aoss 401, fall 2006 lecture 17 october 22, 2007 richard b. rood (room 2525, srb) [email protected]...
TRANSCRIPT
AOSS 401, Fall 2006Lecture 17
October 22, 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
Material from Chapter 4
• Vorticity, Vorticity, Vorticity
– Definition of barotropic and baroclinic– Review– Scaling
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
Vorticity, Vorticity, Vorticity
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.
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.
Want to formalize the representation of the role of rotation
and divergence
Suppose we have some flow
Imagine at the point flow decomposed into two “components”
A “component” that flows into or away from the point.
Imagine at the point flow decomposed into two “components”
A “component” that flows around the point.
Lets consider a spinning skaterMotion is in the (x,y) plane
Axis of rotation is in the vertical plane
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
Vertical Component of Vorticity
aUk
vorticityofcomponent
vertical with theconcernedonly generally
In what plane is the motion?In what direction is the vorticity?
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
Planetary Vorticity
f
fearth
)sin(2vorticityplanetary
)sin(2
uk
Absolute (or total) Vorticity
fy
u
x
veartha
vorticityabsolute
vorticityrelativevorticityplanetary vorticityabsolute
ukukUk
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
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?
The scaled horizontal momentum equation in z coordinates
fuy
p
dt
dv
fvx
p
dt
du
1
1
no viscosity
Relative Vorticity
y
u
x
v
(u,v,w)
vorticityrelative
velocity relative u
Take derivatives
fuy
p
dt
dv
x
fvx
p
dt
du
y
1
1
Take derivatives
)(1
)(1
fuxy
p
xdt
dv
x
fvyx
p
ydt
du
y
Take derivatives
)(1
)(
)(1
)(
fuxy
p
xv
t
v
x
fvyx
p
yu
t
u
y
u
u
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
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?
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
Scale of relative vorticity
1510
sL
U
y
u
x
v
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.
Time rate of change of relative vorticity
211
2102
2
10
10,,
sHL
WU
zw
sL
U
yv
xu
t
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
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
Consider divergence term
)())((y
v
x
uf
y
v
x
uf
We saw that the relative vorticity is less than the planetary vorticity. So
So for quasi-nondivergent flow
21010)(
sy
v
x
uf
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.
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
))(()(
y
v
x
uf
Dt
fD
y
fv
Dt
Df
horizontal
returned relative vorticity to equation
A simple version of potential vorticity
)()(
)(
1
y
v
x
u
Dt
fD
fhorizontal
what’s this?
A simple version of potential vorticity
z
w
Dt
fD
fhorizontal
)(
)(
1
Assume constant density and temperature.
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.
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?
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
What happens when the vortex meets the mountain?
Surface with a hill.
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.
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.
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
Scaled vorticity equation
))(()(
y
v
x
uf
Dt
fDhorizontal
Pure constant vorticity flow.
Pure divergent flow
Divergence influence on vorticity
Divergence influence on vorticity
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.
Possible development of a surface low.
Earth’s surface
DIVERGENCE
warming
LOW
Changes in depth of vortex
H H CONVERGENCE
And a little more realistically
Concentrates relative vorticity
Dilutes relative vorticity
What if the wind is geostrophic?
Scaled vorticity equation
))(()(
y
v
x
uf
Dt
fDhorizontal
What happens if the wind is geostrophic?
Or returning to potential vorticity
0)( H
f
Dt
D ghorizontal
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 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?
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