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Exercise 1

Dynamics 2

Lecturer: Prof. Dr. G. Lohmann

Due date: 22.04.2011

Exercise 1

Summer semester 2013

Exercise 1

15.04.2011

April 16, 2013

1. Ekman spiral

Show the Ekman equation and solution which was shown in the lecture. Draw a

figure.

4 points

2. Wind-driven ocean circulation

When the windstress is only zonal, the Sverdrup transport is

ρ 0 βV = curl τ = − ∂ ∂y τ x (1)

and Ekman transports and Ekman pumping velocity of the surface layer are

Assume furthermore

ρ 0 fV E = −τ x (2)

ρ 0 w E = − ∂ ∂y τ x . (3)

τ x = −τ ∗ cos(πy/B) (4)

for an ocean basin 0 < x < L, 0 < y < B. Westerlies are in the northern part and

easterlies are in the equatorward part.

a) Calculate V, V E , w E .

a) At which latitudes are |V | and |V E | at their maximum? Calculate their magnitudes.

Take constant f = 10 −4 s −1 and β = 1.8 · 10 −11 m −1 s −1 and B =

5000 km, τ ∗ /ρ 0 = 10 −4 m 2 s −2 .

b) calculate the maximum of w E for constant f (value see above).

4 points

1

Dynamics 2

Lecturer: Prof. Dr. G. Lohmann

Due date: 22.04.2011

Summer semester 2013

Exercise 1

15.04.2011

3. The Stommel model: of the wind-driven circulation in a homogeneous ocean of

constant depth h is described by

R∇ 2 ψ + β∂ x ψ =

1

ρ 0

(∂ x τ y 0 − ∂ y τ x 0 ) (5)

where R is a coefficient of bottom friction, β the derivative of the Coriolis frequency

at a central latitude, and the 2-dimensional τ 0 the windstress vector. The depth

integrated velocity is

(U, V ) =

Finally, ψ is the related streamfunction

∫ 0

−h

(u, v)dz .

U = −∂ y ψ, V = ∂ x ψ .

a) Derive this equation from the conservation of momentum (linearized) and mass

(volume!) assuming w = 0 at the mean surface z = 0 and at the bottom z = −h.

For simplicity take Cartesian coordinates for the horizontal, β = df/dy. Boundary

condition for the flux of momentum are τ(z = 0) = τ 0 and τ(z = −h) = R(−V, U).

b) in the boundary layer the terms on the left hand side of (5) get large. Show by

scaling that the width of the layer is W = R/β.

c) how large must R be to get a width W = 100 km? (β = 2 × 10 −11 m −1 s −1 ).

6 points

Notes on submission form of the exercises: Working in study groups is encouraged,

but each student is responsible for his/her own solution. The answers to the questions

shall be provided at the tutorium at the due date.

2

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