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 formulationTrenberth formulationQ-Vector formulation

standard coordinatesnatural coordinates

3. The Semi-Geostrophic Equations

Sawyer-Eliassen Equation

4. Instability mechanisms

Conditional and Potential InstabilityInertial InstabilityConditional 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)

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

History

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

Vfdt

fd

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;

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:

V

p

g

p

gdt

d

11

Let’s let

p

g

1(inverse of static stability)

dt

dV

ln

Then:

But from vorticity equation:

Vf

dt

fd

dt

d

dt

fd lnln

Therefore:

lnln dfd

dt

d

dt

fd lnln

d

f

fd

00

d

f

fdf

f

Integrate from anInitial to final value:

00

lnln

f

f

00

f

f

00

f

f

0

0

ff

This equation implies that the quantity:

p

gf

1

is a constant.

p

fg

is the isentropic potential vorticity, which is conserved in adiabatic, inviscid flow

Cp

fg

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 knownb) 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

0

Vtdt

d

kV ˆ

Barotropic vorticity equation

Geostrophically balanced atmosphere

2

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.

Consider a simpler invertibility problem

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

p

f

exceeds local average

Either: 1) the relative vorticity is larger than average2) the static stability is larger than average3) 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 outsideTherefore; Isentropes must slope across warm-cold boundary, and must be packed in anomaly

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”

N

fLH

Penetration depth f =Coriolis parameter L=characteristic length scale N= Brunt-Vaisala frequency

Traditional view using QG forcing Alternate view using PV forcing

Are these views mathematically equivalent?

2

2202

002

2202 1

pV

ff

fVf

tp

fgg

Start with QG heighttendency equation

2

220

02

2

2202

p

fffV

p

f

t g

Combine tendency

And advection terms

2

220

02

2

220

02

p

fffV

p

fff

t g

00 t

ffAdd to LHS

Divide both sides by f0

2

202

02

202

0

11

p

ff

fV

p

ff

ft g

In pressure coordinates, with QG assumptions, potential vorticity is given by

2

202

0

1

p

ff

fPVg

2

202

02

202

0

11

p

ff

fV

p

ff

ft g

ggg PVVPVt

0gg

PVdt

d

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 examiningThe same physical processes

Low level PV anomalies

Note fake isentropes below ground(positive static stability!)

Positive vorticity

Warm anomaly at surface associated with low

pressure system

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, sincecold 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

Cyclogenesis from a PV perspective

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 lowerlevel anomalies

1. Upper positive anomaly is associated with a tropospheric circulation below it2. Circulations advects thermal field inducing a low level anomaly to its east3. Low level anomaly is associated with a tropospheric circulation above it4. 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 lowerlevel 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

p

fgPV

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

p

fgdt

PVd

where dt

d

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 exampleThink of a hurricane!

In an extratropical cyclone

diabatic heatingbuilds ridge aloft

and strengthenscyclone 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

PVVPVP

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 perturbationGray: Positive perturbation

Perturbation heights Perturbation associated with upper PV

Perturbation associated with diabatic PV Perturbation associated with near surface PV

Diabatic heating along fronts and enhancement of shear

Diabatic heating…. …leads to increase in PVand creation of cyclonic shear

along front

PV and occlusions1 PVU = 10-6 m2 K kg-1 s-1

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