Exam in GEF2500 - Geophysical Fluid Mechanics
Day of exam: Friday, 5 th
June, 2009
Exam hours: 9.00-12.00
This examination paper consists of 3 pages
Appendices: None
Permitted materials: None
Make sure that your copy of this examination paper
is complete before answering.
Problem 1
By combining the first and second law of thermodynamics, we obtain
TDs
= c p
DT
−
T
T
Dp (1.1) a) Explain the various symbols in (1.1). b) Show for an ideal gas that
T
=
1 / T . c) What do we mean by an adiabatic process? d) Derive an expression for the (dry) adiabatic lapse rate.
Problem 2 a) What causes an inertial oscillation? b) How does the velocity vector move during one inertial period in the Northern
Hemisphere? (explain without calculations). c) A particle that moves with constant speed u in a circular orbit has a centripetal
0 acceleration u 2
0
/ R directed towards the centre. Calculate the radius R in the circle if this is an inertial oscillation, and sketch the direction of the particle motion.
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Problem 3 a) What do we mean by hydrostatic balance? b) What do we mean by geostrophic balance? c) We have a pressure field in the Northern Hemisphere: p
= −
gz
+
1
2
y
2 +
P
0
, (3.1) where
(the density),
α
, and P are positive constants. Find the geostrophic velocity.
0 d) Sketch the geostrophic velocity together with the isobars in the ( x , y ) -plane. Indicate (with arrows) the balance of forces on a fluid particle. e) Consider the planetary Ekman layer. We assume that it extends to the ground z
=
0 .
Define the Ekman-layer thickness D
E and explain the symbols. f) Sketch the balance of forces on a fluid particle in the planetary Ekman layer. g) Integrate the continuity equation
∂ u /
∂ x
+ ∂ v /
∂ y
+ ∂ w /
∂ z
=
0 from the horizontal ground to the top of the Ekman layer, and show that w
E
= −
∂
U
∂ x
−
∂
V
∂ y
. (3.2)
Here w
E is the vertical velocity at the top of the Ekman layer, and ( U , V ) are the volume transports defined by
U
=
0
D
E udz , V
=
D
E
0
vdz .
(3.3) h) If the geostrophic velocity is directed along the x -axis (= u ), the volume transports (3.3) g in the Ekman layer can be written
U
= ⎜
1
1
2
D
E u g
,
(3.4)
V
=
1
2
D
E u g
.
Explain this result on the basis of the discussion in question (3f).
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i) Assume that u g is the geostrophic velocity in question (3c). Find w
E
in this case. j) Sketch the vertical motion above the Ekman layer if w is zero near the tropopause. k) Can this have any consequences for the geostrophic current as time goes on?
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