Potter T.D., Colman B.R. (co-chief editors). The handbook of weather, climate, and water: dynamics, climate physical meteorology, weather systems, and measurements
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CHAPTER 15
PHYSICAL ATMOSPHERIC SCIENCE*
GREGORY TRIPOLI
1 OVERVIEW
Perhaps the oldest of the atmospheric sciences, physical meteorology is the study of
laws and processes governing the physical changes of the atmospheric state. The
underlying basis of physical meteorology is the study of atmospheric thermo-
dynamics, which describes relationships between observed physical proper ties of
air including pressure, temperature, humidity, and water phase. Generally, thermo-
dynamics 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 statis-
tics of the microscale state and processes. Temperature and the air density are two
examples of such statistics representing the mean kinet ic energy of a population of
molecules and the mass of molecules of air per unit volume, respectively.
A second major branch of physical meteorology is the study of the growth of
water and ice precipitation called cloud microphysics. Here theories of how liquid
and ice hydrometeors first form and subsequently evolve into rain, snow, and hail are
sought. The interaction of water with atmospheric aeroso ls is an important par t of
atmospheric microphysics, which leads to the initial formation of hydrometeors. The
quantification of these theories leads to governing equations that can be used to
simulate and predict the evolution of precipitation as a function of changing atmo-
spheric conditions. By studying the processes that lead to the formation of clouds,
Handbook of Weather, Climate, and Water: Dynamics, Climate, Physical Meteorology, Weather Systems,
and Measurements, Edited by Thomas D. Potter and Bradley R. Colman.
ISBN 0-471-21490-6 # 2003 John Wiley & Sons, Inc.
*The derivations, concepts, and examples herein are based partially on course notes from the late
Professor Myron Corrin of Colorado State University. Other derivations are taken from the Isbane and
Godson text, the Emanuel (1994: Atmospheric Convection, Oxford University Press) text, and the Dutton
(The Ceaseless Wind ) book.
177
clouds themselves must be categorized by the processes that form them, giving rise
to the study of cloud types. From the discipline of microphysics emerges the study of
weather modification, which studies the possi bility of purposefully modifying
natural precipitation formation processes to alter the precipitation falling to the
surface. Another discipline associated with microphysics is atmospheric electricity,
which studies the transfer of electrical energy by microphysical processes.
A third major branch of physical meteorology is the study of atmospheric radia-
tion. This area of study develops theories and laws governing the transfer of energy
through the atmospher e by radiative processes. The se include the adsorbtion, trans-
mission, and scattering of both solar radiation and terrestrial radiation. A primary
goal of atmospheric radiation studies is to determine the net radiative loss or gain at
a particular atmospheric location that leads to thermodynamic change described by
thermodynamics theory. Since radiative transfer is affected by details of the atmo-
spheric 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 effect 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.
Finally, boundary layer meteorology studies the transfer of heat, radiation, moist-
ure, and momentum between the surface and the atmosphere. The transfer of energy
occurs on a wide range of scales from the molecular scale all the way to the scales of
thermal circulations. The challenge is to quantify these transfers on the basis of
observable or modelable atmospheric quantities. This also gives rise to the need to
represent the evolution of the soil, vegetation, and water surfaces. 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 efforts on laying down the foundations of
basic thermodynamics, microphysics, and radiative transfer and surface layer theory.
2 ATMOSPHERIC THERMODYNAMICS
Atmospheric thermodynamics is the study of the macro scopic physical properties of
the atmosphere for which temperature is an important variable. The ther modynamic
behavior of a gas, such as air, results from the collective effects and interactions of
the many molecules of which it is composed. However, because these explicit
microscale processes are too small, rapid, and numerous to be observable and too
chaotic to be predictable, their effects are not described by the Newtonian laws
governing the molecular processes. Instead, classical thermodynamics consists of
laws governing the observed macroscopic statistics of the behavior of the micro-
scopic system of air and the suspended liquids and solids within. As a consequence,
178 PHYSICAL ATMOSPHERIC SCIENCE
thermodynamic laws are only valid on the macroscale where a sufficient population
of molecular entities is present to justify the statistical approach to their behavior.
Classical atmospheric thermodynamics seeks to:
Develop an understanding for a statistically significant mass of air called a
parcel.
Classify the interacting forms of energy contained within a parcel.
Never ask ‘‘why’’ in terms of the first 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 thermo-
dynamics begins with a classification of energy into thermal and nonthermal
forms as is observed.
Create an efficient book-keeping system in which the interacting forms of
energy and mass are defined in a convenient and nonredundant form.
Quantify the amount and rate of energy transfer from one form to another.
Provide a means of calculating a true equilibrium state among these energies as
well as defining the conditions for constrained equilibrium.
The atmospheric application of classical thermodynamics differs in that particular
attention is paid to the properties of air and forms of the thermodynamic laws and
relationships that are most useful for application to the atmosphere. We define air to
be the gas presen t in Earth’s lower atmosphere where air is sufficiently dense for the
empirical laws to apply. This only excludes regions on the order of 50 km and higher
above the surface.
This section will begin by defining the basic concepts needed, then move into the
thermodynamics of gasses and then specifically into the thermodynamics of air.
Finally the ther modynamics of the mixed-phase system, including liquid and ice
processes, will be described in Chap. 16 .
Basic Concepts, Definitions, and Systems of Units
We begin by defining a system to represent a portion of the universe selected for
study. It may or may not contain matter or energy. Systems are class ified as open,
isolated,orclosed. 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 defined 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 define 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,
and energy. Properties may be classified as intensive or extensive. An intensive
property is a characteristic of the system as a whole and not given as the sum of
2 ATMOSPHERIC THERMODYNAMICS 179
these properties for portions of the system. Pressure, temperature, density, and
specific heat are intensive properties. We will adopt a naming convention that
uses capital letters for variables describing an ‘‘intensive’’ property. An extensive
property is dependent on the dimensions of the system and is given by the sum of the
extensive properties of each portion of the system. Examples are mass, volume, and
energy. Extensive variables will be named with lowercase letters.
A homogeneous system is one in which 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 ¼ mz ð1Þ
where z is the specific property.
We define phases to be subsets of a syst em that are homogeneous in themselves
but are physically different from other portions. A heterogeneous system is defined
to be a system composed of two or more phases. In this case, any extensive proper ty
will be the sum of the mass-weighted phases of the system, i.e.,
Z ¼
P
a
m
a
z
a
ð2Þ
where a refers to the particular phase.
An inhomogeneous system is one in which 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 homogeneous or heterogeneous as an approximation.
The study of the atmosphere requires the definition 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 infinitesimal and
homogeneous but large enough for the definitions of thermodynamic and microphy-
sical statistics to be valid. In an atmosphere, where air movements are not confined to
a rigid container and therefore expand and contract against local forces of pressure
gradient, the parcel quantification is a natural choice for defining an air sample.
Energy is the ability to perform work and is a property defined in the first law of
thermodynamics. Energy is an extensive variable that is normally defined as a
product of an intensive and extensive variable. The intensive term defines, by differ-
encing, 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 inten-
sive 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 defi nes chemical energy and so
on.
The state of a system is defined by a set of system properties. A certain minimum
number of such properties are required to specify the stat e. For instance, in an ideal
gas, any three of the four properties—pressure, volume, number of moles, and
180 PHYSICAL ATMOSPHERIC SCIENCE
temperature—will define the state of the system. The so-called equation of state
states an empirical relationship between the state properties that define 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 nonequilibrium. Independence of the state to time is a
necessary but not sufficient 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 that if slightly perturbed in any way will accelerate
away from equilibrium, it is referred to an unstable equilibrium. Metastable equili-
brium 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 on which to grow. Introduce
that crystal and growth will begin, but without it the system remains time indepen-
dent even to other perturbations such as a temperature or pressure perturbation.
A process is the description of the manner by which a change of state occurs.
Here are some process definitions:
Change to the System Any difference produced in a system’s state. The refore
any change is defined only by the initial and final 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 sur roundings) is returned to its initial state.
Reversible or Quasi-Static Process A special, idealized thermodynamic
process that can be reversed without change to the universe. It will be
shown that this is possible only when the conditions differ infinitesimally
from equilibrium. Hence at any given point during a reversible process the
system is nearly in equilibrium. One can think of a reversible process as being
composed of small irreversible steps, each of which have only a small
departure from equilibrium (see Fig. 1).
In association with these concepts of processes, we also define the internal deriva-
tive (d
i
) to be the change in a system due to a purely adiabatic process. Conversely,
the external derivative (d
e
) is defined 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, i.e.,
d ¼ d
i
þ d
e
ð3Þ
2 ATMOSPHERIC THERMODYNAMICS 181
where d is the total derivative. Now, it follows from the definition of adiabatic that
þ
d ¼
þ
d
e
þ
d
i
¼ 0
Zeroth Principle of Thermodynamics: Definition of Temperature (T )
We define a diathermic wall as one that allows thermal interaction of a system with
surroundings but not mass exchange. The zeroth principle of thermodynamics states:
If some system A is in thermal equilibrium with another system B separated by a
diathermic wall, and if A is 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 that describes
this thermal equilibrium and that property is defined to be temperature. The so-
called reference body defines this property and is called a thermometer. A number is
assigned to the property by creating a temperature scale. This is accomplished by
finding a substance that has a property that changes in correspondence with different
thermal states. For instance, Table 1 defines five thermometric substances and corre-
sponding properties.
A thermometer is calibrated to changes in its observed thermometric property,
obeying the zeroth principle. The thermometer must be much smaller than the
substance measured so that the changes brought to the substance by exchanges
from the thermometer can be neglected.
Figure 1 Reversible process as a limit of irreversible processes (from Iribarne and Godson,
1973).
182
PHYSICAL ATMOSPHERIC SCIENCE
A linear temperature scale may be defined generally as:
T
X
¼ aX þ b ð4Þ
where T
X
is an arbitrary temperature scale, a represents 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-defined thermal states and an
approximate linear relationship of T to changes in the substance’s thermometric
property in order to determine a and b. The Celsius, or Centigrade, temperature
scale is defined 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
Fahrenhe it temperature scale, used primarily in the United States for nonscientific
measurements and unofficially 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.
A more general scale can be derived using the thermometric properties of gasses
listed as the first two examples in Table 1. 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 define the lowest possible thermo-
dynamic state to be that where pressure decreases to zero (at const ant volume).
Letting the tem perature be zero at that point, defines the intercept to also be zero.
Now only a single well-defined thermodynamic state is necessar y to define the scale.
This is chosen to be 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 its choice defines the value of the slope a. We can define 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 ¼273:15
C. Then defining
the freezing point at 1 atm to be 273.15
and the triple point of water to be 273.16
of the absolute temperature, 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. Similarly, an
absolute temperature scale is also defined for Fahrenheit units and is called the
TABLE 1 Empirical Scales of Temperature
Thermometric Substance Thermometric Property X
Gas, at constant volume Pressure
Gas, at constant pressure Specific Volume
Thermocouple, at constant pressure and tension Electromotive force
Pt wire, at constant pressure and tension Electrical resistance
Hg, at constant pressure Specific Volume
Source: Irabarne and Godson (1973).
2 ATMOSPHERIC THERMODYNAMICS 183
Rankine scale having units of degrees Rankine (R). Generally, atmospheric scientists
work with the Celsius and Kelvin scales.
The relationships between the four major scales are
T ¼ T
C
þ 273 :15
T
R
¼ T
F
þ 459 :67
T
F
¼
5
9
T
C
þ 32
where T; T
C
; T
F
, and T
R
are the Kelvin, Celsius and Fahrenheit and Rankine
temperatures, respectively.
Ideal Gas and Equation of State for the Atmosphere
A gas is composed of molecules moving randomly about, spinning, vibrating and
occasionally colliding with each other. If held within a container, the gas molecules
will also collide with the container wall, exerting a force on the walls. The nature of
a molecule’s movement and the movement of the entire population of molecules is
chaotic and unpredictable on a time scale of only a fraction of a second. We make no
attempt to quantify a physical description of this process with Newtonian mechanics.
The statistical properties of a gas that are 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.
Newtonian mechanics tells us that force will be proportiona l to the average
change in momentum of the molecules as they reflect off the surface, which is
then proportional to the mass and normal velocity component of each molecule
and the total number of molecules hitting the surface.
The mass of the gas is defined to be
m nM ð5Þ
where n is the number of moles and M is the molecular weight (in grams per mole).
It is often advantageous to express an extensive ther modynamic quantity as the ratio
of that quantity to its total mass. That ratio is referred to as the specific value of that
quantity. For instance, the specific volume is given by:
a
V
m
1
r
ð6Þ
where V is the volume and r is the density K=g
3
.
We can now define the equation of state for the atmosphere. Observations suggest
that under normal tropospheric pressures and temperatures, air gases exhibit nearly
identical behavior to an ideal gas. To quantify this observation, we define an ideal
184 PHYSICAL ATMOSPHERIC SCIENCE
gas to be one that obeys the empirical relationship; also called the ideal gas law or
the equation of state.
pV ¼ nR*=T ¼ mR*T = M ¼ mRT ð 7Þ
where n is the number of moles, M is the molecular weight of the gas, R* is the
universal gas constant, and R ¼ R*=M is the specific gas constant valid only for a
particular gas of molecular weight M. Observations show that, R* ¼ 8:3143 J=
mol=K and is the same for all gases having pressures and temperatures typical of
tropospheric conditions.
Employing these definitions, the equation of state for an ideal gas can be written
as:
p ¼
RT
a
¼ rRT ð8Þ
It may be shown, by comparison with measurements, that for essent ially all tropo-
spheric 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 ( p
j
) is defined to be the pressure that the jth individual gas constituent
would have if the same gas at the same temperature occupied an identical volume by
itself. We can also define the partial Volume (V
j
) to be 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:
p ¼
P
j
p
j
ð9Þ
where p is the total pressure and the sum is over all components ( j ) of the mixture.
Since each partial pressure independently obeys the ideal gas law, volume can be
written
V ¼
P
V
j
ð10Þ
where V is the total volume. Also, it follows that
P
j
¼ n
j
R*T=V ð11Þ
and
p ¼ðR*T=V Þ
P
j
n
j
ð12Þ
2 ATMOSPHERIC THERMODYNAMICS 185