# Atmosphere and Greenhouse Gas Primer

W. A. van Wijngaarden  Department of Physics and Astronomy, York University,
Canada  W. Happer  Department of Physics, Princeton University, USA

###### Abstract

We discuss the basic ways greenhouse gases affect radiation transfer in
Earth’s atmosphere. We explain how greenhouse gases like water vapor, H2O, or
carbon dioxide, CO2, differ from non-greenhouse gases like nitrogen, N2, or
oxygen, O2. Using simple thermodynamics and fluid mechanics, we show that the
atmosphere of a planet with sufficiently high concentrations of greenhouse
gases must develop a convecting troposphere between the surface and the
tropopause altitude. The planet must also develop a non-convecting
stratosphere for altitudes above the tropopause. In the simplest approximation
of an atmosphere that is transparent to sunlight and has frequency-independent
opacity for thermal radiation (an infrared gray atmosphere), one can find
simple formulas for the tropopause altitude, and for the altitude profiles of
pressure and temperature. The troposphere is nearly isentropic and the
stratosphere is nearly isothermal. The real atmosphere of the Earth is much
more complicated than the simple model, but it does have a troposphere and a
stratosphere. Between the surface and the tropopause the entropy per kilogram
of real tropospheric air increases slowly with altitude. The entropy increases
much more rapidly with altitude in the stratosphere. The stratosphere has a
nearly isothermal lower part and a hotter upper part due to absorption of
solar ultraviolet radiation by ozone. The thermal opacity of the real
atmosphere has a complicated frequency dependence due to the hundreds of
thousands of vibration-rotation transitions of its greenhouse molecules.
Unlike the simple model where nearly all radiation to space originates at the
tropopause altitude, radiation to space from Earth’s real atmosphere
originates from both the surface and all altitudes in the troposphere. A small
additional amount of radiation originates in the stratosphere. When these
complications are taken into account, model calculations of the thermal
radiation spectrum at the top of the atmosphere can hardly be distinguished
from those observed from satellites.

##  1 Introduction

Worldwide industrialization and the associated combustion of fossil fuels have
increased the concentrations of carbon dioxide (CO2) and methane (CH4) since
1750. These gases along with nitrous oxide (N2O) and assorted lesser players
like halocarbon refrigerants are examples of “greenhouse gases”. It should be
noted that by far the most abundant greenhouse gas in the atmosphere is water
vapor. There is little that one can do about water vapor on our watery planet
Earth, with 70% of its surface covered by oceans.

Greenhouse gases were first discovered by John Tyndall in the course of
brilliant experimental work in the 1850’s [1]. Tyndall recognized that
greenhouse gases warm Earth’s surface. Some 50 years later Svante Arrhenius
made the first theoretical estimates of how much surface warming would result
if atmospheric concentrations of carbon dioxide were doubled [2]. The
atmosphere and oceans are so complicated that to this day no one knows what
the exact warming will be. But basic physics and the geological record
indicate that the warming will be small and probably good for life on Earth.
The additional carbon dioxide of the past century has already benefitted
agriculture, forestry and photosynthetic life in general [3, 4]. Both Tyndall
and Arrhenius thought that greenhouse warming was a good thing.

##  2 Earth’s Atmosphere

We begin with the basic physics of heat transfer in Earth’s atmosphere. Recall
that the largest part of Earth’s atmosphere is a gas, a state of matter where
molecules or atoms spend most of their time hurtling through free space. The
molecules have occasional collisions with others. At sea level each molecule
experiences a collision about once a nanosecond (a billion collisions per
second). A collision sends a colliding pair of molecules careening off in
different directions, like the collisions of billiard balls on a pool table.
If the molecule consists of two or more atoms, like most molecules of Earth’s
atmosphere, some of the energy of internal vibrations and rotations can be
exchanged. But the sum of the translational, rotational and vibrational
energies of a colliding pair of molecules is the same before and after the
collision. The total energy of the colliding pair is conserved, even though
the energy of one of the colliding molecules may be bigger and that of the
other smaller after the collision.

If one were to follow an individual molecule over the billion or so collisions
it experiences every second, it would turn out that the molecule has a well-
defined probability to be found in any vibration-rotation state of quantized
energy $E_{j}$. This is an example of the famous ergodic theorem of
statistical physics, and it describes a system in local thermodynamic
equilibrium. The probability of finding the molecule in a state of energy
$E_{j}$ is proportional to the Boltzmann factor, $e^{-E_{j}/k_{B}T}$. Here $T$
is the absolute temperature of the gas, and Boltzmann’s constant has the value

$k_{B}=1.38\times 10^{-23}{\rm J/K}.$ (1)

Austrian Ludwig Boltzmann, for whom Boltzmann’s constant $k_{B}$ is named, and
his thesis advisor, Joseph Stefan, played very important roles in our
understanding of thermal radiation, statistical mechanics and thermodynamics.

In addition to molecules, Earth’s atmosphere also contains radiation. Sunlight
is present during the day. But thermal radiation is present both day and
night. Like a hot coal from a campfire, Earth glows in the dark with thermal
infrared radiation. There is much more thermal radiation near the warm equator
than near the colder poles or near the surface than at the cold tropopause,
around 11 km in altitude for temperate latitudes.

It is often convenient to think of the radiation as photons, which are a bit
like air molecules, and which have quantized energies $E=h\nu$, where $h$ is
Planck’s constant and $\nu$ is the photon frequency. Inside a dense, nighttime
cloud, the radiation can be very close to thermal equilibrium with the air
molecules and cloud particulates, all of which will have nearly the same
temperature. Then the radiation is well described by Planck’s formula for
blackbody radiation, the very first formula of quantum physics. But most of
the time, atmospheric radiation is far from thermodynamic equilibrium because
the photon mean free paths are much longer than the length scales for
atmospheric temperature changes.

The energy density of radiation in Earth’s atmosphere is many orders of
magnitude smaller than the translational, rotational and vibrational energy
densities of the molecules. Radiation has relatively little heat capacity. It
is ludicrous to hear about “heat trapped” as photons. Molecules account for
almost all of atmospheric heat.

A substantial fraction of thermal-radiation frequencies is in the “infrared
window” where there is negligible clear-sky opacity between the surface and
outer space. Here, there is extremely efficient “ballistic” heat transfer by
radiation.

Greenhouse gases can efficiently absorb thermal radiation with frequencies in
opaque spectral regions. But greenhouse gases almost never scatter thermal
radiation. This is unlike radiation transport of visible light through clouds
or the Rayleigh scattering of blue and ultraviolet light that makes the blue
sky of a sunny day. Visible photons can be scattered hundreds of times with
little absorption.

Although greenhouse gases do not scatter thermal radiation, they do emit it
spontaneously. So thermal radiation in opaque spectral regions is absorbed and
independently reemitted as though it were being isotropically scattered. The
kinetic theory of gases tells us that the diffusion coefficient $D$ of a gas
of molecules (or of a gas of photons) of mean free path $l$ and velocity $v$,
is $D=vl/3$. For quasi-diffusional transport of thermal radiation in opaque
spectral regions, the mean free paths, $l$, of photons between absorptions and
reemissions are much longer than those of molecules. This greatly increases
the heat-transport efficiency of “diffusing” photons, compared to that of
diffusing molecules. Photons also move at the speed of light $c$, about a
million times faster than the fluctuating molecular speed $v$, with an average
value close to the speed of sound in air. So both $l$ and $v$ are orders of
magnitude larger for photons than for molecules.

In summary heat transport by thermal radiation in Earth’s atmosphere is orders
of magnitude faster than heat transfer by molecular diffusion. Heat transfer
by conduction in air (that is, by molecular diffusion) is so small that it is
normally irrelevant compared to heat transfer by radiation or heat transport
by convection. Heat convection by moist air, which can carry lots of latent
heat, as well as sensible heat, is especially important.

###  2.1 Greenhouse Gases

What is a greenhouse gas? It is a gas that is almost transparent to sunlight,
and allows the Sun to heat the ground on a cloud-free day. But greenhouse
gases have significant opacity for the thermal radiation that is constantly
emitted by Earth’s surface and by the warm greenhouse gases themselves. The
nitrogen molecules, N2, and oxygen molecules, O2, which make up 99% of Earth’s
atmosphere, are not greenhouse gases, since they are nearly transparent to
both sunlight and thermal radiation.

Vibrating or rotating electric dipole moments are the most efficient molecular
“antennas” for emitting or absorbing electromagnetic radiation. The electric
dipole moment of a molecule is the product of the spatial separation between
the center of negative electron charge and the center of positive nuclear
charge, and the magnitude of the positive charge. By symmetry, the centers of
both positive and negative charges of N2 and O2 are at the center of the
molecule (and at the center of mass). There is no separation of the charge
centers, so the dipole moment is zero, no matter how the molecules vibrate or
rotate. The diatomic molecule carbon monoxide, CO, has much the same
electronic structure as the nitrogen molecule N2, but because of the lack of
symmetry, the O end of the CO molecule has a negative charge and the C end has
a positive charge. Therefore, CO is a greenhouse gas, but isoelectronic N2 is
not. The greenhouse effects of CO are so much smaller than those of H2O and
CO2 that CO gets little attention in discussions of climate.

The situation with polyatomic molecules is more complicated and interesting.
To illustrate the basic ideas, we will discuss the triatomic CO2 molecule in
some detail. Fig. 1 shows the main features of the CO2 molecule, as they were
sketched by Enrico Fermi in his classic paper on mode mixing and “Fermi
resonances,” published in the year 1931 [5]. Similar considerations apply to
other polyatomic molecules like methane or nitrous oxide. The unexcited CO2
molecule, with neither rotational nor vibrational energy, is linear. The C
atom in the center has a slight positive charge because of the strongly
electronegative O atoms on each end. The CO2 molecule has no electric dipole
moment because the centers of positive and negative charge both coincide with
the center of symmetry.

Figure 1: The three vibrational modes of the CO2 from Fermi’s classic paper on
mode mixing [5]. For the most abundant isotopolgue, 16O 12C 16O, the mode
frequencies are: symmetric stretch (b) $\nu_{1}=1388$ cm-1; (c) bending
$\nu_{2}=667$ cm-1; and (d) asymmetric stretch, $\nu_{3}=2349$ cm-1. Only the
bending mode has a large direct effect on global warming of Earth, but the
other two modes have indirect effects since they have “Fermi resonances” with
overtones of the bending mode.

A CO2 molecule can bend and vibrate after being hit by another molecule during
a collision, much like a xylophone bar will vibrate after being hammered by a
musician. Just as the vibrating xylophone bar emits a sound wave into the air,
the vibrating CO2 molecule emits thermal radiation. A bent CO2 molecule has an
electric dipole moment that points from the center of mass of the two,
slightly negative O atoms toward the slightly positive C atom. The vibrations
of this bending mode make the biggest contribution to the absorption and
emission of thermal radiation by CO2 molecules.

As sketched in Fig. 1, the CO2 molecule also has two modes of vibration that
keep the straight, linear alignment of the three atoms in the unexcited
molecule. In the symmetric stretch mode, the C atom remains fixed as the O
atoms vibrate with equal and opposite displacements along the molecular axis.
For this mode of vibration, there is no vibrating dipole moment, since the
centers of positive and negative charge remain at the center of the molecule.
So the symmetric stretch mode of CO2 does not contribute directly to the
greenhouse properties of the molecule, although it can have an indirect
influence through “mode mixing.” For the asymmetric stretch mode of CO2, the
carbon atom moves in one direction along the symmetry axis, and the two O
atoms on either side move in the opposite direction. The vibration frequency
of the asymmetric stretch mode is higher than the frequencies of most thermal
photons, so it has little direct effect on the greenhouse properties of the
molecule. However, the resulting strong absorption band, centered at a
wavelength of 4.3 micrometers, is used for most non-dispersive infrared
monitors that measure CO2 concentrations in ambient air.

Molecules in Earth’s atmosphere not only vibrate but rotate. This adds
rotational sidebands to the vibration-frequencies of molecules. Because of the
transverse nature of electromagnetic waves, vibrating molecules do not easily
emit nor absorb radiation propagating along the direction of the vibrational
axis. The vibrating electric dipole produces an anisotropic radiation pattern
with most radiation emitted at right angles to the axis of vibration. Rotating
molecules produce a rotating radiation pattern, much like the light beam of a
rotating searchlight or the radar beam from a rotating airport antenna. This
causes the observed intensity to blink on and off at the rotation frequency.
As in amplitude modulation of “carrier” frequencies of radio transmitters,
upper and lower sidebands, displaced by the rotational frequency, are added to
the vibrational carrier frequency of the radiation. The low-frequency sideband
is called the P branch and the high-frequency side band is called the R
branch.

Electronic or nuclear excitations are needed to permit linear molecules like
N2, O2, CO or CO2 to rotate around their axial symmetry axis. The energies are
much too high to be thermally excited in Earth’s atmosphere. So the rotation
axis for linear molecules is always perpendicular to the symmetry axis. This
causes the radiation patterns from axial vibrations of CO or CO2 to split into
the lower and upper sidebands that make up the P and R branches. There is no
radiation at the vibration frequency of the non-rotating molecule. A CO2
molecule vibrating in the asymmetric stretch mode radiates in much the same
way as a “double-sideband carrier-suppressed” radio transmitter, sometimes
used by ham radio operators.

Rotations of CO2 molecules with bending-mode vibrations are more interesting.
The rotation axis must still be perpendicular to the molecular symmetry axis.
But the molecule is vibrating at right angles to the symmetry axis, so the
vibration axis may be perpendicular or parallel to the rotation axis. If the
rotation and vibration axes happen to be perpendicular, as is always the case
for asymmetric stretch vibrations, P and R sidebands are produced by the
spinning antenna pattern. But if the vibrations happen to be parallel to the
rotation axis, the radiation pattern will be unaffected by rotations, and the
molecule will emit radiation at the unshifted vibrational frequency. This is
called the Q branch of the bending-mode band, and it is at the frequency of
the bending mode. Slightly displaced sidebands are produced by centrifugal
stretching of the rotating molecule, but the strongly displaced P and R
branches are missing. The Q branch of the bending mode of CO2 can be seen
clearly in high-resolution, spectral measurements of thermal radiation from
Earth’s atmosphere.

The vibration-rotation spectrum of nitrous oxide, N2O, another linear molecule
like CO2 has properties similar to those discussed above for CO2. But the
analog of CO2’s symmetric stretch mode for the less symmetric molecule N2O can
emit and absorb radiation, unlike CO2 where this mode is “infrared inactive”.
The tetrahedral molecule methane, CH4, is somewhat more complicated but like
CO2 it has several vibrational modes, some of which can absorb and emit
radiation, and some of which cannot. Like the CO2 bending mode, the methane
bands have P, Q and R branches.

Figure 2: To calculate greenhouse effects in detail, one must include the
opacity of hundreds of thousands of individual line intensities, shown here as
colored dots from the HITRAN data base. More details of how line intensities
are used can be found in references [6, 7].

###  2.2 A Model Atmosphere

How greenhouse gases affect Earth’s climate is a complicated issue, where
atmospheric thermodynamics and convection are intimately involved. We will
simplify the discussion as much as possible, but we will also try to adhere to
Einstein’s admonition: “Everything should be made as simple as possible, but
not simpler”. We hope our readers will take the trouble to understand the
basic thermodynamics that we review below.

Solar heating drives Earth’s climate. At the mean distance of Earth from the
Sun, sunlight carries an energy flux of about 1,360 Watts per square meter
(W/m2). We are familiar with this flux, part of which warms us when we
sunbathe at the beach on a cloud-free summer day. The flux at the top of
Earth’s atmosphere actually varies a little bit over the year, since Earth’s
orbit around the Sun is slightly elliptical. Earth is about 3.3% closer to the
Sun in early January than in early July. Since solar flux decreases as the
square of the distance from the Sun, the solar flux at the top of the
atmosphere is about 6.7% or 91 W/m2 greater in January than in July. As we
will discuss in more detail below, for cloud-free temperate latitudes,
doubling the concentration of CO2 would decrease thermal radiation to space by
about 3 W/m2. Note 3 W/m2 is much less than the planet-wide, winter-summer
difference of 91 W/m2. And we are a long way from doubling CO2 .

Launching into our maximally simplified discussion of the greenhouse effect,
we consider a hypothetical Earth with a transparent atmosphere that is 80%
nitrogen and 20% oxygen, and with the same mass as today’s atmosphere. But we
assume no greenhouse gases at all, no CO2, no H2O and no clouds. To be
consistent with no clouds, this hypothetical Earth must have no oceans, from
which water vapor could evaporate. Oxygen, O2, actually does absorb a very
small amount of sunlight and also thermal radiation, but we will ignore that
absorption and assume the atmosphere is completely transparent. To further
simplify the problem, we assume that the Sun shines steadily with equal
intensity on every part of Earth’s surface, from the tropics to the poles.
Earth’s rotation provides daily averaging of insolation over longitude but not
latitude. We will neglect the Coriolis force which is caused by the Earth’s
rotation, that is so important for dynamics of the real atmosphere.

With its radius, $r=6378$ km, the Earth intercepts the solar flux with an
equivalent disk area of $\pi r^{2}$. But the area of the spherical surface of
the Earth is $4\pi r^{2}$. So the average flux per unit area of Earth’s
surface is one quarter of that at the top of the atmosphere or

$F=1360/4{\rm\ W/m}^{2}=340{\rm\ W/m}^{2}.$ (2)

We assume the Earth is a perfect “blackbody” for both sunlight and thermal
radiation, in the sense that the surface absorbs all radiation incident on it,
irrespective of the frequency or direction of the incident light. We assume
negligible heat exchange between the surface and the interior of the Earth.
Then heat can leave the surface in two ways:

• Heat can leave by thermal radiation through the transparent atmosphere to
space;

• Heat can flow to or from the atmosphere, either by conduction or convection,
but not by radiation since we assume the atmosphere has no greenhouse gases
and is completely transparent.

###  2.3 An Isothermal Atmosphere

We assume that the Earth has come to complete thermal equilibrium with an
absolute surface temperature $T_{0}$, which is also equal to the temperature
of the hypothetical, completely transparent atmosphere above. The flux of
thermal radiation by the Earth to space is equal to the absorbed solar flux
(2) so the surface neither heats nor cools. Then the Stefan-Boltzmann law of
thermal radiation by blackbodies requires the thermal radiation emitted to
space be

$F=340{\rm\ W/m}^{2}=\sigma T_{0}^{4}.$ (3)

Here the Stefan Boltzmann constant is

$\sigma=5.67\times 10^{-8}{\rm\ W/(m^{2}K^{4})}.$ (4)

The temperature that solves equation (3) is

$T_{0}=278.3{\rm\ K}.$ (5)

Smaller temperatures for a greenhouse-free planet Earth are often quoted
because clouds of condensed water reflect about 30% of the solar radiation in
an atmosphere paradoxically devoid of Earth’s most important greenhouse gas,
water vapor. In discussions of the most basic physics of greenhouse warming,
clouds are an unnecessary distraction. But clouds are very important for a
detailed understanding of greenhouse effects.

Once the surface and the atmosphere have reached the same temperature, the
temperature (5) of the surface will be the same value the Earth would have if
there were no atmosphere at all. But the Earth’s atmosphere is massive, with a
surface pressure of about 10 tons per square meter, the weight of air
molecules in a one square meter column of air rising from the surface to
infinity. For convenience, we assume the hypothetical atmosphere produces the
“standard” surface pressure

$p_{0}=10^{5}{\rm\ Pa}=1000{\rm\ hPa}.$ (6)

The MKS unit of pressure, Pa = Pascale, is one Newton per square meter.
Besides mass, the atmosphere also contains lots of energy, both thermal energy
and gravitational potential energy. For computational convenience and physical
clarity we will consider the energy per atmospheric molecule. The average
thermal energy of a molecule of temperature $T_{0}$ is

$u=c_{v}k_{B}T_{0}.$ (7)

Here the per-molecule heat capacity at constant volume, in units of
Boltzmann’s constant (1), is very nearly

$c_{v}=5/2.$ (8)

Recall from the kinetic theory of gases that the 5 in the numerator of (8) is
the sum of 3, the number of translational degrees freedom, and 2, the number
of rotational degrees of freedom for linear molecules like O2 or N2.
Atmospheric temperatures are not high enough for vibrational degrees of
freedom to make a significant contribution to the heat capacity.

For a static atmosphere, the air pressure at any altitude $z$ is simply the
weight, per unit area, of air at higher altitudes. The differential expression
of this fact is the equation of hydrostatic equilibrium,

$dp=-\ {{mg}\over v}dz,$ (9)

where the per-molecule gas volume is $v$ and $m$ is the mass per molecule.
According to (9), the decrease $dp$ in pressure for an incremental increase
$dz$ of altitude is the weight per unit area of an atmospheric slab of width
$dz$. Using the per-molecule ideal gas formula

$pv=k_{B}T,$ (10)

we can integrate (9) at constant temperature, $T=T_{0}$, to find the
barometric formula for an isothermal atmosphere,

$p=p_{0}e^{-z/{z_{T}}}.$ (11)

The average altitude of an atmospheric molecule in an isothermal atmosphere is

$z_{T}={{k_{B}T_{0}}\over{mg}}.$ (12)

The mass per molecule of our model atmosphere of 80% N2 and 20% O2 by mole
fraction is

$m=4.81\times 10^{-26}{\rm\ kg}.$ (13)

The average acceleration of gravity at Earth’s surface is

$g=9.8{\rm\ ms^{-2}}.$ (14)

So the average gravitational potential energy of a molecule in an isothermal
atmosphere is

$\epsilon_{g}=mgz_{T}=k_{B}T_{0}.$ (15)

Summing the thermal energy (7) and the gravitational potential energy (15), we
find the total, per-molecule energy of an isothermal atmosphere,

$\epsilon=u+\epsilon_{g}=(c_{v}+1)k_{B}T_{0}=c_{p}k_{B}T_{0}.$ (16)

Here the per-molecule heat capacity at constant pressure is

$c_{p}=c_{v}+1=7/2.$ (17)

The per-molecule enthalpy is

$h=u+k_{B}T_{0}=(c_{v}+1)k_{B}T_{0}=c_{p}k_{B}T_{0},$ (18)

the same as the formula (16) for the sum of the kinetic energy and
gravitational potential energy. The identity of (16) and (18) is not
accidental. The enthalpy increase $dH=Ndh$ of a gas of $N$ molecules due to an
increment $dQ$ of absorbed heat is the sum of the increase of thermal energy,
$dU=Ndu$ and the work, $dW=pdV=Nk_{B}dT$, done when the gas expands by a
volume increment $dV=Ndv$, at constant pressure $p$, into the surrounding
environment. A fraction $c_{v}/c_{p}=5/7$ of the heat added to an isothermal
atmosphere goes into increasing the thermal energy of the molecules. A
fraction $1/c_{p}=2/7$ goes into lifting the molecules to higher altitudes.

###  2.4 Non-Isothermal Atmospheres

For many reasons, the Earth’s atmosphere is not isothermal. Here we want to
focus on the most important reason: greenhouse gases force Earth’s lower
atmosphere, the troposphere, to convect. The convection drives the troposphere
toward an adiabatic atmosphere, for which the dry adiabatic lapse rate is a
temperature drop of $dT/dz=-9.8$ K/km.

To understand the essence of convection we use the concept of a fluid
“parcel.” A parcel is a fixed mass $M$ of the fluid in a volume $V$. Noting
that the number of air molecules in a mass $M$ is $N=M/m$, we can use the mass
per molecule (13) and the ideal gas law (10) to show that a parcel with a mass
$M=1$ kg at the temperature (5) and pressure (6) would occupy a volume of
$V=0.78$ cubic meters (m3). We assume that the parcel is small enough that all
parts of it have nearly the same temperature $T$ and pressure $p$.

The parcel is surrounded by other gas at nearly the same pressure $p$. If the
parcel volume $V$ changes reversibly by the increment $dV$, the parcel will do
an increment of work

$dW=pdV$ (19)

on the surroundings. The parcel can also reversibly exchange increments of
heat,

$dQ=TdS,$ (20)

with its surroundings. Heat is absorbed, $dQ>0$ if the entropy increment of
the parcel is positive, $dS>0$, and heat is released, with $dQ<0$ if the
entropy change of the parcel is negative, $dS<0$.

The first law of thermodynamics is a statement of the conservation of energy.
It can be written as

$dU=dQ\ –\ dW=TdS-pdV,$ (21)

The expression after the first equal sign is true in general, even for a non-
reversible process where entropy increases inside the parcel with no addition
of heat. This often involves explosive processes like the ignition of the gas-
air mixture of an internal combustion engine. The expression after the second
equal sign is true for a reversible process like those of (19) and (20), where
the “entropy of the universe” remains constant.

###  2.5 Entropy

Entropy is one of the most profound and poorly understood concepts of physics.
It is also very helpful in understanding how greenhouse gases work, so we
briefly review it here. The most fundamental definition of the thermodynamic
temperature follows from (20) and is,

${1\over T}={{dS}\over{dQ}}.$ (22)

The ratio of the small entropy increase $dS$ of a parcel in thermal
equilibrium to the small quantity of heat, $dQ$, that was absorbed to cause
the entropy increase, is the inverse of the thermodynamic temperature. Adding
heat to a system can increase its temperature, so the heat increment $dQ$ must
be small enough to cause a negligible increase of the temperature $T$. In (22)
we have assumed that no water contributes to the energy change of the parcel,
$dW=0$.

Like energy, entropy is additive, so we can compute the total entropy of the
atmosphere by adding up the entropies of all of its parcels. But unlike
energy, which is rigorously conserved, “the entropy of the universe” is a
steadily increasing quantity. This is the essence of the Second Law of
Thermodynamics. The German physicist Clausius, who is credited, along with the
Scottish physicist William Thomson (Lord Kelvin), with discovering the
significance of entropy, summarized all of thermodynamics in the laconic
words: Die Energie der Welt ist konstant. Die Entropie der Welt strebt einem
Maximum zu. The energy of the universe is constant. Its entropy tends to a
maximum [8].

For example, suppose two air parcels are in contact, the first with absolute
temperature $T_{1}$ and the second with a different temperature $T_{2}$.
Transferring a small, positive quantity $dQ$ of heat from parcel 1 to parcel 2
will change the entropy of parcel 2 by $dS_{2}=dQ/T_{2}$ and will change the
entropy of parcel 1 by $dS_{1}=-dQ/T_{1}$. Since energy is conserved, the
increment of heat $dQ$ gained by parcel 2 is equal to the increment of heat
lost by parcel 1. As a result of this exchange, the entropy of the universe
changes by

$dS=dS_{1}+dS_{2}=dQ\bigg{(}{{-1}\over{T_{1}}}+{1\over{T_{2}}}\bigg{)}.$ (23)

We have assumed a positive heat increment, $dQ>0$, so to ensure that $dS>0$,
(22) implies that $1/T_{2}>1/T_{1}$; that is, $T_{1}>T_{2}$. Heat can flow
spontaneously from a hot parcel to a cold parcel, but never in the opposite
direction. An exception is exceptionally small parcels where “fluctuations”
allow heat to flow temporarily from cool surroundings into a warmer, small
parcel. An extreme example is a “parcel” consisting of a single CO2 molecule
which regularly gets so much hotter than the surrounding gas that it can emit
a photon at the frequency of a vibrational mode, even though the photon energy
is several times larger than $k_{B}T$.

The concept of entropy originated in classical thermodynamics, in the mid-19th
century. But it played a key role in Boltzmann’s development of classical
statistical physics, and in Planck’s invention of quantum mechanics toward the
end of the century. It was Boltzmann who first recognized the close connection
between entropy $S$ and the thermodynamic probability, $W$. Here $W$ stands
for Wahrscheinlichkeit = probability in German, not for work. Boltzmann
considered the connection so important, and rightly so, that engraved on his
tombstone in Vienna’s Zentralfriedhof is the equation

$S=k{\rm\ }logW.$ (24)

Here $k=k_{B}$ is Boltzmann’s constant of (1), the natural unit of entropy.

In statistical mechanics, entropy is a measure of disorder of the physical
system. For example, one kg of boiling water at a temperature of 100 C, a
pressure of 1 atmosphere and volume of 1 liter, has far less entropy or
disorder than one kg of steam, which has the much larger volume of 1244 liters
at the same temperature and pressure.

Entropy is an extensive state function of thermodynamic systems, much like the
volume $V$ or the internal energy $U$. The units of entropy are J/K, Joules
per degree Kelvin, the same units as Boltzmann’s constant (1). For an air
parcel with $N$ molecules and total entropy $S$, we will define the entropy
per molecule as

$s={S\over N}.$ (25)

Then the first law of thermodynamics (21) for an ideal gas can be written in
the per-molecule form

$c_{v}k_{B}dT=Tds\ –\ pdv.$ (26)

Integrating (26) with the aid of the per-molecule ideal gas law (10), we find
the per-molecule entropy

$s=s_{o}+c_{p}k_{B}ln(T/T_{0})\ –\ k_{B}ln(p/p_{0}).$ (27)

The per-molecule entropy at the Earth’s surface is $s_{o}$. The per-molecule
heat capacity at constant pressure $c_{p}$ was given by (17). The pressure and
temperature at the surface are $p_{0}$ and $T_{0}$.

For an isothermal atmosphere, $T=T_{0}$ at all altitudes $z$, the pressure is
given by the barometric formula (11). So the per-molecule entropy (27) for an
isothermal atmosphere increases linearly with altitude and is given by

$s=s_{o}+{{mgz}\over{T_{0}}}.$ (28)

For a general, non-isothermal atmosphere we can differentiate (27) with
respect to altitude $z$, and use the equation of hydrostatic equilibrium (9)
for $dp/dz$, to find

${{ds}\over{dz}}={{c_{p}k_{B}}\over T}\bigg{(}{{dT}\over{dz}}+L\bigg{)},$ (29)

In (29) we have introduced the “adiabatic lapse rate” for dry air

$L={{mg}\over{c_{p}k_{B}}}=9.8{\rm\ K/km}.$ (30)

In atmospheric physics the entropy per molecule is often defined in terms of a
function of temperature and pressure called the “potential temperature,”

$\theta=T\bigg{(}{{p_{0}}\over p}\bigg{)}^{1/c_{p}}.$ (31)

In terms of the potential temperature (31) the expression (27) of the per-
molecule entropy simplifies to

$s=s_{o}+k_{B}ln\bigg{(}{{\theta}\over{T_{0}}}\bigg{)}^{c_{p}}.$ (32)

Molecules with the same potential temperature have the same entropy $s$.

###  2.6 Atmospheric Stability

The linear increase of entropy with altitude of an isothermal atmosphere,
which is described by (28) gives the atmosphere “stability,” in the sense that
air parcels try to remain at their original altitudes. To get a more
quantitative understanding of atmospheric stability we note that the mass
density of environmental air is

$\rho={m\over v}={{mp}\over{k_{B}T}}.$ (33)

According to the Archimedes Principle, the per-molecule force $f$ on a parcel
p of mass density $\rho_{\rm p}=m/v_{\rm p}$ and per-molecule volume $v_{\rm
p}$, that has displaced an equal volume $v=v_{\rm p}$ of environmental air of
mass density $\rho$ is

$f=-gv_{\rm p}(\rho_{\rm p}-\rho).$ (34)

The first term of (34), $-gv_{\rm p}\rho_{\rm p}=-gm$, is the negative force
of gravity per parcel molecule. The second term, $gv_{\rm p}\rho$, is the
upward buoyant force of displaced environmental air. The buoyant force will
not be big enough to support the parcel’s weight if $\rho<\rho_{\rm p}$ and
the parcel will sink down. Or the buoyant force may exceed the parcel’s weight
if $\rho>\rho_{\rm p}$ and the parcel will float up. Suppose that at altitude
$z$, the parcel is a sample volume of environmental air, so that $\rho_{\rm
p}(z)=\rho(z)$. Then from (34), $f(z)=0$ and the parcel will be in mechanical
equilibrium with no net force acting on it.

Now imagine lifting a parcel of environmental air by an altitude increment
$dz$, where the environmental air density is $\rho(z+dz)=\rho(z)+d\rho$. Since
the parcel is neutrally buoyant, negligible work is needed to lift it by a
small amount. The lifted parcel will expand and do an increment $dW$ of work
on the lower-pressure surrounding air. From the first law of thermodynamics
the energy for the work comes from heat $dQ$ that flows into the parcel, and
from $-dU$, the decrease of internal energy. Because of the very small thermal
conductivity of air, it takes a very long time for appreciable heat to flow
into or out of a parcel of reasonable size. For example, temperature
equilibration by molecular heat conductivity alone in a parcel with a diameter
$d=1$ meter would require a time of order, $t=d^{2}/D$, or about a day for a
sea-level molecular diffusion coefficient of $D=0.1$ cm2 s-1. But the pressure
equilibration times are on the order of $d/v$, the parcel diameter divided $d$
by the speed of sound, $v$, around 300 m/s. The pressure equilibration time
for the parcel would only be a few milliseconds. We will therefore consider
lifting processes where there is not enough time for appreciable heat flow
into or out of a parcel, and we can set $dQ=0$. Then the lifted parcel retains
the same per-molecule entropy as it had at the altitude $z$, and the lifting
will be adiabatic.

At the altitude $z+dz$ the force may no longer be zero since the lifted parcel
may have a different density from the environmental air it displaced. The net
per-molecule force at the altitude $z+dz$ will be

$df=-gv(d\rho_{\rm p}-d\rho).$ (35)

The density increment of the adiabatically lifted air is $d\rho_{\rm p}$ and
the density increment of the displaced air is $d\rho$. Differentiating the
expression (33) and using the condition (9) for hydrostatic equilibrium as
well as the ideal gas law (10), we find that the density increment for the
environmental air is

$d\rho=-{m\over{Tv}}\bigg{(}{{mg}\over{k_{B}}}+{{dT}\over{dz}}\bigg{)}dz.$
(36)

To order $dz$ we get the density increment for the adiabatically lifted parcel
from (36) by replacing $dT/dz$ by the negative of the adiabatic lapse rate
(30),

$d\rho_{\rm p}=-{m\over{Tv}}\bigg{(}{{mg}\over{k_{B}}}-L\bigg{)}dz.$ (37)

Substituting (36) and (37) into (35), as well as using (29) and (30), we find
the force per parcel molecule

$df=-{{mg}\over
T}\bigg{(}L+{{dT}\over{dz}}\bigg{)}dz=-{{mg}\over{c_{p}k_{B}}}\
{{ds}\over{dz}}\ dz=-Lds.$ (38)

If a parcel is adiabatically lifted by an increment $dz$ in altitude, the
increment $df$ of the buoyant force per molecule is simply the negative
product of the adiabatic lapse rate $L$ of (30) and the per-molecule entropy
increment, $ds$, of environmental air between altitudes $z+dz$ and $z$. If the
per-molecule entropy increases with increasing altitude, $ds/dz>0$, there will
be a net restoring force that will try to push the parcel back down to its
original altitude, or push it back up if $dz<0$.

For a stable atmosphere, with $ds/dz>0$, the molecules behave as if they were
bound to their initial altitudes by springs of force constants $k=Lds/dz$. We
recall from elementary mechanics that a particle of mass $m$ held by a spring
of force constant $k$ will oscillate with a frequency $\Omega=(k/m)^{1/2}$. So
a stable atmosphere has a buoyancy frequency $\Omega$ given by

$\Omega^{2}={L\over m}\ {{ds}\over{dz}}={g\over{c_{p}k_{B}}}\
{{ds}\over{dz}}.$ (39)

The buoyancy frequency, $\Omega$ defined by (39), is called the Brunt-Vaisala
frequency [9].

For an isothermal atmosphere, with $dT/dz=0$, we can use the expression (29)
for $ds/dz$ to write the square of the buoyant frequency as

$\Omega^{2}={{mg^{2}}\over{c_{p}k_{B}T_{0}}}={{gL}\over{T_{0}}}.$ (40)

Using the acceleration of gravity $g$ of (14), the adiabatic lapse rate $L$
and the surface temperature $T_{0}$ of (5) in (40), we find that the buoyancy
frequency of a hypothetical, isothermal atmosphere is

$\Omega=0.0186{\rm\ s}^{-1}={{2\pi}\over{338{\rm\ s}}}$ (41)

The oscillation period is 338 s, or between 5 and 6 minutes. This period is
comparable to the fluctuation times of the internal structure of clouds,
drifting through a blue sky. We recall that the squared frequency of a
pendulum of length $l$ is

$\Omega^{2}={g\over l}.$ (42)

Comparing (40) and (42), we see that the buoyancy oscillations of an
isothermal atmosphere have the same frequency as a pendulum of length

$l={{T_{0}}\over L}=z_{a}=28.4{\rm\ km}.$ (43)

As we will discuss in the following section, the length $z_{a}$ defined by
(43) is the height of an adiabatic atmosphere of ideal gas, with a surface
temperature $T_{0}$. The numerical value, 28.4 km, would be in the mid
stratosphere of Earth’s atmosphere.

###  2.7 An Adiabatic Atmosphere

In addition to the isothermal atmosphere discussed above, another important
limit is the adiabatic atmosphere, where the per-molecule entropy $s$ is
independent of altitude $z$. For parcels in Earth’s atmosphere, we will use
the adjectives adiabatic and isentropic interchangeably, but under some
conditions, adiabatic processes can differ dramatically from isentropic
processes. An isentropic change in the volume of a parcel is one where the
parcel entropy is the same before and after the change. An adiabatic change of
the volume of a parcel is one in which no heat is exchanged between the parcel
and the environment. But the moving boundaries of the parcel can do positive
or negative work on the parcel.

Adiabatic volume changes cannot be too fast or they will increase the parcel
entropy. For example, if the containing boundaries move at supersonic speeds,
converging boundaries can produce shock waves, and expanding boundaries can
produce expansion fans in the parcel gas. Both phenomena increase the parcel
entropy, even though no heat has been added. This is an important issue for
laser-fusion work, where the parcel is the gas in a small pellet that is
imploded by intense laser pulses. Most natural atmospheric changes are slow
enough that isentropic and adiabatic mean the same thing.

From inspection of (29) we see that for an adiabatic atmosphere, with
$ds/dz=0$, the rate of change of temperature with altitude must be equal and
opposite to the adiabatic lapse rate,

${{dT}\over{dz}}=-L=-9.8{\rm\ K/km}.$ (44)

The temperature profile of an adiabatic atmosphere must therefore be

$T=T_{a}-zL.$ (45)

The temperature, $T_{a}$, at the bottom of the adiabatic atmosphere need not
be the same as the equilibrium surface temperature $T_{0}$ of (5) for an
atmosphere without greenhouse gases. The thickness, $z_{a}$, of an adiabatic
atmosphere,

$z_{a}={{T_{a}}\over L}={{c_{p}k_{B}T_{a}}\over{mg}}$ (46)

is the altitude at which the absolute temperature (45) goes to zero, $T=0$. We
already mentioned in connection with (43) that a representative thickness is
$z_{a}=28.4$ km.

Using the adiabatic temperature profile (45) with the formula (27) for the
per-molecule entropy and replacing the surface temperature $T_{0}$ with
$T_{a}$, we find that the pressure in an isentropic atmosphere must be given
by the formula

$p=p_{0}(1-z/z_{a})^{c_{p}}.$ (47)

The fraction of atmospheric molecules between altitudes with pressure $p$, and
altitudes with pressure $p+dp$, is $dp/p_{0}$. So the average temperature of a
molecule in an adiabatic atmosphere is

$<T>={1\over{p_{0}}}\int_{0}^{p_{0}}Tdp.$ (48)

Noting from (45) and (47) that

$T=T_{a}\bigg{(}{p\over{p_{0}}}\bigg{)}^{1/c_{p}},$ (49)

we can integrate (48) to find

$<T>=T_{a}{{c_{p}}\over{c_{p}+1}}.$ (50)

We can replace $T_{0}$ with $<T>$ in (7) to find that the per-molecule thermal
energy of an adiabatic atmosphere is

$u=c_{v}k_{B}T_{a}{{c_{p}}\over{c_{p}+1}}.$ (51)

In like manner, one can show that the mean altitude of a molecule is

$<z>={1\over{p_{0}}}\int_{0}^{p_{0}}zdp={{z_{a}}\over{c_{p}+1}}.$ (52)

Then the mean per-molecule gravitational energy becomes

$\epsilon_{g}=mg<z>=k_{B}T_{a}{{c_{p}}\over{c_{p}+1}}.$ (53)

For an adiabatic atmosphere with a surface temperature $T_{a}$ equal to the
surface temperature $T_{0}$ of an isothermal atmosphere, both the thermal and
gravitational energies of the adiabatic atmosphere are smaller than those of
the isothermal atmosphere by the factor $c_{p}/(c_{p}+1)$ or 7/9 for the
hypothetical atmosphere of diatomic molecules with negligible vibrational
excitation. For the same surface temperature, molecules of an adiabatic
atmosphere have a lower average altitude and a lower average temperature than
those of an isothermal atmosphere.

Finally, we note that since $ds/dz=0$ for an adiabatic atmosphere, the
buoyancy frequency for the atmosphere, given by (39), is zero,

$\Omega=0.$ (54)

Air parcels in an adiabatic atmosphere have no tendency to return to their
original altitude if they are displaced upward or downward. There is no
restoring force, as there is for an isothermal atmosphere.

In the absence of greenhouse gases, the isothermal atmosphere will not change
with time, since there is no thermal gradient to drive heat flow. However,
without greenhouse gases, a thermally isolated adiabatic atmosphere would
slowly evolve to an isothermal atmosphere because of conductive heat flow from
the warmer lower atmosphere to the colder upper atmosphere.

As we will discuss in the next section, atmospheres with sufficient
concentrations of greenhouse gases can maintain an approximately adiabatic
troposphere and an approximately isothermal stratosphere indefinitely, because
greenhouse gases facilitate convective transfer of heat from the solar heated
surface through the troposphere.

###  2.8 Atmospheres with Greenhouse Gases

The physics we have outlined above changes dramatically if we add greenhouse
gases to our model. For simplicity, we will consider a well-mixed greenhouse
gas, like CO2 in Earth’s atmosphere. In keeping with our earlier discussion of
greenhouse gases, our model atmosphere with greenhouse gas lets the Sun
continue to heat the Earth’s surface with the flux (3), but the greenhouse
gases attenuate thermal radiation such that the fraction of monochromatic
vertical radiation that reaches outer space without absorption is

$f=e^{-\tau_{o}}.$ (55)

Here $\tau_{o}$, the vertical optical depth between the surface and outer
space, will be proportional to the partial pressure of the greenhouse gases.
For Earth and other planets of the Sun, the optical depth can greatly exceed
unity at the centers of strong absorption bands. Negligible surface radiation
reaches outer space for these strongly absorbed frequencies, and most escaping
thermal radiation is emitted by greenhouse gases at altitudes well above the
surface.

The radiation to space by greenhouse gases cools air parcels. The cooling is
most pronounced at higher altitudes where the radiation has a good chance to
reach outer space without being absorbed by still higher-altitude greenhouse
gases. Therefore, greenhouse gases cause high-altitude air parcels to cool. As
the parcels cool, contract and become heavier than the surrounding air, they
sink and are replaced by rising parcels of warmer air that float up from
below.

Since greenhouse gases do not absorb solar radiation, the Sun continues to
heat the surface at the rate (3). The rising and sinking air parcels exchange
little heat with the surrounding air, so they behave like an adiabatic
conveyor belt that carries the solar energy that has been absorbed by the
surface to sufficiently high altitudes where radiating greenhouse gases can
release the energy to space. The most dramatic effect of sufficiently high
concentrations of greenhouse gases is to create a nearly adiabatic troposphere
of convecting air.

Because of the complicated vibrations and rotations of real greenhouse
molecules, the optical depth $\tau_{o}$ of (55) depends strongly on the
frequency of the thermal radiation. There are infrared “window” frequencies
where there is almost no attenuation and one can set $\tau_{o}=0$. But for
frequencies in the center of the Q branch of CO2, one finds optical depths of
order $\tau_{o}=500,000$.

##  3 A Gray Atmosphere

A model for the frequency dependence of atmospheric opacity (which pushes the
limits of Einstein’s admonition – that models should not be simpler than
possible) is a gray atmosphere, with no frequency dependence at all. Then the
optical depth is the same for all thermal radiation frequencies. We will also
assume that as one descends into the atmosphere from outer space, where the
pressure is $p=0$, to an altitude of pressure $p$, the optical depth grows to
the value

$\tau=\tau_{o}{p\over{p_{0}}}.$ (56)

The surface pressure $p_{0}$ was given by (6). For an adiabatic atmosphere, we
can use (47) to write the optical depth (56) as

$\tau=\tau_{o}\bigg{(}1-{z\over{z_{a}}}\bigg{)}^{c_{p}}.$ (57)

For optically thick atmospheres, with $\tau_{o}>>1$, most of the monochromatic
radiation from the atmosphere to space comes from altitudes $z$, or pressures
$p$, near those for which the optical thickness to space is $\tau=1$. So we
can use (56) to define emission pressure as

$p_{e}={{p_{0}}\over{\tau_{o}}}$ (58)

and we can use (57) to define an emission altitude as

$z_{e}=z_{a}\bigg{(}1-{1\over{{{\tau_{o}}^{1/c_{p}}}}}\bigg{)}.$ (59)

At altitudes above (59) so little greenhouse gas remains that the radiative
flux remains constant versus increasing altitude. The radiation flux below the
emission altitude rapidly drops to nearly zero because the intensity becomes
isotropic, with nearly as much downward radiative flux from the atmosphere
above as upward flux from the atmosphere below. The emission level is a site
of intense radiative cooling of the atmosphere. The greenhouse molecules at
the emission altitude supply the model planet’s thermal radiation to space.
Energy to replace that radiated away at the emission altitude is provided by
convection of warm air parcels floating up from below. Having cooled, the
parcels sink back to the surface to complete the convection cycle.

For the model of a gray atmosphere, the emission altitude (58) can be
considered the tropopause, below which there is a convective troposphere, and
above which there is a nearly isothermal stratosphere, with negligible
convection. For this maximally simplified model, heat transport from the
surface to the tropopause is almost all convective and heat transport above
the troposphere is all radiative.

According to (45) and (46), the temperature at the emission altitude is

$T_{e}=T_{a}\bigg{(}1-{{z_{e}}\over{z_{a}}}\bigg{)}.$ (60)

Here $T_{a}$ is the surface temperature of an adiabatic atmosphere. The
emission altitude $z_{e}$ of (59) is where the gray adiabatic atmosphere sends
the absorbed solar flux (3) back to space as thermal radiation. So the
emission temperature $T_{e}$ must be the same as the surface temperature
$T_{0}$ of (5) for an Earth with no greenhouse gases. So (60) implies that

$T_{0}=T_{a}\bigg{(}1-{{z_{e}}\over{z_{a}}}\bigg{)}.$ (61)

From (61) and (59) we find the surface temperature of the gray atmosphere is

$T_{a}=T_{0}{\tau_{o}}^{1/c_{p}}.$ (62)

Differentiating this equation yields

${{dT_{a}}\over{T_{a}}}={{d\tau_{o}}\over{c_{p}\tau_{o}}}.$ (63)

All but 1% of Earth’s atmosphere is comprised of diatomic gases N2 and O2 for
which $c_{p}=3.5$. Hence, the relative change in surface temperature is 29% of
the relative change of the atmosphere’s optical depth. From (46), (61) and
(62) we see that the emission (or tropopause) altitude of the gray atmosphere
is

$z_{e}=({\tau_{o}}^{1/c_{p}}-1){{c_{p}k_{B}T_{0}}\over{mg}}.$ (64)

The pressure at Earth’s tropopause varies from around 100 hPa in the tropics
to 300 hPa at the poles. Taking a midlatitude pressure of $p_{e}=200$ hPa with
the surface pressure $p_{0}=1000$ hPa of (6), we see from (58) that the
optical depth of our gray atmosphere would have to be

$\tau_{o}=5.$ (65)

According to (62), for the heat capacity $c_{p}=3.5$ of (17) the optical depth
(65) would require the surface to warm by a factor

${\tau_{o}}^{1/c_{p}}=5^{1/3.5}=1.58.$ (66)

For the surface temperature $T_{0}=278.3$ K of a model planet with no
greenhouse gases, the temperature increase would then be

$\Delta T=T_{a}-T_{0}=({\tau_{o}}^{1/c_{p}}-1)T_{0}=162{\rm\ K}.$ (67)

Figure 3: The simplest example of greenhouse warming. Greenhouse gases in a
gray atmosphere, which is uniform in the horizontal direction, produce a
convecting, adiabatic troposphere below the tropopause, at the emission
altitude $z_{e}$ of (68). The emission temperature $T_{0}$ is the same as the
surface temperature (5) of the surface if there were no greenhouse gases. In
the adiabatic troposphere, the heat flux $Z$ is convective and is equal to the
solar heating flux $F=\sigma T_{0}^{4}$ of (3). The heat flux in the
troposphere is entirely convective while in the isothermal stratosphere the
flux is entirely radiative.

Using (66) in (64) we find a tropopause altitude for the gray atmosphere

$z_{e}=16.6{\rm\ km}.$ (68)

The height of the troposphere in the real atmosphere of the Earth ranges from
as low as 6 km over the midwinter poles to almost 20 km over the equator.

The results of this section are summarized in Fig. 3. The simple “one-
dimensional” model of a gray atmosphere is only semi-quantitative, but it
leads us to a number of important insights on how greenhouse gases work.

• Greenhouse gases cool the upper atmosphere by radiating heat to space.

• For a gray atmosphere, surface radiation to space is attenuated by factor
$e^{-\tau_{o}}$ where $\tau_{o}$ is the vertical optical depth from the
surface to outer space. Absorbed surface radiation is replaced by radiation
emitted by greenhouse gases higher in the atmosphere.

• For optical depths, $\tau_{o}>>1$, atmospheres develop a lower troposphere
where convection by parcels of warm air floating upward and parcels of cold
air sinking downward transport most of the solar energy absorbed by the
surface. Greenhouse gases radiate heat to space from altitudes close to the
tropopause. Radiation to space from the surface and from greenhouse molecules
in the lower troposphere is negligible.

• Radiative heat transport is negligible compared to convective heat transport
below the tropopause.

• Convective heat transport is negligible compared to radiative heat transport
above the tropopause.

• Without greenhouse gases, the adiabatic temperature profile of the
troposphere would evolve into an isothermal profile with the same temperature
as the surface.

• Greenhouse gases increase the temperature of the surface, compared to the
surface temperature in an atmosphere without greenhouse gases.

##  4 Realistic Model of Atmosphere

Earth’s real atmosphere responds to greenhouse gases in a more complicated way
than the simple model discussed above. But as the model predicts, the Earth’s
atmosphere is partitioned into a convecting troposphere between the surface
and the tropopause, and a stratosphere above the tropopause where there is
very little vertical convection. There is greenhouse warming of the real
Earth’s surface, but much less than for the model discussed above. We briefly
discuss some of the most important details of the atmosphere that must be
accounted for to make a realistic model.

The Earth is not uniformly heated by the Sun, as was assumed in the model. The
midday heating is greatest at the subsolar latitudes, where the Sun is
directly overhead at noon. The subsolar latitude reaches its most northern
bound, the Tropic of Cancer at 23.5o north around 21 June, the northern summer
solstice. The most southern bound, the Tropic of Capricorn at 23.5o south, is
attained on the southern summer solstice, about 21 December.

Figure 4: The continuous blue curve is the yearly average of incoming short-
wave solar flux (net visible, near infrared and ultraviolet) absorbed by the
Earth. The dashed red curve is the yearly average of the outgoing thermal flux
(net longwave infrared) radiated to space by the Earth. Excess solar energy
absorbed in the tropics is transported to the poles by mass flux in the
atmosphere and oceans. The data is from satellite observations [10].

As shown in Fig. 4, the yearly averaged solar insolation at the top of the
atmosphere is about six times larger at the equator than at either pole. A
significant part of the solar energy absorbed in the tropics is convected by
the atmosphere and oceans toward the poles before it is radiated to space by
Earth’s surface and greenhouse gases. In the tropics the Earth radiates less
energy to space than the solar energy it receives, since lots of heat is
exported north and south. Near the poles, the Earth radiates more heat to
space than the solar energy it receives. Heat convected toward the poles from
the tropics makes up the difference.

The intertropical convergence zone is where trade winds from the north and
south converge. There, they loft huge quantities of moist air up to the
stratosphere and produce heavy rainfall. The cloud tops in the tropopause of
the intertropical convergence zone can be up to 18 km high and with
correspondingly low temperatures, down to 200 K or less. The weak thermal
emission to space from the high, cold cloud tops of the tropics are
responsible for the dip in the thermal flux at the top of the atmosphere,
which can be seen in Fig. 4. The intertropical convergence zone migrates north
and south along with the subsolar latitude. This leads to the dramatic
monsoons of tropical and subtropical latitudes. The complicated distribution
of land and oceans on Earth’s surface means that for a given longitude, the
well-defined subsolar latitude may be greater or less than the latitude of the
intertropical convergence zone.

Earth’s winds are strongly affected by the rotation of the Earth, which was
also neglected in the simple model. The Coriolis forces for the rotating Earth
push moving currents of atmospheric air (or ocean water) to the right in the
northern hemisphere and to the left in the southern hemisphere. A dramatic
result is the formation of jet streams in the upper atmosphere.

The preferential heating of the tropics causes atmospheric isobars, surfaces
of constant pressure, to slope downward from the tropics to the poles. The
resulting “baroclinicity,” is a key driving factor for the Rossby waves of
temperate latitude weather fronts and for tropical cyclones like hurricanes.

Figure 5: Left. A standard midlatitude atmospheric temperature profile [11].
The altitudes of the tropopause, stratopause and mesopause are shown. Higher
parts of the atmosphere have negligible effects on radiative transfer. The
Earth’s mean surface temperature is 288.7 K. Right. Observed concentrations,
$C_{\rm sd}^{\\{i\\}}$ for greenhouse molecules versus altitude [12]. The sea
level concentrations are 7750 ppm of H2O, 1.8 ppm of CH4 and 0.32 ppm of N2O.
The O3 concentration peaks at 7.8 ppm at an altitude of 35 km. The CO2
concentration is 400 ppm at all altitudes.

Unlike the model, which assumed dry air, the real atmosphere of the Earth is
moist. The evaporation of water at the surface, the condensation of water to
clouds in the troposphere, and the precipitation of rain and snow, transport
large amounts of heat both vertically and horizontally. The condensation of
moisture in upwardly convecting air parcels releases large amounts of heat to
the surrounding N2 and O2 molecules that make up most of the atmosphere. As a
result, the dry adiabatic lapse rate of (30), $L=9.8$ K/km, is only observed
near the top of the troposphere, where little water vapor remains to condense.
The lapse rate can be half the dry rate or less for very moist air near the
surface. A typical average lapse rate between the surface and the tropopause
is about 6.5 K/km, as shown in the “standard” temperature profile of the
troposphere, shown in the left panel of Fig. 5.

Unlike the simple model, the per-molecule entropy, $s$, of the real
troposphere increases slightly with increasing altitude, so the troposphere is
not fully adiabatic and is marginally stable to vertical convection. The model
stratosphere is isothermal, but the temperature of the real stratosphere shown
in Fig. 5 increases due to heating of the ozone layer by solar ultraviolet
radiation. The rising temperature increases the restoring force $df$ of (38)
for displaced air parcels. This increases the stability of the stratosphere to
vertical convection.

Figure 6: Upward spectral flux $\tilde{Z}$, for representative frequencies,
$\nu$: (a) $\tau_{o}=2.30$, $z_{e}=2.9$ km, $\tilde{B}_{o}=134.2$ (i.u.); (b)
$\tau_{o}=51687$, $z_{e}=85.3$ km, $\tilde{B}_{o}=131.8$ (i.u.); (c)
$\tau_{o}=0.016$, $z_{e}=0$ km, $\tilde{B}_{o}=86.4$ (i.u.); (d)
$\tau_{o}=7.11$, $z_{e}=34.0$ km, $\tilde{B}_{o}=79.3$ (i.u.). Here the
spectral intensity unit is 1 i.u. = 1 mW m-2 cm sr-1. The dashed red lines
mark the emission altitudes.

###  4.1 Clear Skies

One can accurately calculate thermal radiation transfer for skies without
clouds using the Schwarzschild equation

$\cos\theta\ {{\partial\tilde{I}}\over{\partial\tau}}=\tilde{B}-\tilde{I}.$
(69)

A detailed discussion of how to solve this equation is given in reference
[13], to which we refer readers who are interested in more details. We give
only a brief summary of the meaning of (69) here. The monochromatic intensity
of radiation with a spatial frequency (wave peaks per cm) $\nu$ and making an
angle $\theta$ to the vertical, is $\tilde{I}=\tilde{I}(\nu,\tau,\theta)$. The
vertical optical depth $\tau$ is the number of e-foldings by which vertically
propagating surface radiation of frequency $\nu$ is attenuated when it reaches
the altitude $z$. Note that in (69) the optical depth is measured from the
bottom of the atmosphere, but in (57) it was measured from the top. The Planck
intensity for isotropic radiation of frequency $\nu$ in thermal equilibrium at
an absolute temperature $T$ is

$\tilde{B}(\nu,T)={{2hc^{2}\nu^{3}}\over{e^{x}-1}}.$ (70)

Here $c$ is the speed of light, $h$ is Planck’s constant and the photon
energy, in units of $k_{B}T$ is $x=\nu ch/(k_{B}T)$.

Figure 7: Effects of changing concentrations of carbon dioxide, CO2 on the
thermal radiative flux to space from the top of a midlatitude standard
atmosphere at altitude $z_{\rm mp}=86$ km, with the temperature profile of
Fig. 5. The smooth blue line is the spectral flux from a surface at the
temperature $T_{0}=288.7$ K for a transparent atmosphere with no greenhouse
gases. The green line is the flux if all CO2 were removed but with all the
other greenhouse gases at their standard concentrations. The black line is for
all greenhouse gases at their standard concentrations. The red line is for
twice the standard concentration of CO2 but with all the other greenhouse
gases at their standard concentrations. Doubling the standard concentration of
CO2 from 400 to 800 ppm increases the forcing (the area between the black and
red lines) by 3.0 W m-2.

The net upwards flux, $\tilde{Z}$, the “upwelling” minus “downwelling,” is the
sum of the projections of the intensities $\tilde{I}$ onto the vertical
directions

$\tilde{Z}(\nu,z)=2\pi\int_{-1}^{1}\tilde{I}(\nu,\tau,\theta)\ \cos\theta\
d\cos\theta.$ (71)

Calculations of $\tilde{Z}$ at various frequencies as a function of altitude
are shown in Fig. 6. These were done for the temperature profile and
greenhouse altitudinal dependences shown in Fig. 5. The model included the 5
most important naturally occurring greenhouse gases: H2O, CO2, O3, N2O and
CH4.

For the gray atmosphere, there is a single emission altitude $z_{e}$, given by
(64), where most radiation is released to space. The complicated line
intensities of Fig. 2 were replaced by a cross section that is independent of
the thermal radiation frequencies, and independent of temperature and
pressure. All thermal radiation frequencies are attenuated equally for the
gray atmosphere.

For the real atmosphere, Fig. 2 implies an extremely complicated dependence of
the emission altitude on frequency, due to the hundreds of thousands of
vibrational-rotational lines of the greenhouse molecules. The emission
altitude varies dramatically across the thermal radiation spectrum. Emission
altitudes are indicated by dashed-horizontal red lines in Fig. 6. The green
lines show the heat flux carried by frequencies given at the tops of the
panels.

Fig. 6a shows that for a frequency $\nu=500.5$ cm-1, in the pure rotational
band of water vapor, the spectral flux $\tilde{Z}$, “breaks out” near the
emission altitude $z_{e}=2.9$ km, in the lower troposphere. The energy for the
flux is radiated by H2O molecules, which extract heat from convecting air
parcels. The atmosphere is cooler at an altitude of 2.9 km than at the
surface, so the spectral flux $\tilde{Z}$ that is radiated to space, the value
of the green line at the top of the atmosphere, is only about 80% of the
surface Planck value $\pi\tilde{B}_{o}$.

Fig. 6b shows that for a frequency $\nu=667.4$ cm-1, in the middle of the
strong Q branch of the CO2 bending-mode band, the emission altitude is near
the top of the atmosphere, $z_{e}=85.3$ km. Here the energy for the emission
is not from convecting air parcels. Most of the emitted energy is supplied by
the absorption of solar ultraviolet radiation. Heavily absorbed frequencies,
like $\nu=667.4$ cm-1, make a negligible contribution to radiative heat
transport in the troposphere.

Fig. 6c shows that for a frequency $\nu=971$ cm-1, in the middle of the
atmospheric window, greenhouse gases are so nearly transparent that the
surface radiates directly to space. This provides extremely efficient,
ballistic heat transfer from the surface to cold space. It is one of the
factors that leads to dew or frost on calm, dry, cloud-free nights. The
spectral flux $\tilde{Z}$ is equal to the spectral Planck flux
$\pi\tilde{B}_{o}$ at the surface.

Fig. 6d shows that for a frequency of $\nu=1016.2$ cm-1, in the middle of the
O3 absorption band, the emission altitude $z_{e}=34.0$ km is at the same
altitude where the O3 concentration is a maximum as is shown in Fig. 5. There
is little ozone in the troposphere, so the spectral flux $\tilde{Z}$ in the
troposphere is not much smaller than the surface Planck flux
$\pi\tilde{B}_{o}$. Most of the flux from the surface and troposphere is
absorbed by O3 in the lower stratosphere, to be replaced by emission from the
warmer, upper parts of the stratosphere, where the temperature maximizes
because of absorption of solar ultraviolet light by ozone. The absorbed
ultraviolet light provides the energy that is radiated back to space at the
frequency $\nu=1016.2$ cm-1.

The effect of the greenhouse gases on the spectral flux $\tilde{Z}$ emitted at
the top of the atmosphere, located at $z_{\rm mp}=86$ km, is shown in Fig. 7.
Absorption by pure rotational transitions of H2O at frequencies below 550
cm-1, absorption and emission by CO2 near its bending mode frequency of 667
cm-1, absorption and emission by O3 at 1016 cm-1 and absorption and emission
by bending modes of H2O at frequencies above 1200 cm-1 dominate the spectrum.
Integrating the monochromatic flux over all frequencies gives the total
infrared flux

$Z(z)=\int_{0}^{\infty}d\nu\ \tilde{Z}(\nu,z).$ (72)

Figure 8: Upward flux $Z$ or forcing $F$ calculated from the Schwarzschild
equation for a standard atmosphere having 400 ppm CO2, double that amount, 800
ppm, and half that amount, 200 ppm. The other greenhouse gas concentrations
were kept fixed at their standard values shown in Fig. 5. Values for $Z$ and
$F$ are given at the surface, tropopause altitude of 11 km and top of the
atmosphere altitude, $z_{\rm mp}=86$ km.

One can define the forcing as the difference between the infrared flux emitted
by the surface at temperature $T_{0}$ through a transparent atmosphere and the
infrared actual flux $Z$ which includes the effects emission and absorption by
greenhouse gases

$F(z)=\sigma{T_{0}}^{4}\ –\ Z(z).$ (73)

Figure 9: The “blue marble” a photograph taken by the crew of the Apollo 17
mission to the Moon on December 7, 1972, close to midsummer of the southern
hemisphere. The clouds of the intertropical convergence zone can be seen over
Africa, as well as heavier, low cloud cover further south and over Antarctica
[15].

The upward flux $Z$ and the forcing $F$ are plotted in Fig. 8 as a function of
altitude for three different CO2 concentrations while maintaining the other
greenhouse gases at their standard concentrations as given in Fig. 5. Fig. 8
illustrates the radiative flux, modeled with the Schwarzschild equation (69),
for the Earth’s real atmosphere. This should be compared to the flux for a
gray atmosphere shown in the left panel of Fig. 3. Unlike the gray atmosphere,
which has only convective heat flux in the troposphere, the real atmosphere
has both radiative flux $Z(z)$ at the altitude $z$, which is shown explicitly,
and convective flux, which is not shown explicitly, as it was in Fig. 3. The
convective flux is implicitly given by $Z(11{\rm\ km})–Z(z)$, since it
vanishes at the tropopause altitude $z=11$ km. To the extent that absorption
of solar radiation in the troposphere can be neglected, the sum of the steady-
state radiative and convective fluxes must be constant to conserve energy.

Unlike the gray atmosphere of Fig. 3, where there is negligible radiative flux
in the troposphere, Fig. 8 shows that there is a significant amount of
radiative flux in the troposphere, even at the surface, because of the
atmospheric window, which allows ballistic radiative heat transport to space.

Figure 10: Vertical intensites $I(0)$ at the top of the atmosphere observed
with a Michaelson interferometer in a satellite [17], and modeled with
radiation transfer theory for the Sahara desert, Mediterranean and Antarctica.
The intensity unit is 1 i.u. = 1 mW m-1 cm sr-1. Radiative forcing is negative
over wintertime Antarctica since the relatively warm greenhouse gases in the
troposphere mostly CO2, O3 and H2O, radiate more flux to space than the cold
ice surface at a temperature of $T=190$ K could radiate through a transparent
atmosphere.

For the gray atmosphere of Fig. 3, there is an abrupt increase of radiative
flux at the tropopause, where all of the heat convected from the surface is
released. As shown in Fig. 8, the varied emission altitudes of the real
atmosphere ensure that convected heat continues to be released at all
altitudes between the surface and the tropopause. For the real atmosphere, the
radiative flux increases roughly linearly, as shown in Fig. 8, as convective
heat is released at increasing altitudes $z$.

As shown in Fig. 7 and Fig. 8, the decrease in upwards flux at the top of the
atmosphere, $Z(z_{\rm mp})$, due to doubling the CO2 concentration, 3 W m-2,
is about a 1% decrease of $Z(z_{\rm mp})$. Doubling the other anthropogenic
greenhouse gases, N2O or CH4, results in even smaller changes to the upwards
flux. These flux changes are called the instantaneous forcings. If there were
no change in absorbed solar radiation the instantaneous forcing would cause
heat to build up in the atmosphere and Earth below. This is where the
reliability of climate modeling ends, since no one knows just how the
complicated climate system of Earth’s atmosphere and oceans will respond to
the small forcings. As shown in reference [6], if one assumes no change in
solar heating, and that convective heat transfer keeps the troposphere
approximately adiabatic, the surface temperature will warm by about 1 C from
doubling CO2, and the upper stratosphere will cool by about 8 C.

A surface warming of 1 C is too small a prediction for those who claim that
increasing concentrations of greenhouse gases have already led to a climate
emergency. Various strongly positive feedbacks have been proposed to predict
more surface warming. One of the favorite feedback mechanisms is lofting water
vapor, the most important greenhouse gas to higher, colder emission altitudes.
The decreased emission would have to be compensated for by warmer surface
temperatures at the bottom of a nearly adiabatic atmosphere, just as for the
model gray atmosphere of Fig. 3.

Like greenhouse gases, emission and absorption of thermal radiation by clouds
can modify the release of thermal radiation to space. High clouds with cold
tops are very poor thermal radiators. Increases or decreases in high clouds
could decrease or increase thermal radiation to space. The iris-effect of
Lindzen and Chou [14] is a possible negative feedback due to changes in high
cirrus clouds. The tops of low clouds are not much cooler than Earth’s
surface, so low clouds decrease thermal radiation to space much less than high
clouds.

Thick low clouds also reflect sunlight back to space before it is absorbed and
converted to atmospheric heat. Unlike greenhouse gases, which are nearly
transparent to sunlight, clouds can be very non-transparent due to scattering
and weak absorption of visible light. The beautiful images of Earth taken from
space, “the blue marble”, show the large amount of sunlight reflected from
clouds, land and oceans [15]. Any sunlight you can see has not heated the
Earth. Averaged over its surface, Earth reflects about 30% of sunlight back to
space; in other words, the “Bond albedo” of Earth is about 0.3.

Another positive feedback that has been proposed is a change in the albedo in
polar regions due to melting ice. However, a reduction in summer Arctic ice of
about 2 million km2 as has been observed during primarily from 1990 to 2007
[16] represents less than 0.4% of the Earth’s surface and therefore has
primarily a regional as opposed to a global impact.

An important check on any calculation is to compare observations to modeled
results. Fig. 10 shows intensities emitted by the top of the atmosphere as
measured by an interferometer in a satellite. The modeled results were
obtained by solving the Schwarzschild equation and using the appropriate
temperature profiles. The surface water vapor concentration varied
significantly from 31,000 ppm for the Sahara, 12,000 ppm for the Mediterranean
and only 2,000 ppm for Antarctica. The excellent quantitative agreement with
observations at very different latitudes give confidence that the calculations
of infrared flux are correct.

##  5 Summary

Greenhouse gases are responsible for the most striking feature of Earth’s
atmosphere, a lower troposphere and an upper stratosphere. In the troposphere,
below the tropopause boundary, a large fraction of the energy flux from the
solar heated surface is carried by convection, and not by thermal radiation.
Convection maintains average temperature lapse rates in the troposphere that
are close to adiabatic. In the stratosphere, most of the upward heat flux is
carried by radiation. Greenhouse gases warm the surface because they increase
the “thermal resistance” of the atmosphere to the vertical flow of energy from
the solar-heated surface to space. The larger the thermal resistance between
the surface and the emission altitude, the larger the temperature difference
needed to drive the solar energy absorbed by the surface back to space.
Without the thermal resistance induced by greenhouse gases, Earth’s surface
would be much colder and life as we know it would not be possible.

Increasing carbon dioxide will cause a small additional surface warming. It is
difficult to calculate exactly how much, but our best estimate is that it is
about 1 C for every doubling of CO2 concentration, when all feedbacks are
correctly accounted for. Alarming predictions of dangerous warming require
large positive feedbacks. The most commonly invoked feedback is an increase in
the concentration of water vapor in the upper troposphere. But most climate
models have predicted much more warming than has been observed, so there is no
observational support for strong positive feedbacks [18, 19]. Indeed, most
feedbacks in nature are negative as expressed by Le Chatelier’s Principle
[20]: When any system at equilibrium for a long period of time is subjected to
a change in concentration, temperature, volume or pressure, the system changes
to a new equilibrium, and this change partly counteracts the applied change.

We have barely touched atmospheric dynamics, perhaps the most interesting part
of the grand drama of weather and climate. We are all familiar with
manifestations of atmospheric dynamics: warm fronts, cold fronts, droughts,
floods, hurricanes, tornados etc. Equally fascinating ocean dynamics, like the
El Nino cycles of the tropical Pacific Ocean, also contributes to weather and
climate. Earth’s atmosphere works like an extremely complicated engine that
transforms heat from the Sun into the work that drives the winds, the weather
and ocean dynamics. Greenhouse gases are the heat exchanger which allows the
atmospheric heat engine to dump waste heat into cold space.

## Acknowledgements

The Canadian Natural Science and Engineering Research Council provided
financial support of one of us.

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