Thermocline problem
325
a. Determine the equations that describe the flow outside the boundary layers
at the bottom and at the surface.
b. Derive solutions of the velocity field in the Ekman layer at the bottom and
at the ocean-atmosphere surface.
c. Show by matching of the vertical velocity that the geostrophic zonal velocity
field is given by
u
0 (θ)=C 1 τ
φ (θ)+C 2
| sin θ |
cos θ
and argue that C 2 =0.
d. Choose two simple latitudes θ 0 and θ 1 and sketch the velocity field, including the flow in the Ekman layers.
(13.3) Continuity of the vertical velocity
In this exercise we consider the continuity of the vertical velocity in the
two-layer planetary model over the thermocline.
a. Show that
w 2| z=−h − w 1| z=−h =(u 1 − u 2 ) ·∇h
where
∇h =
1
cos θ
∂h
∂φ
∂h
∂θ
.
b. Next, use the geostrophic balance to show that
u 2 sin θ = e 3 ∧∇(p 1 − γh),
c. Prove that w is continuous at z = −h.
(13.4) Two-layer planetary model
Consider the situation as in exercise (13.1), but now for a two-layer ocean
with g ′ =0 .01 ms −2 and equilibrium layer thicknesses H 1 = 500 ma n d
H 2 = 3500 m.
a. Prove that with a time-independent wind forcing, the lower layer is
motionless.