AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) [email protected]...
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Transcript of AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) [email protected]...
![Page 1: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/1.jpg)
AOSS 401, Fall 2006Lecture 17
October 22, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
![Page 2: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/2.jpg)
Class News
• Final exam will be last day of class
![Page 3: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/3.jpg)
Material from Chapter 4
• Vorticity, Vorticity, Vorticity
– Definition of barotropic and baroclinic– Review– Scaling
![Page 4: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/4.jpg)
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
![Page 5: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/5.jpg)
Vorticity, Vorticity, Vorticity
![Page 6: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/6.jpg)
Two important definitions
• barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature.
• baroclinic – density depends on pressure and temperature.
![Page 7: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/7.jpg)
Attributes of important dynamical features
• There is rotation of wind• Which is due to the rotation of the Earth
• Vertical wind requires divergence of the horizontal wind
• Which requires an ageostrophic part of the wind.
![Page 8: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/8.jpg)
Want to formalize the representation of the role of rotation
and divergence
![Page 9: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/9.jpg)
Suppose we have some flow
![Page 10: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/10.jpg)
Imagine at the point flow decomposed into two “components”
A “component” that flows into or away from the point.
![Page 11: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/11.jpg)
Imagine at the point flow decomposed into two “components”
A “component” that flows around the point.
![Page 12: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/12.jpg)
Lets consider a spinning skaterMotion is in the (x,y) plane
Axis of rotation is in the vertical plane
![Page 13: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/13.jpg)
Vorticity
vorticityrelative Earth theof surface the torelative
motion the todueon contributi a is There
Earth. theof
rotation the todueon contributi a is There
hence, system; coordinate inertialin velocity is
fluid in therotation of measure a isvorticity
,definitionby is, vorticity
a
a
U
U
![Page 14: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/14.jpg)
Vertical Component of Vorticity
aUk
vorticityofcomponent
vertical with theconcernedonly generally
In what plane is the motion?In what direction is the vorticity?
![Page 15: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/15.jpg)
Relative Vorticity
y
u
x
v
y
u
x
v
y
u
x
v
x
w
z
u
z
v
y
w
(u,v,w)
vorticityrelative
,,
velocity relative
uk
u
u
![Page 16: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/16.jpg)
Planetary Vorticity
f
fearth
)sin(2vorticityplanetary
)sin(2
uk
![Page 17: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/17.jpg)
Absolute (or total) Vorticity
fy
u
x
veartha
vorticityabsolute
vorticityrelativevorticityplanetary vorticityabsolute
ukukUk
![Page 18: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/18.jpg)
Vorticity and Divergence
• Related to shear of the velocity field. ∂v/∂x-∂u/∂y
• Related to stretching of the velocity field. ∂u/∂x+∂v/∂y
![Page 19: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/19.jpg)
Full equations of motionRemember these?
1 and
)1
(
)()cos(21v
)v()sin(21v)tan(v
)()cos(2)sin(v21)vtan(
222
22
2
RTp
JDt
Dp
Dt
DTc
Dt
D
wΩugz
p
a
u
Dt
Dw
Ωuy
p
a
w
a
u
Dt
D
uΩwΩx
p
a
uw
a
u
Dt
Du
v
u
How would you calculate of the time rate change of the vertical component of vorticity?
![Page 20: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/20.jpg)
The scaled horizontal momentum equation in z coordinates
fuy
p
dt
dv
fvx
p
dt
du
1
1
no viscosity
![Page 21: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/21.jpg)
Relative Vorticity
y
u
x
v
(u,v,w)
vorticityrelative
velocity relative u
![Page 22: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/22.jpg)
Take derivatives
fuy
p
dt
dv
x
fvx
p
dt
du
y
1
1
![Page 23: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/23.jpg)
Take derivatives
)(1
)(1
fuxy
p
xdt
dv
x
fvyx
p
ydt
du
y
![Page 24: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/24.jpg)
Take derivatives
)(1
)(
)(1
)(
fuxy
p
xv
t
v
x
fvyx
p
yu
t
u
y
u
u
![Page 25: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/25.jpg)
Take derivatives
)(1
)(
)(1
)(
fuxy
p
xv
xt
v
x
fvyx
p
yu
yt
u
y
u
u
Pay attention to details of calculus here
![Page 26: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/26.jpg)
Subtract these equationsConservation of vorticity
x
p
yy
p
xy
fv
z
u
y
w
z
v
x
w
y
v
x
uf
t
11)(
))((u
Where do the definitions barotropic and baroclinic make a difference?
What are physical interpretations of these terms?
![Page 27: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/27.jpg)
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
![Page 28: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/28.jpg)
Scale of relative vorticity
1510
sL
U
y
u
x
v
![Page 29: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/29.jpg)
Compare relative vorticity to planetary vorticity
NumberRossby Ro
10
10
10
0
1
0
140
15
Lf
U
f
sf
sL
U
In general planetary vorticity is larger than relative vorticity.
![Page 30: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/30.jpg)
Time rate of change of relative vorticity
211
2102
2
10
10,,
sHL
WU
zw
sL
U
yv
xu
t
![Page 31: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/31.jpg)
Rest of the terms
21122
210
211
290
1011
: termodynamicthat therm
10 :advectionvorticity planetary
10)( : termtilting
10))(( : termdivergence
sL
p
x
p
yy
p
x
sUy
fv
sHL
UW
z
u
y
w
z
v
x
w
sL
Uf
y
v
x
uf
![Page 32: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/32.jpg)
Consider divergence term
290 10))(( : termdivergence
sL
Uf
y
v
x
uf
This term scales larger than all of the other terms. This suggests that the divergence of the horizontal wind must, in actuality, be small; hence, quasi-nondivergent. (∂u/∂x+∂v/∂y)~<10-6s-1
![Page 33: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/33.jpg)
Consider divergence term
)())((y
v
x
uf
y
v
x
uf
We saw that the relative vorticity is less than the planetary vorticity. So
![Page 34: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/34.jpg)
So for quasi-nondivergent flow
21010)(
sy
v
x
uf
![Page 35: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/35.jpg)
Compare relative vorticity to planetary vorticity and
to divergence
100
10
10
0
0
yv
xuf
yv
xu
f
Again we see the importance of the rotation of the Earth.
![Page 36: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/36.jpg)
Assume balance among terms of 10-10s-2
)(
0)(
y
v
x
uf
y
fv
yv
xu
t
y
fv
y
v
x
uf
yv
xu
t
![Page 37: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/37.jpg)
A nuance on vorticity and the scaled equation: potential vorticity
![Page 38: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/38.jpg)
A simple version of potential vorticity
))(()(
y
v
x
uf
Dt
fD
y
fv
Dt
Df
horizontal
returned relative vorticity to equation
![Page 39: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/39.jpg)
A simple version of potential vorticity
)()(
)(
1
y
v
x
u
Dt
fD
fhorizontal
what’s this?
![Page 40: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/40.jpg)
A simple version of potential vorticity
z
w
Dt
fD
fhorizontal
)(
)(
1
Assume constant density and temperature.
![Page 41: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/41.jpg)
A simple version of potential vorticity
H
)()()(
)(
1
)(
)(
1
122
1
2
1
2
1
zwzwdz
Dt
fD
f
dzz
wdz
Dt
fD
f
z
z
horizontal
z
z
z
z
horizontal
Integrate with height,z1 z2 over a layer of depth H.
![Page 42: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/42.jpg)
A simple version of potential vorticity
0)(
)()( 12
H
f
Dt
D
zwzwDt
DH
horizontal
Integrate with height,z1 z2 over a layer of depth H.
Why can we do this?
![Page 43: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/43.jpg)
A simple version of potential vorticity
vorticitypotentialH
f
This is the potential vorticity under the set of assumptions that we used to derive the equation. Constant density, constant temperature so only in a shallow layer might this be relevant to the atmosphere.
Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.
![Page 44: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/44.jpg)
Relative vorticity with change of depth
![Page 45: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/45.jpg)
What happens when the vortex meets the mountain?
Surface with a hill.
![Page 46: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/46.jpg)
A less simple version of potential vorticity
vorticitypotential))((
p
gf
This is the isentropic potential vorticity which is conserved for isentropic motion.
Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.
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Vorticity and depth
• We can see that there is a relationship between depth and vorticity.
• As the depth of the vortex changes, the relative vorticity has to change in order to conserve the potential vorticity.
• This is the play between relative and planetary vorticity.
![Page 48: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/48.jpg)
How are divergence and vorticity related?
• We have gotten to a situation where we have linked the rotational and irrotational components of the wind. divergence and curl
vorticity and divergence
![Page 49: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/49.jpg)
Scaled vorticity equation
))(()(
y
v
x
uf
Dt
fDhorizontal
![Page 50: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/50.jpg)
Pure constant vorticity flow.
![Page 51: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/51.jpg)
Pure divergent flow
![Page 52: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/52.jpg)
Divergence influence on vorticity
![Page 53: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/53.jpg)
Divergence influence on vorticity
![Page 54: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/54.jpg)
Vorticity and divergence
• As a mid-latitude large scale flow diverges (converges) it causes a change in absolute vorticity, primarily, acting on planetary vorticity.
• The relative vorticity has to change in order to conserve the potential vorticity.
• This is the play between relative and planetary vorticity.
![Page 55: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/55.jpg)
Possible development of a surface low.
Earth’s surface
DIVERGENCE
warming
LOW
Changes in depth of vortex
H H CONVERGENCE
![Page 56: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/56.jpg)
And a little more realistically
Concentrates relative vorticity
Dilutes relative vorticity
![Page 57: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/57.jpg)
What if the wind is geostrophic?
![Page 58: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/58.jpg)
Scaled vorticity equation
))(()(
y
v
x
uf
Dt
fDhorizontal
What happens if the wind is geostrophic?
![Page 59: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/59.jpg)
Or returning to potential vorticity
0)( H
f
Dt
D ghorizontal
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An observation
• The vorticity is dominated by the geostrophic component of the wind.
• The divergence requires the wind to be away from geostrophic balance.
• Generally vg/va >= 10
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Let’s consider the geostrophic potential vorticity
0)()(
such that , function, stream a define
22
ft
xv
yu
g
g
k
Do you recognize this type of equation?
![Page 62: AOSS 401, Fall 2006 Lecture 17 October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu.](https://reader034.fdocuments.in/reader034/viewer/2022051401/5697bfbe1a28abf838ca2d13/html5/thumbnails/62.jpg)
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