
The meridional and the zonal overturning cells are shown in Fig. 16. a) and b). In the vertical
direction, there are two strong baroclinic gyres circulate in the opposite direction.
Fig. 16. Time-mean circulation patterns, depth in km.
a) Meridional streamfunction in Sv. Note that contour interval is 20 Sv for the unlabeled curves. b)
Longitudinal streamfunction. c) Zonally mean meridional salinity distribution (deviation from the
22
mean salinity). d) Meridionally mean zonal salinity distribution (deviation from the mean salinity).
Interval is 0.05 for the solid lines and 0.5 for the dashed lines.
In this idealized numerical experiment, surface freshwater flux is the only driving force.
Nevertheless, the model demonstrated that the salinity distribution in the oceans in the results of the
freshwater flux condition specified on the sea surface, Fig. 17.
Fig. 17. Salinity (deviation from the mean) distribution at 15, 302, 604, and 1230 m.
Under the freshwater forcing, there is a complicated three-dimensional haline circulation,
Fig. 18. Precipitation tends to be very irregular in nature. Thus, in human’s mind’s eyes, there is not
a clear connection between the tiny bits of rain drops in precipitation and large-scale organized
haline circulation in the ocean. However, precipitation and evaporation are responsible for creating
the salinity difference in the oceans, and thus regulating the haline component of the oceanic
general circulation.
23
Fig. 18. Schematic structure of the saline circulation driven by precipitation at high latitudes and
evaporation at low latitudes.
In some previous studies, the salt flux diagnosed from models driven by the virtual salt flux
has been misinterpreted as real salt flux in the oceans. As shown in Fig. 19, the net meridional salt
flux approaches a quasi-equilibrium, as the circulation itself approaches the final equilibrium, Fig.
19a. The large advective salt flux diagnosed from models based on the virtual salt flux condition
should be re-interpreted, and after such a reinterpretation the meridional salt flux diagnosed from
models based on the virtual salt flux should also reaches a quasi-equilibrium. Note that for timedependent problem, the so-called salt flux diagnosed from models based on virtual salt flux
condition is meaningless.
24
Fig. 19. Time evolution of the meridional salt fluxes at 28o N , in 106 kg / s : a) The new model; b)
the old model based on virtual salt flux condition.
5. Salinity contribution to stratification and horizontal pressure gradient in the oceans
Density is a nonlinear function of temperature, salinity, and pressure. In particular, the
thermal expansion coefficient strongly depends on temperature. As a result, density at warm
temperature is primarily controlled by temperature; however, at high altitudes, thermal expansion
coefficient is so small (Fig. 20) that density is primarily controlled by salinity.
25
Fig. 20. Thermal expansion coefficient at the sea surface in the world ocean, in units of 10−3 / o C
At low temperature salinity can made important contribution to the stratification in the
ocean, and this can be illustrated through the following diagnosis based on climatological mean T,
S properties in the world oceans. First, we will study how much salinity contributes to the vertical
stratification in the water column. As an example, we shown the stratification ratio, which is
defined as follows.
The commonly used stratification can be extended as
g d ρθ
g d ρθ (T , S0 )
N2 = −
, NT2 = −
, DN 2 = N 2 − NT2
(26)
dz
ρ0 dz
ρ0
where NT2 is defined as the equivalent stratification due to the vertical distribution of temperature
only, with the salinity set to a constant value of S0 = 35 . The difference between these two terms is
denoted as DN 2 , thus DN 2 / N 2 indicates salinity’s contribution to the stratification. For example,
the zero contour indicates the place where vertical salinity gradient does not contribute to the local
stratification. As shown in Fig. 21a), in the upper ocean, stratification is primarily due to density
gradient associated with temperature profile, while salinity gradient tends to weaken the
stratification. At the level of nearly 1.5km below the sea level, the stratification is primarily
controlled by the relatively fresh water of the Antarctic Intermediate Water, Fig. 21b).
26
Fig. 21. Contribution of temperature and salinity to stratification along 30.5oW, inferred from
climatology. Left panel: N 2 (black line), N 2 due to temperature only (red line), and N 2 due to
salinity only (blue line); right panel stratification ratio.
In the Atlantic Ocean, salinity control of the stratification can be clearly identified from
depth below the upper kilometer, Fig. 22b. In particular, both salinity gradient associated with
masses of Antarctic Intermediate Water and the North Atlantic Deep Water clearly plays major role
in shaping up the stratification at middle depth.
27
Fig. 22. Contribution of temperature and salinity to stratification along 30.5oW, inferred from
climatology. Left panel: N 2 (black line), N 2 due to temperature only (red line), and N 2 due to
salinity only (blue line); right panel stratification ratio.
Another way to illustrate salinity’s contribution is through its contribution to the horizontal
pressure gradient. Since the sea surface elevation is unknown, we will omit the potential
contribution due to difference in sea surface elevation and focus on the baroclinic pressure gradient
associated with the T,S distribution only. Furthermore, we will also subtract the horizontal mean
pressure, and present the deviation from the horizontal mean pressure at each level only. Similar to
the discussion above, we will separate the pressure into two components as follows.
p = ∫ ρ dz , pT (T , S0 ) = ∫ ρ (T , S0 ) dz, pS = p − pT
0
0
z
z
(27)
Thus, p is the commonly used pressure, pT is the pressure due to temperature distribution in the
water column only, and pS is the pressure due to salinity distribution only. Since density is a
nonlinear function of temperature, salinity and pressure, the above definition does not provide an
exact partition between temperature and salinity.
As shown in Fig. 23a, at the middle level, there is a southward pressure gradient force in the
Atlantic basin, which is responsible for maintaining the southward flow at the middle level as the
return component of the current meridional overturning in the Atlantic Ocean. By examining the
pressure distribution in the North Atlantic, it is readily seen that temperature and salinity
contributions have opposite signs at middle depth. In fact, the southward pressure force at the
middle level is mostly provided by the salinity distribution (Fig. 23d), but not the temperature (Fig.
28
23c). This suggests that meridional overturning in the Atlantic Ocean may be rather sensitive to
changes in salinity, which is in turn related to the hydrological cycle. This connection will be
explored shortly in the discussion about thermohaline catastrophe associated with freshening of the
subpolar North Atlantic.
On the other hand, the strong fronts associated with the Antarctic Circumpolar Current are
primarily due to the temperature front, with a relatively small contribution form the salinity
distribution.
o
c) Δ Pbc due to temperature (db)
0
0
0.3
−4000
−5000
−5000
−40
−20
0
20
40
60
−80
0
−60
−40
.1
00..43
0.1
.2
.8
−4000
0
0.1
0.2
−0
0.4
0.3
4
−0
00..4
3
−2000
0
0.1
0
0
0
0
−3000
−4000
60
0.3 0
0.4 0.6
−1000
2
−0.
.4 −0.6
−3000
40
−0
0
0.4 0.3
2
0.1
0
−2000
0.
20
0
0.2
0.2
−1000
−0.4
−0.3
−0.2
0
d) Δ Pbc due to salt (db)
b) Δ P(db)
0
−20
0.2
−60
0.3
0.6
0.8
−80
−0.6
−0.9
−0.6
−3000
6
−0 .
−0.3
0.30
0.6
0.9
−4000
−0.3
−0.6
−0.3
0.9
−3000
−0.6
−2000
0
−2000
−1000
0
−0.3
0.3
0.6
−1000
−0.
3
−0.3
0
0. 3
a) Δ Pbc(db), along 30.5 W
0
−5000
−80
−5000
−60
−40
−20
0
20
40
60
−80
0
−60
−40
−20
0
20
40
60
Fig. 23. Baroclinic pressure deviation from the horizontal mean, averaged over 3o zonally with the
central line taking along 30.5oW. Panel b indicates the pressure difference, using 2000m as the
reference level, i.e., assume no horizontal pressure gradient at this level.
The situation in the Pacific is different from that in the Atlantic. In the Pacific basin,
pressure distribution at upper and middle levels is roughly symmetric with respect to the equator,
except south of 45oS (Fig. 24a, b). The symmetric baroclinic pressure implies no cross equator flow
in the Pacific. This symmetric pattern in baroclinic pressure is primarily due to the temperature
distribution, as shown in upper and middle panels in Fig. 24. Salinity distribution implies a
northward pressure gradient force at the middle and deep level, which seems to linked to the
northward movement of water mass at these levels.
29
o
a) Δ Pbc(db), along 169.5 W
c) Δ Pbc due to temperature (db)
3
−0.
0
−0.3
0.3
0.6
.6
−0
0.9
0.3
−0 .
6
−0.6
−0.3
−0.6
−0.6
−0.6
0
0
−5000
0
0
−5000
.9
−0
−4000
3
−0.
0.3
3
−0 .
−4000
−3000
0.6.9
0
−3000
−2000
−0.6
0.60.9
−2000
−0.9
−1000
0.3
−1000
0
0
−0.3
0.3
−0.
6
0
0
0
−40
−20
0
20
40
60
−80
−60
0.1
0
0.
10.2
0.4
0.3
−
−00..3
2
0. 2
0
20
0
0
40
60
0
0.2
−1000
0.3
−2000
0.3
.1
−0
0
−1000
4
0.4
0.2 0.1
.3
.4
0.1
−0
.
−20
d) Δ Pbc due to salt (db)
b) Δ P(db)
0
−40
−2000
0
−0.4
−60
−0.2
−80
0
0
0.1
0.2
−5000
−80
−5000
−60
−40
−20
0
20
40
0.3
−4000
−20
0
0.2
0.2
0.3
0
0.2
0.3
60
−80
−60
−40
0
0.4
0.2
−4000
−3000
0
0.1
−3000
20
40
60
Fig. 24. Baroclinic pressure deviation from the horizontal mean, averaged over 3o zonally with the
central line taking along 179.5oW.
6. The feedback between the hydrological cycle and the meridional thermal circulation
One of the possible reasons for neglecting the freshwater flux in heat flux analysis could be
the misconception that freshwater flux in the ocean is so small that it seems to play such a
minuscule role in controlling the heat flux. As will be shown shortly, however, the freshwater flux
through the oceans plays an important role in controlling the strength of the meridional overturning
and poleward heat flux.
Due to evaporation at low latitudes and precipitation at high latitudes, there is a meridional
salinity gradient at the sea surface. As shown in the previous section, the meridional density
gradient due to the salinity difference is opposite to the meridional density gradient due to the
thermal forcing in the upper oceans. Thus, in the present climate setting, the hydrological cycle is a
brake for the poleward heat transportation carried by the meridional thermally-forced circulation in
the oceans. In other words, the poleward latent heat flux associated with the water vapor cycle has a
negative feedback on the oceanic sensible heat flux.
If there were no hydrological cycle, the meridional overturning cell and the associated
poleward heat flux in all five basins would be intensified. As an example, in the North Atlantic the
surface density difference between the equator and high latitudes is about Δσ = 3.39 , based on the
Levitus and Boyer (1994) data, Fig. 25a. If there were no evaporation and precipitation, there
would be very little salinity difference in the oceans, so we set salinity to a constant value of 35.
Assuming the surface temperature distribution remains unchanged -- a very bold assumption that is
30
unlikely true in the real world--the corresponding surface density difference would be increased
to Δσ = 5.39 , Fig. 24b. Due to the increase of the north-south density difference both the
meridional overturning rate and poleward heat flux should increase.
Fig. 25. Two models for the North Atlantic: a) the standard model, forced by observed sea surface
temperature and salinity; b) a conceptual model with uniform salinity.
We carried out four numerical experiments for the Atlantic in order to explore the dynamic
role of freshwater flux in the meridional circulation. The MOM2 (Pacanofsky, 1995) was used with
a horizontal resolution of 2.5o × 2o and 15 levels vertically, with a constant diapycnal mixing rate of
0.4 × 10-4m-2s-1. The model's temperature and salinity were initialized with the Levitus and Boyer
(1994) data, and in each experiment the model was run for 1000 years to reach a quasi equilibrium.
In experiment A both the surface temperature and salinity were relaxed toward the monthly
mean climatology, along with the Hellermann and Rosenstein (1983) monthly mean wind stress
data. In experiment B salinity was set to a constant value of 35, but all parameters remained the
same as in experiment A.
In experiments C and D the density effect due to temperature was ignored. Thus, the salinity
difference was the only driving force for the circulation. In order to calculate the poleward heat
flux, temperature was treated as a passive tracer, with an upper boundary condition such that the
sea surface was relaxed toward the monthly mean climatology. In experiment C the surface salinity
was subject to the surface relaxation condition, assuming that the climatological monthly mean sea
surface salinity remained unchanged. In experiment D, the model's salinity was subject to the
virtual salt flux diagnosed from the steady circulation obtained from experiment A.
The meridional overturning circulation (MOC) in the second experiment was 68% higher
than in experiment A (Table 2), quite close to our scaling analysis. The poleward heat flux (PHF) in
the second experiment was 21% higher than experiment A, somewhat lower than the prediction
from the simple scaling.
In experiments C and D, the meridional overturning cell reversed, and the associated
meridional heat flux was equatorward, as shown in Table 2. The amount of equatorward heat flux
was about 0.2 PW for the case with virtual salt flux. Thus, our numerical experiments demonstrated
that the haline circulation is an important factor controlling the poleward heat flux.
Temperature
A
SST relaxation
B
SST relaxation
Salinity
MOC(Sv)
SSS relaxation
12.58
Set to S=35
21.10(+68%)
C
Treated as a
tracer
SSS relaxation
-13.27
D
Treated as a
tracer
Virtual salt Flux
-18.70
31
Poleward heat
0.72
0.87(+21%)
-0.07
flux (PW)
Table 2. Four numerical experiments for the model Atlantic.
-0.23
It is much more important to examine the poleward heat flux profile and the associated heat
flux divergence. It is clear that in experiment B, the ocean transported much more heat to high
latitudes. In particular, the poleward heat flux divergence was much larger for latitudes higher than
30oN (Fig. 26). Thus, in the case without a hydrological cycle, the oceanic current will transport
more heat poleward and release it to higher latitudes, partially compensating for the decline of
poleward heat flux caused by the lack of latent heat flux associated with the hydrological cycle.
Fig. 26. Poleward heat flux (upper panel) and its divergence (lower panel) (due to advection)
diagnosed from the standard model with both temperature and salinity relaxed (thin line) and the
model with uniform salinity (S=35, heavy line).
We also ran other experiments for the global oceans. The model had a low horizontal
resolution of 4o × 4o and 15 layers. The Indonesian Throughflow is included in these low-resolution
simulations. The model's temperature and salinity were initialized with the Levitus and Boyer
(1994) data and in each experiment. The model was run for 1000 years to reach a quasi
equilibrium.
In the standard experiment both the surface temperature and salinity were relaxed toward
the monthly mean climatology, with using the Hellermann and Rosenstein (1983) monthly mean
wind stress data. In the second experiment salinity was set to a constant value of 35, but all
remained the same as in the standard experiment. The poleward heat flux is calculated, using the
definition discussed in the Appendix, so the heat flux in both the South Pacific and Indian is
uniquely defined.
32
Fig. 27. Poleward flux in the world oceans, diagnosed from a model based on observed SST and
SSS (thin lines) and a model based on relaxation to observed SST and with constant salinity (heavy
lines).
It is clear that by turning off the salinity effect, the total northward heat flux in the Northern
Hemisphere increases, while it declines in the Southern Hemisphere, Fig. 27. Most interestingly,
the northward heat flux in the North Pacific increases substantially. This increase of heat flux at
mid latitudes is probably associated with the newly-created thermal mode of the meridional
circulation in the North Pacific, caused by the lack of salinity effect that opposes the thermal mode.
The northward heat flux in the Atlantic declines due to the diminishing effect of the global
conveyor belt that is driven, at least partially, by the salinity difference between the Pacific and
Atlantic. As a result of the decline of the global conveyor belt, the poleward heat flux in the South
Indian Ocean declines. Note that the heat flux in the South Atlantic is now southward, as required
by the thermal mode in this basin.
A major assumption we made use is that vertical mixing coefficient and wind stress remain
the same for all these experiments. This is a highly idealized assumption. Since thermohaline
circulation is so closely related to climate, changes in the circulation must affect both the wind
stress and mixing coefficient; however, a more comprehensive study is difficult and this time, and
we will leave it to the readers who are interest in pushing along this line of thought.
References
Bryden, H. L. and S. Imawaki, 2001: Ocean heat transport. In G. Siedler, J. Church, and J. Gould
(Eds.): "Ocean Circulation and Climate", International Geophysical Series, Academy Press,
New York, pp455-474.
Da Silva, A., Young, A. C., and S. Levitus, 1994: Atlas of surface marine data 1994. Volume 1.
Algorithms and Procedures. National Oceanographic and Atmospheric Administration.
33
Gaffen,D. J., R. D. Rosen, D. A. Salstein and J. S. Boyle, 1997: Evaluation of tropospheric water
vapor simulations from the atmospheric Model Intercomparison Project. J. Climate, 10,
1648-1661.
Hellerman, S. and M. Rosenstein, 1983: Normal monthly wind stress over the world ocean with
error estimates. J. Phys. Oceanogr., 13, 1093-1104.
Huang, R. X. and R. W. Schmitt, 1993. The Goldsbrough--Stommel circulation of the world
oceans. J. Phys. Oceanogr., 23, 1277-1284.
Huang, R. X., 1993. Real freshwater flux as a natural boundary condition for the salinity balance
and thermohaline circulation forced by evaporation and precipitation. J. Phys. Oceanogr.,
23, 2428-2446.
Huang, R. X., and X.-Z. Jin, 2002. Sea surface elevation and bottom pressure anomalies due to
thermohaline forcing, Part I: Isolated perturbations. J. Phys. Oceanog., 32, 2131-2150.
Pacanofsky, R. C., 1995: MOM2 documentation user's guide and reference manual, version 2.
GFDL Ocean Technical Report #3.
Schmitt, R. W., 1995: The ocean component of the global water cycle. U.S. National Report to
International Union of Geodesy and Geophysics 1991-1994. Rev. Geophs., suppl., 13951409.
Trenberth, K. E. and J. M. Caron, 2001: Estimates of meridional atmosphere and ocean heat
transport, J. Climate, 14, 3433-3443.
Warren, B. A., 1980: Deep circulation of the world ocean. In B. A. Warren and C. Wunsch (Eds.):
"Evolution of physical oceanography", The MIT Press, Cambridge, Massachusetts, pp6401.
Wijffels, S. E., 2001: Ocean transport of fresh water. in G. Siedler, J. Church, and J. Gould (Eds.):
"Ocean Circulation and Climate", International Geophysical Series, Academy Press, New
York, pp475-488.
Zhang, K. Q. and J. Marotzke, 1999: The importance of open-boundary estimation for an Indian
Ocean GCM-data system, J. Mar. Res., 57, 305-334.
Appendix. Definition of the oceanic sensible heat flux.
Traditionally, the poleward oceanic heat flux is defined as
Ho = ∫
∫
ρ c p vθ dxdz
where ρ is the in-situ temperature, c p is the specific heat under constant pressure, v is the
meridional velocity, and θ is the potential temperature. It is well known that if the system has a net
mass flux through the section, the heat flux calculated from this formula may depend on the choice
of the temperature scale. In general, heat flux itself has no direct physical meaning, and the most
important quantity associated with the heat flux is its divergence. There are two criteria in defining
heat flux: First, the divergence of heat flux should satisfy the local heat balance equation. Second,
the heat flux should depend on the local properties only.
The best-known example is the circulation in the South Pacific and Indian. Due to the
Indonesian Throughflow, the poleward heat flux in the South Pacific and Indian is not uniquely
defined. To over come the complication due to the net mass flux through the sections, many
different approaches have been used. For example, Zhang and Marotzke (1999) have proposed a
procedure in which the heat flux associated with the westward flow in the strait is used to evaluate
the poleward heat flux. Although the heat flux definition they proposed satisfy the first criterion, it
does not satisfy the second criterion. In fact, their definition includes a term that depends on the
34
thermal condition in the cross section of the Throughflow. If the condition within the strait changes,
the heat flux evaluated will also change, even if the circulation and water properties at the given
section south of the strait does not change.
To over come such a problem, thus, a much simpler definition is recommended:
Ho = ∫
∫
ρ c p v(θ − θ 0 )dxdz
where θ 0 is a reference potential temperature, which can be chose arbitrarily, and the most obvious
choice is the global-mean potential temperature, about 2o C . This definition satisfies the two
criteria list above. We will analyze the meaning of the heat flux defined in this way as follows.
In the oceans, there is indeed net mass flux through any given section due to either
evaporation/precipitation or throughflow.
H o = H o ,0 + H o ,1
H o ,0 = ∫
∫ ρ c v(θ − θ )dxdz
H = ∫ ∫ ρ vc (θ − θ )dxdz
where ρ v = ∫ ∫ ρ vdxdz / ∫ ∫
0
p
o ,1
0
p
dxdz is the mass flux averaged over the section. The first component,
H o ,0 , is the oceanic sensible heat flux through this section, carried by the circulation without the
net mass flux. By its definition, it is independent of the choice of the reference temperature. On the
other hand, the second component, H o ,1 , clearly depends on the choice of the reference
temperature, so it should be isolated from the first component. In the oceans, the net mass flux
through a given section is due to either the evaporation/precipitation or inter-basin loop current,
such as the Indonesian Throughflow. We will separate the net mass flux through each section into
two parts:
ρ v = ρ vloop + ρ v emp
where ρ v loop is the zonally-mean meridional mass flux associated with the loop current going
through the whole basin and the total amount of mass flux associated with this component should
constant at any zonal section, such as the Indonesian Throughflow; ρ v emp is the zonally-mean
meridional mass flux associated with evaporation and precipitation. This component of the net
mass flux through the section is associated with the water vapor cycle in the atmosphere, so it
should not be counted as part of the oceanic sensible heat flux.
Accordingly, poleward sensible heat flux in the Atlantic can be defined as
∫ ρ c v (θ − θ ) dxdz − H
= c ∫ ∫ ρ v ( y ) ⎡⎣θ ( y ) − θ
H oA,sen = ∫
H oA,emp
0
p
A
A
o ,emp
⎤dxdz = c p M emp , A ( y ) ⎡θ A ( y ) − θ 0 ⎤
⎦
⎣
⎦
is the zonally-integrated meridional mass flux due to evaporation/precipitation.
p
A
emp
A
0
where M emp , A
Similarly, the poleward heat flux in the Pacific and Indian are defined
H oP,sen = ∫
P
ρ c p v (θ − θ 0 ) dxdz − H oP,emp , H oP,emp = c p M emp ,P ( y ) ⎡⎣θ P ( y ) − θ 0 ⎤⎦
H oI,sen
I
ρ c p v (θ − θ 0 ) dxdz − H oI,emp , H oI,emp = c p M emp ,I ( y ) ⎡⎣θ I ( y ) − θ 0 ⎤⎦
∫
=∫ ∫
35
The sum of water flux due to evaporation and precipitation, including the river run-off, in all
oceans should equal to the water vapor flux in the atmosphere. Thus, the sum of the evaporation
and precipitation in three basins
M emp = M emp , A + M emp ,P + M emp ,I
The heat flux associated with this mass flux is
H o ,emp = c p ⎡⎣ M emp , Aθ A ( y ) + M emp ,Pθ P ( y ) + M emp ,I θ I ( y ) ⎤⎦ − c p M empθ 0
This heat flux is independent on the choice of the temperature scale.
This example demonstrates that a working definition of heat flux should be independent of
the specific choice of the temperature scale. In order to achieve this goal, one has to separate the net
mass flux through a given section, and link this net mass flux with the corresponding return flow in
the climate system, i.e., the water vapor flux in the atmosphere.
air − sea
H emp
= − M emp Lh + c p ⎡⎣θ atmos ( y ) − θ 0 ⎤⎦ + H o ,emp
where the first term indicates the heat content flux in the atmospheric branch of the water vapor
loop, and the second term is the heat content flux in the oceanic branch of the water cycle. This
definition is, of course, independent of the choice of the temperature scale.
{
}
36