Lecture 13. The Ventilated Thermocline
3/6/2006 9:46 PM
I. Introduction
A major conceptual difficulty in getting subsurface layer in motion is that they are not
directly in contact with the atmospheric forcing locally. As discussed in Lecture 4, one way to get
these subsurface layers in motion is the close geostrophic contours induced by strong forcing upon
the surface layer. However, there is another way through which the subsurface layer can be put into
motion, this is the ventilation process. Looking at a sea surface density distribution map, one
realizes that many isopycnals outcrop in poleward part of the oceans. When a layer outcrops, it is
exposed to the atmospheric forcing directly. Thus, the outcropping layer is in motion under the
wind stress forcing. Intuitively, a layer, which is in motion under the direct wind forcing, should
continue its motion, even after it is subducted under the other layer.
Iselin (1939) made a link between the T-S relation found in a vertical section and the
wintertime mixed layer at higher latitudes. His schematic picture for this ventilation process is
shown in Fig. 1. The speculated motion is indicated by arrows. Iselin's model, was the first
prototype for water mass formation in the oceans; however, it is almost a surprise that such a
simple and important dynamical idea had not been pursued further in the ensuing decades.
Fig. 1. Schematic representation of water mass formation due to water sinking along isopycnal
surfaces (Iselin, 1939).
In modern terminology the basic idea is that within the subtropical gyre water is pushed
downward into the thermocline by Ekman pumping and then downwells along isopycnals as it
moves southward induced by the Sverdrup dynamics. The particles' motion after their ejection from
the base of the mixed layer is confined within the corresponding isopycnal surface because mixing
is relatively weak within the main thermocline. The weakness of mixing in the upper ocean and
below the mixed layer has been confirmed by observations, such as the recent tracer release
experiments. This process is called ventilation by Ekman pumping*. Thus, outcropping functions
as windows for water in the subsurface layers to go through, escaping the blockage by the eastern
boundary.
*There are other types of ventilation. For example, water mass can be ventilated through
mode water formation or deepwater formation, where convective overturning induced by cooling or
salt ejection due to ice formation leads to ventilation to great depth, on the order of a few hundreds
meters or all the way to the sea floor. Water mass can also be ventilated through the western or
eastern boundary currents.
1
One of the conceptual difficulties in building up a model for the annual mean wind-driven
circulation is the strong seasonal cycle in wind stress, heat and freshwater fluxes at the sea surface.
As a result, mixed layer properties vary greatly over the annual cycle, as shown in Fig. 2. It seems
obvious that the seasonal variability in the surface conditions may affect the wind-driven
circulation in forms of waves and time-varying currents; thus, making simple analytical model
almost impossible.
In a highly innovative paper, Stommel (1979) proposed that a demon is working so that
only the late winter properties are selected, i.e., water that actually enters the permanent
thermocline is the water formed at the time when mixed layer is deepest and the density is the
highest. It is indeed a highly idealization of the complicated processes taking place in the upper
ocean. It is such a bold assumption, that Hank himself was not sure even after the paper was
published. (I was told by Hank himself that he destroyed all the reprints because he thought the
paper was wrong.) Since then Stommel's idea of selecting the late winter properties as that for the
water mass formed during winter time remains the backbone for the theory of wind-driven
circulation, including ventilation, subduction and obduction.
Therefore, if we want to simulate the annual mean wind-driven circulation, we should
select the late winter thermohaline boundary conditions at the sea surface, including mixed
layer density and depth; but, the wind stress forcing, or the Ekman pumping rate, should the
annual mean because we are concerned about the annual-mean movement of the particles in
the thermocline.
For example, if you use the annual mean temperature and salinity condition on the surface,
your model will be unable to form the right type of deep water or mode water --- annual mean
density on the surface is much smaller than that of winter time (such as late March), so that deep
water formed in your model is too light.
Stommel's idea can be illustrated as follows. Each month a water parcel is released at the
base of the mixed layer. Assuming a constant equatorward velocity of 10-2m/s and vertical velocity
of w = 0.5 ×10−6 m / s , the trajectories of these parcels can be calculated, as shown in Fig. 2. It is
clear that water parcels released at 40oN in February will be overtaken by the mixed layer on its
way down south in March. However, water parcels released in March and April can enter the
permanent thermocline, and thus can be counted as effective subduction and water mass formation.
Fig. 2. Mixed layer properties along 160.5oE section for the North Pacific: a) Mixed layer depth
and water particle's trajectories; b) Surface density deviation from the annual mean; the heavy line
for March, the thin solid line for February, and the dashed thin line for April.
Tracer release experiments in the 1990s provided strong support for the ideal-fluid
thermocline theory because it was found that diapycnal mixing is very small in the main
2
thermocline, on the order of 0.1cm2/s. Thus, to a very good approximation the structure of the
thermocline and the wind-driven circulation can be treated as ideal-fluid motions. It is also very
important to note that wind stress is one of the important sources of energy supporting mixing in
the ocean interior; thus, although diapycnal mixing is very small, but not infinitesimal. In some
theoretical studies, people search for the limit of wind-driven circulation with an extremely small
and non-zero mixing rate; however, some of such studies may be physically meaningless.
II. The ventilated thermocline (Luyten, Pedlosky, and Stommel, 1983)
In a most innovative and remarkable paper, Luyten, Pedlosky and Stommel (1983, hereafter
LPS) formulated a multi-layer model for the ventilated thermocline and applied the model to the
North Atlantic. The model is formulated for the ocean interior, excluding the western
boundary region and the Ekman layer. The upper ocean is divided into several layers of constant
density which outcrop at different latitudes, Fig. 3. Remember that layer models are based on
density coordinates, although they are highly truncated for the most cases. The uppermost layer is
directly driven by the Ekman pumping. After subduction water particles retain their potential
vorticity and continue their southward motion. Along the eastern boundary, the model predicts that
water moving in a subsurface layer has to depart from the wall in order to maintain its potential
vorticity. Our notations are based on dimensional variables, so the figures shown are also in
dimensional units, which give the reader a clear view of the circulation calculated from the model,
in comparing with the real oceans.
W e=0
θ=θ0
θ=θ1
We
h
h1
ρ1
h2
ρ2
ρ3
ρ4
Fig. 3. Sketch of a two-moving layer model of the thermocline in the subtropical basin. The Ekman
pumping vanishes at θ = θ 0 and is downward for θ < θ 0 .
A. The ventilated zone
In the ocean interior, the basic equations are: geostrophy for the horizontal momentum
equations, hydrostatic approximation in the vertical direction, and mass conservation. IN this
3
section, we will assume that water below the second layer is stagnant. Thus, north of θ1 , only the
second layer is in motion and its momentum equations are
∂h
2ω sin θ v2 = g '
(1)
a cosθ∂φ
∂h
2ω sin θ u2 = − g '
(2)
a∂θ
where g ' = g Δρ / ρ 0 , Δρ = ρ3 − ρ 2 , and h = h2 for θ > θ1 . The mass conservation equation is
⎤
1 ⎡ ∂
∂
( h2v2 cosθ ) + ( h2u2 )⎥ + we = 0
⎢
a cosθ ⎣ ∂θ
∂φ
⎦
Subtracting Eqs. (1) and (2) into (3) gives the vorticity equation
cosθ h2v2 = a sin θ we
Substituting this relation into (1) and integrating in the zonal direction gives
h22 = h22e + D02
where h2 e is the constant layer thickness along the eastern boundary and
(3)
(4)
(5)
φ
4ω a 2 sin 2 θ e
D =−
∫ we (θ ,φ ')dφ '
g'
φ
2
0
(6)
is the layer thickness (square) deviation from that at the eastern boundary. Note that thickness of
each layer has to be constant along the eastern boundary of the model basin in order to satisfy the
boundary condition that there is no geostrophic flow across the boundary. Within the formulation
of ideal-fluid thermocline, we cannot determine h2 e . In fact, h2 e is an external parameter of the
ventilated thermocline model, and different h2 e would give rise to different solutions. The basic
philosophy of the ideal fluid thermocline is to assume that h2 e is set up by some process, such as
the thermohaline circulation, not directly simulated in the model. When h2 e is known, both u2 and
v2 can be determined geostrophically.
As water moves southward crossing latitude θ1 , the second layer continues to flow
southward under the first layer. South of the outcropping line both layers are in motion, the
momentum equations for the second layer have the same form as (1) and (2), although now
h = h1 + h2 ; thus, we have
∂(h + h )
2ω sin θ v2 = g ' 1 2 ;
(1’)
a cosθ∂φ
∂(h + h )
2ω sin θ u2 = − g ' 1 2 .
(2’)
a∂θ
⎤
1 ⎡∂
∂
(3’)
( h2v2 cosθ ) + ( h2u2 )⎥ = 0
⎢
∂φ
a cosθ ⎣ ∂θ
⎦
Accordingly, the flow in the second layer follows the constant h line, so a h-contour is also a
potential vorticity contour. Denote potential vorticity in the second layer sin θ / h2 should be a
function of h only, which we denote as of G ( h ) , i.e., sin θ / h2 = G ( h ) . Since the second layer is
now shielded from the Ekman pumping, along a streamline a water parcel maintains its potential
4
vorticity function G ( h ) is set along the line θ = θ1 (the outcrop line for layer 1 and also the
subduction line for the second layer):
sin θ
sin θ1 sin θ1
G(h) =
=
=
.
(7)
h θ =θ1 h |θ =θ1
h
The last equal sign above is due to the fact that h is constant along trajectories in the second layer
after subduction. Thus, following each streamline started from the outcropping line the potential
vorticity conservation law give us the following relation
sin θ
sin θ1
= G ( h) =
(8)
h2
h
Accordingly, thickness for these two moving layers obey
sin θ
f
h2 =
h= h
(9)
sin θ1
f1
⎛ sin θ ⎞
⎛
f ⎞
(10)
h1 = ⎜1 −
⎟ h = ⎜1 − ⎟ h
f1 ⎠
⎝ sin θ1 ⎠
⎝
Notice that layer thickness ratio is entirely determined by the planetary potential vorticity. This
simple relation gives rise to simple analytical solution for the ventilated thermocline. Were the
outcrop line not zonal, the solution would be in a much more complicated form, and this is the
beauty of the ventilated thermocline model, as proposed by Luyten et al..
To determine the still unknown total layer depth h in this zone we can use the Sverdrup
relation for the barotropic mass flux. The momentum equations for the first layer are
∂
(11)
2ω sin θ v1 = g '
(γ h1 + h2 )
a cosθ∂φ
∂
(12)
2ω sin θ u1 = − g '
(γ h1 + h2 )
a∂θ
ρ − ρ1
. The mass conservation equation is
where γ = 3
ρ3 − ρ 2
∂h u ⎤
1 ⎡ ∂
(13)
(h1v1 cosθ ) + 1 1 ⎥ + we = 0
⎢
∂φ ⎦
a cosθ ⎣ ∂θ
Following the same approach in deriving the vorticity (4), substituting Eqs. (1’, 2’) into (3’), and
Eqs. (11, 12) into (13) leads to two relations, and adding these two relations, one obtains the
Sverdrup relation for the barotropic mass flux
cosθ (h1v1 + h2v2 ) = a sin θ we .
(14)
Substituting Eqs. (1’) and (11) nto (14) and integration leads to an equation for the layer thickness
(15)
(γ − 1)h12 + h 2 = D02 + (γ − 1)h12e + he2
Along the eastern boundary the no-zonal flow condition requires both h1 and h2 be constant. Since
h1 = 0 north of θ1 , h1 must be identically zero along the entire eastern boundary. Similarly, h2 = h2e
is constant along the entire eastern boundary. Note that this is one of the major difficulties in the
ventilated thermocline theory that will be discussed late.
Using (10), one obtains the sum of layer thickness
5
h2 =
D02 + h22e
1 + (γ − 1) (1 − f / f1 )
2
, for θ ≤ θ1
(16)
B. The shadow zone
The no-zonal flux condition requires layer thickness be constant along the eastern boundary.
The constant layer thickness implies that the potential vorticity f / hi is non-constant along the
eastern boundary. Since potential vorticity should be conserved along streamlines in
subsurface layers, the eastern boundary cannot be a streamline in these layers. Thus, the
region next to the eastern boundary should be a shadow zone where fluid is stagnant, and the
motion must be confined to the uppermost layer that is directly exposed to the Ekman pumping.
Because this layer is directly forced, so potential vorticity is not conserved along streamlines. For
the present model, the upper layer has a zero thickness along the eastern boundary, so all motion is
confined into a singular line at the junction between the upper surface and the eastern boundary.
(This is a weakness in the original ventilated thermocline model, and it can be improved as will be
shown in later lectures.)
To find out the boundary between the ventilated thermocline and the shadow zone, we
notice that total depth h should be constant along streamlines in the second layer. Thus, Eq. (16)
can be used to describe this boundary in the following parametric form
D02 + h22e
(17)
= const .
1 + (γ − 1)(1 − f / f ) 2
Since the boundary between the shadow zone and the ventilated thermocline passes the boundary
point (φe ,θ1 ) where f = f1 and D02 = 0 , this boundary line φ = Φ (θ ) is determined by
D02 (Φ,θ ) = h22e (γ − 1)(1 − f / f1 ) 2 .
(18)
This boundary can be identified in numerical calculation as following. Note that h is flat
within the shadow zone, so h = h2e is valid within the shadow zone up to its western boundary.
Therefore, the western boundary of the shadow zone can be identified as the place where the total
depth h of this two-moving layer solution, calculated from Eq. (16), equal to the thickness of the
second layer along the eastern boundary, h2e .
It is easy to show that layer thickness is continuous across this line; however, tangential
velocities are discontinuous. Within the shadow zone the second layer is stagnant and the Sverdrup
flux is concentrated in the first layer. Thus, the meridional velocity can be described by
cosθ h1v1 = a sin θ we ,
(19)
and the layer thickness is
1/ 2
⎛ ρ − ρ2 ⎞
h1 = ⎜ 3
⎟
⎝ ρ 2 − ρ1 ⎠
D0 (θ ,φ ) .
(20)
C. The basic structure of the ventilated thermocline
The typical structure of the solutions of the model is shown in Figs. 4. The model is driven
by a simple sinusoidal Ekman pumping we = w0 sin (π ( y − ys ) / ( yn ) − ys ) , w0 = 1.0 ×10−4 cm / s ;
g ' = 1cm / s 2 , and ρ3 − ρ 2 = ρ 2 − ρ1 ; the lower layer thickness along the eastern boundary is 500m,
and the upper layer outcrops along 44oN. Note that south of the outcrop line and near the western
6
boundary there is water coming out from the western boundary currents whose potential vorticity
cannot be determined by the interior model alone, and this is called the pool zone. Strictly
speaking, we do not really know the solution within the pool region. The solution drawn in Fig. 4 is
obtained under the assumption that south of the outcrop line and in the pool zone, potential
vorticity function in the second layer remains in the same form as that east of the pool zone. In fact,
such a solution can be obtained, if one extends the model basin beyond the current western
boundary, and using part of the solution for this enlarged basin which falls within the boundary of
the current model basin.
According to the model, therefore, there are different dynamic regions in the second layer:
the region north of the outcrop line where water in the second layer is directly exposed to external
forcing; south of the outcrop line, there are three regions in the second layer: the pool region where
water comes from the western boundary region, the ventilated region where water continues its
southward motion, started from the open window north of the outcrop line where the second layer
is directly exposed to Ekman pumping, and an eastern region which consists of shadow zone below
the moving uppermost layer.
a) Upper layer thickness
b) Lower layer thickness
50
50
6
45
40
40
7
6
1
Latitude
Latitude
5
45
1
35
7
35
6
30
2
4
25
2
0
1
10
20
30
40
Longitude
3
1
50
3
25
20
5
4
5
30
3
60
20
0
4
5
10
20
30
40
Longitude
4
5
50
60
Fig. 4. Layer thickness (in units of 100m) of a two-layer ventilated thermocline. The upper layer
outcrops along 44oN, indicated by the heavy dashed line, and the shadow zone is depicted by the
thin dashed line in the south-east part of the basin.
Note that although thickness contours in both layers are continuous, their slope is
discontinuous across the boundary of the shadow zone, as shown in Fig. 4a and 4b; however,
streamlines in the upper layer are continuous across this line, in indicated by 2h1 + h2 contours in
Fig. 5, but the velocity is discontinuous across this line.
7
Upper layer geostropic flows
50
45
Latitude
40
35
30
25
20
0
10
20
30
Longitude
40
50
60
Fig. 5. Geostrophic flows in the upper layer, with the outcrop line and the boundary of the shadow
zone depicted by dashed lines.
The vertical structure of solution is shown in Fig. 6, where the shadow zone, corresponding
to the places where the lower interface is flat, is clearly seen.
b) Meridional section
0
−1
−1
−2
−2
Depth (100m)
Depth (100m)
a) Zonal section
0
−3
−3
−4
−4
−5
−5
−6
0
10
20
30
40
Longitude
50
60
−6
20
25
30
35
Latitude
40
45
50
Fig. 6. Zonal section along 26oN and meridional section along 45oE of the two-ventilated-layer
model.
The ventilated thermocline theory provides a solid dynamical structure for the schematic
picture of ventilation process proposed by Iselin (1939). The existence of a shadow zone required
by the ventilated thermocline is consistent with observations that a large zone of oxygen minimum
water exists at mid-depth in both the North Atlantic and North Pacific.
D. Potential vorticity maps for unventilated layers
We have assumed that the third layer is stagnant in the analysis above. Although a
motionless third layer is a consistent assumption, there is a possibility that the third layer is in
8
motion yet all the dynamical constraints are not violated. In the past, studies based on layer models
have been concentrated on cases when the interfacial deformation is small and can be treated as a
small perturbation. Consequently, potential vorticity contours in the subsurface layers are
dominated by planetary vorticity, the β term. In the thermocline theory what we are dealing is
characterized by large layer thickness change; thus, the interfacial deformation is no more
negligible.
Let us use the same Ekman pumping field and assume that along the eastern boundary both
the lower layer and bottom layer thickness equal to 500 m. If we assume that third layer is still
motionless and calculated the potential vorticity contours in the third layer, Fig. 7.
sin θ
Π3 =
(21)
h2 e + h3e − D02 + h22e
Near the eastern boundary D02 → 0 , so potential vorticity is controlled by the planetary vorticity
sin θ / h3e . As a result, some potential vorticity contours emerge from the eastern boundary, along
which geostrophic motion is forbidden. However, there are contours steaming from the western
boundary as well. According to the ideal fluid theory, water is free to travel along these contours.
Bottom layer gestropic contours
P
50
3
3
45
2
4
40
Latitude
4
3
35
2
3
1.7
2
30
1.7
2
1.4
1.7
1.4
1.4
25
1.1
20
0
10
1.1
20
30
Longitude
1.1
40
50
60
Fig. 7. Isolines of potential vorticity in the bottom layer. Although east of the dashed line potential
vorticity contours is blocked by the eastern boundary, west of the dashed line potential vorticity
contours disconnected from the eastern boundary are closed through the western boundary.
The essential feature of this vorticity map is that the meridional vorticity gradient has
different signs on the two sides of point P, which location is determined by
d Π3
= 0, at θ = θ 0
(22)
dθ
Note that near the inter-gyre boundary,
obtains
φe − φ p =
cos θ 0 g ' h2 e h3e
⎛ ∂w ⎞
4ω a 2 sin 3 θ 0 ⎜ e ⎟
⎝ ∂θ ⎠ θ =θ0
D02 + h22e ≈ h2 e . After some algebraic manipulations, one
(23)
9
Point P is some time called the Rossby repellor where characteristics from the eastern boundary
and the western boundary meet, and it is a very important singular point in thermocline theory. The
characteristic stated from P separates the basin into two regions: the eastern region where vorticity
contours start from the eastern boundary and geostrophic motion is forbidden, and the western
region where vorticity contours start from the western boundary and fluid is free to travel along
these contours, assuming no friction in the western boundary current --- an idealization widely used
to construct solution in the ocean interior in ideal-fluid thermocline theory.
Note that the existence of the western region depends on the smallness of h2e and h3e and
strong forcing we . If layers are too thick and forcing is not so strong, P would be located to the
west of the western wall, so there would be no close geostrophic contours. This is the case which
has been explored many times before. It is also interesting to notice that for a given forcing we and
h2e , we can always choose a h3e so small that the Rossby repellor is located within the basin
interior. Therefore, close potential vorticity contours always exist if the layer thicknesses in the
model are chosen properly.
E. Potential vorticity homogenization for unventilated layers
The vorticity balance in the third layer can be written as
G
∇ ⋅ (u3h3Π 3 ) = D3
(24)
where D3 is the vorticity dissipation term. Integrating (24) over a closed area gives
∫ ∫ D r cosθ dθ dφ = 0
3
(25)
A3
because the advection terms integrate to zero. We also assume a special form of vorticity
dissipation
D3 = ∇ H ⋅ F3 and F3 = −κ∇ H Π 3
(26)
thus, the integral is reduced to
∂Π
(27)
vC∫ ∂n3 dl = 0
3
Since dissipation is assumed to be very weak, vorticity is nearly conserved along streamlines. Thus,
Π 3 = Π 3 ( p3 ) , and (27) is further reduced to
G G
∂Π 3
κ
u
(28)
3 H ⋅ tdl = 0
∂p3 Cv∫3
If the fluid is in motion, the integral is no-zero, so
∂Π 3
= 0, on C3
(29)
∂p3
By repeating the same argument for all closed contours, one comes to a conclusion that potential
vorticity is uniform within the domain defined by the out most closed potential vorticity contour.
Finally, we notice that if the water does not move, vorticity is not homogenized.
Notice that the potential vorticity homogenization depends on the assumption of small
down-gradient dissipation. The solutions with an unventilated layer where potential vorticity is
homogenized can be found easily following the analysis in section B. East of Φ p the third layer is
motionless, so there is only one moving layer. West of Φ p potential vorticity is homogenized in
10
the third layer. Since potential vorticity should be a constant along the line started from Φ p , this
constant is equal to the potential vorticity along the northern boundary. Accordingly, the third layer
thickness is
f
h3 = h3e
(30)
f0
The zonal momentum equations for the second and third layers are
∂
2ω sin θ v2 = g ' 3
(γ 3 h2 + h3 )
(31)
a cos θ∂φ
∂
2ω sin θ v3 = g ' 3
(h2 + h3 )
(32)
a cos θ∂φ
where
ρ − ρ3
ρ − ρ2
g '3 = g 4
, γ3 = 4
(33)
ρ0
ρ 4 − ρ3
The Sverdrup relation for these two layers is
cosθ (h2v2 + h3v3 ) = a sin θ we
(34)
Combining these three equations, one can obtain an integral relation for the layer thickness
(γ 3 − 1)h22 + (h2 + h3 ) 2 = D02 + (γ 3 − 1)h22e + (h2 e + h3e ) 2
(35)
Combining with (30), one can calculate the layer thickness h2 and h3 . The velocity field can be
calculated from the geostrophy.
To simplify the problem, we will discuss the case when layer 1 outcrop south of the
southern boundary of the model basin; thus, there is actually no layer 1 within our model domain.
In the following discussion, we will call layer 2 the upper layer, which now covers the whole model
basin, and layer 3 will be called lower layer. Thicknesses of both layers along the eastern boundary
are set to 500 m. The density stratification and Ekman pumping field remain the same as discussed
above.
The structure of the solution is shown in Fig. 8. It is readily seen that the solution consists of
two dynamics domains separated by the eastern boundary of the potential vorticity homogenization
region in the lower layer, which can be easily identified as the domain where the lower layer
thickness is independent of latitude (Fig. 8b). The construction of the solution is the following.
Using Eqs. (30) and (35), one can calculate the solution within the domain of potential vorticity
homogenization in the lower layer. Using this method, one can formally calculate the distribution
of total layer thickness h = h2 + h3 , and the eastern boundary of the domain of potential vorticity
homogenization is the line where h = h2 e + h3e because east of this line the lower layer is stagnant,
so the lower interface should flat.
When the third layer is allowed to move, part of the Sverdrup mass flux is now distributed
into the third layer. Consequently, over the potential vorticity homogenization zone of the third
layer, the mass flux in the second layer is reduced, so is the interfacial deformation. For example, in
the present case, layer 2 has a thickness about 600m at 46oN and near the western boundary (Fig.
8a); however, the corresponding layer thickness for the case with no close contour in layer 3 is
slightly larger than 700m (Fig. 4b). East of the closed geostrophic contours in the lower layer, layer
thickness in these two layers are exactly compensating each other, indicating a flat interface below
layer 3. The layer thickness compensation can be seen easily in zonal and meridional section, Fig.
9.
11
a) Upper Layer thickness
b) Lower layer thickness
50
50
5.5
5.5
6
6.5
Latitude
7
4.5
4
4
5.
6.5
3.5
30
5
30
35
5
35
4.5
40
6
40
Latitude
45
3.5
6
25
4
25
4.5
5.5
20
4.
45
0
10
20
30
40
Longitude
50
20
60
0
10
20
30
40
Longitude
50
60
Fig. 8. Layer thickness (in units of 100m) for a two-and-a-half layer ventilated thermocline model.
b) Meridional section
0
−1
−1
−2
−2
−3
−3
−4
−4
Depth (100m)
Depth (100m)
a) Zonal section
0
−5
−6
−7
−8
−5
−6
−7
−8
−9
−9
−10
−10
−11
0
10
20
30
40
Longitude
50
60
−11
20
25
30
35
Latitude
40
45
50
Fig. 9. Zonal section along 48oN and meridional section along 45oE of the two-layer unventilated
model, with potential vorticity homogenized in the lower layer within the closed geostrophic
contours.
III. Basic assumptions made in the ventilated thermocline model (LPS)
1) Sea surface density ρ s is specified. Comment: It is difficult to calculate ρ s at the same time.
2) Ekman pumping is specified. Comment: One can also specify the wind stress.
3) Ideal fluid assumptions. Therefore, density, potential vorticity, Bernoulli function are conserved
along the trajectories after subduction.
4) No interfacial mass flux. Comment: Interfacial mass flux can be included; however, the problem
is more complicated, this will be discussed shortly.
5) No mixed layer and the upper surface is put at z=0. Comment: The mixed layer can be added
easily, we will discuss this issue.
6) Stratification along the eastern boundary must be specified. Comment: This is one of the basic
assumptions of the ideal-fluid thermocline theory.
7) The western boundary current is excluded. Comment: Inertial western boundary current can be
matched with the inertial flow under some constraints.
12
8) No seasonal cycle. Comment: The model is based on the Stommel Demon, i.e., the model is
forced by late winter mixed layer properties. Coupling the model with seasonal cycle is still
one of the most challenging questions.
IV. Basic structure of the ventilated thermocline.
1) The ventilated region where potential vorticity is conserved along trajectory.
2) The shadow zone where subsurface water is stagnant. Thus, the stratification is whatever
we specified and the potential vorticity cannot be determined by the model. Comment: As the
number of layer increases, the number of different shadow zone may increases exponentially. What
can we do about it?
3) The pool region was left untouched in the original LPS model. However, it can be shown
easily that as the number of layer increases, the area occupied by the pool region also increases
quickly, shown in Fig. 10. Thus, it is desirable to find out whether continuously stratified model
can overcome such problem.
Shadow
zone
The pool
Fig. 10. The expansion of the shadow zone and the pool region, as the outcrop line moves
northward.
Reference:
Luyten, J., J. Pedlosky, and H. M. Stommel, 1983: The ventilated thermocline. J. Phys. Oceanogr.,
13, 292-309.
Iselin, C. O'D., 1939: The influence of vertical and lateral turbulence on the characteristics of the
waters at mid-depths. Trans. Amer. Geophys. Union, 20, 414-417.
Stommel, H. M., 1979. Determination of watermass properties of water pumped down from the
Ekman layer to the geostrophic flow below. Proc. Natl. Acad. Sci. U.S., 76, 3051-3055.
13