Post on 29-Jan-2016
AOSS 401, Fall 2007Lecture 23
November 05, 2007
Richard B. Rood (Room 2525, SRB)rbrood@umich.edu
734-647-3530Derek Posselt (Room 2517D, SRB)
dposselt@umich.edu734-936-0502
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/day0-7loop.html
• Weather Underground– http://www.wunderground.com/cgi-bin/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?Init=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
)(0 fp
ft ggg
V
Advection of vorticity
Let’s take this to the atmosphere
Advection of planetary vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
vg > 0 ; β > 0 vg < 0 ; β > 0
Advection of planetary vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
-vg β < 0 -vg β > 0
Advection of relative vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Advection of ζ> 0
Advection of ζ< 0
Advection of vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Advection of ζ> 0
Advection of f< 0
Advection of ζ< 0
Advection of f> 0
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
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Advection of ζ tries to propagate the wave this way
Advection of f tries to propagate the wave this way
Geopotential Nuanced
Assume that the geopotential is a wave
yx Lland
Lk
ay
lykxpAfypUfpyx
2
2
)(
cossin)()()(),(
0
000
Remember the relation to geopotential
)cossin)()()((1
)cossin)()()((11
;
windcgeostrophi of Definition
0000
00000
00
lykxpAfypUfpyf
u
lykxpAfypUfpxfxf
v
yuf
xvf
g
g
gg
Remember the relation to geopotential
'
0000
'
0
00000
sinsin)()(
)cossin)()()((1
coscos)(1
)cossin)()()((11
gg
g
gg
g
uUlykxplApUu
lykxpAfypUfpyf
u
vlykxpkAxf
v
lykxpAfypUfpxfxf
v
Advection of relative vorticity
lykxpAlkUkx
U
yv
xuU
g
gg
gggg
coscos)()(
)(
22
''
V
Advection of planetary vorticity
lykxpkAvg coscos)(
Compare advection of planetary and relative vorticity
))2
()2
((
coscos)()(
coscos)(
22
22
yx
gg
g
gg
g
LLU
v
lykxpAlkUk
lykxpkAv
V
V
Advection of vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Advection of ζ tries to propagate the wave this way
Advection of f tries to propagate the wave this way
Compare advection of planetary and relative vorticity
))2
()2
(( 22
yx
gg
g
LLU
v
V
Short waves, advection of relative vorticity is larger
Long waves, advection of planetary vorticity is larger
Advection of vorticity
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Short waves
Long waves
Go to the real atmosphere
An estimate of the January mean zonal wind
northwinter
southsummer
--u
Advection of relative vorticity for our idealized wave
lykxpAlkUkx
U
yv
xuU
g
gg
gggg
coscos)()(
)(
22
''
V
An estimate of the January mean zonal wind
northwinter
southsummer
What is the difference in
the advection of vorticity at
the two levels?
An estimate of the January mean zonal wind
lykxpAlkUkx
U g coscos)()( 22
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
gg
gaag
ggg
gaagg
vfp
fDt
D
vfy
v
x
uf
Dt
D
fy
u
x
v
vy
v
x
uf
Dt
D
02
02
02
02
2
0
0
)(
1
)(
Another interesting way to rewrite vorticity equation
)1
(1
)1
(1
2
00
2
0
2
00
2
0
ffp
ftf
ffp
fft
g
g
V
V
(Flirting with) An equation for geopotential tendencyAn equation in geopotential and omega. (2 unknowns, 1 equation)
Quasi-geostrophic
)1
(1
)1
(1
2
00
2
0
2
00
2
0
ffp
ftf
ffp
fft
g
g
V
V
Geostrophic
ageostrophic
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
pg
aa
g
gagg
c
R
p
J
pt
py
v
x
u
f
yfDt
D
;
0
1
0
0
V
kV
VkVkV
Now let’s use this equation
pg
aa
g
gagg
c
R
p
J
pt
py
v
x
u
f
yfDt
D
;
0
1
0
0
V
kV
VkVkV
Rewrite the thermodynamic equation to get geopotential
tendency
p
J
ptp
p
J
ppt
p
J
pt
g
g
g
V
V
V
Rewrite this equation to relate to our first equation for
geopotential tendency.
p
J
pf
pf
p
f
ptp
f
p
p
Jff
p
f
tp
f
p
J
ptp
g
g
g
0000
00
00
)()( V
V
V
Scaled equations of motion in pressure coordinates
)1
(1
)()()(
2
00
2
0
0000
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
g
g
V
V
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
)1
(1
)()()(
2
00
2
0
0000
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
g
g
V
V
Note that this looks something like the time rate of change of static stability
Explore this a bit.
)1
()1
()(
)()()(
000
0000
t
T
Spf
p
T
tpRf
tp
f
p
p
RT
p
p
J
pf
pf
p
f
ptp
f
p
p
g
V
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)
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
Vorticity Advection
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
Thickness Advection
How do you interpret this figure in terms of geopotential?
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
ζ < 0; anticyclonic
ζ > 0; cyclonicζ > 0; cyclonic
Short waves
Long waves
2
0
1
fg
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((2
02
00
202
p
f
pf
ff
tp
f
p gg
VV
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)
))(()1
())((2
02
00
202
p
f
pf
ff
tp
f
p gg
VV
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
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
Thickness Advection
Weather
• National Weather Service– http://www.nws.noaa.gov/– Model forecasts:
http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html
• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/getForecast?
query=ann+arbor
– Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US
Cold and warm advection
cold
warm
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 WARM
Question
• What happens the warm air?– Tell me at least two things.
COOL
WARM
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
Thickness Advection
Lifting and sinking
Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency
(assume J=0)
))(()1
())((
)1
(1
)()()(
202
00
202
2
00
2
0
0000
p
f
pf
ff
tp
f
p
ffp
ftf
p
J
pf
pf
p
f
ptp
f
p
gg
g
g
VV
V
V
Thickness Advection
A nice schematic
• http://atschool.eduweb.co.uk/kingworc/departments/geography/nottingham/atmosphere/pages/depressionsalevel.html
More in the atmosphere(northern hemisphere)
SouthNorth
WarmCool
Temperature
What can you say about the wind?
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
CloudSat Radar
Norwegian Cyclone Model
What’s at work here?
Mid-latitude cyclone development
Mid-latitude cyclones: Norwegian Cyclone Model
• http://www.srh.weather.gov/jetstream/synoptic/cyclone.htm
Below
• Basic Background Material
Tangential coordinate system
Ω
R
Earth
Place a coordinate system on the surface.
x = east – west (longitude)y = north – south (latitude)
z = local vertical orp = local vertical
Φ
a
R=acos()
Tangential coordinate system
Ω
R
Earth
Relation between latitude, longitude and x and y
dx = acos() dis longitudedy = ad is latitude
dz = drr is distance from center of a “spherical earth”
Φ
a
f=2Ωsin()
=2Ωcos()/a
Equations of motion in pressure coordinates(using Holton’s notation)
written)explicitlynot (often
pressureconstant at sderivative horizontal and time
; )()
re temperatupotential ; velocity horizontal
ln ;
0)(
Dt
Dp
ptDt
D( )
vu
pTS
p
RT
p
c
JST
t
TS
y
Tv
x
Tu
t
T
ppy
v
x
u
fDt
D
pp
p
ppp
p
V
jiV
V
V
VkV
Scale factors for “large-scale” mid-latitude
s 10 /
m 10
m 10
! s cm 1
s m 10
5
4
6
1-
-1
UL
H
L
unitsW
U
1-1-11-
14-0
2
3-
sm10
10
10/
m kg 1
hPa 10
y
f
sf
P
Scaled equations of motion in pressure coordinates
pg
aa
gagg
g
c
R
p
J
pt
py
v
x
u
yfDt
D
f
;
0
1
0
0
V
VkVkV
kV Definition of geostrophic wind
Momentum equation
Continuity equation
ThermodynamicEnergy equation