Upload
thais
View
41
Download
0
Tags:
Embed Size (px)
DESCRIPTION
AOSS 401, Fall 2007 Lecture 22 November 02 , 2007. Richard B. Rood (Room 2525, SRB) [email protected] 734-647-3530 Derek Posselt (Room 2517D, SRB) [email protected] 734-936-0502. Class News November 02 , 2007. Homework 5 (Due Monday) Posted to web - PowerPoint PPT Presentation
Citation preview
AOSS 401, Fall 2007Lecture 22
November 02, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News November 02, 2007
• Homework 5 (Due Monday)– Posted to web– Computing assignment posted to ctools under
the Homework section of Resources
• Next Test: November 16
Seminars Today
• Professor Cecilia Bitz is giving the Dept. of Geological Sciences’ Smith Lecture on Friday (tomorrow, Nov. 2) from 4-5pm. The lecture is held in room 1528 in C.C. Little and is followed by a reception. Cecilia is an expert in high-latitude climate, climate change and variability. The title of her lecture will be:– “Future thermohaline collapse and its impact
are unlike the past”
Seminars Today
• Dr. Guy Brasseur - Professor and Associate Director, National Center for Atmospheric Research and Director of the Earth and Sun Systems Laboratory will visit U of M on Thursday/Friday. – Impact of solar variability and anthropogenic
forcing on the whole atmosphere: Simulations with the HAMMONIA Model
• Friday, November 2, 3:30 pm -- refreshments at 3 pm North Campus AOSS Auditorium, Room #2246
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
Material from Chapter 6
• Quasi-geostrophic theory
• Quasi-geostrophic vorticity– Relation between vorticity and geopotential
Going way back
Mathematics
• Remember the idea that mathematics is a language to use to help us explore a complex system.– Verb: equal, – Qualification: not equal, greater than, less than,
approximately
• We own the equations and can do to them what we want, as long as we remember equal and not equal.
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
Approximate horizontal momentum equation
gagg yf
Dt
DVkVk
V 0
This equation states that the time rate of change of the geostrophic wind is related to1. the coriolis force due to the ageostrophic
wind and 2. the part of the coriolis force due to the
variability of the coriolis force with latitude and the geostrophic wind.
Both of these terms are smaller than the geostrophic wind itself.
Derived a vorticity equation
• Provides a “suitable” prognostic equation because need to include div(ageostrophic wind) in the prognostics.
• Remember the importance of divergence in vorticity equations.
Scaled horizontal momentum in pressure coordinates
0
0
0
0
0
gagg
gagg
gagg
yuufDt
vD
yvvfDt
uD
yfDt
D
VkVkV
Use definition of vorticity vorticity equation
gaagg
gagg
gagg
vy
v
x
uf
Dt
D
yuufDt
vD
x
yvvfDt
uD
y
)(
0)(
0)(
0
0
0
One interesting way to rewrite this equation
gaa
ggg
gaagg
vy
f
y
v
x
uf
t
y
f
vy
v
x
uf
Dt
D
)(
)(
0
0
V
Expand material derivative
One interesting way to rewrite this equation
)(0
0
fp
ft
vy
f
pf
t
ggg
gggg
V
V
Equation of continuity
Understand how this is equivalent
One interesting way to rewrite this equation
)(0 fp
ft ggg
V
Advection of vorticity
One interesting way to rewrite this equation
ggggg vf VV )(
Advection of vorticity
Advection of relative vorticity
Advection of planetary vorticity
Let’s take this to the atmosphere
Geopotential Map (Northern Hemisphere)
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
Where is geostrophic approximation valid?What other force balance is important?What is the sign of the geostrophic wind?
Geostrophic wind
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
vg > 0 vg < 0
Geopotential Map (Northern Hemisphere)
Φ0 - ΔΦ
Φ0 + ΔΦ
Φ0
ΔΦ > 0
A
B
C
٠
٠
٠
x, east
y, north
L LH
What is the sign of planetary vorticity?What is the sign of the relative vorticity?
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
β > 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 > 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
ζ from >0 to <0vg > 0; ug > 0
ζ from <0 to >0vg < 0; ug > 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.
Summary: Vorticity Advection in Wave
• What does this do to the wave.
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
Remember the relation to geopotential
2
02
2
2
2
0
00
1)(
1
;
windcgeostrophi of Definition
fyxfy
u
x
v
yuf
xvf
gg
gg
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
)(
Barotropic fluid
gg
gg
vfDt
D
vfp
fDt
D
02
02
02
barotropic
Perturbation equation
xxu
t
vvuuu
vfDt
D
g
ggggg
gg
2
02
)(
equation of formon perturbati
;
Wave like solutions
0))((
)Re(
)(
22
)(0
2
klkuk
e
xxu
t
g
tlykxi
g
Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.
Stationary wave
g
g
tlykxi
ulk
klkuk
e
22
22
)(0
0))((
0
)Re(
Wind must be positive, from the west, for a wave.
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
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
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
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 adapt a forecast to a particular region
given, for instance, local information.
• See you Monday