Antarctic Circumpolar Current
349
14.4. Exercises on chapter 14
(14.1) Scaling of the ACC transport
In section 14.2, we derived that in the case of lateral friction, the zonal transport through the zonal channel scales as 1/A H . Consider now the case where
only bottom friction is the dissipation mechanism.
a. Determine an expression for the bottom shear stress with help of the
(planetary) bottom Ekman solution in chapter 13.
b. Determine the scaling of the ACC transport versus A V , in case the bottom
friction balances the wind stress.
(14.2) Depth averaged vorticity balance
Consider a constant density flow in a zonal channel as in section 14.2 and
neglect the bottom stress term.
a. Use the depth-averaged equations (14.9) to derive the vorticity equation for
this flow, i.e.,
2Ω¯ v cos
2 θ =
1
ρ 0 r 0
J(H, p b )+
1
ρ 0
(
∂τ θ
∂φ
−
∂(τ φ cos θ)
∂θ
),
where J is the Jacobian.
b. Compare the result above for the flat bottom case with the Sverdrup balance
(13.33) derived in section 13.2. Why are the results different?
(14.3) Geostrophic contours
a. Determine the geostrophic contours for the bottom topography given by
h b (x, y)=h 0 e
−(
x
L
) 2 ,
where h 0 > 0, x ∈ [−L, L] and y ∈ [y 0 ,y 1 ].
Assume that the flow is driven by a constant wind stress τ x = τ 0 , τ y =0,that
the density is constant and neglect bottom friction.
349
14.4. Exercises on chapter 14
(14.1) Scaling of the ACC transport
In section 14.2, we derived that in the case of lateral friction, the zonal transport through the zonal channel scales as 1/A H . Consider now the case where
only bottom friction is the dissipation mechanism.
a. Determine an expression for the bottom shear stress with help of the
(planetary) bottom Ekman solution in chapter 13.
b. Determine the scaling of the ACC transport versus A V , in case the bottom
friction balances the wind stress.
(14.2) Depth averaged vorticity balance
Consider a constant density flow in a zonal channel as in section 14.2 and
neglect the bottom stress term.
a. Use the depth-averaged equations (14.9) to derive the vorticity equation for
this flow, i.e.,
2Ω¯ v cos
2 θ =
1
ρ 0 r 0
J(H, p b )+
1
ρ 0
(
∂τ θ
∂φ
−
∂(τ φ cos θ)
∂θ
),
where J is the Jacobian.
b. Compare the result above for the flat bottom case with the Sverdrup balance
(13.33) derived in section 13.2. Why are the results different?
(14.3) Geostrophic contours
a. Determine the geostrophic contours for the bottom topography given by
h b (x, y)=h 0 e
−(
x
L
) 2 ,
where h 0 > 0, x ∈ [−L, L] and y ∈ [y 0 ,y 1 ].
Assume that the flow is driven by a constant wind stress τ x = τ 0 , τ y =0,that
the density is constant and neglect bottom friction.