Download Earth – The Water Planet

PHYS-575/CSI-655
Introduction to Atmospheric Physics and Chemistry
Atmospheric Thermodynamics – Part 2
1. Thermodynamics Review/Tutorial
- Ideal Gas Law
- Heat Capacity
- 1st & 2nd Laws of Thermodynamics
- Adiabatic Processes
- Energy Transport
2.
3.
4.
5.
6.
Hydrostatic Equilibrium
Adiabatic Lapse Rate – DRY
Adiabatic Lapse Rate - WET
Static Stability
SLT and the Atmosphere
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Role of Water in the Atmosphere
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Evaporation and Condensation
Equilibrium
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Dry Adiabatic Lapse Rate
For the Earth:
DALR ~ -7-8 K/km
If we know the temperature of
the atmosphere are any level,
and we know that the heat flux
is zero, i.e. adiabatic, then we
can deduce the temperature at
any other level.
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Role of Water Vapor in Atmospheric Thermodynamics
of the Troposphere
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http://www.auf.asn.u/metimages/lapseprofile.gif
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4. Adiabatic Lapse Rate - Wet
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Water Vapor in the Atmosphere:
The Wet (Moist) Adiabatic Lapse Rate
Γd = -g/Cp = DALR
The Wet Adiabatic Lapse Rate is smaller than the DALR, because the
effective heat capacity of a wet atmosphere is larger than that of a dry
atmosphere. The phase change of water is an effective heat reservoir.
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What is Evaporation?
Evaporation is one type of vaporization that occurs at the
surface of a liquid. Another type of vaporization is boiling, that
instead occurs throughout the entire mass of the liquid.
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Importance of Evaporation
Evaporation is an essential part of the water cycle.
Solar energy drives evaporation of water from
oceans, lakes, moisture in the soil, and other sources
of water.
Evaporation is caused when water is exposed to air
and the liquid molecules turn into water vapor which
rises up and can forms clouds.
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What is Humidity?
Humidity is the amount of water vapor in the air.
Relative humidity is defined as the ratio of the partial
pressure of water vapor to the saturated vapor pressure of water
vapor at a prescribed temperature.
Humidity may also be expressed as specific humidity.
Relative humidity is an important metric used in forecasting
weather. Humidity indicates the likelihood of precipitation, dew,
or fog.
High humidity makes people feel hotter outside in the summer
because it reduces the effectiveness of sweating to cool the body
by reducing the evaporation of perspiration from the skin.
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Saturation Conditions
At saturation, the flux of water molecules
into and out of the atmosphere is equal.
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Saturation
Vapor
Pressure of
Water Vapor
over a Pure
Water Surface
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Moisture Parameters
The amount of water vapor in the atmosphere may be expressed in
a variety of ways, and depending upon the problem under consideration,
some ways of quantifying water are more useful than others.
es = Saturation Partial Pressure
mv
w=
Mass Mixing Ratios
md
Where mv is the mass of water vapor in a given parcel, and md is the
mass of dry air of the same parcel. This is usually expressed as grams
of water per kilogram of dry air. w typically varies from 1 to 20 g/kg.
mv
w
q

mv  md 1  w
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Specific Humidity (typically a few %)
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Moisture Parameters for Saturation
es = Saturation Partial Pressure
mvs
ws =
md
Saturation Mixing Ratio
vs'
es
( p  es )
ws  ' 
/
 d ( RvT ) ( Rd T )
For Earth’s Atmosphere:
ws  0.622
es
e
 0.622 s
p  es
p
ρ’vs is the mass density of water
required to saturate air at a given T.
p = total pressure
Relative Humidity
w
e
RH  100
 100
ws
es
The dew point, Td, is the temperature to which air must be cooled at constant
pressure for it to become saturated with pure water.
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Relative Humidity
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Saturation of Air
Air is Saturated if the abundance of
water vapor (or any condensable) is
at its maximum Vapor Partial
Pressure.
In saturated air, evaporation is
balanced by condensation. If water
vapor is added to saturated air,
droplets begin to condense and fall
out.
Under equilibrium conditions at a
fixed temperature, the maximum
vapor partial pressure of water is
given by its Saturated Vapor
Pressure Curve.
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http://apollo.lsc.vsc.edu/classes/met130/notes/chapter5/graphics/sat_vap_press.free.gif
Relative Humidity is the ratio of the
measured partial pressure of vapor
to that in saturated air, multiplied
by 100.
The relative humidity in clouds is typically
about 102-107%, in other words, the
clouds are Supersaturated.
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Saturation Vapor Pressure:
Clausius-Clapeyron Equation of State
Psv(T) = CL e-Ls/RT
Psv(T) = Saturation vapor pressure
at temperature T
CL = constant (depends upon
condensable)
Ls = Latent Heat of Sublimation
R = Gas constant
Phase Diagram of Water
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Vertical Motion and Condensation
Upward motion leads to cooling, via the FLT. Cooling increases the
relative humidity. When the relative humidity exceeds 100%, then
condensation can occur.
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Adiabatic Motion of Moist Parcel
As a parcel of air moves upwards, it expands and cools. The cooling leads
to an increase in the relative humidity. When the vapor pressure exceeds
the saturation vapor pressure, then condensation can occurs.
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Saturation Profile and Temperature
Amounts of water
necessary for
super-saturation,
and thus condensation.
Is it possible to have snow when the atmospheric temperature is below – 30oC?
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Water/Ice Transition
 Water Triple Point
The saturation vapor pressure of water over ice is higher than that
over liquid water. This leads to small, but measurable change is
the relative humidity.
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Liquid/Ice Transition
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Lifting Condensation Level
The Lifting Condensation Level (LCL) is defined as the level to which an
unsaturated (but moist) parcel of air can be lifted adiabatically before it
becomes saturated with pure water.
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Wet (Moist) Adiabatic Lapse Rate
Γd = -g/Cp = Dry Adiabatic Lapse Rate
In determining the moist adiabatic lapse
rate, we must modify the First Law of
Thermodynamics to include the phase
change energy.
Let μs = mass of liquid water.
dQ = CpdT + gdz
(FLT for a parcel)
dQ = – Lsdμs (Heat added from water
condensation)
Here we assume that the water which condenses drops out of the parcel. Thus
this process is strictly irreversible.
Together this implies that the FLT becomes:
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CpdT + gdz + Lsdμs = 0
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Wet Lapse Rate - continued
CpdT + gdz + Lsdμs = 0
(FLT for a saturated parcel)
The mass of water depends upon the degree of saturation:
μs = Є (es/p) and by the chain rule dμs/μs = des/es – dp/p
des = (des/dT) dT
(1/es) des/dT = Ls/RT2 (Differential form of Clausius-Clapeyron Eqn.)
dp = -gdz/RT
(Hydrostatic Law)
This gives us
dμs/μs = LsdT/RT2 + gdz/RT
Using this equation and the FLT form at the top of this page we get:
(Cp + Ls2μs/RT2) dT + g(1+Lsμs/RT) dz = 0
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Wet Lapse Rate - Continued
Γw = dT/dz = -(g/Cp) ((1+Lsμs/RT) / (1 + Ls2μs/CpRT2))
Note that when μs = 0, this reduces to Γd
The factor ((xx)) is always less or equal to1. So Γd < Γw
Thus, water acts as an
agent to increase the
effective heat capacity
of the atmosphere.
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5. Static Stability
Archimedes Principle:
The upward force (buoyancy)
is equal to the weight of the
displaced air.
The net force on a parcel
is equal to the difference
between weight of the air
in the parcel and the
weight of the displaced air.
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Vertical Stability
dT/dz = -g/Cp = dry adiabatic lapse rate (neutrally stable)
dT/dz < -g/Cp  Unstable
dT/dz > -g/Cp  Stable
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Static Stability
Stable
Unstable
Γd = -g/Cp
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Lifting Condensation Level
The Lifting Condensation Level (LCL) is defined as the level to which an
unsaturated (but moist) parcel of air can be lifted adiabatically before it
becomes saturated with pure water.
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Stability and the Effects of Condensation
Moisture leads to conditional stability in the atmosphere.
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Analogs for Stability
Under stable atmospheric conditions, an air parcel that is displaced in the vertical
direction will return to its original position.
Neutral stability occurs when the air parcel will remain at it’s displaced position
without any additional forces acting on it.
For unstable conditions, an air parcel that is displaced in the vertical will continue
to move in the direction of the displacement.
Conditional instability occurs when a significant displacement of the air parcel
must occur before instability can occur.
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Regions of Convective Instability
Convective instability may occur in only a small portion of
the vertical structure. Temperature inversions therefore
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Atmospheric Waves
http://weathervortex.com/images/sky-ri87.jpg
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Waves in Clouds
http://weathervortex.com/images/sky-ri39.jpg
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Mountain Waves
http://www.siskiyous.edu/shasta/map/mp/bswav.jpg
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Archimede’s Principle
When an object is immersed in water, it feels lighter. In a cylinder filled with
water, the action of inserting a mass in the liquid causes it to displace upward.
In 212 B.C., the Greek scientist Archimedes discovered the following
principle: an object is immersed in a fluid is buoyed up by a force equal to the
weight of the fluid displaced by the object.
This became known as Archimede's principle. The weight of the displaced fluid
can be found mathematically. The fluid displaced has a weight W = mg. The
mass can now be expressed in terms of the density and its volume, m = pV.
Hence, W = pVg.
It is important to note that the buoyant force does not depend on the weight or
shape of the submerged object, only on the weight of the displaced fluid.
Archimede's principle applies to object of all densities. If the density of the
object is greater than that of the fluid, the object will sink. If the density of the
object is equal to that of the fluid, the object will neither sink or float. If the
density of the object is less than that of the fluid, the object will float.
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Atmospheric Oscillations:
Gravity Waves in Stable Air
Consider the force on a parcel of air that has been displaced vertically
by a distance z from its equilibrium altitude. Assume that the air is dry
and that displacements occur sufficiently slow that we can assume that
they are adiabatic. Primed quantities will denote parcel variables.
By Archimedes Principle, the force on the parcel is the buoyancy force
minus the gravitational force. The net force is:
F  (   ' ) g
Acceleration:
2
d
z
'
 2  (   ' ) g
dt
Substituting from IGL:
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d 2z F    ' 
 g
 '  
2
'
dt
   
OR
1 1

d 2z T T '

g
2
1
dt
T'
OR
 T ' T 
d 2z

 g 
2
dt
 T 
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Atmospheric Oscillations - continued
If we assume a linear atmospheric temperature profile with rate of change
with altitude of Г, then the temperature profile may be written (z’ = displacement)
T  T0  z '
The parcel moves adiabatically in the vertical, so its temperature is:
T '  To  d z '
Which gives:
T '  T  d   z '
The equation of motion becomes:
d 2z'
g
'







z
d
dt 2
T
d 2z'
2 '

N
z 0
Which can be written:
2
dt
g
Brunt-Väisälä Frequency: N 2     
d
T
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Atmospheric Oscillations - continued
The equation of motion for the parcel is
Brunt-Väisälä Frequency:
d 2z'
2 '

N
z 0
2
dt
g
N  d   
T
2
If the air is stably stratified, i.e., Гd > Г,
then the parcel will oscillate about its
starting position with simple harmonic
motion.
These are called buoyancy oscillations.
Typical periods are about 15 minutes.
Here Гe = Г in notes
For winds of ~ 20 ms-1, the wavelength
is ~10-20 km.
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Lee Waves Observed from Space
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Mountain Winds
Mountain regions display many interesting weather patterns. One example
is the valley wind which originates on south-facing slopes (north-facing in the
southern hemisphere). When the slopes and the neighboring air are heated
the density of the air decreases, and the air ascends towards the top following
the surface of the slope. At night the wind direction is reversed, and turns into
a down-slope wind. If the valley floor is sloped, the air may move down or up
the valley, as a canyon wind. Winds flowing down the leeward sides of
mountains can be quite powerful:
Examples are the Foehn in the Alps in Europe, the Chinook in the Rocky Mountains,
and the Zonda in the Andes. Examples of other local wind systems are the Mistral
flowing down the Rhone valley into the Mediterranean Sea, the Scirocco, a southerly
wind from Sahara blowing into the Mediterranean sea.
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Mountain Winds and Climate
Hawaii
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Mountain (Lee) Waves
Buoyancy Oscillations:
g
N  d   
T
2
Observed from ground
Lee Waves
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6. The Second Law of Thermodynamics
The Second Law of Thermodynamics states that it is impossible to
completely convert heat energy into mechanical energy. Another way to
put that is to say that the level of entropy (or tendency toward
randomness) in a closed system is always either constant or increasing.
Implications of the Second Law



It is impossible for any process (engine), working in a
cycle, to completely convert surrounding heat to work.
Dissipation will always occur.
Entropy will always increase.
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Second Law of Thermodynamics
and Atmospheric Processes
The Entropy of an isolated system increases when the system
undergoes a spontaneous change.
Entropy is the heat added (or subtracted) to a system divided by
its temperature.
dS = dQ/T
Second Law of
Thermodynamics
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The Carnot Cycle
The First Law of Thermodynamics is a statement about conservation of energy.
The Second Law of Thermodynamics is concerned with the maximum fraction of
a quantity of heat that can be converted into work. There is a theoretical limit
to this conversion that was first demonstrated by Nicholas Carnot.
A cyclic process is a series of operations by which the state of a substance
(called the working substance) changes, but is finally returned to its original
state (in all respects).
If the volume changes during the cycle, then work is done (dW = PdV).
The net heat that is absorbed by the working substance is equal to the work
done in the cycle. If during one cycle a quantity of heat Q1 is absorbed and
a quantity Q2 is rejected, then the net work done is Q1 – Q2.
The efficiency is:  
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Q1  Q2 Work _ done _ by _ engine

Q1
Heat _ abosrbed
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Carnot’s Ideal Heat Engine
T1>T2
1. AB Adiabatic Compression
Work done on substance
2. B C Isothermal Expansion
Work done on environment
3. C D Adiabatic Expansion
Work done on environment
4. D A Isothermal Compression
Work done on substance
Incremental work done: dW = PdV
So the area enclosed on the
P-V diagram is the total Work.
Only by transferring heat from a
hot to a cold body can work be
done in a cyclic process.
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Isotherms and Adiabats
Isothermal Process: T = constant, dT = 0
P-V diagram
Adiabatic: dQ = 0
Isentropic: dS = 0
T-S diagram
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Saturation Vapor Pressure:
The Clausius-Clapeyron Equation
By application of the ideas of a
cyclic process changing water
from a liquid to a gas, we can
derive the differential form of
the Clausius-Clapeyron equation:
Lv
Lv M w
1 des


2
es dT RvT
1000 R*T 2
In its integrated form:
es (T )  Ce Lv / RT
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Water Vapor and the Carnot Cycle
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Ambient Pressure and Boiling Point
Water boils at a temperature TB
such that the water vapor pressure
at that temperature is equal to the
ambient air pressure, i.e.,
es(TB) = Patmos
The change in boiling point, TB,
as a function of temperature is
given by a form of the ClausiusClapeyon equation:
TB
TB ( 2  1 )

patmos
Lv
Because α2 > α1, TB increases with increasing patmos. Thus if the ambient
atmospheric pressure is less than sea level, the TB will be lower.
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Generalized Statement of the Second Law of
Thermodynamics
Q
S 
T
If the system is reversible, no dissipation occurs.
For a reversible transformation there is no change in the entropy of the
universe (system + surroundings).
The entropy of the universe increases as a result of irreversible transformations.
“The Second Law of Thermodynamics cannot be proved. It is believed because
it leads to deductions that are in accord with observations and experience.”
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Questions for Discussion
1.
2.
3.
4.
5.
6.
How does one define energy, apart from what
it does or is capable of doing?
What is Thermodynamics?
Why is Thermodynamics relevant to
atmospheric science?
Why is Thermodynamics a good starting point
for discussing atmospheric science?
What causes energy transport?
Is it possible to perform work with an
isothermal system?
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Questions for Discussion
7.
8.
9.
10.
11.
12.
13.
Why is entropy an important concept in
atmospheric physics?
Does an atmospheric “parcel” really exist?
Is the atmosphere in thermal equilibrium?
Is the atmosphere in dynamical equilibrium?
What is the difference between steady state and
equilibrium?
In what ways are the Earth’s atmosphere like a
heat engine?
Why is it impossible to prove the Second Law of
Thermodynamics?
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Pseudoadiabatic
Chart
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Normand’s Rule
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