AOSS 401, Fall 2007
Lecture 23
November 05, 2007
Richard B. Rood (Room 2525, SRB)
rbrood@umich.edu
734-647-3530
Derek Posselt (Room 2517D, SRB)
dposselt@umich.edu
734-936-0502
Class News
November 05, 2007
• Homework 6 (Posted this evening)
– Due Next Monday
• Important Dates:
– November 16: Next Exam (Review on 14th)
– November 21: No Class
– December 10: Final Exam
Weather
• National Weather Service
– http://www.nws.noaa.gov/
– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day07loop.html
• Weather Underground
– http://www.wunderground.com/cgibin/findweather/getForecast?query=ann+arbor
– Model forecasts:
http://www.wunderground.com/modelmaps/maps.asp
?model=NAM&domain=US
Couple of Links you should know
about
• http://www.lib.umich.edu/ejournals/
– Library electronic journals
• http://portal.isiknowledge.com/portal.cgi?In
it=Yes&SID=4Ajed7dbJbeGB3KcpBh
– Web o’ Science
Material from Chapter 6
• Quasi-geostrophic theory
• Quasi-geostrophic vorticity
– Relation between vorticity and geopotential
• Geopotential prognostic equation
• Relationship to mid-latitude cyclones
One interesting way to rewrite this equation
 g

 f0
 Vg  ( g  f )
t
p
Advection of vorticity
Let’s take this to the atmosphere
Advection of planetary vorticity
ζ < 0; anticyclonic
٠
ΔΦ > 0
vg > 0 ; β > 0
Φ0 - ΔΦ
vg < 0 ; β > 0
B
L
Φ0
H
٠
Φ0 + ΔΦ
L
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Advection of planetary vorticity
ζ < 0; anticyclonic
٠
ΔΦ > 0
-vg
-vg
β<0
B
Φ0 - ΔΦ
L
Φ0
L
H
٠
Φ0 + ΔΦ
β>0
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Advection of relative vorticity
ζ < 0; anticyclonic
ΔΦ > 0
Advection of ζ
>0
٠
Advection of ζ
<0
B
Φ0 - ΔΦ
L
Φ0
H
٠
Φ0 + ΔΦ
L
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Advection of vorticity
ΔΦ > 0
Φ0 - ΔΦ
Advection of ζ
>0
Advection of f
<0
ζ < 0; anticyclonic
٠
B
L
Φ0
L
H
٠
Φ0 + ΔΦ
Advection of ζ
<0
Advection of f
>0
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Summary: Vorticity Advection in Wave
• Planetary and relative vorticity advection in
a wave oppose each other.
• This is consistent with the balance that we
intuitively derived from the conservation of
absolute vorticity over the mountain.
Advection of vorticity
ζ < 0; anticyclonic
 Advection of ζ tries to propagate the wave this way 
ΔΦ > 0
٠
Φ0 - ΔΦ
B
L
Φ0
L
H
 Advection of f tries to propagate the wave this way 
٠
Φ0 + ΔΦ
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Geopotential Nuanced
Assume that the geopotential is a wave
( x, y )   0 ( p)  f 0U ( p) y  f 0 A( p) sin kx cos ly
y  a(  0 )
2
2
k
and l 
Lx
Ly
Remember the relation to geopotential
Definition of geostrophi c wind


f 0vg 
; f 0u g  
x
y
vg 
1  1 

( 0 ( p)  f 0U ( p) y  f 0 A( p) sin kx cos ly )
f 0 x
f 0 x
1 
ug  
( 0 ( p )  f 0U ( p) y  f 0 A( p) sin kx cos ly )
f 0 y
Remember the relation to geopotential
vg 
1  1 

( 0 ( p)  f 0U ( p ) y  f 0 A( p ) sin kx cos ly )
f 0 x
f 0 x
vg 
1 
 kA( p ) cos kx cos ly  v g'
f 0 x
ug  
1 
( 0 ( p )  f 0U ( p ) y  f 0 A( p ) sin kx cos ly )
f 0 y
u g  U ( p )  lA( p ) sin kx sin ly  U  u g'
Advection of relative vorticity
 Vg   g  (U  u )
'
g
 U
 g
x
 g
x
v
'
g
 g
y
 kU (k  l ) A( p) cos kx cos ly
2
2
Advection of planetary vorticity
 v g   kA( p) cos kx cos ly
Compare advection of planetary and relative vorticity
 vg   kA( p) cos kx cos ly
 Vg   g  kU (k 2  l 2 ) A( p) cos kx cos ly
vg
Vg   g


2 2 2 2
U (( )  ( ) )
Lx
Ly
Advection of vorticity
ζ < 0; anticyclonic
 Advection of ζ tries to propagate the wave this way 
ΔΦ > 0
٠
Φ0 - ΔΦ
B
L
Φ0
L
H
 Advection of f tries to propagate the wave this way 
٠
Φ0 + ΔΦ
٠
y, north
A
x, east
ζ > 0; cyclonic
C
ζ > 0; cyclonic
Compare advection of planetary and relative vorticity
vg
Vg   g


2 2 2 2
U (( )  ( ) )
Lx
Ly
 Short waves, advection of relative vorticity is larger 
 Long waves, advection of planetary vorticity is larger 
Advection of vorticity
ζ < 0; anticyclonic
 Short waves 
ΔΦ > 0
٠
Φ0 - ΔΦ
B
L
Φ0
H
٠
Φ0 + ΔΦ
L
Long waves 
y, north
A
x, east
ζ > 0; cyclonic
٠
C
ζ > 0; cyclonic
Go to the real atmosphere
An estimate of the January mean zonal wind
-u
south
summer
north
winter
Advection of relative vorticity for our idealized wave
 Vg   g  (U  u )
'
g
 U
 g
x
 g
x
v
'
g
 g
y
 kU (k  l ) A( p) cos kx cos ly
2
2
An estimate of the January mean zonal wind
What is the
difference in
the
advection of
vorticity at
the two
levels?
south
summer
north
winter
An estimate of the January mean zonal wind
 U
 g
x
 kU (k 2  l 2 ) A( p) cos kx cos ly
Vertical Structure
• The waves propagate at different speeds
at different altitudes.
• The waves do not align perfectly in the
vertical.
• (This example shows that there is vertical
structure, but it is only a (small) part of the
story.)
A more general equation for geopotential
An equation for geopotential tendency
Dg  g
ua va
  f0 (

)  vg
Dt
x
y
v g u g
1 2
g 

 
x
y
f0
ua va
    f0 (

)  f 0 vg
Dt
x
y
Dg 2
2 
   f0
 f 0 vg
Dt
p
Dg
2
2
Another interesting way to rewrite vorticity equation
 1 2

1 2
   f0
 Vg  (    f )
t f 0
p
f0
1 2 

1 2

 f0
 Vg  (    f )
f0
t
p
f0
(Flirting with) An equation for geopotential tendency
An equation in geopotential and omega. (2 unknowns, 1 equation)
Quasi-geostrophic
 1 2

1 2
   f0
 Vg  (    f )
t f 0
p
f0
1 2 

1 2

 f0
 Vg  (    f )
f0
t
p
f0
ageostrophic
Geostrophic
Previous analysis
• In our discussion of the advection of
vorticity, we completely ignored the term
that had the vertical velocity.
• Go back to our original vorticity equation
– Tilting
– Divergence
– Thermodynamic ... (solenoidal, baroclinic)
• Which still exist after our scaling and assumptions?
We used these equations to get previous equation for
geopotential tendency
Dg Vg
Dt
  f 0k  Va  yk  Vg
1
Vg  k  
f0
ua va 


0
x y p
J
R

 
   
;
  Vg   
p
cp
 t
 p
Now let’s use this equation
Dg Vg
Dt
  f 0k  Va  yk  Vg
1
Vg  k  
f0
ua va 


0
x y p
J
R

 
   
;
  Vg   
p
cp
 t
 p
Rewrite the thermodynamic equation to get geopotential
tendency
J

 
   
  Vg   
p
 t
 p
 

J
 Vg  
  
t p
p
p
 

J
 Vg  
  
p t
p
p
Rewrite this equation to relate to our first equation for
geopotential tendency.
 

J
  Vg  
  
p t
p
p
f 0  
f0
f 0 J

  Vg  
 f 0 
 p t

p
 p
 f 0  
 f0


 J
(
)   ( Vg  
)  f0
 f0
p  p t
p 
p
p
p p
Scaled equations of motion in pressure coordinates
 f 0  
 f0


 J
(
)   ( Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1 2

 f0
 Vg  (    f )
f0
t
p
f0
Note this is, through continuity, related to the divergence of
the ageostrophic wind
Note that it is the divergence of the horizontal wind, which is
related to the vertical wind, that links the momentum
(vorticity equation) to the thermodynamic equation
Scaled equations of motion in pressure coordinates
 f 0  
 f0


 J
(
)   ( Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1 2

 f0
 Vg  (    f )
f0
t
p
f0
Note that this looks something like the time rate of change of
static stability
Explore this a bit.
 f 0  
 f0


 J
(
)   ( Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p

RT

p
p
 f 0  
 1  T
 1 T
(
)   f0 R (
)   f0
(
)
p  p t
p  t p
p S p t
So this is a measure of how far the atmosphere moves away
from its background equilibrium state
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
Vorticity Advection
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
Thickness Advection
How do you interpret this figure in terms of geopotential?
1 2
g   
f0
ζ < 0; anticyclonic
 Short waves 
ΔΦ > 0
٠
Φ0 - ΔΦ
B
L
Φ0
H
٠
Φ0 + ΔΦ
L
Long waves 
y, north
A
x, east
ζ > 0; cyclonic
٠
C
ζ > 0; cyclonic
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p 
p
2
This is, in fact, an equation that given a geopotential
distribution at a given time, then it is a linear partial
differential equation for geopotential tendency.
Right hand side is like a forcing.
You now have a real equation for forecasting the height
(the pressure field), and we know that the pressure
gradient force is really the key, the initiator, of motion.
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p 
p
2
An equation like this was very important for weather
forecasting before we had comprehensive numerical
models. It is still important for field forecasting, and
knowing how to adapt a forecast to a particular region
given, for instance, local information.
Think about thickness advection
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
Thickness Advection
Weather
• National Weather Service
– http://www.nws.noaa.gov/
– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day07loop.html
• Weather Underground
– http://www.wunderground.com/cgibin/findweather/getForecast?query=ann+arbor
– Model forecasts:
http://www.wunderground.com/modelmaps/maps.asp
?model=NAM&domain=US
Cold and warm advection
Question
• What happens when warm air is advected
towards cool air?
COOL
WARM
Question
• What happens when warm air is advected
towards cool air?
COOL
Question
• What happens the warm air?
– Tell me at least two things.
COOL
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
Thickness Advection
Lifting and sinking
Add these equations to eliminate omega and we have a
partial differential equation for geopotential tendency
(assume J=0)
 f 0  
 f


 J
(
)   ( 0 Vg  
)  f0
 f0
( )
p  p t
p 
p
p
p p
1 2 

1

 f0
 Vg  (  2   f )
f0
t
p
f0
f 02
 f 02  
1 2


(  (
))
  f 0 Vg  (    f )  (
Vg  (
))
p  p t
f0
p

p
2
Thickness Advection
A nice schematic
• http://atschool.eduweb.co.uk/kingworc/dep
artments/geography/nottingham/atmosphe
re/pages/depressionsalevel.html
More in the atmosphere
(northern hemisphere)
What can you say about the wind?
Temperature
Cool
Warm
North
South
Idealized vertical cross section
Increasing the pressure gradient force
Relationship between
upper troposphere and surface
divergence over
low enhances
surface low
//
increases vorticity
Relationship between
upper troposphere and surface
vertical stretching
//
increases vorticity
Relationship between
upper troposphere and surface
vorticity advection
thickness advection
Relationship between
upper troposphere and surface
note tilt with height
Mid-latitude cyclones:
Norwegian Cyclone Model
Fronts and Precipitation
Norwegian Cyclone Model
CloudSat Radar
What’s at work here?
Mid-latitude cyclone development
Mid-latitude cyclones:
Norwegian Cyclone Model
• http://www.srh.weather.gov/jetstream/syno
ptic/cyclone.htm
Below
• Basic Background Material
Tangential coordinate system
R=acos()
Place a coordinate
system on the surface.
Ω
R a
Φ
Earth
x = east – west
(longitude)
y = north – south
(latitude)
z = local vertical
or
p = local vertical
Tangential coordinate system
f=2Ωsin()
Relation between
latitude, longitude and x
and y
Ω
=2Ωcos()/a
R a
Φ
Earth
dx = acos() dl
l is longitude
dy = ad
 is latitude
dz = dr
r is distance from center
of a “spherical earth”
Equations of motion in pressure coordinates
(using Holton’s notation)
DV
 fk  V  
Dt
u v


(  )p 
 V 
0
x y
p
p
T
T
T
T
J
u
v
 S p 
 V   T  S p 
t
x
y
t
cp

RT
 ln 
   
; S p  T
p
p
p
V  ui  vj  horizontal velocity ;   potential temperatu re
D( ) 

Dp
 ) p  (V  ) p  
;
Dt
t
p
Dt
time and horizontal derivative s at constant pressure
(often not explicitly written)
Scale factors for “large-scale” mid-latitude
U  10 m s
-1
P  10 hPa
W  1 cm s units!
  1 kg m
L  10 m
 /   10  2
-1
6
H  10 m
-3
4
L / U  10 s
5
f 0  10-4 s 1
f

 10-11 m -1s -1
y
Scaled equations of motion in pressure coordinates
1
Vg  k  
f0
Definition of
geostrophic wind
Dg Vg
Momentum
equation
Dt
  f 0k  Va  yk  Vg
ua va 


0
x
y p
Continuity
equation
J
R

 
   
;
  Vg   
p
cp
 t
 p
Thermodynamic
Energy equation