Download Contents - Atmospheric and Oceanic Sciences

Contents
1 Overview
2 Atmospheric Thermodynamics
2.1 Basic Concepts, Denitions and Systems of Units . . . . .
2.2 Temperature and the Zeroth Principle of Thermodynamics
2.3 Ideal Gas and Equation of State for the Atmosphere . . . .
2.3.1 Composition of Air . . . . . . . . . . . . . . . . . .
2.4 Work by Expansion . . . . . . . . . . . . . . . . . . . . . .
2.5 First Law (or Principle) of Thermodynamics . . . . . . . .
2.5.1 Heat Capacities . . . . . . . . . . . . . . . . . . . .
2.5.2 Enthalpy . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Reversible Adiabatic Processes for an Ideal Gas . .
2.6 Second Law (or Principle) of Thermodynamics . . . . . . .
2.6.1 Mathematical Statement of the Second Law . . . .
2.6.2 The Carnot Cycle . . . . . . . . . . . . . . . . . . .
2.6.3 Restatement of First Law with Entropy . . . . . . .
2.6.4 The Free Energy Functions . . . . . . . . . . . . . .
2.6.5 Concept of Equilibrium . . . . . . . . . . . . . . . .
3 Moist Thermodynamics
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
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Latent Heats - Kircho's Equation . . . . . . . . . . . . . . . . . . . . . .
Gibbs Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase Equilibrium For Water . . . . . . . . . . . . . . . . . . . . . . . . .
Clausius-Clapeyron Equation . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Equilibrium between liquid and solid-Clapeyron Equation . . . . . .
3.4.2 Equilibrium between liquid and vapor-Clausius-Clapeyron Equation
3.4.3 Equilibrium between Ice and Vapor . . . . . . . . . . . . . . . . . .
3.4.4 Computation of Saturation Vapor Pressure . . . . . . . . . . . . . .
General Theory For Mixed Phase Processes Within Open Systems . . . . .
Enthalpy Form of the First-Second Law . . . . . . . . . . . . . . . . . . . .
Humidity Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Vapor Pressure (ev ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.2 Mixing Ratio and Specic Humidity . . . . . . . . . . . . . . . . . .
3.7.3 Relative Humidity . . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.1 Virtual Temperature . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.2 Dew Point Temperature (Td ) . . . . . . . . . . . . . . . . . . . . . .
3.8.3 Wet Bulb Temperature (Tw ) . . . . . . . . . . . . . . . . . . . . . .
Entropy variables for Moist air . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.1 Potential Temperature for Moist Air . . . . . . . . . . . . . . . . .
3.9.2 Equivalent Potential Temperature . . . . . . . . . . . . . . . . . . .
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3.10
3.11
3.12
3.13
3.9.3 Ice-Liquid Water Potential Temperature
Simplifying Approximations . . . . . . . . . . .
3.10.1 Pseudo-adiabatic Process . . . . . . . . .
3.10.2 Other Approximations . . . . . . . . . .
Hydrostatic Balance . . . . . . . . . . . . . . .
Static Energy . . . . . . . . . . . . . . . . . . .
Methods of Solving Thermodynamic Systems . .
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4 Thermodynamic Analysis of the Atmosphere
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4.1 Atmospheric Thermodynamic Diagrams . . . . . . . . . . . . . . . . . . . .
4.1.1 Classical Thermodynamic Diagrams . . . . . . . . . . . . . . . . . . .
4.1.2 The Emagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 The Tephigram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 The Skew T-log P Diagram . . . . . . . . . . . . . . . . . . . . . . .
4.1.5 The Stuve Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.6 Summary of the Attributes of Thermodynamic Diagrams . . . . . . .
4.2 Atmospheric Static Stability and Applications of Thermodynamic Diagrams
to the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Environmental Structure and Parcel Path . . . . . . . . . . . . . . .
4.2.2 Dry Adiabatic Lapse Rate . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Moist Adiabatic Lapse Rate . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Diagnosis of Atmospheric Stability . . . . . . . . . . . . . . . . . . .
4.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Microphysics, Rauber
Clouds, Fog and Haze, ?
Atmospheric Electricity, Lyons
Radiative Transfer in the Atmosphere, Ackerman
Surface Layer Processes, Stull
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PHYSICAL ATMOSPHERIC SCIENCE
1 Overview
Perhaps the oldest of the atmospheric sciences, physical meteorology is the study of laws
and processes governing the physical evolution of the atmosphere. The underlying basis of of
physical meteorology is the study of atmospheric thermodynamics which formulate equations
governing the relationships between observed physical properties of air including pressure,
temperature, humidity and water phase. Generally, thermodynamics as well as most of
physical meteorology does not seek to develop explicit models of the processes it studies,
such as the evolution of individual air molecules or individual water droplets. Instead, it
formulates relationships governing the statistics of these processes. Temperature and air
density are two primary examples of such statistics representing the mean kinetic energy
and number of molecules.
Thermodynamics studies the statistics of molecular scale processes under prescribed mechanical forcings and thermal energy transfer into the system. Mechanical transfers of energy
occur through dynamical processes and are studied by dynamic meteorology . Thermal energy transfers occur through convection, conduction and radiation. All but convection are
molecular scale process and are treated by branches of Physical Atmospheric Science.
The study of the growth of water and ice precipitation is a major discipline of physical
meteorology called microphysics . Here theories of how liquid and ice hydrometeors rst
form and subsequently evolve into the rain, snow and hail. The interaction of water with
atmospheric aerosols is an important part of atmospheric microphysics which leads to the
initial formation of hydrometeors. The quantication of these theories leads to governing
equations that can be used to simulate and predict the evolution of precipitation as a function
of changing atmospheric condition.
Another major branch of physical meteorology is the study of atmospheric radiation.
This area of study develops theories and laws governing the transfer of energy through the
atmosphere by radiative processes. This includes the adsorbtion, transmission and scattering
of both solar radiation and terrestrial radiation. The rst goal of atmospheric radiative
transfer is to determine the net radiative loss or gain at a particular atmospheric location that
leads to thermodynamic change described by thermodynamics theory. Since the radiative
transfer is aected by details of the atmospheric thermal, humidity and chemical structure,
it is possible to recover some details of those structures by observing the radiation being
transferred. This has given rise to a major branch of atmospheric radiation called remote
sensing , the goal of which is to translate observations of radiation to atmospheric structure.
The study of atmospheric optics is another branch of atmospheric radiation that applies
concepts of radiative transfer to study the eect of atmospheric structure on visible radiation
passing through the atmosphere. Phenomena such as blue sky, rainbows, sun-dogs and so
on are subjects of this area.
3
The study of atmospheric electricity is a branch of physical meteorology closely related
to microphysics. This area of study seeks to develop theories related to the formation of
electrical charges and the release of these charges in the form of lightning. Generally, the
formation of charge separation occurs in thunder storms and is related to the evolution of
microphysics.
Finally, the study of conduction between the surface and the air is within the realm of
physical atmospheric science. The transfer of heat occurs on the molecular scale and is generally formulated statistically by relationships dened empirically. These transfer processes
are forced by humidity or temperature dierences between the surface and the air and so
the evolution of the surface is of primary concern. They may involve a changing surface of
soil, water or vegetation. This has given rise to branches of atmospheric and other Earth
Sciences aimed at developing models of the soil, water and vegetation.
In this chapter we will concentrate our eorts on laying down the foundations of basic
thermodynamics, microphysics and radiative transfer. Discussions of atmospheric optics,
electricity and surface energy transfer will be presented.
2 Atmospheric Thermodynamics
Atmospheric thermodynamics is the study of the macroscopic physical properties of the atmosphere for which temperature is an important variable. The thermodynamic behavior of
a gas, such as air, results from the collective eects and interactions of the many molecules
of which it is composed. However, because these microscale processes are highly nonlinear, chaotic and too numerous, small and too rapid to be observable, their eects are not
described by the laws governing the microscale processes themselves. Instead, classical thermodynamics consists of laws governing the observed macroscopic statistics of the behavior
of the microscopic system of air and the the suspended liquids and solids within. As a consequence, thermodynamic laws are only valid on the macroscale where a sucient population
of molecular entities is present to justify the statistical approach to their behavior.
Classical thermodynamics seeks to:
classify the forms of energy
not to ask \why" in terms of the rst principles of atomic and molecular structure and
the energetics of interactions between atoms and molecules
Derive all principles from observables . For example, the First Law of Thermodynamics begins with a classication of energy into thermal and non-thermal forms as is
observed.
serve an an ecient book keeping device in which data are stored in convenient and
non redundant form
provides a means of calculating the true equilibrium state as well as dening the conditions for constrained equilibrium
4
Atmospheric thermodynamics diers from classical thermodynamics in that particular
attention is paid to the properties of air, and forms of the thermodynamic laws and relationships which are most useful for application to the atmosphere. This section will start by
dening the basic concepts needed, then move into the thermodynamics of gasses and the
specically into the thermodynamics of air. Air refers to the gas present in the Earth's lower
atmosphere where air is suciently dense for the empirical laws to apply. This only excludes
regions on the order of 50 km and higher above the surface. Finally the thermodynamics of
the mixed phase system including liquid and ice processes, will be described.
2.1 Basic Concepts, Denitions and Systems of Units
We begin by dening a system to represent a portion of the universe selected for study. It
may or may not contain matter or energy. Systems are classied as open , isolated or closed
. In contrast, surroundings refer to the remainder of the universe outside the system. An
isolated system can exchange neither matter nor energy with its surroundings. The universe
is dened by the First Law as an isolated system! A closed System can exchange energy but
not matter with its surroundings. An open System , on the other hand, can exchange both
energy and matter with its surroundings.
Next we dene a property of the system to be an observable. It results from a physical or
chemical measurement. A system is therefore characterized by a set of properties. Typical
examples are mass, volume, temperature, pressure, composition, energy, etc. Properties
may be classied as intensive or extensive. An intensive property is a characteristic of the
system as a whole and not given as the sum of the property for portions of the system.
Pressure, temperature, density an specic heat are intensive properties. We will try to stick
to a naming convection which uses capital letters for variables describing an \intensive"
property. An extensive property is dependent on the extent of the system and is given by
the sum of the extensive properties of portions of the system. Examples are mass, volume
and energy. We will use a naming convention with variables having lower case letters if they
refer to an extensive property.
A homogenous System is a system where each intensive variable has the same value
at every point in the system. Then any extensive property Z , can be represented by the
mass-weighted intensive properties:
Z = mz
(1)
where z is the specic property.
We dene phases to be subsets of a system that are homogenous in themselves but
dierent from other portions. A heterogenous system is dened to be a system composed
of two or more phases. In this case, any extensive property will be the sum of the mass
weighted phases of the system, ie:
X
Z = m z
(2)
where refers to the particular phase.
5
An inhomogeneous System is dened to be a system where intensive properties change
in a continuous way, such as how pressure or temperature change from place to place in the
atmosphere. Our thermodynamic theory will be applied only locally where the system can
be considered homogenous or heterogenous as an approximation.
The study of the atmosphere requires the denition of an air parcel. This is a small
quantity of air whose mass is constant but whose volume may change. The parcel should
be considered to be small enough to be considered innitesimal and homogeneous, but large
enough for the denitions of thermodynamic and microphysical statistics to be valid. In an
atmosphere, where air movements are not conned to a rigid container and therefore expand
and contract against local forces of pressure gradient, the parcel quantication is a natural
choice.
Energy is the ability to perform work and is a property dened in the First Law of
Thermodynamics. Energy is an extensive variable that is normally dened as a product of
an intensive and extensive variable. The intensive term denes, by dierencing, the direction
of the energy transfer. The extensive term is also called the capacity term. An example is
mechanical energy in the air which is written as a product of a pressure (intensive) and
volume change (extensive). When other intensive variables are substituted other forms of
energy are represented. For instance replace pressure in the example above with temperature
and the energy is thermal energy, replace it with chemical potential and one denes chemical
energy and so on.
The state of a system is dened by a set of system properties. A certain minimum number
of such properties are requires to specify the state. For instance, in an ideal gas, any three
of the four properties - pressure, volume, number of moles and temperature will dene the
state of the system. The so called equation of state states an empirical relationship between
the state properties which dene the system.
If the state of a system remains constant with time if not forced to change by an outside
force, the system is said to be in a state of equilibrium , otherwise we say the system is in a
state of non-equilibrium . Independence of the state to time is a necessary but not sucient
condition for equilibrium. For instance, the air within an evaporating convective downdraft
may have a constant cool temperature due to the evaporation rate equaling a rate of warming
by convective transport. The system is not in thermodynamic equilibrium but is still not
varying in time. In that case we say the system is steady state .
When there is an equilibrium which if slightly perturbed in any way will accelerate
away from equilibrium, it is referred to an unstable equilibrium . Metastable equilibrium is
similar to unstable equilibrium except with respect to perhaps only a single process. For
instance, when the atmosphere becomes supersaturated over a plain surface of ice, no ice will
grow unless there is a crystal to grow onto. Introduce that crystal and growth will begin,
but without it the system remains time independent even to other perturbations such as a
temperature or pressure perturbation.
A process is the the description of the manner by which a change of state occurs. Here
are some process denitions:
- Change to the System Any dierence produced in a system's state. Therefore any
6
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change is dened only by the initial and nal states.
Isothermal Process A change in state occurring at constant temperature.
Adiabatic Process A change of state occurring without the transfer of thermal energy
or mass between the system and its surroundings.
Diabatic Process A process resulting from the exchange of mass or energy with the
surroundings. An example is radiative cooling.
Cyclic Process A change occurring when the system (although not necessarily its surroundings) is returned to its initial state.
Reversible or Quasi-Static Process A special, idealized thermodynamic process during
which the conditions dier only innitesimally from equilibrium. Hence at any given
point during the reversible process the system is nearly in equilibrium. We can think
of a reversible process as being composed of small irreversible steps, each of which have
only a small departure from equilibrium (see gure 1)
In association with these concepts of processes, we also dene the internal derivative (di)
to be the change in a system due to a purely adiabatic process. Conversely, the external
derivative (de) is dened to be the change in a system due to a purely diabatic process
We can say that the total change of a system is the sum of its internal and external
derivatives, ie:
d = di + de
(3)
where d is the total derivative. Now, it follows from the denition of adiabatic that:
I
I
d = de
I
di = 0
2.2 Temperature and the Zeroth Principle of Thermodynamics
We dene a diathermic wall to allow thermal interaction of a system with surroundings but
not mass exchange. The zeroth principle of thermodynamics then states: , if some
system \A" is in thermal equilibrium with another system \B" separated by a diathermic
wall, and if \A" is also in thermal equilibrium with a third system \C", then systems \B"
and \C" are also in thermal equilibrium. It follows that all bodies in equilibrium with
some reference body will have a common property which describes this thermal equilibrium
and that property is dened to be temperature. The so called reference body denes this
property and is called a thermometer. A number is assigned to the property by creating a
temperature scale. This is accomplished by nding a substance which as a property that
changes in correspondence with dierent thermal states. For instance, the following are
thermometric substances and corresponding properties:
7
Figure 1: Reversible process as a limit of irreversible processes (From Iribarne and Godson,
1973)
Thermometric Substance
Gas, at constant volume
Gas, at constant pressure
Thermocouple, at constant pressure and tension
Pt wire, at constant pressure and tension
Hg, at constant pressure
Thermometric Property X
Pressure
Specic Volume
Electromotive force
Electrical resistance
Specic Volume
Table 1: Empirical Scales of Temperature (From Irabarne and Godson, 1973)
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A thermometer is calibrated to changes in its observed thermometric property, obeying
the zeroth principle. The thermometer must much smaller than the substance measured
so that the changes brought to the substance by exchanges from the thermometer can be
neglected.
The temperature scale my be dened generally as:
TX = aX + b
(4)
where TX is an arbitrary temperature scale, a is a constant representing the slope b is a
constant representing the intercept of the linear scale, and \X" is the thermometric property
of the substance. Use of this scale requires two well dened thermal states in order to
determine a and b. The Celsius, or Centigrade, temperature scale is dened in this way by
the freezing point of pure water at one atmosphere to be 0 C and the boiling point of pure
water at one atmosphere to be 100 C. The Fahrenheit temperature scale, used primarily in
the United States for non-scientic measurements and unocially in the United Kingdom is
calibrated by assigning the boiling point of pure water at one atmosphere to be 212 F and
the freezing point of pure water at one atmosphere to be 32 F.1
A more general scale can be derived using the thermometric properties of gasses listed
as the rst two examples in the table above. Using gas as the thermometric substance and
pressure as the thermometric property, it is observed that pressure decreases as the thermal
state of the gas falls. We then can dene the lowest possible thermodynamic state to be
that where pressure decreases to zero (at constant volume). Letting the temperature also
be zero at that point, denes the intercept to also be zero. Now only a single well-dened
thermodynamic state is necessary to dene the scale. This is chosen to be the the triple
point for pure water where all three phases of water can coexist. The actual value of this
\absolute" temperature scale at the triple point is arbitrary and it choice denes the value of
the slope \a". We can dene this scale to be such that the thermal increments for one degree
of temperature equal that of the Celsius scale. We do this by solving for the intercept of the
Celsius scale, using pressure as the thermometric property. This yields b = ;273:15C . Then
dening the freezing point at 1 atm to be 273.15 degrees and the triple point of water to be
273.16 degrees of the absolute temperature scale the scale is set to have equal increments
to Celsius temperature but have an absolute scale beginning at absolute zero. This scale is
called the Kelvin scale. The Kelvin temperature is detonated by \K".
The relationship between the three major scales are:
T = TC + 273:15
(5)
5
(6)
TF = 9 TC + 32
where T TC TF are the Kelvin, Celsius and Fahrenheit temperatures respectively.
1 This odd assignment of scale dates back to Medieval times when the scale was dened less precisely
to be 100F for the temperature of the human body and 0F for the freezing point of bodily uids, which
are basically a saline water solution. The original temperature denition is largely preserved in the modern
denition which is based on the more well-dened thermal states of pure water.
9
2.3 Ideal Gas and Equation of State for the Atmosphere
A gas is composed of molecules moving randomly about, colliding with and bouncing of
each other and the walls of any container in which the gas is located. The nature of a
molecule's individual evolution and the evolution of the entire population of molecules is
chaotic and unpredictable on an individual basis. We make no attempt to quantify a physical
description of this process with Newtonian mechanics. Instead we invoke the discipline
of thermodynamics which treats these processes statistically. In other words, we dene
statistical properties of the gasses and then study these properties observationally, looking
for robust relationships that govern how these statistical properties behave. We can then
use these relationships to predict future behavior as well. It must be remembered, however,
that the method only is valid for statistically signicant populations of molecules under a
limited range of conditions, dened by the observations used to dene these properties.
The statistical properties of a gas that can be measured include its volume, its temperature and its pressure. The temperature is a measure of the average kinetic energy of the
molecules, which is proportional to the average of the square of the speed of the molecules.
The pressure is proportional to force per unit area exerted on a plane surface resulting from
the collisions of gas molecules with that surface. It is intuitively obvious from Newtonian
mechanics that that force will be proportional to the average momentum of the molecules,
which is then proportional to the mass and velocity of each molecule and the total number
of molecules hitting the surface.
The mass of the gas is dened to be:
m = nM
(7)
where n is the number of moles and M is the molecular weight (g mole;1 ). It is often
advantageous to express express an extensive thermodynamic quantity as the ratio of that
quantity to its total mass. That ratio is referred to as the specic value of that quantity.
For instance, the specic volume is given by:
V =1
= m
(8)
where is the density.
We can now dene the equation of state for the atmosphere. Observations suggest that
under normal tropospheric pressures and temperatures, most gases exhibit almost identical
relationships between changes in their volume, pressure and temperature. To quantify this
observation, we dene an ideal gas to be one which obeys the empirical relationship:
pV = nR T = mR T=M = mRT
(9)
where n is the number of moles, M is the molecular weight of the gas, R is the universal
gas and R = R=M is the specic gas constant valid only for a particular gas of molecular
weight M . Observations show that, R = 8:3143 Jmol;1K ;1
10
Employing these denitions, the equation of state can be written as:
p = RT = RT (10)
It may be shown, by comparison with measurements, that for essentially all meteorological conditions, the ideal gas law is valid within 0:1%. Under stratospheric conditions the
departure from ideal behavior may be somewhat higher.
We now consider multiple constituent gases such as air. In such a case, the partial
pressure (pj ) is dened to be the pressure that the ith individual gas constituent would have
if the same gas at the same temperature occupied an identical volume by itself. We can also
dene the partial Volume (Vj ) to be to the volume that a gas of the same mass and pressure
as the air parcel would occupy by itself.
Dalton's law of partial pressure states that the total pressure is equal to the sum of the
partial pressures:
X
p = pj
(11)
j
where p is the total pressure and the sum is over all components of the mixture. Similarly
for volume,
X
V = Vj
(12)
where V is the total volume.
If each constituent behaves as an ideal gas, then:
Pj = nj R T=V
and,
p = (RT=V )
X
j
(13)
nj :
(14)
If we divide through by the total mass of the gas (mj ) and note that nj = mj =Mj , where
Mj is the molecular weight of the species i, and total mass is:
X
m = mj
(15)
then we can sum the equations of state for all constituents to obtain:
P mj
p
V
Mj
:
p P m = = p = R T P m
j
j
(16)
where the specic volume is dened = 1=. To permit the statement of the ideal gas law
(eq. of state) for a mixture of gases, we would need to write:
p = R T = RT
(17)
M
11
where M is the mean molecular weight of the mixture which is dened:
P
j
M = Pm
mj :
Mj
or, since mj = Mj nj ,
PM n
M = P nj j
j
X nj
=
Mj n
X
=
Mj Nj
where Nj is the molar fraction of a particular gas. Alternatively, we can write:
Nj = ppj = VVj :
(18)
(19)
(20)
2.3.1 Composition of Air
The original atmosphere of the newly formed Earth is believed to have been composed
almost entirely of hydrogen and helium, which are the two most abundant gases found in
the universe. Hydrogen compounds were believed to also have been abundant in the early
atmosphere, including methane and ammonia. Because of their low density, it is believed
that most of the early atmosphere escaped to space, both from its own energy absorbed from
the suns radiation, and by the stripping eects of the solar wind.
As the early atmosphere escaped to space, a second atmosphere formed by the processes
of volcanic out-gassing, which is believed to have vented gasses much as today's volcanoes
do. Typically, today's volcanic gasses are composed of 80% water vapor, 10% carbon dioxide
and a few percent nitrogen. The early atmosphere was very warm and of extremely high
pressure. After all, it was able to contain much of the water, now in our oceans in a vapor
state! Eventually much of the water vapor was converted to rain where it fell to form the
oceans, lakes and streams of the world.
It is believed that some of the vapor was broken into molecular oxygen and hydrogen
by the radiative eects of the sun. This process is called photodissociation. The stripped
hydrogen, most likely rose to the top of the atmosphere over time where it escaped to space,
much as did the rst atmosphere. In addition, much of the carbon dioxide became dissolved
into the oceans. Over time, this built an atmosphere rich in oxygen and nitrogen as well as
carbon dioxide and water vapor.
As life forms evolved, especially plants, carbon dioxide became an essential part of the
photosynthesis process which eectively broke the carbon dioxide into molecular oxygen and
carbons. Much of the resulting carbons were taken out of the system and stored in the
Earth's crust as plant and animal carcasses. Today we can nd that stored carbon in the
form of limestone and fossilized carbons such as coal and oil.
The current atmosphere of the Earth is composed of \air" which we will nd to contain:
12
Gas
Symbol Molecular Molar (or
Weight
Volume)
Fraction
Nitrogen
N2
28.013
0.7809
Oxygen
O2
31.999
0.2095
Argon
Ar
39.948
0.0093
Carbon Dioxide
CO2
44.010
0.0003
Helium
He
0.0005
Methane
CH4
0.00017
Hydrogen
H
0.00006
Nitrous Oxide
N2 O
0.00003
Carbon Monoxide CO
0.00002
Neon
Ne
0.000018
Xenon
X
0.000009
Ozone
O3
0.000004
Krypton
Kr
0.000001
Sulfur Dioxide
SO2
0.000001
Nitrogen Dioxide NO2
0.000001
Chloroorocarbons CFC
0.00000001
Total
1.0000
Mass
Specic Gas
Fraction Constant
(J kg;1 K;1 )
0.7552 296.80
0.2315 259.83
0.0128 208.13
0.0005 188.92
mj Rj =m
(R)
(J kg;1 K;1 )
224.15
60.15
2.66
0.09
1.0000
R=287.05
Table 2: Composition of Dry Air near the Earth's surface
(1) Dry Air, which is a mixture of gases described below
(2) Water, which can be in any of the three states of liquid, solid or vapor
(3) Aerosols, which are solid or liquid particles of small sizes
The chemical composition of \Air" is given in Table 2.
Water vapor and the liquid and solid forms of water vary in its volume fraction from 0%
in the upper atmosphere to as high as 3% near the surface under humid conditions. Because
of this variation, dry air is treated separately from vapor in thermodynamic theory.
Among the trace constituents are carbon dioxide, ozone, Chloroorocarbons and Methane
which despite their small amounts have a very large impact on the atmosphere because of
their interaction with terrestrial radiation passing through the atmosphere, or in the case of
CFCs because of their impact on ozone.
Note from table 2 that the molecular weighted average gas constant for air is Rd =
R=287.05 Jkg;1K ;1 , which we also call the dry air gas constant because the eects of
moisture are not considered.
13
Figure 2: Work of expansion (From Iribarne and Godson, 1973)
2.4 Work by Expansion
If a system is not in mechanical equilibrium with its surroundings, it will expand or contract.
Assume that is the surface to the system which expands innitesimally to 0 in the direction
ds (see gure 2 ) Then the surface element d has performed work against the external
pressure Psurr . The work performed is:
(dW )d = Psurr dd cos = Psurr dV
where dV is the change in volume.
For a nite expansion, then :
W=
Zf
j
(21)
pdV
(22)
where i, f stand for the initial and nal states. Since the integrand is not a total derivative,
total work over a cyclic process can be non-zero. Therefore:
Z b
! Z b
!
I
W = pdV =
PdV ;
PdV
(23)
a
1
a
2
which is dened to be positive if integrated in the clockwise sense. This is the area enclosed
by the trajectory in the graph given in gure 3.
The work by expansion is the only kind of work that we shall consider in our atmospheric
systems. Assuming the pressure of the system is homogeneous and in equilibrium with the
surroundings, we can adopt the notation
dW = PdV
where W is the work performed on the surroundings by the system.
14
(24)
Figure 3: Work of expansion in a cycle (From Iribarne and Godson, 1973)
2.5 First Law (or Principle) of Thermodynamics
The First Law of Thermodynamics can be simply stated: The energy of the universe is
constant. The law makes no statement telling us exactly how much total energy there is nor
does it even tell us exactly what energy is. It merely states, that there is a budget of energy
in the universe and it balances exactly.
If we dare to ask: \What is energy?", the answer is very dicult to dene precisely. We
know that if we apply force to an object over some distance and as a result we accelerate the
object, we will have performed work on the object, measured in the energy units of Joules
. That work will precisely equal the increase in kinetic energy of the object, the total of
which is measured by half its mass multiplied by the square of its new velocity. Moreover, we
understand that our eort expends energy from the working substance. Depending on the
method we used to accelerate the object, the energy was previously stored in some other form
ranging from perhaps the potential energy of a compressed spring, chemical energy stored in
the muscle cells of our arm, or perhaps the kinetic energy of an object thrown at the object
that was accelerated. There are obviously many possibilities for working substances featuring
dierent types of stored energy. We also know that mass itself is a highly concentrated store
of energy equal to half the mass multiplied by the square of the velocity of light. When you
cut mass up into smaller and smaller pieces you tend to nd in the end that there are only
collections of forces competing against other collections of forces. Energy, in eect, quanties
a potential to force and so perform work.
The First Law requires, that for any thermodynamic system, an energy budget must exit
that requires any net energy owing into a system to be accounted for by changes in the
15
external energy of or the internal energy stored within the system. The external energy of
the system includes its kinetic energy of motion and its energy of position, relative to forces
outside the system such as gravitational, electrical, magnetic, chemical and many other forms
of potential energy between the system and its surroundings. The same energies can also
exist within the system between the molecules, atoms, and subatomic particles composing
the system and are internal energies. Kinetic energy of molecular motions relative to the
movement of the center of mass of the system is measured by the system's temperature
and represent the thermal internal energy. The other internal energies are potential internal
energies and may include a potential energy against the intermolecular forces of attraction
(latent heat ), chemical potential energy (for instance a gas composed of oxygen and hydrogen can potentially react) and so on. Because we probably are not even aware of all of the
forms of internal forces that exist in a system, we are unable to evaluate the total internal
potential energy. Instead, for the purposes of classical thermodynamics, we need only consider changes in internal energy occurring in allowable processes. For instance, we normally
choose to ignore nuclear reactions, and even most chemical reactions. But we cannot neglect
the changes in internal potential energy due to inter-molecular forces of liquid and ice water
phases in the atmosphere. The intermolecular forces are substantial, and the energy needed
to over come them is the latent heat of vaporization and melting. Thermodynamic energy
transfers between thermal and these internal potential energies drive the general circulation
of the Earth! Our thermodynamic discussion will ignore the treatment of some important
potential internal energies such as surface energy of droplets, chemical energy in photochemical processes, electrical energy in thunderstorms and the upper atmosphere, chemical
processes involving chloroorocarbons, and other processes which are known to have important secondary impacts on the evolution of the atmospheric thermodynamic system. These
processes can be added to the system when needed following the methods described in this
chapter.
In most texts of classical thermodynamics, closed thermodynamic systems are usually
assumed. A closed system allows no mass exchange between the parcel (system) and the
surroundings. This is considered to be a good approximation if a parcel is chosen to be
innitesimal and therefore small enough to ignore turbulent uctuations relative to the gas.
Ignoring molecular heat transfer is generally considered to be a minor approximation to
thermodynamic systems applied to macro scales. It is not a forgivable approximation to
the micro scales where it has to be considered. Although it is often deemed important to
the smallest of the turbulent scales, molecular transfer is usually represented as an outside
thermal heat source in the First Law energy budget and not a diabatic mass ux across the
parcel boundary.
One open mass ux must be accounted for with an open system, and that is the mass
ux of precipitation. We ax our parcel coordinate relative to the center of mass of the
dry air, which will be assumed to be the same for vapor. The liquid and ice components
of the system, however, may attain a terminal velocity which allows it to ow into and out
of the parcel. As a result, we will derive thermodynamic relationships for an open system
, or one where external uxes of mass into and out of the system are allowed. In order to
16
retain some simplicity, however, we will make the assumption that external mass uxes into
and out of the system will be of constituents having the same state as those internal to the
system. Hence we will allow rain to fall into our parcel, but it will be assumed to have the
same temperature as the system. Although the eect of this assumption can be scaled to
be small, there are instances where it can be important. One such instance is the case of
frontal fog, where warm droplets falling into a cool air mass result in the formation of fog.
As demanded by the First Law, we form an energy budget rst for a simple onecomponent system of an ideal gas:
Energy Exchanged with Surroundings = ChangeXin Energy Stored
Q ; W = U ; Ak
k
(25)
where Q represents the ow of thermal energy into the system including radiative energy
and energy transferred by conduction, W is the work performed on the surroundings by the
system, U is the thermal internal energy, and Ak is the kth component of potential internal
energy, summed over all of the many possible sources of internal energy. The work, as
described in the previous section, is the work of expansion by the parcel. It applies only to our
gas system, and does not exist in the same form for our liquid and ice systems. Presumably in
those systems, it should be reected as a gravitational potential term, however, it is typically
neglected.
In general, we treat the air parcel as a closed system, whereby there are no mass exchanges
with the surroundings. This is generally a good approximation as molecular transfers across
the boundaries can usually be neglected or included other ways. One exception occurs and
that is when we are considering liquid and ice hydrometeors. If we x our parcel to the
center of mass of the dry air, as we do, then there can be a substantial movement of liquid
or ice mass into and out of the parcel. Clearly this process must be considered with an open
thermodynamic system, where at least uxes of liquid and ice are allowed with respect to the
center of mass of the dry air parcel. Hence we will consider the possibility that Ak may change
in part because of exchanges of mass between the parcel and the environment, particular due
to falling precipitation in the application to liquid and ice constituents. Combining equation
25 with 24 for an innitesimal process we obtain the form of the First Law:
X
Q = dU + PdV ; dAk
(26)
k
where is should be noted that P is the pressure of the surroundings, as PdV denes work
performed on the surroundings and ;dV is the change in volume of the surroundings.
Note that for the adiabatic case, Q = 0 by denition.
Because we require conservation of energy, it is evident that the change in both internal
thermal energy (dU ) and internal potential energy (dAk ) must be exact dierentials, only
dependent on the initial and nal states of the process and not the path that the process
takes. Hence U and Ak must also be state variables.
17
2.5.1 Heat Capacities
Heat capacity is a property of a substance and is dened to be the rate at which the substance absorbs(loses) thermal energy (Q) compared to the rate at which its temperature (T)
rises(falls) as a result. It is a property usually dened in units of dQ=dT ( energy per temperature change) or in units of dq=dT (energy per mass per temperature change) in which case
it is called specic heat capacity . If heat is passed into a system it may be used to either
increase the internal thermal or potential energy of the system or can be used to perform
work. Hence there an innite number of possible values for heat capacity depending on the
processes allowable. It is useful to determine the heat capacity for special cases where the
allowable processes are restricted to only one. Hence we neglect the storage of potential
internal energy and restrict the system to constant volume or constant pressure processes.
Hence,
!
!
Q
Cv =
cv = q dT
dT
V
!
!
(27)
Cp = Q
cp = q
dT p
dT p
where Cv and Cp are the heat capacities at constant volume and pressure respectively and
where cv and cp are the specic heat capacities at constant volume and pressure respectively.
It can be expected that Cp > Cv , since in the case of constant pressure, the parcel can use
a portion of the energy to expand, and hence perform work on the surroundings, reducing
the rise in temperature for a given addition of heat.
Since the First Law requires that the change in internal energy , U be an exact dierential, U must also be a state variable, ie its value is not dependent upon path. Hence
U = u(P T ). If we accept that air can be treated as an ideal gas, then the equation of
state eliminates one of the variables, since the third becomes a function of the other two.
Employing Euler's rule,
!
!
@U
(28)
dU = @V dV + @U
@T dT:
T
V
Substituting equation 28 into the rst aw (equation 25), we get:
!
!
X
@U
@U
Q = @T dT + @V + P dV ; dAk k
V
T
For a constant volume process,
!
!
Q
= @U
Cv = dT
@T V
V
Hence, returning to 28,
!
@U dV
dU = Cv dT + @V
T
18
(29)
(30)
(31)
Monotonic gas Cv = 23 R Cp = Cv + R = 52 R
Diatomic gas Cv = 25 R Cp = 72 R
Table 3: Relationship between specic heats and molecular structure of gas
@U It has been shown experimentally that @V
= 0, indicating that air behaves much as an
T
ideal gas. Hence the internal energy of a gas is a function of temperature only, provided that
v
Cv is a function of temperature only. If we assume that possibility, @C
@V = 0. Experiments
show that for an ideal gas the variation of Cv with temperature is small, and since we hold
air to be approximately ideal, it follows that Cv is a constant for air. The First Law (eq. 29)
is therefore written as:
q = C dT + PdV ; P dA (32)
v
k
k
One can also obtain an alternate form of the rst law by replacing dV with the ideal gas
law, where:
pdV + V dP = RdT
(33)
which when combined with equation 32 yields:
X
Q = (Cv + R )dT ; V dP ; dAk (34)
k
For an isobaric process, (dP = 0), one nds:
!
q = C + R
Cp = @T
v
p
or
(35)
Cp ; Cv = R:
Similarly for dry air, the specic heat capacities are:
(36)
cpd ; cvd = Rd :
(37)
Hence equation 36 clearly demonstrates that Cp > Cv . Statistical mechanics show that:
Since dry air is nearly a diatomic gas,
cpd = 1004Jkg;1K ;1 cvd = 717Jkg;1K ;1
(38)
Substituting equation 36 into equation 34
Q = CpdT ; V dP ; Pk dAk (39)
This form of the rst law is especially useful because changes of pressure and temperature
are most commonly measured in atmospheric science applications. Note that CpdT is not
purely a change in energy, nor is V dp purely the work.
19
2.5.2 Enthalpy
For convenience, we introduce a new state variable called enthalpy, dened as:
H = U + PV
Dierentiating across we obtain:
dH = dU + pdV + V dP
and substituting equation 26, to eliminate dU :
X
Q = dH ; V dP ; dAk and using Euler's rule with equation 39, we obtain:
@H @T p = Cp
k
(40)
(41)
(42)
(43)
Enthalpy diers only slightly from energy, but arises as the energy variable when the work
term is broken into a pressure change and enthalpy change part. The enthalpy variable exists
purely for convenience and is not demanded or dened by any thermodynamic law.
2.5.3 Reversible Adiabatic Processes for an Ideal Gas
This type of process is of great importance to the atmospheric scientist. Ignoring radiative
eects, or heating eects of condensation, the dry adiabatic process represents the thermal
tendencies that an air will experience as it rises and the pressure lowers or as it sinks and the
pressure rises. In the absence of condensation, this represents the bulk of the temperature
change a parcel will experience if its rise rate is fast enough to ignore radiative eects. We
dened an adiabatic process as one for which q = 0. Then for an adiabatic process, the First
law of thermodynamics written as equation 39
CpdT ; V dP = 0:
(44)
Substituting the equation of state for V and dividing by R T , we obtain:
(Cp=R)d ln T = dlnP:
(45)
Integrating eq. 45, we obtain:
T = kP (46)
where = R =Cp = 0:29 and k is a constant of integration which can be determined from a
solution point.. Equation 46 is known to atmospheric scientists as the Poisson equation , not
to be confused with the second order partial dierential equation also known as \Poisson's
equation". The Poisson equation states that, given a relationship between temperature and
pressure at some reference state, the value of temperature can be calculated for all other
pressures the system might obtain through reversible adiabatic processes.
20
2.6 Second Law (or Principle) of Thermodynamics
The Zeroth Principle of Thermodynamics dened temperature to be a quantity which determined from two bodies were in thermal equilibrium. The First Law stated that principle
of energy conservation during any thermodynamic process. The Second Law deals with the
direction of energy transfer during a thermodynamic process occurring when two bodies are
not in thermodynamic equilibrium. It is again an empirical statement, not derived in any
way from rst principles but rather from observations of our perceived universe. The Second
Law can be stated in two ways, that can be shown to be equivalent:
1. Thermal energy will not spontaneously ow from a colder to a warmer object.
2. A thermodynamic process always acts down gradient in the universe to reduce dierentiation overall and hence mix things up and reduce order overall. If we dene entropy
to be a measure of the degree of disorder, then the Second Law states: The entropy
of the universe increases or remains the same as a result of any process. Hence, the
entropy of the universe is constantly increasing.
Just as the First Law demanded that we dene the thermal variable temperature, the
Second Law requires a new thermodynamic variable entropy. In order to form a mathematical
treatment for entropy, we consider several special thermodynamic processes which reveal the
nature of entropy behavior and provide guidance to its form.
2.6.1 Mathematical Statement of the Second Law
It is illustrative to examine the amount of work performed by a parcel on its surroundings
under isothermal conditions, implying that the internal energy of the system is held constant.
This will demonstrate an interesting behavior addressed by the Second Law.
Consider a xed mass of an ideal gas conned in a cylinder tted with a movable piston of
variable weight. The weight of the piston and its cross sectional area determine the pressure
on the gas. Let the entire assembly be placed in a constant temperature bath to maintain
isothermal conditions. Let the initial pressure of the gas be 10 atmospheres and the initial
volume be 1 liter. Now consider three isothermal processes:
Process I The weight on the piston is changed so as to produce an eective pressure of 1 atmosphere and since PV = nRT , its volume becomes 10 liters. The work of expansion is
given by:
Wexp = Psurr V:
(47)
Since Psurr is constant at 1 atmosphere, the work is 1(10 ; 1) = 9 latm. This work is
performed on the surroundings. With respect to the system, the energy transfer (
Q)
is therefore -9 latm.
Process II This will occur in two stages: (1) Decrease the pressure exerted by the piston to 2.5
atm so that the specic volume will be 4 liters and then (2) reduce the pressure to 1
21
Figure 4: Illustration of three isothermal processes.
22
atm with a specic volume of 10 l. The work of expansion becomes the sum:
wexp = P1V1 + P2V2
(48)
= 7:5 + 6:0
(49)
= 13:5lkg;1atm:
(50)
The mechanical energy transfer from the system (
Q) is thus -13.5 latm.
Process III Let the pressure exerted by the piston be continuously reduced such that the pressure
of the gas is innitesimally greater than that exerted by the piston (otherwise no
expansion would occur). Then the pressure of the gas is essentially equal to that of
the surroundings and since dw = PdV ,
V Z V2
Wexp = Pd = R T ln V2 :
(51)
V1
1
In each of the above three cases, the system responded to the removal of external weight
to the piston by changing state and performing a net work on the surroundings while maintaining isothermal conditions within the system by absorbing heat from the attached heat
reservoir, the reservoir being part of the surroundings. In each case, the total change in the
system state was identical, although the energy absorbed by the system from the surroundings, equal also to the work performed on the surroundings by the process varied greatly.
Note that Process III diered from the rst two processes in that it was an equilibrium
process, or one where the dierence between the intensive driving variable (pressure in this
case) of the system diers only innitesimally from the surroundings. The maximum work
possible for the given change in system state occurs in such an equilibrium processes. All
other processes, not in equilibrium, produce less net work on the surroundings and thus also
absorb less energy from the heat reservoir!
It is important to note that, unlike the non-equilibrium processes, the net work for the
equilibrium process is dependent only on the initial and nal states of the system. Hence
if the equilibrium process is run in reverse, the same amount of work is performed on the
system as the system performed on the surroundings in the forward direction. In the case of
the non equilibrium process, this is not the case and so the amount of energy released to the
surroundings for the reverse process is unequal to that absorbed in the forward direction.
Hence the system is irreversible . Hence we can equate an irreversible process to one which
is not in equilibrium.
For an isothermal reversible process involving an ideal gas, dU = 0 and hence Q =
pdV = dWmax. Thus Q is dependent only on the initial and nal states of the system.
Hence, for a reversible process, Q and W behave like state functions.As a consequence, for
either the irreversible or reversible processes we can state:
In general:
Q = W:
(52)
Qrev = Wmax
(53)
23
Since the Qrev is related to a change is state, then the isothermal reversible result described
by process III must apply for the general isothermal case, ie :
Qrev = RdlnV
(54)
T
where the right-hand-side of equation 54 is an exact dierential. By this reasoning it is
concluded that the left-hand-side of equation 54 must also be an exact dierential of some
variable that we will dene to be the entropy (S ) of the system (JK ;1 ) . Since the change
in entropy of the system is always an exact dierential, its value after any thermodynamic
process is independent of the path of that process and so entropy, itself, is also a state
function. We can write:
dS = QTrev :
(55)
From equation 53 we can also show that:
Wmax = TdS
(56)
which states that the maximum work which can be performed by a change in state is equal
to the temperature multiplied by the total change in entropy from the initial to the nal
state of the system. Since we showed that the net work performed on the surroundings as a
result of the isothermal process must be less than or equal to dWmax , then it is implied that:
dS > Q
(57)
T
for an irreversible process. It is also reasoned that by virtue of the Second Law, and obvious
from the exam,ple described above, that a process where
dS < Q
(58)
T
would be a forbidden process. Since the entropy of the surroundings must also satisfy these
constraints, the entropy of the universe can either remain constant for a reversible process
or increase overall for an irreversible process. This is the mathematical statement of the
Second Law for the case of isothermal processes.
We must now expand this concept to a system not constrained to isothermal conditions. To build a theory applicable to a general process, we consider the following additional
particular processes:
1. Adiabatic reversible expansion of an ideal gas.For a reversible adiabatic expansion,
Q = 0 and so dS = 0. This is an isentropic process . Since a reversible dry adiabatic
process is one where is conserved, a process with constant is also called an isentropic
process.
24
2. Heating of a gas at constant volume. For a reversible process at constant volume, the
work term vanishes:
(59)
dS = dQTrev = Cv dT
T = Cv d ln T
3. Heating of an ideal gas at constant pressure. For a reversible process:
dS = dQTrev = Cp dT
(60)
T = Cpd ln T
2.6.2 The Carnot Cycle
We demonstrated earlier that a net work can also be performed on the surroundings as a
system goes through a cyclic process. That work, we showed, was equal to the area traced
out on a P-V diagram by that cyclic process. The maximum amount of work possible by any
process was that performed by a reversible process . We can call our system which performs
work on the surroundings as the working substance .
Now we ask, Where does the energy for the work performed originate?. For a cyclic process
the initial and nal states of the working substance are the same so any work performed over
that process must come from the net heat absorbed by the system from the surroundings
during that process. We say net , because the system typically absorbs heat and also rejects
heat to the surroundings. If the system converted all of the heat absorbed to work without
rejecting any we would say that the system is 100% ecient. That can not happen typically.
For instance, the engine in a car burns gasoline to heat air within the cylinder that expands
to do work. But much of the heat given o from the gas actually ends up warming the engine
and ultimately the surroundings rather than making the engine turn. The degree to which
the energy warms the surroundings compared to how much is used to turn the engine and
ultimately move the car (more energy is lost to friction in the engine and transmission and
so on) is a measure of how ecient the car is.
For any engine, we can quantify its eciency as:
= MechanicalWorkPerformedbyEngine
= Q1 ; Q2
(61)
HeatAbsorbedByengine
Q1
where Q1 is the heat absorbed by the engine and Q2 is the heat rejected by the engine to
the surroundings. In terms of specic heat:
(62)
= Q1 Q; Q2 :
1
The maximum eciency that any engine can ever achieve, would be that of an engine powered
by a reversible process. Processes like friction in the engine or transmission, not perfectly
insulated walls of the pistons and so on, would all be irreversible parts of the cyclic process
in a gasoline engine. But lets imagine an engine where the thermodynamic cyclic process
is carried out perfectly reversibly. One simple reversible thermodynamic process which ts
this bill is the Carnot Cycle which powers an imaginary engine called a Carnot Engine. No
25
one can build such a perfect frictionless engine, but we study it because it tells us what the
maximum eciency possible is for cyclic process. Yes, even the Carnot engine is not 100%
ecient and below, we will see why.
First lets describe the Carnot cycle. Simply put, the Carnot cycle is formed by two
adiabatic legs and two isothermal legs so that the net energy transfers can be easily evaluated
with simple formulations for adiabatic or isothermal processes. Figure 5 shows such a process.
The four legs of the process are labeled I-IV beginning and ending at point \A". To visualize
this physically, we can imagine some working substance contained within a cylinder having
insulated walls and a conducting base. and a frictionless insulated beginning piston. The
cycle is explained as:
I The cylinder is placed on a heat reservoir held at T = T1 . Beginning a T = T1 , and
V = VA, the working substance is allowed to slowly expand maintaining isothermal
conditions.
II The cylinder is placed on an insulated stand and allowed to expand further to volume
VC and temperature T2 . Since the working substance is totally insulated for stage II,
the process is adiabatic.
III The cylinder is taken from the stand and placed on a heat reservoir where the working
substance is held at T = T2 and slowly compressed (isothermally) to specic volume
VD .
IV The cylinder is replaced on the insulated stand and the working substance is slowly
compressed (perhaps adding weight to piston slowly) causing the substance to warm
adiabatically back to temperature T1 and specic volume VA
This process is illustrated graphically in gure 5. The cycle is reversible and consists of two
isotherms at temperatures T1 and T2 , where T2 < T1 and two adiabats, ie 1 and 2 .
The net work performed by the system is the area formed by the intersection of the four
curves. We can compute the work \A" and the heat \Q" for the four steps of the process
(direction indicated by arrows) in the following table:
RB
VB
(I)
UI = 0 QI = ;AI
=
A PdV = R T1 ln VA
(II) QII = 0 ;AII = ;Uii = CV (T1 ; T2 )
(III) UIII = 0 Q2 = ;AIII = ;R T2 ln VDC
(IV) QIV = 0 ;AIV = ;UIV = ;CV (T1 ; T2 )
Since this is a cyclic process, the sum of the four terms u is zero. The work terms on
the two adiabats cancel each other. The gas eectively absorbs the quantity of heat Q1 > 0
from the warmer reservoir and rejects it Q2 < 0 in the colder reservoir. The total work is
then is W = AI + AI II = ;(Q1 + Q2).
We can now relate q1 and q2 to the temperatures of the two heat reservoirs. From
Poisson's equation,
T1 = VC R =Cp = VD R =Cp
(63)
T2
VB
VA
26
Figure 5: Carnot Cycle (from Iribarne and Godson, 1973)
and so:
V B = VC
(64)
VA VD
which , substituting into the expressions for q1 and q2 gives:
Q1 + Q2 = 0
(65)
T1 T2
or
jQ2 j
T2
=
(66)
Q1 T1
Now we can calculate the thermodynamic eciency of the Carnot engine:
(67)
= T1 T; T2
1
Hence, the Carnot cycle achieves its greatest eciencies when the temperature dierence
between the two heat reservoirs is the greatest!.
It is useful to generalize this result to the case of irreversible processes. To do so, we
consider the Second Law, and its implication that heat can only ow from a warmer toward
a cooler temperature. As a rst step, it is noted that equation 64 is valid for any reversible
cycle preformed between two heat sources T1 and T2, independent of the nature of the cycle
and of the systems. This is Carnot's Theorem .
27
Just as equation 65 holds for a reversible cycle, it can be shown that for any irreversible
cycle, the Second Law requires that: (Q1=T1 ) + (Q2 =T2) < 0, ie heat must ow from warm
to cold and not vise versa. It can also be shown that any reversible cycle can be decomposed
into a number of Carnot cycles (reversible cycles between two adiabats and two isotherms).
Therefore, for any irreversible process it can be shown that:
I Q
(68)
T 0:
which implies equation 57.
The Carnot cycle can also be viewed in reverse, in which case it is a refrigerating machine.
In that case a quantity Q2 of heat is taken from a cold body ( the cold reservoir) and heat
Q1 is given to a hot reservoir. For this to happen, mechanical work must be performed on
the system by the surroundings. In the case of a refrigerator, an electric motor supplies the
heat.
2.6.3 Restatement of First Law with Entropy
The Second Law is a statement of inequality regarding the limits of entropy behavior. When
combined with the First Law, the resulting relationships become inequalities, ie
X
(69)
Tds dU + PdV ; dAk
k
X
TdS dH ; V dP ; dAk
(70)
k
for the internal energy and enthalpy forms respectively. Alternatively, we may write the
special case,
X
dU = TdS ; PdV + dAk
(71)
k
X
dH = TdS + V dP + dAk
(72)
k
Note this diers from previous forms of the rst law in that we have an equation for the
parcel (or system) in terms of state functions only. The dierence between the general
equation 69 and 70 and the special case equations 71 and 72 is a positive source term for
entropy resulting from a non-equilibrium reaction that has not yet been determined. In the
world of dry atmospheric thermodynamics, we can live with assuming the dry system is in
equilibrium and so the second form is sucient. However, when we begin to consider moist
processes, the system is not always in equilibrium, and so the sources for entropy from nonequilibrium processes will have to be evaluated. The framework for the evaluation of these
eects involves the creation of Free energy relations described in the next section.
28
2.6.4 The Free Energy Functions
In their present form, equations 69 and 70 relate thermodynamic functions that involve
dependent variables that include the extensive variable S . It is again convenient to make a
variable transformation to convert the dependence to T instead of S . The transformation
is made by dening the so-called free energy functions called the Helmholtz free energy (F )
and the Gibbs free energy (G) are dened as:
F
G
U ; TS
H ; TS:
(73)
(74)
In derivative form, they are written:
dF = dU ; TdS ; SdT
dG = dH ; TdS ; SdT:
(75)
(76)
For our application to atmospheric problems, we will work primarily with the Gibbs free
energy, because of its reference to a constant pressure process.
Combining equation 76 with equation 72 we obtain:
X
X
dG = ;SdT + V dP + dAk + dGk
(77)
k
k
where Gk refers to Gibbs energies for each constituent of the system (k) necessary to form
an equality of equation 70. The restrictions imposed
by the Second Law and reected by
those inequalities result in the requirement that Pk dGk 0. Because G = G(T V nj ) is an
exact dierential, then
!
!
!
@G
@G
@G
dG = @T
dT + @P
dP + @n
dnk
(78)
Pnk
Tnk
k PTnj
We can evaluate the potential internal energy and Gibbs free energy terms of equation 77
as:
!
X
@U
dAk = @n
dnk
k TP
k
!
X
@G
dGk = @n
dnk :
k TP
k
where G is the Gibbs free energy resulting from chemical potential. The terms of equation
78 are then evaluated:
!
!
!
@G
@U
@G
@nk TP = @nk TP dnk + @nk TP dnk
29
!
@G
@T Vnk =
!
@G
@V Tnk =
;S
;P
(79)
2.6.5 Concept of Equilibrium
Equilibrium is not an absolute, but dened in terms of permitted processes. In statics,
equilibrium is described as the state when the sum of all forces is zero. This may be stated
as a principle of work, ie, If a system is displaced minutely from an equilibrium state by a
change in one of the system properties, the sum of all the energy changes is zero. Neglecting
internal potential energies (Ak ), this leads to the statement that for a single component
system of ideal gas, undergoing a constant temperature, volume process, dF = 0 while for
a constant temperature and constant pressure dG = 0 at equilibrium. Hence if one were to
plot G (or F) as a function of system properties, equilibrium will appear as a maximum or
minimum , and precisely a minimum in the function.
In general, we dene a spontaneous process as one in which the system begins out of
equilibrium and moves toward equilibrium, resulting in an increase in entropy. The rate
of the process is undetermined thermodynamically. Consider the Gibbs free energy for a
constant temperature and constant pressure process. Since equilibrium occurs at a minimum
in Gibbs free energy, and can dene:
A Spontaneous Process G < 0
A Equilibrium Process G = 0
A Forbidden Process G > 0
Dene Molar chemical Potential (of species \k" (k ) as:
!
@G
j @n
:
(80)
k T:pnj
For a system of only one component of a set on non-interacting components:
k dnk = Gk (m) = Hk ; TSk
in which Gk (m) is the molar free energy of species \k". We now write:
X
X
dG = ;SdT + V dP + dAk + k dnk
k
k
(81)
(82)
Hence the third term on the RHS would represent free energies resulting from the potential
for interaction between components in the system, while the last term on the RHS represents
the non interacting free energy of each component, for example surface free energy.
30
Consider a system composed of of a single ideal gas and held in a state of equilibrium
where dG = 0. Ignoring internal potential energies, equation 82 becomes:
d = SdT ; V dP
(83)
Using the equation of state and integrating we obtain:
= o(T ) + nR T ln P
(84)
where o(T ) is the standard chemical potential at unit pressure (usually taken to be 1 atm).
Hence, for an ideal gas with two constituents:
G = ;Wmax = (2 ; 1)(n2 ; n1) = nRT ln p2
(85)
p1
We can also look at the chemical potential of a condensed phase, using the concept of
equilibrium. Consider liquid water in equilibrium with water vapor at a temperature T and
a partial pressure of saturated vapor es. Applying the principle of virtual work, we transfer
dn moles of water from one phase to the other under the condition:
liq dn = vapdn
liq = vap = oliq + Rv T ln es:
(86)
This is a general statement and demonstrates that the chemical potential of any species is
constant throughout the system (if the equilibrium constraints permit transfer of the species).
We can also see that for a non equilibrium phase change:
G = v ; l = Rv T ln eev
(87)
s
Hence if ev < es, then G < 0 for a process of condensation and hence that is a forbidden
process. By the same token, evaporation becomes a spontaneous process. When v = l ,
the system is in equilibrium and at saturation.
These concepts can be extended to include other free energies aecting condensation or
sublimation such as solution eects, curvature eects, to process of chemical equilibrium and
so on.
31
3 Moist Thermodynamics
3.1 Latent Heats - Kircho's Equation
Consider the eect resulting from mass uxes of constituents between constituents of the gas
within the system. In particular, mass uxes between the three possible phases of water are
considered. We can express these phase transformations by modifying equation 44 to be:
X @Hk !
dH = Q + V dP +
ydnk
(88)
k @nk
k
where @H
@nk represents the energy per mole of of the nk constituent of the system. We let the
system of moist air be composed of nd moles of dry air, nv moles of vapor gas, nl moles of
liquid water and ni moles of ice. Then:
@Hl dn + @Hi dn i
dn
dH = Q + V dP + @H
v+
i
@n!l l @n
@nv
v
!
@H
@H
v @Hl
i @Hl
= Q + V dP +
(89)
@nv ; @nl dnv + @ni ; @nl dni
We dene latent heats to be:
!
v @Hl
Lvl = ; @H
;
@n
@nl !
v
i @Hl
Lil = + @H
@ni ; @nl
Liv = Lil + Llv
Hence,
(90)
dH = Q + V dP ; Liv dnv + Lil dni
(91)
In order to derive Kircho's equation, write equation 91 for three adiabatic homogenous
systems consisting of vapor, liquid and ice held at constant pressure:
v
Cpv T = @H
@nv
l
ClT = @H
@nl
@H
CiT = @n i
(92)
i
where Cpv Cl and Ci are the heat capacity of vapor at constant pressure, the heat capacity
of liquid and the heat capacity of ice. Taking dierences among the three equations we nd
32
Phase 1
vapor
vapor
ice
to Phase 2
, liquid
, ice
, liquid
Name
Latent heat of vaporization (condensation)
Latent heat of sublimation
Latent Heat of freezing (melting)
Energy/mole Energy/mass
Lvl
Liv
Lil
lvl
liv
lil
Table 4: Summary of denitions of latent heat. Subscripts i,l,v refer to the phases of ice,liquid
and vapor and the double subscripts exchanges between phases. Note that the notation here
is in keeping with the case conventions of the text. In the literature, however, meteorologists
often express latent heat as an energy per mass but use the upper case L.
the three forms of Kircho's equation:
Llv = C ; C
pv
l
T
Lil = C ; C
l
i
T
Liv = C ; C
(93)
pv
i
T
These expressions are known as Kircho's equation. They can also be written in dierential
form as:
@Llv = C ; C
pv
l
@T
@Lil = C ; C
l
i
@T
@Liv = C ; C
(94)
pv
i
@T
Given heat capacities determined observationally as a function of temperature we can
determine the variation of latent heat with temperature. Kircho' law holds true not only
for phase changes of water, but for the reaction heats of of chemical changes. A similar
relationship can be derived for reactions at constant volume.
Latent heat is dened to be the dierence in enthalpy per mole between any two phases
of water at constant pressure. It may be expressed either in units of energy per mole or
energy per mass:
L = Hn
(95)
l = h:
(96)
For water substance in the atmosphere, atmospheric scientists use the notation and terminology described in table 4.
33
3.2 Gibbs Phase Rule
For the case of a simple homogenous gas, we found that there were three independent variables, being the pressure, temperature and volume assuming the number of moles was specied. Once we require that the gas behave as an ideal gas, the imposition of the equation
of state (and the implicit assumption of equilibrium) reduced the number of independent
variables by one to two.
If we now look at a system consisting entirely of water, but allow for both liquid and gas
forms, and again assume equilibrium, then not only must the vapor obey the ideal gas law,
but the chemical potential of the liquid must equal that of the vapor. This means that if
we know the pressure of the vapor, only one water temperature can be in true equilibrium
with that vapor, ie the temperature which makes the liquid exert a vapor pressure equal to
the vapor pressure in the air. Hence by requiring equilibrium with two phases, the number
of degrees of freedom is reduced to one.
If we allow for all three phases, ie vapor, liquid and ice, then there is only one temperature
and pressure where all three states can exist simultaneously, called the triple point.
Now consider a two component system such as one where we have water and air mixed.
Consider a system allowing only liquid but not the ice phase of water. Now we have to
consider the equilibrium as is applied to the dry air by the ideal gas law in addition to the
equilibrium between the liquid and ice water. For this case, the water alone had one independent variable and the dry air had two, for instance its partial pressure and temperature.
If the vapor pressure is specied, than the temperature of the water is specied and for
equilibrium, so is the air temperature. Hence adding the additional component of air to the
two phase water system increased the number of independent variables to two.
A general statement concerning the number of independent variables of a heterogenous
system is made by Gibbs Phase Rule which states:
= c ; + 2
(97)
where is the number of independent variables (or degrees of freedom) , c is the number
of independent species, and is the number of phases total. Hence for a water-air mixture,
allowing two phases of water and one phase of dry air, there is 1 degree of freedom, ie if we
specify vapor pressure, than liquid temperature is known, air temperature is known, and so
air pressure is xed assuming the volume of the system is specied.
3.3 Phase Equilibrium For Water
Figure 6 shows where equilibrium exists between phases for water. Note that equilibrium
between any two phases is depicted by a line, suggesting one degree of freedom, while all three
phases are only possible at the triple point. Note that the curve for liquid-ice equilibrium is
nearly of constant temperature but weakly slopes to cooler temperatures at higher pressures.
This is attributable to the fact that water increases in volume slightly as it freezes so that
freezing can be inhibited some by applying pressure against the expansion.
34
Figure 6: Phase-transition equilibria from Iribarne and Godson, 1973
Note also that at temperatures below the triple point there are multiple equilibria, ie
one for ice-vapor and another (dashed line) for liquid vapor. This occurs because of the
free energy necessary to initiate a freezing process is high and local equilibrium between the
vapor and liquid phase may exist.
Note that at high temperatures, the vapor pressure curve for water abruptly ends. This
is the critical point (pc) where there is no longer a discontinuity between the liquid and
gaseous phase. As we showed earlier, the latent heat of evaporation lvl decreases with
increasing temperature. It becomes zero at the critical point.
Since the triple point is a well dened singularity, we dene thermodynamic constants
for water at that point and these values are given in table 6. At the critical point, liquid
and vapor become indistinguishable. Critical point statistics are given in table 7. These
phase relationships can also be seen schematically with a Amagat-Andrews diagram (Figure
7). Note how the isotherms follow a hyperbola-like pattern at very high temperatures as
would be expected by the equation of state for an ideal gas. However once the temperature
decrease to values less than TC an isothermal process goes through a transition to liquid as
the system is compressed isothermally. The latent heat release and the loss of volume of the
system keep the pressure constant while the phase change occurs. After the zone of phase
transition is crossed, only liquid (or ice at temperatures below Tt) remains which has very
low compressibility and so a dramatic increase in pressure with further compression.
Phase equilibrium surfaces can be displayed also in three dimensions as a p-V-T surface
and seen in gure 8. Here we see that at very high temperatures, the region where phases
coexist is lost.
Note how the fact that water expands upon freezing results in the kink backward of the
liquid surface. To see this eect, contrast gure 8 with the same gure for a substance which
contracts upon freezing (gure 9). This is contrasted with a 3d Note that we can attain our
35
Variable
Symbol Value
Temperature
Tt
273.16 K
Pressure
pt
610.7 Pa , 6.107 mb
Ice Density
it
917 kg m;3
Liquid Density
lt
1000 kg m;3
Vapor Density
vt
0.005 kg m;3
Specic Volume Ice
it
1.091 x 10;3 m3 kg;1
Specic Volume Liquid
lt
1.000 x 10;3 m3 kg;1
Specic Volume Vapor
vt
2.060 x 102 m3 kg;1
Latent Heat of Condensation lvlt
2.5008 x 106 J kg;1
Latent Heat of Sublimation lvit
2.8345 x 106 J kg;1
Latent Heat of Melting
lilt
0.3337 x 106 J kg;1
Table 5: Triple point values for Water
Variable
Symbol Value
Temperature
Tc
647 K
Pressure
pc
2.22 x 107 Pa (218.8 atm)
Specic Volume Vapor ct
3.07 x 10;3 m3 kg;1
Table 6: Critical point values for Water
Figure 7: Amagat-Andrews diagram (from Iribarne and Godson, 1973)
36
Figure 8: Thermodynamic (p-V-T) Surface for Water Vapor (Sears,1959)
Figure 9: Thermodynamic (P-V-T) Surface for a substance that expands upon freezing.
(Sears,1959)
37
Figure 10: Projection of the p-V-T surface on the p-T and p-V planes. (Sears,1959)
previous saturation curves or phase diagrams by cutting cross-sections through the p-V-T
diagram and constant volume or temperature (gure10).
We can use these gures to view how the phase of a substance will vary under various
conditions, since the phases of the substance must lie on these surfaces. Figure 11 shows one
such system evolution beginning at point \a". Note the liquid system exists at point "a"
under pressure p1. The pressure presumably is exerted by an atmosphere of total pressure
p1 above the surface of the liquid. Holding this pressure constant and heating the liquid,
we see that the system warms with only a small volume increase to point "b" where the
system begins to evolve to a vapor phase at much higher volume as the temperature holds
constant. The evolution from "b" to "c" is commonly called boiling and occurs when the
vapor pressure along the vapor-liquid interface becomes equal to the atmospheric pressure.
Hence we can also view the saturation curves as boiling point curves for various atmospheric
pressures.
Alternatively, if the system cools from point a the volume very slowly decreases until
freezing commences at point "d". There we see the volume more rapidly decrease as freezing
occurs. Of course for water, the volume change is reversed to expansion and is not depicted
for the substance displayed on this diagram.
3.4 Clausius-Clapeyron Equation
We will now express mathematically the relation between the changes in pressure and temperature along an equilibrium curve separating two phases. We know from the assumption
of equilibrium:
ga = gb
a = b
38
(98)
(99)
Figure 11: Projection of the p-V-T surface on the p-T and p-V planes. (Sears,1959)
Ta = Tb
ua = ub
(100)
(101)
where a and b are the two phases. For equilibrium, we require innitesimal changes in
conditions which all the time preserve equilibrium. Hence,
dga
da
dTa
dua
=
=
=
=
dgb
db
dTb
dub
(102)
(103)
(104)
(105)
The Gibbs free energy was dened as:
g = u + p ; Ts
= h ; Ts
(106)
Therefore, by virtue of equation 106,
dga = (dua) ; sadT + (;Tdsa + pda ) + adp
= (dub) ; sb dT + (;Tdsb + pdb) + bdp = dgb
where the terms in parenthesis drop out (because of equation 105 and the First law to give:
;sa dT
+ a dp = ;sb dT + b dp
39
(107)
Figure 12: Cycle related to Clausius Clapeyron equation (Hess, 1959)
Hence, we make the manipulations:
(sb ; sa)dT = (b ; a)dp
sb ; sa = dp
b ; a
dT
From the equation 106,
gb ; ga = hb ; ha ; T (sb ; sa) = 0
and since, lab = hb ; ha , we can write:
(108)
(109)
lab = T (sb ; sa)
(110)
Hence, we can rewrite equation 108 as:
lab
dp =
(111)
dT T (b ; a)
which is the general form of the Clapeyron equation.
The physical meaning of this equation can be illustrated by the process depicted in gure
12. Consider the 4 step process shown. Beginning at T,P in the lower left point of the cycle
we can move to the point T + dT and p + dp in either of the two paths shown. Since g is
a state function, the path does not matter for the change in g and hence the change is g
along both paths can be equated yielding the equation. Hence the equation denes how the
equilibrium vapor pressure must vary with temperature based on the value of latent heat.
3.4.1 Equilibrium between liquid and solid-Clapeyron Equation
Applying the phase equilibrium to a phase transition between liquid and ice yields:
de
dT
= ; T (liil;l)
40
(112)
where we are using the symbol \e" for pressure of the liquid and ice and we will always take
our Latent heat lil as the positive denite dierence in enthalpy between a liquid and frozen
phase where the liquid phase has the greater enthalpy. For a freezing process considered
here, the enthalpy change passing from liquid to ice will be negative. Hence the negative
sign appears on the RHS of eq. 112. Since the volume of the ice is greater than that of the
liquid, the change in vapor pressure with temperature is negative!
3.4.2 Equilibrium between liquid and vapor-Clausius-Clapeyron Equation
Applying the phase equilibrium to a phase transition between liquid and vapor under equilibrium conditions yields:
des = ; lvl
(113)
dT
T ( ; )
l
v
where es is the vapor pressure of the equilibrium state, and the latent heat of vaporization
lvl is dened as the absolute value of the dierence in enthalpies between the liquid and ice
phase. Since the enthalpy of the vapor state is higher than that of the liquid, the negative
sign appears in front of equation 113. To a very good approximation, we can drop the
specic volume of liquid water compared to vapor. Hence, applying the equation of state to
the vapor specic volume:
des = lvl es
(114)
dT Rv T 2
or
d ln es = lvl
(115)
dT
Rv T 2
This is the Clausius-Clapeyron equation and we will use it for a number of applications.
The equation tells us how the saturation vapor pressure varies with temperature. Hence
given an observation point, such as the triple point, we can determine es at all other points
on the phase-equilibrium diagram.
3.4.3 Equilibrium between Ice and Vapor
Similarly, for equilibrium between ice and vapor, we can write:
d ln e
dT
= RlivveTsi2
(116)
where we again take lil to be the absolute value of the dierence in enthalpy between the
vapor and solid phase of water.
3.4.4 Computation of Saturation Vapor Pressure
For a precise integration, one must know how the latent heat varies as a function of T. To a
good rst approximation, let temperature be independent of L and integrating, yield:
Z
Z
l
vl dT
d ln e = R T 2
v
41
ln ee0s =
or we can write it in the form:
s
;
lvl T ; T 0
Rv TT 0
(117)
ln es = ; lvl + const:
(118)
Rv T
This formulation has assumed latent heat to be independent of temperature. This is a
good approximation for sublimation, but not as good for freezing or condensation. We can
improve on this by using our formulation for latent heat variation and specic heat variation
with temperature yielding the series expansion:
"
#
l
1
0
2
(119)
ln es = R ; T + ln T + 2 T + 6 T + + const
v
where the integration constant is determined empirically.
We need not achieve this accuracy for most meteorological considerations, since we can
not measure vapor pressure that precisely anyway. Hence we can make approximations to
that allow us to calculate es or esi with sucient accuracy. Here are two approximations:
1. Solving for the constant at Tt, and assuming Latent heat constant
es = 10(9:4051;2354=T )
(120)
(10:5553;2667=T )
esi = 10
(121)
2. Assuming heat capacity constant and retaining two terms in a series, we get Magnus's
formula (liquid-vapor only)
es = 10(; 2937T :4 ;4:9283 log T +23:5518)
(122)
this has been simplied by Tetens (1930) and later by Murray (1966) to the easily
computed form (for millibars):
es = 6:1078 exp
h a(T ;273:16) i
(T ;b)
(123)
where the constants are dened as:
3. Many atmospheric scientists use the Go-Gratch (1946) formulation given by:
T T s
10
es = 7:95357242x10 exp ;18:1972839 T + 5:02808 ln Ts
2
3
;
26
:
1205253
Ts 5
;70242:1852 exp 4
T
+58:0691913 exp ;8:03945282 TTs
(124)
42
water
ice
a 17.2693882 21.8745584
b 35.86
7.66
Table 7: Constants for Teten's Formula
for saturation over liquid where Ts = 373:16K . For saturation over ice:
T T 0
10
es = 5:57185606x10 exp ;20:947031 T ; 3:56654 ln T0
9
=
2
:
01889049
; T 0
T
(125)
where T0 = 273:16K .
tables 8 and 9 show how the accuracy of Teten's formula compares to Go-Gratch as
reported by Murray (1966):
Both of these forms of saturation vapor pressure are commonly used as a basis to compute saturation vapor pressure. However because of the computational expense of using
exponentials, the values are often computed at 1K intervals to make a table of saturation
with pressure, and then vapor pressures are interpolated linearly from the table.
3.5 General Theory For Mixed Phase Processes Within Open Systems
Often thermodynamic theory is applied to precipitating cloud systems containing liquid and
ice which are not in equilibrium. In many thermodynamic texts, the simplifying assumption
of equilibrium is made which can invalidate the result. Moreover, ice processes are often
neglected, despite the existence of ice processes within the vast majority of precipitating
clouds. We will adopt the generalized approach to the moist thermodynamics problem,
developed by Dutton (1973) and solve for the governing equations for a non-equilibrium
three phase system rst and then nd the equilibrium solution as a special case.
Besides the non-equilibrium phase changes, the formation of liquid and ice hydrometeors
create a heterogenous system having components of gas, solids and liquids, each of which
may be moving at dierent velocities. We must therefore forgo our traditional assumption of
an adiabatic system and instead consider an open system. In doing so, we x our coordinate
system relative to the center of the dry air parcel . Although we can assume that vapor will
remain stationary relative to the dry air parcel, we must consider movements of the liquid
and solid components of the system, relative to the parcel, hence implying diabatic eects.
We must therefore generalize the derivation of equation 91 to include these eects. It will
be convenient to work in the system of specic energies (energy per mass).
43
Table 8: Saturation Vapor pressure over liquid
44
Table 9: Saturation Vapor pressure over ice
We will assume that out system is composed of several constituents, each of mass mj .
Then the total mass is:
X
m = mj :
(126)
j
We account for both internal changes (di) which result from exchanges between phases
internal to the parcel and external changes (de) which result from uxes of mass into and
out of the parcel. The total change in mass is thus:
dmj = dimj + demj :
(127)
By denition and for mass conservation, we require:
X
dim =
dimj = 0
j
X
de m =
demj = deml + demi
j
(128)
(129)
As discussed above, external mass changes are restricted to liquid or ice constituents,
while gas constituents are assumed not to move relative to the system. An exception can be
made, although not considered here, for small scale turbulent uxes relative to the parcel.
From our original discussion of the First Law, we can write the generalized form of equation
91 in a relative mass form as:
X
(130)
dH = Q + V dP + hj dmj j
where
@Hj
hj = @m
j
45
(131)
We found earlier that the terms involving hj eventually result in the latent heat term.
We can split the heating function, Q = Qi + Qe . Here Qi is the diabatic change due to
energy owing into our out of the system that does not involve a mass ow. Qe accounts
for diabatic heat uxes resulting from mass uxes into or out of the system. The diabatic
heat source of conduction between the parcel and an outside source would be accounted for
in Qi if no mass back and forth ow into and out of the parcel is not explicitly represented.
Even the case of turbulent heat transfer into and out of the parcel would aect Qi unless
the explicit uxes of mass in and out were represented. The heat ux associated with the
movement of precipitation featuring a dierent temperature than the parcel, into and out of
the parcel, would also be represented by Qe, as it would appear as a dierent enthalpy for
the precipitation constituent, and be accounted for by the demj term. Hence, we can assume
Qe = 0.
Qi = diH ; V dp ;
0 =
X
hj dimj
j
X
deH ; (hj ; h)demj :
j
(132)
(133)
We can similarly split the enthalpy tendency into its internal and external parts:
X
dHi = Qi + V dp ; hj dimj
(134)
j
X
dHe = ; (hj ; h)demj :
(135)
j
Ignoring the non-interacting free energies, Gibbs' relation for the mixed phase system
(equation 82 is written in \per mass" form as:
X
X
(136)
TdS = dH ; V dp ; hj dmj ; j dmj
j
j
where j is the chemical potential per mass rather than per mole as we rst dened it. We
can now divide the entropy change between internal and external changes, and employing
equation 132
X
X
TdiS = Qi ; hj dimj ; j dimj
(137)
j
j
X
TdeS = ; j demj (138)
j
for the external change. Since, j = gj = hj ; Tsj , and equation 133, then
X
X
TdeS = ; hj demj ; Tsj demj :
j
j
46
(139)
Now we apply our results to the particular water - air system. Consider a mixture
composed of md grams of dry air, mv grams of vapor, ml grams of liquid water, and mi
grams of ice. We require:
dimd = demd = 0
dimv + diml + dimi = 0
demv = 0
(140)
The internal enthalpy equation (equation 134 and internal entropy equation (equation
137) can then be written for this mixed phase system:
X
diH = Qi + V dp + hj dimj
(141)
j
P
P
h
j dimj
Q
j di mj
i
j
diS = T ;
; j
(142)
T
T
For the air water system equation 142 becomes:
diS = ; hTl diml ; hTl dimi ; hTv dimv ; Tl diml ; Tl dimi ; Tv dimv + QTi (143)
= ; hl ; hv dimv + hl ; hi dimi ; l ; v dimv + l ; i dimi + Qi
T
T
T
T
T
l
l
a
a
Q
= ; iv mv + il dimi ; iv dimv + il dimi + i
(144)
di
T
T
T
T
where the avi and ail are the specic anities of sublimation and melting respectively. The
anity terms are nite when the chemical potential between two existing phases diers,
making the system out of equilibrium with respect to phase change. The anity is the
Gibbs free energy available to drive a process, and unavailable to perform work. Notice the
symmetry between the anity declarations and the latent heat declarations.
If we apply equation 144 to adiabatic melting, condensation and sublimation, we obtain:
T (sl ; si) = lil + ail
T (sl ; sv ) = llv + alv
T (sv ; si) = liv + aiv :
(145)
Moreover, we can state that the total entropy is equal to the sum of the entropy in each
system component:
S = mdsd + mv sv + ml sl + misi:
(146)
where the the dry air entropy is:
sd cpd ln T ; Rd ln pd
47
(147)
We have required that the only external uxes are those of precipitation. Hence, we nd:
diS = dS ; deS
= dS ; sl deml ; sidemi
= d(mdsd + mv sv ) + sidiml
+sidimi + ml dsl + mi dsi = ATlv dimv ; ATil dimi + QTi :
Combining equations 145 - 145 and equation 140 to eliminate disv with
obtain:
d mdsd + mTv llv + mv d ATlv
+(mv + ml )dsl ; lTil di mi ; mi d lTil + d ATil ; dsi =
d md sd + mTv llv ; di wTi lil + mv d ATlv
;mid ATil + (mv + ml + mi)dsl =
(148)
equation 148 we
Qi
T
Qi
T
(149)
if we assume that the liquid is approximately incompressible, then
(150)
dsl = cl dT
T:
If we dene mixing ratio of the jth constituent as:
mj
rj = m
(151)
d
and divide equation 149 by md , then we obtain the following general relation:
n
o
d cpd ln T ; Rd ln pd + rvTllv ; di riTlil + rv d ATlv ; ri d ATil + (rv + rl + ri )cl dTT = qTi :
(152)
This rather remarkable equation contains essentially no signicant degree of approximation to
thermodynamic systems in its application to the Earth's lower atmosphere. The dierentials
account for the inter-relationship between temperature change, pressure change and liquid
phase change. In addition, additional diabatic heating tendencies by radiative transfer,
molecular diusion, eddy diusion can be accounted for with the internal heating term qi .
Neglected, are the eects of chemical reaction on entropy, although the heating eect can be
included in the heating term. Generally, these eects are small and can be neglected.
What is included are not only the latent heating eects of the equilibrium reaction, but
the eects on heating resulting from the entropy change in non-equilibrium reactions. Hence,
since entropy change removes energy from that available for work, non-equilibrium heating
from phase change modies the enthalpy change resulting from phase change. The anity
terms account for these eects.
48
Notice that both anity terms are inexact dierentials, suggesting that their eect is
irreversible. The heat storage term (last term on the LHS) is also an exact dierential if we
neglect the variation of cl and we assume the total water,
rT = rv + rl + ri
(153)
is a constant. If there is a change, ie a diabatic loss or gain of moisture by precipitation,
then the dierential is inexact and obviously the process is irreversible. This term also also
contains all of the net eects of diabatic uxes of moisture for systems which are otherwise
in equilibrium between phases. The heating term, qi is purely diabatic and denes a diabatic thermal forcing on the system such as by radiative transfer or molecular diusion and
turbulence.
Note, that although the rst term on the LHS is written as a total derivative, the external
derivative of the quantity in brackets is in fact zero. Hence all reversible terms appear as
internal derivatives while the reversible terms contain both internal derivatives for moist
adiabatic process and external derivatives for diabatic ux terms.
3.6 Enthalpy Form of the First-Second Law
We now derive what is perhaps a more commonly used form of equation 152. Whereas
equation 152 is in a form representing entropy change, it can be rewritten as an enthalpy
change. To do so we make the manipulations:
First, we make the following assumptions:
1. Neglect the curvature eects of droplets
2. Neglect solution eects of droplets
These assumptions are really quite unimportant for the macro system of uid parcels. However the assumptions would be critical when discussing the micro-system of the droplet itself,
since these aects strongly inuence nucleation and the early growth of very small droplets.
Hence microphysical models of droplet evolution will need to consider the contribution of
these eects to chemical potential. With these assumptions, the chemical potential of the
vapor is dened as shown in the previous section as:
l = o + Rv T ln ev
(154)
where ev is the atmospheric partial pressure of the vapor. The chemical potential of the
liquid and ice are dened as the chemical potential of vapor which would be in equilibrium
with a plane pure surface of the liquid or ice and are given by:
l = o + Rv T ln es
i = o + Rv T ln esi
49
(155)
(156)
where es and esi are the saturation vapor pressures of the ice and liquid dened by the
temperature of the liquid or ice particle. This temperature need not be the same as the
vapor temperature T for this equation.
From the equation of state, we write the relationships:
ev = v Rv T = d rv Rv T
pd = dRd T
Now combine to eliminate d T and log-dierentiate:
pd = ev
Rd
rv Rv
pd rv Rv = Rdev
d ln pd + d ln rv = d ln ev
Now combine the two equations of state to get a single equation for total pressure:
p = pd + ev
= d Rd T + v Rv T v
= d Rd T 1 + rv R
Rd d
= d Rd T 1 + rv M
M
v
r
v
= pd 1 + where = Md =Mv = 1:61 Next dierentiate this equation:
r
dp = 1 + v dpd + pd drv
rv dp = 1 + dpd + drv
rv
pd r
v
p
rv
rv d
dp = 1 + d ln p + d ln r
d
v
rv
rv pd
Now substitute for d ln rv :
rv dp = 1 + d ln p + d ln e ; d ln p
d
v
rv
rv pd
dp = 1 + rv d ln p + rv d ln e ; rv d ln p
d
v
d
pd
dp = d ln p + rv d ln e
d
v
pd
50
(157)
(158)
(159)
(160)
(161)
(162)
Now, since:
r
v
p = pd 1 + dp = dp 1 + rv pd
p dp = d ln p 1 + rv
(163)
pd
Hence, substituting in, we attain the identity:
d ln p 1 + rv = d ln pd + rv d ln ev
(164)
This nally leads to the useful identity relating the change in the log of total pressure to the
change in log of partial pressures of vapor and dry air:
d ln p(Rd + rv Rv ) = Rdd ln pd + rv Rv d ln ev
(165)
Now, we have dened our variations in Latent heat with temperature as:
dliv = c ; c
pv
l
dT
dlil = c ; c
l
i
dT
lil = liv + llv :
(166)
(167)
(168)
Combining these and our identity with equation 152, we obtain:
cpm d ln T ; Rm d ln p + lTlv drv ; lTil diri
1
1
q
+(rv + ri) llv d T + Rv d ln es ; ri liv d T + Rv d ln esi = Ti
(169)
where cpm = cp +rv cvp +ri ci +rl cl is the eective heat capacity of moist air and Rm = Rd +rv Rv
is the moist gas constant (not to be confused with the gas constant of moist air.
If we dene the saturation vapor pressure to vary as the equilibrium vapor pressure over
a plane surface of pure water, as we do by denition, then the terms in the brackets vanish
by virtue of the Clausius-Clapeyron relationship! The following more simple form of the
First-Second Law results:
cpmd ln T ; Rm d ln p + lTlv drv ; lTil diri = qTi
(170)
Although equation 170 appears dierent from equation 152, it contains no additional approximations other than the neglect of the curvature and solution eects implicit in the assumed
51
form of chemical potential. It is simpler and easier to solve than the other form, because the
anity terms and latent heat storage terms are gone. Note, however that there are some
subtle inconveniences. In particular, each term is an inexact dierential . which means that
they will not vanish for a cyclic process. Hence it would be dicult to integrate equation
170 analytically. Nevertheless, it is a convenient form for applications such as a numerical
integration of the temperature change during a thermodynamic process.
Some of the eects of precipitation falling into or out of the system are included in
equation 170 implicitly. To see this look at the change in vapor. It is a total derivative
because only internal changes are allowed. The ice change, on the other hand is strictly
written as an internal change. Hence it is the internal change that implies a phase change,
and knowing the ice phase change and liquid phase change, the liquid phase change is
implicitly detrained since the total of all internal phase changes are zero. Since, by virtue
of the assumption that a heterogenous system is composed of multiple homogenous systems,
we assumed that hydrometeors falling into or out of the system all have the same enthalpy
as those in the system itself, there is no explicit aect on temperature.
This assumption can have important implications. For instance, frontal fog forms when
warm droplets fall into a warm parcel, hence providing an external ux of heat. We neglect
this eect implicitly P
when we neglect a variation in the enthalpy of each system component.
By requiring deH = j de(mj hj ) = Pj hj demj , we only considered the external changes due
to an external ux of water with same enthalpy of the parcel. The neglect of these eects
is consistent with the pseudo-adiabatic assumption which is often made. That assumption
assumes condensed water immediately disappears from the system, and so the heat storage
eects within the system and for parcels falling into or out of the system can be neglected.
So far, the pseudo adiabatic assumption has only been partially made, because we still retain
the heat storage terms of the liquid and ice phases within the system. It is unclear whether
there is an advantage of retaining them only partially.
3.7 Humidity Variables
Water vapor, unlike the \dry" gases in the atmosphere exists in varying percentages of the
total air mass. Obviously, dening the amount of vapor is critical to understanding the
thermodynamics of the water-air system. We have developed a number of variables to dene
the vapor, liquid and ice contents of the atmosphere. Below is a list of the variables used to
dene water content.
3.7.1 Vapor Pressure (ev )
Vapor pressure represents the partial pressure of the vapor and is measured in Pascals. The
saturation vapor pressure over a plane surface of pure liquid water is dened to be es while
the vapor pressure exerted by a plane surface of pure ice is esi.
52
3.7.2 Mixing Ratio and Specic Humidity
We have already introduced mixing ratio to be
rv = v
d
Now relate mixing ratio to vapor pressure:
(171)
rv = v
d
ev
= RpvdT
=
=
=
This is sometimes approximated as:
Rd T
ev Rd
pd Rv
ev
pd
ev :
p ; ev
rv epv :
Similarly, we dene specic humidity to be:
qv = v
Now relate specic humidity to vapor pressure:
qv = v
ev
R
vT
=
p
(Rd +rv Rv )T
(172)
(173)
(174)
Rd
= ev
p (Rd + rv Rv )
e
(175)
= v 1 rv
p (1 + )
This is also sometimes approximated as:
qv epv :
(176)
which, with the approximation, would be identical to rv . In general, we relate mixing ratio
to specic humidity as:
qv = v 53
= +v d
v
1
= 1
rv + 1
= 1 +rv r
v
(177)
3.7.3 Relative Humidity
We dene relative humidity to be the ratio of the vapor pressure to the vapor pressure
exerted by a plain surface of pure water. There is a relative humidity for liquid (Hl ) and a
relative humidity over ice (Hi).
Hl = eev
s
e
Hi = e v
si
(178)
(179)
which is approximately equal to:
rv
rs
Hi0 rrv
si
where rs and rsi are the saturation mixing ratios over liquid and ice.
Hl0
(180)
(181)
3.8 Temperature Variables
3.8.1 Virtual Temperature
We now apply the ideal gas law to the mixture of air and vapor similar to how we applied it
to the mixture of oxygen and nitrogen in section 1. The total pressure is given by Dalton's
Law to be:
p = pd + ev
(182)
and applying the equation of state separately to the vapor and dry air components :
R T + R T
p = d M
v
Mv
d
d
v = ( M + M )R T
d
v
d
v Md
R
+M
= (d + v ) M T M
+
1 d r d M v d v
d
= Rd T
+ v
Md Mv 1 + rv
54
1 + MMvd rv
= Rd T
#
" 1 + rv
(1
+
1
:
61
r
v)
)
= Rd T
1 + rv
' Rd T (1 + 1:61rv )(1 ; rv )
= Rd T (1 + 1:61rv ; rv ; 1:61rv2)
' Rd T (1 + 0:61rv )
(183)
It is a practice in atmospheric science to incorporate the eect of vapor into the temperature by dening the virtual temperature (Tv ) which contains this eect. Hence, we dene:
Tv = T (1: + 0:61rv )
(184)
and so the equation of state becomes:
p = Rd Tv
(185)
Note the eect of adding moisture is to increase the virtual temperature over the temperature.
Since the total density is a function of the pressure and virtual temperature, the addition
of moisture then lowers the air density, ie humid air is actually less dense, counter to our
tuition which makes us say how \heavy" the air is on a humid day!
3.8.2 Dew Point Temperature (Td)
This is the temperature to which moist air must be cooled at constant pressure and vapor
mixing ratio in order become saturated over a plane surface of pure water.
Dew point temperature can be calculated with the following algorithm:
:2325
(186)
Td = 35:86lnlne e;s ;234947
:6837
s
where es is in mb and Td is in Celsius.
A Frost Point Temperature is also dened analogously for saturation over a plane surface
of pure ice.
3.8.3 Wet Bulb Temperature (Tw )
The wet bulb temperature (Tw ) may be dened in two ways:
1. Isobaric Process. Tw is the temperature to which air will cool by evaporating water
into it at constant pressure until it is saturated. The latent heat is assumed to be
supplied by the air. Note that rv is not kept constant and so Tw diers from Td
55
When measured by a psychrometer , the air is caused to move rapidly past two thermometer bulbs, one dry and the other shrouded my a water soaked cloth. When
thermal equilibrium is reached on the wet bulb, the loss of heat by air owing past the
wet bulb must equal the the sensible heat which is transformed to latent heat. Hence,
(T ; Tw )(cp + rv cpv ) ; (rs(Tw p) ; rv )lvl
(187)
where T is the temperature of the air approaching the wet bulb.
Given temperature and mixing ratio, and a suitable relation for obtaining rs(Tw ) and
lvl , one can solve for Tw . Alternatively one can measure T and Tw directly with a
psychrometer, and knowing pressure solve for rv .
2. Adiabatic Process. One can nd Tw by :
(a) Begin with pressure and mixing ratio
(b) Reduce pressure dry-adiabatically until saturation is reached to nd temperature
and pressure of Lowest Condensation Level
(c) Increase pressure moist-adiabatically form LCL to original pressure
(d) The temperature at the original pressure is Tw .
This is sometimes called the wet-adiabatic wet-bulb temperature. It diers at most a
few tenths of a degree from the other wet bulb temperature.
56
3.9 Entropy variables for Moist air
3.9.1 Potential Temperature for Moist Air
We earlier derived the Poisson's equation for relative to a dry air parcel. Later we pointed
out that the conservation of in a dry parcel, was equivalent to the conservation of entropy
for a dry adiabatic, reversible process. Hence, we could show that
cpd ln = ds:
(188)
We now extend our denition of to a system containing vapor gas as well as dry air.
To derive this, we return to equation 170 and assume and adiabatic reversible system. We
then obtain:
d(cp + rv cl ) ln T ] ; d(Rd + rv ) ln p] = Tqi
d(cpm ln T ) ; d(Rm ln p) = qTi
cpmd ln m = d(cpm ln T ) ; d(Rm ln p) = qTi
d ln m = d ln T ; cRm d ln p = c qi T
(189)
pm
pm
where we have factors the cpm and Rm because they are constant for an adiabatic process.
Solving for m to be the temperature at 1000 mb pressure, we obtain:
Rm
m = T ppoo cpm
(190)
Now, solve for the form of the diabatic forcing:
qi
dm = Tm cpm
(191)
This denition of diers only about 1% from the previous denition neglecting the moisture
eect. As a result, the dierence is usually neglected in most applications to atmospheric
science.
3.9.2 Equivalent Potential Temperature
We will now nd the potential temperature for an air parcel undergoing phase transition.
We again assume an adiabatic, reversible system with multiple phases. Of course, this
is only possible at the triple point temperature, otherwise the ice and liquid cannot both
be in equilibrium with vapor at the same time. Equation 152 eectively depicts entropy
change for an irreversible system, hence the total temperature change of a process, will be
dierent depending on path. Moreover, there is not a general condition of equilibrium at
any temperature to reduce this equation to an exact dierential. Since entropy is a state
57
variable, the entropy of the nal state is determined by the state parameters of the nal
state, which themselves are dependent on path. Hence we cannot integrate equation 152 to
nd this entropy.
Nevertheless, as previously stated, entropy is an absolute, and according to equation 146,
it is a sum of the entropies of each component, which for specic entropy is written:
s = sd + rv sv + rl sl + risi
(192)
Employing the denition of specic entropy (equation 72) we can write:
sd
sv
svl
svi
=
=
=
=
cp ln T ; Rd ln pd
cpv ln T ; Rv ln ev
cpv ln T ; Rv ln es
cpv ln T ; Rv ln esi:
(193)
(194)
(195)
(196)
where svl is the equilibrium entropy of the liquid surface and svi is the equilibrium entropy
over the ice surface, dened by the entropy of vapor at saturation vapor pressure as given
by the Clausius Clapeyron equation. The entropies for pure liquid and ice are respectively:
sl = cl ln T
si = ci ln T:
(197)
(198)
Now substituting equations (153, 145-145, 193-198) into equation 192 we obtain an equation describing the total entropy of a mixed phase system:
(199)
s = cpd + rl cl + (rT ; rl )] ln T ; Rd ln pd + rvTlvi + rv ATiv :
Although the eects of curvature, solution, chemical changes and other fairly minor considerations have been neglected, this equation accurately denes the total specic entropy of
a parcel. Under equilibrium conditions, we would expect s to be invariant. In fact, if we
dierentiate equation 199, we can seperate ds, within equation 152 to obtain:
!
qi
dsirreversible = de lTil ri ; ATlv drv + ATil dri ; cl dT
d
(200)
e rT +
T
T
where we see the irreversible eect of non equilibrium is properly related to the amount of
phase transition occurring under non equilibrium conditions.
The heat storage term dierence turns out to be related to the external derivative of total
water, hence a diabatic mass tendency. It is missing from the entropy dierential because it
is a diabatic eect. Finally, the qi is the diabatic thermal tendency which is missing because
it is fundamentally diabatic. Equation 152 tells us how the non equilibrium system develops
as a function of path while equationn 199 denes the entropy at any time. The dierence
between the two equations, given by equation 200, tells us the source term of entropy for a
58
process. It does nottell us the change in entropy of the universe, only of the system as a
result of irreversibility.
It will be advantageous to employ equation 199 to solve for an entropy based temperature
for the mixed phase system. Since it is an exact dierential we can use the procedures adopted
for . We rst, however use the assumptions we made for equilibrium with equations 193-196.
We dene We dene the equivalent potential temperature (e ) as:
cpl ln e s + Rd ln poo
(201)
where cpl is given by
(202)
cpl cpd + rl cl + (rT ; rl )ci:
Substituting equation 199 into equation 201, we obtain the dening expression for e :
poo cRpld
; rvcplRv lcivplrTv
e = T pd (Hi)
e :
(203)
In a way analogous to for dry adiabatic reversible processes, e is conserved for all moist
adiabatic processes carried out at equilibrium. We can easily relate the changes of e to the
irreversible changes in entropy:
de = ceT lil deri + Rv T ln Hl (drv + diri) ; Rv T ln Hidiri ; cl T ln Tde(rl + ri) + qi] (204)
pl
Note the inclusion of the external derivative for ice. This demonstrates that for the case of
an adiabatic system in equilibrium, and neglect of heat capacity of liquid, e is conserved.
Note that the more vapor in the air the greater the e. It is proportional to the amount
of potential energy in the air by the eects of temperature, latent heat and even geopotential
energy combined. Hence e tends to be high on humid warm days, and low on cool and/or
dry days. It tends to be higher at high elevations than low elevations. The statement:
\The higher the e is at low levels, the greater the potential for strong moist convection", is
analogous to the statement: \The higher the at low levels, the greater the potential for dry
convection". Where as, we look at a tank of relatively incompressible water and say warm
water on the bottom rises, we look at the dry atmosphere and say warm rises and we look
at the moist atmosphere and we say warm e rises (although the air must be saturated for
the process to be analogous for e).
The eects of irreversibility tend to be less than 1% and so one cold drop the humidity
terms of equation 204 with reasonable accuracy . In fact if we ignore diabatic heating eects
such as radiation, friction and heat conduction at the surface, e is nearly perfectly conserved
in the atmosphere! e acts like a tracer such that one can identify an air parcel from its
e and trace where it came from, even in the midst of moist processes. For instance when
you feel a cool draft of air as a thunderstorm passes, it usually has a lower e which must
have been entrained into the thunderstorm from somewhere. Since e can be low because
of low humidity or because of low temperature (or both) we can see strong e structure in
the atmosphere comprising the storm environment. Normally, we nd the cold downdraft
originated from a layer of dry air, with the same e values as the cool downdraft but up at
mid levels, and we can prove that it came from there because we know e is conserved!
59
3.9.3 Ice-Liquid Water Potential Temperature
Examination of equation 204 will reveal that there are some inconveniences in its application.
In particular, note that the equivalent potential temperature is dependent of vapor and hence
contains similar information to the vapor eld. It could be advantageous to have a conservative temperature variable that does not duplicate the information of the vapor variable.
In addition, notice that the denition makes reference to four mixing ratio quantities, when
there are only three unique mixing ratios. Moreover, in the absence of condensation there
is merit to having the temperature variable reduce to the moist air temperature variable,
which in the absence of vapor reduces to a dry air temperature variable. To do so it is most
convenient to return to equation 152 and eliminate the vapor variable. To do so, we follow
a similar procedure to the one which we used during the development of equation 170. We
will make extensive use of equation 153 and the relationship:
drv = dirv
= dirT ; dirl ; diri
= ;dirl ; diri
(205)
Substituting both into equation 152 we attain:
d(cpdln T ) !+ di(rTcl ln T!) ; Rdd ln pd!
+di rT lTvl ; di rl lTvl ; di ri lTvi
A A q
A vl
vl
+rT di T ; rl di T ; ridi Tvi = Ti :
Regrouping terms, we obtain:
d(cpd ln T )!+ di(rT cl ln !T ) ; Rd d ln p!d
+rT di lvl + Avl ; di rl lvl ; di ri lvi
T T
T
T
A A vl
;rl di
; ri d i
vi
T
T
We now employ equations 154, 155, 156, 194, 197, 195 to get:
= Tqi
d(cpd ln T ) + di(rT cl ln T ) ; Rd d ln pd
+rT di (cl ln T + cpv ln T ; Rv ln es +Rv ln!es ; Rv ln ev!)
lvl ; d r lvi
;di rl
i i
T
T
;rl di (Rv ln es ; ln ev ) ; ri di (Rv ln esi ; Rv ln ev )
60
= qi
T
and with further manipulation:
d(cpd + rT cpv ) ln T ] ; Rddln pd !; rT Rvdi ln e!v
lvl ; d r lvi
;di rl
i i
T
T
Using equation 165, we get:
+rl di (Rv ln Hl ) + ridi (Rv ln Hi) = qTi
d(cpd + rT cpv ) ln T ] ; (Rd d ln pd + rv Rv d ln ev ); (rl!+ ri)Rv d ln e!v
lvl ; d r lvi
;di rl
i i
T
T
and so:
+rl di (Rv ln Hl ) + ridi (Rv ln Hi) = qTi
d(cpd + rT cpv ) ln T ] ; (Rd + rv Rv )d ln p; (rl!+ ri)Rv d ln e!v
lvl ; d r lvi
;di rl
i i
T
T
+rl di (Rv ln Hl ) + ridi (Rv ln Hi) = qTi
!
d(cpd + rT cpv ) ln T ] ; (Rd + rT Rv )d ln p ; di rl lTvl ; di ri lTvi
+di (rl Rv ln Hl ) + di (riRv ln Hi)
= qi + (Rv ln Hl ) dirl + (Rv ln Hi) diri + (rl + ri)Rv d ln ev
T
p
!
(206)
If we now redene Rm and cpm to the more general form:
Rm = Rd + rT Rv
cpm = cpd + rT cpv
(207)
(208)
Then we attain the following form of the First-Second Law:
!
(
r
l lvl + ri lvi )
di(cpm ln T ) ; di(Rm ln p) ; di
+ di (rl Rv ln Hl ) + di (riRv ln Hi)
T
= qi + (Rv ln Hl ) dirl + (Rv ln Hi) diri + (rl + ri)Rv d ln ev
(209)
T
p
Now we see that the exact integrals are conned to the left hand side and the inexact to
the right hand side. Note that we have designed our arrangement so that cpm and Rm are
61
independent of internal changes. Hence, We can dene our il variable from the LHS as:
d ln il = LHS
cpm
!
R
(
r
R
R
m
l lvl + ri lvi )
v
v
= d ln T ; c ln p ; c T + rl c ln Hl + ri c ln Hi
pm
and then integrating:
pm
rl Rv
ri Rv
pm
(lvl rl +liv ri )
pm
(210)
il = m Hl cpm Hicpm e; cpmT
(211)
Notice that as the total condensate vanishes, il reduces to m which is our potential temperature for moist air, and as rT vanishes, it reduces to the dry air potential temperature.
Now we utilize equation 209 to dene the source terms to il :
d ln il = RHS
cpm
! !
!
(
r
R
R
q
l lvl + ri lvi )
v
v
i
+ c ln Hl dirl + c ln Hi diri
= c T ; de
cpmT
pm
pm
pm
+(rl + ri) Rv d ln ev
cpm p
and so:
(
!
(
r
il 1
l lvl + ri lvi )
dil = T c qi ; de
+ Rv T ln Hldirl + Rv T ln Hidiri
T
pm
)
e
v
(212)
+(rl + ri )Rv Td ln p
The variable il now has most of the advantages we sought. Notice that il is invariant
except for the diabatic forcing qi, two terms involving non-equilibrium processes, and the
last term on the RHS of equation 212 which is not insignicant. To examine this term, note
that part of the term is proportional to the d ln ev . Assuming that the system ev is near
saturation in the presence of condensate, we know from the Clausius Clapeyron equation
that :
Rv Td ln ev = ;llv dT
(213)
T
which means that the term increases in importance (in the presence of condensate) exponentially as the temperature decreases. It turns out, that omission of this term can lead to
serious error a low temperatures, as exist in the upper troposphere.
Unfortunately, the need to include this term makes the tendency of il dependent on
the internal change of vapor. That means that the condensation process cannot be treated
implicitly, and knowledge of the internal change of vapor is necessary to evaluate this term.
62
Typically, however, il has been approximated by dropping this term. Initially il was used
only for the liquid phase and in the boundary layer where the term is unimportant. Later,
attempts to extend the use of il to deep convection models were thwarted, as scientists
discovered the importance of this term at cold temperatures.
It would seem that e would be free of such error, but that is a subtle problem. Notice
that is dened with the dry air pressure pd rather than p as is il . Also, note that e
does not have m as the core as does il . Now if you examine the approximate forms of
e used, they usually use the dry as the core , ie from the e denition, they eectively
approximate:
Rd
Rd
T (poopd ) cpl ' T (poop) cp
(214)
which at rst examination seems to be an innocent statement that the dry air pressure is
a good approximation for the total pressure and the moisture eect on the exponentials is
small. However, if we use equation 165 to determine the exact error terms involved, we nd
that a term similar to equation eq:err1appears and also leads to an exponential increase of
error at low temperature! Hence, unless we do not make approximations, e suers from the
same error as il . Nevertheless, there seems to not be a serious error in e , so long as the
partial pressure of dry air is used .
3.10 Simplifying Approximations
3.10.1 Pseudo-adiabatic Process
Because entropy is a function of pressure, temperature, and liquid and ice water, a multiphase process cannot be represented on a two-dimensional thermodynamic diagram. It is
convenient to dene a pseudo-adiabatic process, where the heat capacity of liquid and ice is
neglected.
Omitting the liquid water term (and references to ice) from equation 199, and dierentiating we can write:
!
(215)
dsp = (cpd + rv cl )d ln T ; Rd d ln pd + d rvTlvl ; d(rv Rv ln Hl ):
Since the rst term on the RHS now contains rv instead of rT an exact dierential is not
possible. Bolton integrated numerically and derived the following expression for the pseudoequivalent potential temperature ep , as:
!0:2854(1;0:28rv ) 3376
p
oo
ep = T p
exp rv (1 + 0:81rv )
(216)
Tsat ; 2:54
where Tsat is a saturation temperature dened as one of:
Tsat = 3:5 ln T ;2840
ln e ; 4:805 + 55
63
(217)
or
Tsat =
1
1
T ;55 ;
ln(Hl )
2840
+ 55
(218)
Tsat can also be determined graphically to be the temperature of the lowest Condensation
Level (LCL).
The pseudo adiabatic ep isentropes parallel the adiabatic wet bulb potential temperature
(w ). The ep is related to w by:
ep = w exp rv0 (1 + 0:81rv 0) 3376
; 2:54
(219)
w
where
rv rsl (poo w )
(220)
The pseudo equivalent potential temperature is interpreted as a two step process being
pseudo adiabatic ascent to zero pressure followed by dry adiabatic decent to poo = 1000mb.
This is how we label the moist adiabats on a thermodynamic diagram.
3.10.2 Other Approximations
A common approximation made to the First-Second Law is to
1 cpm ' cp
2 Rm ' Rd .
Hence, equation 170, becomes
cpd ln T ; Rdd ln p + lTlv drv ; lTil diri = qTi
(221)
Generally the eects of this approximation are found to aect the predicted temperature
change by only a percent or so.
p Rcpd
m ' T Poo :
(222)
we can obtain, for a pseudo-adiabatic system:
(223)
d ln ; clivT drv + clil diri = qi
p
p
Similarly, for 's,
! cRp
(224)
m ' T ppoo
e
il
'
'
lvl rv ;lil ri
m e cpT
lvl rl +liv ri
m e; cpT
64
(225)
(226)
where it is further assumed that these variables are approximately conserved over an adiabatic process without phase change for m and with phase change for e . Latent heats are
often assigned to a single value dened at 0C.
Serious errors have been found to arise at very cold temperatures as a result of the error
terms, discussed in the previous section, which have been dropped. An empirically designed
approximation to the above il denition has been found to adequately reduce this error to
acceptable levels:
lvl rl +liv ri
il ' m e; cp max(T253:)
(227)
where latent heats are taken constant dened at 0C. It was shown experimentally by Tripoli
and Cotton (1981) that the error introduced by the approximations of constant latent heat
and the max function, eectively cancel much of the error of the terms neglected resulting
in a greatly reduced error of this approximation.
3.11 Hydrostatic Balance
It is sometimes useful to consider the force balance responsible for suspending an air parcel
above the surface. The force upward results from the pressure change across the parcel per
height change which gives T net force per parcel volume. The downward force per parcel
volume is gravity multiplied by parcel density. Setting these forces equal we obtain the
hydrostatic balance:
dp = ;gdz:
(228)
where z is the height coordinate.
Strict hydrostatic balance is not adhered to in the atmosphere, otherwise there would
not be vertical acceleration. But the balance is usually very close, and nearly exact over a
time average. It is usually reasonable from a thermodynamic point of view to assume this
balance, although some error is possible. Assuming the balance, variations in pressure can
be converted to variations in height.
We can substitute the equation of state for density to obtain the hypsometric equation:
d ln p = ; g
(229)
dz
Rd Tv
The hypsometric can be integrated to give:
;g z
p = poe RdTv
(230)
where the subscript "o" refers to an initial value, the z is the height change from the initial
value and the bar represents an average value during the change from the initial state. This
equation is commonly used to nd the height change between two pressures or the pressure
change between two height. A common use is to use this formula to nd the pressure the
atmosphere would have at sea level given a pressure measured at a surface location above
sea level.
65
3.12 Static Energy
It is sometimes convenient to work with a thermodynamic variable which is directly related
to enthalpy. Recall that we looked at this in the rst section where we dened gravitational
potential energy. It is useful to do the same here further subdividing the contributing
energies. For simplicity, we will neglect the energy contained within the latent heat of fusion.
In most applications of static energy, direct measurement of condensate is not possible and
so we cannot consider the energies stored in the liquid or ice phases.
Since the use of static energies has often been in tropical regions, the energy partitioned
into stored latent heat of condensation is considered, while that of fusion is often ignored.
Ignoring the existence of ice and liquid, the dierences between p and pd, the heat capacity
of water and assuming a reversible adiabatic system, equation 152 can be written:
(
)
r
v llv
d cpd ln T ; Rd ln p + T = 0:
(231)
Now assuming a hydrostatic atmosphere,
dp = ;gdz
= RpT gdz
d
d(Rd ln p) = gTdz
(232)
or, substituting into equation 231:
d(cpdT ; gz + rv llv ) = dhm
(233)
where h is the moist static energy . If we dene hm to be zero at sea level and 0K, we dene
hm as:
hemcpdT ; gz + rv llv
(234)
where the three terms on the RHS are the sensible, geopotential and latent heat contributions
to the total moist static energy. The dry static energy is simply the moist static energy less
the latent heat term
hd = cpdT ; gz:
(235)
We see that the dry static energy is closely related to and the moist static energy is
closely related to e .
3.13 Methods of Solving Thermodynamic Systems
Here I will discuss how we can predict the change in temperature, and other relevant thermodynamic variables, given a predicted changes in state variables. We will assume that the
condensate is divided between the liquid and ice mixing ratio, and the liquid mixing ratio
is divided between rain drops and cloud droplets. We will dene cloud droplets to be very
66
Type Prognostic
Thermo.
I.
T
II.
T
III. IV. V.
e
VI. il
Diagnostic
Moisture Thermo.
rv rc rr ri rT rr ri rv rc rr ri T rT rr ri T rT rr ri T rT rr ri T Moisture
rs(T p)
rs(T p) rv rc
rs(T p)
rs(T p) rv rc
rs(T p) rv rc
rs(T p) rv rc
Table 10: A series of possible prognostic systems. Note that rc and rr are the mixing ratios
of cloud droplets and rain droplets.
small droplets which evaporate and condense on time scales much smaller than the model
integration time step so that we can assume the system maintains equilibrium with respect
to liquid so long as cloud droplets exist. The growth and evaporation or rain and ice will be
predicted by a microphysical prediction scheme which we need not consider here. External
changes, ie precipitation tendencies will also be predicted by this model. We assume that
the model also predicts the change of total water.
As is typical, the model will predict pressure changes also. Another thermodynamic
variable must be predicted by virtue of Gibbs Phase rule.
The remaining variable is the thermodynamic variable. Several possibilities exist:
(a) Predict temperature (T )
(b) Predict potential temperature (), diagnose temperature (T)
(c) Predict equivalent potential temperature (e ), diagnose temperature (T), (d) Predict ice-liquid water potential temperature (il ), diagnose temperature (T), This is summarized in table 10. Note that all but I and III require iterative solutions. That is
because one typically assumes a category of liquid water is composed of very small droplets,
called cloud droplets , (rc) which grow and evaporate very rapidly. Because of their quick
adjustment, the system is assumed to be in approximate equilibrium with the vapor pressure
over liquid when in the presence of cloud droplets. Larger droplets called rain droplets , also
can exit, however the rain droplets evaporate and grow much slower because of their larger
size and much smaller surface to volume ratio. Their slower growth rates prevent them from
maintaining approximate equilibrium by themselves. If the ice crystals are very small, then
the system could conceivably maintain equilibrium over the ice once the cloud droplets have
all been evaporated. We will not explicitly treat that case here. This equilibrium condition
leads to the set of diagnostic relationships:
67
Step
Diagnostic Equation
h
:16) i
1.
es = 6:1078 exp a(T(;T273
;b)
2.
rs = p;eses
3.
rc = max(0 rT ; rr ; ri ; rs
4.
rl = rc + rr
5.
rv = rT ; rl ; ri
Table 11: Five step prognostic procedure for diagnosis of equilibrium liquid based here on
Tetens formula (see equation for denition of constants a and b)
In addition to the above equations, prognostic and diagnostic equations for the thermodynamic variables are necessary. The following table summarizes the various forms through
we have applied the combined First-Second Law, and some approximations:
Together, these tables summarize the tools necessary to integrate the First-Second Laws
in order to predict thermodynamic changes. We have not addressed the issue of prognosing
the growth of liquid or ice water constituents aside for diagnostic procedures where equilibrium is maintained. In other cases where various categories of ice particles and water
droplets interact with each other and grow at predicted rates detailed microphysical prediction schemes are needed. We also have not addressed the issue of the external derivatives of
liquid and ice which are related to precipitation of the microphysical quantity. This will be
discussed later when we treat microphysics.
One more point which can be made now. The thermodynamic system that we have been
dealing with is a Lagrangian system, ie moving with the ow. We can eectively transform
our Lagrangian system to an Eulerian system by, (for example in the case of il and rT as
the prognostic variables):
dil = @il + u @il + v @il + w @il
(236)
dt
@t
@x
@y
@z
drT = @rT + u @rT + v @rT + w @rT
(237)
dt
@t
@x
@y
@z
where x,y and z are the Cartesian coordinates, and u,v and w are the velocity components
along those coordinates respectively. We are yet to dene the form of the external derivatives
relating to the fall velocity of droplets and crystals. We will leave that topic to our treatment
of microphysics.
68
Thermodynamic Variable Denition
Source Term
dT = 1 dp ; llv drv + lil di ri + qit
a. T = T
dt
cpm dt cpm dt cpm dt
cpm
poo cRpmm
d
r
d
dr
m
v
i i
m
b. m = T p
dt = ; cpm T llv dt + lil dt + qit
R
(
r
+
r
)
R
r
v
v
v
i + i llv rv ;lil ri
Rd ;
h
de = e lil de ri ; cl T ln T de rl + de ri
c. e = T ppood cpl Hl cpl Hi cpl e cpl T
dt
cpl T
dt
dt
dt
i
+Rv T ln Hl ( drdtv + ddti ri ) ; Rv T ln Hi ddti ri + qi
de rl +l de ri ) rl Rv ri Rv (lvl rl +liv ri )
vi dt
dil = il 1 ; (lvl dt
d. il = m Hl cpm Hicpm e; cpm T
+ qit
dt
T cpm
T
ev
+(rl + ri )Rv T d lndt p
o
+Rv T ln Hl ddti rl + Rv T ln Hi ddti ri
d = ; llv drv + lil ] di ri + qit
dt
cpmT
dt
dt
h
i
de = e lil (Tt ) de ri + qi
dt
cp T dt
de rl +l (T ) de ri ) vi t dt
dil = il 1 ; (lvl (Tt ) dt
qit
dt
T cp
T
Rd
e. = T ppoo cp
llv r(Tt )s ;lil r(Tt )i
cp T
f. e = e
(lvl r(Tt )l +liv (Tt )ri )
g. il = e; cp max(T253:)]
Table 12: Various forms of the First-Second Law employed to predict the evolution of a
thermodynamic system. Forms f-g contain simplifying assumptions. Tt is the triple point
temperature and qit is the diabatic heating term expressed per unit time .
69
4 Thermodynamic Analysis of the Atmosphere
4.1 Atmospheric Thermodynamic Diagrams
Atmospheric scientists regularly employed graphical techniques for evaluating atmospheric
processes of all kinds including radiation processes, dynamical processes and thermodynamic
processes before computations were made on computers. Over the past four decades, the use
of most of the graphical techniques have diminished except for the use of thermodynamic
diagrams and conserved variable thermodynamic diagrams. These diagrams are still used today because they optimally display thermodynamic processes in such a way that the scientist
can not only view the potential for a process, but can examine the thermodynamic structure
of the air and imply its thermodynamic history. Nevertheless, some will argue that this
too can be accomplished numerically rendering even the thermodynamic diagram obsolete.
It is probably true that thermodynamic diagrams are often used too often in lieu of more
comprehensive three dimensional multiphase analysis through data assimilation. Nevertheless, the thermodynamic diagram is irreplaceable as a tool for understanding atmospheric
vertical structure and how it is related to the formation of clouds and the development of
precipitation processes.
We will begin our discussion with traditional thermodynamic and then look at some new
techniques called \conserved variable diagrams" which are new in the literature, but employ
the use of the properties of conserved variables to graphically study mixed phase processes.
4.1.1 Classical Thermodynamic Diagrams
The thermodynamic diagram is used to graphically display thermodynamic processes which
occur in the atmosphere. The diagram's abscissa and ordinate are designed to represent
two of the three state variables, usually a pressure function on one and a thermodynamic
function on another. Any dry atmospheric state may then be plotted. Unfortunately, any
moist state cannot be plotted as a unique point since that must depend on the values of rv ,
rl , and ri. But vapor content can be inferred by plotting dew point and moist processes can
be accounted for by assuming certain characteristics of the moist process such as assuming
that the process be pseudo-adiabatic.
There are three characteristics of a thermodynamic diagram which are of paramount
importance. They are:
1. Area proportional to the energy of a process or the work done by the process. An
important function of a thermodynamic diagram is to nd the energy involved in a
process. If the diagram is constructed properly, the energy can be implied, as we have
shown earlier with our p vs V diagram, by the area under a curve or the area between
two curves representing a process.
2. Fundamental Lines are Straight. This is for ease of use.
70
Figure 13: Representation of a cycle on and , -p diagram and its equal area transformation
to an A, B diagram.
3. Angle between isotherms and isentropes () lines be as large as possible. A major
function of the thermodynamic diagram is to plot an observed environmental sounding
and then compare its lapse rate to the dry lapse rate. Since small dierences are
important, the larger the angle between an isotherm and an adiabats (isentropes), the
more these small dierences stand out.
As we have seen, work is given by dw = pd which would suggest p and would be
suitable for the coordinates of a thermodynamic diagram. However, on such a diagram, the
angle between T and would be small, violating our third criteria for a suitable thermodynamic diagram. Hence we must nd two other thermodynamic variables which also have
area proportional to energy. We will require that the area enclosed by a process on this
alternate diagram be equal to the area enclosed on a p vs diagram. This is called an equal
area transformation.
Consider two variables, A and B. Let each be an independent function of one or more
thermodynamic variables. Hence if we know and p we can nd A and B. Hence each point
on an , -p diagram corresponds to a point on the A,B diagram and any closed cycle on one
corresponds to a closed cycle of the same area on the other (see gure 13). Thus:
I
I
; pd = AdB
(238)
for any given cyclic process. Therefore,
I
(pd + AdB ) = 0
(239)
For the closed integral to be zero, the integrand must be an exact dierential, so,
pd + AdB = ds
71
(240)
where we can view s as a function of and B, ie
!
!
@s
@s dB
ds( B ) = @ d + @B
(241)
B
Therefore:
!
@s
p = @
(242)
B
!
@s
(243)
A = @B
Cross dierentiating,
!
2 !
@s
@p
=
(244)
@B @@B
!
2 !
@A
@s
=
(245)
@ B
@@B
Therefore, areas will be equal if:
! !
@A = @p
(246)
@ B
@B Hence if we specify what B is , we can determine the A that will give an equal area transformation fro the , p diagram.
4.1.2 The Emagram
Here we let B=T. This is a logical choice since T is the fundamental thermodynamic variable.
From equation 246,
! !
@A = @p
(247)
@
@T
T
From the equation of state, we can then write:
!
RT !
@
@A
=
@
@T !T
@A
R
=
@
! T
@A d = R d
@ T
Then integrating,
A = R ln + F (T )
72
(248)
where F is an unspecied function of T, because in the partial derivative T was held constant.
Hence we can make F any function of T we wish. Taking the log of the equation of state:
ln = ; ln p + ln R + ln T
(249)
and substituting into equation 248 we obtain:
A = ;R ln p + R ln R + R ln T + F (T )]
(250)
We choose F (T ) to make the terms in the brackets cancel completely. Hence:
A = ;R ln p
(251)
B = T
(252)
are the coordinates of the Emagram named by Refsdal as an abbreviation for \energy pre
unit mass diagram". The abscissa is temperature and the ordinate is the log of pressure.
Hence the isobars and isotherms are straight and perpendicular to each other.
Since the pressure is on a logarithmic scale, p=0 is at innity and so the diagram is
terminated at p=400 mb.
The shape of the isentropes (dry adiabats) can be found by taking the log of Poisson's
equation and letting be constant, ie:
1
; ln p = ; ln T + constant
(253)
Since ln p is one of the coordinates, and ln T is not, the isentropes are logarithmic curves
which become steeper with decreasing temperature, but are close to straight lines in the
meteorological range.
The pseudo adiabats, ie ep lines are markedly curved. The saturation mixing ration line,
ie rs are gently curved. The angle between the adiabats and the isotherms can be varied
by changing one of the coordinate scales. In practice, the angle is usually about 45 degrees,
which is much better than the -p, diagram.
4.1.3 The Tephigram
Let B=T as with the Emagram, again giving equation 248. However, this time, introduce
Poisson's equation to attain an expression for ln :
!
T = p p
oo !
= RT
poo
Taking the logarithms:
ln = 1 ln ; ln T ] + ln T + ln R ; ln poo
(254)
73
or
R ln = cp ln + G(T )
where the function G(T) includes the constants poo and R. Hence
A = cp ln + F (T ) + G(T )
(255)
(256)
This time we choose G and F so that F (T ) = ;G(T ), and so the coordinates become:
A = cp ln B =
(257)
(258)
Since cp ln is the entropy of a dry parcel (except for a constant) this diagram was called
by Sir. Napier Shae a T ; diagram or Tephigram as it is normally referenced.
The equation for isobars is obtained by taking the logarithm of Poisson's equation. For
constant p,
ln = ln T + const
(259)
Since one coordinate of the Tephigram is ln and the other is just T, an isobar is a logarithmic
curve sloping upward to the right and decreasing in slope with increasing temperature. In the
range of meteorological observations, the isobars are only gently sloped. The Tephigram is
usually rotated so that the isobars are oriented essentially horizontal with decreasing pressure
upward. The pseudo-adiabats are quite curved, but lines of constantly saturation mixing
ratio tend to be straight. By denition, the angle between the isotherms and isentropes is
90 degrees. Hence the diagram is especially good for looking at variations in stability of an
environmental sounding.
4.1.4 The Skew T-log P Diagram
This diagram attempts to rework the Emagram to increase the angle between an isotherm
and an isentrope. The procedure was rst suggested by Herlofson in 1947. Let B = ;R ln p
identical to the A coordinate of the Emagram. Then equation 246 becomes
!
!
@A
1
@p
= ; R @ ln p
(260)
@
or
ln p
Multiply by d and integrate to yield:
@A
@
A =
=
!
ln p
= ; Rp
p + F (ln p)
R
;T + F (ln p)
;
74
(261)
We can choose the arbitrary function, F, to be
F (ln p) = ;K ln p
(262)
where K is a constant we may choose as we wish. Hence, the coordinates may be written:
A = T + K ln p
B = ;R ln p
(263)
(264)
The diagram is constructed with B as the ordinate (log pressure decreasing upward) and
A as the abscissa. An isotherm has the equation:
or
or
A = const + K ln p
(265)
A = const ; K
RB
(266)
R A + const
B = ;K
(267)
Hence the isotherms are straight parallel lines whose slope depends on K. K is chosen to
make the isotherms form a 90 degree angle with the isentropes, in which case the isotherms
also make about a 45 degree angle with the isobars, and slope upward to the right, while the
isentropes slope upward to the left. The equation for the isentropes, or dry adiabats, is:
ln T = R ln p + const
(268)
cp
The quantity R ln p is one of the coordinates although T is not. Hence the adiabats are not
perfectly straight, although they are nearly straight in the meteorological range, tending to
be concave upward. K is chosen to keep the isotherm-isentrope angle close to 90 degrees in
the meteorological range.
The pseudo-adiabats are distinctly curved. To straighten out the pseudo-adiabats one
would need to sacrice the energy-area proportionality. Saturation mixing ratio itself, does
appear nearly straight on this diagram.
4.1.5 The Stuve Diagram
This diagram is based on using p as the ordinate, increasing downward, and T as the
abscissa. The choice of coordinates insures that the dry adiabats are also straight lines,
although the pseudo-adiabats are also curved. The isotherm-adiabats angle is usually about
45 degrees. Unfortunately, this diagram is not and equal area transformation. This diagram
is clearly inferior to the other energy conserving diagrams and remains in use only because
it was introduced very early in the history of modern meteorology. In fact, we have some in
our Department!
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Attribute
Area Energy
T vs angle
p
T
ep
rs
Emagram
yes
45o
straight
straight
gently curved
curved
gently curved
Tephigram
yes
90o
gently curved
straight
straight
curved
straight
Skew T-Log p
yes
90o
straight
straight
gently curved
curved
straight
Stuve
no
45o
straight
straight
straight
curved
straight
Table 13: Table showing attributes of each type of thermodynamic diagram
4.1.6 Summary of the Attributes of Thermodynamic Diagrams
We can summarize the attributes of thermodynamic diagrams in table 13. It is evident that
the most capable diagrams are the Tephigram and the skew T - log P diagram. Generally,
the Tephigram is most widely used by tropical meteorologists because of its superior ability
to display very small dierences in stability of the environmental temperature. In the tropics,
and even in summer air masses at middle latitudes, convective potential may be modulated
by only small variations in vertical stability and the Tephigram perhaps best captures small
changes.
The skew-T -log P diagram has become the most widely used diagram among scientists
investigating middle latitude atmospheres both in the summer and winter. The major attribute of the skew-T diagram is that the isobars are perfectly straight horizontal lines which
are oriented horizontal and that atmospheric temperature proles appear as close to vertical
lines, giving intuitive appeal over very sloped environmental soundings appearing on the
Tephigram. Ultimately, one chooses a diagram because not only of these attributes, but
because of the ease of its use. This depends also on the color, thickness and type of lines
used. In addition, one tends to prefer the diagram with which they have become familiar.
4.2 Atmospheric Static Stability and Applications of Thermodynamic Diagrams to the Atmosphere
Atmospheric thermodynamic diagrams are widely used by forecasters and scientic investigators of the atmosphere to diagnose not only the current state of the atmosphere, but even the
recent history of the local atmosphere and the likelihood of future evolution. Atmospheres
resident for some over arctic land masses acquire distinctively dierent characteristics to
their thermodynamic proles than do air masses resident over the arctic oceans. Air masses
resident over dry elevated terrain, such as that of the High Plains of North America, acquire distinctive characteristics making those air masses clearly identiable when plotted on
a thermodynamic diagram. In addition air masses, inuenced by motions associated with
approaching weather systems, also acquire characteristics associated with those circulations.
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A trained meteorologist, can read an atmospheric sounding of temperature, humidity and
wind plotted on a thermodynamic diagram like a book, revealing the detailed recent history
of the air mass.
Thermodynamic diagrams are also used to study the potential warmth of the near surface
atmosphere during the day, the potential lowering of temperature at night and the potential
for fog. The potential for cloudiness can be found due to the movements induced by approaching or with drawing weather systems, or by the development of unstable rising motion
associated with cumulonimbus clouds. When combined with the vertical wind structures,
the thermodynamic prole also can give information on the structure and severity of potential cumulus clouds. The sounding, plotted on a thermodynamic diagram can describe the
likelihood of a local thermal circulation such as occurs with sea breezes and some lake eect
storms, and even the structures and dimensions of cloud groups associated with these structures. The sounding can also describe the potential for local circulations such as downslope
wind storms and upslope clouds and precipitation.
Indeed the thermodynamic sounding plotted on a thermodynamic diagram is the most
powerful of the diagnostic tools available to meteorologists for evaluating the current state
and potential future states of the atmosphere. It is far beyond the scope of this discussion
to describe how it is used for all of these applications. Instead, the use of the sounding
plotted on the thermodynamic diagram is discussed in virtually all branches of atmospheric
science with regards to that application. Here, we will discuss some of the basic interpretive
concepts employed most generally.
4.2.1 Environmental Structure and Parcel Path
The sounding, plotted on a thermodynamic diagram represents a particular atmospheric
state. This state may be observed by radiosonde, satellite derived, or perhaps forecast by a
computer atmospheric model. Plotted on the diagram are temperature and dew point as a
function of height. Figure 14 depicts a sample sounding from Oklahoma City, observed by a
radiosonde. Individual observations are plotted as points and then the points are connected
forming the temperature and dew point curves. Note the dew point temperature is always
less than or equal These curves represent the proles of environmental temperature and dew
point as a function of total pressure.
The other humidity variables at any pressure can also be found from this diagram. The
saturation mixing ratio lines represent the saturation mixing ratio for a given pressure and
temperature. Since the dew point represents the temperature at which the air would be
saturated at a given pressure, the saturation mixing ratio line intersecting the dew point
temperature curve at a given pressure represents the actual mixing ratio of the air at that
pressure. One can calculate vapor pressure, relative humidity, specic humidity all from the
mixing ratio, temperature, pressure and saturation mixing ratio at the actual temperature (
the mixing ratio line that intersects the temperature at the given pressure).
Conservative variables can also be diagnosed. The potential temperature () is found
by following the dry adiabats intersecting the given pressure and temperature, to a pressure
77
Figure 14: Skew-T Log-P diagram of a sounding taken prior to a severe weather event.
The heavy solid curve is the temperature sounding, the heavy dashed surve is the dewpoint
sounding, the heavy dash-dotcurve is the parcel path. The Lowest Condensation Level
(LCL), Lefel of Free Convection (LFC) and Equilibrium Level (EL) are labeled.
78
of 1000 mb and then reading the temperature at that point. In fact, the dry adiabats are
labeled by their potential temperature so that the value of the dry adiabats intersecting a
particular observation point is the potential temperature.
The equivalent potential temperature can be approximately found by moving upward on
a thermodynamic diagram from the observation point along the dry adiabat that intersects
the observation point, until an intersection with the mixing ratio line that also intersects the
dew point at the observe pressure is reached. This is the point where a parcel would reach
saturation if forced to rise and would be the cloud base for a parcel rising from the given
observation point. One then moves along the moist adiabat, since the rising parcel would
be saturated, upward until the moist adiabat becomes parallel to the dry adiabats. This is
where all of the moisture would be condensed from a parcel originating at the observation
point. Then one follows the dry adiabat intersecting that point down to the pressure of
1000mb. This would be close to the equivalent potential temperature (e ) not including ice
eects. It is not exact because we assume pseudo adiabatic ascent that neglects the heat
storage by condensed water carried up with the parcel.
4.2.2 Dry Adiabatic Lapse Rate
It is now instructive to introduce the concept of hydrostatic balance. If the gravitational
force downward on a parcel is balanced exactly by the pressure gradient force upward, then
the parcel is in hydrostatic balance. In general, most air parcels are in fairly close hydrostatic
balance, or else they would be rapidly accelerating up or downward. The hydrostatic balance
can be expressed as:
dP = ;g = ; g
(269)
dz
where z is the height coordinate and g is the acceleration of gravity (g = 9:8ms;2 .
tuting the equation of state for density, we obtain:
d ln P = ; RgT dz:
d
But from Poisson equation d ln p = (cp=R)dlnT , so:
cpd d ln T = ; g dz
Rd
Rd T
Rearranging terms, we can write:
g = ;9:72Kkm;1 =
;d = ; dT
=
dz cpd
Substi-
(270)
(271)
(272)
where ;d is the dry adiabatic lapserate dened to be the rate (with height) at which the
temperature of air parcel would decrease as it rises dry adiabatically in a hydrostatic atmosphere.
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A further consequence of the hydrostatic assumption can be seen by its application to
the First law. Since 0 = cpdT ; dp, for adiabatic ascent or decent, the hydrostatic equation
implies:
0 = cpdT + gdz
(273)
Hence the enthalpy change from dp is equivalent to a change in gravitational potential
energy in a hydrostatic atmosphere! As a parcel rises(falls) its enthalpy decreases(increases)
while the gravitational potential energy (gz) increases (decreases). This exchange between
gravitational potential energy and enthalpy is an alternative (and equivalent) explanation
to why a parcel cools at it rises or warms as it sinks.
4.2.3 Moist Adiabatic Lapse Rate
The moist adiabatic lapse rate (;m) is similar to the dry adiabatic lapse rate except that it
applies to adiabatic temperature decrease of a rising parcel which is saturated and in vapor
pressure equilibrium with a plane surface of pure water. Temperature decrease with height is
reduced from the dry rate by latent heat release as condensation occurs. This reduced rate of
temperature decrease is the moist adiabatic lapse rate and has slope approximately parallel
to the moist adiabat and exactly parallel to a line of constant e. The moist adiabat on the
thermodynamic diagram ignores the contribution by ice which is more dicult to represent
because freezing and melting typically do not occur in equilibrium as does condensation.
The moist adiabatic lapse rate can also be derived in a similar fashion to the dry adiabatic
lapse rate with the equivalent form of the rst law for moist (neglecting ice) adiabatic
hydrostatic ascent:
0 = cpdT + gdz + lvl Tdrsl
(274)
Since, from the denition of mixing ratio, rsl esl =p, we can write
drsl desl ; dp :
(275)
rsl esl p
Then, again using the hydrostatic relationship, and substituting back into the First law:
!
g
de
sl
(276)
;Lrsl
esl + RdT dz = cpdT + gdz
Employing the Clausius Clapeyron equation, we can then solve for ;m, the moist adiabatic
lapse rate:
2
lvl rsl 3
1
+
dT
g
;m = ; dz = c 4 lvlRrdsl T
lvl 5
(277)
p 1 + Rd T cpT
Since (lvl )=(cpT ) > 1, we can see that the moist adiabatic lapse rate is always less than the
dry rate and as rsl vanishes at low temperatures and ;m tends toward ;d . As a result ;m
will be near 5:5Ckm;1 in the lower troposphere at temperatures around 20 C and approach
9:8Ckm;1 by -40 C.
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4.2.4 Diagnosis of Atmospheric Stability
A parcel displaced adiabatically from an initial location at the environmental state to a new
pressure will experience a change in temperature consistent with an adiabatic process which
is independent of the environment. On the other hand, dynamic accelerations will maintain
a parcel pressure in equilibrium with the environment. Because we assume the static environment maintains hydrostatic balance, a parcel will fall out of hydrostatic equilibrium as
its temperature becomes dierent from the environment. The diering temperature at environmental pressure dictates a density dierence between the parcel and environment. Since
the downward gravitational force produced by environmental density is in hydrostatic equilibrium with the pressure eld, the parcel density is not in equilibrium and will experience
accelerations upward (downward) if it is relatively warm(cold). If an upward displacement
causes the parcel to warm (cool) relative to the environment, the the parcel will tend to accelerate upward (decelerate upward) and so the environment would be considered statically
unstable (stable ) to vertical displacement. If the displaced parcel maintains a temperature
equal to the environment, the environment would be considered neutral .
To determine the stability of the environment one can compare the environmental lapse
rate (;e) to the parcel lapse rate. If the parcel's lapse rate is greater(less) than that of the
environment the environment is considered stable(unstable) and if the parcel's lapse rate is
equal to that of the environment the environment is considered to be neutral. The parcel
lapse rate can be either ;d or ;m depending on weather the parcel is dry or saturated. We
can then quantify the environmental stability as:
;d
;d
;e > ;d absolutely unstable
> ;e > ;m conditionally stable
> ;e
absolutely stable
Static stability can be determined from a thermodynamic diagram by comparing the
slope of the environmental temperature prole to the moist and dry adiabats, where the
slope of the dry adiabat is ;d, the slope of the moist adiabat is ;m and the slope of the
environmental sounding is ;e.
If the environment is diagnosed to be absolutely unstable, than any displacement will
cause the growth of the displacement. As a result, absolute instabilities tend to be removed
from the atmosphere almost immediately. As a result, any observation plotted on a thermodynamic diagram that exhibits absolute instability is most likely in error. The exception to
this is near the surface, where frictional eects of the surface may locally prevent unstable
overturning allowing a thin layer of absolute instability near the ground during the times of
greatest surface heating.
If the environment is diagnosed to be conditionally unstable, parcel ascent will be unstable
only if the parcel is saturated. An observation of conditionally unstable environment under
saturated conditions is unlikely because of similar reasons to absolute instability. However,
an observation of conditional instability under unsaturated conditions may be shown to
produce unstable overturning if the parcel is forcibly displaced upward until saturated and
81
then a bit more. To show this, one can use the thermodynamic diagram. Take a parcel in a
conditionally unstable environment and forcibly lift along a dry adiabat until its path cross
the saturation mixing ratio line also intersecting the parcels dew point at the parcels initial
pressure. At that point, the parcel becomes saturated and cloud begins to form. That is
called the Lowest Condensation Level (LCL). Since it is conditionally unstable, the parcel
will be colder than the environment at that point, and if lifting were ceased, it would fall
back downward. Continue forcing the parcel up, but now along the moist adiabat, and in a
conditionally unstable environment, the parcel temperature will eventually become greater
than that of the environment and the same pressure. Subsequently, the parcel can continue
to rise unstably on its own propelled by a hydrostatic imbalance of forces called buoyancy.
That point is called the LFC (Level of Free Convection). Unstable ascent will continue until
the environment again becomes warmer than the parcel. That point is the EL (Equilibrium
Level).
On an equal area thermodynamic diagram, the area between the parcel path and the
environmental sounding between the LFC and EL is proportional to the CAPE (Convective
Available Potential Energy, Jkg;1) gained by the parcel during the unstable ascent. It can
also be expressed mathematically by:
Z LCL (p arcel ; e nv )
dp
(278)
CAPE = ;
LFC
If a convective plume were to convert all of its CAPE to kinetic energy of the updraft, the
updraft speed would be:
p
w = 2CAPE
(279)
The area between the parcel path and environmental sounding between the parcel origin
and LFC is the convective inhibition energy, or the amount of energy given up by the lifting
mechanism to lift the parcel to free convection.
CAPE is commonly used to quantify the degree of conditional instability. A convective
plume must be driven by the ascent of a set of atmospheric parcels, comprising a layer of
some minimum thickness. Therefore, CAPE should be computed by starting with a parcel
representative of the mean properties of a layer at least 50 mb thick. It also should represent
the state of the atmosphere expected when convection occurs, typically following afternoon
heating and moistening. Hence CAPE is usually computed from a parcel not necessarily
equal to the state at any point in the environmental sounding. A CAPE value of 1000 Jkg;1
is minimal conditional instability for deep convection while a value of 3000 Jkg;1 is high
conditional instability.
The degree of conditional instability is also sometimes expressed as the LI (Lifted Index).
LI is the dierence between the 500 mb temperature of the environment and the parcel. A
negative lifted index implies conditional instability. An LI of -4 is moderate conditional
instability and a value of -12 is extreme conditional instability.
82
4.3 References
Dutton, J. , 1976: The Ceaseless Wind , McGraw-Hill Emanuel , K. A., 1994: Atmospheric
Convection Oxford University Press Hess, S. L., 1959: Introduction to Theoretical Meteorology, Holt, Reinhart, and Winston Iribarne ,j. V. and W. L.
Godson, 1973: Atmospheric Thermodynamics D. Reidel Publishing Co. Sears,
F. W., 1953: Thermodynamics , Addison-Wesley Wallace and P. V. Hobbs,
1977: Atmospheric Science: An Introductory Survey , Academic Press
83
5
6
7
8
9
Microphysics, Rauber
Clouds, Fog and Haze, ?
Atmospheric Electricity, Lyons
Radiative Transfer in the Atmosphere, Ackerman
Surface Layer Processes, Stull
84