Chapter 3: Atmospheric Thermodynamics
Objectives:
1. Demonstrate quantities used by Atmospheric Scientists to relate
properties of air parcels aloft with those at the surface.
2. Develop increasingly more accurate models for the temperature,
pressure, and density of air in the atmosphere.
3. Stability of air parcels.
Should be very familiar with these topics as we cover this chapter:
a. Ideal gas equation applied to dry and moist air. SEE THIS AWESOME SIMULATION!
b. Virtual temperature.
c. Potential temperature.
d. Hydrostatic equation.
e. Increasingly detailed description of the temperature and pressure distribution in the atmosphere.
f. SkewT logP diagrams.
z. Relative humidity, absolute humidity.
g. Dew point temperature.
h. Wet bulb temperature.
i. Equivalent potential temperature.
j. Latent heat release and absorption in condensation and evaporation of water.
k. Stability of air parcels.
l. Indices on soundings.m. Lapse rate, adiabatic lapse rate, deviations from adiabatic lapse rate,
pseudoadiabats.
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Equation of State for an Ideal Gas: Air
1.
2.
3.
V = volume
molecule size is ignorable.
molecules don’t interact (attract or repel each
other).
molecular collisions are like hard point like
spheres.
Most primitive, intuitive form of the
I.G.L. (ideal gas law):
PV = NkT
P = pressure
N = # molecules
T = absolute temperature (Kelvin)
k = Boltzmann’s constant = 1.38 x 10-23 Joules / (molecule K)
Now we manipulate to find a satisfying form of the I.G.L for analysis:
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Various Equivalent Forms of the I.G.L.
Note the useful bottom line form P=RT:
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We will use this most often.
Partial Pressure and Ideal Gas Mixtures
EACH GAS SEPARATELY OBEYS THE IDEAL GAS LAW.
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Applications of Dalton’s Law of Partial Pressures…
What is the total pressure in the room?
What is the partial pressure due to nitrogen molecules N2?
What is the partial pressure due to oxygen molecules, O2?
What is the partial pressure due to carbon dioxide molecules, CO2?
Wait a minute… how can it be that these molecules apply
pressure according to their number concentration? They don’t
all have the same mass… What is going on?
The fine print from Wikipedia…
Dalton's law is not exactly followed by real gases. Those deviations are considerably large at high pressures. In such conditions, the volume
occupied by the molecules can become significant compared to the free space between them. Moreover, the short average distances between
molecules raises the intensity of intermolecular forces between gas molecules enough to substantially change the pressure exerted by them.
Neither of those effects are considered by the ideal gas model.
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Applications of Dalton’s Law of Partial Pressures…
What is the total pressure in the room? 860 mb on 9/9/2010.
What is the partial pressure due to nitrogen molecules N2?
860 mb * 0.78 = 670 mb. Air is composed of 78% N2 molecules.
What is the partial pressure due to oxygen molecules, O2?
860 mb * 0.21 = 180 mb.
What is the partial pressure due to carbon dioxide molecules, CO2?
860 mb * 0.000385 = 0.34 mb.
For 10 mb water vapor partial pressure, air is about 1% water vapor.
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Kinetic Theory of Pressure (Wikipedia…)
Box of sides L
m vx
Nature is fair …
On average, molecules share
the burden of random kinetic
energy, also known as heat.
K.E.=mv2/2. On average,
molecules with smaller m
move faster than large m
molecules.
Pressure in the kinetic theory of gases …
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Special Case: Partial Pressure of Water Vapor, e
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Virtual Temperature Tv.
dry air
P = PD
Total pressure=
partial pressure
due to dry air.
same for both
T = temperature
P = pressure
V = volume
N = # molecules
moist air
P = PD+e
Total pressure=
partial pressure
due to dry air + water vapor.
dry air
>
moist air
PRT
TWEAK …
Raise the temperature of the dry air on the left to lower its
density so that it is the same as the density of the moist air on
the right. We have to let some of the molecules out of the box.
This raised temperature is the virtual temperature by definition. It
is a useful construct because the I.G.L. for dry or moist air is
written P  = RD Tv .
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Virtual Temperature Tv Calculation.
box 1
dry air
P = PD
Total pressure=
partial pressure
due to dry air.
Tv
same for both
T = temperature
P = pressure
V = volume
PRT
box 2
moist air
P = PD+e
T
Total pressure=
partial pressure
due to dry air + water vapor.
Crank up the temperature of box 1, keeping pressure and volume constant (let
some dry air molecules leak out), until the mass (density) of box 1 is the same as
that of box 2.
From the I.G.L.
General Note:
Tv>T.
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Virtual Temperature Example
Let
e=10 mb
P=1000 mb
T=280 K
Remember
=0.622
Then Tv ≈ T [1+e(1-)/P] = T(1+0.0038) ≈ 281 K
(binomial expansion was used to show an equivalent form)
This gives us a rough idea of the temperature increase needed
to make dry air have the same density as the moist air
described above.
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Potential Temperature
Adiabatic compressional warming (and cooling by expansion) lends itself to
predicting large-scale weather patterns, because air motions in large weather
systems are, for all practical purposes, generally adiabatic in nature.
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sustainable
only with
diabatic
heating!
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CAPE (J/kg): 0-1000 (small) 1000-2500 (moderate)
2500-4000 (large) > 4000 (extreme).
LCL: Lifting condensation level.
LFC: Level of free convection.
EL: Equilibrium level.
CAPE: Convective available potential energy.
CIN: Convective inhibition.
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CIN (J/kg): 0 to -25 (weak) -25 to -50 (moderate)
- 50 to -100 (strong convective inhibition)
Solar heating, surface convergence
promote parcels to the LFC: Must
pass above the inversion in the CIN
area.
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Winds on Skew T Log P Charts
wind is 15 knots
coming from the northeast.
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Moist Adiabats: Trajectories of Saturated Parcels.
Pseudoadiabats because products of condensation may fall
out of the parcel as precipitation. Trace a few of them below.
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The Saturated Adiabatic Lapse Rate
Temperature Changes and Stability Inside Clouds
from
http://geog-www.sbs.ohio-state.edu/courses/G230/hobgood/ASP230Lecture17.ppt
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Temperature Changes Inside Clouds
Two processes occur simultaneously inside clouds that affect the
temperature.
(1) Rising air expands, does work and cools;
(2) Condensation releases latent energy which is then stored
as internal energy and warms the air inside the cloud.
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Temperature Changes Inside Clouds (Cont.)
Normally, the cooling due to the work of expansion is greater than
the warming associated with the release of latent energy and its
conversion to internal energy.
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Temperature Changes Inside Clouds (Cont.)
Thus, as air rises inside a cloud it still gets colder, but it does so at
a slower rate than the Dry Adiabatic Lapse rate.
The rate at which rising air inside a cloud cools is called the
Saturated Adiabatic Lapse Rate (SALR).
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The Saturated Adiabatic Lapse Rate (SALR)
The derivation of the equation for the SALR begins with a form of
the First Law of Thermodynamics
dq = cpdT - αdp
What would we do here to get the dry adiabatic lapse rate?
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The Saturted Adiabatic Lapse Rate (Cont.)
In this case the energy gained, dq, is equal to the latent energy
released when water vapor condenses inside the cloud.
dq = -Lvdws
where
Lv is the latent heat of vaporization, and
dws is the change of specific humidity of the air parcel when water
vapor condenses
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The SALR (Cont.)
Substitute for dq in the First Law of thermodynamics to get
-Lvdws = cpdT – αdp
Add –cpdT + Lvdws to both sides to get
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The SALR (Cont.)
-cpdT = -αdp + Lvdws
Divide by cpdz to get
-cpdT = -αdp + Lvdws
cpdz
cpdz cpdz
Since α = 1/ρ we can write this as
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The SALR (Cont.)
-dT = - 1 dp + Lvdws
dz cpρ dz cpdz
From the hydrostatic approximation
-1 dp = g
ρ dz
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The SALR (Cont.)
Substitution results in
-dT =
dz
SALR =
g
cp
+ Lvdws
cp dz
cooling
due to
work of
expansion
Γs = Γdry + Lv dws
cp dz
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Γdry
+
= Γs
warming due to
latent energy
released during
condensation
The SALR (Cont.)
The saturated adiabatic lapse rate is always less than the dry
adiabatic lapse rate because the cooling caused by adiabatic
expansion is partially offset by the release of latent energy
during condensation.
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The SALR (Cont.)
The saturated adiabatic lapse rate is quite variable!!
The magnitude of the SALR is determined by the amount of
water vapor that condenses.
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The SALR (Cont.)
When warm moist air rises in a cloud, more water vapor condenses
and the SALR is smaller.
When cooler, drier air rises inside a cloud, less water vapor
condenses and the SALR is larger.
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SALR = 0.5°C/ 100 m
SALR =
0.9°C/100m
cooler drier air
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more water
vapor
condenses
less water
vapor
condenses
warmer moister air
Lifting
Condensation
Level (LCL)
Saturated air rises
inside the cloud and
the release of latent
energy during
condensation causes
it to cool at the SALR
Unsaturated air rises and
cools at the DALR
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Stability Cases for Clouds
(1) When the environmental lapse rate (ELR) is greater than the SALR,
then the air inside the cloud is unstable. Unstable air moves
vertically and we tend to get tall, vertical clouds like cumulus and
cumulonimbus.
z
Tlifted > Tenv
Tenv
O
SALR
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ELR
T(z)
Stability Cases for Clouds (Cont.)
cumumlonimbus
cumulus
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Stability Cases for Clouds (Cont.)
(2) When the ELR is equal to the SALR, then the air inside the cloud is
neutral.
(3) When the ELR is less than the SALR, then the air inside the cloud is
stable.
z
Tlifted < Tenv
Tlifted
O
ELR
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SALR
T(z)
Stability Cases for Clouds (Cont.)
There is much less vertical motion when air is neutral or stable.
Thus, when air inside the clouds is neutral or stable, the clouds
tend to have a flat, layered appearance. These types of
layered clouds are called stratus clouds.
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Stability Cases for Clouds (Cont.)
stratus
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Stability Cases for Clouds (Cont.)
There is a special stability case that occurs when the
Environmental Lapse Rate is between the Dry Adiabatic Lapse
Rate and the Saturated Adiabatic Lapse Rate.
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Stability Cases for Clouds (Cont.)
For example, what if
DALR = 1.00°C/100 m
ELR = 0.75°C/100 m
SALR = 0.50°C/100 m
If the air is unsaturated ELR < DALR and the air is stable, but if the
air is saturated, then ELR > SALR and the air is unstable.
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Stability Cases for Clouds (Cont.)
This special case is called conditionally unstable, because the air
must be lifted until it becomes saturated in order for it to become
unstable
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DALR = 1°C/100 m
ELR = 0.75°C/100 m
SALR = 0.5°C/100 m
2000 m
T = 14°C, Td = 14°C
Tenv = 13°C
Air is unstable
1600 m
LCL =
800 m
T = 16°C, Td =
16°C
T = 20C°, Td = 20°C
Tenv = 16°C
Air is neutral
Tenv = 22°C
Air is stable
0000 m
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T = 28°C, Td = 20°C
Tenv = 28°C
Stability Cases in Clouds (Cont.)
When the atmosphere is conditionally unstable it can lead to the
rapid development of thunderstorms when a cold front or other
weather feature lifts warm moist air in the spring.
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Stability Cases in Clouds (Cont.)
The air is stable as long as it isn’t lifted high enough, but if it is lifted
until the parcel is warmer than the environment, then the air
instantaneously becomes unstable and starts rising on its own.
Then thunderstorms can form rapidly.
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The Effect of Topography on Precipitation Patterns
Precipitation patterns in mountainous regions tend to be closely
related to the prevailing wind direction.
Much higher precipitation amounts fall on the side of the mountains
where the air is rising and it is much drier on the side where the
air is sinking.
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The Effect of Topography on Precipitation Patterns
(Cont.)
The process where air is forced to rise up the side of a mountain is
sometimes called orographic lifting.
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Leeward side
Windward side
wind direction
T = 6°C, Td = 6°C
3000 m
Air sinks and warms at
the DALR = 1°C/100 m
Air rises and cools at
SALR = 0.5°C/ 100 m
T = 16°C, Td = 16°C
LCL
Air is warm and dry
1000 m
500 m
T = 26°C, Td = 16°C
0 meters
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Air rises and
cools at
DALR
T = 31°C, Td = 6°C
The Effect of Topography on Precipitation Patterns
(Cont.)
Rising motion causes clouds and precipitation on the windward
side of the mountain range.
Sinking motion causes warm, dry conditions on the leeward side of
the mountain.
The dry area on the leeward side of the mountain is called the rain
shadow.
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The Effect of Topography on Precipitation Patterns
(Cont.)
Since the prevailing wind direction in the middle latitudes is from
the west, the western sides of the mountains along the west
coast of the U.S. are the rainy sides and the rain shadows occur
along the eastern slopes of the mountains.
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Find the LCL for a surface parcel: find Tw, the wet bulb temperature.
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Evaporative Cooler: Swamp Cooler.
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NASA satellite image (MODIS imager on board the Terra satellite)
of a wave cloud forming off of Amsterdam Island in the far
southern Indian Ocean. Image taken on December 19, 2005.
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