Lecture 15. Subduction and obduction
4/17/2006 8:43 PM
1. Introduction
A) Iselin's (1939) model.
A major conceptual difficulty in getting subsurface layer in motion is that they are not
directly in contact with the atmospheric forcing locally. However, by looking at any hydrographic
section, one realizes that most isopycnals are in contact with the atmosphere, primarily at high
latitudes, i.e., they are somehow ventilated through the outcropping window.
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 in Lecture 13. The arrows indicate the speculated motion. 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 observation, such as the recent tracer release experiments.
Fig. 1. Iselin’s conceptual model for water mass formation due to water sinking along isopycnal
surfaces (Iselin, 1939).
Iselin's model was the first prototype for water mass formation in the oceans; however, it
was incomplete in two aspects. First, Iselin neglected the mixed layer that plays a most important
role in water mass formation. Second, since mixed layer depth and density change greatly from
season to season, it was not clear how to make the link between water mass properties and mixed
layer properties at winter as suggested by Iselin.
B) How to calculate the water mass formation rate?
According to Iselin's model, the Ekman pumping rate might serve as the water mass
formation rate. Although, this incorrect concept had dominated for a long time, it turned out that
the Ekman pumping rate is not exactly the rate of water mass formation. A better way is to
calculate the mass flux across the base of the mixed layer. Mixed layer models have been
developed, and they can provide the seasonal cycle of the entrainment/detrainment rate across the
base of the mixed layer. Can we use the annually integrated entrainment/detrainment rate as the
local water mass formation rate? The answer is NO. Water leaving the mixed layer may not enter
the permanent pycnocline; instead, some of it may be re-entrained into the mixed layer
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downstream. Similarly, water entrained into the mixed layer may not come from the permanent
pycnocline; instead it may be water temporarily detrained upstream.
C) Subduction/obduction rate
A modified conceptual model is shown in Fig. 2. The upper ocean is divided into four
layers, the Ekman layer, the mixed layer, the seasonal pycnocline, and the permanent pycnocline.
Water mass exchange between the mixed layer and the seasonal pycnocline is called
entrainment/detrainment, while water mass exchange between the seasonal pycnocline and the
permanent pycnocline is called subduction/obduction. Accordingly, the annual mean subduction
rate is defined as the total amount of water going from the mixed layer, passing through the
seasonal pycnocline, to the permanent pycnocline irreversibly in one year. Similarly, the annual
mean obduction rate is defined as the total amount of water going from the permanent pycnocline,
passing through the seasonal pycnocline, to the mixed layer irreversibly in one year.
Fig. 2. Water mass formation and elimination through subduction and obduction process.
2. The Stommel demon
A major technical difficulty was that water properties and mixed layer depth vary
considerably in a year. By carefully analyzing the processes involved, Stommel (1979) was able to
show that a process is at work that selects only the late winter water for actual subduction into the
permanent pycnocline, Fig. 3. As will be discussed later, the effective detrainment period is marked
by the Lagrangian trajectories of water particles released from the base of the mixed layer. The
basic mechanism is that the mixed layer reaches its annual maximum density and depth at late
winter, so there is a very thick layer of almost vertically homogenized water. When spring comes,
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the mixed layer shoals very quickly (as indicated by the sharp turning of the mixed layer depth in
the upper and lower panels of Fig. 3) and leaves the homogenized water behind, so that water
subducted has properties very close to those of the late winter mixed layer. It is readily seen that if
the time evolution of the mixed layer depth really is like a δ function, i.e., ΔT → 0 , the subducted
water would have the properties of late winter water. Stommel called this process a demon; thus, it
seems appropriate for us to call it the "Stommel demon" to honor his contribution.
Fig. 3. The Stommel demon: how the subduction selects the mixed layer properties at late winter.
Notice that the subduction process in the oceans is a very complicated process involving the
seasonal cycle. In fact, now the challenging problem is to find out the annual mean subduction rate
including the seasonal cycle. In one way, this mean can be considered as some kind of weighted
average of the instantaneous detrainment rate. Choosing the late winter properties is equivalent to
using a δ function as the weight-function. Stommel's suggestion has been used extensively in
almost all theoretical models of the ventilated pycnocline. What Stommel suggested gives an
elegant solution to this rather intricate problem. This can be thought of as the lowest-order solution.
The next step is to find out a weight-function that is better than the δ function. In other words, we
would like to know the next-order correction to the subduction rate calculated according to the
Stommel formula. As such a correction must include the seasonal cycle, it is not an easy problem.
3. Subduction
Let us begin with a layered model without a seasonal cycle. In such a model the ventilation
process can be divided into two steps. First, ventilation occurs when water flows downward from
the mixed layer into a layer below. Second, water in each density layer follows a southward motion
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induced by Sverdrup dynamics. Thus, water within a dense layer will move underneath the next
lighter layer. This process of the submersion of a denser layer under a lighter one is called
subduction. The term subduction has been used in geology to describe the similar process during
the movement of tectonic plates. According to this strict classification, as the number of layers
increases, the first stage (ventilation) becomes shorter and shorter. It is readily seen that for a
continuously stratified ocean, these two stages will merge, and we use the term subduction and
reserve the term ventilation for the general case of either subduction or obduction.
The seasonal cycle plays one of the most important roles in the upper ocean dynamics, so
we must include the seasonal cycle in our subduction model. The most important parameter
describing the subduction/ventilation process is the subduction rate. The instantaneous detrainment
rate is defined as the volume flux of water per unit horizontal area (Cushman-Roisin, 1987)
G
D = − ( wmb + vmb ⋅ ∇h + ∂h / ∂t )
(1)
where wmb = we −
β
∫
0
G
vdz and vmb are the vertical and horizontal velocity at the base of the
f −h
mixed layer, and h is the mixed layer depth, Fig. 4. The first term on the right-hand side is
contribution due to vertical pumping at the base of the mixed layer, which is slightly smaller than
the Ekman pumping rate due to the geostrophic flow in the mixed layer. The second term is due to
the lateral induction. The third term is due to the mixed layer detrainment.
Fig. 4. The definition of instantaneous detrainment rate.
If there were no seasonal cycle, the subduction rate would be the same as the detrainment
rate,
G
S = D = −( wmb + vmb ⋅ ∇h )
(2)
Thus, this equation can be used for calculating the subduction rate, if there is no seasonal cycle;
however, the strong seasonal cycle in the oceans make the subduction rate calculation much more
complicated.
Another parameter commonly used in the description of tracer ventilation is the so-called
ventilation rate of an individual isopycnal or water mass, defined as
( water mass )Volume
(3)
Vr =
S
where S is the subduction velocity defined above. Physically, the ventilation rate determines the
average time (in units of years) it takes to renew the entire water mass through the ventilation
process, or the average time water particles remain in a water mass category, for example, Jenkins
(1987).
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In earlier models, the mixed layer thickness was set to zero for simplicity, so the only term
contributing to subduction was the vertical pumping, which was the same as the Ekman pumping
since the mixed layer thickness was zero. One of the pitfalls in the subduction process is this notion
that the subduction velocity seems to be the same as the Ekman velocity. Actually, the mixed layer
depth is non-zero and it varies with time and location, so each term on the right-hand side of (1)
contributes differently.
First, because the mixed layer has a finite thickness, the vertical velocity at the base of the
mixed layer is slightly smaller than the Ekman pumping velocity. Second, the depth advection term
actually contributes to subduction substantially. In the North Atlantic the wintertime mixed layer
deepens northward from 100 meters to about 400 meters within 3000 kilometers, so we estimate
that the mixed layer slope is about 0.0001. Given that the meridional velocity in the mixed layer is
about 1 cm/sec, we conclude that the contribution from vertical pumping and that from the lateral
induction are the same order. According to a calculation by Huang (1990), the contribution from
the vertical pumping amounts to 12.1 Sv, and that from the lateral induction is about 12.7 Sv.
When the time-dependent term is non-zero, the situation becomes much more complicated. First of
all, there is the seasonal pycnocline between the mixed layer and the permanent pycnocline. The
seasonal pycnocline plays the role of a buffer, i.e., the mass exchange between the mixed layer and
the permanent pycnocline must go through the seasonal pycnocline. Thus, a complete picture must
consist of four layers, as shown in Fig. 2. We will call the mass flux from the seasonal pycnocline
to the permanent pycnocline subduction, while the mass exchange between the mixed layer and the
seasonal pycnocline is called detrainment/entrainment. In general, the subduction rate is different
from the detrainment rate because they represent different processes!
Motions in the mixed layer have a broad spectrum in space and time; there are two
prominent cycles in the mixed layer, i.e. the diurnal cycle and the annual cycle. For simplicity, we
will discuss the seasonal cycle only. Since we have assumed that the flow in the permanent
pycnocline is time independent, the subduction across the base of the seasonal pycnocline does not
vary with time. However, water mass exchange across the base of the mixed layer has a prominent
seasonal cycle. In earlier theory of the mixed layer, the seasonal cycle was separated into two
phases, detrainment and entrainment. However, advancement of the subduction theory suggests that
the seasonal cycle in the subtropical basin can be further divided into three phases (CushmanRoisin, 1987). Between late winter and early fall, detrainment is activated due to the Ekman
pumping and mixed layer shoaling. This period can be further divided into two sub-phases: From
late winter to early spring, water entering the seasonal pycnocline from the mixed layer will
eventually reach the permanent pycnocline; this process is called the effective detrainment. From
early spring to early fall, water entering the seasonal pycnocline will be re-taken by rapid mixed
layer deepening during the winter season, resulting in temporary (ineffective) detrainment (Fig. 5).
From early fall to late winter, the mixed layer deepens rapidly -- the entrainment phase. It appears
that the temporal and spatial inhomogeneity of motions in the mixed layer can create a fairly
complicated detrainment/entrainment process, and comprehensive understanding of the intermittent
and sporadic nature of detrainment/entrainment and its contribution to subduction is yet to come
through observations and theoretical investigations.
4. Subduction rate defined as an integral property
Were the mixed layer to overlay a stagnant ocean, the subduction rate would be a purely
local property. In the oceans the mixed layer overrides currents in the seasonal/permanent
pycnocline. As soon as water particles are left behind the mixed layer, they are carried downstream
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by the currents. There is no chance that water particles could be overtaken at the same location
where they first left the mixed layer. This situation is very similar to a human breathing air. A
person confined in a small box may breathe the same air again and again. A jogger can never inhale
the same air he exhales.
In the oceans, the mixed layer can only overtake the water pumped down from upstream,
but not the water pumped down at the same location. Although a person sitting at a station may
know the local rate of mixed layer entrainment/detrainment as a function of time, that person
cannot be sure how much of this water really reaches the permanent pycnocline. To obtain the
correct answer, one has to check at stations downstream because subduction is a non-local process.
The subduction rate can be defined in different ways depending on the coordinates used. First,
subduction rate can be defined in Lagrangian coordinates (Woods and Barkmann, 1986).
1 T
Δh
(4)
S L = ∫ wdt + L
T o
T
where T is the time of the average, taken as one year because of the seasonal cycle, and ΔhL
means the mixed layer depth change accumulated over a one-year trajectory in Lagrangian
coordinates. Thus, this definition includes both the temporal average and the spatial average over a
one-year trajectory. A schematic picture in Fig. 5 illustrates this definition for a two-dimensional
case. An instrument called a Bobber is released in late winter when effective detrainment starts at a
station. This instrument can be checked continuously by acoustic signals. If we were following the
instrument, we would see that during the first part of the trajectory effective detrainment takes
place, i.e. the mixed layer retreats and leaves stratified water behind. During the second half of the
trajectory mixed layer entrainment takes place and re-takes part of the water which entered the
seasonal pycnocline (at an earlier time). Thus, the seasonal cycle can be divided into three phases,
i.e., the effective detrainment phase during which water that left the mixed layer flows
geostrophically into the seasonal pycnocline and enters the permanent pycnocline irreversibly, the
ineffective detrainment phase during which water that entered the seasonal pycnocline will be retaken later (at a downstream location), and the entrainment phase. Calculation of the subduction
rate requires accurate information about the kinematic structure of the mixed layer and the velocity
field in the pycnocline. Because such detailed information is very difficult to obtain from oceanic
climatology, a simplified formula is
( d − d ) − ( hm,1 − hm,0 )
S L = tr ,1 tr ,0
(5)
T
where d tr ,0 and d tr ,1 are the depth of the trajectory at the beginning and end of one year, hm,0 and
hm,1 the depth of the mixed layer at the beginning and end of one year, T=1 year is the duration of
the motion.
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Fig. 5. The annual mean subduction rate defined in the Lagrangian coordinates.
The annual mean subduction rate can be defined also in Eulerian coordinates. In this case
we are standing by a fixed station. In order to calculate the annual subduction rate, we will monitor
the local mixed layer detrainment and entrainment. In addition, we have to follow the trajectories of
particles released from our station to see whether these particles will eventually enter the permanent
pycnocline or whether they will be overtaken by the mixed layer entrainment downstream (Fig.6).
Similar to the case in Lagrangian coordinates, the seasonal cycle in the mixed layer can be divided
into three phases, effective detrainment, ineffective detrainment, and entrainment (Fig. 6).
The annual mean subduction rate is defined as
1 TE
S E = ∫ Ddt
T TS
indicate the starting and ending times of the effective detrainment. In Fig. 5 we have assumed a
simple case for the northern part of the subtropical basin where the mixed layer slope increases
northward and the Ekman pumping velocity increases southward.
Fig. 6. Finding the critical trajectory that defines the end of the effective detrainment period by
tracing trajectories released at a fixed station and regular time interval.
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The two additional terms are defined as
d
S M = wmb + vmb hmax
(6)
dy
d
S F = wmb + vmb h
(7)
dy
where h is the annual mean mixed layer depth. Note that both of these definitions treat the
subduction in the local sense, so these two subduction rates do not include the average over the
trajectory downstream. As a result, the rates calculated from these two equations are smaller than
the rates calculated from the above defined S E and S L , both of which include the contribution
resulting from the spatial variance of the Ekman pumping velocity and mixed layer depth, Fig. 7.
Thus, one should not use a simple annual mean for calculating the so-called annual mean
circulation; there is always some nonlinear effect, which must be investigated carefully.
Fig. 7. An example of the seasonal cycle of the detrainment/ entrainment and different annual mean
subduction rates.
5. Obduction
Upwelling/entrainment prevails in subpolar basins. Similar to what happens during the ineffective
detrainment period, water entrained into the mixed layer may not actually come from the
permanent pycnocline instead, it may come from the seasonal pycnocline whose water was
detrained from the mixed layer previously (Woods, 1985; Cushman-Roisin, 1987).
In order to clarify the physical processes involved in entrainment, we introduce the term,
obduction. Obduction has been used in geology in describing the process of upward thrusting of a
crystal plate over the margin of an adjacent plate. Here obduction is borrowed to describe the
process in which water from the permanent pycnocline upwells into the mixed layer and flows over
the adjacent layers of water (Fig. 8). Although obduction is basically a continuous process between
the permanent pycnocline and the seasonal pycnocline, effective entrainment from the seasonal
pycnocline to the mixed layer occurs only during part of the entrainment period. Water from the
permanent pycnocline, which has not been exposed to the surface processes, is entrained into the
mixed layer through the pycnocline. During the rest of the entrainment period (i.e. the ineffective
entrainment period), water that has been exposed to air-sea interactions in the past year enters
entering the mixed layer from the seasonal pycnocline.
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Fig. 8. An example of obduction, upwelling with a uniform upwelling of 18 m/year. The mixed
layer depth has a simple sinusoidal cycle.
Using a special term, obduction, helps us to clarify the irreversible mass flux from the
permanent pycnocline to the mixed layer. For example, although mixed layer entrainment takes
place in subtropical basins during a seasonal cycle, in most places water entrained into the mixed
layer actually comes from the seasonal cycle, so there is no obduction. In fact, obduction takes
place only within the subpolar basins and the subtropical-subpolar boundary regions, as will be
shown through processing the climatological data.
The obduction rate can be defined in a way similar to the subduction rate. Though
obduction is an antonym of subduction, obduction is not simply subduction with an opposite sign.
There are two major differences between these two terms. First, the physical processes involved in
subduction and obduction are different. Subduction takes place in the subtropical basins, where
water geostrophically flows down into the permanent pycnocline. As a result, water subducted to
the permanent pycnocline carries late winter mixed layer properties. In comparison, obduction
takes place in the subpolar basin, where water from the permanent pycnocline below flows
geostrophically upward into the seasonal pycnocline and eventually enters the mixed layer. As
water enters the mixed layer, it quickly loses its identity as a result of strong mixing, and it is
impossible to keep track of the trajectory of an individual water parcel afterwards.
The difference in the physics is reflected in the mathematical formulation of the suitable
boundary value problems for the pycnocline structure in the subtropical and subpolar basins. The
pycnocline equation is a nonlinear hyperbolic equation (Huang, 1988a, b). In the subtropical basin,
density is specified as an upper boundary condition because the upper surface is the upstream
boundary. In the subpolar basin, the upper surface density cannot be specified. In fact, the mixed
layer density is determined by the dynamics of the permanent pycnocline and is a part of the
solution.
Second, the times when effective detrainment and effective entrainment take place are
different. It is well known that effective detrainment takes place after late winter when the mixed
layer reaches its annual maximum depth and density and starts to retreat. Effective entrainment
takes place between late fall and early winter when the mixed layer deepens quickly, but before it
reaches its annual maximum depth and density.
In calculating the obduction rate, it is important to trace back to the origin of the entrained
water. To make our presentation clearer, we will assume t=0 at late winter, say March 1. We will
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begin with a simple case where the mixed layer depth has a simple sinusoidal cycle, and its
amplitude is spatially uniform. The upwelling rate is 18m / year , and it is also uniform along the
trajectory. Because the mixed layer is relatively shallow, we will assume that the vertical velocity is
approximately the same as the Ekman upwelling rate everywhere along the trajectory. Below the
base of the mixed layer mixing is negligible, so the identity of water parcels is preserved. As a
result, the particle trajectory can be used to trace the origin of the water before it is entrained into
the mixed layer. As soon as water parcels enter the mixed layer, they lose their identity because of
the strong vertical mixing within the mixed layer, and it is impossible to continue tracking the paths
of water particles.
During spring and early summer the mixed layer retreats and leaves the stratified water
behind, so this is the period of detrainment, although it is only a temporary detrainment in the
present case. Beginning in early fall, the mixed layer deepens and entrainment takes place. The
water entering the mixed layer during the first period does not really come from the permanent
pycnocline. In fact, these water parcels were detrained into the seasonal pycnocline at an earlier
time and at some upstream locations, as indicated by the top five lines in Fig. 8. Such water was
contaminated by the mixed layer in the past year, so this is not real effective entrainment. It is only
within the second phase of entrainment that water from the permanent pycnocline enters the mixed
layer, as indicated by the lower trajectories.
For the situation shown in Fig. 8 a simple calculation shows that for the case with a uniform
upwelling of 10m/year, the effective entrainment starts at TS = 0.8762 and ends at TE = 1.0088 . On
an ordinary calendar, the effective entrainment starts at January 18 and ends at March 4, and the
duration is about 44 days. The annual obduction rate is O(10 m/year) , the same as the Ekman
upwelling rate.
The contrasts between subduction and obduction are the following:
Table 1. Ventilation: Subduction vs. Obduction
Subduction
Obduction
Mass flux
From Mixed Layer to
From Thermocline to Mixed
Thermocline
layer
Water mass
Formation
Elimination
Time
Spring
Winter
Atmospheric forcing
Heating
Cooling
Mixed layer
Shoaling
Deepening
Trajectory tracking
Downstream
Upstream
6. Obduction rate defined as an integral quantity
The obduction rate can be defined slightly differently, depending on the coordinates used.
First, paralleling Woods (1985), the obduction rate can be defined in the Lagrangian sense.
Accordingly, the annual mean obduction rate is
1 0
Δh
(8)
OL = ∫ wtr dt + L
T −T
T
where T is the time of the average, taken as one year because of the seasonal cycle, the subscript tr
indicates the critical trajectory in Fig. 8 (which marks the end of obduction), and ΔhL indicates the
mixed layer depth change accumulated over one year's Lagrangian trajectory. Note that both terms
include the temporal average over the past year and the spatial average over the one-year trajectory.
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Second, paralleling Cushman-Roisin (1987), an instant entrainment rate in Eulerian coordinates can
be defined as
G
E = wmb + umb ⋅ ∇h + ∂h / ∂t
(9)
where the subscript mb indicates the base of the mixed layer. Because the instantaneous
entrainment rate fluctuates greatly during one seasonal cycle, it is not convenient to use it as an
index for the water mass conversion rate; however, it is more meaningful to use the annual mean
obduction rate defined as
1 TE
OE = ∫ Edt
(10)
T TS
where TS and TE are the times when the effective entrainment started and ended, E is the
instantaneous entrainment rate defined in (9). Again, a pitfall in calculating the annual mean
obduction rate is using a simple Eulerian mean over the whole year
1 0
(11)
OE = ∫ Edt
T −T
G
Substituting (9) would lead to a wrong estimation of OE = wmb + umb ⋅∇ h . In general, this
substitution tends to underestimate the annual mean obduction rate.
The major differences between these two definitions of obduction rate follow. In Eulerian
coordinates, water parcels entrained into the mixed layer at one station are monitored, and
trajectories of parcels are traced back upstream for one year to determine whether the water comes
from the permanent pycnocline or not. Accordingly, the beginning and the end of the effective
entrainment are determined, and the annual obduction rate can be calculated by (10). In Lagrangian
coordinates, water parcels entrained into the mixed layer at a station are monitored to determine the
critical trajectory that marks the end of obduction. Given this trajectory, one can trace back along
the trajectory for one year and calculate the obduction rate by (8).
Calculating the annual mean subduction rate according to these two definitions requires accurate
information on the spatial and temporal evolution of the mixed layer, and such detailed information
is almost impossible to obtain from any climatic data. Thus, we propose a simple way of
calculating the annual mean Lagrangian obduction rate by
( d − d ) − ( hm,−1 − hm,0 )
OL = tr ,−1 tr ,0
(12)
T
where d and h are the depth of the trajectory and the mixed layer base, subscript and 0 and -1
indicates the fact that these quantities are calculated at the beginning of the year and the previous
location by upstream-tracing of the critical trajectory for one year.
Although obduction rates calculated according to these definitions are slightly different,
their difference is quite small compared with the errors existing in present-day climatology data.
Therefore, our definition (12) can serve as a convenient and practical tool for calculating the
subduction rate from climatological data with errors comparable to those of other processes.
7. Water mass formation rate balance
Water mass formation and elimination in the oceans take place in the oceans through
subduction and obduction processes discussed in this lecture. With the mixed layer and the
Stommel demon these processes can be schematically viewed as in Fig. 9.
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Fig. 9. Define the subduction and obduction rate in the Lagrangian coordinate.
In a steady state, the water mass volume between two density surfaces must be in balance, Fig. 10:
M Sub − M Ob + M DM + M LF = 0
(13)
For a basin with open boundaries, the contribution of the last term should be balance by the mass
flux through the mixed layer, i.e.,
(14)
∫ ∫ M LF = ∫ ∫ M ML
The contribution due to the diapycnal mixing integrated over all the density range also vanishes
because this term can transform water mass between different density categories. Thus, for a basin
with open boundaries the total amount of subduction should equal to the total amount of obduction
plus the mixed layer inflow from the side boundary
(15)
∫ ∫ M Sub = ∫ ∫ M Ob + ∫ ∫ M ML
For a closed basin, there is no mixed layer inflow through the side boundaries; thus, the total
amount of subduction and obduction should be exactly balanced:
(16)
∫ ∫ M Sub = ∫ ∫ M Ob
Equation (15) or (16) is one of the most important constraints for the water mass formation rates in
a open (closed) basin, and we will show that this constraint is satisfied in our diagnostic calculation
for the Pacific.
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Fig. 10. Water mass balance, where Sub indicates subduction, Ob for obduction, DM for diapycnal
mixing flux, and LF for lateral flux through the side boundary.
8. Examples of subduction/obduction
This is a natural place to examine the ventilation rate at the place where the Kuroshio (or
the Gulf Stream) separates from the western boundary. A prominent maximum in the late winter
mixed layer depth exists off the coast of Japan, where there is strong cooling by cold, dry polar air
from the continent. For simplicity, we assume that the along-current-path distribution of the late
winter mixed layer depth can be represented by the profile shown in Fig. 11a. Also assume the
seasonal cycle of the mixed layer depth, f(t), has the typical saw-tooth profile shown in Fig. 11b
(the thin line). The mixed layer pattern in this case can be expressed as
2
hm ( x, t ) = hmin + ⎡100 × (1 + 0.001x ) + 100e − ( x / Δx ) − hmin ⎤ f (t )
(17)
⎣
⎦
where hmin = 40m m is the mixed layer depth minimum, and Δx is the width of the Gaussian
profile.
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4000
3000
Subduction rate (m/yr)
Effective
Detrainment
2000
Ineffective
1000
Detrainment
Ineffective
T
Tee
Entrainment
1
0
s
Effective
Entrainment
Ted
e
−1000
Mixed layer depth (m)
−2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
Time (yr)
0.6
0.7
0.8
0.9
1
0
−50
−100
−150
−200
200
150
SE
100
−
S
SL
Rate (m/yr)
50
0
−50
OE
−100
L
O
−150
−200
−2.5
−2
−1.5
−1
−0.5
0
X(1000km)
0.5
1
1.5
2
2.5
Fig. 11. An example for subduction/obduction at the Kuroshio Extension. a) Mixed layer depth in
late winter (solid curve) and in summer (dashed line). Trajectories of water particles are depicted
by horizontal arrows. b) Instantaneous ventilation rate at station x=0 as a function of time; unit in
m/year. The thin solid line indicated the seasonal cycle of the local mixed layer depth. c) Annual
mean subduction (positive) and obduction (negative) rates: heavy solid lines indicate the
Lagrangian rates, long-dashed line the Eulerian rates, and the short-dashed line is the rate
calculated using the local winter mixed layer slope as defined in Eq. (18)
Where the Kuroshio separates from the coast, water travels relatively quickly. When
combined with the strong mixed layer depth gradient, it makes a large contribution to the
ventilation rate. In comparison, the contribution due to Ekman pumping is negligible because this is
close to the inter-gyre boundary. To illustrate our basic idea, we will neglect the vertical pumping
and assume water parcels travel 1000 km in one year. Our focus is on the station at the center of the
winter mixed layer trough, x=0. The instantaneous Eulerian subduction/obduction rate calculated
by (1) is shown in Fig. 11b (the thick line). Note that the time axis t=0 corresponds to March 1.
14
Effective detrainment begins at t=0 when the mixed layer starts to shoal rapidly. The end of the
effective detrainment, Teed = 0.37 , is calculated by tracing water particles released from this station
at a time interval of Δt = 0.001 . The ineffective detrainment period is between Teed and T1 = 0.336 ,
because the water subducted during this period is entrained into the downstream mixed layer at a
later time. Entrainment starts at T1 . The beginning of the effective entrainment is calculated by
tracing water particles upstream for one year. It occurs in this case at Tsee = 0.716 . Note that
although detrainment takes place for one-third of a year, the effective detrainment is rather short,
about 13 days. Similarly, entrainment takes place during two-thirds of a year, but only the last onehalf of this period, about 103 days, is effective entrainment.
At x=0 as shown in Fig. 11b, effective entrainment takes place in winter. Near the end of
winter, the entrainment mode slows down and eventually switches to the detrainment mode. The
transition between these two modes is so quick, water that entered the mixed layer remains close to
the location where it leaves the permanent pycnocline and this same water is left behind and
geostrophically flows into the permanent pycnocline thereafter. Using this example, we have shown
that obduction and subduction can take place at the same location, although water masses involved
come through effective detrainment and entrainment during different phases of the seasonal cycle.
The spatial distribution of the annual mean subduction/obduction rates is shown in Fig. 11c, in
which the subduction rate is defined as positive and the obduction rate is defined as negative. The
solid lines denote the rates calculated in Lagrangian coordinates, whereas the long-dashed lines
denote those calculated in Eulerian coordinates. Here, simplified versions of (12), i.e.,
S L = ( hm,0 − hm,1 ) / T and OL = ( hm,0 − hm,−1 ) / T , have been used in computing the Lagrangian rates.
Note that the subduction/obduction rates calculated using these two formulas agree well both in
magnitude and in spatial distribution. The slight difference is because the two definitions use
different weighting on the annual means.
Four dynamically distinct regions appear in Fig. 11c. In the upstream direction, there is a
region of pure obduction because the mixed layer depth increases monotonically downstream. In
the middle of the mixed layer depth trough, there is an ambiductive region where both subduction
and obduction take place and compensate for each other. There is a narrow band downstream where
only subduction occurs. Farther downstream, an insulated region exists where there is neither
effective subduction nor effective obduction.
For comparison, the subduction/obduction rate calculated using just the annual mean
velocity and winter mixed layer depth
G
S = −u ⋅ ∇hw int er
(18)
is shown in Fig. 11c by the short-dashed line. Such a formula is similar to the one used by Marshall
et al. (1993) in their data analysis on subduction in the North Atlantic and the one used in the
analytical study of the subtropical North Pacific by Huang and Russell (1994). Although this
definition is much easier to apply, it inflates the subduction/obduction rate near the mixed layer
depth fronts. In addition, such a definition eliminates possible overlapping and exclusion of
subduction and obduction because it neglects the seasonal cycle and the along-trajectory changes.
Note that calculating the annual mean subduction/obduction rate in Eulerian coordinates requires
detailed information of the seasonal cycle. On the other hand, the simplified formula (12) in
Lagrangian coordinates requires the annual mean velocity and winter mixed layer properties only.
Such a calculation can provide much more accurate information about the ventilation rates,
although caution must be taken to set the rate to zero whenever it is negative.
15
9. Ventilation of the North Atlantic and North Pacific
Ventilation in the North Atlantic and North Pacific is examined by analyzing the Levitus
(1982) climatological data and the Hellerman & Rosenstein (1983) wind stress data (Fig. 12a). The
calculation is based on the geostrophic velocity, using the 2000-m level as the reference, and the
one-year trajectories are shown in Fig. 12b. Note that during one-year period, water particles can
travel over a great distance on the order to 1000 km; thus, using the local gradient of the mixed
layer depth, as in Eq. (6), is inaccurate for the subduction/obduction calculation.
Fig. 12. a) Ekman pumping rate in the North Atlantic, in m/yr; b) one-year trajectories of water
particles released from the base of the Mixed layer in March, with the along isopycnal geostrophic
velocity calculated using the 2000-m depth as the reference level. (Qiu and Huang, 1995).
Ventilation in the North Atlantic and North Pacific can be classified into four types of
different regions: the subductive regions, the obductive regions, the ambiductive regions where
both subduction (in spring) and obduction (in winter) take place, and the insulated regions where
neither subduction nor obduction happens (Figure 13). Although the total subduction rates in these
two oceans are comparable, the obduction rates are considerably different (Table 2). In the North
Atlantic, obduction is strong (23.45 Sv), which is consistent with the notion of the fast
thermohaline circulation and the short renewal time of the subpolar water masses in the Atlantic
basin. Obduction is weak in the North Pacific (7.83Sv), consistent with the sluggish thermohaline
circulation and the slower renewal process of the subpolar water masses there.
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Fig. 13 Annual mean ventilation rate for the North Atlantic. Subduction rate and its two
components are shown in (a) the vertical pumping rate, (b) the lateral induction term, and (c) the
subduction rate. Stippled regions in (c) indicate zero subduction rate. Obduction rate and its two
components are shown in (d) the vertical pumping term, (e) the lateral induction term, and (f) the
obduction rate. Stippled regions in (f) indicate zero obduction rate. The dotted line with crosses in
(d) indicates the southern limit of the obduction zone. (Qiu and Huang, 1995).
The most interesting features of these maps are the ambiductive regions in the oceans,
where the local water mass conversion rate can reach 80 m/year . These local conversion rate
maxima are very closely related to the heat flux maxima from the ocean to the atmosphere, as
indicated by the dashed lines in Figs. 14.
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Fig. 14. Local mass conversion rate in the North Atlantic, defined within the ambiductive region,
with contour interval of 40m/yr. The shaded areas are insulated regions where neither subduction
nor obduction takes place. The dashed lines indicate the annual heat loss from the ocean to the
atmosphere, in units of w / m2 , adapted from Hsiung (1985). (Qiu and Huang, 1995).
Fig. 15. Subduction/obduction rates per 0.2 σ θ interval as a function of density for the North
Atlantic (upper) and the North Pacific (lower). (Qiu and Huang, 1995).
18
The water mass formation (destruction) rate computed as the sum of subduction (obduction)
rate integrated over the corresponding outcropping area is plotted in Fig. 15. The peaks of the
subduction rate correspond to the subtropical mode water in the North Atlantic and North Pacific.
A second peak in the North Atlantic indicates the subpolar mode water, which has no
corresponding part in the North Pacific because shallow marginal seas, such as Sea of Okhotsk,
were not included in the calculation.
Table 2. Ventilation rates for the North Atlantic and the North Pacific, in Sv:
N. Atlantic
N. Pacific
Sum
A) Subduction
Ekman pumping
-22.2
-30.8
-53.0
Vertical pumping
17.5
25.1
42.6
Lateral Induction
9.5
10.1
19.6
Total Subduction
27.0+3.1*
35.2
65.3
B) Obduction
Ekman pumping
2.8
3.6
6.4
Vertical pumping
-0.62
3.1
2.5
Lateral induction
24.1
4.7
28.8
Total Obduction
23.5
7.8
31.3
C)Local Conversion
4.0
3.5
7.5
* 3.1 Sv is the sum of the localized subduction south of Iceland (1.9 Sv) and in the Labrador Sea
(1.2 Sv).
One of the most interesting features from Table 2 is that Obduction rate is 10 times larger than the
total Ekman upwelling rate. This indicates the important dynamical role of the sloping mixed layer
depth. It is clear that using the term “Ekman upwelling” to describing water mass destruction in the
ocean is not acceptable.
Appendix A. On the Eulerian subduction/obduction rate
Subduction rate in Eulerian coordinates can be defined in two ways:
A) Subduction going through the upper surface of the permanent pycnocline at a given location
(our equation (4.5)). This is the definition used by Marshall et al. (1993), Huang (1990), Huang and
Russell (1994). Notice that there is nothing wrong about this definition. In fact, this definition gives
more accurate information of how much water enters the permanent pycnocline at each location.
In addition, the subduction rate calculated in this way is exactly the amount of subduction rate into
a given density layer assuming that density does not vary in the permanent pycnocline.
The shortcoming of this definition is that water subducted into permanent pycnocline at the given
station does not necessary come from the local mixed layer detrainment. In fact, most of the water
entering permanent pycnocline at the given station actually comes from upstream mixed layer
detrainment. Therefore, such a subduction rate has no direct link with the local mixed layer
detrainment.
B) Subduction rate equal to the effective detrainment rate at a local station, this is the definition we
have been used. This definition is defined as how much water is actually leaving the mixed layer
and eventually enters the permanent pycnocline somewhere downstream. The advantage of this
definition is the direct link with the local mixed layer dynamics. The disadvantage is that
subduction rate defined in this way is not really a local property; instead, it is an integrated average
19
over the downstream neighborhood. Furthermore, water subducted at one station is the water
detrained during the effective detrainment period. Although the effective detrainment is relatively
short, about one to three months, water properties, such as temperature, salinity, and density may
very during this period. As a result, water subducted at one station does not belong to a single
density category, i.e., subduction rate's distribution in density coordinates calculate by this
definition is not very accurate!
According to this argument, we may want to be a little careful, when we discuss the
distribution of subduction/obduction rate in the density coordinate, such as our last figure.
1) Subduction/Obduction can happen at the same location in our definition.
2) They may or may not correspond to slightly different density.
3) If we look at what happen at the upper surface of the permanent pycnocline at a given station,
the mass flux can have only one sign, either subduction or obduction, so Marshall et al. (1993) is
right in this sense!
4) If we are only interested in what happen to the permanent pycnocline, subduction's distribution
should be calculated in definition A). However, that Marshall et al. (1993) did not calculated the
basin sum of subduction rate nor the sum in term of density.
In summary, our last figure is not a very accurate account for the mass balance of the
permanent pycnocline; however, it is not entirely wrong. First, we have been standing at the sea
level, and we have been analyzing the mass flux leaving and coming into the mixed layer, which
we defined as subduction/obduction for a station. This figure is what you would see if you are
meteorologist because it is really the mass flux leaving the mixed layer.
In this sense subduction/obduction can still happen at the same station, and at roughly the same
density interval, so this is the way to explain the figure.
In addition, subduction and obduction may still happen at the same density interval (AT
DIFFERENT LOCATIONS!). There are so many different ways of doing the calculation, they
should not conflict with each other; instead they should compromise and enhance each other!
Of course, there is always the problem that the subduction/obduction period is not extremely short,
so the density does vary slightly, make our calculation a little inaccurate, otherwise there is really
nothing wrong. It is a matter of definition!
Appendix B. Potential vorticity in the ventilated thermocline
For simplicity, let us assume a two-dimensional case.
Fig. B1. A sketch for the potential vorticity formed during subduction.
Using the density conservation, we have
20
G
u ⋅ ∇ρ m
Q=
G
ρ 0 w + u ⋅ ∇h
Thus, potential vorticity in the ventilated thermocline is linearly proportional to the meridional
density gradient of the mixed layer density, and inversely proportional to the sum of the vertical
velocity at the base of the mixed layer plus the horizontal increment of the mixed layer depth.
Therefore, low potential vorticity water is formed when:
i) There is low meridional gradient of mixed layer density.
ii) Strong Ekman pumping.
f
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