GEF3450
Exercises for group session: Midterm practice
Ada Gjermundsen
E-mail: ada.gjermundsen@geo.uio.no
September 28, 2015
Exercises from “Atmosphere, Ocean and Climate Dynamics” by Marshall and Plumb
and “Atmospheric Science” by Wallace and Hobbs.
1 For streamfunctions ψ with the following functional forms, sketch the velocity
field:
i) ψ = my
ii) ψ = my + n cos(2πx/L)
iii) ψ = m(x2 + y 2 )
iv) ψ = myx
where m and n are constants.
2 For each of the flows in the previous exercise
i) Calculate the strain and rotation matrices
ii) Describe the distribution of vorticity
3 Within a local region near 40◦ N, the geopotential height contours on a
500 hPa chart are oriented east-west and the spacing between adjacent contours (at 60 m intervals) is 300 km, with a geopotential height decreasing
toward the north. Calculate the direction and speed of the geostrophic wind.
4 During winter in the mid-latitudes, the meridional temperature gradient is
typically on the order of 1◦ C per degree of latitude, while the potential temperature increases with height at a rate of roughly 5◦ C km−1 . What is a
typical slope of the potential temperature surfaces in the meridional plane?
Compare this result to the result of the 500 hPa surface in the previous
exercise.
5 i) From the pressure coordinate thermal wind relationship, (eqns. 3.45-46 in
the gfd notes), and approximating
∂u
∂u/∂z
'
∂p
∂p/∂z
(1)
show that in geometric height coordinates
∂u
g ∂T
'−
∂z
T ∂y
1
(2)
ii) The winter polar stratosphere is dominated by the “polar vortex”, a strong
westerly circulation about 60◦ latitude around the cold pole, as depicted
schematically in the figure on the next page. (This circulation is the subject
of considerable interest, because it is within the polar vortices - especially
that over Antarctica in the southern winter and spring - that most of the
ozone depletion is taking place.)
Assuming that the temperature at the pole is (at all heights) 50 K colder at
80◦ latitude than at 40◦ latitude (and that it varies uniformly in between),
and that the westerly wind speed at 100 hPa pressure at 60◦ latitude is
10 ms−1 , use the thermal wind relation to estimate the wind speed at 1 hPa
pressure at 60◦ latitude. (you need the gas constant for air = 287 Jkg−1 K).
2