quasi-geostrophic theory chapter 4 - uc davis climate and...

Quasi-Geostrophic Theory Chapter 4

Paul A. Ullrich

[email protected]

Part 4: The Geopotential Tendency Equation

Vorticity & Geopotential

@⇣g@t

= f0@!

@p� ug ·r⇣g � vg�

QG Vorticity Equation

Advection Terms

Geostrophic Wind

ug = � 1

f0

@�

@yvg =

1

f0

@�

@x

⇣g =@vg

@x

� @ug

@y

=1

f0

✓@

2�

@x

2+

@

2�

@y

2

◆=

1

f0r2�

Geostrophic Vorticity

Paul Ullrich Quasi-Geostrophic Theory March 2014

Relates geostrophic vorticity and geopotential

QG Geopotential Tendency Starting from here:

Dg⇣gDt

= f0@!

@p� @f

@yvg = f0

@!

@p� �vg

⇣g =@vg

@x

� @ug

@y

=1

f0r2�

Dg

Dt

✓1

f0r2�

◆= f0

@!

@p� �vg

@

@t

✓1

f0r2�

◆+ ug ·r

✓1

f0r2�

◆= f0

@!

@p� �vg


QG Geopotential Tendency

Definition: The geopotential tendency of a flow is the Eulerian rate of change in geopotential with respect to time.

� ⌘ @�

@t

1

f0r2� = �ug ·r

✓1

f0r2�+ f

◆+ f0

@!

@p

Geostrophic Terms Ageostrophic Term



1

f0r2� = �ug ·r

✓1

f0r2�+ f

◆+ f0

@!

@p

ug =1

f0k⇥r�Geostrophic Wind

1

f0r2� = �

✓1

f0k⇥r�

◆·r

✓1

f0r2�+ f

◆+ f0

@!

@p


Only unknowns are Φ and ω


Dgug

Dt= �f0k⇥ ua � �yk⇥ ug

ug =1

f0k⇥r�

@ua

@x

+@va

@y

+@!

@p

= 0

✓@

@t+ ug ·r

◆✓�@�

@p

◆� �! =

J

p

Momentum Equation

Geostrophic Wind

Continuity Equation

Thermodynamic Equation

The previous equation for geopotential tendency used these three equations.

For barotropic systems (ω=0) this leads to a closed prognostic equation for geopotential.


Dgug

Dt= �f0k⇥ ua � �yk⇥ ug

ug =1

f0k⇥r�

@ua

@x

+@va

@y

+@!

@p

= 0

✓@

@t+ ug ·r

◆✓�@�

@p

◆� �! =

J

p

Momentum Equation

Geostrophic Wind

Continuity Equation

Thermodynamic Equation

However, for baroclinic systems (horizontal temperature / density gradients), all four equations must be used.



1

f0r2� = �

✓1

f0k⇥r�

◆·r

✓1

f0r2�+ f

◆+ f0

@!

@p

Replace this term

Starting from the QG thermodynamic equation:

� @

@t

✓@�

@p

◆� ug ·r

✓@�

@p

◆� �! =

J

p

� @

@p

✓@�

@t

◆� ug ·r

✓@�

@p

◆� �! � J

p= 0Flip derivatives

@�

@p



�@�

@p� ug ·r

✓@�

@p

◆� �! � J

p= 0

f0�

@

@p

f0�

@

@p

�@�

@p� ug ·r

✓@�

@p

◆� �! � J

p

�= 0

Assume stability parameter σ is constant with respect to p

@

@p

✓f0�

@�

@p

◆= � @

@p

f0�ug ·r

✓@�

@p

◆�� f0

@!

@p� f0

@

@p

✓J

�p

◆



1

f0r2� = �

✓1

f0k⇥r�

◆·r

✓1

f0r2�+ f

◆+ f0

@!

@p

Geopotential tendency equation from before:

From QG thermodynamic equation:

@

@p

✓f0�

@�

@p

◆= � @

@p

f0�ug ·r

✓@�

@p

◆�� f0

@!

@p� f0

@

@p

✓J

�p

◆

Different sign!

Adding these two equations will eliminate the omega term.


QG Geopotential Tendency Eq’n (Adiabatic)


r2 +

@

@p

✓f20

�

@

@p

◆�� = �f0ug ·r

✓1

f0r2�+ f

◆� @

@p

�f2

0

�ug ·r

✓�@�

@p

◆�

Local geopotential

tendency

Proportional to absolute vorticity

advection

Proportional to (differential) thickness

(temperature) advection

Only the initial distribution of Φ needs to be known!


For wave-like flows, second derivatives of χ are proportional to -χ.


For example, � = sinx @

2�

@x

2= � sinx = ��

r2 +

@

@p

✓f20

�

@

@p

◆�� ⇡ c · (��)


QG Equations

Question: What’s the point?

We want to describe the evolution of two key features of the atmosphere: •  Large-scale waves (in particular, the connection between large-

scale waves and geopotential) •  Midlatitude cyclones (that is, the development of low pressure

systems in the lower troposphere)


Let’s look at the absolute vorticity advection term


r2 +

@

@p

✓f20

�

@

@p

◆�� = �f0ug ·r

✓1

f0r2�+ f

◆

Relative vorticity Planetary vorticity

�� = �@�

@t⇡ �f0ug ·r

✓1

f0r2�+ f

◆

Absolute vorticity advection (times f0)


A

B

C

٠

٠

٠

ζ < 0; anticyclonic

ζ > 0; cyclonic ζ > 0; cyclonic

An Upper Tropospheric Wave

H

L L


�0 ��

�0

�0 +��

�� > 0

Short waves: Advection of ζ is dominant here.

Advection of η > 0.

Advection of ζ > 0��Advection of f < 0

Advection of ζ < 0��Advection of f > 0

Short Waves


Advection of η < 0.

A

B

C

٠

٠

٠




H

L L


�0 ��

�0

�0 +��

�� > 0





Short Waves



@�/@t < 0 @�/@t > 0

A

B

C

٠

٠

٠




H

L L


�0 ��

�0

�0 +��

�� > 0

Long waves: Advection of f is dominant here.




Long Waves



A

B

C

٠

٠

٠




H

L L


�0 ��

�0

�0 +��

�� > 0





Long Waves



@�/@t > 0 @�/@t < 0

A

B

C

٠

٠

٠



H

L L


�0 ��

�0

�0 +��

�� > 0


! Short waves, advection of relative vorticity is larger !

" Long waves, advection of planetary vorticity is larger "

QG Equations

Question: What’s the point?

We want to describe the evolution of two key features of the atmosphere: •  Large-scale waves (in particular, the connection between large-

scale waves and geopotential) •  Midlatitude cyclones (that is, the development of low pressure

systems in the lower troposphere)


QG Geopotential Tendency Let’s look at the differential temperature advection:

� = @�@t ⇡ @

@p

h�ug ·r

⇣�@�

@p

⌘i⇡ @

@p (�ug ·rT )

Differential

Thickness advection

Temperature advection

Question: Why is thickness advection proportional to temperature advection?


QG Geopotential Tendency Differential temperature advection:

� = @�@t ⇡ @

@p

h�ug ·r

⇣�@�

@p

⌘i⇡ @

@p (�ug ·rT )


+ (builds a ridge, increasing geopotential) in case of decreasing warm air advection with height − (deepens a trough, decreasing geopotential) in case of decreasing cold air advection with height

Differential temperature advection below 500 hPa:

� ⇡ � @@z (�ug ·rT )

•  … are broader in shape than cold fronts •  … tend to move more slowly than cold fronts •  … have precipitation spread out over a larger distance

Warm Fronts


Figure 9.6 in The Atmosphere, 8th edition, Lutgens and Tarbuck, 8th edition, 2001.

•  … are vertically steep •  … tend to travel faster than warm fronts •  … are associated with strong storms at boundary

Cold Fronts


Figure 9.6 in The Atmosphere, 8th edition, Lutgens and Tarbuck, 8th edition, 2001.

Paul Ullrich CCWAS: Lecture 05

Extratropical Cyclones

October 16, 2013

Extratropical Cylones are important for driving weather in the mid-latitudes. They are closely related to weather fronts. Particularly strong extratropical systems are responsible for large-scale storm systems.

Figure: Extratropical Cyclones are associated with severe winter storm systems, and are particularly relevant for the US Northwest and Northern Europe.

warm

Atmospheric Wave Motion


Real Baroclinic Disturbances 850 hPa Temperature and Geopotential Thickness

Cold Air Advection west of the surface low enhances the

trough

Warm Air Advection east of

the surface low enhances the ridge


The wave becomes a mature low pressure system, while the cold front, moving faster than the warm front, "catches up" with the warm front. As the cold front overtakes the warm front, an occluded front forms.

Overhead view

Norwegian Cyclone Model

http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm


We see that:

•  Geostrophic advection of geostrophic vorticity causes waves to propagate

•  The vertical difference in temperature (thickness) advection causes waves to amplify

QG Geopotential Tendency / � @

@z [�ug ·rT ]

/ �ug ·r (⇣g + f)

Thickness advection

Vorticity advection


Baroclinic Instability Upper troposphere and surface

Divergence over low enhances

surface low and increases vorticity



Vertical stretching

increases vorticity



Vorticity advection

Thickness advection



Note tilt with height. Conservation of energy

at work here.

It turns out that a disturbance with a

westward tilt leads to poleward transfer of

energy


quasi-geostrophic theory chapter 4 - uc davis climate and...

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