Arctic Ocean Circulation
373
(15.3) Topography in the quasi-geostrophic case
In this exercise the quasi-geostrophic two-layer model with bottom topography is considered. Starting equations are again given by (15.3) on a domain
[x w ,x e ] × [−L, L] which contains the point (0, 0). We will neglect bottom
friction and hence put ǫ 0 =0. The bottom topography is given by
h b (x, y)=h b0
1 −
x 2 + y 2
R 2
when x 2 + y 2 pumping is chosen as
w E (x, y)=
⎧
⎨
⎩
α(y − L):y 0 α(y 0 − L):−y 0 −α(y + L):−L and hence w E is everywhere negative over the domain.
a. Assume that R lower layer is at rest and determine the streamfunction in the upper layer.
b. Determine the conditions under which closed geostrophic contours appear.
Now assume that the conditions under b. are such that no closed geostrophic
contours appear due to the wind-stress forcing and consider the case h b0 > 0.
c. Determine the condition on h b0 such that closed geostrophic contours appear
in the lower layer.
(15.4) Bottom velocities on closed f/H contours
It is possible to derive an approximate formula for the bottom velocities on
closed f/H contours for the model in section 15.4. To do this, we start from
(15.45) here rewritten for convenience as
v b =
1
fρ 0
e 3 ∧∇ψ b
a. Introduce q = f/H as one coordinate and p as a coordinate perpendicular
to q (hence along ∇(f/H)). Show that when velocities perpendicular to f/H
are much larger than those perpendicular to it, the bottom velocities can be
determined from
v b ≈
1
fρ 0
∂ψ b
∂q
e 3 ∧∇q