252
DYNAMICAL OCEANOGRAPHY
The geostrophic velocities (u 0 ,v 0 ) can easily be determined from (11.13a) and
with u 0 (x E ,y)=0, where x E is the eastern boundary, they become
u
0 (x, y)=2 π
2 (x − x E )cos2π(y − y 0 )
(11.15a)
v
0 (x, y)=−
1
2y
(1 + cos 2π(y − y 0 )) − π sin 2π(y − y 0 ) (11.15b)
p
0 (x, y)=y(x E − x)π sin 2π(y − y 0 )+xτ
x (y)
(11.15c)
The profiles of τ x , u 0 , v 0 and p 0 are plotted in Fig. 11.8a for x =1 /2, x E =1
and y 0 =0 .5. Indeed, over an interval in y, the zonal velocity is positive and
in opposite direction to the zonal wind stress. From the meridional velocity, we
see that there is a convergence of mass at y = y 0 and a divergence at y = y 0 +
1/2. The pressure p 0 (which is also the sea surface height) profile indicates that
there the sea surface height decreases over the region where the zonal velocity is
positive in agreement with geostrophic equilibrium.
◭
The example above serves to explain the physics of the ECC. The wind stress
pushes water up to the western continent and hence induces an east-west slope in
the sea surface height. Because the zonal winds vary in meridional direction the
sea level height also changes (Fig. 11.8b). This can be seen from the solution of
the geostrophic pressure (11.15c) which is also the sea surface height (Fig. 11.8a).
Imagine y 0 to be located at about 4 ◦ where the trade winds are still strong. With
the weakening of the trade winds more northward the meridional slope in the sea
level is negative inducing a positive zonal geostrophic velocity. Because of the
variation in the Coriolis parameter, there is a slight shift in the sea level height
variations with the wind stress variations.
Ex. 11.2
We have assumed here that the value of the Coriolis parameter f 0 was so small
that strict (constant f 0 =0 ) geostrophic equilibrium on the f -plane was not possible. At latitudes of the ECC, however, Coriolis effect are not zero and one could
do with a midlatitude β-plane model to compute velocities and sea surface height
with latitude. It appears that this does not change anything in the explanation of
the ECC as given above.
11.4. Equatorial waves
In the reduced gravity model of section 11.2.2, consider the motionless (¯ u ∗ =
¯
v ∗ =0 ) reference state with ¯
h ∗ = H. This is a stationary solution of the unforced, nondissipative equations (11.5). The equations governing small amplitude
motions are obtained by linearizing the equations (11.5) around this reference
state and become
∂u ∗
∂t ∗
− β 0 y ∗ v ∗ = −g
′ ∂h ∗
∂x ∗
,
(11.16a)
DYNAMICAL OCEANOGRAPHY
The geostrophic velocities (u 0 ,v 0 ) can easily be determined from (11.13a) and
with u 0 (x E ,y)=0, where x E is the eastern boundary, they become
u
0 (x, y)=2 π
2 (x − x E )cos2π(y − y 0 )
(11.15a)
v
0 (x, y)=−
1
2y
(1 + cos 2π(y − y 0 )) − π sin 2π(y − y 0 ) (11.15b)
p
0 (x, y)=y(x E − x)π sin 2π(y − y 0 )+xτ
x (y)
(11.15c)
The profiles of τ x , u 0 , v 0 and p 0 are plotted in Fig. 11.8a for x =1 /2, x E =1
and y 0 =0 .5. Indeed, over an interval in y, the zonal velocity is positive and
in opposite direction to the zonal wind stress. From the meridional velocity, we
see that there is a convergence of mass at y = y 0 and a divergence at y = y 0 +
1/2. The pressure p 0 (which is also the sea surface height) profile indicates that
there the sea surface height decreases over the region where the zonal velocity is
positive in agreement with geostrophic equilibrium.
◭
The example above serves to explain the physics of the ECC. The wind stress
pushes water up to the western continent and hence induces an east-west slope in
the sea surface height. Because the zonal winds vary in meridional direction the
sea level height also changes (Fig. 11.8b). This can be seen from the solution of
the geostrophic pressure (11.15c) which is also the sea surface height (Fig. 11.8a).
Imagine y 0 to be located at about 4 ◦ where the trade winds are still strong. With
the weakening of the trade winds more northward the meridional slope in the sea
level is negative inducing a positive zonal geostrophic velocity. Because of the
variation in the Coriolis parameter, there is a slight shift in the sea level height
variations with the wind stress variations.
Ex. 11.2
We have assumed here that the value of the Coriolis parameter f 0 was so small
that strict (constant f 0 =0 ) geostrophic equilibrium on the f -plane was not possible. At latitudes of the ECC, however, Coriolis effect are not zero and one could
do with a midlatitude β-plane model to compute velocities and sea surface height
with latitude. It appears that this does not change anything in the explanation of
the ECC as given above.
11.4. Equatorial waves
In the reduced gravity model of section 11.2.2, consider the motionless (¯ u ∗ =
¯
v ∗ =0 ) reference state with ¯
h ∗ = H. This is a stationary solution of the unforced, nondissipative equations (11.5). The equations governing small amplitude
motions are obtained by linearizing the equations (11.5) around this reference
state and become
∂u ∗
∂t ∗
− β 0 y ∗ v ∗ = −g
′ ∂h ∗
∂x ∗
,
(11.16a)