Download AOSS_401_20070924_L08_Static_Wave_Thermo

AOSS 401, Fall 2006
Lecture 8
September 24, 2007
Richard B. Rood (Room 2525, SRB)
[email protected]
734-647-3530
Derek Posselt (Room 2517D, SRB)
[email protected]
734-936-0502
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/day07loop.html
• Weather Underground
– http://www.wunderground.com/cgibin/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
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )   2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )  2 ( v)
Dt
a
a
 y
Dw u 2  v 2
1 p


 g  2Ωu cos( )  2 ( w)
Dt
a
 z
D
We saw that the first two
    u
Dt
equations were dominated
DT
D 1
cv
p
( )J
by the geostrophic balance.
Dt
Dt 
What do we do for the
1
p  RT and  

vertical motion?
Thermodynamic equation
(Use the equation of state)
(cv  R) DT R Dp J


T
Dt p Dt T
D(ln T )
D(ln p) J
(cv  R)
R

Dt
Dt
T
Definition of potential temperature
  T(
psfc
p
)
R /(cv  R )
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.

0
z
T
g

z
cv  R
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
Cooler
If the parcel moves up and
finds itself cooler than the
environment then it will sink.
(What is its density? larger or
smaller?)
Warmer
Parcel cooler than environment
z
Cooler
If the parcel moves up and
finds itself cooler than the
environment, then it will sink.
(What is its density? larger or
smaller?)
Warmer
Parcel warmer than environment
z
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?)
Warmer
Parcel cooler than environment
z
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.
Warmer
Let’s quantify this.
T
z
So if we go from z to z  Δz, then the change in T of the environmen t is
T  Tsfc  z  constant  lapse rate
T  Tsfc   ( z  z )  (Tsfc  z )  z
Under consideration of T changing with a constant linear
slope (or lapse rate).
Let’s quantify this.
So if we go from z to z  Δz , then the change in T of the parcel is
T  Tparcel( z )  d z  Tparcel( z )  d z
d 
g
 adiabatic lapse rate
cp
Under consideration of T of parcel changing with the dry
adiabatic lapse rate
Stable: temperature of parcel cooler than
environment.
Tparcel  Tenvironment
  d
Unstable: temperature of parcel greater than
environment.
Tparcel  Tenvironment
  d
Stability criteria from physical argument
d   unstable
d   neutral
d   stable
Let’s return to the vertical
momentum equation
What are the scales of the terms?
Dw u  v
1 p


 g  2Ωu cos( )  2 (w)
Dt
a
 z
2
W*U/L
10-7
2
Psfc
U*U/a
10-5
H
g
10
10
Uf
10-3
W
H2
10-15
What are the scales of the terms?
Dw u  v
1 p


 g  2Ωu cos( )  2 (w)
Dt
a
 z
2
W*U/L
10-7
2
Psfc
U*U/a
10-5
H
g
10
10
Uf
10-3
W
H2
10-15
Vertical momentum equation 
Hydrostatic balance
Dw u 2  v 2
1 p


 g  2Ωu cos( )  2 ( w)
Dt
a
 z
hydrostati c balance
1 p
0
g
 z
Hydrostatic balance
environmen t in hydrostati c balance
1 penv
0
g
 env z
But our parcel experiences an acceleration
Dw D z
1 penv




g
2
Dt
Dt
 parcel z
2
Assumption of adjustment of pressure.
Solve for pressure gradient
environmen t in hydrostati c balance
1 penv
0
g
 env z
 g env
penv

z
But our parcel experiences an acceleration
 g env
Dw D 2 z


g
2
Dt
Dt
 parcel
 env   parcel
 env
D2 z
 g(
 1)  g (
) or
2
Dt
 parcel
 parcel
 parcel
 parcel   environment
D z
 g(
 1)  g (
)
2
Dt
 environment
 environment
2
Again, our pressure of parcel and
environment are the same so
 parcel   environment
Tparcel  Tenvironment
D z
 g(
)  g(
)
2
Dt
 environment
Tenvironment
2
So go back to our definitions of temperature
and temperature change above
Tparcel  Tenvironment
D z
 g(
)
2
Dt
Tenvironment
2

g
Tz @ displacement  z
g

(  d ) z
T0  z
(  d ) z
Use binomial expansion
1
1

T0  z T (1  z )
0
T0
for small displaceme nts
z
1
1
 (1  )
T0
T0  z T0
z
T0
is small and
So go back to our definitions of temperature
and temperature change above
2
D z
g

(



)
z
d
2
Dt
T0  z
1
z
 g (1  )(  d ) z
T0
T0
Ignore terms in z2
2
D z g
g

(



)
z


(



)
z
d
d
2
Dt
T0
T0
or
2
D z g

(



)
z

0
d
2
Dt
T0
For stable situation
D2 z g
 (d   ) z  0
2
Dt
T0
g
d   and (d   )  0
T0
Seek solution of the form
z  A cos
2

t  B sin
2

t
For stable situation
Seek solution of the form
z  A cos

2

t  B sin
2
g
(d   )
T0
2

t
Parcel cooler than environment
z
Cooler
If the parcel moves up and
finds itself cooler than the
environment then it will sink.
(What is its density? larger or
smaller?)
Warmer
Example of such an oscillation
For unstable situation
D2 z g
 (d   ) z  0
2
Dt
T0
g
d   and (d   )  0
T0
Seek solution of the form
ze
i t
Parcel cooler than environment
z
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.
Warmer
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
------------------ ~ 2
Mountain
Troposphere
------------------ ~ 1.6 x 10-3
Earth radius
Troposphere: depth ~ 1.0 x 104 m
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
Cooling
Warming
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?
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )  2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )  2 ( v)
Dt
a
a
 y
U*W/a
U*U/L
10-4
U*U/a
10-5
10-8
P
L
10-3
U
Uf Wf
10-3 10-6
H2
10-12
What are the scales of the terms?
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )  2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )  2 ( v)
Dt
a
a
 y
U*W/a
U*U/L
10-4
U*U/a
10-5
10-8
P
L
Uf Wf
10-3
10-3 10-6
Largest Terms
U
H2
10-12
Geostrophic balance
Low Pressure
High 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 p
 x
1 p
0
 y
1 p
0
g
 z

0
t
T
cv
J
t
0
p  RT and  
1

Compare with geostrophic balance.
Specify something for J
T
J
t
J  radiation  latent heat 
cv
thermal conductivi ty  frictional heating
Specify something for J
T
cv
J
t
J  div (radiative flux)    T
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.
Top of Atmosphere / Edge of Space
CLOUD
ATMOSPHERE
heat is moved to poles
cool air moved towards equator
cool air moved towards equator
SURFACE
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
Du uvtan(  ) uw
1 p



 2Ωv sin(  )  2Ωw cos( )  2 (u )
Dt
a
a
 x
Dv u 2 tan(  ) vw
1 p



 2Ωu sin(  )  2 ( v)
Dt
a
a
 y
U*W/a
U*U/L
10-4
U*U/a
10-5
10-8
P
L
Uf Wf
10-3
10-3 10-6
Largest Terms
U
H2
10-12
The geostrophic balance
1 p
0
 2Ωv sin(  )
 x
1 p
0
 2Ωu sin(  )
 y
f  2Ω sin(  )
1 p
fv 
 x
1 p
fu  
 y
The geostrophic balance
1 p
fv 
 x
1 p
fu  
 y
How do we link the
horizontal and
vertical balances?
The geostrophic balance
Take a vertical derivative of the equation.
 
1 p 
 fv 

z 
 x 
 
1 p 
 fu  

z 
 y 
The geostrophic balance
Use equation of state to eliminate density.
v g T v T


z fT x T z
u
g T u T


z
fT y T z
Thermal wind
relationship in height
v g T
(z) coordinates

z fT x
u
g T

z
fT y
Shear? (1)
moving block
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.
Shear? (2)
• Shear is a word used to describe that
velocity varies in space.
moving fluid
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.
Shear? (3)
• Shear is a word used to describe that velocity
varies in space.
z
moving fluid
more slowly moving fluid
u
 vertical shear of zonal wind.
z
The geostrophic balance
What does this equation tell us?
u
g T

z
fT y
Thermal wind
relationship in height
(z) coordinates
Can we start to relate vertical structure and wind?
Troposphere
------------------ ~ 2
Mountain
Troposphere
------------------ ~ 1.6 x 10-3
Earth radius
Troposphere: depth ~ 1.0 x 104 m
An estimate of the January mean temperature
mesosphere
stratopause
note where
the
horizontal
temperature
gradients
are large
stratosphere
tropopause
troposphere
south
summer
north
winter
An estimate of the January mean zonal wind
note the jet
streams
south
summer
north
winter
An estimate of the July mean zonal wind
note the jet
streams
south
winter
north
summer
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.