warming on saturation vapor pressure
dominates the effect of increasing partial
pressure, so that subsidence of initially
saturated air follows the dry adiabat. Assuming
that the displacement causes condensation, we
may replace pc by pc,sat(T) and rc by r
electromagnetic spectrum. The Median
Emission Temperature is the temperature of a
blackbody for which half of the emitted power
is below the given frequency (or equivalently,
wavelength or wavenumber). The Peak-
Temperature is the temperature of a blackbo
the star. The stellar luminosity is the net power
output of a star, and if the stars emission can
be represented as blackbody radiation, the
luminosity is given by L~ = 4r2 ~T 4 ~. 3.3.
RADIATION BALANCE OF PLANETS 115 We are
now equipped to compute the e
supplied to the surface by heat transport from
the deep interior, fed by radioactive decay, tidal
dissipation, or high temperature material left
over from the formation of the planet. Heat flux
from the interior is a major player in the
climates of some g
and familiar ranges of numbers. Wavelengths
themselves will sometimes be measured in m
(microns, or 106m). Figure 3.1 gives the
approximate regions of the electromagnetic
spectrum corresponding to common names
such as Radio Waves and so forth. If a field
Journal of Physical and Chemical Reference
Data and results pertinent to planetary
atmospheres are often reported in the journal
Icarus. Chapter 3 Elementary models of
radiation balance 3.1 Overview Our objective is
to understand the factors governing the
radiation. Since the outgoing radiation has
longer wavelength than the incoming radiation,
the flux of emitted outgoing radiation is often
referred to as outgoing longwave radiation, and
denoted by OLR. For a non-isothermal planet,
the OLR is a function o
in Eq. 2.23. This is accomplished by multiplying
Eq. 2.29 by T to obtain the heat budget,
dropping the terms proportional to rc (which
are small in the dilute limit), and making use of
RaT /pa = 1/a 1/. The hydrostatic relation is
used to rewrite the pres
buoyancy due to compressional heating, but if
one could find a situation where composition
rendered subsiding parcels unstable to descent,
the resulting convection would be very
interesting indeed to study. 102 CHAPTER 2.
THERMODYNAMICS IN A NUTSHELL Howe
component atmosphere is derived in
Problem ?. Note that for a single-component
saturated atmosphere, the supposition that the
atmosphere is saturated is sufficient to
determine the temperature profile, regardless
of the means by which the saturation is
ma
gas/solid,and so the appropriate latent heat to
use in the Clausius-Clapeyron relation is the
latent heat of sublimation. Above the triple
point, the favored transition is gas/liquid,
whence one should use the latent heat of 2.6.
THERMODYNAMICS OF PHASE C
convenient basis for diagnostics of atmospheric
energy flows. For the same reason, it provides a
convenient basis for 104 CHAPTER 2.
THERMODYNAMICS IN A NUTSHELL the
formulation of simplified vertically-averaged
energy balance models of climate. Such mode
or can be computed in terms of ln T and ln pa.
Therefore, the equation defines a first order
ordinary differential equation which can be
integrated (usually numerically) to obtain T as a
function of pa. Usually one wants the
temperature as a function of t
6371km) The bold assumption introduced by
Planck is that electromagnetic energy is
exchanged only in amounts that are multiples
of discrete quanta, whose size depends on the
frequency of the radiation, in much the same
sense that a penny is the quantum of
dimensionless variable u = h/(kT). Recalling
that each degree of freedom has energy 1 2 kT
in the average, we see that u is half the ratio of
the quantum of energy at frequency to the
typical energy in a degree of freedom of the
matter with which the elec
substance to become subsaturated. Does
condensation occur on ascent (which lowers
the total pressure) or descent (which raises the
total pressure)? To answer this question, we
must compare the moist adiabatic slope d ln
T /d ln p computed from Eq. 2.33 wi
scaling in temperature using classical
thermodynamic reasoning. However, classical
physics yields an infinite value for the constant
. The formula for clearly reveals the
importance of quantum effects in determining
this constant, since diverges like 1/h3
a major constituent; at temperatures much
above 320K, it rapidly becomes the dominant
constituent of the atmosphere. Note also that
the distinction between the ice and liquid phase
transitions has a marked effect on the vapor
pressure. Because the latent
temperatures encountered in the atmosphere
under consideration. This might be a mixture of
condensible methane on Titan with 2.7. THE
MOIST ADIABAT 97 100 120 140 160 180 200
220 240 260 10 100 1000 T p (Pa) Condensation
level Figure 2.6: The adiabatic pr
air at the ground in 2.7. THE MOIST ADIABAT 95
tropical conditions (1000mb and 300K)? What is
the mass mixing ratio? What is the massspecific humidity? What is the molar mixing
ratio (in ppm) of water vapor in air at the
tropical tropopause (100mb and 200
effect of the mass of retained condensate on
buoyancy can be significant in some
circumstances. In other planetary atmospheres
the effect of retained condensate could be of
greater importance. The temperature profile
obtained by assuming condensate is rem
with frequency without bound; a body with any
nonzero temperature would emit infrared at a
greater rate than microwaves, visible light at a
greater rate than infrared, ultraviolet at a
greater rate than visible, X-rays at a greater rate
than ultraviolet,
intersect the planet are very nearly parallel to
the line joining the center of the planet to the
center of its star; the sunlight comes in as a
nearly parallel beam, rather than being
isotropic, as would be the case for true
blackbody radiation. The para
that given by a unit vector n. The energy
spectrum (~r, , n) at this point is defined
such that the energy contained in a finite but
small sized neighborhood of the point (~r, , n)
is dV dd, where dV is a small volume of
space, d is the width of the frequ
exceedingly large. An example for present
Summer Martian conditions (specifically, like
the warmest sounding in 2.2) is shown in Figure
2.6. A comparison with the Martian profiles in
Figure 2.2 indicates that something interesting
is going on in the Marti
experience that a sufficiently hot body emits
light hence terms like red hot or white
hot. Once it is recognized that light is just one
form of electromagnetic radiation, it becomes a
natural inference that a body with any
temperature at all should emit s
matter; any radiation impinging on the body
will not travel far before it is absorbed, and in
this sense the body is called black even
though, like the Sun, it may be emitting light.
Nineteenth century physicists found it natural
to seek a theoretical exp
distance from the center equal to 90% of the
visible radius, the temperature is above 600,
000K. However, the Sun is encased in a layer a
few hundred kilometers thick which is
sufficiently dense to act like a blackbody, and
which has a temperature of abou
of some particular kind of measuring apparatus.
A more meaningful quantity can be derived
from the cumulative flux spectrum, value at a
given point in the spectrum is the same
regardless of whether we use wavenumber,
wavelength, log or any other coordinat
pose a considerable challenge. Compositional
effects on Earth at its present temperature are
slight, but for atmospheres in which the
condensible is nondilute, these effects become
more and more important. With regard to the
modelling of convection, this
as the orbital distance is large compared to the
stellar radius. From the displacement law, it
follows that the planet loses energy through
emission at a distinctly lower wavenumber than
that at which it receives energy from its star.
This situation is il
100 1000 104 105 200 220 240 260 280 300 320
Saturation Vapor Pressure for Water Vapor
pressure over liquid Vapor pressure over ice
Vapor pressure (Pa) T Ice Liquid Vapor Figure
2.5: Saturation vapor pressure for water, based
on the constant-L form of the
density just gives the point at which the
cumulative emission function has its maximum
slope. This depends on the coordinate used,
unlike the point of median emission. Figure 3.1
shows the the portion of the spectrum in which
blackbodies with various temp
enclosing area dA (from one chosen side to the
other), placed in the interior of an ideal
blackbody; an equal amount passes through the
hoop in the opposite sense. The way the
angular distribution of the radiation is
described by the Planck function can b
objects. Blackbody radiation is a prime example
of this. Once the quantum assumption was
introduced, Planck was able to compute the
irradiance (flux spectrum) of blackbody
radiation with temperature T using standard
thermodynamic methods. The answer is B(