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DYNAMICAL OCEANOGRAPHY
13.6. Exercises on chapter 13
(13.1) Planetary Ekman layers
Consider the flow in the spherical sector [φ, θ] ∈ [286, 350] × [10, 70] as a
model of the ocean circulation in the North Atlantic Ocean. The density of the
water is constant and the flow is driven by the wind-stress field
τ
φ
∗ (φ, θ)=−
τ 0
2π
cos 2π(
θ − 10
60
); τ
θ =0
(13.102)
where τ 0 =0 .1 Pa. The depth of the basin D = 4000 m and the bottom
is flat. Assume that the mixing coefficients of momentum are constant with
A V =10 −3 m 2 s −1 and A H =10 5 m 2 s −1 .
a. Determine the thickness (in m) of the Ekman layers at the ocean-atmosphere
interface and near the bottom.
b. Determine the pattern and amplitude of the upwelling over the basin.
c. Determine the difference in sea surface height between the subtropical gyre
and the subpolar gyre (in m).
d. Determine the pattern and amplitude (in ms −1 ) of the geostrophic meridional velocity in the basin.
(13.2) Homogeneous planetary circulation
Consider a circumglobal zonal channel for θ ∈ [θ 0 ,θ 1 ] ⊂ (−π/2, 0) bounded
between the latitudes θ 0 en θ 1 . The wind stress at the ocean-atmosphere surface is of the form
τ
φ =sin(π
θ − θ 1
θ 0 − θ 1
); τ
θ =0
The bottom of the channel is flat and the surface deformation of the
ocean-atmosphere surface is negligible. The density of the ocean water
is constant, the flow caused by the wind stress is steady and horizontal
mixing of momentum can be neglected. In this exercise we try to find flows
which are independent of the zonal coordinate, i.e., solutions of the form
(u(θ, z),v(θ, z)w(θ, z),p(θ, z)).
DYNAMICAL OCEANOGRAPHY
13.6. Exercises on chapter 13
(13.1) Planetary Ekman layers
Consider the flow in the spherical sector [φ, θ] ∈ [286, 350] × [10, 70] as a
model of the ocean circulation in the North Atlantic Ocean. The density of the
water is constant and the flow is driven by the wind-stress field
τ
φ
∗ (φ, θ)=−
τ 0
2π
cos 2π(
θ − 10
60
); τ
θ =0
(13.102)
where τ 0 =0 .1 Pa. The depth of the basin D = 4000 m and the bottom
is flat. Assume that the mixing coefficients of momentum are constant with
A V =10 −3 m 2 s −1 and A H =10 5 m 2 s −1 .
a. Determine the thickness (in m) of the Ekman layers at the ocean-atmosphere
interface and near the bottom.
b. Determine the pattern and amplitude of the upwelling over the basin.
c. Determine the difference in sea surface height between the subtropical gyre
and the subpolar gyre (in m).
d. Determine the pattern and amplitude (in ms −1 ) of the geostrophic meridional velocity in the basin.
(13.2) Homogeneous planetary circulation
Consider a circumglobal zonal channel for θ ∈ [θ 0 ,θ 1 ] ⊂ (−π/2, 0) bounded
between the latitudes θ 0 en θ 1 . The wind stress at the ocean-atmosphere surface is of the form
τ
φ =sin(π
θ − θ 1
θ 0 − θ 1
); τ
θ =0
The bottom of the channel is flat and the surface deformation of the
ocean-atmosphere surface is negligible. The density of the ocean water
is constant, the flow caused by the wind stress is steady and horizontal
mixing of momentum can be neglected. In this exercise we try to find flows
which are independent of the zonal coordinate, i.e., solutions of the form
(u(θ, z),v(θ, z)w(θ, z),p(θ, z)).