Conservation of PV

 g    f 
C
p
1 PVU = 10-6 m2 K kg-1 s-1
There is a “potential” to create relative vorticity by changing latitude or by
changing the thickness of isentropic layers
A parcel of air flowing from a cold, statically stable region to a warm, less stable
region will acquire positive relative vorticity through PV conservation (or stretching).
Climatological PV and theta
(zonally averaged, northern winter)
Theta: dashed contours with intervals 30K
PV: solid contours, units: PVU
PV increases northward and upward.
Hoskins 1990
H
fL
N
Penetration depth
f =Coriolis parameter
L=characteristic length scale
N= Brunt-Vaisala frequency
Positive PV anomaly:
- Cyclonic flow
- Magnitude maximum at anomaly level
- Circulation extends above and below anomaly
- Depth of circulation called the “penetration depth”
Negative PV anomaly:
- Anticyclonic flow
- Magnitude maximum at anomaly level
- Circulation extends above and below anomaly
- Depth of circulation called the “penetration depth”
Low level PV anomalies
Warm anomaly at surface
associated with low
pressure system
Positive vorticity
Note fake isentropes below ground
(positive static stability!)
Cyclogenesis from a PV perspective
The nature of propagation of an upper-level PV anomaly
Consider the (x,y) projection of
an upper air PV anomaly
(this is equivalent to a trough, since
cold air is present beneath the anomaly)
The anomaly will propagate
westward with a new (negative)
anomaly developing to the east due
to advection of PV.
This is equivalent to the westward
propagation of Rossby waves due to
advection of planetary vorticity
Cyclogenesis from a PV perspective
The nature of propagation of a low-level PV anomaly
Consider the (x,y) projection of
An lower atmosphere PV anomaly
(a wave in the potential temp field)
The anomaly will propagate
eastward due to thermal advection.
Cyclogenesis from a PV perspective
Upper level PV anomaly
Lower level PV anomaly
Each anomaly has a circulation associated with it that extends some depth through the troposphere
For development of a cyclone, these circulations must come into phase and
reinforce one another – but how, since they propagate in opposite directions?
Consider an
upper-level
positive PV
anomaly
propagating
over a lowlevel baroclinc
zone
WTA
• Upper positive PV anomaly is associated with a cyclonic circulation
below it (determined by the penetration depth);
• The resultant low-level circulation induces a warm anomaly to its east
through warm thermal advection (WTA).
• The warm anomaly acts as a positive PV anomaly near the surface. It
has its own cyclonic circulation, which extends upward to some extent
(determined by the penetration depth)
• The resultant upper-level circulation is to the east of the original positive
PV anomaly.
PPVA
• This circulation induces positive PV advection (PPVA) to its west ,
which is in the eastern half of the original upper-level positive PV
anomaly.
• The effect is to strengthen the upper level positive PV anomaly and
reduce its westward propagation tendency.
• The upper level anomaly has increased influence on lower level thermal
advection, strengthening the lower level anomaly and reducing its
eastward propagation tendency.
Ut
• The upper-level and low-level circulations are phase locked and amplify
each other.
• Cyclone development can not occur unless there is an up-shear vertical
tilt.
Diabatic processes and the PV
perspective (290-292 of Martin)
Diabatic processes are associated with the creation or destruction of PV.

PV   g    f 
p
In pressure coordinates:
d PV 
 g(a  Ý)
dt
d
Where ηa is the 3D absolute vorticity vector, and  
is the heating rate.
dt
Neglecting the horizonal
 components:
d PV 

  g   f 
dt
p
PV is increased when the vertical gradient of diabatic heating is positive.
Maximum Heating Rate
A diabatic heating maximum occurs
downstream of the upper level PV anomaly
where air is rising most vigorously and
in the middle troposphere
where the maximum condensation occurs.
Erodes upper level PV anomaly
Strengthens low-level PV anomaly
In an extratropical cyclone
diabatic heating
builds ridge aloft
and strengthens
cyclone at surface
Stratiform Precipitation
Stratiform Precipitation
d(PV)
0
dt
Height
(km) B

Spin up the mid-level
cyclonic circulation
A

Heating
Schumacher et al. 2004
d(PV)
0
dt
d(PV)
0
dt
Spin down the low-level
cyclonic circulation