Lecture 8. EPS 5
October 07, 2010
•Air may be forced to move up or down in the atmosphere by mechanical forces (wind
blowing over an obstacle, like a mountain) or by buoyancy forces.
•Air that is forced to move up does work on the atmosphere. If no heat is added or
removed, the work that is done comes exclusively from thermal energy of the molecules
in the air. Air moving up "adiabatically" cools at a rate of –9.8 K/km.
•Air that is forced to move down has work done on it by the atmosphere. If no heat is
added or removed, the work that is done goes exclusively into thermal energy of the
molecules in the air which warms at a rate of 9.8 K/km.
•If the change in atmospheric temperature with altitude is less negative than –9.8
K/km, the atmosphere is stable because an air parcel that starts to move vertically will
be pushed by buoyancy forces back to where it started. The parcel will be colder [more
dense] than the environment [the surrounding atmosphere] above where the parcel
started, and warmer below where it started.
•Heating of the ground by the sun during the day time makes air parcels buoyant. They
rise until their temperature is the same as that of the atmosphere. The "atmospheric
mixed layer" ["Planetary Boundary Layer"] develops in the lowest 1-3 km in a few hours.
•As parcels cool when they rise in the atmosphere, water vapor may condense if the
parcel has sufficient water content. Condensation adds heat to an air parcel, tending to
make the lapse rate less negative than –9.8K/km, and the parcel is less dense/more
buoyant than a dry parcel.
Question: Where does the energy come from for an
air parcel to do this work on the atmosphere?
The concept of atmospheric stability
•The atmosphere is stable if air cannot move up or down spontaneously. A parcel that
is displaced a little distance vertically tends to move back to where it started.
•The atmosphere is unstable if air tends to move up or down spontaneously. A parcel
that is displaced a little distance vertically tends to accelerate in the direction it was
moved.
•The atmosphere is neutral ("neutrally stable") if it neither tends to move back to
where it started nor accelerates.
Forces are balanced on both of these
spheres. But one is stable, and the
other unstable.
The environment determines if the
sphere (or an atmospheric air parcel)
is stable, neutral, or unstable.
The sphere is
neutrally stable
on a flat surface.
Ambient lapse
rates and parcel
temperature
changes.
Schematic diagram
showing stable
(Curve A) and
unstable (Curve B)
temperature profiles
for the ambient
atmosphere,
compared to the
lapse rate followed
by a parcel
subjected to
adiabatic vertical
displacement.
km
Stability and lapse rates in the atmosphere
1
A
0
.5
B
290
295
300
ToK
Heating of the
ground and
“dry convection”:
Diagram of the lapse
rate just before
sunrise (stable) and
in mid-afternoon
(neutral or slightly
unstable). Dry
convection driven by
solar heating stirs the
lower atmosphere,
creating the planetary
boundary layer or
mixed layer during
daytime over land.
DIURNAL CYCLE OF SURFACE HEATING/COOLING:
ventilation of urban pollution
z
Subsidence
inversion
MIDDAY
1 km
G
Mixing
depth
0
NIGHT
MORNING
T
NIGHT
MORNING AFTERNOON
Vapor Pressure
of Water
Psat = A exp [B( 1/273.15 – 1/T)] A=6.11
mbar, B= 5308K. A=water vapor pressure
at 0C.
The pressure of
H2O vapor in
equilibrium with
liquid water.
ClausiusClapeyron
relation. Water
vapor pressure
versus T.
Moist pseudo-adiabatic lapse rate
Air is heated by release of latent heat when
water condenses: T will decline less rapidly
than the dry adiabat
Pressure (Mb)
Temperature (C)
-40
0
40
Ambient T
-9.5
-6.4
-3.0
15 (->35)
600 4.2km -9.3
-5.4
1000
-13
200 11.8km -8.6
Dry Adb. -9.8
-58
-9.8
-9.8
G= -g/(cp +λ Dw/DT )
λ= latent heat of vaporization (J/kg); Dw/DT=change in spec humidity/K
dew pt
dew pt
Atmospheric temperature and
dewpoint for a typical summer day
shows the "planetary boundary
layer" or "atmospheric mixed
layer", that develops as the sun
heats the ground in the daytime.
This graph is drawn from actual
data obtained by Harvard's Forest
and Atmosphere Studies group
during an experiment (code name
"COBRA") over North Dakota in
August, 2000.
What you see…
Puffy little clouds, called fair weather cumulus,
occurring over land on a typical afternoon. The
lapse rate in the mixed layer is approximately
adiabatic, and air parcels heated near the ground
are buoyant. Each little cloud represents the top
of a buoyant plume. (Photograph courtesy
University of Illinois Cloud Catalog).
Convective cloud over Amazonia
3
Z
km
latent heat
release
2
1
Tdew
0
283 293 303
Temperature
K
[Photo: S. Wofsy,
Manaus, Brazil, 1987.]
cloud base
Tdew = Tair
Franconia Notch, NH, 09 Oct 2010
Solving a practical problem with EPS 5: What was the temperature at the summit of
Mount Lafayette (but I forgot my thermometer) ?
Franconia Notch, NH, 09 Oct 2010
summit – 150 m
summit – in the clouds
SUBSIDENCE INVERSION
typically
2 km altitude
Atmospheric lapse rates
Altitude (km)
Suppose the atmosphere has the
structure at the left.
1.
What are the scale heights at the
surface and at 1.5 km?
2.
What is the atmospheric lapse
rate from 0 to 1.5 km? Is the
atmosphere stable, unstable or
neutral?
3.
How high can a dry air parcel rise
that is heated to 310 K at the
ground?
4.
How high will a wet air parcel rise
from the ground, heated to 310K?
Assume condensation starts at
290.2 K, and above that level the
parcel lapse rate is -4.8 K/km.
1.5
0
285.3
300
Temperature (k)
An example problem, worked out.
Scale height = kT/mg = 29.34 x T = 8800 m (T=300) and 8370 m (T=285.3)
Lapse rate = ( 285.3 – 300 ) / 1.5 = -9.8 K/km; neutral
How high will a dry parcel rise: The parcel will cool at the rate of –9.8 K/km. When it
cools to 285.3 K, it will have the same temperature as the environment, and this will be its
equilibrium level. ( 285.3 – 310) / (-9.8) = 2.5 km.
How high will a wet parcel rise:
Solve the problem in two steps: first determine how high the parcel rises at the dry
adiabat to the point where condensation starts.
Then determine how high it has to go, cooling at the wet adiabat, to reach the
atmospheric temperature.
The parcel will cool at the rate of –9.8 K/km up to 290.2 K. That's –19.8 K temperature
change, which it will reach at 2 km.
When it cools to 285.3 K, it will have cooled by an additional 4.8 K, which requires 1 km
more rise.
The wet parcel will move up 3 km.
Lecture 8. EPS 5
October 06, 2010
•Air may be forced to move up or down in the atmosphere by mechanical forces (wind
blowing over an obstacle, like a mountain) or by buoyancy forces.
•Air that is forced to move up does work on the atmosphere. If no heat is added or
removed, the work that is done comes exclusively from thermal energy of the molecules
in the air. Air moving up "adiabatically" cools at a rate of –9.8 K/km.
•Air that is forced to move down has work done on it by the atmosphere. If no heat is
added or removed, the work that is done goes exclusively into thermal energy of the
molecules in the air which warms at a rate of 9.8 K/km.
•If the change in atmospheric temperature with altitude is less negative than –9.8
K/km, the atmosphere is stable because an air parcel that starts to move vertically will
be pushed by buoyancy forces back to where it started. The parcel will be colder [more
dense] than the environment [the surrounding atmosphere] above where the parcel
started, and warmer below where it started.
•Heating of the ground by the sun during the day time makes air parcels buoyant. They
rise until their temperature is the same as that of the atmosphere. The "atmospheric
mixed layer" ["Planetary Boundary Layer"] develops in the lowest 1-3 km in a few hours.
•As parcels cool when they rise in the atmosphere, water vapor may condense if the
parcel has sufficient water content. Condensation adds heat to an air parcel, tending to
make the lapse rate less negative than –9.8K/km, and the parcel is less dense/more
buoyant than a dry parcel.
FORCES BEHIND THE ATMOSPHERIC CIRCULATION
1. The pressure-gradient force – application to the sea breeze.
2. The Coriolis force
We want to explain circulation patterns like these, which take place over
large enough scales that the rotation of the earth has an effect on moving
air parcels:
Sea-breeze effect driven by the pressure-gradient force
Coriolis force: fictitious force due to
Earth’s rotation
…must be taken into account because our frame of reference is the rotating Earth!
latitude λ
distance from axis r = Rcos λ
An observer fixed in space sees a point on Earth at latitude λ moving with a
translational velocity V = (2πRcos λ)/t where t = 24 h
At the latitude of Boston (42o ), V = 1250 km h-1 (fast!)
Define angular velocity ω = 2π/t so that V = ωr
THE CORIOLIS FORCE: fictitious force applied
to moving bodies in rotating frame of reference
Consider an observer at the North Pole (O) shooting a ball at a target T:
From the observer’s perspective (rotating frame of reference), the ball has been
deflected to the right while the target has remained fixed; an observer fixed in
space would see the target move and no deflection of the ball
Coriolis Force applied to meridional motion
1
In the absence of torque, a spinning object (here the ball) conserves its angular
momentum L = mvr where v is its velocity and r is its distance from the rotation axis.
1. Consider a ball at rest at latitude λ1 ; its velocity is v1 = V(λ1 )
2. Throw it to latitude λ2 ; it conserves its angular momentum
L1  L2  mv1r1  mv2 r2  v2  V (1 )
cos 1
 V (1 )  V (2 )
cos 2
3. Therefore the ball acquires an eastward velocity in the rotating frame of
reference as it moves northward; it is deflected to the right.
4. Similar deflection to the right takes place if ball is thrown from λ2 to λ1
Coriolis force applied to longitudinal motion
Equilibrium of forces for a ball at rest on Earth’s surface
Earth is not a perfect
sphere!
= mv2 /r
Equatorial diameter
= 12,756 km
Polar diameter
= 12,714 km
1. For a ball at rest, v = V(λ)
2. Throw the ball eastward; v > V(λ) so the centrifugal force increases; ball is
deflected to the right.
3. Throw the ball westward: v < V(λ) so the centrifugal force decreases; ball
is deflected to the right.
GENERAL FORMULATION OF
CORIOLIS FORCE
• Coriolis force is perpendicular to
direction of motion, to R in northern
hemisphere and to L in southern
Coriolis
hemisphere   2 v sin 
c
Acceleration
Angular velocity of
Earth (2/24h)
Latitude
Speed of moving object
In rotating frame of reference
• Coriolis force is zero at equator, strongest at the poles; sign change—acts
to the right of the velocity in the NH, the left in the SH
• Coriolis force is weak and important only for large scales of motion:
examples of a basketball vs. a missile.
GEOSTROPHIC FLOW: equilibrium between
P-gradient force (p) and Coriolis force (c)
Isobar
• steady
• parallel to isobars
• speed ~
pressure gradient
N hemisphere
example
Geostrophy
For air in motion, not on the equator, not
near the surface
•Coriolis Force ≈ Pressure gradient force
•Air motion is ≈ parallel to isobars
The geostrophic approximation is a simplification of complicated
atmospheric motions. This approximation is applied to synoptic scale
systems and circulations, roughly 1000 km. (It is easiest to think about
measuring the pressure gradient at a constant altitude, although other definitions
are more rigorous. )
Circulation of air around regions of
high and low pressures in the Northern
Hemisphere. Upper panel: A region of
high pressure produces a pressure
force directed away from the high. Air
starting to move in response to this
force is deflected to the right (in the
Northern Hemisphere), giving a
clockwise circulation pattern.
Lower panel: A region of low pressure
produces a pressure force directed
from the outside towards the low. Air
starting to move in response to this
force is also deflected to the right,
rotating counter-clockwise.
Directions of rotation of the wind about
high or low centers are reversed in the
Southern Hemisphere, as explained
earlier in this chapter.
Low
P
add mean
W wind
Low pressure (cyclonic) circulation,
embedded in the Jet Stream
Low
P
The effect of friction
around a high pressure
region is to slow the wind
relative to its geostrophic
velocity. This causes the
pressure force to slightly
exceed the Coriolis force.
The three forces add
together as shown in the
figure. Air parcels
gradually drift from
higher to lower pressure,
in the case shown here,
from the center of a high
pressure region outward.
An analogous flow
(inward) occurs in a lowpressure region.
Air converges near
the surface in low
pressure centers, due
to the modification of
geostrophic flow under
the influence of
friction. Air diverges
from high pressure
centers. At altitude, the
flows are reversed:
divergence and
convergence are
associated with lows
and highs respectively,
closing the circulation
through analogous
processes noted in the
sea breeze example
March 02, 2008
http://www.weather.com/maps/maptype/currentweatherusnational/index_large.html
Goes East Movie: Oct 2010