AOSS 401, Fall 2006Lecture 18
October 24, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News
• Final exam will be last day of class
• Derek and I decided to think about good homework problems for another day.– No homework posted today.
Material from Chapter 4
• Vorticity, Vorticity, Vorticity
– Relative and planetary vorticity– Mid-latitude disturbances– Vorticity, divergence, in 3-D
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
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.
Absolute (or total) Vorticity
fy
u
x
veartha
vorticityabsolute
vorticityrelativevorticityplanetary vorticityabsolute
ukukUk
Relative and planetary vorticity
• Planetary vorticity is cyclonic is positive vorticity
• Planetary vorticity, in middle latitudes, is usually larger than relative vorticity
We derived the vorticity equation
x
p
yy
p
x
z
u
y
w
z
v
x
w
y
v
x
uf
y
fv
t
11
)(
))((u
TERMS
DIVERGENCE
TILTING
SOLENOIDAL or
BAROCLINIC
Comments on the terms
• There are important dynamical features in the atmosphere where all of these terms are important.
• Baroclinic terms are due to there being gradients of temperature on pressure surfaces. (Are they explicitly there in pressure coordinates?) Like a thermodynamic “source” of rotation.
Tilting Term
rotation in, say, (y, z)
plane, “vorticity” in x
plane
as the wheel is turned there is a
component of “vorticity” in the z
plane
Divergence influence on vorticity
Divergence influence on vorticity
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
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
A nuance on vorticity and the scaled equation: potential vorticity
A simple version of potential vorticity
0)( H
f
Dt
Dhorizontal
Integrate with height,z1 z2 over a layer of depth H.
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.
Relative vorticity with change of depth
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.
Scaled vorticity equation
))(()(
y
v
x
uf
Dt
fDhorizontal
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
Let’s explicitly map these ideas to the Earth
Local vertical / planetary vorticity
relative vorticity/planetary vorticity
relative vorticity
planetary vorticity
Compare relative vorticity to planetary vorticity
NumberRossby Ro
10
10
10
0
1
0
140
15
Lf
U
f
sf
sL
U
planetary vorticity
is usually larger than
relative vorticity
for large-scale and
middle latitudes
Relative and planetary vorticity
• Planetary vorticity is cyclonic is positive vorticity• Planetary vorticity, in middle latitudes, is usually
larger than relative vorticity• A growing cyclone “adds to” the planetary
vorticity.– Lows intense
• A growing anticyclone “opposes” the planetary vorticity.– Highs less intense
Compare relative vorticity to planetary vorticity and
to divergence
100
10
10
0
0
yv
xuf
yv
xu
f
Flow is rotationally dominated, but divergence is crucial to understanding
flow.
Consider our simple form of potential vorticity
vorticitypotential
0)(
H
fH
f
Dt
Dhorizontal
From scaled equation, with assumption of constant density and temperature.
Fluid of changing depth
Two things that we have learned about vorticity.
• Convergence and divergence in a column of fluid, impacts the vorticity throughout the column.– Specifically, divergence above causes low
pressure at the surface.
• Stretching and shrinking of a column of vorticity will change the relative vorticity.
Possible development of a surface low.
Earth’s surface
pressure surfaces
Lets return to our simple problem
Earth’s surface
pressure surfaces
warming
cooling
Lets return to our simple problem
Earth’s surface
pressure / height
surfacesrisewarming
cooling
pressure / height
surfacessink
Lets return to our simple problem
Earth’s surface
warming
cooling
PGF H
L
Lets return to our simple problem
Earth’s surface
warming
cooling
PGF H
L mass leaves
column / low forms at groundL
Lets return to our simple problem
Earth’s surface
warming
cooling
PGF H
L mass leaves
column / low forms at groundL
mass enters column / high
forms at ground
H
Lets return to our simple problem
Earth’s surface
warming
cooling
PGF H
L mass leaves
column / low forms at groundL
mass enters column / high
forms at ground
H PGF
Mass continuity?
• What are the implications of mass continuity?
• What is your law, your equation, your tool to answer that question?
Temperature
• Assuming the air moves isentropically, what happens to the temperature?
• What is your law, your equation, your tool to answer that question?
Lets return to our simple problem
Earth’s surface
warming
cooling
PGF H
L
LH PGF
Simple Thermal Circulation
• There is the sense of the air moves to counter the heating.
• If the heating ended, then the circulation would end, acting to bring back the original equilibrium situation.
• This sort of low is cause by heating, is called a “thermal” low, warm core. It tends to damp out.
Lets return to our simple problem
Earth’s surface
warmcore
coldcore
PGFHL
LH PGF
Simple Thermal Circulation
• This sort of low is cause by heating, is called a “thermal” low, warm core. It tends to damp out.– Remember the question about the hurricane
being warm core.
• What about the divergence and convergence?
Lets return to our simple problem
Earth’s surface
warmcore
coldcore
PGF HL
LH PGF
DIVERGENCE
CONVERGENCE
CONVERGENCE
DIVERGENCE
Simple Thermal Circulation
• What about the divergence and convergence?– Convergence and Divergence are aligning
over top of each other in the vertical.– Again, in this case there is a tendency for the
circulation to damp out.
Back to the earth again
Still in the atmosphere
Flow over a hill
HILL
Derived a simple form of potential vorticity
vorticitypotential
0)(
H
fH
f
Dt
Dhorizontal
From scaled equation, with assumption of constant density and temperature.
Flow over a hill(long in the north-south)(can’t go around the hill)
west east
Flow over a hill
HILL
west east
Dep
th,
H
Flow over a hill(assume flow is adiabatic)
HILL
west east
Dep
th,
H
θ
θ + Δθ
Flow over a hill(far upstream constant zonal flow)
HILL
west east
Dep
th,
H
θ
θ + Δθ
ζ=0
Derived a simple form of potential vorticity
vorticitypotential
0)(
H
fH
f
Dt
Dhorizontal
From scaled equation, with assumption of constant density and temperature.
What happens as air gets to hill?
HILL
west east
Dep
th,
H
θ
θ + Δθ
ζ=0
What happens as air gets to hill?
HILL
west east
Dep
th,
H
θ
θ + Δθ
ζ=0
Air is lifted. Lifting higher at ground than upper air.(pressure gradient force spreads it out)
What happens as air gets to hill?
HILL
west east
Dep
th,
H +ΔH
θ
θ + Δθ
ζ=0
Air is lifted. Lifting higher at ground than upper air.(pressure gradient force spreads it out)
What happens as air gets to hill?
HILL
west east
Dep
th,
H +ΔH
θ
θ + Δθ
ζ must increase
Air is lifted. Lifting higher at ground than upper air.(pressure gradient force spreads it out)
How does vorticity increase?
What happens in these waves?
Gains cyclonic vorticity
Loses cyclonic vorticitySame as gains
anticyclonic vorticity
Or schematically
Cyclonic Anticyclonic
Rotational
Shear
What happens as air gets to hill?
HILL
west east
Dep
th,
H +ΔH
θ
θ + Δθ
ζ must increase
Air turns cyclonically to increase vorticity.In northern hemisphere turns north.
In the (east-west, north-south) planeD
epth
, H
Dep
th,
H +ΔH
west easts
n
What happens as air goes over hill?
HILL
west east
Dep
th,
H -ΔH
θ
θ + Δθ
Air turns anti-cyclonically to decrease vorticity.In northern hemisphere turns south.
ζ must decrease
In the (east-west, north-south) planeD
epth
, H
Dep
th,
H +ΔH
west easts
n Dep
th,
H -ΔH
What happens as air goes down hill?
HILL
west eastD
epth
, H
+ΔH
θ
θ + Δθ
Air turns cyclonically to increase vorticity.In northern hemisphere turns north.
ζ must increase
In the (east-west, north-south) planeD
epth
, H
Dep
th,
H +ΔH
west easts
n Dep
th,
H -ΔH
Dep
th,
H +ΔH
What is happening with planetary vorticity?(In the (east-west, north-south) plane)
Dep
th,
H
Dep
th,
H +ΔH
west easts
n Dep
th,
H -ΔH
Dep
th,
H +ΔH
f is greater for deflections to north
f is less for deflections to south
What is happening with planetary vorticity?(In the (east-west, north-south) plane)
Dep
th,
H
Dep
th,
H +ΔH
west easts
n Dep
th,
H -ΔH
Dep
th,
H +ΔH
f + ζ is less than earth’s vorticity and wants to turn north.
Arrives here wanting vorticity. “Overshoots”
What is happening with planetary vorticity?(In the (east-west, north-south) plane)
Dep
th,
H
Dep
th,
H +ΔH
west easts
n Dep
th,
H -ΔH
Dep
th,
H +ΔH
What happens if wind is from east?
HILL
west east
θ
θ + Δθ
What is happening with planetary vorticity?(In the (east-west, north-south) plane)
Dep
th,
H
Dep
th,
H +ΔH
west easts
n Dep
th,
H -ΔH
Dep
th,
H +ΔH
Flow from east planetary and relative vorticity interact together, no overshoot or undershoot.
Excursion into the atmosphere
Middle latitude cyclones
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
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