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AOSS 401, Fall 2006Lecture 8
September 24, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News
• Contract with class.– First exam October 10.
• Homework 3 is posted.– Due Friday
• Solution sets for Homework 1 and 2 are posted.
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
Outline
• Vertical Structure Reset
• Stability and Instability– Wave motion
• Balances
• Thermal Wind
• Maps
Full equations of motion
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 We saw that the first two equations were dominated by the geostrophic balance. What do we do for the vertical motion?
Thermodynamic equation(Use the equation of state)
T
J
Dt
pDR
Dt
TDRc
T
J
Dt
Dp
p
R
Dt
DT
T
Rc
v
v
)(ln)(ln)(
)(
Definition of potential temperature
)/()( RcRsfc v
p
pT
This is the temperature a parcel would have if it was moved from some pressure and temperature to the surface.
This is Poisson’s equation.
This is a very important point.
• Even in adiabatic motion, with no external source of heating, if a parcel moves up or down its temperature changes.
• What if a parcel moves about a surface of constant pressure?
Adiabatic lapse rate.
For an adiabatic, hydrostatic atmosphere the temperature decreases with height.
Rc
g
z
Tz
v
0
Another important point
• If the atmosphere is in adiabatic balance, the temperature still changes with height.
• Adiabatic does not mean isothermal. It means that there is no external heating or cooling.
The parcel method
• We are going displace this parcel – move it up and down.– We are going to assume that the pressure adjusts
instantaneously; that is, the parcel assumes the pressure of altitude to which it is displaced.
– As the parcel is moved its temperature will change according to the adiabatic lapse rate. That is, the motion is without the addition or subtraction of energy. J is zero in the thermodynamic equation.
Parcel cooler than environment
z
Warmer
Cooler
If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?)
Parcel cooler than environment
z
Warmer
Cooler
If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?)
Parcel warmer than environment
z
Warmer
Cooler
If the parcel moves up and finds itself warmer than the environment then it will go up some more. (What is its density? larger or smaller?)
Parcel cooler than environment
z
Warmer
Cooler
If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?)
This is our first example of “instability” – a perturbation that grows.
Let’s quantify this.
zzTzzT
Tz z zz
TzTT
sfcsfc
sfc
)()(T
ist environmen theof in change then the,Δ to from go weif So
rate lapse constant
Under consideration of T changing with a constant linear slope (or lapse rate).
Let’s quantify this.
rate lapse adiabatic
T
is parcel theof in change then the,Δ to from go weif So
)()(
pd
dzparceldzparcel
c
g
zTzT
Tz z z
Under consideration of T of parcel changing with the dry adiabatic lapse rate
Stable: temperature of parcel cooler than environment.
d
tenvironmenparcel TT
Unstable: temperature of parcel greater than environment.
d
tenvironmenparcel TT
Stability criteria from physical argument
stable
neutral
unstable
d
d
d
Let’s return to the vertical momentum equation
What are the scales of the terms?
2H
WW*U/L
U*U/a
UfgH
Psfc
10-7
10-5 10
10-3
10 10-15
)()cos(21v 2
22
wΩugz
p
a
u
Dt
Dw
What are the scales of the terms?
2H
WW*U/L
U*U/a
UfgH
Psfc
10-7
10-5 10
10-3
10 10-15
)()cos(21v 2
22
wΩugz
p
a
u
Dt
Dw
Vertical momentum equation Hydrostatic balance
gz
p
wΩugz
p
a
u
Dt
Dw
10
balance chydrostati
)()cos(21v 2
22
Hydrostatic balance
gz
penv
env
1
0
balance chydrostatiin t environmen
But our parcel experiences an acceleration
gz
p
Dt
zD
Dt
Dw env
parcel
1
2
2
Assumption of adjustment of pressure.
Solve for pressure gradient
z
pg
gz
p
envenv
env
env
1
0
balance chydrostatiin t environmen
But our parcel experiences an acceleration
)()1(
or )()1(
2
2
2
2
2
2
tenvironmen
tenvironmenparcel
tenvironmen
parcel
parcel
parcelenv
parcel
env
parcel
env
ggDt
zD
ggDt
zD
gg
Dt
zD
Dt
Dw
Again, our pressure of parcel and environment are the same so
)()(2
2
tenvironmen
tenvironmenparcel
tenvironmen
tenvironmenparcel
T
TTgg
Dt
zD
So go back to our definitions of temperature and temperature change above
zzT
g
zzT
g
T
TTg
Dt
zD
d
dntdisplacemez
tenvironmen
tenvironmenparcel
)(
)(
)(
0
@
2
2
Use binomial expansion
)1(11
and small is ntsdisplaceme smallfor
)1(
11
000
0
00
0
T
z
TzT
T
z
Tz
TzT
So go back to our definitions of temperature and temperature change above
zT
z
Tg
zzT
g
Dt
zD
d
d
))(1(1
)(
00
02
2
Ignore terms in z2
0)(
)()(
02
2
002
2
zT
g
Dt
zD
or
zT
gz
T
g
Dt
zD
d
dd
For stable situation
0)( and
0)(
0
02
2
dd
d
T
g
zT
g
Dt
zD
Seek solution of the form
tBtAz
2
sin2
cos
For stable situation
Seek solution of the form
)(
2
2sin
2cos
0
dTg
tBtAz
Parcel cooler than environment
z
Warmer
Cooler
If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?)
Example of such an oscillation
For unstable situation
0)( and
0)(
0
02
2
dd
d
T
g
zT
g
Dt
zD
Seek solution of the form
tiez
Parcel cooler than environment
z
Warmer
Cooler
If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?)
This is our first example of “instability” – a perturbation that grows.
This is our first explicit solution of the wave equation
• These are called buoyancy waves or gravity gaves.
• The restoring force is gravity, imbalance of density in the fluid.
• We extracted an equation through scaling and use of balances.– This is but one type of wave that is supported by the
equations of atmospheric dynamics.
• Are gravity waves important in the atmosphere?
Near adiabatic lapse rate in the troposphere
Troposphere: depth ~ 1.0 x 104 m
Troposphere------------------ ~ 2Mountain
Troposphere------------------ ~ 1.6 x 10-3
Earth radius
GTQ: What if we assumed that the atmosphere was constant density? Is there a depth the atmosphere cannot exceed?
Looking at the atmosphere
• What does the following map tell you?
Forced Ascent/Descent
WarmingCooling
An Eulerian Map
Let us return to the horizontal motions
Some meteorologist speak
• Zonal = east-west• Meridional = north-south• Vertical = up and down
What are the scales of the terms?
2H
U
)v()sin(21v)tan(v
)()cos(2)sin(v21)vtan(
22
2
Ωuy
p
a
w
a
u
Dt
D
uΩwΩx
p
a
uw
a
u
Dt
Du
U*U/L
U*U/a
U*W/a
Uf WfL
P
10-4
10-5
10-8
10-3 10-310-6 10-12
What are the scales of the terms?
2H
U
)v()sin(21v)tan(v
)()cos(2)sin(v21)vtan(
22
2
Ωuy
p
a
w
a
u
Dt
D
uΩwΩx
p
a
uw
a
u
Dt
Du
U*U/L
U*U/a
U*W/a
Uf WfL
P
10-4
10-5
10-8
10-3 10-310-6 10-12
Largest Terms
Geostrophic balance
High Pressure
Low Pressure
Atmosphere in balance
• Hydrostatic balance• Geostrophic balance• Adiabatic lapse rate
• We can use this as a paradigm for thinking about many problems, other atmospheres. Suggests a set of questions for thinking about observations. What is the rotation? How does it compare to acceleration, represented by the spatial and temporal scales?
Atmosphere in balance
• Hydrostatic balance• Geostrophic balance• Adiabatic lapse rate
• But what we are really interested in is the difference from this balance. And this balance is like a strong spring, always pulling back. It is easy to know the approximate state. Difficult to know and predict the actual state.
Let’s think about another possible balance
Thermodynamic balance(velocity and acceleration = 0)
1 and
0
10
10
10
RTp
Jt
Tc
t
gz
p
y
p
x
p
v
Compare with geostrophic balance.
Specify something for J
heating frictionaltyconductivi thermal
heatlatent radiation
J
Jt
Tcv
Specify something for J
TJ
Jt
Tcv
flux) (radiative div
Where we ignore for latent heat release for convenience (e.g. dry atmosphere). We know frictional heating is zero for no velocity.
We can show
• Horizontal gradients of both pressure and density must equal zero.– Hence horizontal temperature gradient must be zero.
T=T(z)
• If there is a horizontal temperature gradient then there is motion. If differential heating in the horizontal then temperature gradient. Hence motion.
Transfer of heat north and south is an important element of the climate at the Earth’s surface.
Redistribution by atmosphere, ocean, etc.
SURFACE
Top of Atmosphere / Edge of Space
ATMOSPHERECLOUD
heat is moved to poles
cool air moved towards equator cool air moved towards equator
This is a transfer. Both ocean and atmosphere are important!
Hurricanes and heat
Middle latitude cyclones
Thermodynamic Balance
• The atmosphere and ocean are NOT in thermodynamic balance.
• If there is a temperature gradient, then there is motion.
• Temperature gradients are always being forced.
Return to the Geostrophic Balance
The geostrophic balance
2H
U
)v()sin(21v)tan(v
)()cos(2)sin(v21)vtan(
22
2
Ωuy
p
a
w
a
u
Dt
D
uΩwΩx
p
a
uw
a
u
Dt
Du
U*U/L
U*U/a
U*W/a
Uf WfL
P
10-4
10-5
10-8
10-3 10-310-6 10-12
Largest Terms
The geostrophic balance
y
pfu
x
pfv
Ωf
Ωuy
p
Ωx
p
1
1
)sin(2
)sin(21
0
)sin(v21
0
The geostrophic balance
y
pfu
x
pfv
1
1How do we link the horizontal and vertical balances?
The geostrophic balance
y
pfu
z
x
pfv
z
1
1
Take a vertical derivative of the equation.
The geostrophic balance
y
T
fT
g
z
u
x
T
fT
g
z
v
z
T
T
u
y
T
fT
g
z
u
z
T
T
v
x
T
fT
g
z
v
Use equation of state to eliminate density.
Thermal wind relationship in height (z) coordinates
moving block
Shear? (1)
stationary surface
There is force due to fact that there is a velocity and when the moving blocks are in contact the interfaces experience a force – say , friction, the surfaces can distort. One form of distortion is shearing.
moving fluid
Shear? (2)
• Shear is a word used to describe that velocity varies in space.
more slowly moving fluid
There is force due to fact that there is a velocity gradient, and because our fluid is a fluid, the fluid surface responds to this gradient, which is called the shear.
moving fluid
Shear? (3)
• Shear is a word used to describe that velocity varies in space.
more slowly moving fluid
wind.zonal ofshear vertical
z
u
z
The geostrophic balance
y
T
fT
g
z
u
What does this equation tell us?
Thermal wind relationship in height (z) coordinates
Can we start to relate vertical structure and wind?
Troposphere: depth ~ 1.0 x 104 m
Troposphere------------------ ~ 2Mountain
Troposphere------------------ ~ 1.6 x 10-3
Earth radius
An estimate of the January mean temperature
northwinter
southsummer
tropopause
stratopause
mesosphere
stratosphere
troposphere
note where the
horizontal temperature gradients are
large
An estimate of the January mean zonal wind
northwinter
southsummer
note the jet streams
An estimate of the July mean zonal wind
northsummer
southwinter
note the jet streams
Gosh, that’s a lot
• Think about it!
• Do your homework?
• This is new material now?
• From that July wind field, what are the differences between January and July temperatures.
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