128
DYNAMICAL OCEANOGRAPHY
Through the analysis in the previous chapter, an enormous reduction of
the mathematical problem of the wind-driven circulation has been accomplished. At the beginning of section 5.1, we needed to solve a system
of four coupled partial differential equations in three space dimensions,
which has now been reduced to the solution of one scalar partial differential equation (5.91) in two spatial dimensions. But the problem can be
reduced even more! In section 6.1, inner and outer expansion techniques
are used on the barotropic vorticity equation to obtain the famous Sverdrup balance outside continental boundary layers. The structure of the
flow in the boundary layers is presented in section 6.2 for the frictional
cases (the Munk and Stommel boundary layers) and in section 6.3 for the
inertial case. From this theory, the mechanism of the western intensification of midlatitude ocean current such as the Gulf Stream is described.
Finally, in section 6.4 some nonlinear aspects of the circulation are discussed.
6.1. The Sverdrup balance
As we have already seen, the values of L = 1000 km, U =1 0 −2 ms −1 apply
for the gyre-scale flow in the North Atlantic and with β 0 =2.0 × 10 −11 (ms) −1 it
follows that β = O(10 2 ). If we look at the terms in the equation (5.91), then the
first term in the left hand side is O(1) and the last two terms in the right hand side
are both at most O(1). Hence, only the wind stress term can balance the β term
which leads to
O(
αr
2
)=O(β) →O
τ 0
ρf 0 δ E U
E
1/2
V
ǫ
= O
τ 0
ρU D
L
U
= O
β 0 L 2
U
, (6.1)
from which the horizontal velocity U (which was previously not related to the
parameters, but based on observational values) follows as
U =
τ 0
ρDβ 0 L
.
(6.2)
First of all, we need to check whether this gives consistent values of U . With
D =1 0 3 m, ρ =1 0 3 kgm 3 , τ 0 =0 .2 Nm −2 ,a n dβ 0 =2 .0 × 10 −11 (ms) −1 ,
we find U ≈ 10 −2 ms −1 which is indeed consistent. As mentioned before, this
velocity is a depth averaged velocity at the basin scale.
With the choice of U , the barotropic vorticity equation (5.91) becomes
1
β
∂ψ
∂x
∂
∂y
−
∂ψ
∂y
∂
∂x
∇
2 ψ − Fψ + η b
+
∂ψ
∂x
DYNAMICAL OCEANOGRAPHY
Through the analysis in the previous chapter, an enormous reduction of
the mathematical problem of the wind-driven circulation has been accomplished. At the beginning of section 5.1, we needed to solve a system
of four coupled partial differential equations in three space dimensions,
which has now been reduced to the solution of one scalar partial differential equation (5.91) in two spatial dimensions. But the problem can be
reduced even more! In section 6.1, inner and outer expansion techniques
are used on the barotropic vorticity equation to obtain the famous Sverdrup balance outside continental boundary layers. The structure of the
flow in the boundary layers is presented in section 6.2 for the frictional
cases (the Munk and Stommel boundary layers) and in section 6.3 for the
inertial case. From this theory, the mechanism of the western intensification of midlatitude ocean current such as the Gulf Stream is described.
Finally, in section 6.4 some nonlinear aspects of the circulation are discussed.
6.1. The Sverdrup balance
As we have already seen, the values of L = 1000 km, U =1 0 −2 ms −1 apply
for the gyre-scale flow in the North Atlantic and with β 0 =2.0 × 10 −11 (ms) −1 it
follows that β = O(10 2 ). If we look at the terms in the equation (5.91), then the
first term in the left hand side is O(1) and the last two terms in the right hand side
are both at most O(1). Hence, only the wind stress term can balance the β term
which leads to
O(
αr
2
)=O(β) →O
τ 0
ρf 0 δ E U
E
1/2
V
ǫ
= O
τ 0
ρU D
L
U
= O
β 0 L 2
U
, (6.1)
from which the horizontal velocity U (which was previously not related to the
parameters, but based on observational values) follows as
U =
τ 0
ρDβ 0 L
.
(6.2)
First of all, we need to check whether this gives consistent values of U . With
D =1 0 3 m, ρ =1 0 3 kgm 3 , τ 0 =0 .2 Nm −2 ,a n dβ 0 =2 .0 × 10 −11 (ms) −1 ,
we find U ≈ 10 −2 ms −1 which is indeed consistent. As mentioned before, this
velocity is a depth averaged velocity at the basin scale.
With the choice of U , the barotropic vorticity equation (5.91) becomes
1
β
∂ψ
∂x
∂
∂y
−
∂ψ
∂y
∂
∂x
∇
2 ψ − Fψ + η b
+
∂ψ
∂x