Download Thermodynamics - Atmosphere Physics

PHYS-575/CSI-655
Introduction to Atmospheric Physics and Chemistry
Lecture Notes #3 – Part 1: Thermodynamics
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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What is Thermodynamics?
Thermodynamics is the study of heat and its transformation from a
macroscopic point of view.
"Department of Entropy"
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"Now, in the second law of thermodynamics..."
2
1. Thermodynamics Tutorial
Thermodynamics is the study of heat and its transformation to and
from other sources of energy, from a macroscopic point of view.
 Statistical Mechanics connects thermodynamics to the microscopic
world through the statistical description of an ensemble of atoms or
molecules that constitute a macroscopic system.
 The transfer of heat, in turn, is driven by differences in temperature
or potential differences associated with chemical reactions.
In the interest of crafting a brief tutorial for applications to the atmosphere, I have
glossed over some of the finer (but yet important) points of thermodynamics.
For more complete treatment:
General: Fundamentals of Statistical and Thermal Physics (McGraw-Hill
Series in Fundamentals of Physics) by Frederick Reif, 1965.
Atmospheric: Atmospheric Thermodynamics, by C.F. Bohren and
B.A. Albrecht, Oxford University Press, Oxford, 1998.
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Defining Temperature
Temperature is a measure of the mean kinetic energy of gas molecules.
1 2
3
mv  kT
2
2
The temperature of an ideal monatomic gas is related to the average kinetic
energy of its atoms as they move. In this animation, the size of helium atoms
relative to their spacing is shown to scale under 1950 atmospheres of pressure.
These room-temperature atoms have a certain, average speed (slowed down
here two trillion fold).
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Measure of Temperature
Temperature of a measure of the mean kinetic energy of gas molecules.
kT
c 
m
2
2 ( v / c )2
f (v)dv  v e
dv

1 2
1
m 2
3
2
mv  m v   v f v dv  kT
2
2
20
2
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Temperature Scales
Temperature is a measure of the random kinetic energy of atoms and/or molecules




The Fahrenheit Scale:
•
we are most familiar with this one
•
water freezes at 32 degrees F.
•
water boils at 212 degrees F.
•
when we cool to the absolute lowest
temperature we reach -459 degrees
(this is referred to as Absolute Zero)
The Celsius Scale:
•
water freezes at 0 degrees C.
•
water boils at 100 degrees C.
•
Absolute Zero is at -273 degrees C.
The Kelvin Scale
•
Absolute Zero is 0 K
•
a temperature change of 1 degree K is
the same as a temperature change of 1
degree C.
•
water freezes (or melts) at 273 K
•
water boils at 373 K
<1/2 mv2> = 3/2 k T
The Kelvin is scale is the more useful scale
for our course since it refers to Absolute
Zero in a direct way.
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The Ideal Gas Law
Laboratory experiments show that the pressure, volume, and temperature of
any material may be related by an Equation of State (EOS). These variables
are known as State Variables. All atmospheric gasses follow an equation of
state known as the Ideal Gas Law (IGL) to a very high degree of accuracy.
We assume the IGL to be exact in atmospheric science.
The Ideal Gas Law may be written:
pV = mRT
where
p = pressure
V = volume
m = mass
T = temperature (absolute Kelvin; K = oC + 273.15)
R = gas constant
The gas constant R depends upon the particular gas under consideration.
Since m/V = ρ (density of the gas), the IGL may be written:
p = ρRT
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Ideal Gas Law (IGL) - continued
We can also define α = 1/ρ, known as the specific volume,
to write the IGL as:
pα = RT
Boyle’s Law: For fixed temperature, the pressure of a gas is inversely
proportional to its volume, i.e., P ~ 1/V.
Additional forms of the Ideal Gas Law:
A mole (gram-molecular weight) of any substance is the molecular weight M
of the substance expressed in grams. For example, the molecular weight of
water is 18.015 gm, so 1 mole of water is 18.015 gm of water. The number
of moles (N) in a mass m (in grams) of a substance is given by:
N = m/M
The number of molecules in 1 mole of any substance is a universal
constant called Avogadro’s number, NA.
NA = 6.022 x 1023 molecules per mole.
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Ideal Gas Law (IGL) – continued again
The Ideal Gas Law for 1 mole of any gas can be written:
pV = R*T
Where R* is the universal gas constant = 8.3145 J K-1 mol-1.
So for N moles of any gas, the IGL will be:
pV = NR*T
The gas constant for 1 molecule of any gas is also a universal constant
known as Boltzmann’s constant, k = 1.38 x 10-23 J K-1 molecule-1
So for a gas with n gas molecules per unit volume V, the IGL is then
p = nkT
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Ideal Gas Law - Summary
Ideal Gas Law:
P = nkT = ρRT= RT/α
PV = R*T/M
P = pressure
m = mass per gas particle
n = number density of gas particles
ρ = mn = mass density
α = 1/ρ = specific volume
V = volume of one mole of gas
k = Boltzmann’s constant
R = gas constant (gas specific) = R*/Μ
M = molar mass
R* = universal gas constant
T = temperature
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Heat Capacity
The Heat Capacity of a material is a measure of its ability to absorb and retain heat.
More precisely, the Heat Capacity is the energy (dQ) required to increase the
temperature of a unit volume of any substance from T to T+dT (in Kelvin)
 dQ 
C 

 dT 
The Heat Capacity depends upon the nature of the material and its temperature.
The Heat Capacity also depends upon exactly how the energy is added. If the
heat is added to a gas at constant volume the heat capacity is lower than if the
heat is added at constant pressure. The reason is that heat performs work if
the volume changes.
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Heat Capacity - continued
Heat capacity is related to the ability of a substance to store energy.
Energy can be stored in a variety of ways. For a gas, the most obvious
way to store energy is in random kinetic energy of the gas molecules.
1 2
3
mv  kT
2
2
The 1/2mv2 is the kinetic energy of a molecule of mass m moving with a
velocity v. There is ½ kT of energy “per degree of freedom” of the molecule.
For a molecule moving in 3-dimensions, there are 3 degrees of freedom
and thus the average kinetic energy is stored as 3/2kT.
If there are other ways for a molecule to store energy, then the heat capacity
will be higher.
Thus the Heat Capacity depends upon the phase of the substance.
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Heat Capacity of Water
The Heat Capacity of water makes
an excellent example.
When frozen, water molecules do not
have translational kinetic energy and
thus its heat capacity is low. Molecules
can only vibrate.
Thawing requires heat and thus is
a portion of its heat capacity.
Upon thawing, water molecules can
have kinetic energy of translation and
the heat capacity increases with
temperature.
Evaporation requires heat and thus
increases the heat capacity.
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http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Thermal/gifs/HeatCapacity02.gif
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Heat Capacity - continued
As can be seen with water, the Heat Capacity is a function of only temperature.
Thus we define the Internal Energy, U, of a unit volume of material to be the
measure of the amount of thermal energy stored in the material.
The Internal Energy thus depends only upon temperature.
U = ρCvT
For a gas, the distribution of speeds is a
strong function of temperature.
So the internal energy increases as the
temperature increase.
If you add heat (dq) to a gas, you can cause the internal energy (U) to
increase and/or cause the gas to expand and do work on its environment.
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Specific Heats
Suppose a small quantity of heat dq is added to a unit mass of material and
this causes the material to rise in temperature from T to T+dT.
Then
dq
is the specific heat of the material.
dT
If the volume of the material is kept constant, then the specific heat
at constant volume Cv is defined as:
 dq 
Cv  

dT

v const
However, if the volume of the material is kept constant, then dq = du
(heat changes internal energy and does no work on the environment)
and:
 du 
Cv  

dT

v const
For an ideal gas u depends only upon temperature (T),
so Cv depends only upon T.
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Specific Heats - continued
We can also define a specific heat at constant pressure Cp as:
 dq 
cp  

 dT  p const
But when heat is added to a parcel of gas at constant
pressure, some energy can be used in expanding the gas.
So more heat must be added to a given mass of material
at constant pressure to raise it to a given temperature
than if the material was kept at constant volume.
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The 3 Laws of Thermodynamics
 First
Law: You can’t win.
 Second Law: You can’t break even.
 Third Law: You can’t get out of the game.
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The Three Laws of Thermodynamics
1) Conservation of Energy: Energy is neither created
nor destroyed, it is merely converted from one form
to another.
2) The Entropy of an isolated system increases when a
system undergoes a spontaneous change.
3) The Entropy of all substances approaches zero as
the temperature (in Kelvin) approaches zero.
All substances have zero energy at absolute zero.
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The First Law of Thermodynamics
Conservation of Energy: Energy is neither created
nor destroyed, it is only changed from one form to another.
What is Energy?
Forms of Energy:
-- Gravitational Potential
-- Kinetic Energy
-- Chemical Energy
-- Electromagnetic Energy
-- Rest mass energy
For any system (e.g. a specific collection of matter), the change
in energy of the system is equal to the energy transferred by work
plus the energy transferred by heat.
Heat is the transfer of energy to or from a system associated with
a temperature difference.
Work is the transfer of energy to a system by the application of a force.
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The first law of thermodynamics is the
application of the conservation of energy
principle to thermodynamic processes:
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/firlaw.html
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The First Law and Work
http://www.grc.nasa.gov/WWW/K-12/airplane/thermo1.html
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http://www.fas.org/irp/imint/docs/rst/Sect14/stability2.jpg
22
Useful Forms of the
First Law of Thermodynamics
dU  dQ  dW
dU  dQ  PdV
dU  C p dT  PdV
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Second Law of Thermodynamics
The Entropy of an isolated system increases when the system
undergoes a spontaneous change.
Entropy is the heat added (or subtracted), ΔQ, to a system divided
by its temperature in Kelvin (T). It is a measure of the disorder
of a system; a measure of the unavailability of a system’s energy to
do work; a measure of the disorder of the molecules in a system;
a measure of the number of possible states of a system.
Q
S 
T
dQ is the heat absorbed in an isothermal and reversible process.
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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 (e.g., engine), working in
a cycle, to completely convert surrounding heat to work.
 Dissipation will always occur.
 Entropy will always increase.
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Work and Heat Dissipation
No matter how efficient the system (engine) is, dissipation will always occur.
This usually appears as heat released from the system to its surroundings
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Work and Heat
http://physics.uoregon.edu/~courses/dlivelyb/ph161/heat_engine_schem.gif
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Work and Efficiency
The Second Law of Thermodynamics states that it is impossible for
any heat engine to be 100 % efficient:
No process is possible which results in the extraction of an
amount of heat from a reservoir and its conversion to an equal
amount of mechanical work.
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Efficiency of Automobiles
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The Second Law and Heat Dissipation
for a Automobile
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The Atmosphere as an Engine with
Associated Dissipation
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Atmospheric Circulation acts an Engine transferring
heat from a hot region to a cold region.
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Energy Flow in the Biosphere as an Engine with
Dissipation
Visible light contains most energy from the sun (per wavelength
interval) and overlaps the region where the atmosphere is most
transparent, and also is the region where most photosynthesis
occurs in the biosphere.
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Usable Energy
http://trc.ucdavis.edu/biosci10v/bis10v/week2/2webimages/figure-06-03b.jpg
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The Third Law of Thermodynamics
The Entropy of all substances approaches zero
as the temperature (in Kelvin) approaches zero.
All substances have zero energy at absolute zero.
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The Use of
Thermodynamic
Diagrams
A pair of variables:
(P, V) or (P,T) or (V,T) or…
denote a state of the system.
A P-V diagram shows the possible
states that the system can have.
dW = PdV = Force x Displacement
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Parcel Concepts
Below approximately 100 km altitude,
air is relatively well mixed.
Virtually all mixing is accomplished
by the exchange of air “parcels”
which have horizontal dimensions
ranging from mm to the scale of the
Earth.
An air “parcel” of infinitesimal
dimensions is assumed to be:
(1)
(2)
Thermally insulated from the
environment (no energy exchange)
Moving slowly so that kinetic energy
of motion is much smaller than it’s
total energy.
In reality, both of these assumptions are violated to some extent. But for small
displacements over small time intervals they can be excellent approximations.
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Thermodynamic Descriptions of the Atmosphere
During any atmospheric process, the state of a parcel of
atmospheric gas (P, V, T, S, etc) will change. The Laws of
Thermodynamics determine exactly how these changes can
occur. Phase Diagrams describe these changes in the state
variables describing the gas.
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Thermodynamic Descriptions of the Atmosphere
During any atmospheric process, the state of a parcel of
atmospheric gas (P, V, T, S, etc) will change.
Any pair of variables can be used to describe the state of the
system: (P,V) or (T, S) or (P, S), etc.
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Example: Adiabatic Process - No Energy In/Out
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Parcel Concepts: Applications of the laws of
thermodynamics to air parcels
First Law of Thermodynamics:
dQ = dU + PdV
Internal Energy:
dU = CpdT
Second Law of Thermodynamics: dS = dQ/T
Adiabatic means dQ = 0.
dQ implies dS = 0.
Thus an adiabatic process is also an isentropic process.
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Energy Transport
There are three primary ways that energy is
transported in planetary atmospheres.
(1) Conduction: is the transfer
of energy by collisions
between particles (generally
atoms or molecules). Also
known as diffusive transport
of energy.
(2) Convection: is the motion of
a fluid caused by density
gradients which are a result of
temperature differences.
(3) Radiation: is the transport of
energy by photons.
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Diffusion of Mass and Heat
Diffusion can be driven by concentration gradients, temperature gradients,
and pressure gradients. When diffusion is produced by temperature gradients
this is known as thermal conduction and leads to the transfer of heat.
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Thermal Conduction is the transfer of heat by collisions
between particles
Q = heat flux (erg cm-2 s-1)
dT/dz = temperature gradient in z direction
κT = thermal conductivity is a measure of a material’s physical ability to
conduct heat.
Fick’s First Law of Diffusion
dT
Q   T
dz
The rate of change of energy per unit volume is given by:
Fick’s Second Law of Diffusion
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dU dQ

dt
dz
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Thermal Conduction is the transfer of heat by collisions
between particles
U = internal energy = ρCPT, where
ρ = mass density
CP = heat capacity at constant
pressure (it can also occur at
constant v)
T = temperature
dT
 2T
C p
  T 2
dt
z
Or
T
 2T
  D 2
t
z
Where κD = thermal diffusivity
= κT/ρCP
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Convection
Convection is the motion of a fluid caused by density
gradients which result from temperature differences.
Examples:
Boiling water
Cloud formation
Plate tectonics
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Convection in the Atmosphere
Convection is the motion of a fluid caused by density gradients
which result from temperature differences.
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The Atmospheric General Circulation
is a Manifestation of Convective Currents
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Quantifying Convective Energy Transport
The Convective Energy Flux can be quantified analogous to
the thermal conduction flux, if the thermal diffusion
coefficient is replaced by an Eddy Diffusion Coefficient, Ke.
First Law of Diffusion
dT
Q   e
dz
The rate of change of energy per unit volume is given
by:
T
 2T
Second Law of Diffusion t   e z 2
Ke = Eddy Diffusion Coefficient
The key problem in convection and mixing is the choice of Ke. It is usually
determined by observations of tracer motions in the atmosphere.
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Energy Transport by Radiation
c = 2.998 x 108 ms-1 speed of electromagnetic radiation
λ = wavelength (wavenumber = k = 1/λ)
ν = frequency, such that:
h = Planck’s Constant
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c   

k
Energy:
E  h
50
Radiation: Emission
Thermal Blackbody Emission
The Spectrum of Solar Radiation
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The Sun is a near-blackbody at 5770 K
51
Radiation: Absorption and Emission by Matter
Line emission and absorption
Line wavelengths correspond to energy changes in absorbing/emitting atoms.
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Atomic Structure: Photon Absorption & Emission
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Spectroscopy
Each element (and molecule) has a unique spectroscopic signature.
This is due to their unique structure and energy level distributions.
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Atmospheric Absorption by Molecules
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The Greenhouse Effect: A Consequence of
Radiation Absorption and Emission by
Greeenhouse Gases in the Atmosphere
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The Greenhouse Effect and Global Warming
Sunlight brings energy into the climate system;
most of it is absorbed by the oceans and land.
THE GREENHOUSE EFFECT:
Heat (infrared energy) radiates outward from
the warmed surface of the Earth.
Some of the infrared energy is absorbed by
greenhouse gases in the atmosphere, which
emits the energy in all directions.
Some of this infrared energy further warms the Earth.
GLOBAL WARMING:
Increasing concentrations of CO2 and other "greenhouse" gases trap more
infrared energy in the atmosphere than occurs naturally. The additional heat
further warms the atmosphere and Earth’s surface.
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Vertical Structure of the Earth’s Atmosphere:
Illustration of Heat Transport Mechanisms
Conduction: Solar
extreme ultraviolet (EUV)
photons absorbed in the
upper atmosphere deposit
energy which is conducted
downwards.
Radiation: Stratospheric
ozone (O3) absorb solar
ultraviolet photons which
cause local heating.
Convection: The steep
vertical temperature
gradient produces
unstable air parcels.
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Applications of Thermodynamics to
Atmospheric Processes








What is a Storm?
Parcel Concepts
Hydrostatic Equilibrium
Vertical Temperature Profile
Adiabatic Lapse Rate
Dry vs. Wet Atmosphere
Static Stability
The Second Law of Thermodynamics and the
Atmosphere
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What is a Storm?
1. Do all storms have the
same cause?
2. Do all storms have the
same ending?
3. Are there aspects that all
storms have in common?
http://www.noaanews.noaa.gov/stories2005/images/ivan091504-1515zb.jpg
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Storm Definitions:



behave violently, as if in state of a great anger
take by force; "Storm the fort"
rain, hail, or snow hard and be very windy, often with
thunder or lightning; "If it storms, we'll need shelter"
 a violent weather condition with winds 64-72 knots (11
on the Beaufort scale) and precipitation and thunder and
lightning
 blow hard; "It was storming all night"
 a violent commotion or disturbance; "the storms that had
characterized their relationship had died away"; "it was
only a tempest in a teapot"
 attack by storm; attack suddenly
 a direct and violent assault on a stronghold
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What Causes a Storm?
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http://systhread.net/pics/storm/jpegs-orig/10.jpg
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How Long Can a Storm Last?
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Jupiter’s Giant Red Spot
64
Baby Red Spots?
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Review: Parcel Concepts
Below approximately 100 km altitude,
air is relatively well mixed.
Virtually all mixing is accomplished
by the exchange of air “parcels”
which have horizontal dimensions
ranging from mm to the scale of the
Earth.
An air “parcel” of finite dimension is
assumed to be:
(1)
(2)
Thermally insulated from the
environment (no energy exchange)
Moving slowly so that kinetic energy
of motion is much smaller than it’s
total energy.
In reality, both of these assumptions are violated to some extent. But for small
displacements over small time intervals they can be excellent approximations.
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How do you describe air Parcels?
Ideal Gas Law - Summary
Ideal Gas Law: P = nkT = ρRT=RT/α
PV = R*T/M
P = pressure
m = mass per gas particle
n = number density of gas particles
ρ = mn = mass density
α = 1/ρ = specific volume
V = volume of one mole of gas
k = Boltzmann’s constant
R = gas constant (gas specific) = R*/Μ
M = molar mass
R* = universal gas constant
T = temperature
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Parcel Concepts: Applications of the laws of
thermodynamics to air parcels
Internal Energy (kinetic energy):
1 2
3
mv  kT
2
2
Internal Energy (general):
U = ρCpT
Internal Energy Change:
dU = CpdT; Work=PdV
First Law of Thermodynamics:
dQ = dU + PdV
Second Law of Thermodynamics: dS = dQ/T
Hydrostatic Equilibrium:
dP = -ρgdz
Specific Heats (Constant P,V)
 dq 
C p ,v  

dT

 p ,v const
Adiabatic means dQ = 0.
dQ implies dS = 0.
Thus an adiabatic process is also an isentropic process.
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Hydrostatic Equilibrium
Air pressure at any height in the
atmosphere is due to the force per unit
area exerted by the weight of all of the
air lying above that height.
The air is in hydrostatic balance if the
net upward force on a thin slab of air
is equal to the net downward force on
the slab.
Thus the change in pressure between the
top and bottom of the thin slab is equal
to the weight (dM g = ρdV g) of the slab
per unit area, where dV = dAdz.
Hydrostatic Balance:
ρg dAdz = P(z) dA – P(z+dz) dA
P(z+dz) = P(z) + (dP/dz) dz
Thus
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dP/dz = -ρg
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Hydrostatic Equilibrium
Hydrostatic Balance: dP = - ρgdz
Ideal Gas Law: P = nkT = (ρ/m) kT = ρRT
P = pressure
m = mass per gas particle
n = number density of gas particles
ρ = mn = mass density
kT
k = Boltzmann constant
H
R = gas constant
mg
T = temperature
dP   gdz
dP
 mP 
  g  
g
dz
 kT 
dP
 mg 
 
dz
P
 kT 
Z
P

Z  Zo 
dP
 mg 
P P  Z  kT dz  ln  Po    H
o
o
P
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Hydrostatic Equilibrium
P

Z  Zo 
ln    
H
 Po 
P ( z )  P ( z o )e
 ( z  zo ) / H
This means that pressure falls off exponentially with altitude z.
H = kT/mg is the Atmospheric Scale Height, and is also the
equivalent thickness of the atmosphere for constant temperature.
Near the Earth’s surface, H ~ 7-8 km.
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Atmospheric Scale Height
P ( z )  P ( z o )e
 ( z  zo ) / H
where
kT
H
mg
This means that pressure falls off exponentially with altitude z,
with a e-folding “distance” in the vertical of H. For an isothermal atmosphere
(T = constant), density would have the same functional form.
H is also the equivalent “thickness” of the atmosphere. If the entire atmosphere
was compressed to sea level pressure (Po), then the atmosphere would extend to
a height of H.
Near the Earth’s surface, H ~ 7-8 km.
H is also roughly the altitude to which an atom moving vertically can reach.
This can be seen by equating an atom’s kinetic to gravitational potential energy:
½ mv2 = 3/2 kT = 3/2 mgH
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72
Geopotential Height
The Geopotential, Φ, at any point in the Earth’s atmosphere is defined as
the work that must be done against the Earth’s gravitational field to raise
a mass of 1 kg from sea level to that point. Units are J kg-1 or m2s-2.
The force (in newtons) acting on 1kg at height z above sea level is numerically
equal to g. The work (in joules) done in raising 1 kg from z to z+dz is:
d  gdz
From hydrostatic equilibrium (dp = -ρgdz) we get:
d  gdz  dp
The Geopotential at height z is then:
z
( z )   gdz
0
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73
We can also define a quantity called the Geopotential Height, Z, as:
( z ) 1
Z

gdz

g
g0 0
z
Where go is the globally averaged acceleration due to gravity at the Earth’s
surface.
Geopotential Height is used as the vertical coordinate in most atmospheric
applications in which energy plays an important role (e.g., large scale
motions). In the lower atmosphere, there is only a small difference between
the physical height z and the geopotential height Z.
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74
Thickness of Layer Between Pressure Levels
In meteorological applications it is not convenient to deal with density, as it
is difficult to measure. Pressure makes a more convenient vertical variable.
Hydrostatic equilibrium:
p
pg

z
RT
d  gdz   RT
Geopotential:
dp
p
Integrate the geopotential between two pressure levels, p1 and p2.
2
p2
p2
dp
d



RT

p p
1
1
implies:
From the definition of geopotential height:
This is known as the thickness of the layer
between two pressure layers.
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dp
 2  1   R  T
p
p1
p
R 1 dp
Z 2  Z1 
T

g 0 p2 p
75
Scale Height for an Isothermal Atmosphere
p
Thickness:
R 1 dp
Z 2  Z1 
T

g 0 p2 p
If T = constant, OR if the mean
temperature is used in this expression,
then we get:
RT
Z 2  Z1 
g0
p
R 1 dp
Z 2  Z1 
T
g 0 p2 p
p1
dp
p
p2
Which can be integrated exactly:
Z2  Z1  H ln( p1 / p2 )
Or, by raising e to the power of each side:
 ( Z 2  Z1 ) 
p2  p1 exp 

H

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Scale Height: H = RT/go
H ~ 7-8 km in lower atmosphere
76
Thickness and Heights of Constant Pressure Surfaces
Note that there is a unique
relationship between P & Z.
 ( Z  Z1 ) 
p2  p1 exp  2

H

Thus pressure can (and is)
used as a vertical coordinate.
Lines of constant pressure are
known as isobars.
Pressure decreases monotonically with height, thus pressure surfaces
never intersect.
Pressure is a principle driver of atmospheric motions, and thus characterizing
the variation of pressure can provide insight into atmospheric dynamics.
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77
First Law of Thermodynamics, Once Again
dQ = dU + PdV
Heat Flow (in/out) = change in internal energy
+ work done by parcel
Using the specific heat relationship we have
dU = Cv dT
So we can write: dQ = Cv dT + PdV
as equivalent statement of the First Law of Thermodynamics.
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78
Derivation of the Dry Adiabatic Lapse Rate (DALR)
Hydrostatic Equilibrium:
dP = -ρg dz, or VdP = -g dz
(where V = specific volume = 1/ρ)
Ideal Gas Law:
P = ρRT = RT/ V
First Law of Thermodynamics:
dQ = CvdT + P dV
Rewrite IGL
PV = RT
Differentiate the IGL 
P dV + V dP = RdT = (Cp-Cv) dT
(For an ideal gas Cp – Cv = R)
Combine FLT & IGL 
dQ = Cv dT + (Cp-Cv) dT – V dP
dQ = CpdT – V dP
But for an adiabatic process
(no energy flow into or out of the parcel) dQ = 0
So CpdT = V dP = -g dz 
dT/dz = -g/Cp
DALR
Note that the DALR doesn’t say anything about the actual value of T,
but it provides a very strong constraint on how T varies with altitude.
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79
Adiabatic Processes for Parcels
If we slowly move a parcel of dry air vertically, such that there is no energy
flow with its environment (dQ=0), then its temperature will change with
altitude following the Dry Adiabatic Lapse Rate (DALR).
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dT
g
DALR  d 

dz
Cp
80
The Dry Adiabatic Lapse Rate
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81
Dry Adiabatic Lapse Rate and the PV Diagram
The (P,V) curve in a thermodynamic diagram of vertical motion
in the atmosphere following the DALR will be an Adiabat.
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82
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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83
Entropy and Potential Temperature
First Law of Thermodynamics:
dQ = dU + PdV = dU - VdP
Internal Energy:
dU = Cp dT
Second Law of Thermodynamics: dS = dQ/T
S = Entropy of the system
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84
Entropy and Potential Temperature
FLT
dQ  dU  dW  dU  PdV  dU  VdP
SLT
TdS  C p dT  VdP  C p dT  ( RT / P)dP
dT
dP
dS  C p
R
T
P
Integration 
Rewrite:
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T 
P
S  So  C p ln    R ln  
 To 
 Po 

S  C p ln Tp

 S
'
o
Where κ =R/Cp (approximately
2/7 for a diatomic gas)
So is the constant of integration.
85
Adiabatic Processes: dQ = 0
Constant Entropy Processes: dS = 0
From the Second Law of Thermodynamics dS = dQ/T, an adiabatic
process is thus an isentropic process.
dT
dP
dS  C p
R
T
P
dT
dP
Cp
R
T
P
dS=0 

Integrate:
Po
dT
dP
Cp 
 R
T
P
T
P
Potential Temperature
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P 

C p ln    R ln  o 
T 
P
 Po 
  T 
P

86
Slanted blue lines are adiabats
Potential Temperature
 Po 
  T 
P

If we take a parcel of gas at
T,P and change it
adiabatically to standard
pressure Po, it will have a
temperature of θ.
Why do airplanes need
air conditioners?
P(mbar)
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km
http://dewx.easternuswx.com/Figures/Figure_2.jpg
87
Parcel Concepts: Key Ideas
Internal Energy (kinetic energy):
1 2
3
mv  kT
2
2
Internal Energy (general):
U = ρ CpT
Internal Energy Change:
First Law of Thermodynamics:
dU = CpdT; Work=PdV
dQ = dU + PdV
Second Law of Thermodynamics: dS = dQ/T
Hydrostatic Equilibrium:
dP = -ρ gdz
 dq 

Specific Heats (at constant P) C p  
 dT  p const
Specific Heats (at constant V)
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 dq 
Cv  

dT

v const
88