Download 1. Radiative Transfer 2. Spectrum of Radiation 3. Definitions

1. Radiative Transfer
Virtually all the exchanges of energy between the earth-atmosphere system and the
rest of the universe take place by radiative transfer. The earth and its atmosphere
are constantly absorbing solar radiation and emitting their own radiation to space.
Over a long period of time, the rates of absorption and emission are very nearly
equal, thus the earth-atmosphere system is very nearly in equilibrium with the sun.
Radiative transfer also serves as a mechanism for exchanging energy between
the atmosphere and the underlying surface, and among different layers of the atmosphere. Radiative transfer plays an important role in a number of chemical
reactions in the upper atmosphere and in the formation of photochemical smogs.
The transfer properties of visible radiation determine the visibility, the color of the
sky and the appearance of clouds. Radiation emitted by the earth and atmosphere
and intercepted by satellites is the basis for remote sensing of the atmospheric
temperature structure, water vapor amounts, ozone and other trace gases.
2. Spectrum of Radiation
Electromagnetic radiation may be viewed as an ensemble of waves propagating
at the speed of light (c∗ = 2.998 × 108 m/s through vacum). We characterize
radiation in terms of:
frequency ν = c∗ /λ
wavelength λ = c∗ /ν
Radiative transfer in planetary atmosphere involves an ensemble of waves with
a continuum of wavelengths and frequencies. We partition them into bands :
shortwave (λ < 4µm) carries most of the energy associated with solar radiation or longwave (λ > 4µm) which refers to the band that encompasses most of
the terrestrial. The visible region 0.39 − 0.79µm is defined by the range of wavelengths that the human eye is capable of sensing, and subranges of the visible are
discernible as colors.
3. Definitions
Solid Angle ω Consider a cone with its vertex at the origin of a concentric spherical surface. The solid angle is defined as the ratio of the area of the sphere
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Figure 1: Ahrens, Chapter 2
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intercepted by the cone to the square of the radius.
ω=
A
r2
(1)
dω =
dA
r2
(2)
In spherical coordinates
dA = r2 sinθdθdφ
(3)
The unit of the solid angle is the steradian. The area cut out of a sphere by
one steradian is equal to the square of the radius. Integration over the entire
spherical surface then gives ω = 4πsteradians
Flux density F Amount of radiant energy passing through a unit area per unit
time. Expressed in [W m−2 ]. This flux includes energy contributions from
all wavelengthsR between some
R ν2 specified limits λ1 and λ2 (the range cannot
λ2
be zero). F = λ1 Fλ dλ = ν1 Fν dν.
We define monochromatic flux density Fλ as
Fλ = lim∆λ→0
F (λ, λ + ∆λ)
∆λ
(4)
If you mark an area on the ground, the amount of daylight falling on it
can be measured in watts per square meter. This is the incident flux of
solar radiation - and it would decrease with an obstruction, like a cloud
or as the evening comes. The flux makes no distinction concerning where
the radiation is coming from, so in order to completely characterize the
radiance field at a given location, we must know not only the flux but also
the direction from which the radiation is comingthis is the radiant intensity
Intensity I Radiant energy per unit time coming from a specific direction and
passing through a unit area perpendicular to that direction. Total intensity
is
the integral over all wavelengths within given limits I =
R λcalculated
Ras
ν2
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I dλ = ν1 Iν dν. The intensity I tells you in detail both the strength and
λ1 λ
direction of various sources contributing to the incident flux on a surface. If
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you are looking at the sky, some regions have higher radiant intensity (when
you look at the sun for example)
The units are [W m−2 sr−1 ].The monochromatic intensity Iλ and flux density Fλ are related by
dF
(5)
Iλ =
dωcosθ
We can integrate over the solid angle subtended by a hemisphere to determine the monochromatic flux density coming from all directions
Z
Fλ =
2πsr
Z
Iλ cosθdω =
0
2π
Z
dφ
0
π/2
Iλ cosθsinθdφ
(6)
0
4. Blackbody Radiation
A blackbody is a surface that absorbs all incident radiation.
The Planck Function Determined experimentally, the intensity of radiation emitted by a blackbody is
c1 λ−5
Bλ =
(7)
π(ec2 /λT − 1)
where c1 = 3.74 × 10−16 W m2 and c2 = 1.45 × 10−2 mK. Theoretical justification of this empirical relationship led to the development of the theory
of quantum physics.
Wien’s Displacement Law Differentiating Equation 7 and setting the derivative
equal to zero, gives the wavelength of peak emission for a blackbody at
temperature T (HW6).
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(8)
λm =
T
where T in K and λm in µm. An important consequence of Wien displacement law is the fact that solar radiation is concentrated in the visible and
near-infrared parts of the spectrum, while radiation emitted by the planets
and their atmospheres is largely confined to the infrared. The nearly complete absence of overlap between the curves justifies dealing with solar and
planetary radiation separately in many problems of radiative transfer.
Stefan-Boltzmann Law The black body flux density obtained by integrating the
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Planck function Bλ over all wavelengths.
F = σT 4
(9)
Where σ is the Stefan-Boltzmann constant equal to 5.67 × 10−8 W m−2 K −4
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5. Radiative properties of Nonblack materials
Unlike blackbodies, which absorb all incident radiation, nonblack bodies such
as gaseous media can also reflect and transmit radiation. We will give a brief
description of the radiative processes in nonblack bodies. The fate of radiation
depends on wavelength.
1. Transmitted Radiation passes undisturbed. We define the monochromatic
fractional transmissivity as Tλ = IλI(transmitted)
λ (incident)
2. Reflected Radiation. Reflectivity is depicted by albedo. Albedo usually
represents all wavelengths and refers to the earth-atmosphere reflection. We
define the monochromatic fractional reflectivity as Rλ = IIλλ(reflected)
(incident)
3. Absorbed Increase in internal energy of the object. We define the monochromatic fractional absorptivity as αλ = IIλλ(absorbed)
(incident)
(a) Ionization-Dissociation Interactions. In these interactions, an electron
is stripped from an atom or molecule, or a molecule is torn apart.
These interactions occur primarily at ultraviolet and shorter wavelengths. All solar radiation shorter than about 0.1 µm in wavelength
is absorbed in the upper atmosphere by ionizing atmospheric gases,
particularly atomic oxygen. Between 0.1 and 0.2 µm molecular oxygen dissociates into atomic oxygen. Radiation between 0.2 and 0.3
µm is absorbed by dissociation of ozone. These bands are important
for preventing the radiation from reaching the ground and in satellite
meteorology for measuring ozone concentrations.
(b) Electronic Transitions Orbital electron jumps between quantized energy levels. These occur mostly in the UV and visible. Ozone, and
molecular oxygen.
(c) Vibrational transitions A molecule changes vibrational energy states.
These transitions occur mostly in the infrared portion of the spectrum
and are extremely important for satellite meteorology. The two chief
absorbers in the infrared region of the spectrum are carbon dioxide
and water vapor. Symmetric stretching has neither a static or dynamic
electric dipole moment because the symmetry of the molecule is maintained. If a molecule has no electric dipole moment, the electric field
of incident radiation cannot interact with the molecule. (This is why
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N2 and O2 , the two most abundant gases in the atmosphere, are transparent in the infrared.
(d) Rotational transitions a molecule changes rotational states. These occur in the far infrared and microwave portion of the spectrum. They
can occur at the same time as vibrational transitions. Figure 4.7.
The three are related by αλ + Rλ + Tλ = 1. For a black body αλ = 1.
4. We can also define the Monochromatic Emissivity λ as the ratio of the
monochromatic intensity of the radiation emitted by the body to the corresponding blackbody radiation: λ = IλB(emitted)
λ (T )
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5a. Kirchoff’s Law
It can be shown that the radiation emitted by a given material is a function of temperature and wavelength only. Consider an opaque, hollow enclosure with zero
transmissivity into which is placed a slab of finite thickness. In general, this slab
will reflect, absorb and transmit parts of the incident radiation. In addition, it
will emit radiation itself. We now allow the enclosure and the slab to reach thermodynamic equilibrium, such that the slab and the enclosure walls are the same
temperature. Under this condition, the flow of energy in all directions must be the
same. In thermodynamic equilibrium, the amount entering the slab must exactly
equal the amount leaving, or there would be a net flow of heat to or from the walls,
into or out of the slab. Since the slab and the walls are in thermodynamic equilibrium, this would constitute a violation of the Second Law of Thermodynamics.
Therefore, the balance equation is:
Iλ − Rλ Iλ = Tλ Iλ + Eλ
(10)
Where Eλ is the emitted radiance in the same direction as Iλ . But Tλ Iλ = Iλ (1 −
αλ − Rλ ) since αλ + Rλ + Tλ = 1. Therefore,
Iλ (1 − Rλ ) = Iλ (1 − αλ − Rλ ) + Eλ
(11)
Thus, Eλ − αλ Iλ = 0 or Eλ = αλ Iλ
Thus, inside of an opaque, hollow enclosure in thermodynamic equilibrium,
the amount emitted by the slab equals the amount absorbed by the slab. We now
imagine our enclosure to be replaced by a different one, constructed from a different material, and again allow it to come into thermodynamic equilibrium with the
same slab and at the same temperature as before. Consequently, the slab emission
will be the same as before, since it depends only on temperature and wavelength,
neither of which has been changed. Similarly, the slab absorption will not change
because the slab material is the same. Thus we have:
Eλ = αλ Iλ0
(12)
Where Iλ0 is the incident radiation on the slab in the new enclosure, thus it
follows that Iλ = Iλ0 .Thus, the radiation within an opaque, hollow enclosure is
independent of the material from which the walls are made. Re-writing the above
Eλ
= Iλb = f (T, λ) only and Iλb is the radiance inside an
equation we see αl ambda
opaque hollow enclosure at temperature T and wavelength λ.
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This result is known as Kirchhoff’s Law, which states that “The ratio of the
emission to the fractional absorptivity of a slab of any material in a state of thermodynamic equilibrium and at wavelength λ is equal to a constant”.
We may now define the fractional emissivity λ as the ratio of the radiation
emitted at the wavelength λ to that within a hollow enclosure at the same temperature or:
Eλ
(13)
Iλb
From this definition, we see that Eλ = λ Iλb . But from Kirchoff’s Law it then
follows:
λ =
λ = αλ
(14)
Or the fractional emissivity equals the fractional absorptivity
Kirchoff’s Law is fundamental to further development of the subject of radiative transfer, and is frequently applied in a variety of applications. Recalling that
it is strictly valid only under conditions of thermodynamic equilibrium, it is nevertheless generally assumed to be valid for atmospheric problems even though the
atmosphere is not strictly in thermodynamic equilibrium.
We may now carry this thought experiment one step further. Let’s replace
this slab by an ideal black body such that, by definition, it completely absorbs all
radiation falling on it. Inside the hollow enclosure then, the radiation leaving the
black body slab consists entirely of radiation emitted by the slab. The equilibrium
condition becomes
Iλb = Eλ
(15)
leading to the important conclusion that the radiation flowing in any direction
within the hollow enclosure in thermodynamic equilibrium is equal to the energy
emitted in the same direction as an ideal black body. Such radiation is called
black body radiation, and from our earlier arguments is isotropic or equal in all
directions.
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6. Examples
1. Prove that the intensity I of solar radiation is independent of distance from
the sun, provided that the distance is large and that radiation emitted from
each elemental area on the sun is independent of the zenith angle.
2. The average flux density Fe of solar radiation reaching the earth’s orbit is
1368W m−2 . Nearly all the radiation is emitted from the outermost visible
layer of the sun, which has a mean radius of 7×108 m. Calculate the equivalent blackbody temperature or effective temperature of this layer. The mean
distance between the earth and sun is 1.5 × 1011 m.
3. Calculate the equivalent blackbody temperature of the earth assuming a
planetary albedo αp = 0.3 where αp is the fraction of the total incident
solar radiation that is reflected and scattered back to space. Assume that the
earth is in radiative equilibrium.
4. A completely gray flat surface on the moon with an absorptivity of 0.9 is
exposed to direct overhead solar radiation. What is the radiative equilibrium
temperature of the surface? If the actual temperature is 300K, what is the
net flux density above the surface?
5. A flat surface is subject to overhead solar radiation as in the previous example. The absorptivity is 0.1 for solar radiation and 0.8 in the infrared
part of the spectrum, where most of the emission takes place. Compute the
radiative equilibrium temperature.
6. Calculate the radiative equilibrium temperature of the earth’s surface and
atmosphere assuming that the atmosphere can be regarded as a thin layer
with absorptivity of 0.1 for solar radiation and 0.8 for terrestrial radiation.
Assume that the earth’s surface radiates as a black body.
7. Radiative-Convective Equilibrium Temperature Profiles (from Global Physical Climatology by Hartman)
One can solve the radiative transfer equation for global mean terrestrial conditions. This involves construction of appropriate models for the transmission of
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the various band systems of importance in the atmosphere, insertion of these into
a computational model of the radiative transfer equation and iteration to obtain
a steady balance solution. The variables that determine the fluxes of radiant energy in the atmosphere include the atmospheric gaseous components, aerosols and
cloud characteristics, surface albedo and insolation. In a global mean model, the
temperature only depends on altitude. We need to specify:
H2 O Most important gas for radiative transfer. Water vapor has a rotation band
near 6.3µm and a rotation continuum at wavelengths longer than 12 µm. It
also absorbs solar in the troposphere.
CO2 Important because of increased concentrations. Mixing ration can be assumed constant with latitude and altitude up to 100km. Strong vibrationrotation band of CO2 at 15µm makes it important for long wave radiative
transfer. Also absorbs significant amount of solar.
O3 Ozone has fast sources and sinks in the stratosphere. In the surface it is related
to photochemical smog. Ozone has a vibration-rotation band near 9.6 µm
that is important for long wave energy, and has a dissociation continuum
that absorbs solar between 200 and 300 nm. Absorption heats the middle
atmosphere and causes the temperature increase with height that defines the
stratosphere and troposphere.
Aerosols Affect transmission of both solar and terrestrial radiation. Sulfate aerosols
in the troposphere are raditively important.
Surf ace Albedo Highly variable from location to location depending on type
and condition of surface material and vegetation. When the surface is snow
covered, its albedo is generally much higher.
Clouds Clouds vary considerably in amount and type one the globe. They have
try important effects on long wave and solar energy transfer in the atmosphere. The optical properties of the clouds must also be specified. Water
clouds have weak solar absorption, but they seatter solar very effectively.
Thick clouds can be assumed to be black bodies for long wave radiation. A
simple approach is to specify the properties of three types of clouds.
Looking at Figure 3.17 we see that with only water vapor present, a reasonable
approximation to the observed profile is obtained except that the stratosphere
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is absent. Carbon dioxide with a mixing ratio of 300 ppm raises the temperature by 10 K above the equilibrium obtained with only water vapor present. A
sharp tropopause and the increase of temperature with height that characterizes
the stratosphere appear only when solar absorption by ozone is included in the
model.
Figure 3.16 shows a calculated temperature profile that is in radiative equilibrium. Atmospheric temperatures in radiative equilibrium decrease rapidly with
altitude near the surface. In the troposphere, radiative equilibrium temperature
profiles are hydrostatically unstable in the sense that parcels of air that are elevated slightly will become buoyant and continue to rise. In the real atmosphere,
atmospheric motions move heat away from the surface and mix it through the
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troposphere. 60% by latent and sensible heat and 40% by net long wave radiation emission. The global mean temperature profile of the earth is not in radiative
equilibrium, but rather in radiative-convective equilibrium. To obtain a realistic
global-mean vertical energy balance, the vertical flux of energy by atmospheric
motions must be included.
The simplest artifice by which the effect of vertical energy transports by motions can be included in a global-mean radiative transfer model is a procedure
called convective adjustment. Under this constraint the lapse rate is not allowed
to exceed a critical value (6.5Kkm−1 . Where radiative processes would make
the lapse rate greater, a non radiative upward heat transfer is assumed to occur
that maintains the specified lapse rate while conserving energy. This ”adjusted”
layer extends from the surface to the tropopause. A temperature profile that is in
energy balance when radiative transfer and convective adjustment are taken into
account may be called a radiative-convective equilibrium or thermal equilibrium
profile.The thermal equilibrium profile obtained with a lapse rate of 6.5 Kkm−1
is close to the observed global mean temperature profile.
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