Lecture 7. Reduced gravity models for the wind-driven circulation
8/17/2006 11:33 AM
1. Pressure terms and continuity equations in layered models
The simplest way to simulate the ocean circulation is to assume that the ocean is
homogeneous in density. Such a model has no vertical structure. However, there is a prominent
main thermocline in the oceans, so a natural way of simulating the ocean circulation is to treat the
main thermocline as a step function in density. In this way, the stratification in the ocean is
simplified as a two-layer fluid. Fluid below the main thermocline moves much slower than that
above the main thermocline. As a good approximation, one can assume that fluid in the lower layer
is near stagnant. Such a model has one active layer only, and it is called a reduced gravity model.
The advantage of a reduced gravity model is its ability to capture the first baroclinic mode of the
circulation and the structure of the main thermocline. Adding one more layer to the standard
1
reduced gravity model, one obtains the 2 -layer model which will be used in our discussion. The
2
comparison of these models is outlined in Fig. 1.
Fig. 1. Sketch of simple models in density coordinates.
In a sense, a reduced gravity model is equivalent to using just two grids in the density
coordinates. Similarly, multi-layer models are just highly truncated models in the density
coordinates. Most theoretical models are layer models. The reason of favoring the layer model over
the level model is the conception of along isopycnal mixing. In the oceans, mixing is predominately
along isopycnals because along isopycnal mixing requires the least amount of work. In reality,
mixing is not always along isopycnal. For example, baroclinic instability gives rise to crossisopycnal mixing. The issue of along isopycnal versus cross isopycnal mixing will be discussed
late.
Note the standard terminology is: if only the upper layer is in motion, it is called a reduce
gravity model, or a one-and-a-half layer model; if both the first and the second layers are in motion,
but the layers below are stagnant, it is called a two-and-a-half layer model, and so on.
1) Most 2-layer models assume water in each layer is immiscible. Some models allow mass
exchange between layers. It is readily seen that interfacial mass flux can drive the subsurface layer
1
in motion, without the interfacial friction. Thus, interfacial mass flux in layer model is a way to
simulate the thermohaline circulation in multi-layer models.
2) Most 2-layer models assume layer thicknesses are always non-zero. Some models allow
zero thickness in the upper layers – i.e., the outcropping. Outcropping is a strong nonlinear
phenomenon, associated with surface fronts. Handling outcropping properly requires special
numerical schemes.
A. Pressure gradients in multi-layer models
a) Rigid-lid approximation
The essential assumption of the so-called rigid-lid approximation is to move the upper
boundary of the model ocean from the free surface z = η to a flat surface z = 0 , so the original
problem of a moving boundary is reduced to one with a fixed boundary. Using the hydrostatic
relation, one can calculate the pressure in layers beneath. Starting from the upper layer (Fig. 2a),
p1 = pa − ρ1 gz
(1)
Note that pa ≠ const is an unknown pressure at z = 0 . At the base of the upper layer, pressure is
continuous, p2 = p1 = pa + ρ1 gh1 . Belong the interface, pressure in the second and third layers is
p2 = pa + ρ1 gh1 − ρ 2 g ( z + h1 )
(2)
p3 = pa + ρ1 gh1 + ρ gh2 − ρ3 g ( z + h1 + h2 )
Assume the third layer is very thick and motionless, so ∇p3 = 0 . Therefore, we can link ∇pa with
other variables and obtain the pressure gradient terms
∇pa = ( ρ3 − ρ1 ) g∇h1 + ( ρ3 − ρ 2 ) g∇η2
∇p1 = ( ρ3 − ρ1 ) g∇h1 + ( ρ3 − ρ 2 ) g∇h2
∇p2 = ( ρ3 − ρ 2 ) gh1 + ( ρ3 − ρ 2 ) g∇h2
Fig. 2. Multi-layer models with or without a free surface.
2
The pressure gradients can be rewritten as
1
∇p1 = ( g '1 + g '2 )∇h1 + g '2 ∇h2
ρ1
1
ρ2
∇p2 = g '2 ∇h1 + g '2 ∇h2
where g '1 = g
(3)
(4)
ρ 2 − ρ1
ρ − ρ2
are reduced gravity, on the order of 1cm / s 2 .
, g '2 = g 3
ρ0
ρ0
1
For a 1 -layer model, we have
2
1
∇p2 = 0, ∇p1 = g '1 ∇h1
ρ1
(5)
The essential step is assuming no motion in the deep layers, so the unknown pressure pa , which
represents the contribution from the atmosphere pressure and the free surface elevation, can be
eliminated. Without assuming the stagnant deep layer, the rigid-lid pressure pa will remain as a
part of the pressure expression. In such cases one may have to use other approaches to eliminate
pa . For example, in numerical models based on rigid-lid approximation, pa can be eliminated by
cross-differentiating the horizontal momentum equations and the problem is reduced to solve an
elliptic equation for the barotropic streamfunction.
b) Including the free surface elevation
The same expressions can be derived for a model that included the free surface explicitly,
(Fig. 2b). We start from the sea surface z = η and integrate the hydrostatic relation downward
p1 = pa − ρ1 g ( z − η )
p2 = pa + ρ1 g (η + h1 ) − ρ 2 g ( z + h1 )
p3 = pa + ρ1 g (η + h1 ) + ρ 2 gh2 − ρ3 g ( z + h1 + h2 )
where pa is the atmospheric pressure. Assuming pa = const , pressure gradient in each layer can be
written as
∇p1 = ρ1 g∇η
∇p2 = ρ1 g∇η − ( ρ 2 − ρ1 ) g∇h1
∇p3 = ρ1 g∇η − ( ρ3 − ρ1 ) g∇h1 − ( ρ3 − ρ 2 ) g∇h2
Dividing by ρi and using the reduced gravity notation, we obtain
1
∇p1 = g∇η
(6)
ρ1
1
ρ2
1
ρ3
∇p2 = g∇η − g '1 ∇h1
(7)
∇p3 = g∇η − ( g '1 + g '2 )∇h1 − g '2 ∇h2
(8)
These expressions includes the gradient of the free surface, which may be part of the unknown;
however, this unknown can be eliminated by assuming that the lowest layer is very thick, so that
3
pressure gradient in the lowest layer is negligible. If we assume ∇p3 = 0 , ∇η can be written in
term of ∇h1 and ∇h2
1
∇p1 = g∇η = ( g '1 + g '2 )∇h1 + g '2 ∇h2
(9)
ρ1
1
ρ2
∇p2 = g '2 ∇h1 + g '2 ∇h2
These are the same as we have derived above. Thus, we understand better the meaning of "rigid-lid
approximation".
B) The continuity equations
For the i − th layer, mass conservation of incompressible fluid is
uix + viy + wiz = 0
Assuming ui , vi are independent of z within each layer, and integrating from the bottom to the top
of this layer, we obtain
hi (uix + viy ) + wit − wib = 0
∂
∂
∂
hi + u hi + v hi
∂t
∂x
∂y
∂
∂
∂
hi + (hiui ) + (hi vi ) = 0
(10)
∂t
∂x
∂y
Note that for the uppermost layer h1 should include the contribution associated with the free
surface elevation.
wit − wib =
2. Simple reduced gravity models
Wind-driven circulation has been described in terms of the quasi-geostrophic model derived
from the shallow water equation in many text books. However, quasi-geostrophic theory is not
suitable for describing gyre-scale circulation because within the north-south direction the vertical
displacement of isopycnal surfaces is the same order of magnitude as the layer depth. The strong
nonlinearity due to the meridional change of the stratification can be handled much more accurately
by simple reduced gravity models.
The reduced gravity model is probably one of the most used options in oceanic circulation
simulation. The reduced gravity models can simulate the free surface elevation and its time
evolution under realistic wind stress forcing. In addition, the reduced gravity model can provide the
depth of the main thermocline under realistic condition, which is impossible to obtain from the
quasi-geostrophic models. This section is focused on theories for the wind-driven circulation based
on single-moving layer reduced gravity models.
A. Reduced gravity model formulation
The essence in building up a reduced gravity model is to treat the main thermocline (or the
pycnocline) in the oceans as a step function in density, so density in the upper layer equal to a
constant ρ0 and density in the lower layer is ρ0 + Δρ ; furthermore, the lower layer is assumed
infinitely deep, so pressure gradient in the lower layer is infinitely small. Under these assumptions,
the basic equations can be written as follows.
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The Momentum and continuity equations for a reduced gravity model are
hut + h(uu x + vu y ) − fhv = − g ' hhx + τ x / ρ0 + A∇ 2 ( hu ) − κ u ,
(11)
hvt + h(uv x + vv y ) + fhu = − g ' hhy + τ y / ρ 0 + A∇ 2 ( hv ) − κ v ,
(12)
ht + (hu ) x + (hv) y = 0 ,
(13)
where h is the layer thickness, (u, v ) are the horizontal velocity, g ' = g Δρ / ρ0 is the reduced
gravity, which is on the order of 1-2cm/s2, (τ x ,τ y ) are the wind stress, A is the coefficient of
lateral friction, and κ is the coefficient of bottom friction. Note that here wind stress is treated as a
body force for the whole layer, and this concept was first postulated by Charney. The essence of the
reduced gravity model is allowing layer thickness h to vary greatly, including the case of vanishing
layer thickness. The ability of handling finite amplitude perturbation in layer thickness is one of the
most important advantages of using reduced gravity model, in comparison with the quasigeostrophic model. Discussion in this section is limited to the case of a steady state, so that the time
dependent terms in the momentum and continuity equations are omitted.
As will be discussed in this section, wind-driven circulation in the ocean interior can be
described in term of the Coriolis force terms, the pressure gradient term and the wind stress terms.
However, such interior solution cannot satisfy the boundary conditions in a closed basin. In order to
describe a closed circulation in a basin, other high order terms have to be added on, such as the
nonlinear advection terms, the lateral friction terms, and the bottom friction terms. In fact, within
the western boundary, potential vorticity balance is different from that in the ocean interior. For
example, assuming the friction is in forms of bottom friction giving raise to the vorticity balance
between the planetary vorticity advection and the bottom friction torque. Similarly, assuming the
vorticity balance is between planetary vorticity advection and lateral friction or relative vorticity
gives raise to the so-called Munk boundary layer or the inertial boundary layer.
a) Energy conservation law
Multiplying (11) by u and (12) by v and adding these resulted equations lead to
⎞
G ⎛ u 2 + v2
(14)
+ g 'h⎟ = W ,
hu ⋅∇ ⎜
⎝ 2
⎠
where
1
W = (uτ x + vτ y ) − κ (u 2 + v 2 ) + A[u∇ 2 (hu ) + v∇ 2 (hv)] ,
(15)
ρ0
The first term is the work done by the wind stress, the second term is the dissipation due to bottom
friction, and the third term is the dissipation due to lateral friction.
b) Potential vorticity balance
Dividing (11) and (12) by h, then cross-differentiating and subtracting lead to
G
u ⋅∇q + q(u x + v y ) = C ,
(16)
where
q = f + vx − u y ,
is the planetary vorticity plus the relative vorticity, and
⎛ τ y κv A 2
⎞ ⎛ τ x κu A 2
⎞
C =⎜
−
+ ∇ (hv) ⎟ − ⎜
−
+ ∇ (hu ) ⎟ ,
⎝ ρ0h h h
⎠ x ⎝ ρ0h h h
⎠y
(17)
(18)
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is the source of potential vorticity due to wind stress curl and the vorticity sinks due to bottom and
lateral friction.
Using the continuity equation (13), we obtain a concise form of the potential vorticity
equation
f + vx − u y
=C.
(19)
hu ⋅∇
h
Eq. (19) states: potential vorticity advection is balanced by potential vorticity sources and sinks,
including wind stress, bottom and lateral friction. This equation also applies to individual layer in a
multi-layer model. Note potential vorticity in Eq. (19) is a simplified from reduced from the more
general form ( f + vx − u y )ρ z .
c) Solution free of forcing and dissipation
Neglecting forcing and dissipation terms leads to the energy and vorticity conservation laws
2
u + v2
B=
+ g ' h = F (ψ )
(20)
2
f + vx − u y
= G (ψ )
(21)
Q=
h
where ψ is the streamfunction introduced though ψ x = hv, ψ y = −hu . Eqs. (20) and (21) states
that both energy (or the Bernoulli function) and potential vorticity are conserved along streamlines.
Note that F and G are not independent, and this can be shown as follows. Introduce a
streamfunction coordinates dψ = hvdx . As will be discussed shortly, when the layer is shaded from
direct forcing potential vorticity is conserved along streamlines, assuming that mixing and
dissipation along the streamline is negligible. Taking the derivative of Eq. (20) with respect to ψ
and using Eq. (11) with no time dependent term, wind stress and friction terms, one obtain
dF
= G (ψ )
(22)
dψ
c) Dynamics roles of the western boundary current
From both the energy and potential vorticity equations, it is clear that a purely inertial
western boundary current cannot satisfy the energy and vorticity balance in a closed basin. No
matter how small is the friction, it plays an essential role in balancing the energy and vorticity in a
closed basin by dissipating the potential vorticity and energy input from the wind stress in the
basin. This is an extremely important point, and will be discussed in details in following sections.
B. Interior solution
In ocean interior, frictional and inertial terms are negligible. Assume that wind stress is in
simple form τ x = τ x ( y ), τ y = 0 ; thus, the momentum equations are reduced to
− fhv = − g ' hhx + τ x / ρ0
(11')
fhu = − g ' hhy
(12')
and the vorticity equation is reduced to
β hv = −τ yx / ρ 0
(19')
Substituting (19') into (11'), we obtain
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2 f 2 ⎛τ x ⎞
(23)
⎜ ⎟ ( xe − x )
g ' ρ0 β ⎝ f ⎠ y
The eastern boundary is a streamline with ψ = 0 ; thus, the streamfunction is
1 x
ψ=
τ y ( xe − x)
(24)
h 2 = he2 +
ρ0 β
Note that τ x is negative near the equator and reaches a maximum along the inter-gyre boundary,
−τ yx < 0 ; thus, this negative wind stress curl drives an equatorward flow in the interior, and layer
depth increases westward. Although this seems a disadvantage for such a formulation, this
formulation also has an advantage because the Sverdrup flow ψ remains finite along the equator.
On the other hand, the sum of the Ekman flux and the geostrophic flow below the Ekman layer
becomes unbounded because both terms are infinite near the equator.
Since the Ekman volume flux is τ x / f ρ0 , the thermocline depth is controlled by the Ekman
(
)
pumping, i.e., the horizontal convergence of the Ekman flux τ x / f ρ0 ; while the streamfunction
y
is controlled by the wind stress curl (Eq. 24). However, the situation near the inter-gyre boundary is
slightly different from other places. Since
x
⎛ τ x ⎞ τ y βτ x
⎜ ⎟ = − 2 ,
f
⎝ f ⎠y f
near the inter-gyre boundary the second term on the right-hand side may dominate and changes the
sign of the Ekman pumping contribution. In fact, the layer thickness may declines westward near
the inter-gyre boundary. When wind stress is strong enough, the layer thickness calculated from
(23) may be non-positive. In such a case, the interface outcrops, and model used above has to be
modified in order to reconcile the outcropping phenomenon, and this will be discussed in Section 5.
In the discussion above, the wind stress is treated as a body force to the entire upper layer.
There is another approach, in which the dynamic role of the wind stress is explored in terms of the
Ekman pumping rate due to the Ekman flux convergence. Thus, the Ekman layer and the horizontal
mass flux within it are separated from the geostrophic flow below the Ekman layer. The
corresponding momentum equations are reduced to
− fhv = − g ' hhx
(11'')
(12'')
fhu = − g ' hhy
and the continuity equation is
( hu ) x + ( hv ) y = − we
(13'')
where the Ekman pumping rate is related to the wind stress
we = − τ x / f ρ0
(
)
y
The vorticity equation is now
β hv = fwe = − f τ x / f
(
)
y
(21'')
Substituting (21'') into (11''), we recover Eq. (23)
2 f 2 ⎛τ x ⎞
2
2
h = he +
⎜ ⎟ ( xe − x)
g ' ρ0 β ⎝ f ⎠ y
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Since the flow has a source coming from above, the horizontal flow field cannot be
described in terms of a streamfunction alone. Although we can still integrate the meridional
velocity by starting from the eastern boundary, what we obtained should be called the meridional
mass flux rate (or the so-called Sverdrup function)
f ⎛τ x ⎞
m=
(24')
⎜ ⎟ ( x − x)
ρ0 β ⎝ f ⎠ y e
Thus, the layer thickness has an expression similar to that obtained by wind stress formulation,
except that the layer thickness at the eastern boundary may be slightly different, but the integrated
meridional mass flux is differed by the amount equal to the Ekman flux because the mass flux of
the Ekman layer is not included in the geostrophic formulation. This difference is reflected to the
difference in the definition of the inter-gyre boundary in these two formulations.
Note that: First, although any two-dimensional vector field can be separated into two parts:
hu = φ x − ψ y ; hv = φ y + ψ x
The decomposition is not unique. For example, the solution can have an additional component φ ' ,
as long as ∇ 2φ ' = 0 . In addition, the boundary conditions for φ and ψ are not unique. Second,
even if we construct such a streamfunction, it does not represent the streamlines accurately because
the velocity has another component -- ∇φ .
As a compromise, we can use the pressure field or the layer thickness field to plot the
streamlines. Since the pressure or the layer thickness square has a dimension different from the
mass flux, we can use a modified quantity, called the virtual streamfunction, which is defined as
g' 2
f 2 ⎛τ x ⎞
*
2
ψ =
(h − he ) =
⎜ ⎟ ( xe − x)
f 0 ρ0 β ⎝ f ⎠ y
2 f0
where f 0 is the reference latitude where the meridional mass flux is equal to ψ * ; however, away
from the reference latitude, ψ * can be quite different from the value of ψ * at the reference
latitude. The difference between ψ * and m is due to the Ekman pumping in the oceanic interior.
The construction of m will be discussed in Lecture 15 about the communication between the
subtropical and tropical oceans.
The essential points for a reduced gravity model are:
a) The stratification parameters have to be specified a priori, such as layer thickness along the
eastern boundary he and density jump across the interface, Δρ . These parameters are controlled by
some external processes -- the thermohaline circulation. Thus, the reduced gravity model is
essentially a perturbation approach, i.e., treating the wind-driven circulation as a perturbation to a
given stratification profile in the density coordinates.
b) The layer depth is inversely proportional to the density jump, i.e., strong stratification leads to
shallow thermocline and vice versa.
We will find these comments very useful for the multi-layer model or the continuously
stratified model.
c) Energy associated with the wind-driven gyre
The total kinetic energy is
1
1
1
Ek = ρ h(u 2 + v 2 ) = ρ g '2 h(hx2 + hy2 ) / f 2 ≈ ρ g '2 hh '2 / B 2 f 2
2
2
2
where B ≈ 1000km is the north-south length scale of the wind-driven gyre.
8
The total available potential energy is
2
1
1
E p = ρ g ' h 2 − h ≈ ρ g ' h '2
2
2
Therefore, the ratio of these two terms is
Ep B2
=
≈ 1000
Ek λ 2
)
(
where λ = g ' h / f ≈ 30km is the radius of deformation.
C. Stommel boundary layer
Within the western boundary current, scaling analysis indicates that the cross-stream
pressure gradient is in balance with the Coriolis force associated with the down stream velocity,
i.e., the semi-geostrophy
fhv = g ' hhx
(25)
Thus, there is a simple relation for the streamfunction and the layer thickness
g' 2
ψ =ψ I +
(26)
h − hI2
2f
where the subscript I indicates the interior solution at the outer edge of the western boundary
current:
2 f 2 ⎛τ x ⎞
(23')
hI2 = he2 +
⎜ ⎟ ( xe − xw )
g ' ρ0 β ⎝ f ⎠ y
(
ψI =
1
)
τ yx ( xe − xw )
ρ0 β
Since ψ = 0 at the western wall, layer thickness along the western wall is
(24')
2f
ψI
(27)
g'
Because of ψ I > 0 , layer thickness declines toward the wall. This sharp layer thickness slope is
caused by the strong western boundary current required for balancing the model's mass, vorticity,
and energy.
The potential vorticity equation is reduced to
β hv = −τ yx / ρ0 + κ ( u y − vx )
(28)
hw2 = hI2 −
Within the western boundary layer the potential vorticity equation is further reduced to
β hv = −κ vx
(29)
Integrating across the western boundary current leads to
β (ψ −ψ I ) + κψ x / h = 0
(30)
Using (26), we obtain
hx +
β 2 2
( h − hI ) = 0
2κ
with the boundary condition
h(0) = hw
The solution of this equation is
(31)
(32)
9
h = hI
1 − Be −η
1 + Be −η
(33)
where
h −h
hβ
B= I w, η= I x
hI + hw
κ
The scale width of the western boundary layer is
(34)
δS =
(35)
κ
hI β
The streamfunction within the boundary layer can be calculated from (25, 26) accordingly:
2 g ' Be −η
h2
ψ =ψ I −
2 I
−η
f 1 + Be
(
)
For example, we show the structure of the thermocline from such a reduced gravity model (Fig. 3
& Fig. 4). The layer thickness is set to 300 m along the eastern boundary, the reduced gravity
g'=1.5 cm/s , wind stress is (in unit dyn / cm 2 )
⎛ y − ys ⎞
π⎟
⎝ yn − y s ⎠
τ x = −1.5cos ⎜
Fig. 3. Sverdrup function in a model with a Stommel boundary layer in units of Sv.
10
Fig.4. Depth of the thermocline in a model with a Stommel boundary layer, in unit 100 m
Note that the boundary layer solution should match the interior solution at η → ∞ , so at the
finite distance from the western boundary these two solutions do not exactly match. A slightly
elaborate boundary solution can match to the interior solution gradually, but it is beyond the scope
of the discussion here.
The dynamic balance of the circulation is best illustrated in terms of the potential vorticity
change along a streamline. Since the relative vorticity is very small, the potential vorticity can be
approximated by f/h .
Within the basin interior the wind stress curl imposes a source of negative potential
vorticity; thus, potential vorticity declines downstream, see the right panel of Fig. 5. Within the
western boundary, the bottom friction torque produces a positive source of potential vorticity; thus,
potential vorticity of a water parcel increases along the pathway, see the left panel of Fig. 5.
Another interest point is that potential vorticity gradient is mostly confined within the northern
third of the basin, while the potential vorticity gradient is rather small for the most other part of the
basin. Thus, baroclinic instability is mostly active only for the northern third of the basin.
Fig. 5. Potential vorticity (in 10-9/s/cm) and a streamline (heavy line) for a model with a Stommel
boundary layer.
Similarly, the energy balance along the streamlines consists of two stages: First, the wind
stress input mechanical energy into the circulation, primarily near both the northern and southern
boundaries where the zonal velocity and wind stress are large, right panel in Fig. 6. This external
mechanical energy input has to be balanced by energy source, and in the current case it is due to the
bottom friction. As shown in the left panel of Fig. 6, the strong bottom friction due to the strong
meridional current along the western boundary plays the role as the mechanic energy sink in the
model.
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Fig. 6. Source and sink of mechanical work due to wind stress and bottom friction, in cm3 / s 3
D. Munk boundary layer
The momentum equations for the case of lateral friction are
− fhv = − g ' hhx + A∇ 2 (hu ) + τ x / ρ0
(36)
fhu = − g ' hhy + A∇ (hv)
(37)
2
By cross-differentiation and subtraction, we obtain the vorticity equation
βψ x = A∇ (ψ xx + ψ yy ) − τ yx / ρ0
(38)
The next steps are basically the same as in the case of quasi-geostrophic theory. We first separate
the streamfunction into two parts
ψ = ψ I +ψ B
(39)
where ψ I is the interior solution discussed above, and ψ B is the boundary layer correction. Using
scaling analysis, this perturbation part should satisfy a fourth-order equation
(40)
βψ B , x = Aψ B , xxxx
Thus, the scale width of the western boundary layer is
1/ 3
δM = ( A/ β )
Introducing a stretched coordinate
η = x /δM
the vorticity equation is reduced to
ψ B ,ηηηη −ψ B ,η = 0
(41)
(42)
(43)
The first constraint is that the boundary layer correction term should vanish at infinite. In addition,
two types of boundary conditions may apply:
a) ψ B = −ψ I , ψ B ,η = −ψ I ,η , at η = 0, no slip condition
b) ψ B = −ψ I , ψ B ,ηη = −ψ I ,ηη , at η = 0, slip condition
The general solution for (42) is
12
η
⎛ 3 ⎞
⎛ 3 ⎞
−
η ⎟⎟ + c4e 2 sin ⎜⎜ η ⎟⎟
(44)
2
2
⎝
⎠
⎝
⎠
a) Applying the condition at infinite
η → ∞,ψ B → 0 , thus, c1 = c2 = 0 .
b) Applying the boundary condition of ψ = 0 at the wall
η = 0,ψ B = −ψ I ; thus, c3 = −ψ I .
c) Apply the no-slip boundary condition:
∂ψ
η = 0,ψ B ,η = −δ M
, i.e.,ψ x = 0;
∂x
c
2
∂ψ I
c
2
curlτ (0, y )
thus, c4 = 3 −
δM
= 3 −
δM
∂x
βρ0
3
3
3
3
The final solution is
η
⎧⎪
⎛ 3 ⎞ 1
⎛ 3 ⎞ ⎤ ⎫⎪ 2δ M −η2
⎛ 3 ⎞ curlτ (0, y )
− ⎡
(45)
ψ I + ψ B = ψ I ⎨1 − e 2 ⎢ cos ⎜⎜ η ⎟⎟ +
sin ⎜⎜
η ⎟⎟ ⎥ ⎬ −
e sin ⎜⎜
η ⎟⎟
2
2
2
βρ
3
3
⎢
⎥
0
⎠
⎝
⎠ ⎦ ⎭⎪
⎝
⎠
⎣ ⎝
⎩⎪
d) Applying the slip boundary condition
∂ 2ψ I
η = 0,ψ B ,ηη = −δ M2
∂x 2
c
2 2 ∂ 2ψ I
Thus, c4 = − 3 +
δM
∂x 2
3
3
Thus, the final solution is
η
⎛ 3 ⎞ 1
⎛ 3 ⎞ ⎤ ⎫⎪ 2δ 2 M −η2
⎛ 3 ⎞ ∂ 2ψ I
− ⎡
⎪⎧
sin ⎜⎜
e sin ⎜⎜
ψ I + ψ B = ψ I ⎨1 − e 2 ⎢cos ⎜⎜ η ⎟⎟ −
η ⎟⎟ ⎥ ⎬ +
η ⎟⎟ 2 (45')
2
2
2
3
3
⎢
⎥
⎠
⎝
⎠ ⎦ ⎭⎪
⎝
⎠ ∂x
⎣ ⎝
⎩⎪
−
η
ψ B = c1 + c2eη + c3e 2 cos ⎜⎜
Note that the last terms in (45) and (45') is high order term because ε = δ M / L 1 and ε 2 1 ;
thus, they can be neglected.
e) Structure of the western boundary current:
The layer thickness of the boundary current can be obtained from the semi-geostrophic
relation
h 2 = 2 f ψ I + hw2 , hw2 = hI2 − 2 f ψ I
The corresponding velocity field can be calculated accordingly v =
ψx
=
ψ B ,η
, where the
δM h
h
contribution from the partial derivative of ψ I is neglected because it is much smaller in comparison
with the term retained.
E. Inertial western boundary current
The basic equations for the inertial western boundary current are
h(uu x + vu y ) − fhv = − g ' hhx + τ x / ρ 0
(11'')
h(uvx + vv y ) + fhu = − g ' hhy
(12'')
(hu ) x + (hv) y = 0
(13'')
13
Scaling analysis leads to a simpler set of equations:
fhv = g ' hhx , semi − geostrophy
1 2
v + g ' h = F (ψ ), energy conservation
2
f + vx
= G (ψ ), potential vorticity conservation
h
where F (ψ ) and G (ψ ) are functions completely determined from the interior solution at the outer
edge of the western boundary current.
F (ψ I ) = g ' hI (Y )
(46)
f (Y )
(47)
G (ψ I ) =
hI (Y )
where Y is the meridional coordinate at the outer edge of the western boundary layer. From (24'),
x
ψ I = e τ yx (Y )
(48)
ρ0 β
Assuming this function is invertible, we can write
Y = Y (ψ I )
(49)
Thus, both F and G are completely determined from the interior solution.
Note that the one-to-one inversion of (48) breaks down at the latitude where ψ I reaches its
maximum, and this is the northern limit of the purely inertial boundary layer. North of this limit,
other mechanisms are needed in order to maintain a steady boundary current.
The semi-geostrophic relation leads to the relation between the streamfunction and the layer
depth, the same as in the case with bottom friction
g' 2
ψ =ψ I +
(26)
( h − hI2 )
2f
or
2f
(26')
h 2 = hI2 +
(ψ −ψ I )
g'
The meridional velocity can be calculated using the Bernoulli law
v = 2 ⎡⎣ F (ψ ) − g ' h ⎤⎦
(50)
Note that both the layer thickness and meridional velocity are completely determined in the
streamfunction coordinates. In order to determined the solution in terms of the conventional
geometrical coordinates, we can use the coordinate transformation
ψ dψ
(51)
x=∫
0 hv
where hv is the function of ψ , defined in (26') and (50). In fact, the transformation from the
physical coordinates x to the streamfunction coordinates is the well-known Von-Mises
transformation. This coordinate transformation was first used by Charney (1955) to solve the
inertial western boundary current.
a) Special case when F (ψ ) is constant
In such a case, G (ψ I ) = f / hI = const , so the vorticity equation is reduced to
14
f∞ h
h∞
Using the semi-geostrophic condition, this leads to
ff ∞
f2
hxx −
h=−
g ' h∞
g'
The general solution is
h = a ⋅ e − λ x + b ⋅ eλ x + hI
vx + f =
where λ =
(52)
(53)
ff ∞
is the inverse of the deformation radio. Under the following conditions
g ' h∞
h(0) = hw , h(∞) = hI
(54)
the solution is
(55)
h = (hw − hI )e− λ x + hI
where hw is calculated from (26'). The structure of the solution is shown in Fig. 7, including the
inertial western boundary current in the southern half of the western boundary.
Fig. 7. Streamfunction in the model with an inertial western boundary current, in Sv.
Note: 1) A purely inertial western boundary current is allowed for the southern half of the
western boundary only.
2) The inertial western boundary is much narrower than that of a frictional boundary
current, as shown in Fig. 8.
15
Fig. 8. Thermocline depth in the model with an inertial western boundary current, in 100 m.
3. Limitation and extension of the reduced gravity models
A) Layer outcropping and the Parsons's model
A major difference between the reduced gravity models discussed above and the linearized
layer model used in many previous studies is that we do allow the layer thickness vary greatly.
When the forcing is strong, the interface can outcrop. Layer model with outcropping requires
careful treatment. Parsons (1966) discussed such model and made the connection between the Gulf
Stream and the outcropping of the main thermocline in a subtropical basin. The application of
models with outcropping isopycnals to the world oceans has been discussed in many papers, e.g.,
Veronis (1973), Huang and Flierl (1986).
B) Inverse reduced gravity model
For the circulation in the deep basin, flow near the bottom is much faster than flow above,
so we can assume that pressure gradient in water above can be neglected; thus, a reduced gravity
model can be formulated. Such one-layer models have been used in the study of abyssal circulation,
e.g., Stommel and Arons (1960). In addition, because the amount of cold water is finite, the
interface between the cold and warm water can intersect the seafloor; thus, the grounding
phenomenon can be studies, using the same conception first developed by Parsons (1966). As an
example, this technique was used to the study of bottom water circulation by Speer and McCartney
(1991).
Inverse reduced gravity model applies to bottom water circulation (Speer and McCartney,
1991) and deep western boundary current (Stommel and Arron, 1962). The pressure gradient term
in an inverse gravity model can be derived similarly. We can use the pressure expression derived
for the case with a rigid lid. In the present case we assume that the upper layer is vary thick and
motionless, i.e., ∇p1 = 0 . From (1), we have ∇pa = 0 . Taking the gradient of (2) and substituting
these relations to the pressure gradient expression in the second layer, we obtain
∇p2 = − g ( ρ 2 − ρ1 )∇( H − h)
(56)
Therefore, the basic equations for an inverse-reduced gravity model is
− fv = g '( H − h) x
(57)
fu = g '( H − h) y
(58)
16
ht + (hu ) x + (hv) y = wu
(59)
where wu is the vertical velocity leaving the upper surface of lower layer due to detrainment to the
layer above.
For example, along the western boundaries of the oceans, there are deep western boundary
currents. These boundary currents are closely associated with the dense water originated from
deepwater formation sites at high latitudes. The shoaling of the isopycnal surface near the western
boundary (Fig. 3) suggests that (in the Northern Hemisphere) the deep western boundary current
flows equatorward.
Fig. 9. An inverse reduced gravity model.
C) Generalized reduced gravity model
Although most reduced gravity model assume constant density for both the upper and lower
layer, one can also use a model in which the density of the upper layer is not constant
Δρ = Δρ ( x, y ) (Huang, 1991). This additional freedom in the model formulation gives rise to new
useful features. The stratification of the model can be specified a priori, or can be changed with
time, using another prognostic equation (Ripa, 1995).
g
− f ρV = − ⎡⎣( ρ 2 − ρ1 ) h 2 ⎤⎦ + τ x
(60)
x
2
g
f ρU = − ⎡⎣( ρ 2 − ρ1 ) h 2 ⎤⎦
(61)
y
2
where U and V are the vertically integrated volume flux in the upper layer. The corresponding
Sverdrup relation is
ρV = −τ yx / β
(62)
and the layer thickness equation is
( ρ2 − ρ1 ) h
2
= [ ρ 2 − ρ1 ( x, y ) ] h ( xe , y ) + ∫
2
xe
x
2 f 2 ⎛τ x ⎞
⎜ ⎟ dx
gβ ⎝ f ⎠y
(63)
The density distribution in the upper layer can be either specified based on observation, or
calculated from buoyancy conservation
U ρ1, x + V ρ1, y = γ ( ρ * − ρ1 )
(64)
17
To illustrate the basic ideas, an example where the density distribution is calculated be following
the characteristics and solving Eq. (64) is shown in the following figures.
Fig. 10. Vertically integrated mass flux (solid lines, in Sv) and density difference at the interface
(dotted lines, in 10−3 g / cm3 ). Dashed line indicates the northern limit of the model, where the layer
vanishes (Huang, 1991).
Fig. 11. Layer depth, in 100m intervals. a) The generalized gravity model; b) a traditional reduced
gravity model (Huang, 1991).
Fig. 12. Free surface elevation, in cm. a) The generalized gravity model; b) a traditional reduced
gravity model (Huang, 1991).
18
D) Limitation of the reduced gravity models
Since these models represent the vertical stratification with a one-and-a-half layer, they are
highly truncated and thus cannot represent the complicated three-dimensional circulation very
accurately.
The reduced gravity models are based on the assumption that flow underneath the main
thermocline is negligible. For problems associated with time scales shorter than the time scale for
the first baroclinic Rossby waves to move across the basin at the latitude of interest, the errors
implied by the reduced-gravity formulation may not be totally negligible.
Nevertheless, reduced gravity models have been used to simulate seasonal cycle, such as the
free surface elevation and other properties, for the subtropical basin. Furthermore, reduced gravity
models have been extensively used to study equatorial circulation on seasonal time scales.
19