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AOSS 401, Fall 2007Lecture 11
October 1, 2007Richard B. Rood (Room 2525, SRB)
Derek Posselt (Room 2517D, SRB)[email protected]
734-936-0502
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Class NewsOctober 1, 2007
• Ricky will be lecturing again starting Wednesday—I will lecture next on the 17th of October
• There is an exam next Wednesday, but you’re all probably well aware of that…
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Material from Chapter 3(2)
• Balanced flow• Examples of flows in the atmosphere
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Refresher from Friday…
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Geostrophic & observed wind 300 mb
In order to understand the flow on maps that looked like this, we introduced “natural” coordinates.
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The horizontal momentum equation
p
pp
pp
fDtD
fuydt
d
fxdt
du
uku
v
v
Assume no viscosity
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Return to Geopotential (Φ) in upper troposphere
eastwestsouth
north
HIGH t t
tnn nLow
Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle.
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The horizontal momentum equation(in natural coordinates)
nfV
RV
sDtDV
nsfV
RV
DtDV
2
2
formcomponent in and
ntnnt
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nfV
RV
2
Curved flow (Centrifugal Force)
Coriolis Pressure Gradient
One Diagnostic Equation
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Natural Coordinates: Key Points• Velocity is defined to be positive• The n direction always points to the left of the
velocity (remember the right hand rule: k x t = n)• If n points toward the center of curvature, the
radius is positive• If n points away from the center of curvature, the
radius is negative• The pattern of isobars/height lines is assumed to
be fixed in space; no movement of weather systems
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Uses of Natural Coordinates
• Geostrophic balance– Definition: coriolis and pressure gradient in
exact balance.– Parallel to contours straight line R is
infinite
nfV
RV
2
0
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Geostrophic balance
nfV
xf
yfu
p
p
scoordinate p)n,(t, natural inor
v
scoordinate p)y,(x, In
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Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight.
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
northn
fVg
Δn
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How does curvature affect the wind?(cyclonic flow/low pressure)
nfV
RV
2
R
t
n
ΔnΦ0
Φ0+ΔΦ
Φ0-ΔΦ
HIGH
Low
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From Holton
• If Vg/V < 1, geostrophic wind is an overestimate of the actual wind speed
• Since V is always positive, in the northern hemisphere (f > 0) this only happens for R positive
• For typical northern hemisphere large scale flow, R is positive for cyclonic flow; flow around low pressure systems
fRV
VVg 1
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Geostrophic & observed wind 300 hPa
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Geostrophic & observed wind 300 hPa
Observed:95 knots
Geostrophic:140 knots
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How does curvature affect the wind?(anticyclonic flow/high pressure)
nfV
RV
2
R
t
n
Δn
Φ0
Φ0+ΔΦ
Φ0-ΔΦ
HIGH
Low
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From Holton
• If Vg/V < 1, geostrophic wind is an underestimate of the actual wind speed
• Since V is always positive, in the northern hemisphere (f > 0) this only happens for R negative
• For typical northern hemisphere large scale flow, R is negative for anticyclonic flow; flow around high pressure systems
fRV
VVg 1
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Geostrophic & observed wind 300 hPa
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Geostrophic & observed wind 300 hPa
Observed:30 knots
Geostrophic:25 knots
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Uses of Natural Coordinates:Balanced Flows
• Tornados• Hurricanes• General high and low pressure systems
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Cyclostrophic Flow
nfV
RV
sDtDV
nsfV
RV
DtDV
2
2
formcomponent in and
ntnnt
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Cyclostrophic Flow
• A balance in the normal, as opposed to tangential, component of the momentum equation.
• A balance of centrifugal force and the pressure gradient force.
• The following are needed– steady (time derivative = 0)– coriolis force is small relative to pressure
gradient and centrifugal force
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Cyclostrophic Flow
nfV
RV
2
equation momentum ofcomponent normal
Get cyclostrophic flow with either large V small R
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Cyclostrophic Flow
• Radical must be positive: two solutions
nRV
nRV
V:
2
for Solve
0 ,0 .2
0 ,0 .1
nR
nR
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Cyclostrophic Flow
• Tornadoes: 102 meters, 0.1 km• Dust devils: 1 - 10 meters
– Small length scales– Strong winds
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Low
Cyclostrophic Flow
Low
Pressure gradient force
Centrifugal force
0 ,0 .1
n
R 0 ,0 .2
n
R
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Low
Cyclostrophic Flow
Low
0 ,0 .1
n
R 0 ,0 .2
n
R
Counterclockwise Rotation
Clockwise Rotation
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http://www.youtube.com/watch?v=vgbzKF_pSXo
http://www.youtube.com/watch?v=k1dZpW5aFFk
http://www.youtube.com/watch?v=3jQoGm8JEPY
Anticyclonic Tornado (looking up)
Sunnyvale, CA 4 May 1998
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In-Class Exercise: Compute Tornado Wind Speed
• Remember:
nRV
V:
for Solve
P=850 mb
P=750 mb
R = 100 m
np
n
1
(Assume ρ = 1 kg/m3)
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In-Class Exercise: Compute Tornado Wind Speed
1222
2
3
100) 100(
) 100()100(
) 1(1) 100(
1
smsmV
mPa
mkgmV
npRV
nRV
P=850 mb
P=750 mb
R = 100 m
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High
Cyclostrophic FlowAround a High Pressure System?
High
?0 ,0 .1
n
R ?0 ,0 .2
n
R
0
0
n
R
0
0
n
R
n
n
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Gradient Flow(Momentum equation in natural coordinates)
• Balance in the normal, as opposed to tangential, component of the momentum equation
• Balance between pressure gradient, coriolis, and centrifugal force
nfV
RV
sDtDV
2
formcomponent In
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Gradient Flow(Momentum equation in natural coordinates)
Vn
RfRVV
nfV
RV
speed for wind Solve
0
equation momentum ofcomponent normal
2
2
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Gradient Flow(Momentum equation in natural coordinates)
nRfRfRV
nRfRfR
V
nRfRVV
4)(
2
2
4)(
0
2
2
2
Look for real and non-negative solutions
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Gradient FlowSolution must be real
4
04
)(4
)(2
2
2
2
Rfn
nRfR
nRfRfRV
![Page 38: AOSS 401, Fall 2007 Lecture 11 October 1, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)](https://reader035.vdocuments.site/reader035/viewer/2022062503/5a4d1ada7f8b9ab0599747e5/html5/thumbnails/38.jpg)
Low
Gradient Flow
0n
High
Definition of normal, n, direction
n
n
0n
R > 0 R < 0
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Gradient FlowSolution must be real
4
2Rfn
Low∂Φ/∂n < 0
R > 0Always satisfied
High∂Φ/∂n < 0
R < 0Trouble!
pressure gradient MUST go to zero faster than R
![Page 40: AOSS 401, Fall 2007 Lecture 11 October 1, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)](https://reader035.vdocuments.site/reader035/viewer/2022062503/5a4d1ada7f8b9ab0599747e5/html5/thumbnails/40.jpg)
Low
Gradient Flow(Solutions for Lows, remember that square root.)
Low
Pressure gradient force
Centrifugal forceCoriolis Force
V
V
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Low
Gradient Flow(Solutions for Lows, remember that square root.)
Low
Pressure gradient force
Centrifugal forceCoriolis Force
NORMAL ANOMALOUS
V
V
![Page 42: AOSS 401, Fall 2007 Lecture 11 October 1, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)](https://reader035.vdocuments.site/reader035/viewer/2022062503/5a4d1ada7f8b9ab0599747e5/html5/thumbnails/42.jpg)
High
Gradient Flow(Solutions for Highs, remember that square root.)
High
Pressure gradient force
Centrifugal forceCoriolis Force
V
V
NORMAL ANOMALOUS
![Page 43: AOSS 401, Fall 2007 Lecture 11 October 1, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)](https://reader035.vdocuments.site/reader035/viewer/2022062503/5a4d1ada7f8b9ab0599747e5/html5/thumbnails/43.jpg)
Normal and Anomalous Flows
• Normal flows are observed all the time.– Highs tend to have slower magnitude winds
than lows.– Lows are storms; highs are fair weather
• Anomalous flows are not often observed.– Anomalous highs have been reported in the
tropics…– Anomalous lows are strange –Holton “clearly
not a useful approximation.”• But it is possible in tornadoes…
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Compute Wind Speed Around a Hurricane
• R = 100 km• dP = -25 mb• f = 4 x 10-5
• V = 48 m/s = 107 mph = 93 kt• Category 2 hurricane…
nRfRfRV
4
)(2
2
![Page 45: AOSS 401, Fall 2007 Lecture 11 October 1, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)](https://reader035.vdocuments.site/reader035/viewer/2022062503/5a4d1ada7f8b9ab0599747e5/html5/thumbnails/45.jpg)
We have covered a lot of material in a short time!
• Study and think about balances in the natural coordinate system from the point of view of
1. first, pressure gradient, 2. then coriolis force, 3. then the force due to curvature of the lines of
geopotential (or pressure)• Don’t confuse “curvature” in the natural
coordinate system with the curvature terms derived from use of a tangential coordinate system!
![Page 46: AOSS 401, Fall 2007 Lecture 11 October 1, 2007 Richard B. Rood (Room 2525, SRB) 734-647-3530 Derek Posselt (Room 2517D, SRB)](https://reader035.vdocuments.site/reader035/viewer/2022062503/5a4d1ada7f8b9ab0599747e5/html5/thumbnails/46.jpg)
Next time
• Think about adding viscosity to the balance.
• And return to thermal wind balance…
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Weather
• NCAR Research Applications Program– http://www.rap.ucar.edu/weather/
• http://www.aos.wisc.edu/weatherdata/eta_tempest/12UTC/eta_c850_h06.gif
• National Weather Service– http://www.nws.noaa.gov/dtx/