GEF3450
Exercises for group session:
Circulation, Vorticity and Potential Vorticity
Ada Gjermundsen
E-mail: ada.gjermundsen@geo.uio.no
November 5, 2015
Figure 1: A satellite image of Hurricane Patricia before it made landfall in
southwest Mexico on Friday, October 23, 2015. At one point, Patricia had
winds of 200 mph, making it the strongest hurricane on record in the eastern
Pacific Ocean. figure from National Hurricane Center, US
On Earth, even fluids at rest have vorticity. The Planetary vorticity due to
rigid body rotation of the Earth is
f = 2Ω sin(θ)
(1)
Fluid motion is measured relative to the Earth’s rotation, so Relative vorticity is calculated from velocities relative to the Earth:
ζ=
∂v
∂u
−
∂x ∂y
1
(2)
In an inertial frame, fluid motion is both with and relative to the Earth;
Absolute vorticity is the sum of planetary and relative vorticity:
ζa = ζ + f
(3)
Last but not least, we have the Potential vorticity which is a conserved
quantity following fluid parcels
PV =
ζ +f
H
(4)
Figure 2: Potential vorticity map over Europe. Values from 1 PVU
upwards are colored rainbow-wise from dark blue to red, with contour interval 1 PVU, where 1 PVU =10−6 m2 s−1 K kg−1 . figure from
http://www.atm.damtp.cam.ac.uk/mcintyre/papers/ECMWF/
Exercises:
Exercises from “Fluid Mechanics” by Kundu and Cohen, “Atmosphere, Ocean and Climate Dynamics” by Marshall and Plumb, “Atmospheric Science” by Wallace and Hobbs
and“Dynamic Meteorology" by Holton and Hakim.
1 Let’s take a look on the relative vorticity first (i.e. ignore f in this
exercise). Consider a square of 1000 km for an easterly (i.e. westward
flowing) wind that decreases in magnitude toward the north at a rate
of 10 ms−1 pr 500 km.
i) What is the mean relative vorticity in the square?
ii) What is the circulation about the square?
2
2 An air parcel at 30◦ N moves northward, conserving absolute vorticity.
If its initial relative vorticity is 5 × 10−5 s−1 , what is its relative vorticity upon reaching 90◦ N?
3 A water column at 60◦ N with ζ=0 initially stretches from the bottom of
the ocean at 5 km depth to the ocean surface. The water column moves
southwards, towards shallower depth until it reaches 45◦ N, where the
ocean depth is only 3 km. What are the absolute and relative vorticity of the water column at 45◦ N, assuming that the flow satisfies the
barotropic PV equation?
4 A westerly zonal flow at 45◦ is forced to rise adiabatically from the
surface over a north-south-oriented mountain barrier. Before striking
the mountain, the westerly wind increases linearly towards the south
at a rate of 10 ms−1 per 1000 km. The crest of the mountain range is
located at 2000 m, and the tropopause, say at 10 km height, remains
undisturbed.
i) What is the initial relative vorticity of the air?
ii) What is the relative vorticity when the air reaches the crest of the
mountain?
iii) What is the relative vorticity when the air reaches the crest of
the mountain if it is deflected 5◦ latitude towards the south during the
forced ascent?
5 Consider a velocity field that can be represented as
~ = −k̂ × ∇ψ
V
(5)
~ is everywhere
where ψ is called the streamfunction. Prove that ∇H · V
equal to zero and that the vorticity field is given by
ζ = ∇2 ψ
(6)
More challenging:
6 Sketch the response of streamfunction, velocity and temperature to a
ball of positive PV located a small distance above a ball of negative PV.
3
7 Consider barotropic ocean eddies propagating meridionally along a
sloping continental shelf, with depth increasing toward the east (see
figure on the next page; lighter blue shading indicates shallower water), conserving barotropic potential vorticity in accordance with
d f + ζ =0
(7)
dt
H
There is no background flow. In which direction will the eddies propagate? [Hint: consider the vorticity tendency at points A and B.]
Figure 3: From “Atmospheric Science” by Wallace and Hobbs
Solutions:
1 i) The mean relative vorticity
× 10−5 s−1
RR is; ζm = −2
2
ii) The circulation is C =
dA = ζm L = −2 × 107 m2 s−1
A
2 ζf inal = −2.3 × 10−5 s−1
3 ζf inal = −2.72 × 10−5 s−1
4 i) ζ0 = 1 × 10−5 s−1
ii) ζf inal = −1.26 × 10−5 s−1
iii)ζf inal = −3.21 × 10−6 s−1
7 At point A we have a negative vorticity tendency, while at point B
we have a positive vorticity tendency, therefore the eddies will travel
south (in the northern hemisphere)
4