M.Sc. in Computational Science
Fundamentals of
Atmospheric Modelling
Peter Lynch, Met
´
Eireann
Mathematical Computation Laboratory (Opp. Room 30)
Dept. of Maths. Physics, UCD, Belfield.
January–April, 2004.
Lecture 7
Potential Vorticity Conservation
2
Applications of PV Conservation
We consider a few simple applications
of Potential Vorticity conservation.
The treatment is purely qualitative
.
A quantitative treatment will be undertaken in subsequent lectures.
GravityInertia Waves.
Free Rossby Waves.
Forced Rossby Waves.
Leeside Trough.
3
The Potential Vorticity Equation
Recall that the continuity equation may be written:
dh
dt
+
hδ
= 0
The vorticity equation may be written:
d
dt
(
ζ
+
f
) + (
ζ
+
f
)
δ
= 0
.
Taking logarithms, we may write these in the form
d
dt
log(
ζ
+
f
) =

δ ,
d
dt
log
h
=

δ .
We eliminate
δ
between these equations to get:
d
dt
log(
ζ
+
f
)

d
dt
log
h
= 0
.
This may also be put in the following form:
d
dt
ζ
+
f
h
= 0
.
This is the equation of
conservation of potential vorticity
.
4
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Elementary Applications
of PV Conservation
5
Gravity Waves: A First Look
Suppose initially the flow is irrotational and is converging
towards a point.
Assume that
f
is constant.
Consider a
column of fluid.
•
Convergence
induces stretching
•
Stretching implies increased pres
sure at the centre
•
Increasing
h
also implies increas
ing
ζ
•
ζ >
0
implies Cyclonic flow
•
Cyclonic flow around high pres
sure is
unbalanced
•
PGF and Coriolis force act out
wards
•
∴
Divergent
flow is induced.
The restoring forces give rise to
Inertiagravity waves
.
6
Rossby Waves: A First Look
Suppose initially the flow is nondivergent, so there is no
vertical velocity and
h
is constant for a fluid parcel. Then
absolute vorticity is conserved.
d
(
ζ
+
f
)
dt
= 0
.
Suppose a parcel, initially at latitude
y

y
0
with
f
=
f
0
and
ζ
= 0
. Suppose that it moves Northeastward.
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 Spring '07
 Osserman
 Math, Conservation Of Energy, Energy, Kinetic Energy, Potential Energy, vorticity, Potential Vorticity Conservation

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