GEF3450
Exercises for group session: Shallow Water Flows
Ada Gjermundsen
E-mail: ada.gjermundsen@geo.uio.no
October 15, 2015
Exercises from “Fluid Mechanics” by Kundu and Cohen, “Atmosphere, Ocean and
Climate Dynamics” by Marshall and Plumb, “Atmospheric Science” by Wallace and
Hobbs and “Atmospheric and Oceanic Fluid Dynamics” by Vallis.
1 ) Consider an ocean of uniform density ρref = 1000 kg/m3 , as sketched in
Fig. 1. The ocean surface, which is flat in the longitudional direction (x),
slopes linearly with latitude (y) from η = 0.1 m above mean sea level (MSL)
at 40◦ N to η = 0.1 m below MSL at 50◦ N. Using hydrostatic balance, find
the pressure depth H below MSL. Hence show that the latitudional pressure
gradient ∂p/∂y and the geostrophic flow are independent of depth. Determine
the magnitude and the direction of the geostrophic flow at 45◦ N.
Figure 1: From “Atmosphere, Ocean and Climate Dynamics” by Marshall and
Plumb
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2 ) Fig. 2 shows the trajectory of a “champion" surface drifter, which made
one and a half loops around Antarctica between March, 1995, and March,
2000 (courtesy of Nikolai Maximenko). Red dots mark the position of the
float at 30 day interval.
Figure 2: From “Atmosphere, Ocean and Climate Dynamics” by Marshall and
Plumb
i) Compute the speed of the drifter over the 5 years. (Hint: the distance
for one loop around Antarctica at the latitude where the drifter is located is
approximately 22 × 103 km)
ii) Assuming that the mean zonal current at the bottom of the ocean is zero,
use the thermal wind relation (neglecting salinity effects) to compute the
depth-averaged temperature gradient across the Antarctic Circumpolar Current (ACC). Hence estimate the mean temperature drop across the 600 kmwide Drake Passage.
g ∂ρ
(Hint: use the thermal wind equation: f ∂u
∂z = ρref ∂y , the density equation
for fresh water ρ = ρref (1 − αT [T − Tref ]) where αT = 2 × 10−4 K−1 is the
expansion coefficient and an average depth of 4 km)
iii) If the zonal current of the ACC increases linearly from zero at the bottom of the ocean to a maximum at the surface (as measured by the drifter),
estimate the zonal transport of the ACC through Drake Passage, assuming
πy
a meridional velocity profile as in Fig. 3 (u(y, 0) = umax = U0 cos( 2L
))
and that the depth of the ocean is 4 km. The observed transport through
Drake Passage is 130 SV. Is your estimate roughly in accord? If not, why not?
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Figure 3: From “Atmosphere, Ocean and Climate Dynamics” by Marshall and
Plumb
3 ) Consider a straight, parallel, oceanic current at 45◦ N. For convenience, we
define the x- and y- directions to along and across the current, respectively.
In the region −L < y < L, the flow velocity is
πy
z
) exp( )
2L
d
where z is height (note that z = 0 at mean sea level and decreases downwards), L = 100 km, d = 400 m and U0 = 1.5 ms−1 . In the region
|y| > L, u = 0.The surface current is plotted in Fig. 3. Using the geostrophic,
hydrostatic and thermal wind relations:
u = U0 cos(
i) Determine and sketch the profile of surface elevation as a function of y
across the current.
ii) Determine and sketch the density difference, ρ(y, z) − ρ(0, z). (Hint: use
g ∂ρ
the thermal wind equation: f ∂u
∂z = ρref ∂y )
iii) Assuming the density is related to temperature by:
ρ = ρref (1 − αT [T − Tref ])
(1)
determine the temperature difference, T (L, z) − T (−L, z), as a function of
z. Use αT = 2 × 10−4 C−1 . Evaluate the difference at a depth of 500 m.
Compare with Fig. 4.
4 ) An internal Kelvin wave on the thermocline of the ocean propagates along
the west of Australia. The thermocline has a depth of 50 m and has a nearly
discontinuous density change of 2 kg/m3 across it. The layer below the
thermocline is deep. At a latitude 30◦ S, find the direction and the magnitude
of the propagation speed and the decay scale perpendicular to the coast.
(Hint:
the Kelvin waves propagates in the thermocline with the speed c=
√ 0
g H, where g 0 = g∆ρ
ρ is the reduced gravity.)
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Figure 4: Temperature section (in C◦ ) over the top 2 km of the water column crossing the Gulf Stream along 38 ◦ N, between 69 ◦ W and 73 ◦ W. From “Atmosphere,
Ocean and Climate Dynamics” by Marshall and Plumb.
More challenging:
5 ) Fig. 5 shows the pressure and horizontal wind fields in an atmospheric
Kelvin wave that propagates zonally along the equator. Pressure and zonal
wind oscillate sinusoidally with longitude and time, while the meridional velocity is zero everywhere (v = 0). Such waves are observed in the stratosphere
where frictional drag is negligible in comparison with the other terms in the
horizontal equation of motion.
i) Prove that the waves propagate eastward.
ii) Prove that the zonal wind component is in geostrophic equilibrium with
the pressure field.
Figure 5: Distribution of wind and geopotential height on a pressure surface, in an
equatorial Kelvin wave, as viewed in a coordinate system moving with the wave.
From “Atmospheric Science” by Wallace and Hobbs
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