Lecture 14. Thermocline with continuous stratification
4/6/2006 2:45 PM
1. Introduction
There is a prominent vertical temperature gradient maximum in the world oceans, this is
called the (main) thermocline. For a long time scientists have been trying to construct theories to
explain the structure of the thermocline, these are called the thermocline theory. Since the
temperature and density structures are very closely tied to the current, the thermocline theory is also
a theory about the three-dimensional structure of circulation in the upper ocean.
From the very beginning of the development, there have been two approaches to this
problem. In 1959, two papers were published side by side in Tellus. Robinson and Stommel (1959)
proposed a theory of the thermocline in which the vertical diffusion plays a vital role. In this
approach, the main thermocline is viewed as an internal density front or internal thermal boundary
layer; thus, the vertical diffusion term should be retained as an essential component of the
dynamical description. The early development along this line was summed up in the comprehensive
review by Veronis (1969). The most challenging difficulty associated with this approach is the fact
that nobody knows how to formulate suitable boundary value problems for the corresponding
nonlinear equation system, much less how to find the solutions to this problem. In order to
overcome such a difficulty, similarity solutions have been sought. The major disadvantage of the
similarity solutions is that they do not satisfy important boundary conditions, such as the Sverdrup
constraint. A solution, not satisfying the Sverdrup constraint, cannot describe the global structure of
the wind-driven circulation accurately.
On the other hand, Welander (1959) proposed an ideal-fluid theory for the thermocline.
From the beginning, the basic equations for the thermocline seemed so naively simple, that many
people believed that they could be solved easily, so most people did not want to spend time
working on the seemingly incomplete ideal-fluid thermocline theory, except Welander who also
published another very influential paper on the ideal-fluid thermocline (Welander, 1971).
The basic ideas for the ideal-fluid thermocline can be shown clearly in terms of density
coordinates and the Bernoulli function B = p + ρ gz where p is pressure, ρ is density, g is gravity,
and z is the vertical coordinate. Assuming the potential vorticity Q = f ρ z is a linear function of
both the density and the Bernoulli function, Welander (1971) was able to find the first analytical
solution for the thermocline.
Although this is the first elegant solution for the ideal-fluid thermocline, and thus has been
cited in many textbooks, it has some major defects: the solution satisfies neither the Sverdrup
relation nor the eastern boundary condition, and the lower boundary is set at z = −∞ . Since the
Sverdrup constraint is the most important constraint, thermocline solutions that do not satisfy the
Sverdrup relation are less meaningful dynamically.
During the 1960s, similarity solutions remained the mainstream in thermocline theory.
However, the limitation of the similarity approach became clear. Beginning in the early 1980s there
were major breakthroughs in the theory of the dynamic structure of the wind-driven gyre, including
the potential vorticity homogenization theory by Rhines and Young (1982) and the ventilated
thermocline by Luyten et al. (1983). According to these new theories, the wind-driven gyre
includes several regions with different dynamics: the ventilated thermocline where potential
vorticity is set at the surface; the unventilated thermocline where potential vorticity is
homogenized; the shadow zone near the eastern boundary; and the pool region near the western
boundary. These new theories were combined and extended to a theory of the wind-driven gyre in
the continuously stratified oceans by Huang (1988). The dynamic role of the mixed layer, excluded
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from the early theories, was incorporated into the thermocline theory, e.g., Huang (1990), Pedlosky
and Robbins (1991), Williams (1991). The progress made during the past two decades was
reviewed by Huang (1991) and Pedlosky (1996).
A major deficit of thermocline theories is the lack of the western boundary layer and
recirculation. It is clear that in order to explain the structure of the wind-driven circulation,
numerical models must be used in which the dynamic effects of horizontal mixing, the western
boundary layer, and the recirculation are explicitly included. For example, Samelson and Vallis
(1997) studied the thermocline structure in a closed basin, and showed that by using a small
diapycnal mixing rate in the ocean interior, the thermocline does appear in two dynamic regimes,
i.e., the ventilated thermocline for the water entering from the surface layer in the subtropical basin
due to Ekman pumping, and the diffusive thermocline over the density range corresponding to the
subpolar basin. In a more recent publication, Vallis (2000) went through a series of carefully
designed numerical experiments based on a primitive equation model and showed that stratification
below the ventilated thermocline is a result of global dynamics, involving the effect of wind
forcing, geometry of the world oceans, and diffusion. In particular, his study indicated that the
geometry of the Antarctic Circumpolar Current plays a subtle but important role in setting up the
stratification at the mid depth of the world oceans.
2. Ideal fluid thermocline
The so-called thermocline equations look very simple. It consists of geostrophy in the horizontal
directions, hydrostatic approximation in the vertical direction, the incompressibility, and the
continuity equation
− fv = − Px / ρ0
(1)
fu = − Py / ρ 0
(2)
(3)
0 = Pz + ρ g
G
∇ ⋅u = 0
(4)
G
u ⋅∇ρ = 0
(5)
All equations are linear, except the continuity equation; however, the nonlinearity associated with
the advection term is very strong, as we will discuss in the section for the Parsons model. Since
1959 when this equation system was first formulated by Welander, a challenge had been to
formulate suitable boundary value problems for this equation system and to solve them.
A. Conservation quantities
By cross-differentiating, subtracting (1) and (2), and applying (4), one obtains the vorticity equation
β v = fwz
(6)
The z derivative of (5) yields
G
G
u z ⋅∇ρ + u ⋅∇ρz = 0
(7)
Because of the thermal wind relation the vertical shear of the horizontal velocity is perpendicular to
the horizontal density gradient, so the first term on the left-hand side is reduced to
G
β
uz ⋅ ∇ρ = wz ρ z = vρ z
(8)
f
Thus, (7) can be rewritten as
G
u ⋅∇( f ρ z ) = 0
(9)
This is the potential vorticity conservation law, following a streamline.
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Multiplying the momentum equations by v and u, and adding together, one obtains the
mechanical energy balance, noting that the Coriolis force does no work,
G
G
(10)
0 = uPx + vPy = u ⋅∇P − wPz = u ⋅∇P + wρ g
G
Since w = u ⋅∇z , we obtain the Bernoulli conservation law
G
u ⋅∇( P + ρ gz ) = 0
(11)
Thus, density, potential vorticity, and the Bernoulli function are all conserved along streamlines. In
the oceans these quantities are not exactly conserved; however, deviations from the conservation
laws are small enough, so the ideal-fluid thermocline has been rather successful.
3. Simple solutions
A. Reducing to a single ordinary differential equation
Welander (1971) made another important contribution to the ideal fluid thermocline theory by
applying the vorticity conservation and Bernoulli conservation laws to the thermocline problem.
His original formulation was based on the z-coordinate. It is more convenient to formulate the
problem in density coordinates and used the Bernoulli function
B = P + gz
(12)
as the dependent variable in the density coordinates. The first and second derivatives of the
Bernoulli function with respect to the density are
Bρ = gz
(13)
Bρρ = gz ρ =
fg
q ( B, ρ )
(14)
where
q = f ρz
(15)
is the potential vorticity. As stated above, both the Bernoulli function B and the potential vorticity
q are conserved along streamlines.
Welander made a very important step in observing that if the function form of q( B, ρ ) is
given, the equation can be solved either analytically or numerically. In particular, he discussed
several cases.
1) If potential vorticity is a function of the density alone
The equation can be solved easily (Welander, 1959). In fact,
fg
B = B0 + Bρ ,0 + ∫ ∫ d ρ '
(16)
q( ρ ')
2) Assume a linear function q ( B, ρ ) = f ρ z = a ρ + bB + c .
This equation can be easily integrated if we first differentiating it with respect to z and apply the
hydrostatic relation
f ρ zz = (a + bgz ) ρ z
(17)
Integrating twice leads to
ρ z = C1 ( λ ,φ ) e
az + 0.5 bgz 2
f
(18)
z
ρ = ρ 0 (λ , φ ) + C (λ , φ ) ∫ e
0
( z ' + z0 )
D2 f
2
dz '
(19)
where
3
a2
bg
1/ 2
⎛ 2 ⎞
a
,D = ⎜− ⎟
bg
⎝ bg ⎠
Welander added the boundary conditions that density at the sea surface should match the observed
ρ s ( x, y ) , and at the great depth it should approach a uniform value ρ −∞ .
Welander also discussed the general case when q is a function that depends on a linear
combination of B and ρ , say q( B, ρ ) = F (a ρ + bB + c) . Although Welander laid the foundation
for solving the thermocline equation, some very challenge difficulties remained to be overcome.
The most difficult issue had been the conceptual difficulty associated with the boundary
conditions again. Welander's formulation reduces the thermocline problem to a second-order
ordinary differential equation in the density coordinates. A second-order ordinary differential
equation normally can satisfy two boundary conditions only; however, a solution for the
thermocline structure in a basin has to satisfy many boundary conditions. For example, the Ekman
pumping condition requires the solution to fit a two-dimensional array, which seemed to be almost
impossible within the original approach proposed by Welander.
Today, as we review the advance in the past decade, we realize that progress in solving the
thermocline equation has come step by step through the deepening of our understanding of the
thermocline structure. The two contributions of Welander have come primarily from the physical
insight: the neglecting of the diffusion and the introduction of Bernoulli function and vorticity
conservations. The very reason why Welander failed to solve the thermocline problem is again the
lack of enough physical insight. It was only after innovative study of the ventilated thermocline by
Luyten, Pedlosky, and Stommel, we came to understand the physics of the thermocline. As we will
discuss late, similar to the situation for the multi-layered model of the thermocline, the potential
vorticity has to be calculated as the part of the solution; one can only specify the potential vorticity
for the unventilated thermocline. Therefore, Welander's suggestion of specifying the form of the
potential vorticity function for all moving water requires too much information, and it actually
gives rise to an over-determined system. As the result, the solution cannot satisfy more than two
boundary conditions.
C = C1e
−
, z0 =
B. An analytical solution for the ideal-fluid thermocline
An analytical solution for the ideal-fluid thermocline can be found by using an assumption slightly
different from Welander's. Instead of assuming a linear function of the potential vorticity, we
assume that: i) In the ventilated thermocline the potential thickness (which is the inverse of the
potential vorticity) is a linear function of the Bernoulli function, i.e., D = − zρ / f = α 2 B ; ii) In the
unventilated thermocline, potential thickness is constant. Thus, the basic equation for the
thermocline is
Bρρ +
(
fgα
)
2
B=0
(20)
The solution of this equation is in forms of B = a cos b( ρ − ρ s ) , where b = fgα and a = a ( x, y )
can be determined by the boundary conditions, including the Sverdrup constraint (Huang, 2001).
An example is shown in Fig. 1. This solution includes the ventilated thermocline, the unventilated
thermocline, and the shadow zone. Most importantly, the solution satisfies the Sverdrup constraint,
so it provides a rather complete dynamical picture of the thermocline.
It is clearly that the solution resembles the subtropical gyres observed in the oceans. In
some previous studies it was speculated that the main thermocline may appear in form of density
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discontinuity in a truly continuous model. Such a density front is, however, not necessary. In fact,
the simple solution shown in Fig. 1 has a truly continuous structure in a three-dimension space,
including weak discontinuities in potential vorticity.
This model belongs to the category of the solutions which the potential vorticity in the
ventilated thermocline is set as a constant a priori. In the oceans, potential vorticity of the
ventilated thermocline is set by the basin-wide circulation, including the upper boundary
conditions, such as the Ekman pumping rate and the surface density and mixed layer depth. The
mixed layer depth is assumed to be zero in the case presented here; however, it is easy to modify
the formulation to include a mixed layer with a non-zero depth.
Fig. 1. Structure of an analytical solution for the ideal-fluid thermocline. Solid lines are ventilated
thermocline, dashed lines are unventilated thermocline, and the thick lines are the velocity contours
in units cm / s .
4. How to improve the ventilated thermocline models?
A. The singularity in the models
One of the most important problems in these early ventilated thermocline models are due to
the lack of mixed layer. Mixed layer is the major buffer between atmosphere and the permanent
thermocline. Mixed layer depth reaches the annual maximum at late winter, to the order of 200-400
meters. It is readily seen that mass flux in the mixed layer consists a substantial part of the total
mass flux in the wind-driven circulation. Since density is almost vertically uniform in the mixed
layer, its dynamics is quite different from that in the ocean interior. Including the mixed layer is a
vital step toward a more realistic wind-driven circulation. It will be shown in this section that
including a mixed layer with horizontally varying depth can give rise to a subduction rate (this will
be defined in Lecture 15) that is substantially larger than the rate of Ekman pumping. Thus,
including the mixed layer is an essential step in making the thermocline models close to reality.
In early models of the thermocline, the upper surface of the models is set to be z=0. This
choice is apparently made to simplify the models. There are several problems associated with such
an upper boundary condition. First, this upper boundary forces all isopycnals to outcrop at the same
depth, z=0. As a result, all these models have singularity along the eastern, northern and southern
boundary. Second, this boundary condition excludes the contribution to the subduction rate due to
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the mixed layer depth gradient. As will be shown in this section, including the mixed layer is
actually not very difficult at all. In fact, including the mixed layer has substantially improved the
thermocline models. For example, coupling the mixed layer helps to overcome one of the major
problems in thermocline models, i.e., the singularity along the eastern, northern and southern
boundaries.
B. The eastern boundary condition
The eastern boundary has not yet received enough attention. In earlier one-moving layer,
wind-driven theoretical models, the eastern boundary is just a place to start the integration. The
importance of suitable eastern boundary conditions for a stratified model was first encountered in
the Luyten-Pedlosky-Stommel model. A major feature of this model is that all ventilated layers
have zero thickness along the eastern boundary. This is, apparently, inconsistent with observations.
Furthermore, since the calculations have to be started from the eastern boundary, it seemed clear
that if ventilated layers have non-zero thickness along the eastern boundary, the entire solution
might change.
The puzzle about the eastern boundary condition can be seen from the following argument.
Assuming the flow can be described by the ideal-fluid thermocline equation, then the suitable
kinematical condition is u=0 at the eastern boundary x=0. The geostrophy constraint implies
p y = 0, or p = p( z ), at x = 0
(21)
So that u z ≡ 0 . Along the eastern boundary the density conservation law is reduced to
wρ z = 0, at x = 0
(22)
Now if ρ z ≠ 0 on the wall, i.e. the fluid is stratified, then
w ≡ 0, at x = 0
(23)
From the Sverdrup relation β v = fwz , one obtains
v ≡ 0, at x = 0
(24)
By the thermal wind relation, this leads to
ρ x = 0, at x = 0
(25)
If we differentiating the density conservation equation REPEATEDLY, we obtain
(26)
u ≡ v ≡ w ≡ ρx ≡ ρ y ≡ 0
EVERYWHERE in the basin. This seems a very strong mathematical argument; however, as we
noticed earlier, the major flaw in this argument is the assumption that the density and velocity
fields are differentiable as many times as we wanted. In fact, the system is of hyperbolic nature;
thus, these quantities may be non-differentiable, and the whole arguments are no good.
Nevertheless, this argument does shed some light on the problems associated with the eastern
boundary conditions.
To explore suitable eastern boundary conditions for stratified models, Pedlosky (1983)
studied a model of two moving layers. A new eastern boundary condition was used, which required
only the vertically integrated zonal mass flux to be zero and allowed ventilated layers to have nonzero thickness along the eastern wall. Since the model had only two moving layers, the
stratification along the eastern wall could be specified ad-hoc, and the solution in the interior
calculated accordingly. This eastern boundary condition gives rise to the eastern boundary
ventilated thermocline and alters the global structure of the gyre circulation.
The generalized eastern boundary conditions have been extended to the case of a
continuously stratified model by Huang (1989a). It is shown that in a continuously stratified model,
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the stratification along the eastern boundary can no longer be specified ad-hoc; instead, it should be
calculated as a part of the unified gyre-scale circulation. Thus, the eastern boundary conditions are
closely tied to the gyre-scale circulation and cannot be specified arbitrarily.
There are some troubles involved in using the generalized eastern boundary conditions.
First, the model requires some unknown eastern boundary layer that can transfer water vertically.
Second, the stratification along the eastern boundary implies some extra freedom of the system, so
the system becomes highly under-determined. The question is how to find a physically meaningful
solution.
In searching for an answer to the general questions concerning the suitable boundary
conditions at the eastern wall for continuously stratified models raised by Killworth in an
unpublished manuscript, Young and Ierley (1986) studied a thermohaline circulation model with
vertical diffusion. They used a family of similarity solutions to explore the physical meaning of
suitable eastern boundary conditions for the ideal-fluid thermocline. By examining the solutions
obtained when the vertical diffusivity approached zero, they came to a conclusion that the idealfluid thermocline equation has weak solutions, i.e. solutions that have a density discontinuity,
which they interpreted as a thermocline. However, Huang (1988, 1989) showed that truly
continuously stratified solutions of the equations do exist, although potential vorticity would be
discontinuous across the base of the moving water and there was some singularity along the
boundaries of a basin. Thus, it is possible to construct a solution that has a smooth density field in
the interior ocean, although the eastern boundary is always involved with some kind of singularity.
In a model including a mixed layer of horizontally varying density and depth, Huang
(1990b) returned to the old eastern boundary condition of zero zonal velocity below the base of
mixed layer. An implicit assumption of the model is that the onshore geostrophic flow in the
seasonal thermocline is exactly balanced by the offshore Ekman flux due to the southward longshore wind stress. This new formulation successfully overcomes the artificial singularities existing
in many previous theoretical models. Due to the finite depth of the mixed layer, meridional velocity
is finite everywhere. The application of the new eastern boundary condition eliminates the potential
vorticity singularity along the eastern, northern, and southern boundaries in previous models, and
gives rise to shadow zones in the ventilated thermocline.
It is fair to say that the trouble with the eastern boundary condition has not been solved.
Since the local offshore Ekman flux may not always exactly balance the onshore flux in the
seasonal thermocline, there is again some singularity involved that requires further study. In fact,
there is a gap between coastal oceanography and basin-scale oceanography. In coastal
oceanography, stratification in the interior ocean is assumed given, and for most cases this
stratification is taken to be independent of latitude; while for basin-scale oceanography, the
stratification along the eastern boundary is assumed given, presumably determined by some coastal
circulation processes. These two parts should be related through a general circulation of a basin.
5. Coupled with a mixed layer of variable depth
At the early stage of development, the mixed layer was neglected in most ideal-fluid
thermocline models for simplicity, and the upper boundary conditions of these models were that of
specifying we and ρ s at z = 0 . Neglecting the mixed layer leads to many problems in the models.
A close examination reveals that a model with a mixed layer is a fairly easy extension of the
previous models whose upper surface was artificially put at the sea surface.
A. The mixed layer
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Our concern is primarily the dynamics of the main thermocline below the mixed layer. In
addition, it is very difficult to formulate a simple analytical model to incorporate the mixed layer
thermodynamics with the dynamics in the main thermocline. A major challenge involved in such a
goal is the handling of the seasonal cycle in the mixed layer, with all the Rossby waves interacting
with each other and the mean currents. Thus, we will exclude the thermodynamics of the mixed
layer. Instead, we will prescribe the thermodynamic parameters of the mixed layer; the velocity in
the mixed layer is, however, part of the solution.
In our idealization, the Ekman layer is treated as an infinitely thin layer on the surface of the
ocean, where water is collected horizontally by Ekman drift and from which it descends due to
Ekman pumping. Since density is assumed vertically homogenized within the whole depth of the
mixed layer, the Bernoulli function
B = p + ρ gz
(27)
is vertically constant within the mixed layer. At the sea surface, the Bernoulli function is the same
as the pressure, so the horizontal pressure gradient in the mixed layer is
∇p = ∇Bs − gz ρ s
(28)
where the superscript s indicates the sea surface. The first term on the right-hand side is a
barotropic term; the second term is a baroclinic term due to the horizontal density gradient in the
mixed layer.
The horizontal pressure gradient induces a geostrophic flow in the mixed layer
p
p
u g = − y , vg = x
(29)
f ρ0
f ρ0
where f = 2Ω sin θ is the Coriolis parameter, ρ0 is the reference density, subscripts x and y are
the partial derivatives in spherical coordinates
∂
∂
∂
∂
,
=
=
(30)
∂x r cosθ∂λ ∂y r∂θ
The vertically integrated mass flux in the mixed layer is
0
1 ⎛ s
1
s 2⎞
(31)
∫− h udz = − f ρ0 ⎜⎝ By h + 2 g ρ y h ⎟⎠
0
1 ⎛ s
1
⎞
s
(32)
∫− h vdz = f ρ0 ⎜⎝ Bx h + 2 g ρ x hh ⎟⎠
Along the eastern boundary of the model basin, the first term on the right-hand side of (31)
indicates an eastward flow into the eastern boundary, while the second indicates a westward flux
due to the north-south density gradient. We will assume that along the eastern boundary all the
thermocline layers below the base of the mixed layer are stagnant. In reality, along the eastern
boundary water below the mixed layer is in motion, and a solution including such an eastern
boundary ventilated thermocline can be obtained in a multi-layer model or a continuously stratified
model as discussed by Pedlosky (1983) and Huang (1989). As shown in these studies, although
introducing the eastern boundary ventilated thermocline modified the solution, especially in the
southeastern part of the basin, changes in the rest of the basin are relatively small. In addition,
solutions involving the eastern boundary ventilated thermocline are not uniquely defined. Thus, to
ease our analysis we will use a simple no-flow boundary condition and neglect the eastern
boundary ventilated thermocline.
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Z
N
u
ρ1
ρ2
ρ3
Mixed Layer
ρ3
ρ2
ρ1
Fig. 2. The dynamical structure of the eastern boundary. Isopycnals are vertical within the mixed
layer, and horizontal below the mixed layer, implying no motion below the mixed layer.
Accordingly, the eastern boundary condition for our model is that the zonal velocity is identically
zero for water in the thermocline
u ≡ 0, at x = xe
(33)
This implies
p y = 0 for z ≤ − h
(34)
Therefore, the mass flux going into the eastern boundary is
gh 2 s
Mg =∫
ρ y dy
(35)
2 f ρ0
Assuming h = 100m , f = 0.5 × 10−4 / s , Δρ = 0.003 g / cm3 , we obtain M g > 1 Sv . This on-shore
flux of water feeds the upwelling along the eastern boundary. Details of the upwelling along the
eastern boundary are, however, left for further study.
The actual eastern boundary of the model is slightly tilted, so the eastern boundary
condition is
G G
u ⋅ n = 0, along theeasternboundary
The mixed layer plays an important role in setting the stage for subduction. In a steady circulation,
the annual subduction rate (the exact definition of subduction rate will be discussed in Lecture 15)
is defined as
G
S = −( wm + u ⋅∇h)
(36)
where the subscript m indicates the base of the mixed layer; the first term on the right-hand side is
due to vertical pumping at the base of the mixed layer. Because of the geostrophic meridional flux
in the mixed layer, the vertical pumping at the base of the mixed layer is less than the Ekman
pumping
β 0
β ⎛
1
⎞
wm = we − ∫ vdz = we − 2 ⎜ Bxs + g ρ xs h 2 ⎟
(37)
2
f −h
f ρ0 ⎝
⎠
9
The second term on the right-hand side of (2.10) is due to lateral induction. Because the base of the
mixed layer is tilted, horizontal flow gives rise to mass flux into the main thermocline. In fact,
lateral induction is a major contributor to the subduction into the main thermocline. This can be
shown by a very simple estimation as following.
∫ dx ∫ w
b) Horizontal induction: ∫ dx ∫
a) Vertical pumping:
mb
> 6 ×108 × 10−4 × 3 ×108 > 18Sv
vdh > 6 ×108 × 3 × 104 > 18Sv
Thus, these two terms make the same contribution. This estimation will be confirmed by
calculation from our 3-D model.
It is readily seen that calculating the subduction rate requires several variables, including the
mixed layer geometry (the depth), the mixed layer density, and the Bernoulli function. The
horizontal velocity in the mixed layer can be calculated from these variables. Although the
thermodynamic variables of the mixed layer, such as ρ s and h , are specified, the dynamical
G
variables, such as B s and u , are unknown and they are part of the solution we are looking for.
Note that in this section all partial derivatives are defined in the z-coordinates. Because
density is vertically uniform, so density coordinates is meaningless within the mixed layer.
However, our analysis in the following sections is based on density coordinates, so we need to
convert (37) into the density coordinates. When transferring between the z-coordinates and the
density coordinates
∇B |z = ∇p |ρ + Bρ ρ x |z
Thus, at the base of the mixed layer
Bxs |ρ s = Bxs |z + gh ρ xs
Note the terms on the right-hand side are calculated in the z-coordinates because these are defined
in the mixed layer when the density coordinates does not really work. Therefore, in the density
coordinates the vertical velocity at the base on the mixed layer is
βh ⎛
1
dρs ⎞
(37')
wm = we − 2 ⎜ Bxs |ρ + Bρ
⎟
2
f ρ0 ⎝
dx ⎠
B. A free boundary value problem for the ideal-fluid thermocline
The basic formulation of boundary value problems for the ideal-fluid thermocline was
proposed by Huang (1988). This formulation was modified to include the mixed layer with
horizontally varying mixed layer density and depth by Huang (1990). The formulation is based on
(potential) density coordinates ρ . The horizontal momentum equations are reduced to
f ρ 0v = Bx , f ρ 0u = − By
(38)
and the vertical momentum equation degenerates to the hydrostatic relation
Bρ = gz
(39)
where the subscript ρ indicates a partial derivative ρ . The linear vorticity equation is
β vz ρ = fwρ
(40)
Using (38) and (39) to eliminate v and z leads to
g ρ0 f 2
Bρρ Bx =
wρ
β
(41)
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Integrating this equation over [ ρ s , ρ b ], ( ρ s is the given mixed layer density distribution, ρ b is the
unknown free boundary separating the moving water from the stagnant abyssal water) gives
ρb
g ρ0 f 2
wm
(42)
∫ s Bρρ Bx d ρ = −
β
ρ
Using (37') and the condition that Bρ = − gh at the base of the mixed layer, using (37') the righthand side can be rewritten as
g ρ0 f 2
g ρ0 f 2
1 2 dρs
s
wm = −
we − Bρ Bx − Bρ
−
β
β
2
dx
Integrating by parts, the left-hand side of (2.16) can be rewritten as
ρb
∫ρ
ρb
Bρρ Bx d ρ = Bρ Bx |ρ b − Bρ Bx |ρ s − ∫ s Bρ Bρ x d ρ
(44)
ρ
s
(43)
where the first term is zero due to the match boundary condition at the base of the wind-driven
gyre (velocity is zero there). The last integral can be converted into another form
b
s
ρb
d ρb 2
2 dρ
2 dρ
B
d
ρ
=
B
|
−
B
|
+
2
(45)
b
s
ρ
ρ
ρ
∫ρ s Bρ Bxρ d ρ
dx ∫ρ s
dx ρ
dx ρ
By combining equations (42, 43, 44, 45), we obtain
d ρb 2
2 g ρ0 f 2
2 b
Bρ d ρ − Bρ ρ x =
we
(46)
β
dx ∫ρ s
Integration over [ x, xe ] leads to
ρb
ρb
ρ
ρ
ρb
− ∫ s Bρ2 d ρ + ∫ s Bρe d ρ + ∫ s Bρa d ρ =
2
2
ρ
2 g ρ0 f 2
β
∫
xe
x
we dx
(47)
where the superscript e indicates the eastern boundary, and the superscript a indicates the abyss.
Note that B e is a given function derived from the stratification along the eastern boundary, and B a
represents the background stratification. The first and second terms come from the integration of
the first term in (46). This equation is a generalization of the Sverdrup relation often used for a
reduced gravity model. Since we will make use of an additional assumption that along the eastern
boundary the base of moving water is the same as the base of the mixed layer, the second term in
the left-hand side vanishes. As demonstrated by Huang (1990) and Huang and Russell (1994), the
calculation of the ideal-fluid thermocline is reduced to repeatedly solving the following free
boundary value problem in density coordinates
fg
Bρρ =
(48)
Q ( B, ρ )
with the constraints
(49)
Bρ = − gh( x, y ) at ρ = ρ s
B = B a , Bρ = Bρa , at ρ = ρ b ( ρ b is unknown)
ρb
ρb
ρ
ρ
− ∫ s Bρ2 d ρ + ∫ be Bρa d ρ =
2
2 g ρ0 f 2
β
∫
xe
x
we dx
(50)
(51)
Although this approach seems to be a simple extension of the early work by Welander
(1971), there are delicate differences between the new formulation and Welander's old formulation.
In Welander's formulation, it was assumed that potential vorticity Q( B, ρ ) in (48) is a given
function, and the upper and lower boundaries of the integration are fixed. As a result, the ordinary
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differential equation (48) can only satisfy two boundary conditions; thus, for a long time it was not
clear how to find a solution satisfying additional boundary conditions essential for the circulation
physics.
A major problem in Welander's formulation is assuming that potential vorticity is a given
function for the entire thermocline. It took a long time and much effort before it was realized that
the thermocline consists of many regions, such as the ventilated thermocline, the unventilated
thermocline, the shadow zone, and the pool zone, which are dynamically quite different from each
other. It was shown that potential vorticity in the unventilated thermocline is fairly well
homogenized (Rhines and Young, 1982; McDowell et al., 1983). Potential vorticity in the
ventilated thermocline is, however, not homogenized. In fact, potential vorticity in the ventilated
thermocline should be calculated as a part of the solution, as demonstrated by Luyten et al. (1983).
In addition, the base of moving water is not a constant density surface. In light of these new
discoveries, the early model of Welander may be classified as some kind of similarity solution. Our
continuously stratified model has incorporated these new features. As one of the major differences
from the Welander's model, Q in (17) is a given function only for the unventilated thermocline, and
it is unknown for the ventilated thermocline.
This free boundary value problem is solved by a shooting method by starting from a first
guess of the bottom of the moving water ρ s . Integrating upward (toward lower density) to the base
of the mixed layer, we can determine the potential vorticity of the uppermost ventilated layer as
q = f Δρ / Δh , where Δρ is the density increment, and Δh is the thickness of the uppermost layer.
The generalized Sverdrup relation is then checked. If it is not satisfied, the base of the moving
water, ρ b , is adjusted till the integral constraint is met.
C. Application to the North Pacific
This model was applied to the North Pacific, with both h and ρ s specified functions of
geographical location, taken from the climatological mean density and depth data set. The model
ocean is divided into m × n grids, and the calculation of the three-dimensional structure of the
wind-driven subtropical gyre is reduced to repeatedly solving this second-order ordinary
differential equation at each station along individual outcropping line.
The streamlines on four isopycnal surfaces are shown in Fig. 3. According to the model's
results, most part of the wind-driven circulation in the North Pacific is ventilated.
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Fig. 3. Circulation on four different isopycnal surfaces (Huang and Russell, 1994).
The most prominent feature of the model is the strong ventilation due to the inclusion of a
mixed layer of finite, horizontally varying depth. The southern shoaling of the late winter mixed
layer depth gives rise to a strong subduction rate, upper panel of Fig. 4.
Three factors contribute to the mass flux in the ventilated thermocline: the vertical pumping
from the Ekman layer convergence, the lateral induction, and the inter-gyre boundary outflow due
to the northeast-southwest orientation of the zero-Ekman-pumping line. In fact, these three are
equal contributors. The southern shoaling of the mixed layer and the induced lateral induction from
the mixed layer into the main thermocline has very important impact on the wind-driven circulation
and climate.
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Fig. 4. Vertical pumping and subduction rate for the North Pacific, based on the ideal-fluid
thermocline model (Huang and Russell, 1994).
References
Huang, R. X., 1988. On boundary value problems of the ideal-fluid thermocline. J. Phys. Oceanogr.,
18, 619--641.
Huang, R. X., 1990. On the three-dimensional structure of the wind-driven circulation in the North
Atlantic. Dynamics of Atmospheres and Oceans, 15, 117--159.
Huang, R. X., 2001. An analytical solution of the ideal-fluid thermocline. J. Phys. Oceanogr., 31(8),
Part 2, 2441-2457.
Huang, R. X. and S. Russell, 1994: Ventilation of the subtropical North Pacific, J. Phys. Oceanogr., 24,
2589-2605.
Luyten, J., J. Pedlosky, and H. M. Stommel, 1983: The ventilated thermocline. J. Phys. Oceanogr., 13,
292-309.
Pedlosky, J., 1996. Ocean Circulation Theory. Springer-Verlag, Heidelberg, 453 pp.
Pedlosky, J. and P. Robbin, 1991: The role of finite mixed-layer thickness in the structure of the
ventilated thermocline, J. Phys. Oceanogr., 21, 1018-1031.
Rhines, P. B. and W. R. Young, 1982: A theory of the wind-driven circulation. I. Mid-ocean gyres.
Journal of Marine Research, 40 (suppl.), 559-596.
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Robinson, A. R. and H. Stommel, 1959: The oceanic thermocline and the associated thermohaline
circulation. Tellus, 11, 295-308.
Samelson, R. M. and G. K. Vallis, 1997: Large-scale circulation with diapycnal diffusion: the twothermocline limit. J. Mar. Res., 55, 223-275.
Vallis, G. K., 2000: Large-scale circulation and production of stratification: effects of wind, geometry
and diffusion. J. Phys. Oceanogr. , 30, 933-954.
Veronis, G., 1969: On theoretical models of the ocean thermocline circulation. DeepSea Res.,
16(Suppl.,), 301-323.
Welander, 1959: An advective model of the ocean thermocline. Tellus, 11, 309-318.
Welander, P., 1971: Some exact solutions to the equation describing an ideal fluid thermocline. J. Mar.
Res., 29, 60-68.
Williams, R.G., 1991: The role of mixed layer in setting the potential vorticity of the main thermocline.
J. Phys. Oceanogr., 21, 1803-1814.
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