140
DYNAMICAL OCEANOGRAPHY
(a)
(b)
Figure 6.4. (a) Sverdrup flows for the wind stress field τ
x (x, y)=−1/(2π) cos 2πy, τ
y (x, y)=
0. (b) Total flow solution with Stommel western boundary current for δS/L =0.1.
the asymptotic solution deviates substantially from the analytic one, but for
δ S /L =0.01 the difference is very small.
◭
To demonstrate the compensation of the Sverdrup transport by the western
boundary current, we compute the dimensionless meridional velocity, ˆ
v 0 (λ, y)
in the boundary layer and find
ˆ
v
0 (λ, y)=
1
ℓ
∂ ˆ
ψ 0
∂λ
=
L
δ S
∂ ˆ
ψ 0
∂λ
=
L
δ S
Ψ
0 (y)e
−λ .
(6.66)
The total meridional transport in the boundary layer ˆ
Φ y therefore is
ˆ
Φ
y (y)=ℓ
∞
0
ˆ
v
0 (λ, y)dλ =Ψ 0 (y)=−Φ
y (y),
(6.67)
and from this it follows that
ˆ
Φ
y +Φ
y =0,
(6.68)
which demonstrates the compensation.
6.2.3. Physics of the western intensification
Ex. 6.3
As shown by direct calculation, both bottom and lateral friction can provide
provide western boundary currents which compensate for the complete Sverdrup
transport. In this section, we focus on the physics of this result by looking at the
vorticity balance for a subtropical gyre.