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AOSS 401, Fall 2007Lecture 12
October 3, 2007
Richard B. Rood (Room 2525, SRB)[email protected]
734-647-3530Derek Posselt (Room 2517D, SRB)
Class News
• Homework– Homework and some review questions were posted last night.
• Homework due Monday• We will go over the review questions on Friday
– Think about them
• Exam next Wednesday– Today’s lecture is the last fundamentally new material that will
be on the exam • Friday we will talk about vertical velocity some more• Friday and Monday we will look at the material in different ways and
more thoroughly• Also have your questions
• Mid-term evaluation– “students will be notified soon thereafter that they can fill out the
midterm evaluations between October 8 and October 14”
Material from Chapter 3
• Balanced flow
• Examples of flows– Stratospheric Vortex
• Ozone hole
– Surface Flow• Friction
• Thermal wind
Picture of Earth
f=2Ωsin(Φ)
1.4X10-4 s-1
1.0X10-4 s-1
0.0 s-1
Picture of EarthΩ
k
k
k
Ω
Ω
Maximum rotation of vertical column.
No rotation of vertical column.
Rotation
• When a fluid is in rotation, the rotation comes to define the flow field; it provides structure.
• That structure aligns with the vector that defines the angular velocity.– So if the flow is quasi-horizontal, then how the flow
aligns in the vertical is strongly influenced by the rotation and its projection in the vertical.
– On a horizontal surface the curvature of the flow is important
And on the Earth.
• Tropics are more weakly influenced, defined by rotation than middle latitudes.– This also influences the vertical structure of
the dynamical features.
Length scales
• Planetary waves: 107 meters, 10,000 km– Have we seen one of these in our lectures?
• Synoptic waves: Our large-scale, middle-latitude, 106 meters, 1000 km– What’s a synoptic wave? What does synoptic mean?
• Hurricanes: 105 meters, 100 km• Fronts: 104 meters, 10 km• Cumulonimbus clouds: 103 meters, 1 km• Tornadoes: 102 meters, 0.1 km• Dust devils: 1 - 10 meters
Returning to our mid-latitude, large-scale flow.
• We saw last lecture that we could define natural coordinates that were (potentially) useful for determining the motion from maps of thermodynamic fields. That is, the pressure gradient or its analogue, geopotential height.
• We saw that, while a powerful constraint, geostrophy is formally true only when the lines of geopotential are straight.– It’s also a balance, steady state.
• Hence, while seductive, this is not adequate.
How do these natural coordinates relate to the tangential coordinates?
• They are still tangential, but the unit vectors do not point west to east and south to north.
• The coordinate system turns with the wind.
• And if it turns with the wind, what do we expect to happen to the forces?
Ω
Earth
Φ = latitude
a
Looking down from above
Looking down from above
Looking down from above
Looking down from above
Looking down from above
Balanced flows in natural coordinates(balanced, here, means steady)
gradient
hiccyclostrop
cgeostrophi
2
2
nfV
R
V
nR
V
nfV
Low
Cyclostrophic FlowHow do we get this kind of flow?
Low
Pressure gradient force
Centrifugal forceDo we have this balance
around a high?
Low
Gradient FlowWhat forces are being balanced?
0n
High
Definition of normal, n, direction
n
n
0n
Low
Gradient Flow
0n
High
Definition of normal, n, direction
n
n
0n
R>0 R<0
Gradient FlowSolution must be real
4
2Rf
n
Low∂Φ/∂n<0
R>0Always satisfied
High∂Φ/∂n<0
R<0Trouble!
pressure gradient MUST go to zero faster than R
What does this mean physically
• For a high, the pressure gradient weakens towards the center of the high. If pressure weakens, then wind speed weakens. Hence, highs associated with relatively weak winds.
• For a low, there is no similar constraint. Hence lows can spin up into strong storms.
Low
Gradient Flow(Solutions for Lows, remember that square root.)
Low
Pressure gradient force
Centrifugal forceCoriolis Force
NORMAL ANOMALOUS
V
V
High
Gradient Flow(Solutions for Highs, remember that square root.)
High
Pressure gradient force
Centrifugal forceCoriolis Force
V
V
NORMAL ANOMALOUS
Why do we call these flows anomalous?
• Where might these flows happen?
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.”
Balanced flow: an application of all that we know
Geopotential, 50 hPa surfacePressure units:
hPambar
inches of Hg
Length scale?
>1,000 km~10,000 km
What about the wind?
Pressure gradient
Coriolis forceWhat’s the latitude?
Centrifugal force
Wind
Wind
What would happen if I put dye in the low?
So we observe that what happens in this low stays in this low.
tinitialedyedye
dyedt
dyed
HH
dyeHdt
dyed
)()(
0. in, dye puttingQuit source. is
)()(
Ozone, October 23, 2006
Summary from ozone hole
• Ozone hole movie
• Cyclonic polar low isolates air from rest of Earth.
• Extreme cold temperature cause nitric acid and water clouds which changes basic chemical environment of atmosphere.
• Return of sun destroys ozone in isolated air with changed chemical environment.
Let’s move down to the surface.
• At 1000 mb
• How are things different?
• How would we have to modify the equations?
Geostrophic and observed wind 1000 mb (land)
Geostrophic and observed wind 1000 mb (ocean)
Think about this in terms of natural coordinates.
nfV
R
V
sDt
DV
nsfV
R
V
Dt
DV
2
2
erm?friction t some
formcomponent in and
ntnnt
Our geostrophic flow.
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
northn
fVg
Δn
We have said that what’s going on near the surface is related to viscosity.
positive is
- - Friction
draglinear a asfriction Model
motion ofdirection the toopposite actsFriction
2
k
kvku
forceFriction
ji
u
So what does it say if our wind crosses the height contours?
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
north nfVg
Δn ?
So what does it say if our wind crosses the height contours?(Staying in natural coordinates.)
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
north
ΔΦ
tn
So what does it say if our wind crosses the height contours?(Staying in natural coordinates.)
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
north
ΔΦ
tn
u
v
angle, α
Friction force
u
v
k
kvku
tan
o tangent tis
positive is
- - Friction
draglinear a asfriction Model
motion ofdirection the toopposite actsFriction
vt
ji
Friction force
kVvuk
kvku
22Friction
- - Friction
draglinear a asfriction Model
motion ofdirection the toopposite actsFriction
ji
Balance of forces (northern hemisphere)(Staying in natural coordinates.)
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
north
ΔΦ
tn angle, α
Balance of forces (northern hemisphere)(Staying in natural coordinates.)
eastwest
Φ0+ΔΦ
Φ0+3ΔΦ
Φ0
Φ0+2ΔΦ
south
north
ΔΦ
tn angle, α
angle, α, as well?
Angle in terms of forces
f
k
fV
kVvuk
force Coriolis
forceFriction tan
:figure From
force Coriolis
Friction
friction and force Coriolis of balance is angle
22
Can also be derived from
uuku
kfDt
D
kfuydt
d
kufxdt
du
p
pp
pp
v)()v
(
v)()(
Looks like a great homework problem!
Some basics of the atmosphere
Troposphere: depth ~ 1.0 x 104 m
Troposphere------------------ ~ 2Mountain
Troposphere------------------ ~ 1.6 x 10-3
Earth radius
This scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.
Structure of the atmospheric boundary layer
(Vertical length scales)
Viscous sublayerTransition layerInertial sublayer
Atmospheric Surface Layer (ASL)
Planetary (Convective) Boundary Layer (PBL)
Roughness sublayer
~ 10 1~2 m
~ 10 -1~1 m
~ 10 -3 m
~ 10 2-3 m
Free Atmosphere
Wind profile
Blending height
PBL height
Interfacial sublayer
from Bob Su ( www.itc.nl )
k
{
Let’s think about balance on a different scale
• Going back to our equations of motion in the tangential coordinate system.
Equations of motion in pressure coordinates(plus hydrostatic and equation of state)
pp
p
p
c
JS
y
Tv
x
Tu
t
T
py
v
x
u
fDt
D
0)(
uku
Linking thermal field with wind field.
• The Thermal Wind
Geostrophic wind
xfv
yfu gg
1
,1
Hydrostatic Balance
p
RT
p
Geostrophic wind
p
RT
xfp
v
p
RT
yfp
u
pxfp
v
pyfp
u
gg
gg
1 ,
1
1 ,
1
Take derivative wrt p.
Links horizontal temperature gradientwith vertical wind gradient.
Thermal wind
Tf
R
p
x
T
f
R
p
vp
y
T
f
R
p
up
p
RT
xfp
v
p
RT
yfp
u
pg
gg
gg
kU
ln
,
1 ,
1
p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.
A excursion to the atmosphere.Zonal mean temperature - Jan
north (winter)south (summer)
approximate tropopause
A excursion to the atmosphere.Zonal mean temperature - Jan
north (winter)south (summer)
∂T/∂y ?
A excursion to the atmosphere.Zonal mean temperature - Jan
north (winter)south (summer)
∂T/∂y ?
<0
<0
<0
>0
<0
<0
A excursion to the atmosphere.Zonal mean temperature - Jan
north (winter)south (summer)
∂T/∂y ?
<0
<0
<0
>0
<0
<0
> 0
<0
<0
>0
>0
>0
∂ug/∂p ?
A excursion to the atmosphere.Zonal mean wind - Jan
north (winter)south (summer)
Relation between zonal mean temperature and wind is strong
• This is a good diagnostic – an excellent check of consistency of temperature and winds observations.
• We see the presence of jet streams in the east-west direction, which are persistent on seasonal time scales.
• Is this true in the tropics?
Thermal wind
p
p
p
pU
pU
g
pg
pg
pTdf
Rd
pTdf
Rd
Tf
R
p
00
ln
ln
ln
@
@
kU
kU
kU
Thermal wind
p
pT
f
Rpp
pdTf
Rpp
T
pTdf
Rd
pgg
p
p
pgg
p
p
p
pU
pU
g
00
0
@
@
ln)()(
ln)()(
average andby drepresente
islayer ain T y)(x,any at that assume
ln
0
00
kUU
kUU
kU
Thermal wind
p
p
x
T
f
Rv
p
p
y
T
f
Ru
p
pT
f
Rpp
pT
p
T
pgg
0
0
00
ln
ln
ln)()(
kUU
Thermal wind
)(1
)(1
ln)()(
0
0
00
xfv
yfu
p
pT
f
Rpp
T
T
pgg kUU
?
From Previous LectureThickness
1
2
ln
)(
012
0
p
ppTd
g
RZZ
g
zZ
Z2-Z1 = ZT ≡ Thickness - is proportional to temperature is often used in weather forecasting to determine, for instance, the rain-snow transition. (We will return to this.)
Note link of thermodynamic variables, and similarity to scale heights calculated in idealized atmospheres above.
Similarity of the equations
p
p
pgg
p
p
pdTf
Rpp
pdTg
RZZ
0
1
2
ln)()(
ln
0
012
kUU
There is clearly a relationship between thermal wind and thickness.
Schematic of thermal wind.
from Brad Muller
Thickness of layers related to temperature. Causing a tilt of the pressure surfaces.
Another excursion into the atmosphere.
850 hPa surface 300 hPa surface
XX X
from Brad Muller
Another excursion into the atmosphere.
850 hPa surface 300 hPa surface
X
X X
from Brad Muller
Another excursion into the atmosphere.
850 hPa surface 300 hPa surface
from Brad Muller
Another excursion into the atmosphere.
850 hPa surface 300 hPa surface
from Brad Muller
A summary of ideas.
• In general, these large-scale, middle latitude dynamical features tilt westward with height.
• The way the wind changes direction with altitude is related to the advection of temperature, warming or cooling in the atmosphere below a level.– This is related to the growth and decay of these
disturbances. – Lifting and sinking of geopotential surfaces.
Balance and rotation
• We keep making a big deal of rotation and the balance of the coriolis force and the pressure gradient force, e.g. the geostrophic balance.
• We have all of these equations and scale analysis, and they keep leading use to these notions of geostrophic and hydrostatic balance.
• Let’s examine some of these ideas in a more visual way.
Rotation
• When a fluid is in rotation, the rotation comes to define the flow field; it provides structure.
• That structure aligns with the vector that defines the angular velocity.– So if the flow is quasi-horizontal, then how the
flow aligns in the vertical is strongly influenced by the rotation and its projection in the vertical.
And on the Earth.
• Tropics are more weakly influenced, defined by rotation than middle latitudes.– This also influences the vertical structure of
the dynamical features.
Some things that we learned (1)
• Organizing structure provided by rotation.• Rotation is less important in the tropics, which is
clearly observable in the atmosphere.• There is a theoretical limit on pressure gradients
associated with high pressure systems.– Highs tend to be smeared out; they tend to have
moderate wind speeds.
• There is not such a limit for low pressure systems.– Lows can be very intense; The highest wind speeds
are associated with lows.
Some things that we learned (2)
• There is the possibility of “anomalous” circulations.– Possibility of cyclonic highs– Possibility of anti-cyclonic lows
• We can estimate frictional dissipation based on the angle between lines of constant pressure, or height, and the observed wind.
Some things that learned (3)
• Dynamical features can isolate air and allow the evolution of extraordinary chemical processes.
Where do we need to go next?
• We need to understand the role of vertical motion in large-scale dynamics.
• We need to understand the role of thermodynamic variables in the dynamical balances.
Analysis of Hurricane
Let’s take a stab at a hurricane.(Northern hemisphere)
• What balance might we use?
Let’s take a stab at a hurricane.(Northern hemisphere)
L
Let’s take a stab at a hurricane.(Northern hemisphere)
L
r = radial coordinate
Gradient balance for hurricane
rfV
r
V
nfV
R
V
2
2
scoordinate lcylindricain formally
scoordinate naturalin balancegradient Our
Rewrite gradient wind for hurricane
r
rf
r
M
frVrM
4
well)asequation momentum the
sthat'(remember balance indgradient w
theintoput and conserved assume2
hurricane around momentumAngular
2
3
2
2
Define angular momentum(You’ve seen this before.)
r
T
H
R
z
M
r
H
RT
z
zrz
M
r
2
3
2
3
1
H,height scale with studied,
not have wesystem coordinate ain
1
z wrt toDerivative
Some analysis
then0, is you tellI If
1 2
3
z
Mr
T
H
R
z
M
r
Some analysis
center.at max is Hence, 0
then0, is you tellI If
1 2
3
Tr
Tz
Mr
T
H
R
z
M
r
Some analysis
K10:
10,50,100,7:
1
151
2
3
Tyields
sfmsUkmLkmHScales
r
T
H
R
z
M
r
Near ground we have friction.
-kV
or
kfuydt
d
kufxdt
du
pp
pp
ForceFriction
v)()v
(
v)()(
And hurricanes are observed to maintain themselves!
Latent heat from warm ocean water.
Hurricane Heat Engine
• Hurricanes are maintained by latent heat release from water that is evaporated from the ocean.– ~ 27o C is threshold.
• Bring in thermodynamic equation.
• Hurricanes are an efficient heat engines.