Wind-driven circulation
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5.5. Exercises on chapter 5
(5.1) Gulf Stream
In this exercise, we consider the dominant balances in the horizontal momentum equations for the Gulf Stream. Characteristic length scales for the Gulf
Stream are L x =5 0km and L y = 500 km. The characteristic velocity in
east-west direction is about U =0 .1 m/s and that in north-south direction
V =1 .0 m/s. Assume furthermore that the Gulf Stream is a steady current
and take A H =10 5 m 2 s −1 and A V =0.1 m 2 s −1 .
a. Show that in the meridional direction the Gulf Stream is not quite in
geostrophic balance. Which other terms contribute to this momentum
balance?
b. Show that the zonal momentum equation is dominantly geostrophic.
(5.2) An alternative approach to the bottom Ekman layer
Consider an ocean with water of constant density ρ above a flat bottom (at
z ∗ =0 ). The horizontal velocities outside the Ekman layer (where the flow
is hydrostatic and geostrophic) are indicated by (u g∗ ,v g∗ ). The total velocity
can be written as
u ∗ = u g∗ + u E∗ ;
v ∗ = v g∗ + v E∗
where u E∗ en v E∗ are the horizontal Ekman velocities.
a. Show that the Ekman velocities can be determined from
−f 0 v E∗ = A V
∂ 2 u E∗
∂z 2
f 0 u E∗ = A V
∂ 2 v E∗
∂z 2
b. Determine the Ekman transport M E∗ =( M x
E∗ ,M
y
E∗ ) through vertical
integration and show that the direction of this transport is perpendicular to the
bottom shear stress.