Western intensification
131
such that
∂ψ
∂x
0
= v
0 (x, y)=∇·(T ∧ e 3 )=− sin 2πy.
(6.14)
The geostrophic streamfunction ψ 0 (x, y) is
ψ
0 (x, y)=−x sin 2πy +Ψ
0 (y).
(6.15)
Note that ψ 0 is also the shape of the ocean-atmosphere boundary since p 0 = η 0 =
ψ 0 . The zonal geostrophic velocity is
u
0 (x, y)=−∂ψ
0 /∂y = −2πx cos 2πy + U (y).
(6.16)
If we try to satisfy the boundary condition u =0at x =1,then
U (y)=2 π cos 2πy → u
0 (x, y)=−2π(1 − x)cos2πy (6.17a)
ψ
0 (x, y)=( 1 − x)sin2πy + ψ 0 ,
(6.17b)
where ψ 0 is an arbitrary constant which can be taken as zero. The solution is
plotted in Fig. 6.1a and it cannot satisfy the condition u =0at x =0 .T oc l o s e
the circulation, a boundary layer is needed near the western boundary.
If we try to satisfy the boundary condition u =0at x =0 ,wefind
ψ
0 (x, y)=−x sin 2πy + ψ 0 .
(6.18)
Again, with ψ 0 =0the Sverdrup flow is plotted in Fig. 6.1b and this solution
cannot satisfy the condition u =0at x =1 . To close the circulation, a boundary
layer is needed near the eastern boundary.
Considering only Sverdrup dynamics, both solutions in Fig. 6.1 are equally
possible and we have to resolve the boundary layer structure (as will be done in the
next section) to determine which one is dynamically correct. For both solutions,
the total meridional Sverdrup transport Φ y is given by
Φ
y (y)=
1
0
− sin 2πy dx = − sin 2πy.
(6.19)
With values of τ 0 =0.1 Pa, L = 1000 km and D = 1000 m, it follows from (6.2)
that U =1 0 −2 ms −1 and we find a maximum transport in each gyre of about 10
Sv. ◭
6.2. Continental boundary layers
To satisfy the boundary conditions (6.9) we have to investigate the flow behavior near the continents where the length scale L is not the characteristic length of
131
such that
∂ψ
∂x
0
= v
0 (x, y)=∇·(T ∧ e 3 )=− sin 2πy.
(6.14)
The geostrophic streamfunction ψ 0 (x, y) is
ψ
0 (x, y)=−x sin 2πy +Ψ
0 (y).
(6.15)
Note that ψ 0 is also the shape of the ocean-atmosphere boundary since p 0 = η 0 =
ψ 0 . The zonal geostrophic velocity is
u
0 (x, y)=−∂ψ
0 /∂y = −2πx cos 2πy + U (y).
(6.16)
If we try to satisfy the boundary condition u =0at x =1,then
U (y)=2 π cos 2πy → u
0 (x, y)=−2π(1 − x)cos2πy (6.17a)
ψ
0 (x, y)=( 1 − x)sin2πy + ψ 0 ,
(6.17b)
where ψ 0 is an arbitrary constant which can be taken as zero. The solution is
plotted in Fig. 6.1a and it cannot satisfy the condition u =0at x =0 .T oc l o s e
the circulation, a boundary layer is needed near the western boundary.
If we try to satisfy the boundary condition u =0at x =0 ,wefind
ψ
0 (x, y)=−x sin 2πy + ψ 0 .
(6.18)
Again, with ψ 0 =0the Sverdrup flow is plotted in Fig. 6.1b and this solution
cannot satisfy the condition u =0at x =1 . To close the circulation, a boundary
layer is needed near the eastern boundary.
Considering only Sverdrup dynamics, both solutions in Fig. 6.1 are equally
possible and we have to resolve the boundary layer structure (as will be done in the
next section) to determine which one is dynamically correct. For both solutions,
the total meridional Sverdrup transport Φ y is given by
Φ
y (y)=
1
0
− sin 2πy dx = − sin 2πy.
(6.19)
With values of τ 0 =0.1 Pa, L = 1000 km and D = 1000 m, it follows from (6.2)
that U =1 0 −2 ms −1 and we find a maximum transport in each gyre of about 10
Sv. ◭
6.2. Continental boundary layers
To satisfy the boundary conditions (6.9) we have to investigate the flow behavior near the continents where the length scale L is not the characteristic length of