Lecture 14. Equations of Motion
Currents With Friction
Sverdrup, Stommel, and Munk Solutions
Remember that Ekman's solution for wind-induced transport is
which can also be written as
(14.1)
i.e.,
#Q
x,y
= Mx,y. Equations (14.1) represent wind-induced transports per unit length of coastline
2
and have units of L /T.
Now, instead of assuming spatially uniform wind as Ekman did, if we
take into account horizontal shears in the wind field, then we will arrive at the solution obtained
by Sverdrup.
Consider then the following situations of an eastward wind stress that is sheared
in the N-S direction in the northern hemisphere:
)
x
)
Qy
x
Qy
N
E
Note that the situation on the left causes transport convergence, and that on the right yields
transport divergence.
These effects are elucidated from equations (14.1) by differentiating the
first with respect to y to represent the N-S shear of an E-W wind, and by differentiating the y
component of (14.1) with respect to x to represent the E-W shear of the N-S wind.
differentiating procedure yields
1
This
which, upon addition yield
(14.2)
Note that
3
,
is the change of the Coriolis parameter with latitude ( f /
considered negligible for now.
,y),
which will be
We will return to the full form of this equation, which is one
form of Sverdrup's equation.
Then, equation (14.2) with
3
= 0 describes transport
convergences/ divergences (first term on the left hand side) as produced by the curl of the wind
,)y / ,x = 0. Then,
,)x / ,y is positive and (14.2) describes a negative divergence, i.e.,
Similarly, for the situation on the right, ,)x / ,y is negative and (14.2) describes
stress (term on the right hand side). From the diagram above, it is seen that
for the situation on the left,
convergence.
a divergence.
Convergences and divergences cause sea level slopes, which in turn set up pressure gradients.
Convergences tend to pile up water, while divergences set-down the sea surface. These sea level
slopes will support the development of a geostrophic flow (ug) in the waters between the surface
and bottom Ekman layers, as illustrated below
z
›
)y
V
Qx
š
Press.Grad. Force
DE
ug
Note that the geostrophic flow produced by the sea level slope is in the same direction as the
wind stress, and that the net transport in the bottom Ekman layer is to the left of the geostrophic
flow.
Equation (14.2) can be simplified with the use of continuity (first term equals zero) to obtain
2
Sverdrup's meridional transport (parallel to meridians) Qy:
(14.3)
This is the familiar form of Sverdrup's equation. To obtain the zonal transport Qx, Sverdrup used
the principle that allowed the development of the expression for Qy (14.3), i.e., continuity:
With continuity and (14.3), Sverdrup obtained the following relationship:
(14.4)
by neglecting the change of meridional winds with latitude (
,)y /,y 0).
Equation (14.4) says
that the zonal changes of the zonal transport are given by the curvature of the zonal wind with
respect to latitude.
To solve (14.4), Sverdrup integrated it from an eastern boundary where Qx
equals zero, to a distance -x away from the boundary, or
(14.5)
which describes the zonal transport induced by wind stress. Equation (14.5) indicates that the
transport is related to the meridional curvature of the zonal wind stress.
illustrated in the figure presented in class.
This concept is
As seen, the transports predicted by (14.5) are
qualitatively similar to those associated with the equatorial current system. Equation 14.5 also
indicates that transports increase with distance from the coast.
Sverdrup's circulation model allows the presence of a boundary and of horizontal shears in the
wind stress (in contrast to Ekman's). However, Sverdrup's solution is limited to the eastern side
of the oceans, i.e., nothing is known about the western parts.
Also, Sverdrup's model depicts
depth-integrated flows, i.e., nothing is known about the vertical structure of the currents.
Stommel explained what happens in western portions.
Stommel Solution
3
Stommel elucidated why currents are fast and narrow in the west and broad and relatively slow
in the east.
Stommel used Sverdrup's equation and included bottom friction. He calculated
vertically integrated flow patterns over an idealized ocean (process-oriented study) with a
prescribed wind forcing, which was a function of latitude, under a variety of conditions: no
rotation (f = 0), constant Coriolis parameter f, and Coriolis parameter varied constantly as a
function of latitude, i.e.,
3 constant (3-plane approximation). His results were similar for the
3 effect, his results yielded a westward
first two situations (f = 0, and f = constant). With the
intensification of currents.
Then, the change of f with latitude is the main responsible for
westward intensification of oceanic currents as seen in the Figure presented in class.
Stommel used wind patterns between 15° and 45°N.
His addition of friction allowed a closed
circulation across the entire basin, in contrast to Sverdrup's solution (restricted to eastern
boundary). Stommel's result can be understood in terms of vorticity tendencies. We will digress
to discuss vorticity and then we will come back to discuss Stommel's results in terms of vorticity.
Vorticity
Vorticity is the tendency for portions of a fluid to rotate. Vorticity is related to horizontal shears
in the flow.
We will consider the following types of vorticity: relative vorticity, planetary
vorticity, absolute vorticity, and potential vorticity.
Consider the following situations of flow
horizontal shears that induce vorticity
œ
œ
N
E
u
´
´
v
Relative vorticity
is defined as the curl of the horizontal flow, i.e.,
(14.6)
4
Note from the above diagram that in the upper two cases, the relative vorticity
the upper left,
,u/,y
corresponds with (14.6).
rotation.
(14.6).
is positive.
On
is negative and induces a counterclockwise or positive vorticity, which
On the upper right,
,v/,x is positive, thus inducing counterclockwise
Similarly, the lower cases represent negative or clockwise vorticity as explained by
On the lower left
,u/,y is positive, and on the lower right ,v/,x is negative.
Planetary vorticity is that arising from the Earth's rotation and equals the Coriolis parameter
f.
A stationary object on the surface of the Earth has planetary vorticity, which varies with
latitude
.
Absolute vorticity is the addition of planetary plus relative vorticities. The changes in absolute
vorticty are useful to help us understand the tendencies for parcels of water to rotate.
To
describe changes in absolute vorticity, we have to take the curl of the frictionless equations of
motion
The curl is obtained through cross-differentiation (u component with respect to y and v
component with respect to x) and subtraction, to obtain
(14.7)
Equation (14.7) describes changes of absolute vorticity in time.
These are related to
convergences or divergences, as expressed in the right hand side of the equation.
Imagine a
column of water that is subject to stretching or squashing. The stretching situation is illustrated
below
5
Gain of Absolute Vorticity
Column Stretching
f +
œ
Plan View
f
Side View
In the stretching situation (probably due to increase in local depth), the particles at the perimeter
of a given water column, will tend to move towards the center of the column (convergence) as
illustrated in the plan view.
Due to Coriolis accelerations, the particles will acquire cyclonic or
positive relative vorticity and thus the gain of absolute vorticity.
The divergence or column squashing situation is represented in the diagram below. The particles
at the perimeter of the water column will now move outward.
Due to Coriolis effects, the
particles will acquire anticyclonic or negative relative vorticity. This negative relative vorticity
results in the loss of absolute vorticity caused by the water column moving to shallow water.
6
Loss of Absolute Vorticity
Column Squashing
f
´
f -
Plan View
Side View
Now consider a layer of thickness D, and whose equation of continuity is
(14.8)
The changes of the layer thickness in time are given by convergences or divergences. This is
a reiteration of the fact that convergences cause increases in D, and divergences cause decreases
in D. If we combine this equation (14.8) with the equation for absolute vorticity (14.7), we get
(14.9)
This relationship says that the quantity ( +f)/D =
, which is called potential vorticity (absolute
vorticity over layer thickness), remains constant.
potential vorticity.
This is the statement of conservation of
This is a useful and powerful concept because it allows description of
vorticity tendencies in the ocean.
Consider the following situations in the northern hemisphere.
1) A column of water with constant thickness D. If the column moves zonally (along colatitude
lines), then f remains constant and so does
.
If the water column moves meridionally, then if
7
f increases (column moves to the North),
to keep
must decrease (column acquires anticyclonic motion)
constant. Under the same meridional motion, if the water column moves to the south,
f decreases and
must increase by acquiring cyclonic motion.
2) Increased water column thickness (moving towards deep regions).
If the column moves
must increase, i.e., the column acquires cyclonic vorticity (vortex
Then, if the water column moves meridionally to the South, f decreases and must
zonally, f remains constant so
stretching).
increase. If the column moves meridionally to the North, f increases and it is not obvious what
happens to
because it could increase or decrease, depending on the increases of both D and f.
3) Decreased water column thickness (moving towards shallow regions).
zonal motions (vortex squashing).
decrease.
must decrease in
must
For northward meridional motion, f increases and
It is not obvious what may happen in southward meridional motion.
These situations derived from the concept of conservation of potential vorticity can be used to
explain the westward intensification of the ocean currents. Now we return to explain why the
oceanic currents are intensified to the west as Stommel proposed.
Consider the area of the
westerlies and the trade winds in the ocean. The distribution of the wind stress tends to cause
anticyclonic (negative vorticity), as illustrated below
)
Westerlies
´
Easterlies
Then, on the western part of the oceans, the wind-induced relative vorticity is the same as that
in the eastern part.
f increase.
On the western side, however, the currents move northwards, which make
To conserve
,
must decrease, i.e., northward motion on the western part of the
oceans acquires anticyclonic (negative) vorticity. Analogously, southward motion in the eastern
part of the oceans, produce cyclonic (positive) vorticity. Note that on the eastern part, the windinduced vorticity and the relative vorticity arising from southward flow are of opposite sign and
tend to balance each other.
On the other hand, over the western part, the vorticity induced by
northward currents and the wind-induced vorticity are of the same sign.
Then, to maintain
vorticity balances at the east and west, there has to be a large source of positive vorticity on the
8
west.
This source comes from strong lateral shears in the northward flow (flow increasing
eastward), i.e.,
Western Part
Eastern Part
Wind Stress ( )) ´
Wind Stress ( )) ´
p
p
´
as f increases
) + p + f 0
œ
as f decreases
Lateral Friction ( f) œ to balance
....
) + p c 0 ;
) + p 0
Flow looks like Lateral friction on the western part is the necessary ingredient to have vorticity balances in the
ocean.
That is why currents intensify to the west.
Munk's Solution
Munk extended the domain of Stommel to include a larger range of latitudes.
He used more
realistic wind patterns than those idealized by Stommel and also added horizontal shears
(horizontal friction) to obtain his solution. As seen in the figure in class, Munk's solution depicts
ocean gyres that approximate the actual circulation.
9