Potential Vorticity and its application to mid-latitude weather systems
We have developed the following tools necessary to diagnose the
processes that lead to the development of fronts and cyclones:
1. The Sutcliff equations
2. The Quasi-Geostrophic Equations
Traditional formulation
Trenberth formulation
Q-Vector formulation
standard coordinates
natural coordinates
3. The Semi-Geostrophic Equations
Sawyer-Eliassen Equation
4. Instability mechanisms
Conditional and Potential Instability
Inertial Instability
Conditional and Potential Symmetric Instability
We will now develop yet another equivalent, but distinct and
useful perspective, based on the quantity called
Isentropic Potential Vorticity (IPV)
History
The concept of isentropic potential vorticity was first introduced in the
literature by Hans Ertel, a German meteorologist, in 1942. For this
reason, potential vorticity is often called “Ertel Potential Vorticity”.
The use of IPV in diagnosing mid-latitude weather systems began in
earnest following the publication of “On the use and significance of
isentropic potential vorticity maps” by Hoskins et al. in QJRMS in 1985
The use piecewise diagnosis of IPV in mid-latitude weather systems
was introduced with the publication of “Potential vorticity diagnostics of
cyclogenesis” by Davis and Emanuel in MWR in 1991
Assume we have a flow that is adiabatic and we will consider the flow in
isentropic coordinates:
The vorticity equation in isentropic coordinates is given by;

d    f 
    f    V
dt


Consider a column of air between two isentropic surfaces:
Since mass in the column is conserved, and the isentropic surfaces bound
the column, the isentropes must rise and/or fall to accommodate the mass
The continuity equation is isentropic coordinates is given by:


d  1 p   1 p 
 
   
   V
dt  g    g  
Let’s let   
Then:
1 p
g 


(inverse of static stability)


d ln 
   V 
dt

d    f 
    f    V
But from vorticity equation:
dt

Therefore:
d ln    f  d ln 

dt
dt

d ln    f  d ln 

dt
dt
d ln   f   d ln 
d    f  d

   f  
Integrate from an
Initial to final value:
d    f 
d



   f 0    f 
0 
  f

ln
   f 
   f 0
 ln
   f 
   f 0

0


0
   f 
   f 0


0
   f      f 0

0
This equation implies that the quantity:
   f 

1 p
g 
is a constant.
 g    f 

p
is the isentropic potential vorticity, which is conserved in adiabatic, inviscid flow
 g    f 

C
p
Equation says that there is a “potential” to create relative vorticity by
changing latitude or by changing the thickness of isentropic layers
Two important characteristics of IPV make it so useful in synoptic meteorology:
1. IPV is conservative in adiabatic frictionless flow
-If flow is adiabatic, any change in IPV must be due to the
advection of IPV
-If flow is not adiabatic, IPV can be used to diagnose where
and when diabatic processes are acting to influence the flow
2. IPV is invertible
-the distribution of u, v, ø, T,  and other variables can be
derived from the IPV distribution provided that
a) the domain boundary conditions are known
b) a balance condition (e.g. geostrophic, gradient)
is assumed to exist in the domain.
A simple example of why boundary conditions are required for inversion of IPV
Consider a simpler invertibility problem
Barotropic vorticity equation
d  

 V    0
dt
t

V  kˆ  
  2
Geostrophically balanced atmosphere
All solutions for  have no shear or curvature and thus satisfy the equation and
the balance condition. There is no unique solution without knowing the
boundary conditions.
PV anomalies
Anomalies in the average (long time and space scale) PV distribution are of
interest in synoptic meteorology because they have associated with them
identifiable and discrete circulations.
Consider a positive PV anomaly extending into the upper troposphere
    f 
Either:

p
exceeds local average
1) the relative vorticity is larger than average
2) the static stability is larger than average
3) both are larger than average
It must be both
(anomalies must be characterized by vorticity and static stability anomalies of the same sign)
Assume it is only vorticity: then max velocity of air must be at level of anomaly
Therefore: geostrophic shear must exist (panel a)
Therefore: it must be cold under anomaly and warm outside, and warm above anomaly and cold outside
Therefore; Isentropes must slope across warm-cold boundary, and must be packed in anomaly
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”
Traditional view using QG forcing
Alternate view using PV forcing
Are these views mathematically equivalent?
Start with QG height
tendency equation
Combine tendency
And advection terms
ff 0
Add
 0 to LHS
t
Divide both sides by f0

1 2
 2 f 02  2  
   
  



f
V


0 g
2 


p

t


 f0
 f 02  
  2  
Vg   2  
f  
 p  







 2
f 02  2 
f 02  2 
 2
   
  Vg      ff 0 

2 
2 
t 
 p 
 p 


 2
f 02  2 
f 02  2 
 2
    ff 0 
  Vg      ff 0 

2 
2 
t 
 p 
 p 


1 2
f 0  2 
f 0  2 
1 2
    f 
  Vg      f 

2 
2 
t  f 0
 p 
 p 
 f0
In pressure coordinates, with QG assumptions, potential vorticity is given by
1 2
f 0  2 

PVg      f 
2 
 p 
 f0

1 2
f 0  2 
f 0  2 
1 2
    f 
  Vg      f 

2 
2 
t  f 0
 p 
 p 
 f0


PV g   Vg  PV g 
t
d
PVg   0
dtg
This statement of conservation of QG potential vorticity is
identical to the physics in the QG Height tendency equation!
This means that the QG and PV viewpoints are alternate ways of examining
The same physical processes
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 upper and lower PV anomalies
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 upper and lower PV anomalies
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?
The process of cyclogenesis occurs as a feedback between the upper and lower
level anomalies
1.
2.
3.
4.
Upper positive anomaly is associated with a tropospheric circulation below it
Circulations advects thermal field inducing a low level anomaly to its east
Low level anomaly is associated with a tropospheric circulation above it
This circulation advects positive PV northward east of upper PV anomaly
and negative PV west of the upper PV anomaly
- the effect is to reduce the tendency for the upper PV to propagate westward and
strengthens the upper level anomaly
- upper level anomaly has increased influence on lower level thermal advection,
increasing strength of lower level anomaly and causing it to reduce tenedency to
propagate eastward
The process of cyclogenesis occurs as a feedback between the upper and lower
level anomalies
In a more traditional view:
When a trough migrates over a baroclinic zone, the circulation associated with the
trough leads to advection of warm air northward and cold air southward east and
west of the trough axis respectively.
These advective processes deepen the trough and upstream ridge.
Diabatic processes and the PV perspective
Since diabatic processes are associated with the creation or destruction of PV,
We will need to develop an expression for the Lagrangian rate of change of PV

PV   g    f 
p
The mathematics are easier if PV is expressed in pressure coordinates. However,
The coordinate transformation is quite involved (see p. 290-292 of M-L AD)
I will go to the answer, and you should go through the equations as laid out in M-L AD
d PV 

  g   f 
dt
p
where
 
d
dt
PV is increased when the vertical gradient of diabatic heating is positive.
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
To understand this in a common example
Think of a hurricane!
In an extratropical cyclone
diabatic heating
builds ridge aloft
and strengthens
cyclone at surface
Piecewise Inversion of PV
A primary application of PV in synoptic meteorology is called “Piecewise Inversion”
The idea is to divide the existing perturbation PV
PV   PV   PV
into logical partitions, such as the upper and lower PV anomalies, or the
upper, lower, and middle PV anomalies when diabatic heating occurs
One then inverts the partition (non-trivial), and determines what part of
the flow is associated with that anomaly.
Example: Isolate the role of diabatic heating on the development of a low
pressure center
Example: Perturbation geopotential height at 950 mb in an intense Pacific cyclone
Black: Negative perturbation
Gray: Positive perturbation
Perturbation heights
Perturbation associated with diabatic PV
Perturbation associated with upper PV
Perturbation associated with near surface PV
Diabatic heating along fronts and enhancement of shear
Diabatic heating….
…leads to increase in PV
and creation of cyclonic shear
along front
1 PVU = 10-6 m2 K kg-1 s-1
PV and occlusions
PV notch (trowal axis)
Cold airmass beneath PV max
PV and Lee-Cyclogenesis
Cyclogenesis frequency in January
Note maximum to lee of
Canadian and US Rockies
Conservation of PV requires that positive
vorticity increase as column is stretched
Superposition principle and PV anomalies
PV anomalies can be superimposed (or added together) as one anomaly
A PV feature in a deformation field
A PV feature in a deformation field