Online: http://dust.ess.uci.edu/facts Updated: Mar 5, 2013 ,


Radiative Transfer in the Earth System



by Charlie Zender
University of California, Irvine
Department of Earth System Science zender@uci.edu
University of California Voice: (949) 891-2429
Irvine, CA  92697-3100 Fax: (949) 824-3256


Copyright © 1998-2012, Charles S. Zender
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Notes for Students of ESS 223,
Earth System Physics:



This monograph on Radiative Transfer provides some core and some supplementary reading material for ESS 223. We will discuss much of the material in the first twenty pages, and the figures at the end. The Index beginning on page  is also helpful.


Notes for Students of ESS 236,
Radiative Transfer and Remote Sensing:



Yada yada yada.

Contents

1  Introduction
    1.1  Planetary Radiative Equilibrium
    1.2  Fundamentals
2  Radiative Transfer Equation
    2.1  Definitions
        2.1.1  Intensity
        2.1.2  Mean Intensity
        2.1.3  Irradiance
        2.1.4  Actinic Flux
        2.1.5  Actinic Flux Enhancement
        2.1.6  Energy Density
        2.1.7  Spectral vs. Broadband
        2.1.8  Thermodynamic Equilibria
        2.1.9  Planck Function
        2.1.10  Hemispheric Quantities
        2.1.11  Stefan-Boltzmann Law
        2.1.12  Luminosity
        2.1.13  Extinction and Emission
        2.1.14  Optical Depth
        2.1.15  Geometric Derivation of Optical Depth
        2.1.16  Stratified Atmosphere
    2.2  Integral Equations
        2.2.1  Formal Solutions
        2.2.2  Thermal Radiation In A Stratified Atmosphere
        2.2.3  Angular Integration
        2.2.4  Thermal Irradiance
        2.2.5  Grey Atmosphere
        2.2.6  Scattering
        2.2.7  Phase Function
        2.2.8  Legendre Basis Functions
        2.2.9  Henyey-Greenstein Approximation
        2.2.10  Direct and Diffuse Components
        2.2.11  Source Function
        2.2.12  Radiative Transfer Equation in Slab Geometry
        2.2.13  Azimuthal Mean Radiation Field
        2.2.14  Anisotropic Scattering
        2.2.15  Diffusivity Approximation
        2.2.16  Transmittance
    2.3  Reflection, Transmission, Absorption
        2.3.1  BRDF
        2.3.2  Lambertian Surfaces
        2.3.3  Albedo
        2.3.4  Flux Transmission
    2.4  Two-Stream Approximation
        2.4.1  Two-Stream Equations
        2.4.2  Layer Optical Properties
        2.4.3  Conservative Scattering Limit
    2.5  Solar Heating
    2.6  Chapter Exercises
3  Remote Sensing
    3.1  Rayleigh Limit
    3.2  Anomalous Diffraction Theory
    3.3  Geometric Optics Approximation
    3.4  Single Scattered Intensity
    3.5  Satellite Orbits
    3.6  Aerosol Characterization
        3.6.1  Measuring Aerosol Optical Depth
        3.6.2  Aerosol Indirect Effects on Climate
        3.6.3  Aerosol Effects on Snow and Ice Albedo
        3.6.4  Angström Exponent
4  Gaseous Absorption
    4.1  Line Shape
        4.1.1  Line Shape Factor
        4.1.2  Natural Line Shape
        4.1.3  Pressure Broadening
        4.1.4  Doppler Broadening
        4.1.5  Voigt Line Shape
5  Molecular Absorption
    5.1  Mechanical Analogues
        5.1.1  Vibrational Transitions
        5.1.2  Isotopic Lines
        5.1.3  Combination Bands
    5.2  Partition Functions
    5.3  Dipole Radiation
    5.4  Two Level Atom
    5.5  Line Strengths
        5.5.1  HITRAN
    5.6  Line-By-Line Models
        5.6.1  Literature
6  Band Models
    6.1  Generic
        6.1.1  Beam Transmittance
        6.1.2  Beam Absorptance
        6.1.3  Equivalent Width
        6.1.4  Mean Absorptance
    6.2  Line Distributions
        6.2.1  Line Strength Distributions
        6.2.2  Normalization
        6.2.3  Mean Line Intensity
        6.2.4  Mean Absorptance of Line Distribution
        6.2.5  Transmittance
        6.2.6  Multiplication Property
    6.3  Transmission in Inhomogeneous Atmospheres
        6.3.1  Constant mixing ratio
        6.3.2  H-C-G Approximation
    6.4  Temperature Dependence
    6.5  Transmission in Spherical Atmospheres
        6.5.1  Chapman Function
7  Radiative Effects of Aerosols and Clouds
    7.1  Single Scattering Properties
        7.1.1  Maxwell Equations
    7.2  Separation of Variables
        7.2.1  Azimuthal Solutions
        7.2.2  Polar Solutions
        7.2.3  Radial Solutions
        7.2.4  Plane Wave Expansion
        7.2.5  Boundary Conditions
        7.2.6  Mie Theory
        7.2.7  Resonances
        7.2.8  Optical Efficiencies
        7.2.9  Optical Cross Sections
        7.2.10  Optical Depths
        7.2.11  Single Scattering Albedo
        7.2.12  Asymmetry Parameter
        7.2.13  Mass Absorption Coefficient
    7.3  Effective Single Scattering Properties
        7.3.1  Effective Efficiencies
        7.3.2  Effective Cross Sections
        7.3.3  Effective Specific Extinction Coefficients
        7.3.4  Effective Optical Depths
        7.3.5  Effective Single Scattering Albedo
        7.3.6  Effective Asymmetry Parameter
    7.4  Mean Effective Single Scattering Properties
        7.4.1  Mean Effective Efficiencies
        7.4.2  Mean Effective Cross Sections
        7.4.3  Mean Effective Specific Extinction Coefficients
        7.4.4  Mean Effective Optical Depths
        7.4.5  Mean Effective Single Scattering Albedo
        7.4.6  Mean Effective Asymmetry Parameter
    7.5  Bulk Layer Single Scattering Properties
        7.5.1  Addition of Optical Properties
        7.5.2  Bulk Optical Depths
        7.5.3  Bulk Single Scattering Albedo
        7.5.4  Bulk Asymmetry Parameter
        7.5.5  Diagnostics
8  Global Radiative Forcing
9  Implementation in NCAR models
10  Appendix
    10.1  Vector Identities
    10.2  Legendre Polynomials
    10.3  Spherical Harmonics
    10.4  Bessel Functions
        10.4.1  Spherical Bessel Functions
        10.4.2  Recurrence Relations
        10.4.3  Power Series Representation
        10.4.4  Asymptotic Values
    10.5  Gaussian Quadrature
    10.6  Gauss-Lobatto Quadrature
    10.7  Exponential Integrals
Index

List of Figures

    1  Cross Section and Quantum Yield of Nitrogen Dioxide
    2  Vertical Distribution of Photodissociation Rates
    3  Climatological Mean Absorbed Solar Radiation
    4  Climatological Mean Emitted Longwave Radiation
    5  ENSO Temperature and OLR
    6  Seasonal Shortwave Cloud Forcing
    7  Zonal Mean Shortwave Cloud Forcing
    8  Seasonal Longwave Cloud Forcing
    9  Zonal Mean Longwave Cloud Forcing
    10  Climatological Mean Net Cloud Forcing
    11  ENSO Cloud Forcing

List of Tables

    FACTs
    2  Wave Parameter Conversion Table
    3  Actinic Flux Enhancement
    4  Surface Albedo
    5  Temperature Dependence of α p
    6  Pressure-Broadened Half Widths
    7  Mechanical Analogues of Important Gases
    HITRAN database
    9  Full-range Gaussian quadrature
    10  Half-range Gaussian quadrature

0  Introduction

This document describes mathematical and computational considerations pertaining to radiative transfer processes and radiative transfer models of the Earth system. Our approach is to present a detailed derivation of the tools of radiative transfer needed to predict the radiative quantities (irradiance, mean intensity, and heating rates) which drive climate. In so doing we begin with discussion of the intensity field which is the quantity most often measured by satellite remote sensing instruments. Our approach owes much to ] (particle scattering), ] (band models), and ] (nomenclature, discrete ordinate methods, general approach). The nomenclature follows these authors where possible. These sections will evolve and differentiate from their original sources as the manuscript takes on the flavor of the researchers who contribute to it.

0.0  Planetary Radiative Equilibrium

The important role that radiation plays in the climate system is perhaps best illustrated by a simple example showing that without atmospheric radiative feedbacks (especially, ironically, the greenhouse effect), our planet's mean temperature would be well below the freezing point of water. Earth is surrounded by the near vacuum of space so the only way to transport energy to or from the planet is via radiative processes. If \nrg is the thermal energy of the planet, and \flxabssw and \flxolr are the absorbed solar radiation and emitted longwave radiation, respectively, then
∂\nrg

∂\tm
=
\flxabssw − \flxolr
(1)
On timescales longer than about a year the Earth as a whole is thought to be in . That, is, the global annual mean planetary temperature is nearly constant because the absorbed solar energy is exactly compensated by thermal radiation lost to space over the course of a year. Thus
\flxabssw
=
\flxolr
(2)
The total amount of solar energy available for the Earth to absorb is the incoming solar flux (or ) at the top of Earth's atmosphere, \flxslrtoa (aka the ), times the intercepting area of Earth's disk which is \mpi \rdsrth2. Since Earth rotates, the total mean incident flux \mpi \rdsrth2\flxslrtoa is actually distributed over the entire surface area of the Earth. The surface area of a sphere is four times its cross-sectional area so the mean incident flux per unit surface area is \flxslrtoa / 4. The fraction of incident solar flux which is reflected back to space, and thus unable to heat the planet, is called the a or , \rfl. Satellite observations show that \rfl ≈ 0.3. Thus only (1 − \rfl) of the mean incident solar flux contributes to warming the planet and we have
\flxabssw
=
(1 − \rfl) \flxslrtoa / 4
(3)
Earth does not cool to space as a perfect blackbody (
string :autorefequation41a
) of a single temperature and emissivity. Nevertheless the spectrum of thermal radiation \flxolr which escapes to space and thus cools Earth does resemble blackbody emission with a characteristic temperature. The \tptffc of an object is the temperature of the blackbody which would produce the same irradiance. Inverting the (
string :autorefequation73
) yields
\tptffc
( \flxolr / \cststfblt )1/4
(4)
For a perfect blackbody, \tpt = \tptffc. For a planet, the difference between \tptffc and the mean surface temperature \tptsfc is due to the radiative effects of the overlying atmosphere. The insulating behavior of the atmosphere is more commonly known as the .
Substituting (
string :autorefequation3
) and (
string :autorefequation4
) into (
string :autorefequation2
)
(1 − \rfl) \flxslrtoa / 4
=
\cststfblt \tptffc4
(5)
\tptffc
=

(1 − \rfl) \flxslrtoa

4 \cststfblt



 
(6)
For Earth, \rfl ≈ 0.3 and \flxslrtoa ≈ 1367 . Using these values in (
string :autorefequation6
) yields \tptffc = 255 K. Observations show the mean surface temperature \tptsfc = 288 K.

0.0  Fundamentals

The fundamental quantity describing the electromagnetic spectrum is , \frq. Frequency measures the oscillatory speed of a system, counting the number of oscillations (waves) per unit time. Usually \frq is expressed in cycles-per-second, or Hertz. Units of Hertz may be abbreviated Hz, hz, cps, or, as we prefer, . Frequency is intrinsic to the oscillator and does not depend on the medium in which the waves are travelling. The carried by a photon is proportional to its frequency
\nrg
=
\cstplk \frq
(7)
where \cstplk is . Regrettably, almost no radiative transfer literature expresses quantities in frequency.
A related quantity, the \frqngl measures the rate of change of wave phase in radians per second. Wave phase proceeds through 2\mpi radians in a complete cycle. Thus the frequency and angular frequency are simply related
\frqngl
=
2\mpi \frq
(8)
Since radians are considered dimensionless, the units of \frqngl are . However, angular frequency is also rarely used in radiative transfer. Thus some authors use the symbol ω to denote the element of , as in \dfrω. The reader should be careful not to misconstrue the two meanings. In this text we use ω only infrequently.
Most radiative transfer literature use or . Wavelength, \wvl (m), measures the distance between two adjacent peaks or troughs in the wavefield. The universal relation between wavelength and frequency is
\wvl \frq
=
\cstspdlgt
(9)
where \cstspdlgt is the . Since \cstspdlgt depends on the medium, \wvl also depends on the medium.
The \wvn , is exactly the inverse of wavelength
\wvn
1

\wvl


 
= \frq

\cstspdlgt


 
(10)
Thus \wvn measures the number of oscillations per unit distance, i.e., the number of wavecrests per meter. Using (
string :autorefequation9
) in (
string :autorefequation10
) we find \wvn = \frq/\cstspdlgt so wavenumber \wvn is indeed proportional to frequency (and thus to energy). Historically spectroscopists have favored \wvn rather than \wvl or \frq. Because of this history, it is much more common in the literature to find \wvn expressed in CGS units of  than in SI units of . The CGS wavenumber is used analogously to frequency and to wavelength, i.e., to identify spectral regions. The energy of radiative transitions are commonly expressed in CGS wavenumber units. The relation between \wvn expressed in CGS wavenumber units () and energy in SI units (J) is obtained by using (
string :autorefequation10
) in (
string :autorefequation7
)
\nrg
=
100 \cstplk \cstspdlgt \frq
(11)
There is another, distinct quantity also called . This secondary usage of wavenumber in this text is the traditional measure of spatial wave propagation and is denoted by \wvnbr.
\wvnbr
2\mpi
~
\frq
 

 
(12)
The wavenumber \wvnbr is set in Roman typeface as an additional distinction between it and other symbols 1.
Table 
string :autoref 2
summarizes the relationships between the fundamental parameters which describe wave-like phenomena.
Table 2: Wave Parameter Conversion Table23
Variable
Units m s
- 2 2 1
- 1 2 2
100 1 - 200 1
2 - 2
2 - 2
1 1 2 2 -

0  Radiative Transfer Equation

0.0  Definitions

0.0.0  Intensity

The fundamental quantity defining the radiation field is the of radiation. Specific intensity, also known as , measures the flux of radiant energy transported in a given direction per unit cross sectional area orthogonal to the beam per unit time per unit solid angle per unit frequency (or wavelength, or wavenumber). The units of \ntnwvl are Joule meter−2 second−1 steradian−1 meter−1. In SI dimensional notation, the units condense to . The SI unit of power (1 Watt ≡ 1 Joule per second) is preferred, leading to units of . Often the specific intensity is expressed in terms of spectral frequency \ntnfrq with units  or spectral wavenumber (also \ntnwvn) with units .
Consider light travelling in the direction \nglhat through the point \psnvct. Construct an infinitesimal element of surface area \dfr\sfc intersecting \psnvct and orthogonal to \nglhat. The radiant energy \dfr\nrg crossing \dfr\sfc in time \dfr\tm in the solid angle \dfr\ngl in the frequency range [\frq,\frq+\dfr\frq] is related to \ntnfrq(\psnvct,\nglhat) by
\dfr\nrg = \ntnfrq(\psnvct,\nglhat,\tm,\frq)  \dfr\sfc  \dfr\tm  \dfr\ngl  \dfr\frq
(13)
It is not convenient to measure the radiant flux across surface orthogonal to \nglhat, as in (
string :autorefequation13
), when we consider properties of radiation fields with preferred directions. If instead, we measure the intensity orthogonal to an arbitrarily oriented surface element \dfr\xsx with surface normal \nrmhat, then we must alter (
string :autorefequation13
) to account for projection of \dfr\sfc onto \dfr\xsx. If the angle between \nrmhat and \nglhat is \plr then
cos\plr = \nrmhat ·\nglhat
(14)
and the projection of \dfr\sfc onto \dfr\xsx yields
\dfr\xsx = cos\plr  \dfr\sfc
(15)
so that
\dfr\nrg = \ntnfrq(\psnvct,\nglhat,\tm,\frq) cos\plr  \dfr\xsx  \dfr\tm  \dfr\ngl  \dfr\frq
(16)
The conceptual advantage that (
string :autorefequation16
) has over (
string :autorefequation13
) is that (
string :autorefequation16
) builds in the geometric factor required to convert to any preferred coordinate system defined by \dfr\xsx and its normal \nrmhat. In practice \dfr\xsx is often chosen to be the local horizon.
The radiation field is a seven-dimensional quantity, depending upon three coordinates in space, one in time, two in angle, and one in frequency. We shall usually indicate the dependence of spectral radiance and irradiance on frequency by using \frq as a subscript, as in \ntnfrq, in favor of the more explicit, but lengthier, notation \ntn(\frq). Three of the dimensions are superfluous to climate models and will be discarded: The time dependence of \ntnfrq is a function of the atmospheric state and solar zenith angle and will only be discussed further in those terms, so we shall drop the explicitly dependence on \tm. We reduce the number of spatial dimensions from three to one by assuming a which is horizontally homogeneous and in which physical quantities may vary only in the vertical dimension \zzz. Thus we replace \psnvct by \zzz. This approximation is also known as a atmosphere, and comes with at least two important caveats: The first is the neglect of horizontal photon transport which can be important in inhomogeneous cloud and surface environments. The second is the neglect of path length effects at large solar zenith angles which can dramatically affect the mean intensity of the radiation field, and thus the atmospheric photochemistry.
With these assumptions, the intensity is a function only of vertical position and of direction, \ntnfrq(\zzz,\nglhat). Often the τ (defined below), which increases monotonically with \zzz, is used for the vertical coordinate instead of \zzz. The angular direction of the radiation is specified in terms of the polar angle \plr and the azithumal angle \azi. The polar angle \plr is the angle between \nglhat and the normal surface \nrmhat that defines the coordinate system. The specific intensity of radiation traveling at polar angle \plr and azimuthal angle \azi at optical depth level τ in a plane parallel atmosphere is denoted by \ntnfrq(τ,\plr,\azi). Specific intensity is also referred to as intensity.
Further simplification of the intensity field is possible if it meets certain criteria. If \ntnfrq is not a function of position (τ), then the field is . If \ntnfrq is not a function of direction (\nglhat), then the field is .

0.0.0  Mean Intensity

The is an integrated measure of the radiation field at any point \psn. Mean intensity \ntnmnfrq is defined as the mean value of the intensity field integrated over all angles.
\ntnmnfrq =



 
\ntnfrq  \dfr\ngl




 
\dfr\ngl



 
(17)
The solid angle subtended by \ngl is the ratio of the area \AAA enclosed by \ngl on a spherical surface to the square of the radius of the sphere. Since the area of a sphere is 4\mpi\rds2, there must be 4\mpi steradians in a sphere. It is straightforward to demonstrate that the differential element of area in is \rds2 sin\plr  \dfr\plr  \dfr\azi. Thus the element of solid angle is
\ngl
=
\AAA / \rds2
\dfr\ngl
=
\rds−2  \dfr\AAA
=
\rds−2  \rds2 sin\plr  \dfr\plr  \dfr\azi
=
sin\plr  \dfr\plr  \dfr\azi
(18)
The of an instrument, e.g., a telescope, is most naturally measured by a solid angle.
The definition of \ntnmnfrq (
string :autorefequation17
) demands the radiation field be integrated over all angles, i.e., over all 4\mpi steradians. Evaluating the denominator demonstrates the properties of angular integrals. The denominator of (
string :autorefequation17
) is



 
\dfr\ngl
=



 



 
sin\plr  \dfr\plr  \dfr\azi
=
[ \azi ]02\mpi


 
sin\plr  \dfr\plr
=
2 \mpi


 
sin\plr  \dfr\plr
=
2 \mpi [ − cos\plr ]0\mpi
=
2 \mpi [−(−1) − (−1)]
=
4 \mpi
(19)
As expected, there are 4\mpi steradians in a sphere, and 2\mpi steradians in a hemisphere.
It is convenient to return briefly to the definition of . Isotropic radiation is, by definition, equal intensity in all directions so that the total emitted radiation is simply 4\mpi times the intensity of emission in any direction.
Applying (
string :autorefequation19
) to (
string :autorefequation17
) yields
\ntnmnfrq
=
1

4 \mpi


 



 
\ntnfrq  \dfr\ngl
(20)
\ntnmnfrq has units of radiance, . If the radiation field is azimuthally independent (i.e., \ntnfrq does not depend on \azi), then
\ntnmnfrq
=
1

2


 



 
\ntnfrq sin\plr  \dfr\plr
(21)
Let us simplify (
string :autorefequation21
) by introducing the change of variables
\plru
=
cos\plr
(22)
\dfr\plru
=
−sin\plr  \dfr\plr
(23)
This maps \plr ∈ [0, \mpi] into \plru ∈ [1, −1] so that (
string :autorefequation21
) becomes
\ntnmnfrq
=
1

2


 



 
\ntnfrq  \dfr\plru
\ntnmnfrq
=
1

2


 



 
\ntnfrq  \dfr\plru
(24)
The or are simply the up- and downwelling components of which the full intensity is composed (,) = (,,) = { r@   :   ll (,,) 0 < < /2
(,,) /2 < < .
(,) = (,,) = { r@   :   l (,,) 0 u < 1
(,,) -1 < u < 0 .
(,,) = (,+,) = (,0 < < /2,) = (,0 < < 1,)
(,,) = (,+,) = (,/2 < < ,) = (,-1 < < 0,)

0.0.0  Irradiance

The spectral irradiance \flxfrq measures the radiant energy flux transported through a given surface per unit area per unit time per unit wavelength. Although it is somewhat ambiguous, "flux" is used a synonym for irradiance, and has become deeply embedded in the literature ]. Consider a surface orthogonal to the \nglhatprm direction. All radiant energy travelling parallel to \nglhatprm crosses this surface and thus contributes to the irradiance with 100% efficiency. Energy travelling orthogonal to \nglhatprm (and thus parallel to the surface), however, never crosses the surface and does not contribute to the irradiance. In general, the intensity \ntnfrq(\nglhat) projects onto the surface with an efficiency cosΘ = \nglhat ·\nglhatprm, thus
\flxfrq
=



 
\ntnfrq cos\plr  \dfr\ngl
=



 



 
\ntnfrq cos\plr  sin\plr  \dfr\plr  \dfr\azi
(25)
In a plane-parallel medium, this defines the net specific irradiance passing through a given vertical level. Note the similarity between (
string :autorefequation20
) and (
string :autorefequation27
). The former contains the zeroth moment of the intensity with respect to the cosine of the polar angle, the latter contains the first moment. Also note that (
string :autorefequation27
) integrates the cosine-weighted radiance over all angles. If \ntnfrq is isotropic, i.e., \ntnfrq = \ntnfrqnot, then \flxfrq = 0 due to the symmetry of cos\plr.
Let us simplify (
string :autorefequation27
) by introducing the change of variables u = cos\plr, du = −sin\plr  \dfr\plr. This maps \plr ∈ [0, \mpi] into \plru ∈ [1, −1]:
\flxfrq
=



 



 
\ntnfrq \plru  (− \dfr\plru)  \dfr\azi
=



 



 
\ntnfrq \plru  \dfr\plru  \dfr\azi
(26)
The irradiance per unit frequency, \flxfrq, is simply related to the irradiance per unit wavelength, \flxwvl. The total irradiance over any given frequency range, [\frq,\frq+\dfr\frq], say, is \flxfrq  \dfr\frq. The irradiance over the same physical range when expressed in wavelength, [\wvl,\wvl−\dfr\wvl], say, is \flxwvl  \dfr\wvl. The negative sign is introduced since −\dfr\wvl increases in the same direction as +\dfr\frq. Equating the total irradiance over the same region of frequency/wavelength, we obtain
\flxfrq  \dfr\frq
=
− \flxwvl  \dfr\wvl
\flxfrq
=
− \flxwvl   \dfr\wvl

\dfr\frq


 
=
− \flxwvl   d

\dfr\frq


 

\cstspdlgt

\frq


 



 
=
− \flxwvl  
\cstspdlgt

\frq2


 



 
\flxfrq
=
\cstspdlgt

\frq2


 
\flxwvl = \wvl2

\cstspdlgt


 
\flxwvl
(27)
\flxwvl
=
\cstspdlgt

\wvl2


 
\flxfrq = \frq2

\cstspdlgt


 
\flxfrq
(28)
Thus \flxfrq and \flxwvl are always of the same sign.

0.0.0  Actinic Flux

A quantity of great importance in photochemistry is the total convergence of radiation at a point. This quantity, called the , \flxact, determines the availability of photons for photochemical reactions. By definition, the intensity passing through a point \pnt in the direction \nglhat within the solid angle \dfr\ngl is \ntnfrq \dfr\ngl. We have not multiplied by cos\plr since we are interested in the energy passing along \nglhat (i.e., \plr = 0). The energy from all directions passing through \pnt is thus
\flxactfrq
=



 
\ntnfrq  \dfr\ngl
=
4\mpi \ntnmnfrq
(29)
Thus the actinic flux is simply 4\mpi times the mean intensity \ntnmnfrq (
string :autorefequation20
). \flxactfrq has units of  which are identical to the units of irradiance \flxfrq (
string :autorefequation27
). Although the nomenclature "actinic flux" is somewhat appropriate, it is also somewhat ambiguous. The "flux" measured by \flxactfrq at a point \pnt is the energy convergence (per unit time, frequency, and area) through the surface of the sphere containing \pnt. This differs from the "flux" measured by \flxfrq, which is the net energy transport (per unit time, frequency, and area) through a defined horizontal surface. Thus it is safest to use the terms "actinic radiation field" for \flxactfrq and "irradiance" for \flxfrq. Unfortunately the literature is permeated with the ambiguous terms "actinic flux" and "flux", respectively.
The usefulness of actinic flux \flxactfrq becomes apparent only in conjunction with additional, species-dependent data describing the probability of photon absorption, or . Photo-absorption is the process of molecules absorbing photons. Each absorption removes energy (a photon) from the actinic radiation field. The amount of photo-absorption per unit volume is proportional to the number concentration of the absorbing species \cncA [], the actinic radiation field \flxactfrq, and the efficiency with with each molecule absorbs photons. This efficiency is called the , , or simply . The absorption cross-section is denoted by \xsxabs and has units of []. In the literature, however, values of \xsxabs usually appear in CGS units []. To make explicit the frequency-dependence of \xsxabs we write \xsxabsoffrq. If \xsxabs depends significantly on temperature, too (as is true for ozone), we must consider \xsxabs(\frq,\tpt).
The probability, per unit time, per unit frequency that a single molecule of species  will absorb a photon with frequency in [\frq,\frq+\dfr\frq] is proportional to \flxactfrq(\frq) \xsxabsoffrq4. Thus \xsxabsoffrq is the effective cross-sectional area of a molecule for absorption. The absorption cross-section is the ratio between the number of photons (or total energy) absorbed by a molecule to the number (or total energy) per unit area convergent on the molecule. Let \flxabsfrq [] be the energy absorbed per unit time, per unit frequency, per unit volume of air. Then
\flxabsfrq(\frq)
=
\cncA \flxactfrq(\frq) \xsxabsoffrq
(30)
where \cncA  is the number concentration of .
Photochemists are interested in the probability of absorbed radiation severing molecular bonds, and thus decomposing species  into constituent species  and . Notationally this process may be written in any of the equivalent forms + +
+ ^ > +
+ ^ < + Both forms indicate that the efficiency with which reaction (
string :autorefequation33
) proceeds is a function of photon energy \hnu. The second form makes explicit that the photodissociation reaction does not proceed unless \frq < \frqnot, where \frqnot is the . In any case, photon energy is conventionally written \cstplk\frq, rather than the less convenient \cstplk\cstspdlgt/\wvl.
The probability that a photon absorbed by  will result in the photodissociation of , and the completion of (
string :autorefequation33
), is called the or and is represented by \qntyld. The probability \qntyld is dimensionless5. In addition to its dependence on \frq, \qntyld depends on temperature \tpt for some important atmospheric reactions (such as ozone photolysis). We explicitly annotate the \tpt-dependence of \qntyld only for pertinent reactions. Measurement of \qntyldoffrq for all conditions and reactions of atmospheric interest is an ongoing and important laboratory task.
The specific for the photodissociation of a species  is the number of photodissociations of  occuring per unit time, per unit volume of air, per unit frequency, per molecule of . In accord with convention we denote the specific photolysis rate coefficient by \prcfrq. The units of \prcfrq are .
\prcfrq
=
\flxactfrq(\frq) \xsxabs(\frq) \qntyld(\frq)

\cstplk \frq


 
=
4 \mpi \ntnmnfrq(\frq) \xsxabs(\frq) \qntyld(\frq)

\cstplk \frq


 
(31)
The photon energy in the denominator converts the energy per unit area in \flxactfrq to units of photons per unit area. The factor of \xsxabs turns this photon flux into a photo-absorption rate per unit area. The final factor, \qntyld, converts the photo-absorption rate into a photodissociation rate coefficient. Note that each factor in the numerator of (
string :autorefequation34
) requires detailed spectral knowledge, either of the radiation field or of the photochemical behavior of the molecule in question. This complexity is a hallmark of atmospheric photochemistry.
The total \prc is obtained by integrating (
string :autorefequation34
) over all frequencies that may contribute to photodissociation
\prc
=



 
\prcfrq(\frq)  \dfr\frq
=



 
\flxactfrq(\frq) \xsxabs(\frq) \qntyld(\frq)

\cstplk \frq


 
 \dfr\frq
=
4\mpi

\cstplk


 



 
\ntnmnfrq(\frq) \xsxabs(\frq) \qntyld(\frq)

\frq


 
 \dfr\frq
(32)
As mentioned above, evaluation of (
string :autorefequation35
) requires essentially a complete knowledge of the radiative and photochemical properties of the environment and species of interest.
\prc is notoriously sensitive to uncertainty in the input quantities. Integration errors due to the discretization of (
string :autorefequation35
) are quite common. To compute \prc with high accuracy, regular grids must have resolution of  ∼ 1 nm in the ultraviolet, ]. Much of the difficulty is due to the steep but opposite gradients of \flxactfrq and \qntyld that occur in the ultraviolet. High frequency features in \xsxabs worsen this problem for some molecules.
The utility of \prc has motivated researchers to overcome these computational difficulties by brute force techniques and by clever parameterizations and numerical techniques . It is common to refer to photolysis rate coeffients as "J-rates", and to affix the name of the molecule to specify which individual reaction is pertinent. Another description for \prc is the first order rate coefficient in photochemical reactions. For example, \prcNOd is the first order rate coefficient for + ^ < 420 + If [] denotes the number concentration of   in a closed system where photolysis is the only sink of , then
\dfr[\NOd]

\dfr\tm


 
=
− \prcNOd [\NOd] + \SSS\NOd
(33)
where \SSS\NOd represents all sources of . The terms in (
string :autorefequation37
) all have dimensions of . The first term on the RHS is the of  in the system.
Figure 
string :autoref 1
shows the spectral distribution of actinic flux in a clear mid-latitude summer atmosphere, and the absorption cross-section and quantum yield of .
Figure Figure Figure
Figure 1: (a) Spectral distribution of actinic flux \flxact [] at TOA and at the surface for a (MLS) atmosphere with a unit optical depth of dust or sulfate in the lowest kilometer. (b) Absorption cross section of , \xsxabsNOd []. (c) Quantum yield of , \qntyldNOd (
string :autorefequation36
).
Figure 
string :autoref 2
shows the vertical distribution of \prcNOd [] for the conditions shown in Figure (
string :autoref 1
).
Figure Figure
Figure 2: Vertical distribution of \prcNOd [] (
string :autorefequation36
) for the conditions shown in Figure (
string :autoref 1
). (a) Absolute rates. (b) Rates normalized by clean sky rates.
In this section we have assumed the quantities \flxactfrq, \xsxabs, and \qntyld are somehow known and therefore available to use to compute \prc. Typically, \xsxabs and \qntyld are considered known quantities since they usually do not vary with time or space. Models may store their values in lookup tables or precompute their contributions to (
string :autorefequation35
). The essence of forward radiative problems is to determine \ntnwvn so that quantities such as \prcfrq and \flxabs may be determined. In inverse radiative transfer problems, which are encountered in much of remote sensing, both \prc and the species concentration are initially unknown and must be determined. We shall continue describing the methods of forward radiative transfer until we have tools at our disposal to solve for \prcfrq. At that point we shall re-visit the inverse problem.

0.0.0  Actinic Flux Enhancement

The actinic flux \flxactfrq (
string :autorefequation31
) is sensitive to the angular distribution of radiance \ntnfrq. Nearly all scattering processes diffuse the radiation field, i.e., convert collimated photons to more isotropic photons. Such diffusion causes . It is instructive to examine how the relationship between downwelling flux and actinic flux changes in the presence of scattering. Four limiting cases may be identified and are summarized in Table 
string :autoref 3
.
Table 3: Actinic Flux Enhancement by Scattering6
Description
Collimated, non-reflecting 0 0
Isotropic, non-reflecting 0 rcl = /
= 0 0 2
Collimated, reflecting 1 rcl =
= / 3 3
Isotropic, reflecting 1 / 4
The scenarios differ in the isotropicity of the downwelling radiance ( or ) and the  \rfllmboffrq (0 or 1) of the lower boundary, taken to be a . All scenarios are driven by the same downwelling irradiance, taken to be the direct solar beam. The scenarios are arranged in order of increasing actinic flux \flxactfrq, shown in the final column. \flxactfrq increases with both the number and the brightness of reflecting surfaces. The reflectivity of natural surfaces is no more than 90% nor less than 5%7, so that the \flxactfrq in Table 
string :autoref 3
represent bounds on realistic systems.
The Collimated-Non-Reflecting scenario assumes all light travels unidirectionally in a tightly collimated direct beam with irradiance \flxfrq = \flxslrfrq. In this case the radiation field is the delta-function in the direction of the solar beam \ntnfrqofngl = \flxslrfrq\dltfncofnglhatmnglhatnot. The closest approximation to this scenario in the natural environment is the pristine atmosphere high above the ocean in daylight. The actinic flux \flxactfrq = \flxslrfrq follows directly from (
string :autorefequation31
). We define the actinic flux enhancement \flxactfct of a medium as the ratio of the actinic flux to the actinic flux of a collimated beam with the same incident irradiance
\flxactfct
\flxactfrq/\flxdwnfrq
(34)
Table 
string :autoref 3
shows \flxactfct ranges from one for the direct solar beam to a maximum of four in a completely isotropic radiation field. Thus a collimated beam is the least efficient configuration of radiant energy for driving photochemistry. Scattering processes diffuse the radiation field, and, as a result, always enhance the photolytic efficiency of a given irradiance. Absorption in a medium (i.e., in the atmosphere or by the surface) always reduces \flxactfrq and may lead to \flxactfct < 1.
The Isotropic-Non-Reflecting scenario assumes isotropic downwelling radiance above a completely black surface. Under these conditions, the actinic flux is twice the incident irradiance because the photons are evenly distributed over the hemisphere rather than collimated. Moderately thick clouds (τ >~3) over a dark surface such as the ocean create a radiance field approximately like this. However, because clouds are efficient at diffusing downwelling irradiance, they are also efficient reflectors and this significantly reduces the incident irradiance \flxdwnfrq relative to the total extraterrestrial irradiance \flxslrfrq. Whether photochemistry is enhanced or diminished beneath real clouds depends on whether the actinic flux enhancement factor \flxactfrq = 2 compensates the reduced sub-cloud insolation due to photons up-scattering off the cloud and back to space.
The Collimated-Reflecting scenario is a very important limit in nature because the net effect on photochemistry can approach the theoretical photochemical enhancement of \flxactfct = 3. In this limit, collimated downwelling radiation and diffuse upwelling radiation combine to drive photochemistry from both hemispheres. A clear atmosphere above bright surfaces (clouds, desert, snow) approaches this limit. These conditions describe a large fraction of the atmosphere, which is 50-60% cloud-covered. Moreover, the incident flux is not attenuated by cloud transmission, so \flxdwnfrq ≈ \flxslrfrq. The relatively high frequency of occurance of this scenario, combined with the large photochemical enhancement of \flxactfct = 3, are unequalled by any other scenario.
We may also define the actinic flux efficiency \flxactfsh of a medium as the actinic flux relative to the actinic flux of an isotropic radiation field with the same incident irradiance
\flxactfsh
\flxactfrq

4 \flxdwnfrq


 
(35)
Clearly 0 ≤ \flxactfsh ≤ 1. Table 
string :autoref 3
shows \flxactfsh = 0.25 for a collimated beam, and \flxactfsh = 1 for isotropic radiation. Just as scattering of the solar beam is required to increase \flxactfsh above 0.25, absorption must be present to reduce \flxactfsh beneath 0.25.
The Isotropic-Reflecting scenario shows that an isotropic radiation field is most efficient for driving photochemistry. The maximum four-fold increase in efficiency relative to the collimated field arises from the radiation field interacting with the particle from all directions, rather than from one direction only. This is geometrically equivalent to multiplying the molecular cross section by a factor of four, the ratio between the surface and cross-sectional areas of a sphere. Note that photochemistry itself is driven by molecular absorption which reduces \flxactfrq from the values in Table 
string :autoref 3
.
In summary, we have learned that a reactant molecule with a spherically symmetric field of influence receives photochemical radiation much like Earth receives solar irradiance. In both cases the collimated beam intercepts one fourth of the total area of matter (molecule or Earth), while an equal flux of diffuse (or diurnal average) irradiance impinges on four times as much area. Since, in the geometric limit, absorption probability depends upon area, not direction, collimated beams have one fourth the photochemical potential as isotropic radiation.

0.0.0  Energy Density

Another quantity of interest is the density of radiant energy per unit volume of space. We call this quantity the \nrgdnsfrq. The energy density is the number of photons per unit volume in the frequency range [\frq,\frq + \dfr\frq] times the energy per photon, \cstplk \frq. \nrgdnsfrq is simply related to the \flxactfrq (
string :autorefequation31
) and thus to the mean intensity \ntnmnfrq.
\nrgdnsfrq
=



 
 \dfr\nrgdnsfrq
=
4\mpi

\cstspdlgt


 
\ntnmnfrq
(36)
The units of \nrgdnsfrq are .

0.0.0  Spectral vs. Broadband

Until now we have considered only spectrally dependent quantities such as the spectral radiance \ntnfrq, spectral irradiance \flxfrq, spectral actinic flux \flxactfrq, and spectral energy density \nrgdnsfrq. These quantities are called and are given a subscript of \frq, \wvl, or \wvn because they are expressed per unit frequency, wavelength, or wavenumber, respectively. Each spectral radiant quantity may be integrated over a frequency range to obtain the corresponding radiant quantity. Band-integrated radiant fields are often called or Depending on the size of the frequency range, radiant are obtained by integrating over all frequencies:
\ntn
=



 
\ntnfrq(\frq)  \dfr\frq
\flx
=



 
\flxfrq(\frq)  \dfr\frq
\flxact
=



 
\flxactfrq(\frq)  \dfr\frq
\flxabs
=



 
\flxabsfrq(\frq)  \dfr\frq
\prc
=



 
\prcfrq(\frq)  \dfr\frq
\nrgdns
=



 
\nrgdnsfrq(\frq)  \dfr\frq
\htr
=



 
\htrfrq(\frq)  \dfr\frq

0.0.0  Thermodynamic Equilibria

Temperature plays a fundamental role in radiative transfer because \tpt determines the population of excited atomic states, which in turn determines the potential for . Thermal emission occurs as matter at any temperature above absolute zero undergoes quantum state transitions from higher energy to lower energy states. The difference in energy between the higher and lower level states is transferred via the electromagnetic field by photons. Thus an important problem in radiative transfer is quantifying the contribution to the radiation field from all emissive matter in a physical system. For the atmosphere the system of interest includes, e.g., clouds, aerosols, and the surface.
To develop this understanding we must discuss various forms of energetic equilibria in which a physical system may reside. Earth (and the other terrestrial planets, Mercury, Venus, and Mars) are said to be in because, on an annual timescale the solar energy absorbed by the Earth system balances the thermal energy emitted to space by Earth. Radiation and matter inside a constant temperature enclosure are said to be in , or TE.
Radiation in thermodynamic equilibrium with matter plays a fundamental role in radiation transfer. Such radiation is most commonly known as . Kirchoff first deduced the properties of blackbody radiation.
Thermodynamic equilibrium (TE) is an idealized state, but, fortunately, the properties of radiation in TE can be shown to apply to a less restrictive equilibrium known as , or LTE.

0.0.0  Planck Function

The Planck function \plkfrq describes the intensity of blackbody radiation as a function of temperature and wavelength (,) = 2 ^3
(,) = 2 ^2 Blackbodies emit isotropically-this considerably simplifies thermal radiative transfer. The correct predictions (
string :autorefequation41a
) resolved one of the great mysteries in experimental physics in the late 19th century. In fact, this discovery marked the beginning of the science of quantum mechanics.
The relations (
string :autorefequation41a
) and (
string :autorefequation41b
) predict slightly different quantities. The former predicts the blackbody radiance per unit frequency, while the latter predicts the blackbody radiance per unit wavelength. Of course these quantities are related since the blackbody energy within any given spectral band must be the same regardless of which formula is used to describe it. Expressed mathematically, this constraint means
\plkfrq  \dfr\frq
=
− \plkwvl  \dfr\wvl
(37)
Once again, the negative sign arises as a result of the opposite senses of increasing frequency versus increasing wavelength. We may derive (
string :autorefequation41b
) from (
string :autorefequation41a
) by using (
string :autorefequation9
) in (
string :autorefequation42
)
\plkfrq
=
− \plkwvl   \dfr\wvl

\dfr\frq


 
=
\plkwvl \cstspdlgt

\frq2


 
=
\plkwvl \cstspdlgt
\wvl2

\cstspdlgt2


 



 
\plkfrq
=
\wvl2

\cstspdlgt


 
\plkwvl = \cstspdlgt

\frq2


 
\plkwvl
(38)
\plkwvl
=
\frq2

\cstspdlgt


 
\plkfrq = \cstspdlgt

\wvl2


 
\plkfrq
(39)
These relations are analogous to (
string :autorefequation30
).
The Planck function (
string :autorefequation41
) has interesting behavior in both the high and the low energy photon limits. In the high energy limit, known as , the photon energy greatly exceeds the ambient thermal energy
\cstplk \frqmshmax >> \cstblt \tpt
(40)
In Wien's limit, (
string :autorefequation41
) becomes (,) = 2 ^3 ^- /
(,) = 2 ^2 ^- /
In the very low energy limit, known as the , the photon energy is much less than the ambient thermal energy
\cstplk \frqmshmax
<<
\cstblt \tpt
(41)
Thus in the Rayleigh-Jeans limit the arguments to the exponential in (
string :autorefequation41
) are less than 1 so the exponentials may be expanded in Taylor series. Starting from (
string :autorefequation41a
)
\plkfrq(\tpt,\frq)
2 \cstplk \frq3

\cstspdlgt2


 

1 + \cstplk \frq

\cstblt \tpt


 
−1


 
=
2 \cstplk \frq3 \cstblt \tpt

\cstspdlgt2 \cstplk \frq


 
=
2 \frq2 \cstblt \tpt

\cstspdlgt2


 
(42)
Similar manipulation of (
string :autorefequation41b
) may be performed and we obtain (,) 2 ^2
(,) 2
The frequency of extreme emission is obtained by taking the partial derivative of (
string :autorefequation41a
) with respect to frequency with the temperature held constant
∂\plkfrq

∂\frq


 



 
=
2 \cstplk

\cstspdlgt2


 
× 1

( \me\cstplk \frq / \cstblt \tpt − 1)2


 
×
3 \frq2 ( \me\cstplk \frq / \cstblt \tpt− 1) −\frq3 \cstplk

\cstblt \tpt


 
\me\cstplk \frq / \cstblt \tpt


 
To solve for the , \frqmshmax, we set the RHS equal to zero so that one or more of the LHS factors must equal zero
2 \cstplk \frqmshmax2
3 ( \me\cstplk \frqmshmax / \cstblt \tpt − 1 ) − \cstplk \frqmshmax

\cstblt \tpt


 
\me\cstplk \frqmshmax / \cstblt \tpt


 

\cstspdlgt2 ( \me\cstplk \frqmshmax / \cstblt \tpt − 1)2



 
=
0
3 ( \me\cstplk \frqmshmax / \cstblt \tpt− 1 ) − \cstplk \frqmshmax

\cstblt \tpt


 
\me\cstplk \frqmshmax / \cstblt \tpt
=
0
\cstplk \frqmshmax

\cstblt \tpt


 
\me\cstplk \frqmshmax / \cstblt \tpt
=
3 ( \me\cstplk \frqmshmax / \cstblt \tpt− 1 )
\cstplk \frqmshmax

\cstblt \tpt


 
=
3 ( 1 − \me−\cstplk \frqmshmax / \cstblt \tpt)
(43)
An analytic solution to (
string :autorefequation50
) is impossible since \frqmshmax cannot be factored out of this transcendental equation. If instead we solve \xxx = 3 ( 1 − \me−\xxx) numerically we find that \xxx ≈ 2.8215 so that
\cstplk \frqmshmax

\cstblt \tpt


 
2.82
\frqmshmax
2.82 \cstblt \tpt / \cstplk
5.88 ×1010  \tpt       \hz
(44)
where the units of the numerical factor are . Thus the frequency of peak blackbody emission is directly proportional to temperature. This is known as .
A separate relation may be derived for the wavelength of maximum emission \wvlmshmax by an analogous procedure starting from (
string :autorefequation41b
). The result is
\wvlmshmax
2897.8 / \tpt       \um
(45)
where the units of the numerical factor are . Note that (
string :autorefequation51
) and (
string :autorefequation52
) do not yield the same answer because they measure different quantities. The wavelength of maximum emission per unit wavelength, for example, is displaced by a factor of approximately 1.76 from the wavelength of maximum emission per unit frequency.
It is possible to use (
string :autorefequation51
) to estimate the temperature of remotely sensed surfaces. For example, a satellite-borne tunable spectral radiometer may measure the emission of a newly discovered planet at all wavelengths of interest. Assuming the wavelength of peak measured emission is \wvlmshmax . Then a first approximation is that the planetary temperature is close to 2897.8 / \wvlmshmax.

0.0.0  Hemispheric Quantities

In climate studies we are most interested in the irradiance passing upwards or downwards through horizontal surfaces, e.g., the ground or certain layers in the atmosphere. These irradiances measure the radiant energy transport in the vertical direction. These hemispheric irradiances depend only on the corresponding . Let us assume the intensity field is azimuthally independent, i.e., \ntnfrq = \ntnfrq(\plr) only. Then the azimuthal contribution to (
string :autorefequation28
) is 2\mpi and
\flxfrq
=
2 \mpi


 
\ntnfrq \plru  \dfr\plru
=
2 \mpi



 
\ntnfrq \plru  \dfr\plru +


 
\ntnfrq \plru  \dfr\plru

 
(46)
We now introduce the change of variables \plrmu = |\plru| = |cos\plr|. Referring to (
string :autorefequation23
) we find - - = { r@   :   ll 0 < < /2
- /2 < < .
- u - = { r@   :   l u 0 u < 1
-u -1 < u < 0 . Most formal work on radiative transfer is written in terms of \plrmu rather than \plru or \plr. Substituting (
string :autorefequation54
) into (
string :autorefequation53
)
\flxfrq
=
2 \mpi



 
\ntnfrq (−\plrmu)  (−\dfr\plrmu) +


 
\ntnfrq \plrmu  \dfr\plrmu

 
=
2 \mpi



 
\ntnfrq \plrmu  \dfr\plrmu +


 
\ntnfrq \plrmu  \dfr\plrmu

 
=
2 \mpi



 
\ntnfrq \plrmu  \dfr\plrmu +


 
\ntnfrq \plrmu  \dfr\plrmu

 
=
−\flxdwnfrq + \flxupwfrq
=
\flxupwfrq − \flxdwnfrq
(47)
where we have defined the or = 2 _0^1 (+)  
= 2 _0^1 (-)   The hemispheric fluxes are positive-definite, and their difference is the net flux.


\flxfrq = \flxupwfrq − \flxdwnfrq
(48)


\flxfrq = \flxupwfrq − \flxdwnfrq
(49)
The superscripts + and denote (towards the upper hemisphere) and (towards the lower hemisphere) quantities, respectively. The net flux \flxfrq is the difference between the upwelling and downwelling hemispheric fluxes, \flxupwfrq and \flxdwnfrq, which are both positive-definite quantities.
The hemispheric flux transport in isotropic radiation fields is worth examining in detail since this condition is often met in practice. It will be seen that isotropy considerably simplifies many of the troublesome integrals encountered. When \ntnfrq has no directional dependence (i.e., it is a constant) then \flxupwfrq = \flxdwnfrq (
string :autorefequation58
). Thus the net radiative energy transport is zero in an isotropic radiation field, such as a cavity filled with blackbody radiation.
Let us compute the upward transport of radiation \plkfrqtpt (
string :autorefequation41a
) emitted by a perfect blackbody such as the ocean surface. Since \plkfrqtpt is isotropic, the intensity may be factored out of the definition of the upwelling flux (
string :autorefequation56a
) and we obtain
\flxplkfrq
=
2 \mpi


 
\plkfrq \plrmu  \dfr\plrmu
=
2 \mpi \plkfrq


 
\plrmu  \dfr\plrmu
=
2 \mpi \plkfrq \plrmu2

2


 



 
=
2 \mpi \plkfrq  ([1/2] − 0)
=
\mpi \plkfrq
(50)
Thus the upwelling blackbody irradiance tranports \mpi times the constant intensity of the radiation. Given that the upper hemisphere contains 2\mpi steradians, one might naively expect the upwelling irradiance to be 2\mpi \plkfrq. In fact the divergence of blackbody radiance above the emitting surface is 2\mpi \plkfrq. But the vertical flux of energy is obtained by cosine-weighting the radiance over the hemisphere and this weight introduces the factor of [1/2] difference between the naive and the correct solutions.

0.0.0  Stefan-Boltzmann Law

The frequency-integrated hemispheric irradiance emanating from a blackbody of great interest since it describes, e.g., the radiant power of most surfaces on Earth. Although we could integrate the \plkfrq (
string :autorefequation41a
) directly to obtain the broadband intensity \plkfnc ≡ ∫0 \plkfrq(\frq)  \dfr\frq, it is traditional to integrate \plkfrq first over the hemisphere (
string :autorefequation59
). By proceeding in this order, we shall obtain the total hemispheric blackbody irradiance \flxupwplk in terms fundamental physical constants and the temperature of the body.
\flxupwplk
=
\mpi


 
\plkfrq  \dfr\frq
=
\mpi


 
2 \cstplk \frq3

\cstspdlgt2 ( \me\cstplk \frq / \cstblt \tpt − 1)


 
 \dfr\frq
=
2 \mpi \cstplk

\cstspdlgt2


 



 
\frq3

\me\cstplk \frq / \cstblt \tpt − 1


 
 \dfr\frq
(51)
To simplify (
string :autorefequation60
) we make the change of variables
\xxx
=
\cstplk \frq

\cstblt \tpt


 
\frq
=
\cstblt \tpt \xxx

\cstplk


 
\dfr\frq
=
\cstblt \tpt

\cstplk


 
 \dfr\xxx
\dfr\xxx
=
\cstplk

\cstblt \tpt


 
 \dfr\frq
This change of variables maps \frq ∈ [0,∞) to \xxx ∈ [0,∞). Substituting this into (
string :autorefequation60
) we obtain
\flxupwplk
=
2 \mpi \cstplk

\cstspdlgt2


 



 

\cstblt \tpt

\cstplk


 



 
\xxx3

\me\xxx − 1


 
  \cstblt \tpt

\cstplk


 
 \dfr\xxx
=
2 \mpi \cstblt4 \tpt4

\cstspdlgt2 \cstplk3


 



 
\xxx3

\me\xxx − 1


 
 \dfr\xxx
(52)
The definite integral in (
string :autorefequation61
) is \mpi4/15. Proving this is a classic problem in mathematical physics which involves the (and thus prime number theory), the , and contour integration. The procedure used to obtain this result is interesting so we briefly summarize it here. The \rmnztafnc(\xxx) for real \xxx > 1 may be defined as
\rmnztafnc(\xxx)
1

\gmmfnc(\xxx)


 



 
\uuu\xxx−1

\me\uuu −1


 
 \dfr\uuu
(53)
Comparing (
string :autorefequation61
) with the Riemann zeta function definition (
string :autorefequation62
), we see that \xxx = 4, i.e,
\flxupwplk
=
2 \mpi \cstblt4 \tpt4

\cstspdlgt2 \cstplk3


 
\gmmfnc(4)\rmnztafnc(4)
(54)
The integral (
string :autorefequation62
) is analytically solvable for the special case of integers \xxx = \nnn. We may transform the integrand from a rational fraction into a using algebraic manipulation:
\uuu\xxx−1

\me\uuu −1


 
=
\uuu\xxx−1

\me\uuu −1


 
× \me−\uuu

\me−\uuu


 
=
\uuu\xxx−1\me−\uuu

1−\me−\uuu


 
=
\uuu\xxx−1\me−\uuu × 1

1−\me−\uuu


 
(55)
The integration limits in (
string :autorefequation62
) are [0,∞) so it is always true that \me−\uuu < 1 in (
string :autorefequation64
), and thus in the integrand of (
string :autorefequation62
).
Recall that the sum of an infinite with initial term \aaa0 and ratio \rrr is


 
\aaa0\rrr\kkk
=

lim
 
\aaa0 + \aaa0\rrr + \aaa0\rrr2 + …+ \aaa0\rrr\kkk−1 + \aaa0\rrr\kkk
=
\aaa0/(1−\rrr)     for  |\rrr| < 1
(56)
Hence the last term in (
string :autorefequation64
) is the sum of a power series (
string :autorefequation65
) with initial term \aaa0 = 1 and ratio \rrr = \me−\uuu.
1

1−\me−\uuu


 
=


 
\me−\kkk\uuu
(57)
Substituting (
string :autorefequation66
) into (
string :autorefequation64
) we obtain
\uuu\xxx−1

\me\uuu −1


 
=
\uuu\xxx−1\me−\uuu

 
\me−\kkk\uuu
=


 
\uuu\xxx−1 \me−(\kkk+1)\uuu
=


 
\uuu\xxx−1 \me−\kkk\uuu
(58)
where the last step shifts the initial index to from zero to one.
Using the inifinite series representation (
string :autorefequation67
) for the integrand of (
string :autorefequation62
) yields
\rmnztafnc(\xxx)
=
1

\gmmfnc(\xxx)


 



 



 
\uuu\xxx−1 \me−\kkk\uuu

 
 \dfr\uuu
Integration and addition are commutative operations. Interchanging their order yields
\rmnztafnc(\xxx)
=
1

\gmmfnc(\xxx)


 


 




 
\uuu\xxx−1 \me−\kkk\uuu  \dfr\uuu

 
(59)
We change variables from \uuu ∈ [0,+∞] to \yyy ∈ [0,+∞] with
\yyy
=
\kkk\uuu
\dfr\yyy
=
\kkk  \dfr\uuu
\uuu
=
\yyy/\kkk
\dfr\uuu
=
\kkk−1  \dfr\yyy
(60)
so that (
string :autorefequation68
) becomes
\rmnztafnc(\xxx)
=
1

\gmmfnc(\xxx)


 


 




 

\yyy

\kkk


 



 
\me−\yyy \kkk−1  \dfr\yyy


 
=
1

\gmmfnc(\xxx)


 


 
\kkk−(\xxx−1) \kkk−1



 
\yyy\xxx−1 \me−\yyy  \dfr\yyy

 
=
1

\gmmfnc(\xxx)


 


 
\kkk−\xxx [ \gmmfnc(\xxx) ]
=


 
\kkk−\xxx
(61)
where we replaced the integral in brackets with the it defines. Hence the Riemann zeta function of a positive integer \nnn is the sum of the reciprocals of the postive integers to the power \nnn.
Contour integration in the complex plane gives analytic closed-form solutions to Σ1 \kkk−\nnn (
string :autorefequation70
) for positive, even integers \nnn. Our immediate concern is \nnn = 4. Consider the complex function
\fnc(\zzz)
=
\mpicot\mpi\zzz

\zzz4


 
(62)
This function
I
s throughout the complex plane
H
as first order poles at all integer values on the real axis (except the origin)
H
as a fifth order pole at the origin
S
atisfies lim|\RRR| → ∞ \fnc(\zzz) = 0 where \zzz = \RRR\me\mi\plr
Therefore (
string :autorefequation71
) obeys the for suitably chosen contours. In other words, fxm. Having shown
\rmnztafnc(4)
=
\mpi4/90
(63)
Using \gmmfnc(4) = 3! = 3 ×2 ×1 = 6 and \rmnztafnc(4) = \mpi4/90 from (
string :autorefequation72
), Equation (
string :autorefequation63
) becomes
\flxupwplk
=
2 \mpi \cstblt4

\cstspdlgt2 \cstplk3


 
×6 × \mpi4

90


 
×\tpt4
=
2 \mpi5 \cstblt4

15 \cstspdlgt2 \cstplk3


 
\tpt4
This is known as the of radiation, and is usually written as
\flxupwplk
=
\cststfblt \tpt4        where
(64)
\cststfblt
2 \mpi5 \cstblt4

15 \cstspdlgt2 \cstplk3


 
(65)
\cststfblt is known as the and depends only on fundamental physical constants. The value of \cststfblt is 5.67032 ×10−8 . The thermal emission of matter depends very strongly (quartically) on \tpt (
string :autorefequation73
). This rather surprising result has profound implications for Earth's climate.
We derived \flxupwplk (
string :autorefequation73
) directly so that the Stefan-Boltzmann constant would fall naturally from the derivation. For completeness we now present the broadband blackbody intensity \plkfnc
\plkfnc
=



 
\plkfrq(\frq)  \dfr\frq
=
2 \mpi4 \cstblt4

15 \cstspdlgt2 \cstplk3


 
\tpt4
=
\flxupwplk / \mpi = \cststfblt \tpt4 / \mpi
(66)
The factor of \mpi difference between \plkfnc and \flxupwplk is at first confusing. One must remember that \plkfnc is the broadband (spectrally-integrated) intensity and that \flxupwplk is the broadband hemispheric irradiance (spectrally-and-angularly-integrated).

0.0.0  Luminosity

The total thermal emission of a body (e.g., star or planet) is called its , \lmn. The luminosity is thermal irradiance integrated over the surface-area of a volume containing the body. The luminosity of bodies with atmospheres is usually taken to be the total outgoing thermal emission at the top of the atmosphere. If the time-mean thermal radiation field of a body is spherically symmetric, then its luminosity is easily obtained from a time-mean measurement of the thermal irradiance normal to any unit area of the surrounding surface. This technique is used on Earth to determine the intrinsic luminosity of stars whose distance \dstslr is known (e.g., through parallax) including our own.
\lmnslr
=
4\mpi\dstslr2 \flxslrtoa
(67)
The meaning of the \flxslrtoa is made clear by (
string :autorefequation76
). \flxslrtoa is the solar irradiance that would be measured normal to the Earth-Sun axis at the top of Earth's atmosphere. The mean Earth-Sun distance is  ∼ 1.5 ×1011 m and \flxslrtoa ≈ 1367  so \lmnslr = 3.9 ×1026 W. Note that \lmn has dimensions of power, i.e., .
An independent means of estimating the luminosity of a celestial body is to integrate the surface thermal irradiance over the surface area of the body. For planets without atmospheres, \lmn is simply the integrated surface emission. Assuming the (
string :autorefequation4
) of our Sun is \tptffc, the radius \rdsslr at which this emission must originate is defined by combining (
string :autorefequation73
) with (
string :autorefequation76
)
4\mpi\rdsslr2 \cststfblt \tptslr4
=
4\mpi\dstslr2 \flxslrtoa
\rdsslr2
=
\dstslr2 \flxslrtoa

\cststfblt \tptslr4


 
\rdsslr
=
\dstslr

\tptslr2


 
  ⎛


\flxslrtoa

\cststfblt


 
 


 
(68)
For the Sun-Earth system, \tptslr ≈ 5800 K so \rdsslr = 6.9 ×108 m. Solar radiation received by Earth appears to originate from a portion of the solar atmosphere known as the .

0.0.0  Extinction and Emission

Radiation and matter have only two forms of interactions, and . Lambert8, first proposed that the extinction (i.e., reduction) of radiation traversing an infinitesimal path \dfr\pth is linearly proportional to the incident radiation and the amount of interacting matter along the path
\dfr\ntnfrq

\dfr\pth


 
= −\extcffoffrq \ntnfrq       Extinction only
(69)
Here \extcffoffrq is the , a measurable property of the medium, and \pth is the absorber path length. \extcffoffrq is proportional to the local density of the medium and is positive-definite.
The term and the exact definition of \extcffoffrq are somewhat ambiguous until their physical dimensions are specified. Path length, for example, can be measured in terms of column mass path \mpc [], number path of molecules \nbr [], and geometric distance \pth []. Each path measure has a commensurate extinction coefficient: \extcffmss [] (i.e., optical cross-section per unit mass), \extcffnbr [] (i.e., optical cross-section per molecule), \extcffvlm []=[] (i.e., optical cross-section per unit concentration). These coefficients are inter-related by
\extcffmss [\mSxkg]
=
\extcffvlm [\xm]

\dns [\kgxmC]


 
\extcffnbr [\mSxmlc]
=
\extcffmss [\mSxkg] ×\mmw [\kgxmol]

\cstAvagadro [\mlcxmol]


 
\extcffvlm [\xm]
=
\extcffnbr [\mSxmlc] ×\nbrcnc [\mlcxmC]
(70)
We will develop the formalism of radiative transfer in this chapter in terms using the geometric path and volume extinction coefficient (\pth, \extcffvlm) formalism. Our intent is to be concrete, rather than leaving the choice of units unstated. However, there is nothing fundamental about (\pth, \extcffvlm). In Section  we state our preference for working in mass-path units (\mpc, \extcffmss).
Extinction includes all processes which reduce the radiant intensity. As will be described below, these processes include absorption and scattering, both of which remove photons from the beam. Similarly the radiative emission is also proportional to the amount of matter along the path
\dfr\ntnfrq

\dfr\pth


 
= \extcffoffrq \srcfrq       Emission only
(71)
where \srcfrq is known as the . The source function plays an important role in radiative transfer theory. We show in §
string :autorefsubsubsection2.2.1
that if \srcfrq is known, then the full radiance field \ntnfrq is determined by an integration of \srcfrq with the appropriate boundary conditions. Emission includes all processes which increase the radiant intensity. As will be described below, these processes include thermal emission and scattering which adds photons to the beam. Determination of \extcffoffrq, which contains all the information about the electromagnetic properties of the media, is the subject of active theoretical, laboratory and field research.
Extinction and emission are linear processes, and thus additive. Since they are the only two processes which alter the intensity of radiation,
\dfr\ntnfrq

\dfr\pth


 
=
−\extcffvlm \ntnfrq + \extcffvlm \srcfrq
1

\extcffvlm


 
\dfr\ntnfrq

\dfr\pth


 
=
−\ntnfrq + \srcfrq
(72)
Equation (
string :autorefequation81
) is the in its simplest differential form.

0.0.0  Optical Depth

We define the \tautld between points \pnt1 and \pnt2 as
\tautld(\pnt1,\pnt2)
=



 
\extcffvlm  \dfr\pth
\dfr\tautld
=
\extcffvlm  \dfr\pth
(73)
The optical path measures the amount of extinction a beam of light experiences traveling between two points. When \tautld > 1, the path is said to be optically thick.
The most frequently used form of optical path is the . The optical depth τ is the vertical component of the optical path \tautld, i.e., τ measures extinction between vertical levels. For historical reasons, the optical depth in planetary atmospheres is defined τ = 0 at the top of the atmosphere and τ = \taustr at the surface. This convention reflects the astrophysical origin of much of radiative transfer theory. Much like pressure, τ is a positive-definite coordinate which increases monotonically from zero at the top of the atmosphere to its surface value. Consider the optical depth between two levels \zzz2 > \zzz1, and then allow \zzz2 → ∞
τ(\zzz1,\zzz2)
=



 
\extcffvlm  \dfr\zzzprm
τ(\zzz,∞)
=



 
\extcffvlm  \dfr\zzzprm
(74)
=



 
\extcffvlm  \dfr\zzzprm
(75)
Equation (
string :autorefequation84
) is the integral definition of optical depth. The differential definition of optical depth is obtained by differentiating (
string :autorefequation84
) with respect to the lower limit of integration and using the fundamental theorem of differential calculus
\dfrτ
=
[\extcffvlm(∞) − \extcffvlm(\zzz)]  \dfr\zzz
=
−\extcffvlm(\zzz)  \dfr\zzz
(76)
where the second step uses the convention that \extcffvlm(∞) = 0. By convention, τ is positive-definite, but (
string :autorefequation85
) shows that \dfrτ may be positive or negative. If this seems confusing, consider the analogy with atmospheric pressure: pressure increases monotonically from zero at the top of the atmosphere, and we often express physical concepts such as the temperature lapse rate in terms of negative pressure gradients.

0.0.0  Geometric Derivation of Optical Depth

The optical depth of a column containing spherical particles may be derived by appealing to intuitive geometric arguments. Consider a concentration of \cnc  identical spherical particles of radius \rds residing in a rectangular chamber measuring one meter in the \xxx and \yyy dimensions and of arbitrary height. If the chamber is uniformly illuminated by a beam of sunlight from one side, how much energy reaches the opposite side?
For this thought experiment, we will neglect the effects of scattering9. Moreover, we will assume that the particles are partially opaque so that the incident radiation which they do not absorb is transmitted without any change in direction. Finally, assume the particles are homogeneously distributed in the horizontal so that the radiative flux \flx(\zzz) is a function only of height in the chamber. Let us denote the power per unit area of the incident collimated beam of sunlight as \flxslrtoa. Our goal is to compute \flx(\zzz) as a function of \cnc and of \rds.
Since each particle absorbs incident energy in proportion to its geometric cross-section, the total absorption of radiation per particle is proportional to \mpi \rds2 \fshext, where \fshext = 1 for perfectly absorbing particles. If \fshext < 1, then each particle removes fewer photons than suggested by its geometric size10. Conversely, if \fshext > 1 each particle removes more photons than suggested by its geometric size. Each particle encountered removes \mpi \rds2 \fshext\flx(\hgt)  from the incident beam. The maximum flux which can be removed from the beam is, of course, \flxslrtoa.
In a section of height \hgtdlt, the collimated beam passing through the chamber will encounter a total of \cnc \hgtdlt particles. The number of particles encounted times the flux removed per particle gives the change in the radiative flux of the beam between the entrance and the rear wall
\flxdlt
=
− \mpi \rds2 \fshext \flx \cnc \hgtdlt
\flxdlt

\flx


 
=
− \mpi \rds2 \fshext \cnc \hgtdlt
(77)
If we take the limit as \hgtdlt → 0 then (
string :autorefequation86
) becomes
1

\flx


 
 \dfr\flx
=
− \mpi \rds2 \fshext \cnc  \dfr\hgt
\dfr(ln\flx)
=
− \mpi \rds2 \fshext \cnc  \dfr\hgt
(78)
Let us define the \kkk as
\kkk ≡ \mpi \rds2 \fshext \cnc
(79)
Finally we define the optical depth τ in terms of the extinction
τ
\kkk \hgtdlt
(80)
τ
=
\mpi \rds2 \fshext \cnc \hgtdlt
(81)
The dimensions of \kkk are inverse meters [], or, perhaps more intuitively, square meters of effective surface area per cubic meter of air []. Therefore τ is dimensionless. It can be helpful to remember that All quantities which compose τ are positive by convention, therefore τ itself is positive-definite.
We are now prepared to solve (
string :autorefequation87
) for \flx(τ). From the theory of first order differential equations, we know that \flx must be an exponential function whose solution decays from its initial value with an e-folding constant of τ
\flx(τ) = \flxslrtoa \me−τ
(82)
Thus, in the limit of geometrical optics, the optical depth measures the number of e-foldings undergone by the radiative flux of a collimated beam passing through a given medium. This result is one form of the .
The exact value of \fshext(\rds,\wvl) depends on the composition of the aerosol. However, there is a limiting value of \fshext as particles become large compared to the wavelength of light.

lim
 
\fshext = 2
(83)
Thus particles larger than 5-10  extinguish twice as much visible light as their geometric cross-section suggests.
To gain more insight into the usefulness of the optical depth, we can express τ (
string :autorefequation90
) in terms of the aerosol mass \mss, rather than number concentration \cnc. For a monodisperse aerosol of density \dns, the mass concentration is
\mss = 4

3


 
\mpi \rds3 \dns \cnc
(84)
If we substitute
\mpi \rds2 \cnc = 3 \mss

4 \rds \dns


 
(85) into (
string :autorefequation90
) we obtain
τ = 3 \mss \fshext \hgtdlt

4 \rds \dns


 
(86)
Typical cloud particles have \rds  ∼ 10  so that for visible solar radiation with \wvl  ∼ 0.5  we may employ (
string :autorefequation92
) to obtain
τ = 3 \mss \hgtdlt

2 \rds \dns


 
(87)
Thus τ increases linearly with \mss for a given \rds. Note however, that a given mass \mss produces an optical depth that is inversely proportional to the radius of the particles!

0.0.0  Stratified Atmosphere

We obtain the radiative transfer equation in terms of optical path by substituting (
string :autorefequation82
) into (
string :autorefequation81
)
\dfr\ntnfrq

\dfr\tautld


 
=
−\ntnfrq + \srcfrq
(88)
A is one in which all atmospheric properties, e.g., temperature, density, vary only in the vertical. As shown in the non-existant Figure, the photon path increment \dfr\pth at polar angle \plr in a stratified atmosphere is related to the vertical path increment \dfr\zzz by \dfr\zzz = cos\plr  \dfr\pth or \dfr\pth = \plru−1  \dfr\zzz. In other words, the optical path traversed by photons is proportional to the vertical path divided by the cosine of the trajectory. = ^-1  
=   Substituting (
string :autorefequation97a
) into (
string :autorefequation96
) we obtain
\plru \dfr\ntnfrq

\dfrτ


 
=
−\ntnfrq + \srcfrq
(89)
This is the differential form of the radiative transfer equation in a plane parallel atmosphere and is valid for all angles. The solution of (
string :autorefequation98
) is made difficult because \srcfrq depends on \ntnfrq. A more tractable set of equations may be obtained by considering the form of the boundary conditions. For many (most) problems of atmospheric interest, we know \ntnfrq over an entire hemisphere at each boundary of a "slab". Considering the entire atmosphere as a slab, for example, we know that, at the top of the atmosphere, sunlight is the only incident intensity from the hemisphere containing the sun. Or, at the surface, we have constraints on the upwelling intensity due to thermal emission or the surface reflectivity. When combined, these two hemispheric boundary conditions span a complete range of polar angle, and are thus sufficient to solve (
string :autorefequation98
). However, in practice it is difficult to apply half a boundary condition. Moreover, we are often interested in knowing the hemispheric flows of radiation because many instruments (e.g., pyranometers) are designed to measure hemispheric irradiance and many models (e.g., climate models) require hemispheric irradiance to compute surface exchange properties. For these reasons we will decouple (
string :autorefequation98
) into its constituent upwelling and downwelling radiation components.
Using the definitions of the (
string :autorefequation25
) in (
string :autorefequation98
) we obtain - = -        0 < < /2, > 0
- = -        /2 < < , < 0 where we have simply multiplied (
string :autorefequation98
) by −1 in order to place the negative sign on the LHS for reasons that will be explained shortly. The definitions of \srcdwnfrq and \srcupwfrq are exactly analogous to (
string :autorefequation25
).
We now change variables from \plru to \plrmu (
string :autorefequation54
). Replacing \plru by \plrmu in (
string :autorefequation99a
) is allowed since \plrmu = \plru > 0 in this hemisphere. In the upwelling hemisphere where \mpi/2 < \plr < \mpi, \plru < 0 so that \plrmu = −\plru (
string :autorefequation54
). This negative sign cancels the negative sign on the LHS of (
string :autorefequation99b
), resulting in - = -
= - These are the equations of radiative transfer in slab geometry for downwelling (0 < \plr < \mpi/2) and upwelling (\mpi/2 < \plr < \mpi) intensities, respectively. The only mathematical difference between (
string :autorefequation100a
) and (
string :autorefequation100b
) is the negative sign. A helpful mnemonic is that the negative sign on the LHS is associated with \ntndwnfrq while the implicit unary positive sign is associated with \ntnupwfrq. Of course the \ntnfrq and \srcfrq terms on the RHS are prefixed with opposited signs since they represent opposing, but positive definite, physical processes (absorption and emission).
Equation (
string :autorefequation99
) states that \ntnupwdwn depends explicitly only on \ntnupwdwn and on \srcupwdwn, but has no explicit dependence on \ntndwnupw or on \srcdwnupw. Thus it may appear that \ntnupw and \ntndwn are completely decoupled from eachother. However, we shall see that in problems involving scattering, \srcupwdwn depends explicitly on \ntndwnupw because scattering may change the trajectory of photons from upwelling to downwelling and visa versa. By coupling \ntnupw to \ntndwn, scattering allows the entire radiance field to affect the radiance field at every point and in every direction (modulo the speed of light, of course). Thus scattering changes the solutions to (
string :autorefequation100
) from being locally-dependent to depending on the global radiation field.
As a special case of (
string :autorefequation98
), consider a stratified, non-scattering atmosphere in thermodynamic equilibrium. Then the source function equals the Planck function \srcfrq = \plkfrq = \plkfrqtpt and we have
\plru \dfr\ntnfrq

\dfrτ


 
= −\ntnfrq + \plkfrq
(90)
Equation (
string :autorefequation101
) is the basis of radiative transfer in the thermal infrared, where scattering effects are often negligible. The solution to (
string :autorefequation101
) is described in §
string :autorefsubsubsection2.2.2
.

0.0  Integral Equations

0.0.0  Formal Solutions

It is useful to write down the formal solution to (
string :autorefequation98
) before making additional assumptions about the form of the source function \srcfrq.
\plru \dfr\ntnfrq

\dfrτ


 
=
− \ntnfrq + \srcfrq
\plru  \dfr\ntnfrq
=
− \ntnfrq  \dfrτ+ \srcfrq  \dfrτ
\plru  \dfr\ntnfrq + \ntnfrq  \dfrτ
=
\srcfrq  \dfrτ
\dfr\ntnfrq + \ntnfrq

\plru


 
 \dfrτ
=
\plrurcp \srcfrq  \dfrτ
(91)
Multiplying (
string :autorefequation102
) by the \exptauou
\exptauou  \dfr\ntnfrq + \exptauou \ntnfrq

\plru


 
 \dfrτ
=
\plrurcp \exptauou \srcfrq  \dfrτ
\dfr( \exptauou  \ntnfrq )
=
\plrurcp \exptauou \srcfrq \dfrτ
\dfr( \exptauou  \ntnfrq )

\dfrτ


 
=
\plrurcp\exptauou \srcfrq
(92)
The LHS side of (
string :autorefequation103
) is a complete differential. The boundary condition which applies to this first degree differential equation depends on the direction the radiation is traveling. Thus we denote the solutions to (
string :autorefequation103
) as \ntnfrqoftaupmu and \ntnfrqoftaummu for upwelling and downwelling radiances, respectively. We shall assume that the upwelling intensity at the surface, \ntnfrq(\taustr,+\plrmu), and the downwelling intensity at the top of the atmosphere, \ntnfrq(0,−\plrmu), are known quantities. Since \plrmu is positive-definite (
string :autorefequation54
), +\plrmu and −\plrmu uniquely specify the angles for which these boundary conditions apply.
The solution for upwelling radiance is obtained by replacing \plru in (
string :autorefequation103
) by −\plrmu because \plru < 0 for upwelling intensities.
\dfr( \expmtauomu  \ntnupwfrq )

\dfrτ


 
=
−\plrmurcp\expmtauomu \srcupwfrq
(93)
We could have arrived at (
string :autorefequation104
) by starting from (
string :autorefequation100b
), and proceeding as above except using \expmtauomu as the integrating factor. We now integrate from the surface to level τ (i.e., along a path of decreasing \tauprm) and apply the boundary condition at the surface
\expmtaupomu \ntnfrq(\tauprm,+\plrmu) |\tauprm = \taustr\tauprm = τ
=
− \plrmurcp


 
\expmtaupomu \srcupwfrq  \dfr\tauprm
\expmtauomu \ntnfrqoftaupmu −\expmtausomu \ntnfrq(\taustr,+\plrmu)
=
\plrmurcp


 
\expmtaupomu \srcupwfrq  \dfr\tauprm
\expmtauomu \ntnfrqoftaupmu
=
\expmtausomu \ntnfrq(\taustr,+\plrmu) + \plrmurcp


 
\expmtaupomu \srcupwfrq  \dfr\tauprm
\ntnfrqoftaupmu
=
\me(τ− \taustr)/\plrmu \ntnfrq(\taustr,+\plrmu) + \plrmurcp


 
\me(τ− \tauprm)/\plrmu \srcupwfrq  \dfr\tauprm
Note that \taustr > τ and \tauprm > τ so that both of the transmission factors reduce a beam's intensity between its source (at \taustr or \tauprm) and where it is measured (at τ). The physical meaning of the transmission factors is more clear if we write all transmission factors as negative exponentials.
\ntnfrqoftaupmu = \me−(\taustr − τ)/\plrmu \ntnfrq(\taustr,+\plrmu) + \plrmurcp


 
\me−(\tauprm − τ)/\plrmu \srcfrq(\tauprm,+\plrmu)  \dfr\tauprm
(94)
The first term on the RHS is the contribution of the boundary (e.g., Earth's surface) to the upwelling intensity at level τ. This contribution is attenuated by the optical path of the radiation between the ground and level τ. The second term on the RHS is the contribution of the atmosphere to the upwelling intensity at level τ. The net upward emission of each parcel of air between the surface and level τ is \srcupwfrq(\tauprm), but this internally emitted radiation is attenuated along the slant path between \tauprm and τ. The \plrmu−1 factor in front of the integral accounts for the slant path of the emitting mass in the atmosphere.
The solution for downwelling radiance is obtained by replacing \plru in (
string :autorefequation103
) by \plrmu because \plrmu = \plru in the downwelling hemisphere. The resulting expression must be integrated from the upper boundary down to level τ, and a boundary condition applied at the top.
\dfr( \exptauomu  \ntndwnfrq )

\dfrτ


 
=
\plrmurcp\exptauomu \srcdwnfrq
\exptaupomu \ntnfrq(\tauprm,−\plrmu) |\tauprm = 0\tauprm = τ
=
\plrmurcp


 
\exptaupomu \srcdwnfrq  \dfr\tauprm
\exptauomu \ntnfrqoftaummu − \ntnfrq(0,−\plrmu)
=
\plrmurcp


 
\exptaupomu \srcdwnfrq  \dfr\tauprm
\exptauomu \ntnfrqoftaummu
=
\ntnfrq(0,−\plrmu) +\plrmurcp


 
\exptaupomu \srcdwnfrq  \dfr\tauprm
\ntnfrqoftaummu
=
\expmtauomu \ntnfrq(0,−\plrmu) +\plrmurcp


 
\me(−τ+ \tauprm)/\plrmu \srcdwnfrq  \dfr\tauprm
\ntnfrqoftaummu
=
\expmtauomu \ntnfrq(0,−\plrmu) +\plrmurcp


 
\me−(τ− \tauprm)/\plrmu \srcfrq(\tauprm,−\plrmu)  \dfr\tauprm
(95)
The upwelling and downwelling intensities in a stratified atmosphere are fully described by (
string :autorefequation105
) and (
string :autorefequation106
). Such formal solutions to the equation of radiative transfer are of great heuristic value but limited practical use until the source function is known. Note that we have assumed a source function and boundary conditions which are azimuthally independent, but that the derivation of (
string :autorefequation105
) and (
string :autorefequation106
) does not rely on this assumption. It is straightforward to relax this assumption and replace \ntnfrqoftaumu by \ntnfrq(τ,\plrmu,\azi) and \srcfrqoftaumu by \srcfrq(τ,\plrmu,\azi) in the above.

0.0.0  Thermal Radiation In A Stratified Atmosphere

Consider a purely absorbing, stratified atmosphere in thermodynamic equilibrium. Then the source function equals the Planck function \srcfrq = \plkfrq = \plkfrqtpt (
string :autorefequation41
) and the radiative transfer equation is given by (
string :autorefequation101
). It is important to remember that \plkfrq is the complete source function only because we are explicitly neglecting all scattering processes. Thus we need only define the boundary conditions in order to use (
string :autorefequation105
) and (
string :autorefequation106
) to fully specify \ntnfrq. We assume that the surface emits blackbody radiation into the upper hemisphere
\ntnfrq(\taustr,+\plrmu) = \plkfrq[\tpt(\taustr)]
(96)
For brevity we shall define \plkfrqstr = \plkfrq[\tpt(\taustr)]. At the top of the atmosphere, we assume there is no downwelling thermal radiation.
\ntnfrq(0,−\plrmu) = 0
(97)
The solutions for upwelling and downwelling intensities are then obtained by using \srcfrq(\tauprm,\plrmu) = \plkfrq(\tauprm) (the Planck function is isotropic) and substituting (
string :autorefequation107
) and (
string :autorefequation108
) into (
string :autorefequation105
) and (
string :autorefequation106
), respectively
\ntnfrqoftaupmu
=
\me−(\taustr − τ)/\plrmu \plkfrq(\taustr) + \plrmurcp


 
\me−(\tauprm − τ)/\plrmu \plkfrq(\tauprm)  \dfr\tauprm
(98)
\ntnfrqoftaummu
=
\plrmurcp


 
\me−(τ− \tauprm)/\plrmu \plkfrq(\tauprm)  \dfr\tauprm
(99)
The first term on the RHS of (
string :autorefequation109
) is the thermal radiation emitted by the surface, attenuated by absorption in the atmosphere until it contributes to the upwelling intensity at level τ. The second term on the RHS contains the upwelling intensity arriving at τ contributed from the attenuated atmospheric thermal emission from each parcel between the surface and τ. The \plrmu−1 factor in front of the integral accounts for the slant path of the thermally emitting atmosphere. The RHS of (
string :autorefequation110
) is similar but contains no boundary contribution since the vacuum above the atmosphere is assumed to emit no thermal radiation. The upwelling and downwelling intensities in a stratified, thermal atmosphere are fully described by (
string :autorefequation109
) and (
string :autorefequation110
).

0.0.0  Angular Integration

Once the solutions for the hemispheric intensities are known, it is straightforward to obtain the hemispheric fluxes by performing the angular integrations (
string :autorefequation56a
)-(
string :autorefequation56b
).
\flxdwnfrqoftau = 2 \mpi


 
\ntnfrq(τ,−\plrmu) \plrmu  \dfr\plrmu
(100)
Consider first the downwelling flux in a non-scattering, thermal, isotropic, stratified atmosphere obtained by substituting (
string :autorefequation110
) into (
string :autorefequation111
) and interchanging the order of integration
\flxdwnfrqoftau
=
2 \mpi


 

\plrmurcp


 
\me−(τ− \tauprm)/\plrmu \plkfrq(\tauprm)  \dfr\tauprm

 
\plrmu  \dfr\plrmu
=
2 \mpi


 



 
\me−(τ− \tauprm)/\plrmu \plkfrq(\tauprm)  \dfr\plrmu  \dfr\tauprm
=
2 \mpi


 
\plkfrq(\tauprm)



 
\me−(τ− \tauprm)/\plrmu  \dfr\plrmu  

 
 \dfr\tauprm
(101)
Notice that two factors of \plrmu cancelled each other out: The reduction in irradiance due to non-normal incidence (\plrmu) exactly compensates the increased irradiance due to emission by the entire slant column which is \plrmurcp times greater than emission from a vertical column.
In terms of defined in Appendix , the inner integral in parentheses in (
string :autorefequation112
) is \xpn2(τ− \tauprm) ().
\flxdwnfrqoftau = 2 \mpi


 
\plkfrq(\tauprm) \xpn2(τ− \tauprm)  \dfr\tauprm
(102)
Similar terms arise when we consider the horizontal upwelling flux obtained by substituting (
string :autorefequation109
) into (
string :autorefequation56a
) and we obtain
\flxupwfrqoftau
=
2 \mpi


 
\ntnfrq(τ,+\plrmu) \plrmu  \dfr\plrmu
=
2 \mpi


 

\me−(\taustr − τ)/\plrmu \plkfrq(\taustr) + \plrmurcp


 
\me−(\tauprm − τ)/\plrmu \plkfrq(\tauprm)  \dfr\tauprm

 
\plrmu  \dfr\plrmu
=
2 \mpi \plkfrq(\taustr)


 
\me−(\taustr − τ)/\plrmu \plrmu  \dfr\plrmu+ 2 \mpi


 
\plkfrq(\tauprm)



 
\me−(\tauprm − τ)/\plrmu  \dfr\plrmu

 
 \dfr\tauprm
=
2 \mpi \plkfrq(\taustr) \xpn3(\taustr − τ) + 2 \mpi


 
\plkfrq(\tauprm) \xpn2(\tauprm−τ)  \dfr\tauprm
(103)
Subtracting (
string :autorefequation113
) from (
string :autorefequation114
) we obtain the net flux at any layer in a non-scattering, thermal, stratified atmosphere
\flxfrqoftau
=
\flxupwfrqoftau − \flxdwnfrqoftau
=
2 \mpi
\plkfrq(\taustr) \xpn3(\taustr − τ) +


 
\plkfrq(\tauprm) \xpn2(\tauprm − τ)  \dfr\tauprm −


 
\plkfrq(\tauprm) \xpn2(τ− \tauprm)  \dfr\tauprm

 
(104)
Equations (
string :autorefequation113
) and (
string :autorefequation114
) may not seem useful at this point but their utility becomes apparent in §
string :autorefsubsubsection2.3.4
where we define the .

0.0.0  Thermal Irradiance

Assume a non-scattering planetary surface at temperature \tpt emits blackbody radiation such that \ntnfrq(\taustr,+\plrmu) = \plkfrq(\tpt) (
string :autorefequation107
). What is the total upwelling thermal irradiance from the surface? From (
string :autorefequation56a
) we have
\flxupwfrq
=
2 \mpi


 
\plkfrqtpt \plrmu  \dfr\plrmu
=
2 \mpi \plkfrqtpt


 
\plrmu  \dfr\plrmu
=
2 \mpi \plkfrqtpt
\plrmu2

2


 



 
=
2 \mpi \plkfrqtpt
1

2


 
− 0


 
\flxupwfrq
=
\mpi \plkfrqtpt
1

\mpi


 
\flxupwfrq
=
\plkfrqtpt
(105)
Notice the isotropy of the Planck function allows the factor of 2 from the azimuthal integration to cancel the mean value of the cosine weighting function over a hemisphere. Moving the remaining factor of \mpi from the azimuthal integration to the LHS conveniently sets the RHS equal to the Planck function.
We integrate (
string :autorefequation116
) over frequency to obtain the total upwelling thermal irradiance
1

\mpi


 



 
\flxupwfrq  \dfr\frq
=



 
\plkfrqtpt  \dfr\frq
1

\mpi


 
\flxupw
=



 
\plkfrqtpt  \dfr\frq
1

\mpi


 
\flxupw
=
\cststfblt \tpt4

\mpi


 
\flxupw
=
\cststfblt \tpt4
(106)
Thus the factor of \mpi from the azimuthal integration nicely cancels the factor of \mpi from the . Equation (
string :autorefequation117
) applies to any surface whose emissivity is 1. Consider, e.g., a thick cloud with cloud base and cloud top temperatures \tpt(\zzz\btmsbs) = \tpt\btmsbs and \tpt(\zzz\topsbs) = \tpt\topsbs, respectively. Then the upwelling thermal flux at cloud top and the downwelling flux at cloud bottom will be \flxdwn(\zzz\btmsbs) = \cststfblt \tpt\btmsbs4 and \flxupw(\zzz\topsbs) = \cststfblt \tpt\topsbs4, respectively.

0.0.0  Grey Atmosphere

Consider an atmosphere transparent to solar radiation and partially opaque to thermal radiation governed by (
string :autorefequation101
). The exact solution, including angular dependence, is given in §
string :autorefsubsubsection2.2.2
. The hemispheric fluxes and net flux may only be obtained exactly by accounting for the angular dependence as in §
string :autorefsubsubsection2.2.3
. We may eliminate the angular dependence of the net flux by making the simplifying assumption that the hemispheric up and downwelling irradiances equal a constant times the corresponding intensity. A number of methods exist to determine this constant, called the (e.g., §
string :autorefsubsubsection2.2.15
). These methods are all related to the .
One such method is to identify an effective inclination \plrmubar along which all radiation is assumed to travel. With this assumption, the contribution to upwelling irradiance from the lower boundary, the first term on the RHS of
string :autorefequation115
), is
2 \xpn3(\taustr − τ)
=
exp
\taustr − τ

\plrmubar


 



 
(107)
Inspection (or differentiation) shows that the atmosphere within one optical depth makes the dominant contribution to (
string :autorefequation118
). For \dltτ = \taustr − τ = 1, the diffusivity factor
\plrmubar−1
5

3


 
(108)
The hypothetical collimated beam of radiation is inclined to the zenith by arccos(3/5) ≈ 53.13\dgr. This is equivalent to a collimated beam of radiation travelling vertically through an optical depth equal to five thirds the vertical optical depth traversed by the diffuse radiation.
Using this assumption (
string :autorefequation118
), the upwelling hemispheric irradiance (
string :autorefequation56a
) for blackbody radiation is
\flxupwfrq
=
2 \mpi


 
\ntnfrq (+\plrmu) \plrmu  \dfr\plrmu
2 \mpi


 
\ntnfrq (+\plrmubar) \plrmu  \dfr\plrmu
=
2 \mpi \plrmubar \ntnfrq(+\plrmubar)


 
\plrmu  \dfr\plrmu
=
2 \mpi \ntnfrq (+\plrmubar)
\plrmu2

2


 



 
=
2 \mpi \ntnfrq (+\plrmubar)
1

2


 
− 0


 
=
\mpi \ntnfrq (+\plrmubar)
(109)
An analogous relationship holds for the downwelling irradiance.
Based on (
string :autorefequation119
) and (
string :autorefequation120
), the irradiance structure of the thermal atmosphere may approximated by performing a direct angular integration of (
string :autorefequation100
). With our approximation, radiances \ntn integrate directly to irradiances \flx modulo the diffusivity factor \plrmubar. (
string :autorefequation101
) - = -
= -
It is instructive to examine an idealized , where the fluxes of interest have no spectral dependence. Although this is far from true in Earth's atmosphere, the solution is straightforward and sheds light on the and the . We simplify (
string :autorefequation121
) in two ways. First, we introduce a scaled optical depth \dfr\tautld = \plrmubar−1\dfrτ = [5/3]\dfrτ. Second, we drop the frequency dependence, which is equivalent to integrating over a broad range of frequencies. For heuristic purposes, think of this integration as being over the relatively narrow λ = 5-20  range where most of Earth's terrestrial radiative energy resides. - = -
= -
In the absence of dynamical, chemical, and latent heating, the energy deposition in a parcel of air is entirely radiative. Under these conditions the idealize grey atmosphere described by (
string :autorefequation122
) will adjust to a temperature profile determined by . Let the time rate of change of temperature \tpt of a parcel be denoted by \htr [], the parcel 11 or . The warming rate is the rate of net energy deposition divided by the \heatcpcspcprs [] times the density \dnsatm []
\htr ≡ \dfr\tpt

\dfr\tm


 
=
1

\dnsatm \heatcpcspcprs


 
\dfr\flx

\dfr\hgt


 
(110)
K

s


 
=

kg

\mC


 
× J

kg K


 



 
× J

\mSs


 
× 1

m


 
The forcing term on the RHS, \dfr\flx/\dfr\hgt is the , the vertical gradient of net radiative flux. Absorption and emission are the only mechanisms which contribute to the flux divergence. In terms of hemispheric fluxes,
\dfr\flx

\dfr\hgt


 
=
\dfr

\dfr\hgt


 
(\flxdwn − \flxupw)
(111)
By definition, the time variation of net radiative heating vanishes (\dfr\tpt/\dfr\tm = 0) at all levels of an atmosphere in radiative equilibrium. By (
string :autorefequation124
), we see that the vertical gradient in net radiative flux must also vanish in radiative equilibrium. Setting (
string :autorefequation124
) to zero and integrating we obtain
\flxdwn − \flxupw
=
\flxnot
(112)
The net radiative flux \flx(\hgt) = \flxnot is constant in radiative equilibrium.
Adding and subtracting (
string :autorefequation122
), we obtain ( - ) = + - 2
( + ) = - Defining ψ = \flxupw − \flxdwn and ϕ = \flx = \flxupw − \flxdwn, = - 2
= Since ϕ = \flxupw − \flxdwn is constant, (
string :autorefequation127b
) shows that \dfrϕ/\dfr\tautld = 0 and thus
ψ = 2\mpi\plkfnc
(113)
Substituting this into (
string :autorefequation127b
),
\dfr

\dfr\tautld


 
(2\mpi\plkfnc)
=
\flxnot
\dfr\plkfnc

\dfr\tautld


 
=
\dpysty \flxnot

2\mpi


 
\plkfnc(\tautld)
=
\dpysty \flxnot\tautld

2\mpi


 
+ \cstone
(114)
We evaluate the constant of integration by using the boundary condition at the top of the atmosphere. By definition \flxdwn = 0 and \tautld = 0 at . This implies that ϕ = ψ(0) = \flxupw(0) = \flxnot. We must therefore have \plkfnc(0) = ϕ/2\mpi by (
string :autorefequation128
). Using this result in (
string :autorefequation133
),
\plkfnc(0)
=
\dpysty ϕ

2\mpi


 
= \dpysty \flxnot(0)

2\mpi


 
+ \cstone
\cstone
=
\dpysty ϕ

2\mpi


 
= \flxnot

2\mpi


 
(115)
Substituting (
string :autorefequation130
) back into (
string :autorefequation133
) shows
\plkfnc(\tautld)
=
\dpysty \flxnot\tautld

2\mpi


 
+ \flxnot

2\mpi


 
=
\dpysty \flxnot

2\mpi


 
(\tautld + 1)
(116)
The thermal absorption and emission in a grey atmosphere increase linearly with optical depth from TOA to the surface.
The upwelling irradiance at the surface is \flxupw(\tautldstr) = \mpi\plkfnc(\tptsfc) where \tptsfc [] is the surface skin temperature. The atmospheric temperature just above the surface is given by (
string :autorefequation131
) as \plkfnc(\tautldstr) = [\flxnot /(2\mpi )](\tautldstr + 1). Thus there is a temperature discontinuity between the near-surface air and the ground.
Climate models typically express \htr (
string :autorefequation124
) in terms of the flux gradient with respect to pressure by invoking the hydrostatic equilibrium condition ()
\htr ≡ \dfr\tpt

\dfr\tm


 
=
1

\dnsatm \heatcpcspcprs


 
\dfr\flx

[−(\dnsatm\grv)−1\dfr\prs]


 
=
\grv

\heatcpcspcprs


 
\dfr\flx

\dfr\prs


 
(117)
=
\grv

\heatcpcspcprs


 
(\flxdwn\kkk − \flxupw\kkk )−(\flxdwn\kkk+1 − \flxupw\kkk+1)

\prs\kkk −\prs\kkk+1


 
=
\grv

\heatcpcspcprs


 
(\flxdwn\kkk − \flxdwn\kkk+1) + (\flxupw\kkk+1 − \flxupw\kkk )

\prs\kkk+1−\prs\kkk


 
(118)
where the subscript denotes the \kkkth vertical interface level in the atmosphere.

0.0.0  Scattering

Energy interacting with matter undergoes one of two processes, or absorption. Scattering occurs when a photon reflects off matter without absorption. The direction of the photon after the interaction is usually not the same as the incoming direction. The case where the scattered photons are homogeneously distributed throughout all 4\mpi steradians is called . In general, the angular dependence of the scattering is described by the of the interaction. The phase function \phzfnc is closely related to the probability that photons incoming from the direction \nglhatprm = (\plrprm,\aziprm) will (if scattered) scatter into outgoing direction \nglhat = (\plr,\azi). It is usually assumed that \phzfnc depends only on the \nglsct between incident and emergent directions.
cos\nglsct
=
\nglhatprm ·\nglhat
(119)
The case where incident and emergent directions are equal, i.e., \nglprm = \ngl corresponds to \nglsct = 0. When the scattered direction continues moving in the forward hemisphere (relative to the plane defined by \nglhatprm), it is called , and corresponds to \nglsct < \mpi/2. When scattered radiation has been reflected back into the hemisphere from whence it arrived, it is called , and corresponds to \nglsct > \mpi/2. The case where incident and emergent directions are opposite, i.e., \nglhatprm = −\nglhat, corresponds to \nglsct = \mpi.
The Cartesian components of \nglprm and \ngl are straigtforward to obtain in .
\nglhat
=
sin\plr cos\azi  \ihat + sin\plr sin\azi  \jhat +cos\plr  \khat
(120)
\nglhatprm
=
sin\plrprm cos\aziprm  \ihat + sin\plrprm sin\aziprm  \jhat + cos\plrprm  \khat
(121)
The scattering angle \nglsct is simply related to the inner product of \nglhatprm and \nglhat by the cosine law
cos\nglsct
=
\nglhatprm ·\nglhat
=
sin\plrprm cos\aziprm sin\plr cos\azi + sin\plrprmsin\aziprm sin\plr sin\azi + cos\plrprm + cos\plr
=
sin\plrprm sin\plr  ( cos\aziprm cos\azi + sin\aziprmsin\azi ) + cos\plrprm cos\plr
=
sin\plrprm sin\plr cos(\aziprm − \azi) + cos\plrprm cos\plr
(122)

0.0.0  Phase Function

Accurate treatment of the angular scattering of radiation, i.e., the phase function, is, perhaps, makes rigorous demands of radiative transfer applications. A correspondingly large body of literature is devoted to this topic. Essential references include , , , , , and .
The \phzfnc(cos\nglsct) is normalized so that the total probability of scattering is unity 1 _ ( )   = 1
1 _^ _^ (,;,)       = 1 The dimensions of the phase function are somewhat ambiguous. If the (4\mpi)−1 factor in (
string :autorefequation138
) is assumed to be steradians, then \phzfnc is a true probability and is dimensionless. However, if the (4\mpi)−1 factor is considered to be numeric and dimensionless (i.e., a probability), then \phzfnc has units of (dimensionless) probability per (dimensional) steradian, . The latter convention best expresses the physical meaning of the phase function and is adopted in this text. It is therefore important to remember that factors of (4\mpi)−1 which multiply the scattering integral in the radiative transfer equation are considered to be dimensionless in the formulations which follow, e.g., (
string :autorefequation152
). Furthermore, the units of \phzfnc are probability per steradian, .
Scattering may depend on the absolute directions \nglhatprm and \nglhat themselves, rather than just their relative orientations as measured by the angle \nglsct between them. This might be the case, for example, in a broken sea-ice field. For the time being, however, we shall assume that the phase function depends only on \nglsct.
In atmospheric problems, the phase function may often be independent of the azimuthal angle ϕ, and depend only on \plr. In this case the phase function normalization (
string :autorefequation138b
) simplifies to
1

2


 



 
\phzfnc(\plrprm;\plr)  sin\plr  \dfr\plr
=
1
(123)
In accord with the discussion above, the factor of 1/2 is dimensionless, as is the RHS.

0.0.0  Legendre Basis Functions

The phase function (
string :autorefequation138
) specifies the angular distribution of scattering. Solution of any radiative transfer equation involving scattering depends on it. Numerical approaches aim for a suitable approximation or discretization which represents \phzfncofnglsct to some desired level of accuracy. The optimal basis functions for representing \phzfncofnglsct are the .
The Legendre polynomial expansion of the phase function is
\phzfnc(cos\nglsct)
=


 
(2\plridx+1) \lgnxpncffplr \lgnplrofcosnglsct
(124)
An expansion of order \NNN contains \NNN+1 terms. The zeroth order polynomial and coefficient are identically 1.
The Legendre polynomials are on the interval [−1,1]. The factor of 2\plridx+1 that appears in the numerator of the Legendre expansion (
string :autorefequation146
) also appears in the denominator of the Legendre polynomial orthonormality property: 1 _^ _()   = 1
1 _^ _()  () = 1 Some other properties of Legendre polynomials are discussed in §.
The \lgnxpncffplr are defined by the projection of the corresponding Legendre polynomial onto the phase function. = 1 _^ ()  
= 1 _^ ()  ()
= 1 _^ ()   map the phase function into a series of Legendre polynomials. The \lgnxpncffplr are also called the . Radiative transfer programs like [] usually require that directional information be specified as a Legendre polynomial expansion.
The Legendre expansion for simple, smoothly varying and symmetric phase functions is highly accurate with just few terms. For instance, a single moment Legendre expansion exactly describes Rayleigh scattering. However, the expansions of asymmetric phase functions converge much more slowly. The strongly peaked forward scattering lobes which appear at large size parameter may require hundreds of moments for accurate representation.
Computing the \lgnxpncffplr using quadrature methods significantly increases accuracy and reduces time. recommends quadrature for highly asymmetric phase functions:
\lgnxpncffplr
=
1

2


 


 
\wgtlbbidx \lgnplr(cos\nglsctlbb) \phzfnc(cos\nglsctlbb) sin\nglsctlbb
(125)
where \lbbnbr is the number of Gauss-Lobatto abscissae. Lobatto quadrature is discussed more thorought in § .

0.0.0  Henyey-Greenstein Approximation

Computation of the exact phase function of scatterers is laborious but desirable when the objective is to predict directional radiances. However, phase function approximations often suffice when hemispheric fluxes are the objective. The approximates the full phase function in terms of its first Legendre moment, i.e., its  \asmprm.
\phzfnc(cos\nglsct)
=
1−\asmprm2

(1+\asmprm2−2\asmprmcos\nglsct)3/2


 
(126)
The \plridx'th moment in the Legendre expansion of (
string :autorefequation144
) is, conveniently, \asmprm\plridx.
\lgnxpncffplr
=
\asmprm\plridx
(127)
Applying (
string :autorefequation145
) in (
string :autorefequation146
) gives
\phzfnc(cos\nglsct)
=


 
(2\plridx+1) \asmprm\plridx \lgnplrofcosnglsct
(128)
This convenient property (
string :autorefequation145
) makes numerical quadrature of the Legendre expansion coefficients unnecessary once the first moment, \asmprm = \lgnxpncff1, is known.

0.0.0  Direct and Diffuse Components

When working with it is convenient to decompose the downwelling intensity \ntndwn into the sum of a , \ntndwndrcoftaungl, and a , \ntndwndffoftaungl such that
\ntndwnfrqoftaungl
=
\ntndwndrcoftaungl + \ntndwndffoftaungl
(129)
where we have suppressed the ν subscript on the RHS to simplify notation. The direct component refers to any photons contributing from a collimated source which have not (yet) been scattered. Typically the collimated source is solar radiation, and so we subscript the direct component with \drcsbs for "solar". There may be a corresponding direct component of intensity in the upwelling direction \ntnupwdrc if the reflectance at the lower surface is , e.g., ocean glint. In such situations it is straightforward to define
\ntnupwfrqoftaungl
=
\ntnupwdrcoftaungl + \ntnupwdffoftaungl
(130)
Of course there exist planetary atmospheres somewhere which are illuminated by multiple stars. We shall neglect upwelling solar beams for the time being. For consistency, though, we shall shall use \ntnupwdff rather than \ntnupw in equations in which \ntndwndff also appears.
It is plain that \ntndwndrcoftaungl is zero in all directions except that of the collimated beam. Moreover, the intensity in the direct beam is, by definition, subject only to extinction by Bougher's law (i.e., there is no emission). Thus the direct beam component of the downwelling intensity is the irradiance incident at the top of the atmosphere, attenuated by the extinction law (
string :autorefequation91
),
\ntndwndrcoftaungl
=
\flxslrtoa \me−τ/\plrmunot \dltfncofnglhatmnglhatnot
(131)
=
\flxslrtoa \me−τ/\plrmunot \dltfnc(\plrmu − \plrmunot) \dltfnc(\azi − \azinot)

0.0.0  Source Function

The source function for thermal emission \srcemsfrq is
\srcemsfrq
=
\vlmemscff(\frq)

\xsxext(\frq)


 
=
\xsxabs(\frq)

\xsxext(\frq)


 
\plkfrq(\tpt)
=
\xsxext(\frq)−\xsxabs(\frq)

\xsxext(\frq)


 
\plkfrq(\tpt)
=
(1−\ssa)\plkfrq(\tpt)
(132)
With knowledge of the phase function \phzfncofnglprmngl as well as the scattering and absorption coefficients of the particular extinction process, we can determine the contribution of scattering to the total source function \srcfrq. Consider the change in radiance \dfr\ntnfrqofngl occuring over a small change in path \dfr\tautld due to a scattering process with phase function \phzfncofnglprmngl. The scattering contribution to \dfr\ntnfrqofngl from every incident direction \nglhatprm is proportional the radiance of the incident beam, \ntnfrqofnglprm, the probability that extinction of \ntnfrqofnglprm at location \tautld is due to scattering (not to absorption), and to the normalized phase function \phzfncofnglprmngl/4\mpi (
string :autorefequation138
).
\dfr \ntnfrqofngl

\dfr\tautld


 
=
\ntnfrqofnglprm \xsxsctoffrq

\xsxsctoffrq + \xsxabsoffrq


 
\phzfncofnglprmngl

4\mpi


 
=
\ssa

4\mpi


 
\ntnfrqofnglprm \phzfncofnglprmngl
(133)
The source function for scattering \srcsctfrq is obtained by integrating (
string :autorefequation151
) over all possible incident directions \nglhatprm that contribute the exiting radiance in direction \nglhat
\srcsctfrq
=
\ssa

4\mpi


 



 
\ntnfrq(\tautld,\nglhatprm) \phzfnc(\tautld,\nglhatprm,\nglhat)  \dfr\nglprm
(134)
The LHS and RHS of (
string :autorefequation152
) must have equal dimensions. This is easily verified: \ssa is dimensionless, (4\mpi)−1 is dimensionless (cf. §
string :autorefsubsubsection2.2.7
), \ntnfrq and \srcsctfrq both have dimensions of intensity, the dimensions of \phzfnc are  and the cancel the dimensions of \dfr\nglprm which are .
All terms in (
string :autorefequation152
) are functions of position and scattering process. If there are \nnn distinct scattering processes (e.g., Rayleigh scattering, aerosol scattering, cloud scattering) then we must know the properties of each individual process (e.g., \xsxsct\iii, \phzfnc\iii) and sum their contributions to obtain to total scattering source function \srcsctfrq = ∑\iii = 1\iii = \nnn\srcfnc\frq,\iii\sctsbs. We neglect such details for now and merely remind the reader that applications need to consider multiple extinction processes at the same time.
The total source function is obtained by adding the source functions for thermal emission (
string :autorefequation150
) and for scattering (
string :autorefequation152
)
\srcfrq
=
\srcemsfrq + \srcsctfrq
\srcfrq
=
(1−\ssa)\plkfrq(\tpt) + \ssa

4\mpi


 



 
\ntnfrq(\tautld,\nglhatprm)\phzfnc(\tautld,\nglhatprm,\nglhat)  \dfr\nglprm
(135)
Although (
string :autorefequation153
) is complete for sources within a medium, additional terms may need to be added to account for boundary sources, such as surface reflection. Boundary sources are discussed in §
string :autorefsubsection2.3
.

0.0.0  Radiative Transfer Equation in Slab Geometry

Inserting (
string :autorefequation153
) into (
string :autorefequation96
) we obtain the radiative transfer equation including absorption, thermal emission and scattering
\dfr\ntnfrq

\dfr\tautld


 
=
−\ntnfrq + \srcfrq
=
−\ntnfrq + (1−\ssa)\plkfrq(\tpt) + \ssa

4\mpi


 



 
\ntnfrq(\tautld,\nglhatprm) \phzfnc(\tautld,\nglhatprm,\nglhat)  \dfr\nglprm
(136)
This is a general form for the equation of radiative transfer in one dimension, and the jumping off point for our discussion of various solution techniques. The physics of thermal emission, absorption, and scattering are all embodied in (
string :autorefequation154
). The most appropriate solution technique for (
string :autorefequation154
) depends on the boundary conditions of the particular problem, the fields required in the solution (e.g., if irradiance is required but radiance is not), and the required accuracy of the solution.
We are most interested in solving the radiative transfer equations in a slab geometry. To do this we transform from the path optical depth coordinate \tautld to the vertical optical depth coordinate τ. Applying the procedure described in (
string :autorefequation96
)-(
string :autorefequation100
) to (
string :autorefequation154
) we obtain the radiative transfer equation for the half range intensities in a slab geometry - (,) = (,) - (1-) - _ (,) (+,-)  
- _ (,) (-,-)  
(,) = (,) - (1-) - _ (,) (+,+)  
- _ (,) (-,+)   If we substitute (
string :autorefequation147
) into (
string :autorefequation155a
) we obtain
− \plrmu \dfr\ntndwndff(τ,\nglhat)

\dfrτ


 
− \plrmu \dfr\ntndwndrc(τ,\nglhat)

\dfrτ


 
=
\ntndwndff(τ,\nglhat) + \ntndwndrc(τ,\nglhat) − (1−\ssa) \plkfnc − \ssa

4\mpi


 



 
\ntndwndrc(τ,\nglhatprm) \phzfnc(−\nglhatprm,−\nglhat)  \dfr\nglprm
\ssa

4\mpi


 



 
\ntnupwdff(τ,\nglhatprm) \phzfnc(+\nglhatprm,−\nglhat)  \dfr\nglprm− \ssa

4\mpi


 



 
\ntndwndff(τ,\nglhatprm) \phzfnc(−\nglhatprm,−\nglhat)  \dfr\nglprm
The direct beam (
string :autorefequation150
) satisfies the extinction law (
string :autorefequation91
) so that −\plrmu  \dfr\ntndwndrc/\dfrτ = \ntndwndrc. Thus the second terms on the LHS and the RHS of (
string :autorefequation156
) cancel eachother and we are left with
− \plrmu \dfr\ntndwndff(τ,\nglhat)

\dfrτ


 
= \ntndwndff(τ,\nglhat) − (1−\ssa) \plkfnc − \srcstr(τ,−\nglhat)
\ssa

4\mpi


 



 
\ntnupwdff(τ,\nglhatprm) \phzfnc(+\nglhatprm,−\nglhat)  \dfr\nglprm− \ssa

4\mpi


 



 
\ntndwndff(τ,\nglhatprm) \phzfnc(−\nglhatprm,−\nglhat)  \dfr\nglprm
(137)
where \srcstr is called the . \srcstr is defined by the integral containing the direct beam on the RHS of (
string :autorefequation156
)
\srcstr(τ,−\nglhat)
=
\ssa

4\mpi


 



 
\ntndwndrc(τ,\nglhatprm) \phzfnc(−\nglhatprm,−\nglhat)  \dfr\nglprm
=
\ssa

4\mpi


 



 
\flxslrtoa \me−τ/\plrmunot \dltfnc(\nglhatprm,\nglhatnot)\phzfnc(−\nglhatprm,−\nglhat)  \dfr\nglprm
=
\ssa

4\mpi


 
\flxslrtoa \me−τ/\plrmunot\phzfnc(−\nglhatnot,−\nglhat)
(138)
Note that (
string :autorefequation156
) is an equation for the diffuse downwelling radiance, not for the total downwelling radiance. The relation between the diffuse and total radiance is given by (
string :autorefequation147
). The direct component is always known (
string :autorefequation150
) once the optical depth has been determined.
Likewise, inserting (
string :autorefequation147
) and (
string :autorefequation150
) into (
string :autorefequation155b
), we obtain the equation for the diffuse upwelling intensity
\plrmu \dfr\ntnupwdff(τ,\nglhat)

\dfrτ


 
= \ntnupwdff(τ,\nglhat) − (1−\ssa) \plkfnc − \srcstr(τ,+\nglhat)
\ssa

4\mpi


 



 
\ntnupwdff(τ,\nglhatprm) \phzfnc(+\nglhatprm,+\nglhat)  \dfr\nglprm− \ssa

4\mpi


 



 
\ntndwndff(τ,\nglhatprm) \phzfnc(−\nglhatprm,+\nglhat)  \dfr\nglprm
(139)
where
\srcstr(τ,+\nglhat)
=
\ssa

4\mpi


 



 
\ntndwndrc(τ,\nglhatprm) \phzfnc(−\nglhatprm,+\nglhat)  \dfr\nglprm
=
\ssa

4\mpi


 



 
\flxslrtoa \me−τ/\plrmunot \dltfnc(\nglhatprm,\nglhatnot)\phzfnc(−\nglhatprm,+\nglhat)  \dfr\nglprm
=
\ssa

4\mpi


 
\flxslrtoa \me−τ/\plrmunot\phzfnc(−\nglhatnot,+\nglhat)
(140)

0.0.0  Azimuthal Mean Radiation Field

Often we are not concerned with the azimuthal dependence of the radiation field. In some cases this is because the azimuthal dependence is very weak. For example, heavily overcast skies, or diffuse reflectance from a uniform surface. In other cases the azimuthal dependence may not be weak, but there insufficient information to fully determine the solutions. For example, the full surface or the shapes of clouds are not available. The azimuthal dependence of a radiative quantity \XXX(\azi) is removed by applying the
-
\XXX
 

 
=
1

2\mpi


 



 
\XXX(\azi)  \dfr\azi
(141)
For future reference we present also the polar angle integration operator which will be used to formulate the .
-
\XXX
 

 
=



 
\XXX(\plr) sin\plr  \dfr\plr
-
\XXX
 

 
=



 
\XXX(\plrmu)  \dfr\plrmu
(142)
Applying (
string :autorefequation160
) to (
string :autorefequation25
) and (
string :autorefequation138
), we obtain the azimuthal mean , , and respectively,
\ntnupwdwn(τ,\plrmu)
=
1

2\mpi


 



 
\ntnupwdwn(τ,\plrmu,\azi)  \dfr\azi
(143)
\phzfnc(±\plrmuprm;±\plrmu)
=
1

2\mpi


 



 
\phzfnc(±\plrmuprm,\aziprm;±\plrmu,\azi)  \dfr\azi
(144)
\srcstr(τ,±\plrmu)
=
\ssa

4\mpi


 
\flxslrtoa \me−τ/\plrmunot\phzfnc(−\plrmunot,±\plrmu)
(145)
No additional symbols are used to indicate azimuthal mean quantities. The presence of only the polar angle (\plr or \plrmu) on the LHS of (
string :autorefequation162
) indicates that azimuthal mean quantities are involved.
Applying (
string :autorefequation160
) to the full radiative transfer equations for slab geometry, (
string :autorefequation156
) and (
string :autorefequation158
), we obtain the radiative transfer equations for the azimuthal mean intensity in slab geometry
- _ (,) (+,-)   - _ (,) (-,-)  

- _ (,) (+,+)   - _ (,) (-,+)   The azimuthal mean equations (
string :autorefequation165
) are very similar to the full equations (
string :autorefequation156
) and (
string :autorefequation158
). While the remainder of the terms in (
string :autorefequation165
) contain only one azimuthally dependent intensity, the scattering integrals contain two, \phzfnc and \ntnupwdwn. This causes the factor of \ssa/4\mpi in front of the scattering integrals has to become a factor of \ssa/2.
We apply the two stream formalism introduced in §
string :autorefsubsection2.4
to the radiative transfer equation for the azimuthal mean radiation field (
string :autorefequation165
). Recall that the two stream formalism is obtained by applying three successive approximations. The approximation is achieved in three stages. First, we operate on both sides of (
string :autorefequation165a
) with the hemispheric averaging operator (
string :autorefequation161
). The procedure for (
string :autorefequation165a
) is identical.



 
\plrmu \dfr\ntndwndff(τ,\plrmu)

\dfrτ


 
 \dfr\plrmu =


 
\ntndwndff(τ,\plrmu)  \dfr\plrmu − (1−\ssa)


 
\plkfnc  \dfr\plrmu − \ssa

4\mpi


 
\flxslrtoa \me−τ/\plrmunot


 
\phzfnc(−\plrmunot,−\plrmu)  \dfr\plrmu
\ssa

2


 



 



 
\ntnupwdff(τ,\plrmuprm) \phzfnc(+\plrmuprm,−\plrmu)  \dfr\plrmuprm  \dfr\plrmu− \ssa

2


 



 



 
\ntndwndff(τ,\plrmuprm) \phzfnc(−\plrmuprm,−\plrmu)  \dfr\plrmuprm  \dfr\plrmu
(146)
Further approximations are required to simplify equations (
string :autorefequation173
). These approximations allow us to decouple products of functions with \plrmu dependence. The first approximation, (
string :autorefequation167
), replaces the continuous value \plrmu on the LHS of (
string :autorefequation165
) by a suitable hemispheric mean value \plrmubar.



 
\plrmu \dfr\ntndwndff(τ,\plrmu)

\dfrτ


 
 \dfr\plrmu
\plrmubar \dfr

\dfrτ


 



 
\ntndwndff(τ,\plrmu)  \dfr\plrmu = \plrmubar \dfr \ntndwndff(τ)

\dfrτ


 
(147)
Extracting \plrmubar from the integral on the LHS of (
string :autorefequation167
) is an approximation whose validity worsens as the correlation between \plrmu and \ntnupwdwndff(τ,\plrmu) increases.
The first two terms on the RHS of (
string :autorefequation166
) are simple hemispheric integrals of hemispherically-varying intensities. We will replace \ntnupwdwn(τ,\plrmu) by the hemispheric mean intensity \ntnupwdwn(τ) (
string :autorefequation205
). The Planck function is isotropic and so may be pulled outside the hemispheric integral which promptly vanishes since ∫01  \dfr\plrmu = 1. Thus the hemispherically integrated intensities which result explicitly are replaced by the hemispheric mean intensity (
string :autorefequation205
) associated with \plrmubar.
The final three terms on the RHS of (
string :autorefequation166
) involve products of the phase function and the intensity. We approximate the scattering integrals by applying the hemispheric integral over \plrmu to the intensities, and then extracting the intensities from original integrals over \plrmuprm.



 



 
\phzfnc(+\plrmuprm,−\plrmu) \ntnupwdff(τ,\plrmuprm) \dfr\plrmuprm



 
\ntnupwdff(τ,\plrmuprm)  \dfr\plrmu


 
\phzfnc(+\plrmuprm,−\plrmu)  \dfr\plrmuprm
=
\ntnupwdff(τ)


 
\phzfnc(+\plrmuprm,−\plrmu)  \dfr\plrmuprm
(148)
These terms define the backscattered fraction of the radiation field. Equation (
string :autorefequation168
), for example, represents the fraction of upwelling energy backscattered into the downwelling direction. This is one of four similar terms that appear in the coupled equations (
string :autorefequation166
). We define the azimuthal mean \bckcffofplrmu as
\bckcffofplrmu
1

2


 



 
\phzfnc( −\plrmuprm, \plrmu )  \dfr\plrmuprm = 1

2


 



 
\phzfnc( \plrmuprm, −\plrmu )  \dfr\plrmuprm
(149)
The complement of the backscattering function is the \fsfcff.
\fsfcffofplrmu = 1 − \bckcffofplrmu
1

2


 



 
\phzfnc( −\plrmuprm, −\plrmu )  \dfr\plrmuprm = 1

2


 



 
\phzfnc( \plrmuprm, \plrmu )  \dfr\plrmuprm
(150)
Pairs of the terms in (
string :autorefequation169
)-(
string :autorefequation170
) are identical due to . Reciprocity relations state that photon paths are reversible.
The hemispheric mean \bckcff is the hemispheric integral of (
string :autorefequation169
)
\bckcff



 
\bckcffofplrmu  \dfr\plrmu
(151)
=
1

2


 



 



 
\phzfnc( −\plrmuprm, \plrmu )  \dfr\plrmuprm  \dfr\plrmu
(152)
The hemispheric mean forward scattering coefficient is defined analogously.
The result of these two steps is
- _ () (+,-)   - _ () (-,-)  

- _ () (+,+)   - _ () (-,+)   These are the hemispheric mean, azimuthal mean, two stream equations for the radiation field in slab geometry. It should be clear that these equations (
string :autorefequation173
) are not derived simply by performing a hemispheric integral (
string :autorefequation161
) on (
string :autorefequation165
). A strict hemispheric averaging operation would applies to the product of the phase function and and the intensity, not to each separately, so that extracting \ntnupwdwndff from the scattering integrals is an approximation, as is the definition of \plrmubar.

0.0.0  Anisotropic Scattering

Armed with techniques introduced in solving the two stream equations for an isotropically scattering medium §
string :autorefsubsubsection2.4.1
, we now solve analytically the coupled two stream equations for an anisotropic medium. First we recast () into a simpler notation = - (1 - ) - - (1-) -
- = - (1 - ) - - (1-) - For the remainder of the derivation we drop the \frq subscript and the explicit dependence of \ntn± on τ. Adding and subtracting we obtain ( + ) = -( - ) ( - )
( - ) = -( + ) ( + ) where we have defined -[ 1 - ( 1 - ) ] /
/

0.0.0  Diffusivity Approximation

The angular integration required to convert the intensity field into a hemispheric irradiance is a time consuming aspect of numerical models and should be avoided where possible. According to (
string :autorefequation56a
)-(
string :autorefequation56b
) 1 = 2 _0^1 (+)  
1 = 2 _0^1 (-)   The factor of \mpi has been moved to the LHS so that, when the source function is thermal, the RHS is the integral Planck function (
string :autorefequation116
). These angular integrals may be reduced to a calculation of the intensity at certain quadrature points (zenith angles) in each hemisphere with surprising accuracy. The location of the optimal quadrature points may be arrived at through both theoretical and empirical methods.
Two point (§) tells us the optimal angles to evaluate (
string :autorefequation177
) at are \plru = ±3−1/2, \plr = ±54.7356\dgr. 1 2 (+3^-1/2)
1 2 (-3^-1/2) The relation in Equation (
string :autorefequation178
) is exact when \ntnfrqofplrmu is a polynomial of degree < 2. For two point Gaussian quadrature the quadrature weights happen to be unity, but not so for three point (or higher order) Gaussian quadrature. Here is the four point quadrature version of (
string :autorefequation178
)

\plru1 = −\plru4
=
0.339981
\plru2 = −\plru3
=
0.861136
\AAA1 = \AAA4
=
0.652145
\AAA2 = \AAA3
=
0.347855
1

2 \mpi


 
\flxupwfrq
\AAA1 \plru1 \ntnfrq(\plru1) + \AAA2 \plru2 \ntnfrq(\plru2)
1

2 \mpi


 
\flxdwnfrq
\AAA3 \plru3 \ntnfrq(\plru3) + \AAA4 \plru4 \ntnfrq(\plru4)



(153)
What is often used to evaluate (
string :autorefequation177
) instead of Gaussian quadrature is a . Instead of choosing optimal quadrature angles \plrmu\kkk, the diffusivity approximation relies on choosing the optimal effective absorber path. Thus the , \dff, is defined by 1 (^-1)
1 (^-1) Comparing (
string :autorefequation180
) to (
string :autorefequation178
) and (
string :autorefequation177
) shows that \dff replaces, simultaneously, the quadrature angle, its weight, the factor of \plrmu in (
string :autorefequation177
), and a factor of 2 from the azimuthal integration. The angle \plrdff = arccos\dff−1 may be interpreted as the mean slant path of radiation in an isotropic, non-scattering atmosphere.
The reasons for subsuming so many factors into \dff are historical. Nevertheless, (
string :autorefequation180
) is much more common than (
string :autorefequation178
) in the literature. Heating rate calculations in isotropic, non-scattering atmospheres show that using \dff = 1.66 results in errors < 2% .
In applying (
string :autorefequation180
), \dff only affects the optical depth factor of the intensity. Thus computing the intensity in (
string :autorefequation180
) at the angle \plr = \plrdff is formally equivalent to computing the intensity in the vertical direction \plr = 0 but through an atmosphere in which the absorber densities have been increased by a factor of \dff.

0.0.0  Transmittance

The between two points measures the transparency of the atmosphere between those points. The transmittance \trntld from point \pnt1 to point \pnt2 is the likelihood a photon traveling in direction \nglhat at \pnt1 will arrive at \pnt2 without having interacted with the matter in between.
\trntld(\pnt1,\pnt2) = exp[−\tautld(\pnt1,\pnt2)]
(154)
We have left implicit the dependence on frequency. In a stratified atmosphere, we are interested in the transmission of light traveling from layer \zzzprm to layer \zzz at angle \plr from the vertical, thus
\trnbm(\zzzprm,\zzz;\plrmu) = exp[−τ(\zzzprm,\zzz)/\plrmu]
(155)
Thus, in common with absorptance \absbm and reflectance \rflbm, the transmittance \trnbm ∈ [0,1]12
The vertical gradient of \trnbm is frequently used to formulate solutions of the radiative transfer equation in non-scattering, thermal atmospheres.

∂\zzzprm


 
\trnbm(\zzzprm,\zzz;\plrmu)
=

∂\zzzprm


 
exp[−τ(\zzzprm,\zzz;\plrmu)/\plrmu]
∂\trnbm(\zzzprm,\zzz;\plrmu)

∂\zzzprm


 
=
1

\plrmu


 
exp[−τ(\zzzprm,\zzz;\plrmu)/\plrmu] ∂τ(\zzzprm,\zzz;\plrmu)

∂\zzzprm


 
=
1

\plrmu


 
\trnbm(\zzzprm,\zzz;\plrmu) ∂τ

∂\zzzprm


 
(156)
Inverting the above, we obtain
\plrmurcp \trnbm(\zzzprm,\zzz;\plrmu) = − ∂\trnbm(\zzzprm,\zzz;\plrmu)

∂\zzzprm


 
∂\zzzprm

∂τ


 
(157)
The integral solutions for the upwelling and downwelling intensities in a non-scattering, thermal, stratified atmosphere (
string :autorefequation109
)-(
string :autorefequation110
) may be rewritten in terms of \trnbm. (
string :autorefequation110
) from τ to \zzz. To do this we use
\zzz(τ = \taustr)
=
0
\zzz(τ = 0)
=
\trnbm(0,\zzz;\plrmu)
=
\me−(\taustr − τ)/\plrmu
\trnbm(\zzzprm,\zzz;\plrmu)
=
\me−(\tauprm − τ)/\plrmu
\trnbm(\zzz,\zzzprm;\plrmu)
=
\me−(τ− \tauprm)/\plrmu
∂\trnbm(\zzzprm,\zzz;\plrmu)

∂\zzzprm


 
=
∂\trnbm(\zzz,\zzzprm;\plrmu)

∂\zzzprm


 
(158)
The last relation simply states that the transmission decreases as the distance between \zzzprm,\zzz increases and visa versa. Substituting the above into (
string :autorefequation109
) and (
string :autorefequation110
) yields
\ntnfrqofzzzpmu
=
\trnbm(0,\zzz;\plrmu) \plkfrq(0) + \plrmurcp


 
\trnbm(\zzzprm,\zzz;\plrmu) \plkfrq(\zzzprm)  (−\dfr\zzzprm)
\ntnfrqofzzzmmu
=
\plrmurcp


 
\trnbm(\zzz,\zzzprm;\plrmu) \plkfrq(\zzzprm)  (−\dfr\zzzprm)
Rearranging terms we obtain = (0,;) (0) + _0^ (,;) ()  
= _^ (,;) ()   Substituting (
string :autorefequation184
) in the above leads to = (0,;) (0) + _0^ (,;) ()  
= - _^ (,;) ()   If \trnbm is known, then Equations (
string :autorefequation187a
)-(
string :autorefequation187b
) are suitable for use in quadrature formulae to obtain irradiances.

0.0  Reflection, Transmission, Absorption

Perhaps the most useful metrics of the radiative properties of entire systems are the quantities reflection, transmission, and absorption. These metrics describe normalized properties of a system and thus are somewhat more fundamental than absolute quantities (like transmitted irradiance), which may, for example, change depending on time of day. We shall define the reflection, transmission, and absorption first of the radiance field, and then integrate these definitions with the appropriate weighting to obtain the \rfl, \trn, and \abs pertinent to fluxes. We shall introduce a painful but necessary menagerie of terminology to describe the various species of \rfl, \trn, and \abs.
] show how remotely sensed differs significantly from that derived from snow-age models. Using the more realistic albedos improved models of snowmelt by providing more accurate net surface shortwave radiation estimates. This term, \flxnetswsfc, dominates the snow-melt energy budget.
] derived a simple functional fit to the desert surface albedo observed by .

0.0.0  BRDF

The (BRDF) \brdf is the ratio of the reflected intensity to the energy in the incident beam. As such, \brdf is a function of frequency, incident angle, and scattered angle
\brdfoffrqmnglhatprmnglhat
\dfr\ntnupwfrqrfl(\nglhat)

\ntndwnfrq(\nglhatprm) cos\plrprm  \dfr\nglprm


 
(159)
The dimensions of \brdf are  and they convert irradiance to intensity. The reflected intensity in any particular direction \nglhat is the sum of contributions from all incident directions \nglhatprm that have a finite probability of reflecting into \nglhat.
\ntnupwfrqrfl(\nglhat)
=



 
 \dfr\ntnupwfrqrfl(\nglhat)
=



 
\ntndwnfrq(\nglhatprm) \brdfoffrqmnglhatprmnglhat cos\plrprm \dfr\nglprm
(160)
The dependence of \brdfoffrqmnglhatprmnglhat on two directions and on frequency makes it a difficult property to measure.

0.0.0  Lambertian Surfaces

Fortunately many surfaces found in nature obey simpler reflectance properties first characterized by Lambert. A is one whose reflectance is independent of both incident and reflected directions. The reflectance of a Lambertian surface depends only on frequency
\brdfoffrqmnglhatprmnglhat
=
\rfllmboffrq
(161)
The intensity reflected from a Lambertian surface is given by inserting (
string :autorefequation190
) into (
string :autorefequation188
) and then using (
string :autorefequation56b
)
\ntnupwfrqrfl(\nglhat)
=



 
\ntndwnfrq(\nglhatprm) \rfllmboffrq cos\plrprm \dfr\nglprm
=
\rfllmboffrq


 
\ntndwnfrq(\nglhatprm) cos\plrprm \dfr\nglprm
=
\rfllmboffrq \flxdwnfrq
(162)
As expected, the intensity reflected from a Lambertian surface depends only on the incident irradiance, and not at all on the details of the angular distribution of the incident intensity field.
The irradiance reflected from a Lambertian surface \flxupwfrqrfl is the cosine-weighted integral of the reflected intensity (
string :autorefequation191
) over all reflected angles
\flxupwfrqrfl
=



 
\ntnupwfrq(\nglhat) cos\plr  \dfr\ngl
=



 
\rfllmboffrq \flxdwnfrq cos\plr  \dfr\ngl
Neither \flxdwnfrq and \rfllmboffrq depend on the emergent angle \nglhat so the integral reduces to the familiar integral of cos\plr over the hemisphere (
string :autorefequation59
) which is \mpi.
\flxupwfrqrfl
=
\mpi \rfllmboffrq \flxdwnfrq
(163)
The reflected irradiance may not exceed the incident irradiance or the requirement of energy conservation will be violated. Therefore (
string :autorefequation192
) shows that
\rfllmboffrq
\mpi−1
(164)
with equality holding only for a perfectly reflective (non-absorbing) Lambertian surface.
Many researchers prefer the Lambertian BRDF to have an upper limit of 1, not \mpi−1 (
string :autorefequation193
). Thus it is common to encounter in the literature BRDF defined as \rrr = \mpi \brdf.

0.0.0  Albedo

The reflectance of a surface illuminated by a collimated source such as the Sun or a laser is of great interest in planetary studies and in remote sensing. For an incident collimated beam of intensity \flxslrtoa, the diffusely reflected intensity is
\ntnupwfrqrfl(\nglhat)
=



 
\flxslrtoa \dltfncofnglhatprmmnglhatnot \brdfoffrqmnglhatprmnglhat cos\plrprm \dfr\nglprm
=
\flxslrtoa \brdfoffrqmnglhatnotnglhat cos\plrnot
(165)
Integrating (
string :autorefequation194
) over all reflected angles we obtain the diffusely reflected hemispheric irradiance
\flxupwfrqrfl
=



 
\flxslrtoa \brdfoffrqmnglhatnotnglhat cos\plrnot cos\plr \dfr\ngl
=
\flxslrtoa cos\plrnot


 
\brdfoffrqmnglhatnotnglhat cos\plr  \dfr\ngl
(166)
The or is the ratio of the reflected (diffuse) irradiance (
string :autorefequation195
) to the incident (collimated) irradiance (
string :autorefequation150
)
\brdfoffrqmnglhatnottwopi
=
\flxupwfrqrfl

\flxdwnfrq


 
=



 
\brdfoffrqmnglhatnotnglhat cos\plr  \dfr\ngl
(167)
The 2\mpi indicates that the plane albedo pertains to the entire reflected hemisphere.
When the reflecting body is of finite size then the corresponding ratio of reflected to incident fluxes is more complicated because it contains "edge effects", e.g., diminishing contributions from planetary limbs. First we consider the reflection, called the , or , of an entire planetary disk illuminated by collimated sunlight.
The \brdfoffrqmtwopinglhat is the
\flxupwfrqrfl(\nglhat)
=



 
\flxdwnfrq(−\nglhatprm) \brdfoffrqmtwopinglhat cos\plrprm  \dfr\nglprm
(168)
The flux reflectance of a to collimated light is found by subsituting (
string :autorefequation190
) in (
string :autorefequation196
)
\brdfoffrqmnglhatnottwopi
=



 
\rfllmboffrq cos\plr  \dfr\ngl
=
\mpi \rfllmboffrq
(169)
which agrees with (
string :autorefequation192
).
The BRDF must be integrated over all possible angles of reflectance in order to obtain the , \brdfoffrqmnglhatprmtwopi. Representative flux reflectances of various surfaces in the Earth system are presented in Table 
string :autoref 4
.
Table 4: Surface Albedo13
Surface Type Reflectance
Glacier 0.9
Snow 0.8
Sea-ice 0.6
Clouds 0.2-0.7
Desert 0.2-0.3
Savannah 0.2-0.25
Forest 0.05-0.1
Ocean 0.05-0.1
Planetary 0.3

0.0.0  Flux Transmission

The \trnflxfrq between two layers is (twice) the cosine-weighted integral of the spectral transmission
\trnflxfrq(\zzzprm,\zzz)
=
2


 
\trnbm(\zzzprm,\zzz;\plrmu) \plrmu  \dfr\plrmu
\trnflxfrq(\zzzprm,\zzz)
=
2


 
exp[−τ(\zzzprm,\zzz)/\plrmu] \plrmu  \dfr\plrmu
(170)
We can rewrite (
string :autorefequation199
) in terms of exponential integrals ()
\trnflxfrq(\zzzprm,\zzz) = 2 \xpn3[τ(\zzzprm,\zzz)]
(171)
The factor of 2 ensures that expressions for the hemispheric fluxes in non-scattering, thermal atmospheres closely resemble the equations for intensities, as will be shown next.
The vertical gradient of the flux transmission is frequently used to formulate solutions of the radiative transfer equation in non-scattering, thermal atmospheres. It is obtained by differentiating (
string :autorefequation200
) then applying ()

∂\zzzprm


 
\trnflxfrq(\zzzprm,\zzz)
=
2

∂\zzzprm


 
\xpn3[τ(\zzzprm,\zzz)] ∂\trnflxfrq(\zzzprm,\zzz)

∂\zzzprm


 
∂\trnflxfrq(\zzzprm,\zzz)

∂\zzzprm


 
=
2 [−\xpn2(\zzzprm,\zzz)] ∂τ(\zzzprm,\zzz;\plrmu)

∂\zzzprm


 
=
− 2 \xpn2(\zzzprm,\zzz) ∂τ

∂\zzzprm


 
(172)
The integral solutions for the upwelling and downwelling fluxes in a non-scattering, thermal, stratified atmosphere (
string :autorefequation113
) and (
string :autorefequation114
) may be rewritten in terms of \trnflxfrq. The procedure for doing so is analogous to the procedure applied to intensities in §
string :autorefsubsubsection2.2.16
. First, we change the independent variable in (
string :autorefequation113
) and (
string :autorefequation114
) from τ to \zzz.
\flxupwfrqofzzz
=
2 \mpi \plkfrq(0) \xpn3(\zzz) + 2 \mpi


 
\plkfrq(\zzzprm) \xpn2(\zzz−\zzzprm)  \dfr\zzzprm
\flxdwnfrqofzzz
=
2 \mpi


 
\plkfrq(\zzzprm) \xpn2(\zzzprm − \zzz)  \dfr\zzzprm
Substituting (
string :autorefequation201
) into the above yields 1 = (0) (0,) + _^ () (,)  
1 = - _^ () (,)   The analogy between (
string :autorefequation202a
)-(
string :autorefequation202b
) and (
string :autorefequation187a
)-(
string :autorefequation187b
) is complete. \trnflxfrq plays the same role in former that \trnbm plays in the latter.
Equations (
string :autorefequation202a
)-(
string :autorefequation202b
) are particularly useful because they contain no explicit reference to the angular integration. The polar integration, however, is still implicit in the definition of \trnflxfrq (
string :autorefequation199
). The hemispherical irradiances may be determined without any angular integration if the diffusivity approximation is applied to (
string :autorefequation202a
)-(
string :autorefequation202b
). This is accomplished by replacing \trnflxfrq by the vertical spectral transmission \trnbm through a factor \dff (
string :autorefequation180
) times as much mass
\trnflxfrq(\zzzprm,\zzz) ≅ \trnbm(\zzzprm,\zzz;arccos\dff−1)
(173)
Determination of \trnbm requires the use of a spectral gaseous extinction database in conjunction with either a model, a narrow band model, or a broadband emissivity approach.

0.0  Two-Stream Approximation

The to the radiative transfer equation assumes that up- and down-welling radiances travel at mean inclinations \plrmubarupw and \plrmubardwn, respectively. With this assumption, the hemispheric intensities \ntnfrq± in the azimuthally averaged radiative transfer equation (
string :autorefequation165
) lose their explicit dependence on \plrmu and depend solely on optical depth. For an isotropically scattering atmosphere in a slab geometry, the two-stream approximation may be written = - - - (1 - )
- = - - - (1 - ) These equations are no longer exact, but only approximations. Nevertheless, and as we shall show, the two-stream approximation yields remarkably accurate solutions for a wide variety of scenarios including the most important ones in planetary atmospheres. We emphasize that the solutions to the two-stream equation, \ntnupwdwnfrq, are hemispheric mean intensities each associated with a hemispheric mean direction \plrmubarupwdwn. In this interpretation of \ntnupwdwnfrqoftau,
\ntnupwdwnfrqoftau =



 
\ntnupwdwnfrq(τ,\plrmu) \dfr\plrmu




 
 \dfr\plrmu



 
=


 
\ntnupwdwnfrq(τ,\plrmu) \dfr\plrmu
(174)
As emphasized by (
string :autorefequation205
), \ntnupwdwnfrq are not isotropic intensities in each hemisphere.
We now re-define the other important properties of the radiation field to be consistent with the two-stream approximation. Perhaps most important is the hemispheric irradiance. Starting from (
string :autorefequation56
) we obtain
\flxupwdwnfrq(τ)
=
2 \mpi


 
\ntnupwdwnfrq (τ,\plrmu) \plrmu  \dfr\plrmu
2 \mpi \plrmubar \ntnupwdwnfrq (τ)
(175)
where ≈ indicates that (
string :autorefequation206
) is an approximation. Equation (
string :autorefequation206
) reveals one interesting physical interpretation of the two-stream approximation: the radiance field comprises only two discrete streams travelling at angles \plrmubarupwdwn. Mathematically, this radiance field could be represented by the union of the two delta-functions representing collimated beams travelling in the \plrmubarupwdwn directions. This interpretation (presumably) gives the "two stream" method its name. Physically, it is more appropriate to interpret the two-stream radiance field as being continuously distributed in each hemisphere (
string :autorefequation205
), such that the angular moments (e.g., in the definition of irradiance) have the properties in (
string :autorefequation206
).
A common mis-understanding of the two-stream approximation is that the radiance field is isotropic in each hemisphere. If two-stream radiance were isotropic in each hemisphere then the appropriate measure of hemispheric irradiance would be \flxupwdwnfrq = \mpi \ntnupwdwnfrq (
string :autorefequation59
) as opposed to \flxupwdwnfrq = 2 \mpi \plrmubarupwdwn \ntnupwdwnfrq (
string :autorefequation206
). In fact, assuming the intensity field varies linearly with \plrmu (and is thus anisotropic) is a reasonable interpretation of the two-stream approximation.
The mean intensity \ntnmnfrq (
string :autorefequation20
) for an azimuthally independent, isotropically scattering radiation field is
\ntnmnfrqoftau
=
1

2


 



 
\ntnfrq(τ,\plru)  \dfr\plru
=
1

2


 



 
\ntnupwfrq(τ,\plrmu) +\ntndwnfrq(τ,\plrmu)  \dfr\plrmu
1

2


 
[\ntnupwfrqoftau + \ntndwnfrqoftau]
(176)
Although \ntnmnfrqoftau (
string :autorefequation207
) appears independent of \plrmubar, the two-stream intensities in (
string :autorefequation207
) do depend on \plrmubar.
Continuing to write the two-stream approximations for radiative quantities of interest, we turn now to the source function. The source function \srcfrq (
string :autorefequation153
) for an azimuthally independent, isotropically scattering (\phzfnc = 1) radiation field is
\srcfrqoftau
=
(1−\ssa)\plkfrq[\tpt(τ)] + \ssa

2


 



 
\ntnfrq(τ,\plru)  \dfr\plru
(1−\ssa)\plkfrq[\tpt(τ)] + \ssa

2


 
[\ntnupwfrqoftau + \ntndwnfrqoftau]
(177)
The final term on the RHS is the two-stream mean intensity ntnmnfrq (
string :autorefequation207
). \ntnmnfrq appears in \srcfrq (
string :autorefequation208
) due to the assumption of isotropic scattering. For more general phase functions, \srcfrq will not contain \ntnmnfrq.
A comment on vertical homogeneity is appropriate here. The Planck function in (
string :autorefequation208
) varies continuously with τ, whereas \ssa is assumed to be constant in a given layer over which (
string :autorefequation208
) is discretized. Thus τ-dependence appears explicitly for \plkfrq, and is absent for \ssa. More generally, the radiation field (intensity, irradiance, source function, etc.) varies continuously with τ. The discretization of τ (or, in general, all spatial coordinates) is required in order to solve for the continuously varying radiation field. The discretization determines the scale over which the properties of the matter in the medium may change.
The radiative heating rate \htrfrq in the two-stream approximation may be obtained in at least two distinct forms.
\htrfrqoftau
=
∂\flxfrq

∂\zzz


 
=
4\mpi \ntnmnfrqoftau \xsxabs − 4\mpi \xsxabs \plkfrq[\tpt(τ)]
2 \mpi \xsxabs [ \ntnupwfrqoftau + \ntndwnfrqoftau ] − 4\mpi \xsxabs \plkfrq[\tpt(τ)]
(178)
The first term on the RHS represents the rate of absorption of radiative energy, while the second term is the rate of radiative emission. Note that (
string :autorefequation209
) depends on the absorption cross section α rather that the single scattering albedo \ssa. This emphasizes that scattering has no direct effect on heating.
Together the relations for \flxfrq, \ntnmnfrq, \srcfrq, and \htrfrq in (
string :autorefequation206
)-(
string :autorefequation209
) define a complete and self-consistent set of radiative properties under the two-stream approximation. We turn now to obtaining the two-stream intensities \ntnupwdwnfrq(τ) from which these quantities may be computed.

0.0.0  Two-Stream Equations

We shall simplify both the physics and nomenclature of (
string :autorefequation204
) before attempting an analytical solution. First, for algebraic simplicity we set \plrmubarupw = \plrmubardwn = \plrmubar. Also, we neglect the thermal source term \plkfrq[\tpt(τ)]. With these assumptions, (
string :autorefequation204
) becomes = - -
- = - -
For the remainder of the derivation we drop the \frq subscript and the explicit dependence of \ntn± on τ. Adding and subtracting the equations in (
string :autorefequation210
) we obtain ( - ) = + - ( + ) = (1 - ) ( + )
( + ) = - These may be rewritten as first order, coupled equations in \YYYpm where
= 1 ( ) Using (
string :autorefequation212a
) in (
string :autorefequation211
) = (1 - )
= 1 We differentiate (
string :autorefequation213
) with respect to τ to uncouple the equations, 3 = (1 - ) = (1 - ) 1 = 1 -
= 1 = 1 (1 - ) = 1 - The uncoupled quantities \YYYpm satisfy the same second order ordinary differential equation
\dfrsqr\YYYpm

\dfr\tausqr


 
=
Γ2 \YYYpm
(179)
where
Γ2
(1 − \ssa)

\plrmubar2


 
(180)
Elementary differential equation theory teaches us that the solutions to (
string :autorefequation214
) are
\YYYpm(τ)
=
\cst±1 \meΓτ + \cst±2 \me−Γτ
(181)
The general forms of \ntnupwdwn are obtained by inserting (
string :autorefequation216
) into (
string :autorefequation212b
) = ^ + ^-
= ^ + ^-
Equation (
string :autorefequation217
) suggests that there are four unknown constants of integration \cstone-\cstfour. This is a consequence of solving for the hemispheric intensities (
string :autorefequation25
) separately in two equations (
string :autorefequation210
) rather solving the full radiative transfer equation (
string :autorefequation154
) for the full domain intensity \ntn. As discussed in §
string :autorefsubsubsection2.1.16
, the motivation for solving the half-range equations is that the physical boundary condition (at least in planetary science applications) is usually known for half the angular domain at both the top and the bottom of the atmosphere. These two independent boundary conditions for the first order equations for the hemispheric intensities (
string :autorefequation210
) mean there must be two additional relationships among the unknowns \cstone-\cstfour. Note that the boundary conditions themselves are still unspecified.
To obtain the relationships among \cstone-\cstfour we substitute the general solution forms (
string :autorefequation217
) into the governing equations (
string :autorefequation210
). It can be shown (after much tedious algebra) that
\csttwo
=
\rflinf \cstfour
\cstthree
=
\rflinf \cstone
where
\rflinf
1 −

 

1 − \ssa
 

 

1 +

 

1 − \ssa
 

 



 
(182)
With these additional constraints, (
string :autorefequation217
) becomes = ^ + ^-
= ^ + ^- where now only the constants \cstone and \cstfour remain to be determined by the particular boundary conditions of the problem.
The most relevant two-stream problem that is easily tractable is that of an isotropic, downwelling hemispheric radiation field of intensity \ntncal illuminating a homogeneous slab which overlies a black surface (so that upwelling intensity at the lower boundary is zero). () = 0
() =
Substituting (
string :autorefequation220
) into the solutions for the case of an isotropically scattering medium (
string :autorefequation219
) we obtain 3 () = 0 = ^ + ^-
() = = + which lead to 2 = ^- =
= ^ = It will prove convenient to define the denominators of (
string :autorefequation221b
) as D = ^ - ^-
D\strsbs = 1 - ^-2 where D\strsbs = \me−Γ\taustr D. The initial forms of \cstone and \cstfour contain both positive and negative exponentials and are pleasingly symmetrical. These forms have been divided top and bottom by \me−Γ\taustr to obtain the final forms on the RHS for reasons that will be discussed shortly.
Substituting (
string :autorefequation221
) and (
string :autorefequation222
) into (
string :autorefequation219
) we can finally write the intensities at any optical depth τ in terms of the known quantities \taustr and \ssa 2 = [ ^( - ) - ^- ( - ) ] = [ ^- - ^-2 ( - /2) ]
= [ ^( - ) - ^- ( - ) ] = [ ^- - ^-2 ( - /2) ] Two forms of the solution are presented. The initial forms on the RHS are differences between positive and negative exponential terms. In this form, the solutions may appear to depend only on the distance from the lower boundary (\taustr − τ) but notice that the D term (
string :autorefequation222a
) depends on the absolute layer thickness (\taustr) as well. However these forms are not recommended for computational implementation because the positive exponentials are difficult to handle for large optical depths. Moreover, the difference between the exponential terms will quickly lead to a loss of numerical precision as τ→ ∞.
The final forms on the RHS of (
string :autorefequation223
) contain only negative exponentials and are numerically well-behaved as τ→ ∞. Since 0 < τ < \taustr, it is clear that all exponentials in (
string :autorefequation223
) and (
string :autorefequation222a
) are negative exponentials and thus well conditioned for computational applications. There are no physical processes which would lead to positive exponential terms in the solutions, so it is always worthwhile doublechecking for accuracy or stability any formulae which contain positive exponentials.
As expected, the solutions (
string :autorefequation223
) are proportional to the incident intensity \ntncal, which is the only source of energy in the problem since we neglected the thermal source term while constructing the governing equations (
string :autorefequation210
). The upwelling radiance is proportional to the parameter \rflinf. The physical meaning of \rflinf is elucidated by noting that \ntnupw → \rflinf \ntncal as \taustr → ∞. Thus \rflinf is the maximum reflectance of semi-infinite layer of matter with the same optical properties as the layer in question.

0.0.0  Layer Optical Properties

The solutions for the intensities (
string :autorefequation223
) allow us to derive the other optical properties of the layer using (
string :autorefequation206
)-(
string :autorefequation209
). The \brdfoffrqmtwopitwopi is the ratio of reflected irradiance to all incident irradiance.
\rfloffrq
\brdfoffrqmtwopitwopi ≡ \flxupwfrq(τ = 0)

\flxdwnfrq(τ = 0)


 
2 \mpi \plrmubar \ntnupwfrq(τ = 0)

2 \mpi \plrmubar \ntndwnfrq(τ = 0)


 
= \ntnupwfrq(τ = 0)

\ntndwnfrq(τ = 0)


 
=
\rflinf \ntncal

D\strsbs


 
( 1 − \me−2Γ\taustr
\ntncal

D\strsbs


 
(1 − \rflinfsqr \me−2Γ\taustr)


 
=
\rflinf

D\strsbs


 
(1 − \me−2Γ\taustr) = \rflinf

D


 
(\meΓ\taustr − \me−Γ\taustr)
=
\rflinf(1 − \me−2Γ\taustr )

1 − \rflinfsqr \me−2Γ\taustr


 
= \rflinf(\meΓ\taustr − \me−Γ\taustr )

\meΓ\taustr − \rflinfsqr \me−Γ\taustr


 
(183)
where we have used the definition of D\strsbs (
string :autorefequation222b
) in the last step.
The \trnoffrq is the ratio of transmitted irradiance to incident irradiance. 14
\trnoffrq
\trn(\frq,−2\mpi,−2\mpi) ≡ \flxdwnfrq(τ = \taustr)

\flxdwnfrq(τ = 0)


 
2 \mpi \plrmubar \ntndwnfrq(τ = \taustr)

2 \mpi \plrmubar \ntndwnfrq(τ = 0)


 
= \ntndwnfrq(τ = \taustr)

\ntndwnfrq(τ = 0)


 
=
\ntncal

D\strsbs


 
( \me−Γ\taustr − \rflinfsqr \me−Γ\taustr
\ntncal

D\strsbs


 
( 1 − \rflinfsqr \me−2 Γ\taustr )


 
=
( 1 − \rflinfsqr ) \me−Γ\taustr

D\strsbs


 
= 1 − \rflinfsqr

D


 
=
( 1 − \rflinfsqr ) \me−Γ\taustr

1 − \rflinfsqr \me−2 Γ\taustr


 
= 1 − \rflinfsqr

\meΓ\taustr − \rflinfsqr \me− Γ\taustr


 
(184)
The first expressions in (
string :autorefequation224
) and (
string :autorefequation225
) were derived using the D\strsbs forms of the solutions (
string :autorefequation223
) and contain no positive exponential terms which blow up as \taustr → ∞. The second expressions, on the other, hand, are numerically ill-conditioned as \taustr → ∞.
Conservation of energy requires that the \absoffrq is one minus the sum of the transmittance and the reflectance
\absoffrq
\abs(\frq,−2\mpi) ≡ \flxdwnfrq(0)−\flxdwnfrq(\taustr)+\flxupwfrq(\taustr)−\flxupwfrq(0)

\flxdwnfrq(0)


 
\ntndwnfrq(0)−\ntndwnfrq(\taustr)+\ntnupwfrq(\taustr)−\ntnupwfrq(0)

\ntndwnfrq(0)


 
=
1 − \trnoffrq − \rfloffrq
(185)

0.0.0  Conservative Scattering Limit

The preceding section evaluated \rfloffrq and \trnoffrq for the entire layer for a non-conservative scattering (\ssa ≠ 1) atmosphere. When \ssa = 1 the two-stream equations take a simpler form with the result:
\rfloffrq
\brdfoffrqmtwopitwopi ≡ \ntnupwfrq(τ = 0)

\ntndwnfrq(τ = 0)


 
=
\ntncal(\taustr − τ)

2\plrmubar + \taustr


 
×
\ntncal(2\plrmubar + \taustr − τ)

2\plrmubar + \taustr


 



 
=
\taustr − τ

2\plrmubar + \taustr − τ


 
(186)
=
\taustr

2\plrmubar + \taustr


 

\trnoffrq
\trn(\frq,−2\mpi,−2\mpi) ≡ \ntndwnfrq(τ = \taustr)

\ntndwnfrq(τ = 0)


 
=
\ntncal (2\plrmubar + \taustr − τ)

2\plrmubar + \taustr


 
× 1

\ntncal


 
=
2\plrmubar + \taustr − τ

2\plrmubar + \taustr


 
(187)
=
2\plrmubar

2\plrmubar + \taustr


 

0.0  Example: Solar Heating by Uniformly Mixed Gases

In hydrostatic balance, the density of a uniformly mixed species \gasidx is
\dnsidx(\hgt)
=
\dnsidx(\hgt = 0) \me−\hgt/\hgtscl
(188)
(189)
The optical depth of a purely absorbing species with volume absorption coefficient \abscffvlm is (
string :autorefequation84
)
fxm
(190)

0.0  Exercises for Chapter 
string :autorefsection2

C
onsider a vertically homogeneous stratiform liquid water cloud extending from \zzz = 1-2 km above the surface, i.e., the cloud is 1 km thick. Assume the sun is directly overhead and let \flx(\zzz) denote the downwelling flux (in ) in the direct solar beam. At the top of the cloud \flx(\zzz = 2) = \flxslrtoa. In the middle of the cloud it is found that \flx(\zzz = 1.5) = \flxslrtoa/2.
  1. What causes \flx(\zzz) to decrease from the top to the middle of the cloud?
    Answer: Absorption and scattering by cloud droplets.
  2. What is \flx(\zzz = 1), i.e., the downwelling flux exiting the bottom of the cloud? Answer: \flxslrtoa/4
  3. What is the extinction optical depth of the cloud \tauext? Answer: ln2
  4. Assuming the cloud droplets are uniform in size, what additional information about the cloud is needed to estimate the droplet radius? Answer: The mass or number concentration of cloud droplets and their density (i.e., the density of water, 1 ). Also, the extinction efficiency \fshext must be known, but it can be assumed to be 2.

0  Remote Sensing

0.0  Rayleigh Limit

The \szprm is the ratio of particle circumference to wavelength, times the real part of the refractive index of the surrounding medium \idxrfrmdmrl
\szprm
=
\wvnbr \rds \idxrfrmdmrl
(191)
=
2 \mpi \rds \idxrfrmdmrl / \wvl
(192)
For particles in air, \idxrfrmdmrl ≈ 1 so \szprm ≈ 2\mpi\rds/\wvl. The is the regime where the wavelength is large compared to the particle size, \wvl >> \rds, or, equivalently, where \szprm → 0. Note that this regime applies equally to visible light interacting with nanoparticles, and to microwave radiation interacting with cloud droplets. The absorption and scattering efficiencies reduce to simple, closed form expressions in the Rayleigh limit _ 0 = -4   ( ^2-1 )
_ 0 = 8^4 - ^2-1 - ^2 where \im(\zzz) is the imaginary part of \zzz. We see that \fshabs > \fshsct as \szprm→ 0. Thus we may rewrite () as
\xsxextffc(\hgt)
=



 
−4
2\mpi \rds

\wvl


 



 
 \im
\idxrfr2−1

\idxrfr2+2


 



 
\mpi \rds2 \dstnbr(\rds,\hgt)  \dfr\rds
=
8\mpi2

\wvl


 
 \im
\idxrfr2−1

\idxrfr2+2


 



 



 
\rds3 \dstnbr(\rds,\hgt)  \dfr\rds
=
8\mpi2

\wvl


 
 \im
\idxrfr2−1

\idxrfr2+2


 



 
3

4 \mpi \dns


 



 
4 \mpi \dns

3


 
\rds3 \dstnbr(\rds,\hgt)  \dfr\rds
=
6\mpi

\dns \wvl


 
 \im
\idxrfr2−1

\idxrfr2+2


 



 
\mssttl(\hgt)
(193)
where in the last step we have replaced the integrand by the total mass concentration, \mssttl . If the cloud particles are liquid droplets then the density \dns and index of refraction \idxrfr are well known and \mssttl is referred to as the . Assuming the extinction by cloud droplets dominates the volume extinction coefficient then we may integrate over the thickness of the cloud to determine the total optical depth. Referring to () we have
\tauextffc
=
6\mpi

\dns \wvl


 
 \im
\idxrfr2−1

\idxrfr2+2


 



 



 
\mssttl(\hgt)  \dfr\hgt
=
6\mpi

\dns \wvl


 
 \im
\idxrfr2−1

\idxrfr2+2


 



 
\msspth
(194)
where \msspth is the path-integrated water content in the cloud, called the . It is notable that \tauextffc explicitly depends on the total condensed water mass, but not on the cloud droplet size. In contrast, (
string :autorefequation95
) shows that \tauextffc in visible wavelengths depends on both \msspth and \rds. If the microwave optical depth of the cloud is known, through any independent measurement, then we may use it to obtain \msspth by inverting (
string :autorefequation237
). Fortunately \tauextffc(\wvl) may be independently estimated through estimating the microwave of clouds.

0.0  Anomalous Diffraction Theory

The full solutions are amenable to approximations in certain limits, the Rayleigh approximation (§
string :autorefsubsection3.1
) for example. Scattering and absorption by particles with indices of refraction near unity (\idxrfr ≈ 1) may be treated with () in the regime of large \szprm >> 1. , described in ], estimates particle-radiation interaction in this regime by considering the effect refraction has on the phase delay between scattered and diffracted waves. Consider incident plane waves of the form
\lctvctcpx(\drcvct,\tm)
=
\lctvctnot exp[\mi (\wvnbrvct ·\drcvct − \frqngl \tm ) ]
=
\lctvctnot exp[\mi (\wvnbrvct ·\drcvct − \mi \frqngl \tm +\nglphz) ]
\lctvct(\drcvct,\tm)
=
\re(\lctvctcpx)
=
\lctfldnot cos(\wvnbrvct ·\drcvct − \frqngl \tm + \nglphz)
=
\lctfld0,\xxx cos(\wvnbrvct ·\drcvct − \frqngl \tm + \nglphz\xxx) \ihat +\lctfld0,\yyy cos(\wvnbrvct ·\drcvct − \frqngl \tm + \nglphz\yyy) \jhat +\lctfld0,\zzz cos(\wvnbrvct ·\drcvct − \frqngl \tm + \nglphz\zzz) \khat
(195)
where the phase angle \nglphz is an arbitrary angular offset. Of course plane waves which explicitly include \nglphz satisfy (). \nglphz is often omitted from plane wave solutions since, being a constant offset, it has no effect on the physical properties of the solution. We include \nglphz in (
string :autorefequation238
) because it will prove useful to examine the phase delay which accrues in plane waves due to particle scattering.
After particle scattering, the electric field is comprised of two waves: the incident plane wave \lctvctnot which travels unrefracted through the particle (because \idxrfr ≈ 1), and the scattered wave, \lctvctsct which has been diffracted by the edge of the particle.
\lctfld
=
\lctfldnot + \lctfldsct
(196)
We assume that rays suffer no reflection or deviations at the particle boundaries since \idxrfr ≈ 1. Thus the particle changes \lctvctsct only in phase, not in amplitude. The change in phase may be determined by considering the geometric path length through the particle. Collimated rays passing through the particle of radius \rds a distance \mpcprm (the ) from the center define chords of length 2\mpcprm sin\nglntr from the entry point to the exit point. The angle \nglntr lays between the radius to the point where the light ray enters the sphere, and the line segment (of length \mpcprm) joining the center of the sphere to the chord the ray travels through the sphere.
\nglntr
=
cos−1 \mpcprm / \rds
(197)
Thus the angle between the entry and exit points and the center of the particle is 2\nglntr. The phase lag \phzdlt is determined by the phase of the electric field measured on a plane behind the particle but normal to the propagation of the incident wave. The electric field measured within the geometric shadow of the particle has suffered a relative to the field outside the geometric shadow, which has not interacted with the particle. The path length traveled within the particle is \raycrd
\raycrd
=
2 \rds sin\nglntr
(198)
The phase lag \phzdlt suffered during particle transit is \raycrd times the difference in propagation speeds times the spatial wavenumber \wvnbr = 2\mpi / \wvl (
string :autorefequation12
). We assume the particle is suspended in air so the propagation speeds differ by a factor of \idxrfr − 1.
\phzdlt
=
\raycrd ×(\idxrfr − 1 ) ×\wvnbr
=
2 \rds sin\nglntr ×(\idxrfr − 1 ) ×\wvnbr
=
\phzdlyctr sin\nglntr
(199)
where \phzdlyctr is the phase delay suffered by a ray crossing the full diameter of the particle
\phzdlyctr
=
2 \rds (\idxrfr − 1 ) (2 \mpi / \wvl)
=
2 \szprm (\idxrfr − 1 )
(200)
The phase delay (
string :autorefequation242
) causes a radially varying reduction the electric field amplitude behind the particle, so that
\lctfldsct
=
\lctfldnot \me−\mi \phzdlt − 1
(201)
We shall determine the extinction efficiency due to the particle by examining the scattering \sctfnc(\nglsct) at the \nglsct = 0. The amplitude function \sctfnc(0) is the integral over the geometric shadow region of the ratio of the scattered wave to the incident wave.
\sctfnc(0)
=
\wvnbr2

2\mpi


 



 
\lctfldsct

\lctfldnot


 
 \dfr \xsa
Following ], we employ polar coordinates within the geometric shadow region and begin by integrating over the impact parameter (
string :autorefequation240
)
\mpcprm
=
\rds cos\nglntr
from the center of the particle to the edge. The element of area is a ring of radius \mpcprm about the center of the particle. The elemental area is 2\mpi\mpcprm \dfr\mpcprm and area
\sctfnc(0)
=
\wvnbr2

2\mpi


 



 
(1 − \me−\mi \phzdlyctr sin\nglntr)(2 \mpi \mpcprm)  \dfr \mpcprm
=
\wvnbr2


 
(1 − \me−\mi \phzdlyctr sin\nglntr) \mpcprm  \dfr \mpcprm
(202)
The first term in the integrand represents the incident wave. The exponential term which represents the scattered wave is more difficult to integrate. We now change integration variables from impact parameter to entry angle using (
string :autorefequation245
),
\mpcprm
=
\rds cos\nglntr
\nglntr
=
cos−1 (\mpcprm / \rds)
\dfr \mpcprm
=
− \rds sin\nglntr  \dfr \nglntr
The change of variables maps \mpcprm ∈ [0,\rds] to \nglntr ∈ [0,\mpi/2]. Thus
\sctfnc(0)
=
\wvnbr2


 
(1 − \me−\mi \phzdlyctr sin\nglntr)(\rds cos\nglntr)(\rds sin\nglntr)  \dfr \nglntr
=
\wvnbr2 \rds2


 
(1 − \me−\mi \phzdlyctr sin\nglntr)cos\nglntr sin\nglntr  \dfr \nglntr
\wvnbr2 \rds2 \KKK(\www)
(203)
where \www ≡ \mi \phzdlyctr. The first term in the integrand in \KKK(\www) (
string :autorefequation246
) is a simple trigonometric function. The second term may be with the formula ∫\uuu  \dfr \vvv = \uuu \vvv − ∫\vvv  \dfr \uuu where
\uuu
=
sin\nglntr
\dfr \vvv
=
\me−\www sin\nglntr cos\nglntr  \dfr \nglntr
\vvv
=
\me−\www sin\nglntr

\www


 
\dfr \uuu
=
cos\nglntr  \dfr\nglntr
(204)
Using these standard techniques we obtain
\KKK(\www)
=



 
sin\nglntr cos\nglntr  \dfr\nglntr− (\uuu \vvv −


 
\vvv  \dfr \uuu)
=



 
sin2 \nglntr

2


 
 \dfr\nglntr−
− sin\nglntr \me−\www sin\nglntr

\www


 



 
+


 
\me−\www sin\nglntr

\www


 
cos\nglntr  \dfr\nglntr
=

cos2 \nglntr

4


 



 

\me−\www

\www


 
− 0


 
+ 1

\www


 

\me−\www sin\nglntr

\www


 



 
=
−(−1)−[−(1)]

4


 
+ \me−\www

\www


 
+ \me−\www − 1

\www2


 
=
1

2


 
+ \me−\www

\www


 
+ \me−\www − 1

\www2


 
(205)
With \KKK(\www) known we can finally determine the scattering function and the extinction efficiency in the anomalous diffraction limit
\sctfnc(0)
=
\szprm2 \KKK(\www)
\fshext
=
4

\szprm2


 
 \re  [\sctfnc(0)]
=
4  \re  [\KKK(\www)]
(206)
For non-absorbing spheres, \idxrfr is real so (
string :autorefequation249
) becomes
\fshext
=
2 − 4

\phzdlyctr


 
sin\phzdlyctr + 4

\phzdlyctr2


 
( 1 − cos\phzdlyctr )
(207)
Following a similar procedure ] leads to the ADT approximation for absorption efficiency
\fshabs
=
1 + 2

\bbb


 
\me−\bbb + 2

\bbb2


 
(\me−\bbb−1)
(208)
where \bbb = 4\szprm\idxrfrimg.

0.0  Geometric Optics Approximation

] estimate the fraction of absorption due to Mie resonance effects. First, they summarize previous studies which show Mie scattering in transparent spheres results in quasi-periodic resonances with approximate period \prdszprm. For water-based liquid aerosols typical of the atmosphere, the period is of order unity in size parameter space. A more precise estimate is
\sqrtidxrfrsqrmnsone


 

\idxrfrrl2−1
 

 
(209)
\prdszprm
arctan\sqrtidxrfrsqrmnsone

\sqrtidxrfrsqrmnsone


 
(210)
] estimate the absorption due to resonance effects based on () theory. Let \sttdnsrsnfrc be the fractional contribution of resonances to the total mean density of states. In the asymptotic limit as \szprm >> 1,
\sttdnsrsnfrc
=
(\sqrtidxrfrsqrmnsone/\idxrfrrl)3,     for \szprm << 1
(211)
For , \idxrfrrl = 1.33 so \sttdnsrsnfrc  ∼ 29%. Let \fshabsrsnavg be the mean absorption efficiency due to resonances across the approximate resonance period \prdszprm. argues that, since each resonance adds to the absorption efficiency, \sttdnsrsnfrc is a good estimate of \fshabsrsnavg/\fshabs, the fractional contribution of resonances to the overall absorption efficiency.
A second estimate of \fshabsrsnavg/\fshabsrsn can be obtained from the (). A fundamental result of GOA ] is that
\fshabsgoa
=
8

3


 
\idxrfrrl2 (1−\sttdnsrsnfrc)\idxrfrimg \szprm,     for \idxrfrimg\szprm << 1
(212)
] combines (
string :autorefequation255
) with () theory to show
\fshabsrsnavg

\fshabsgoa


 
=
3

4


 
arctan\sqrtidxrfrsqrmnsone

\idxrfrrl3−\sqrtidxrfrsqrmnsone3


 


\sqrtidxrfrsqrmnsone

arctan\sqrtidxrfrsqrmnsone


 



 
−1


 
    for \szprm >> 1, \idxrfrimg << 1
(213)
For \idxrfrrl = 1.33, (
string :autorefequation256
) yields \fshabsrsnavg/\fshabsgoa  ∼ 16%, similar to the estimate based on \sttdnsrsnfrc (
string :autorefequation254
).
Dave recommended a size parameter resolution for Mie integrations of ∆\szprm ≈ 0.1. ] shows this leads to an underestimate of the absorptance of soot in water spheres by about a factor of two.

0.0  Single Scattered Intensity

In clear sky conditions it is reasonable to assume that most of the majority photons measured by Downward looking satellite instruments measure upwelling radiation reflected by the surface plus any photons scattered by the atmosphere into the satellite viewing geometry. In clear sky conditions it is reasonable to approximate the solar radiance measured by a satellite as consisting entirely of singly-scattered photons.
The upwelling intensity measured by a downward-looking instrument mounted, e.g., on an aircraft or satellite, is formally described by the exact solution for upwelling radiance presented in (
string :autorefequation105
). The full, multiple scattering source function with thermal emission \srcupwfrq (
string :autorefequation153
) is difficult to obtain and we shall instead make the . In the single-scattering approximation, we replace \srcupwfrq(τ,\nglhat) by the \srcstr(τ,\nglhat) (
string :autorefequation159
).
In the single-scattering approximation, the upwelling intensity leaving the surface consists solely of reflected photons (
string :autorefequation194
) from the direct downwelling beam (
string :autorefequation150
)
\ntnupwfrq(\taustr,\nglhat)
=



 
\ntndwndrc(\taustr,\nglhatprm) \brdfoffrqmnglhatprmnglhat cos\plrprm  \dfr\nglprm
=



 
\flxslrtoa \me−\taustr/\plrmunot \dltfncofnglhatprmmnglhatnot\brdfoffrqmnglhatprmnglhat \plrmuprm  \dfr\nglprm
=
\plrmunot \flxslrtoa \me−\taustr/\plrmunot \brdfoffrqmnglhatnotnglhat
(214)

0.0  Satellite Orbits

Assume a satellite is in a circular orbit about Earth so that its Cartesian position \psnbld at radial distance \rdl as a function of time \tm is
\psnbld
=
\rdl (sin\vlcngl \tm \ihat + cos\vlcngl \tm \jhat)
(215)
where \vlcngl is the . As its name implies, \vlcngl measures the rapidity with which the angular coordinate changes (in radians per second, denoted ). Let \tauorb denote the of the satellite. We assume the orbital speed \vlc of a satellite in a circular orbit is a constant. Circular geometry dictates that
\tauorb
=
2 \mpi \rds / \vlc
(216)
By definition
\vlcngl
=
2\mpi \tauorb−1
=
2\mpi
2\mpi \rdl

\vlc


 



 
=
\vlc / \rdl
(217)
The acceleration \xclbld of the satellite is the second derivative of the position.
\vlcbld
=
\rdl \vlcngl (cos\vlcngl \tm \ihat − sin\vlcngl \tm \jhat)
(218)
\xclbld
=
− \rdl \vlcngl2 (sin\vlcngl \tm \ihat + cos\vlcngl\tm \jhat)
(219)
=
− \vlcngl2 \psnbld
(220)
It is easy to show that \vlcbld ·\psnbld = 0 because they are orthogonal. Thus the velocity of an object in a circular trajectory is tangential to the radial vector. In plane polar coordinates, \psnbld = \rdl \rdlhat so
\xclbld
=
− \rdl \vlcngl2 \rdlhat
(221)
Moreover, (
string :autorefequation264
) shows that the acceleration of an object in a circular trajectory is opposite in direction to the radial vector.
Since the orbit is circular, the magnitudes of \psnbld, \vlcbld, and \xclbld are all constant with time and only their direction changes. This can be verified by computing the vector magnitudes, e.g.,
\xcl ≡ |\xclbld|
=


 

\xclbld ·\xclbld
 

 
=
\rdl \vlcngl2

 

sin2 \vlcngl \tm + cos2 \vlcngl \tm
 

 
=
\rdl \vlcngl2
(222)
\xcl is known as the . Substituting (
string :autorefequation260
) into (
string :autorefequation265
) we obtain
\xcl
=
\rdl (\vlc / \rdl)2
=
\vlc2 / \rdl
\xclbld
=
\vlc2

\rdl


 
\rdlhat
(223)
According to (), the gravitational force \frcbld () holding the satellite in orbit must exactly balance (
string :autorefequation266
). Let \mssstl and \mssrth denote the mass of the satellite and of Earth, respectively. Then
\frcbld
=
\mssstl \xclbld
\cstgrv \mssstl \mssrth

\rdl2


 
\rdlhat
=
− \mssstl \vlc2

\rdl


 
\rdlhat
\cstgrv \mssstl \mssrth

\rdl2


 
=
\mssstl \vlc2

\rdl


 
\cstgrv \mssrth

\rdl2


 
=
\vlc2

\rdl


 
\vlc2
=
\cstgrv \mssrth

\rdl


 
\vlc
=
  ⎛


\cstgrv \mssrth

\rdl


 
 


 
(224)
We note that \rdl is the distance from Earth's center of mass, not its surface, to the satellite. Hence satellite's speed increases slowly as its distance from Earth decreases. The orbital period \tauorb is determined by substituting (224) into (
string :autorefequation259
)
\tauorb
=
2 \mpi \rds   ⎛


\rdl

\cstgrv \mssrth
 


 


 
\tauorb2
=
4 \mpi2 \rds3

\cstgrv \mssrth


 
(225)
Hence the satellite period squared is proportional to the cube of the orbital size. This relationship is known as in honor of its discoverer, the astronomer Johannes Kepler.

0.0  Aerosol Characterization

The wavelength and size dependence of aerosol optical properties are generally well-predicted if the aerosol composition and shape are known. However, in situ aerosol composition and size are often difficult or impossible to obtain in the field. Thus indirect estimates of particle size and composition are often derived from the measurements that are available. Many field sites and experiments are equipped to measure the spectrally resolved, column-integrated, aerosol extinction optical depth \tauaerext(\wvliii) where \iii denotes the spectral channel. The following sections discuss techniques for inferring aerosol optical depth from surface measurements, and the subsequent uses of \tauaerext(\wvliii) to calibrate radiometry, and to infer information about aerosol size distribution and composition.

0.0.0  Measuring Aerosol Optical Depth

Surface radiometry is incapable of directly measuring (AOD)15 because aerosol is vertically distributed in the atmosphere. We now describe how AOD is inferred from surface measurements. This will enable us to qualify what is usually meant by the phrase "measured optical depth" as it appears in the literature. Upward pointing radiometers may measure one or more parts of the radiance field. Downwelling total solar irradiance \flxdwn is traditionally measured by . The downwelling direct beam irradiance \flxdwndrc is most accurately measured by a (NIP). Downwelling diffuse irradiance \flxdwndff is measured by . Instruments that measure all three irradiances also exist. A class of instruments called do this by periodically blocking the detector from the direct solar beam using an occluding device called a shadowband.

0.0.0  Aerosol Indirect Effects on Climate

Greater than the direct radiative effect of aerosols on climate is the uncertainty are the effects on climate of changes in cloud due to aerosols. These changes are collectively known as (AIE). The first AIE is the or described by ]. The second AIE is the so-called or . Other AIEs include the .
Dust may suppress precipitation ,] including tropical storms and hurricanes 0.0cm, [-0.0cm] or enhance precipitation ,].

0.0.0  Aerosol Effects on Snow and Ice Albedo

Snow and ice always contain a number of particulate and bubble inclusions. The effect on reflectance is non-negligible when the particles are substantially darker than the snow or ice. ] and ] show that Mie theory is an appropriate, but not exact, tool with which to model these effects. ] present updated data on these effects.

0.0.0  Ångström Exponent

The sensitivity of the particle extinction efficiency to wavelength i.e., |∂\fshext(\rds,\wvl)/∂\wvl|, generally increases with decreasing particle size. This sensitivity holds true any paritcle composition, and may be easily demonstrated using Mie codes. An empirical measure of this sensitivity is obtained by defining the or \angxpn.
\angxpn
=
ln[ \tauext(\wvlone) / \tauext(\wvltwo) ]

ln( \wvlone / \wvltwo )


 
(226)
Typical values of \angxpn are \angxpn > 2 for small carbonaceous aerosols (smoke) and urban pollution (sulfate) to \angxpn ≈ 0 for large dust particles.

0  Gaseous Absorption

0.0  Line Shape

Three fundamental processes interact to determine the observed shape of spectral lines. These are the natural line width, pressure broadening, and Doppler broadening. , which arises from the Heisenberg uncertainty principle, is unimportant in planetary atmospheres. Pressure- and Doppler-broadening effects dominated line shapes here. However, the processes determining natural line widths and of lines are statistically identical, so these two processes have the same analytical form. In planetary atmospheres, is intermediate in importance between natural line shape and or . At pressures weaker than about 1  collisions become infrequent enough that the line shapes transition from Lorentzian to Doppler. In this transitional regime the line shape is a convolution of Lorentzian and Doppler shapes. This convolution is known as the .

0.0.0  Line Shape Factor

The fundamental property of a discrete line transition is the , \xsxabsoffrq, which measures the strength of the transition as a function of frequency (or wavelength, etc.). As discussed in Section 
string :autorefsubsubsection2.1.13
, the units of \xsxabsoffrq depend on the application, and they must be consistent with the units of the absorber path, \abspth. The absorber mass path is integral of the absorber amount \abscnc along the path.
\abspth(\pnt1,\pnt2) =


 
\abscnc(\pth)  \dfr\pth
(227)
Thus, if \xsxabsoffrq is expressed in units of , then \abspth and \dnsabs should be expressed in units of  and , respectively. We adopt the convention that, when referrring to gaseous absorption, \xsxabsoffrq is the and is expressed in . Closely related is the \abscffvlm []
\abscffvlmoffrq
=
\xsxabsoffrq \cncttl
(228)
where \cncttl is the number concentration [] of the species.
The value of \xsxabsoffrq depends on the absorber type. For a single spectral line, \xsxabsoffrq is obtained by converting the parameters in the compilation ] to the local temperature and pressure. For a continuum absorption process, e.g.,  Chappuis bands, \xsxabsoffrq is usually obtained from a standard table (which may include a parameterized temperature dependence). The JPL compilation ] is the standard reference for absorption cross sections of photochemical species.
Given these definitions, the optical depth due to absorption between points \pnt1 and \pnt2 is
\tauabs(\pnt1,\pnt2)
=



 
\xsxabsoffrq\abscnc(\pth)  \dfr\pth
=
\xsxabsoffrq


 
\abscnc(\pth)  \dfr\pth
=
\xsxabsoffrq \abspth(\pnt1,\pnt2)
(229)
Comparing this to (
string :autorefequation82
) we see that \xsxabsoffrq is the extinction due to absorption.
The integrated strength of a transition is called the or and is denoted by \lnstr
\lnstr =


 
\xsxabsoffrq  \dfr\frq
(230)
The dimensions of \lnstr are thus the dimensions of the absorption coefficient times the dimensions of frequency (or wavelength), e.g.,  or . The relation between \xsxabsoffrq and the line strength is called the \lnshpofdltfrq =
= /
= / where \frqnot is the central frequency of the unperturbed transition. Thus \lnshpofdltfrq relates the frequency-integrated absorption amount (i.e., line strength) to the specific absorption a distance \frq −\frqnot from the line center. The shape factor for each transition (i.e., line) is normalized to unity



 
\lnshpofdltfrq  \dfr\frq
=
1
(231)
The dimensions of \lnshp are inverse frequency or wavelength, e.g., , , or .
In this section we do not need to specify the which set of self-consistent dimensions we choose for \abscnc, \abspth, and \xsxabsoffrq. Nevertheless, for concreteness we list the three most common choices. First, we may employ dimensions relative to absorber number concentration

\abscnc
=
\nbrcnc       \mlcxmC
\abspth
=
\nbrpth       \mlcxmS
\xsxabsoffrq
=
\abscffvlmoffrq       \mSxmlc



(232)
where \abscffvlmoffrq is a . Alternatively, we may employ dimensions relative to absorber mass concentration

\abscnc
=
\msscnc       \kgxmC
\abspth
=
\msspth       \kgxmS
\xsxabsoffrq
=
\abscffmssoffrq       \mSxkg

where \abscffmssoffrq is a .

(233)
Finally, we may employ dimensions relative to absorber number

\abscnc
=
\msscnc       \kgxmC
\abspth
=
\msspth       \kgxmS
\xsxabsoffrq
=
\abscffmssoffrq       \mSxkg

where \abscffmssoffrq is a .

(234)
We prefer to work in units absorption per unit mass (233) because, at least in models, the mass of absorbers is more readily available than the number of absorbers.

0.0.0  Natural Line Shape

Radiative transitions from an upper to a lower state may occur spontaneously, in a process known as or . The time intervals \tm between spontaneous emission are described by a Poisson distribution
\pdfpss(\tm)
=
1

\tauxct


 
\me−\tm / \tauxct
(235)
where \tauxct is the mean lifetime of the excited state.
The time-dependent Schroedinger equation predicts that the relation between the time intervals between such decays and the energy of the decays gives rise to a continuous, rather than discrete, profile of absorption. The resulting profile is called the and is described by
\lnshpntr(\frq−\frqnot)
=
\hwhmntr

\mpi [ (\frq − \frqnot)2 + \hwhmntr2 ]


 
(236)
where \hwhmntr is the natural line shape half width at half-maximum (HWHM). This is seen by noting that \lnshpntr(\frq = \frqnot) = (\mpi\hwhmntr)−1 is the full maximum value of \lnshpntr(\frq), while \lnshpntr(\frq = \frqnot + \hwhmntr) = (2\mpi\hwhmntr)−1.
Natural broadening is one of two important broadening processes that are described by the , \lnshplrnofdltfrq. The Lorentz line shape is defined identically to (236) as
\lnshplrnofdltfrq
=
\hwhmlrn

\mpi [ (\frq − \frqnot)2 + \hwhmlrn2 ]


 
(237)
The Lorentz profile (237) may also be written in terms of the full width at half-maximum \fwhmlrn = 2\hwhmntr as
\lnshplrnofdltfrq
=
\fwhmlrn/2

\mpi [ (\frq − \frqnot)2 +(\fwhmlrn/2)2 ]


 
(238)
The HWHM of a Lorentz profile is often written in terms of a parameter \Hwhmlrn such that
\lnshplrnofdltfrq
=
\Hwhmlrn/4\mpi

\mpi [ (\frq − \frqnot)2 +(\Hwhmlrn/4\mpi)2 ]


 
(239)
The full meaning of \Hwhmlrn is described below in §.
The Lorentz line shape describes the relative absorption as a function of distance from line center. A separate parameter, the line strength, \lnstr (230), defines the abolute absorption. Thus \lnshplrnofdltfrq must be normalized such that its integral over all frequencies is unity.



 
\lnshplrnofdltfrq  \dfr\frq
=



 
\hwhmntr

\mpi [ (\frq − \frqnot)2 + \hwhmntr2 ]


 
 \dfr\frq
=
\hwhmntr

\mpi


 



 
1

(\frq − \frqnot)2 + \hwhmntr2


 
 \dfr\frq
=
\hwhmntr

\mpi


 
1

1/\hwhmntr2


 
1/\hwhmntr2

1


 



 
1

(\frq − \frqnot)2 + \hwhmntr2


 
 \dfr\frq
=
1

\mpi \hwhmntr


 



 
1

1

\hwhmntr2


 
(\frq − \frqnot)2 + 1



 
 \dfr\frq
We now make the change of variable
\yyy
=
(\frq − \frqnot)/\hwhmntr
\frq
=
\frqnot + \hwhmntr \yyy
\dfr\yyy
=
\dfr\frq / \hwhmntr
\dfr\frq
=
\hwhmntr  \dfr\yyy
This change of variables maps \frq ∈ [0,∞) to \yyy ∈ [−\frqnot/\hwhmntr,∞). Since \frqnot >> 1, and the integrand is peaked near the origin, we will replace the lower limit of integration by −∞.



 
\lnshplrnofdltfrq  \dfr\frq
=
1

\mpi \hwhmntr


 



 
1

\yyy2 + 1


 
\hwhmntr  \dfr\yyy
=
1

\mpi


 



 
1

\yyy2 + 1


 
 \dfr\yyy
=
1

\mpi


 



 
tan−1 \yyy


 
=
1

\mpi


 

\mpi

2


 

\mpi

2


 



 



 
=
1
(240)
Thus the Lorentz line shape (237) is correctly normalized.
Some insight into the origin of natural broadening may be obtained from simple examination of the Heisenberg uncertainty principle which may be expressed as the fundamental uncertainty relating two conjugate coordinates such as position and momentum, or, in our case, time and energy:
\dlttm \dltnrg
 ∼ 
ħ
(241)
When a molecule is in an excited state it has a probability of spontaneously decaying to the lower state. Its actual state is uncertain on the timescale \dlttm, during which it may be in a superposition of allowed states. Corresponding to this uncertainty, then, is an uncertainty in energy \dltnrg = \cstplk \dltfrq so that
\dlttm  \cstplk \dltfrq
 ∼ 
\cstplk/2\mpi
\dltfrq
 ∼ 
1

2\mpi \dlttm


 
(242)
If we identify \dlttm with \tauxct, the mean lifetime of the excited state, and \dltfrq with \hwhmntr, the natural half width, then
\hwhmntr
=
1

2 \mpi \tauxct


 
(243)
For spontaneous emission, \tauxct is the reciprocal of the Einstein \AAA coefficient ]. A typical value of \tauxct is 10−8 s. This is an extremely long time relative to the typical mean time between (optical) collisions in a planetary atmosphere, order 10−10 s. In practice, therefore, the line shapes we are concerned with are dominated by collisional effects.

0.0.0  Pressure Broadening

Collisions between molecules are the most important cause of line broadening in the lower atmosphere. Statistically, collisions in a gas of molecules of uniform speeds take place at random time intervals about a mean value. As with spontaneous decay, the probability distribution of time intervals between collisions is therefore a (235). In reality, molecular velocities are not uniform but obey the
\pdfmxw(\vlc)  \dfr\vlc
=

\mss

2\mpi \cstblt \tpt


 



 
exp
\mss \vlc2

2 \cstblt \tpt


 



 
 \dfr\vlc
(244)
\pdfmxw(\vlc) is the probability that a molecule of mass \mss at temperature \tpt has a velocity in [\vlc,\vlc+\dfr\vlc]. In all other respects, collision-induced line broadening, or of lines is equivalent to (236) and is thus governed by a Lorentz line shape (237).
\lnshpprs(\frq−\frqnot)
=
\hwhmprs

\mpi [ (\frq − \frqnot)2 + \hwhmprs2 ]


 
(245)
where the half width at half-maximum intensity due to pressure (collision) broadening, \hwhmprs is called the . Because \hwhmprs charaterizes collisional interactions, its value depends on the concentrations and masses of all the molecular species which are present. Thus its determination is somewhat involved.
Molecular collisions occur in two senses, the kinetic and the optical. A collision might change the kinetic energy of one or both of the molecules, but this is not neccessary. Collisions which do alter the kinetic energy of one or both of the particpants are . Collisions may also cause a radiative excitation or de-excitation of the radiative energy levels of either or both of the molecules involved. Collisional de-excitation, for example, might release one vibrational quanta of energy without changing the kinetic energy of either of the molecules. Collisions which alter the rotational or vibrational quanta of one or both of the particpants are called . Rotational energy levels are, in general, much smaller than translational energy levels and thus optical collisions may occur more frequently than kinetic collisions.
Many methods of determining the pressure broadened line shape have been proposed. A particularly satisfying model is that of molecules radiating with constant intensity between elastic collisions that change the phase of the emitted wavetrain randomly. The radiation is a delta-function in frequency with harmonic time dependence
\fnc(\tm)
=
\me\mi \frqnot \tm
(246)
\frqnot is the center of the unbroadened line. We may restate (246) in frequency space by taking its Fourier transform
\Fnc(\frq)



 
\fnc(\tm) \me−\mi \frq \tm  \dfr\tm
(247)
=



 
\me\mi \frqnot \tm \me−\mi \frq \tm  \dfr\tm
=

\me\mi (\frqnot − \frq) \tm

\mi (\frqnot − \frq)


 

=
\me\mi (\frqnot − \frq) \taucll

\mi (\frqnot − \frq)
(248)
The phase of the oscillation, and thus of \fnc(\tm) is reset randomly by each collision. The waiting period between collisions determines the length of each wavetrain. The probability of a collision occuring between time \tm and \tm + \dfr\tm follows a
\pdfpss(\tm)  \dfr\tm
=
\me\tm/\taucll  \dfr\tm
(249)
Let \taucll and \tauopt denote the mean times between kinetic collisions and optical collisions, respectively. By definition the mean collision frequency \frqcll is the inverse of \taucll. The relationship between distance, rate, and time may be formulated to obtain \taucll
\taucll
=
\frqcll−1 = \mfp

\vlcmlcavg
(250)
where \vlcmlcavg is the and \mfp is the , both of which are known properties of the thermodynamic state of the atmosphere. First, let us recall that prescribe a square-root dependence of \vlcmlcavg on \tpt
\vlcmlcavgA
=
  ⎛


8 \gascst \tpt

\mpi
 
=   ⎛


8 \gascstunv \tpt

\mpi \mmw
 
(251)
where \mmw is the mean molecular weight of the gas, \gascstunv = \mmw \gascst is the universal gas constant, \gascst is the specific gas constant, and \tpt is the ambient temperature. Kinetic theory tells us []p. 457]Sep97 that the mean free path \mfpAA of molecular species \AAA in a gas of molecular species \AAA is inversely related to the total concentration of molecules and to their cross-sectional area for collisions, \mpi\dmtcllA2/4
\mfpAA
=
1

\mpi √2 \cncA \dmtcllA2
(252)
=
\mmw

\mpi √2 \dns \cstAvagadro \dmtcllA2
=
\mmw \gascst \tpt

\mpi √2 \prs \cstAvagadro \dmtcllA2
=
\gascstunv \tpt

\mpi √2 \prs \cstAvagadro \dmtcllA2
(253)
where16 we have re-expressed \mfpAA in terms of \prs and \tpt by using \cnc = \dns \cstAvagadro / \mmw and then applying the ideal gas law \dns = \prs / (\gascst \tpt) () to (252). There are many subtle assumptions embedded in (252) that should be clarified before proceeding: First is that \mfpAA is the mean free path of molecular species \A in a gas of molecular species \A, i.e., of a homogeneous gas of \A. Often \A is a trace species (e.g., ) in air, represented by species \B. The mean free path of \A in \B incorporates properties of both \A and \B []p. 457]Sep97:
The pressure of an inhomogeneous gaseous mixture of \A and \B is due, mainly, to the presence of the bulk medium which, for our purposes is air represented by species \B. Although the temperatures \tpt of both gases in a mixture are equal in thermodynamic equilibrium, their densities and partial pressures are unequal. Hence the expressions for pressure-broadened line shapes of \A will contain molecular properties of \A as well as bulk thermodynamics state (temperature, pressure) due mainly to \B.
For a homogeneous gas of \A, substituting (251) and (253) into (250) yields
\hwhmprs
=
\frqcll = \vlcmlcavg

\mfp
=
  ⎛


8 \gascstunv \tpt

\mpi \mmwA
 
× \mpi √2 \prs \cstAvagadro \dmtcllA2

\gascstunv \tpt
=
  ⎛


8 \gascstunv \tpt

\mpi \mmwA
 
× \mpi √2 \prs \cstAvagadro \dmtcllA2

\gascstunv \tpt
=
4 \cstAvagadro \dmtcllA2   ⎛


\mpi

\gascstunv \mmwA
 
\prs




\tpt
=
4 \cstAvagadro \dmtcllA2

\mmwA
  ⎛


\mpi

\gascstA
 
\prs




\tpt
(254)
We see the interesting result that \hwhmprs decreases as \tpt increases. This is because although thermal speed increases as √{\tpt } (251), the mean free path \mfp between collisions increases linearly as \tpt (253). The net result of increasing \tpt is therefore to decrease collision frequency \frqcll, and \hwhmprs (254).
If we define
\hwhmprsnot
4 \cstAvagadro \dmtcllA2   ⎛


\mpi

\gascstunv \mmwA
 
\prsnot




\tptnot
4 \cstAvagadro \dmtcllA2

\mmwA
  ⎛


\mpi

\gascstA
 
\prsnot




\tptnot
(255)
then
\hwhmprs(\prs,\tpt)
=
\hwhmprsnot \prs

\prsnot

\tptnot

\tpt



 
(256)
where \hwhmxpn = [1/2]. The exponent \hwhmxpn determining the temperature dependence in (256) is called the . Two of the key line parameters measured in laboratory experiments are \hwhmprsnot and \hwhmxpn. Both are tabulated in databases such as (§). Table  shows the mean, median, and range of the temperature-dependent exponent \hwhmxpn for the pressure-broadened half width \hwhmprs of various optically active gases.
Table 5: Temperature Dependence of \hwhmprs17
Molecule Min Mean Median Max
0.28 0.66 0.97
0.49 0.75 0.78
0.76 0.76 0.76 0.76
0.64 0.78 0.82
0.75 0.75 0.75
0.63 0.71 0.74
0.50 0.66 0.66
0.50 0.60 0.75
For most optically active gases, 0.6 < \hwhmxpnavg < 0.8. This range differs considerably from \hwhmxpn = [1/2] derived in (256), which is known as the of \hwhmxpn. This discrepancy arises becauses the optical cross section \xsxopt of the molecules have a temperature dependence.
Table  shows the mean, median, and range of the pressure-broadened half width \hwhmprs of various optically active gases.
Table 6: Pressure-Broadened Half Widths18
Molecule Min Mean Median Max
0.0077 0.071 0.11
0.055 0.071 0.095
0.049 0.069 0.084
0.069 0.075 0.097
0.018 0.054 0.16
0.028 0.043 0.060
0.040 0.044 0.095
0.10 0.11 0.15
Pure kinetic theory must be combined with the \xsxopt to determine \hwhmprs19
\hwhmprs
=


 
\cnc\iii2

2 \cstblt \tpt

\mpi


 



 

1

\mss


 
+ 1

\mss\iii


 



 



 
(257)
where the summation is over all perturber species \iii (i.e., , , ...) and \mss is the mass of the absorber. Thus \hwhmprs depends quadratically (fxm: typo, must be linear) upon the number density of collision partners and linearly upon the velocity of the molecules (note the factor of √{\cstblt \tpt }). A scaling approximation is therefore frequently used to extrapolate \hwhmprs(\tpt) based on some measured or tabulated value \hwhmprs(\tptnot)
\hwhmprs(\tpt,\prs)
\hwhmprs(\tptnot,\prsnot) \prs

\prsnot


 

\tpt

\tptnot


 



 
(258)
Atmospheric pressure changes by three orders of magnitude, from 1-1000 mb, from 50 km to the surface. Temperature changes by only a factor of 2 over the same altitude range. Since line widths (258) depend linearly on \prs, but only on the square root of \tpt, pressure variation dominates vertical changes in \hwhmprs.
The mean time between optical collisions \tauopt is then related to \hwhmprs by
\tauopt
=
1

2 \mpi \hwhmprs


 
\hwhmprs
=
1

2 \mpi \tauopt


 
\hwhmprs
=
\frqcll

2 \mpi


 
(259)
This is exactly analogous to the relation between \tauxct and \hwhmntr (243) because both processes are described by the Lorentz line shape.
When natural and collision broadening are considered simultaneously, it can be shown that the combined line shape is also a Lorentzian with HWHM \Hwhmlrn/4\mpi (239). The parameter \Hwhmlrn is related to both the natural line width and the mean frequency of collisions \frqcll = \taucll−1 (259) by
\Hwhmlrn
=
4\mpi\hwhmntr + 2 ν
(260)
These relations may be proved by examining the power spectrum of a sinusoidal electric field which is randomly interrupted \frqcll times per second. In practice in the lower atmosphere \hwhmprs >> \hwhmntr so the Lorentz line width is nearly equivalent to the pressure-broadened line width. Therefore, and the will hereafter refer to the Lorentzian line shape (237) due to the convolution of natural and collision broadening. With reference to (260) and (259)
\hwhmlrn
=
\Hwhmlrn/4\mpi
\hwhmlrn
=
\hwhmntr + \frqcll / 2\mpi
\hwhmlrn
=
\hwhmntr + \hwhmprs
(261)
When discussing a given transition, it is convenient to translate the origin of the frequency axis to the line center. Rather than defining a new variable to do this, we continue to use \frq. Thus we replace \frq − \frqnot by \frq alone. The intended meaning of \frq should be clear from the context. Using this notation (237) becomes
\lnshplrnoffrq
=
\hwhmlrn

\mpi(\frq2 + \hwhmlrn2)


 
(262)
The absorption cross-section (0.0.1) for Lorentzian lines may now be written
\xsxabsoffrq
=
\lnstr \hwhmlrn

\mpi(\frq2 + \hwhmlrn2)


 
(263)

0.0.0  Doppler Broadening

Brownian motion causes molecules to constantly change their velocity relative to the photons which may interact with them. While the frequencies of resonant absorption are constant in the frame of motion of the molecule, this random motion broadens the range of resonant frequencies in the frame of a stationary observer. This form of line broadening is called . The probability that the molecule and a stationary reference frame (an "observer") have relative velocity in [\vlc,\vlc+\dfr\vlc] is given by the Maxwell distribution (244)
\pdfmxw(\vlc)  \dfr\vlc
=

\mss

2\mpi \cstblt \tpt


 



 
exp
\mss \vlc2

2 \cstblt \tpt


 



 
 \dfr\vlc
(264)
We convert this Maxwellian PDF directly to the PDF \lnshpdpp(\frq − \frqnot) by noting that each value of \vlc describes a shift in line center \dltfrq ≡ \frq − \frqnot. For \vlc/\cstspdlgt << 1, the relation between \vlc and the Doppler shift \dltfrq is
\dltfrq
=
\frqnot \vlc / \cstspdlgt
\vlc
=
\cstspdlgt \dltfrq / \frqnot
\dfr\frq
=
\dfr\dltfrq = \frqnot

\cstspdlgt


 
 \dfr\vlc
\dfr\vlc
=
\cstspdlgt

\frqnot


 
 \dfr\frq
When we re-express \pdfmxw(\vlc) (264) in terms of \dltfrq by imposing the condition
\pdfmxw(\vlc)  \dfr\vlc
=
\pdfmxw(\dltfrq)  \dfr\dltfrq
(265)
This relation between \pdfmxw(\vlc) and \pdfmxw(\dltfrq) is like that between \plkwvl(\wvl) and \plkfrq(\frq) (
string :autorefequation42
). Inserting (265) into (264) leads to
\pdfmxw[\vlc(\frq)]  \dfr\vlc
=

\mss

2\mpi \cstblt \tpt


 



 
exp
\mss \cstspdlgt2 \dltfrq2

2 \frqnot2 \cstblt\tpt


 



 
  \dfr\vlc

\dfr\frq


 
 \dfr\frq
=
\cstspdlgt

\frqnot


 

\mss

2\mpi \cstblt \tpt


 



 
exp
− \dltfrq2
2 \frqnot2 \cstblt \tpt

\mss \cstspdlgt2


 



 



 
 \dfr\frq
(266)
The RHS of (266) is now strictly a function of \frq. We name this function the , \lnshpdpp(\frq) and define the \hwemdpp as the square root of the portion of the exponential in parentheses
\hwemdpp
=
\frqnot

\cstspdlgt


 
  ⎛


2 \cstblt \tpt

\mss


 
 


 
(267)
so that (266) becomes
\lnshpdpp(\frq)  \dfr\frq
=
\cstspdlgt

\frqnot


 

\mss

2\mpi \cstblt \tpt


 



 
exp
\dltfrq2

\hwemdpp2


 



 
 \dfr\frq
=
1



 

\mpi
 

 


 
\cstspdlgt

\frqnot


 

\mss

2 \cstblt \tpt


 



 
exp
\dltfrq2

\hwemdpp2


 



 
 \dfr\frq
=
1

\hwemdpp

 

\mpi
 

 


 
exp

\frq−\frqnot

\hwemdpp


 



 



 
 \dfr\frq
(268)
The normalization of (268) follows from the fact that it is simply a re-expression of the Maxwell distribution (264), which is already known to be normalized. The full maximum value of \lnshpdpp is \lnshpdpp(\frq = \frqnot) = (\hwemdpp √{\mpi })−1, and \lnshpdpp(\frq = \frqnot + \hwemdpp) = (\me \hwemdpp√{\mpi })−1, so that \hwemdpp is the half width at \me−1 of the maximum of \lnshpdpp. Care should be taken not to confuse \hwemdpp with a half-width at half-maximum. The half width at half-maximum of \lnshpdpp(\frq) is
\hwhmdpp
=
\hwemdpp

 

ln2
 

 
(269)
It is more appropriate to compare \hwhmlrn (255) to \hwhmdpp (269), since they are identical measures of line width, than to \hwemdpp (267).
We recognize that (268) is a form of , although it is not expressed in the canonical form of a Gaussian
\pdfgss(\frq)  \dfr\frq
=
1

\stddvn

 

2\mpi
 

 


 
exp

\frq−\frqnot

\stddvn √2


 



 



 
 \dfr\frq
(270)
in which \stddvn represents the standard deviation and \frqnot the mean value. Comparing (270) to (268) we see that the standard deviation of the Doppler line shape \stddvndpp = \hwemdpp/√2. Thus Doppler broadening alone allows absorptions to occur within \hwemdpp/√2 and √2\hwemdpp of line center with efficiencies (relative to line center) of 1−\me−1 = 0.683, and 1−\me−2 = 0.954, respectively.

0.0.0  Voigt Line Shape

Line broadening processes occur simultaneously in nature. In particular, occurs in tandem with . If we assume these two processes are independent but occur simulateously, then the net line shape of the total process will be the collision-broadened line shape, shifted by the Doppler shift (265), and averaged over the Maxwell distribution (264). The resulting line shape is call the , \lnshpvgt(\frq).
\lnshpvgt(\frq−\frqnot)
=



 

\mss

2\mpi \cstblt \tpt


 



 
exp
\mss \vlc2

2 \cstblt \tpt


 



 
\hwhmlrn

\mpi [ (\frq − \frqnot + \frqnot \vlc / \cstspdlgt )2 + \hwhmlrn2 ]


 
 \dfr\vlc
(271)
=
  ⎛


\mss

2\mpi \cstblt \tpt


 
 


 



 
exp
\mss \vlc2

2 \cstblt \tpt


 



 
\hwhmlrn

\mpi [ (\frq − \frqnot + \frqnot \vlc / \cstspdlgt )2 + \hwhmlrn2 ]


 
 \dfr\vlc
(272)
Analytic approximations to and asymptotic behavior of \lnshpvgt(\frq−\frqnot) are discussed in ,].
With appropriate definitions
\ttt
=


 

ln2
 

 
(\frq − \frqprm)/\hwemdpp
\xxx
=


 

ln2
 

 
(\frq − \frqnot)/\hwemdpp
\hwhmdpp
=


 

ln2
 

 
\hwemdpp
\yyy
=


 

ln2
 

 
\hwhmprs / \hwhmdpp
(273)
we may re-express (272) as
\lnshpvgt(\frq−\frqnot)
=
1

\hwhmdpp


 
  ⎛


ln2

\mpi


 
 


 
\KKK(\xxx,\yyy)
(274)
where the \KKK(\xxx,\yyy) is defined
\KKK(\xxx,\yyy)
=
\yyy

\mpi


 



 
\me−\ttt2

\yyy2+(\xxx−\ttt)2


 
 \dfr\ttt
(275)
Efficient algorithms for evaluating \KKK(\xxx,\yyy) have been developed ,,] to reduce computational expense in detailed line-by-line calculations.

0  Molecular Absorption

We now attempt to develop an introductory understanding of the location of absorption lines in gases. The science of atomic and molecular is extremely detailed and requires a quantum mechanical treatment for a satisfactory explanation of all phenomenon. Nevertheless, important insights as to the spectral location, temperature dependence, and relative population of radiative energy levels may be gained by resorting to a semi-classical treatment of the radiation field. In essence, our first task is to characterize the distribution of observed molecular transitions in the atmosphere, i.e., to characterize \dltnrg from Planck's relation
\dltnrg
=
\cstplk \frq
(276)
To begin, we enumerate the contributions to a molecule's total energy \nrgttl. The total molecular energy comprises the ngrtrn, the \nrglct, the \nrgvbr, the \nrgrtt, and the \nrgncl.
\nrgttl
=
\nrgtrn + \nrglct + \nrgvbr + \nrgrtt + \nrgncl
(277)
These energy components have been listed in order of decreasing magnitude. We are mainly interested in describing vibrational and rotational energy transitions in the next sections.

0.0  Mechanical Analogues

In the quantum mechanical view, radiative absorption occurs when an incident photon of energy level \cstplk \frq encounters a molecule with energy \nrg1. If \nrg1 + \cstplk \frq is "close enough" to an available energy state \nrg2 of the molecule, then absorption may occur. If absorption does occur, the energy of the photon is transferred to to the corresponding energetic mode or modes of the molecule. These molecular modes, e.g., spin, vibration, rotation, take their names from mechanical systems which form their classical analogues. To understand the distribution of energy levels in these modes it is thus useful to describe classical mechanical analogues which are known to behave like molecular systems.
Molecules may be thought of as rotating structures. When the molecular structure is fixed, i.e., the separation between the atoms does not change, we call the molecule obeys the model. Diatomic molecules, for example, may be visualized as barbell-shaped, with an atom at each end, separated by a massless but rigid rod. The separation of the atoms means their charges are separated. If the center of mass between the atoms does not coincide with the center of charge, then the molecule has a permanent . Molecules comprised of a single element, e.g.,  or , are called . Homonuclear molecules are perfectly symmetric rigid rotators, and thus have no permanent dipole moment.
To continue with the quantum mechanical behavior of the rigid rotator molecule, we must determine the available energy states of the system. Let the atoms of masses \mss1 and \mss2 be separated by a distance \rds. Then the distances of each from the center of mass of the system are
\rds1
=
\mss1 \rds

\mss1 + \mss2


 
= \mssrdc

\mss2


 
\rds
(278)
\rds2
=
\mss2 \rds

\mss1 + \mss2


 
= \mssrdc

\mss1


 
\rds
(279)
where the \mssrdc of the system is
\mssrdc
\mss1 \mss2

\mss1 + \mss2


 
(280)
The \mmnrsh of the system about the center of mass is the total mass-weighted mean square distance from the axis of rotation
\mmnrsh
=
\mss1 \rds12 + \mss2 \rds22
(281)
The \mmnngl of the system is the total mass-weighted product of the distance from the axis of rotation times the velocity. Denoting the angular frequency of the rotation by \frqngl, the linear velocity of the atoms are \frqngl \rds1 and \frqngl \rds2.
\mmnngl
=
\mss1 \rds1 \vlc1 + \mss2 \rds2 \vlc2
=
\mss1 \frqngl \rds12 + \mss2 \frqngl \rds22
=
\frqngl (\mss1 \rds12 + \mss2 \rds22)
The classical energy of a rigidly rotating dumbbell may be expressed in terms of \mmnrsh and \mmnngl
\nrgrtt
=
[1/2] \mmnrsh \frqngl2
\nrgrtt
=
\mmnngl2

2 \mmnrsh


 
(282)
The quantum mechanical angular momentum operator, \mmnnglopr, quantizes the angular momentum component of the energy Hamiltonian. The eigenvalues of the square of the angular momentum operator, \mmnnglopr2, are
\mmnnglopr2 \wvfnc
=
ħ2 \qntrtt (\qntrtt + 1) \wvfnc
(283)
where \wvfnc is the wavefunction and \qntrtt, an integer, is the of the system. Substituting (283) into (282) we obtain the quantum mechanical energy of a rigid rotator
\nrgrtt
=
ħ2

2\mmnrsh


 
\qntrtt (\qntrtt + 1)
=
\cstplk2

4\mpi2


 
1

2\mmnrsh


 
\cstspdlgt

\cstspdlgt


 
\qntrtt (\qntrtt + 1)
=
\cstplk

8\mpi2 \cstspdlgt \mmnrsh


 
\cstplk \cstspdlgt \qntrtt (\qntrtt + 1)
=
\cstplk \cstspdlgt \cstrtt \qntrtt (\qntrtt + 1)
(284)
where
\cstrtt
=
\cstplk

8\mpi2 \cstspdlgt \mmnrsh


 
(285)
\cstrtt is the of the species. \cstrtt is purely a function of the atomic masses and geometry of the species through \mmnrsh (281). The \vbrsbs subscript20 indicates that \cstrtt is a function of vibrational state.
Spectroscopists have adopted a variety of equivalent but usually unintuitive notations to describe the lower (lower energy) and upper (higher energy) states of transitions. The most common convention is that lower and upper state quantum numbers are superscripted by a double prime "′′" and a single prime "", respectively. Thus lower and upper vibrational and rotational quantum states are denoted by (\qntvbrlwr,\qntrttlwr) and (\qntvbrupr,\qntrttupr), respectively. Examples of equivalent representations of emission include
ν
ν
ν
ν
(\qntvbrupr,\qntrttupr)
(\qntvbrlwr,\qntrttlwr) +\cstplk \frq
(\qntvbrlwr,\qntrttlwr) + \cstplk \frq
(\qntvbrupr,\qntrttupr)
(286)
Due to destructive interference patterns between quantum mechanical wavefunctions, not all conceivable vibrotational transitions are . Rotational transitions are subject to the that
∆\qntrtt
=
\qntrttupr ±\qntrttlwr = ±1
(287)
Transitions which break selection rules like (287) are called .
Applying this rule to (284) we obtain the energy of photons released in purely rotational emission
∆\nrgrtt
=
\cstplk \cstspdlgt \cstrtt (\qntrtt + 1) (\qntrtt + 2)− \cstplk \cstspdlgt \cstrtt \qntrtt (\qntrtt + 1)
\cstplk \frq
=
\cstplk \cstspdlgt \cstrtt [ \qntrtt2 + 3\qntrtt + 2− ( \qntrtt2 + \qntrtt ) ]
\frq
=
\cstspdlgt \cstrtt ( 2 \qntrtt + 2 )
=
2 \cstspdlgt \cstrtt ( \qntrtt + 1 )
(288)
Thus lines in pure rotational bands are spaced linearly in frequency. Linear, symmetric molecules such as , , and  have no permanent dipole moments in their ground vibrational states because the center of mass coincides with the center of charge. These molecules therefore have no rotational bands in their ground vibrational state.
The factor of \cstspdlgt on the RHS of (288) disappears if we work in wavenumber rather than frequency units. Using \wvn = \frq / \cstspdlgt results in
\wvn
=
∆\nrgrtt = 2 \cstrtt ( \qntrtt + 1 )
(289)
This direct relation between wavenumber and quantum number is one reason spectroscopists prefer to work in wavenumber space.

0.0.0  Vibrational Transitions

A molecule composed of \nclnbr atoms has \modvbrnbr vibrational modes where
\modvbrnbr
=




3\nclnbr −3
   :    Nonlinear molecules
3\nclnbr −2
   :    Linear molecules
(290)
Often modes are degenerate, and so have fewer distinct vibrational quantum numbers than indicated by (290). As shown in Table , linear molecules include all (, , ), as well as  and . Non-linear molecules include , , , and .
Table 7: Mechanical Analogues for Radiatively Important Atmospheric Gases21
Moments of Inertia Class Members
_ = 0, _ = _ 0 22Linear , , , , , , , ,
_ 0, _ = _ 0 Symmetric top , , , ,
_ = _ = _ Spherical top
_ _ _ Asymmetric top , , , ,
The equation of motion for the position \psnvct of a classical oscillator of mass \mss oscillating with restoring force \cstrst is
\mss \dfr2 \psnvct

\dfr \tm2


 
=
−\cstrst \psnvct
\psnvctddot + \cstrst

\mss


 
=
0
\psnvct
=
\AAA sin\frqngl \tm + \BBB cos\frqngl \tm
\frqngl
=
2\mpi \frq =   ⎛


\cstrst

\mss


 
 


 
\frq
=
1

2\mpi


 
  ⎛


\cstrst

\mss


 
 


 
(291)
where \cstrst is the constant of the linear restoring force, known as the or the . Since \cstrst is the restoring force divided by the displacement distance, its SI units are  (Newtons per meter) and its CGS units are  (dynes per centimeter). The spring constant \cstrst depends only on the total particle mass, not on the mass distribution within the particle. The solution to a two particle sysytem where particle equilibrium separation is \psnvct is identical to (291) with the mass \mss replaced by the reduced mass \mssrdc (280)
\frq
=
1

2\mpi


 
  ⎛


\cstrst(\mss1 + \mss2)

\mss1 \mss2


 
 


 
(292)
The bond strength \cstrst depends on the exact arrangement of nuclei within a molecule. For diatomic molecules \cstrst approximately fits the following progression: Single bond, double bond, and triple bond diatomic molecules have \cstrst of approximately 500, 1000, and 1500  (5 ×105, 10 ×105, and 15 ×105 ) respectively.
Let us consider the aborption spectrum due to stretching of the molecule. is approximately a triply bonded diatomic molecule with a bond strength of about 1900 . We will use (292) to predict the fundamental frequency \frq\CO of the absorption spectrum. We shall include the details of the calculation to demonstrate the equivalence of the results obtained using SI and CGS units.
\frq\CO
1

2\mpi


 
  ⎛


1900 \nxm ×(12.0 ×10−3 + 16.0×10−3) \kgxmol

12.0 ×10−3 \kgxmol ×16.0 ×10−3 \kgxmol


 
 


 
1

6.28


 
  ⎛


53.2 \nxm

192 ×10−6 \kgxmol


 
 


 
≈ 0.160   ⎛


2.77 ×105 kg m

s2


 
× 1

m


 
× \cstAvagadro

kg


 
 


 
\frq\CO
84.2   ⎛


1

s2


 
× 6.02 ×1023

1


 
 


 
≈ 65.3 ×1012 \xs = 65.3 \Thz
\wvl\CO
=
\cstspdlgt

\frq\CO


 
3.00 ×108 \mxs

65.3 ×1012 \xs


 
= 4.59 ×10−6 m = 4.59 \um
\wvn\CO
=
\frq\CO

100\cstspdlgt


 
65.3 ×1012 \xs

100 \cmxm ×3.00 ×108 \mxs


 
≈ 2182 \xcm
(293)
The conversion of \frq\CO to \wvn\CO and \wvl\CO is straightforward, and shows that the stretching mode of the  molecule produces an absorption spectrum in the infrared. For pedagogical reasons, let us repeat the derivation of \frq\CO using CGS units to demonstrate that neither system is clearly superior in terms of its simplicity or ease of manipulation.
\frq\CO
1

2\mpi


 
  ⎛


1.9 ×106 \dxcm ×(12.0 + 16.0) \gxmol

12.0 \gxmol ×16.0 \gxmol


 
 


 
1

6.28


 
  ⎛


53.2 ×106 \dxcm

192 \gxmol


 
 


 
≈ 0.160   ⎛


27.7 ×104 g cm

s2


 
× 1

cm


 
× \cstAvagadro

g


 
 


 
and the rest of the computation is identical to (293). It appears that CGS units, historically the system of choice for radiative transfer, are slightly more simple to work with than SI units because of the CGS-centric definition of atomic weights. However, in this era of automated computations, the slight advantage of CGS does not appear to outweigh the usefulness of consistently using SI units (293). All other things being equal, we recommend using SI units wherever practical in radiative transfer, a convention we follow in this text.
The vibrational energy of a quantum harmonic oscillator is
\nrgvbr
=


 
\cstplk \frqmod(\qntvbrmod + [1/2])
(294)
where \frqmod is the frequency of the \modvbridxth mode, \cstplk \frqmod is the of the mode, and \qntvbrmod is the of the mode. Like other quantum numbers, \qntvbr is an integer \qntvbr ∈ [0, 1, 2, …]. The term of [1/2] in (294) is called the . The of a molecule is reached when \qntvbrmod = 0 for all \modvbrnbr vibrational modes. The linear relation between \qntvbrmod and \nrgvbr (294) is an approximation. In reality cause oscillations to deviate from linear behavior. This causes vibrational energy levels to become more closely spaced as \qntvbrmod increases.
Allowed transitions must match applicable . The selection rule for vibrational transitions is
∆\qntvbr
=
\qntvbrupr ±\qntvbrlwr = ±1
(295)
The refers to transitions between the first excited state (\qntvbr = 1) and the vibrational ground state (\qntvbr = 0). Using ∆\qntvbr = ±1 in (294) leads to
∆\nrgvbr
=
\cstplk \frqmod (\qntvbr + 1 + [1/2])− \cstplk \frqmod (\qntvbr + 1

2


 
)
=
\cstplk [ \frqmod ( \qntvbr + 3

2


 
) − \frqmod (\qntvbr + 1

2


 
) ]
=
\cstplk \frqmod
(296)
For simpler molecules such as diatomic molecules, the relation between the type of vibrational mode \modvbridx (e.g., stretching, bending) and the fundamental energy of the mode (296) may depend on only a few parameters.

0.0.0  Isotopic Lines

The quantum mechanical analogue of (291) is
\frqmod
=
1

2\mpi


 
  ⎛


\cstrstmod

\mssrdc


 
 


 
(297)
where now \cstrstmod is the force constant of vibrational mode \modvbridx which depends on the distribution of charge in the molecule but is completely independent of the mass distribution, and thus independent of isotope. Two distinct isotopes \iii and \jjj of the same species will thus have differing equilibrium frequencies \frq\iii and \frq\jjj for the same vibrational mode:
\frq\iii

\frq\jjj


 
=
  ⎛


\mssrdc\jjj

\mssrdc\iii


 
 


 
(298)
For instance, the frequency shift for \chm13C16O relative to the 2140  band of \chm12C16O is 47.7 .

0.0.0  Combination Bands

Transition selection rules for many important molecules (including all diatomic molecules), require that ∆\qntvbr ≠ 0 and ∆\qntrtt ≠ 0, i.e., vibrational transitions must occur simultaneously with rotational transitions. Thus simultaneous transitions of both vibrational and rotational states are the norm, not the exception, in the atmosphere. These simultaneous transitions form what are called or . The energy of such transitions is obtained by subtracting the lower energy state from the upper energy state
\wvn\RRR
=
\wvn\modvbridx + \cstrttupr \qntrttupr (\qntrttupr + 1) −\cstrttlwr \qntrttlwr (\qntrttlwr + 1) \qntrtt2
(299)
where \wvn\modvbridx is the energy of the pure vibrational transition (294). We have not cancelled like terms, as in the idealized case of (288), since \cstrttupr ≠ \cstrttlwr in combination bands. Substituting ∆\qntrtt = ±1 into (299) leads to the energy spacings between the transitions and the transitions, respectively.
\wvn\RRR
=
\wvn\modvbridx + 2 \cstrttupr + (3\cstrttupr − \cstrttlwr) \qntrtt + (\cstrttupr − \cstrttlwr)\qntrtt2
(300)
\wvn\PPP
=
\wvn\modvbridx − (\cstrttupr + \cstrttlwr) \qntrtt + (\cstrttupr − \cstrttlwr) \qntrtt2
(301)
There may be a number of absorption bands in addition to the fundamental bands of a molecule. These include , , , and . Overtone bands occur at frequencies which are multiples of the fundamental frequencies. Combination bands are due to the interaction of two fundamental vibration bands. The combined frequency ... Vibration-rotation bands have already been discussed. Harmonic coupling bands occur when interactions among closely spaced oscillation frequencies produces distinct, unexpected bands. This is relatively uncommon.

0.0  Partition Functions

The \sttwgt measures the number of available states in a given quantum configuration. Quantum theory teaches us that rotational energy levels are 2\qntrtt + 1 degenerate owing to 2\qntrtt + 1 indistinguishable orientations of the component of angular momentum in a fixed direction in space. The statistical weight for rotational levels is therefore
\sttwgtrtt
=
2\qntrtt + 1
(302)
In LTE, rotational states are populated according to , so that the probability of occupancy is proportional to \me−\nrgrtt/\cstblt \tpt. At temperature \tpt, the ratio of molecules in state \qntrttupr to those in state \qntrttlwr is therefore
\cnc(\qntrttupr)

\cnc(\qntrttlwr)


 
\sttwgt\qntrttupr

\sttwgt\qntrttlwr


 
\me−\nrgrtt(\qntrttupr)/\cstblt\tpt

\me−\nrgrtt(\qntrttlwr)/\cstblt\tpt


 
=
2\qntrttupr + 1

2\qntrttlwr + 1


 
exp

\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
[ \qntrttupr ( \qntrttupr + 1 ) − \qntrttlwr ( \qntrttlwr + 1 )]



 
(303)
where now \qntrttupr and \qntrttlwr refer to any values of \qntrtt, not just those satisfying selection rules. To examine the fractional abundance of molecules in state \qntrtt relative to those in all other rotational states, we must sum (303) over all \qntrttlwr while holding \qntrttupr = \qntrtt fixed. This leads to
\cnc(\qntrtt)

\cnc


 
=
2\qntrtt + 1

\prtrtt


 
exp
\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
\qntrtt ( \qntrtt + 1 )


 
(304)
where we have defined the \prtrtt as the total, probability-weighted number of available rotation states in the system
\prtrtt
=


 
(2\qntrtt + 1)exp
\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
\qntrtt ( \qntrtt + 1 )


 
(305)
For atmospheric cases of interest, \cstblt \tpt >> \cstplk \cstspdlgt \cstrtt so the exponential term becomes vanishingly small. To express and evaluate (305) as an integral, we make a change of variables
\xxx
=
\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
\qntrtt (\qntrtt + 1)
\dfr\xxx
=
\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
(2\qntrtt + 1) \dfr\qntrtt
(2\qntrtt + 1) \dfr\qntrtt
=
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
\dfr\xxx
(306)
This maps \qntrtt ∈ [0,∞) to \xxx ∈ [0,∞). Thus \xxx is considered a continuous function of \qntrtt. Substituting (306) into (305) results in
\prtrtt



 
(2\qntrtt + 1)exp
\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
\qntrtt ( \qntrtt + 1 )


 
 \dfr\qntrtt
=



 
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
\me−\xxx  \dfr\xxx
=
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
[ −\me−\xxx ]0
=
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
[ − 0 − ( − 1 ) ]
=
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
(307)
Substituting (307) into (304) we obtain
\cnc(\qntrtt)

\cnc


 
=
\cstplk \cstspdlgt \cstrtt(2\qntrtt + 1)

\cstblt \tpt


 
exp
\cstplk \cstspdlgt \cstrtt

\cstblt \tpt


 
\qntrtt ( \qntrtt + 1 )


 
(308)
Examination of (308) shows that \cnc(\qntrtt)/\cnc is small for \qntrtt → 0 and for \qntrtt → ∞ and maximal in between. The derivation of (308) neglected any vibrational-dependence of \cstrtt.
The \prtvbr is defined by a procedure analogous to (303)-(307). Contrary to rotational states (302), vibrational states all have equal statistical weights
\sttwgtvbr
=
\sttwgt0
(309)
In LTE, , (309) and (294) lead to
\cnc(\qntvbrupr)

\cnc(\qntvbrlwr)


 
\sttwgt\qntvbrupr

\sttwgt\qntvbrlwr


 
\me−\nrgvbr(\qntvbrupr)/\cstblt\tpt

\me−\nrgvbr(\qntvbrlwr)/\cstblt\tpt


 
=
\sttwgt0

\sttwgt0


 
exp



\cstplk \frqnot (\qntvbrupr + 1

2


 
)

\cstblt \tpt



 
\cstplk \frqnot (\qntvbrlwr + 1

2


 
)

\cstblt \tpt



 





 





 
=
exp
\cstplk \frqnot

\cstblt \tpt


 
(\qntvbrupr − \qntvbrlwr)


 
(310)
Notice that the statistical weight and the contribution of \cstplk \frqnot /2 factor out of the relative abundance of molecules in a given vibrational level (310). Nevertheless, the zero-point energy does contribute to the total internal energy of the system and so is properly included in the definition of the vibrational partition function \prtvbr
\prtvbr
=


 
exp
\cstplk \frqnot

\cstblt \tpt


 
( \qntvbr + [1/2] )


 
=
\me−\cstplk \frqnot/2\cstblt \tpt

 
( \me−\cstplk \frqnot / \cstblt \tpt )\qntvbr
=
\me−\cstplk \frqnot/2\cstblt \tpt

1−\me−\cstplk \frqnot / \cstblt \tpt


 
(311)
where in the last step we used the solution for an infinite geometric series whose ratio between terms \rrr < 1,


 
\aaa0 \rrr\nnn
=
\aaa0

1 − \rrr


 
(312)
The fractional abundance of molecules in vibrational state \qntvbr is thus
\cnc(\qntvbr)

\cnc


 
=
1

\prtvbr


 
exp
\cstplk \frqnot

\cstblt \tpt


 
(\qntvbr + [1/2])


 
=
1−\me−\cstplk \frqnot / \cstblt \tpt

\me−\cstplk \frqnot/2\cstblt \tpt


 
\me−\cstplk \frqnot/2\cstblt \tpt\me−\qntvbr \cstplk \frqnot /\cstblt \tpt
=
(1−\me−\cstplk \frqnot / \cstblt \tpt)\me−\qntvbr \cstplk \frqnot /\cstblt \tpt
(313)
Vibrational levels are so widely spaced that most molecules in the lower atmosphere are in \qntvbr = 0 or \qntvbr = 1 states.
It is common to approximate the total internal partition function of atmospheric transitions \prtfnc as the product of the vibrational and the rotational partition functions. Using (311) and (307) we obtain
\prtfnc
\prtvbr \prtrtt
\me−\cstplk \frqnot/2\cstblt \tpt

1−\me−\cstplk \frqnot / \cstblt \tpt


 
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
In the high temperature limit where \cstblt \tpt >> \cstplk \frqnot then we may use the behavior \me−\xxx ≈ 1 − \xxx so this becomes
\prtfnc
1 − \cstplk \frqnot

2\cstblt \tpt


 

1 − 1 + \cstplk \frqnot

\cstblt \tpt


 





 
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
2\cstblt \tpt − \cstplk \frqnot

2\cstblt \tpt


 

\cstplk \frqnot

\cstblt \tpt


 





 
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 
2\cstblt \tpt − \cstplk \frqnot

2 \cstplk \frqnot


 
\cstblt \tpt

\cstplk \cstspdlgt \cstrtt


 

0.0  Dipole Radiation

The probability \prbif of absorption of a photon leading to a change in molecular state from state \iii to state \fff is
\prbif
=
\tm
\chglct2

\cstplk \cstspdlgt3 \msslct2


 



 


 

(⎜)



 
[ \frqngl \pplphtplr(\wvnbrvct) |〈\wvfncf | \me\mi \wvnbrvct ·\psn \eeeplr(\wvnbrvcthat)·\mmnvct | \wvfnci 〉|2 ]fi  \dfr\ngl
(314)
The 〈\wvfncf | \me\mi \wvnbrvct ·\psn\eeeplr(\wvnbrvcthat) ·\mmnvct | \wvfnci 〉 determines the absorption probability. The retains only the first term in the expansion
\me\mi \wvnbrvct ·\psn
=
1 + \mi \wvnbrvct ·\psn +…
so that the matrix element simplifies to
〈\wvfncf | \me\mi \wvnbrvct ·\psn\eeeplr(\wvnbrvcthat) ·\mmnvct | \wvfnci 〉
=
\eeeplr ·〈\wvfncf | \mmnvct | \wvfnci 〉
(315)
This matrix element is related to the \lnstr as follows ....

0.0  Two Level Atom

Einstein created an elegant paradigm for analyzing the interaction of matter and radiation. His tool is called the . It consists of an ensemble of molecules with two discrete energy levels, \nrgone and \nrgtwo > \nrgone. The energy levels have statistical weights \sttwgtone and \sttwgttwo, respectively. There are \pplone molecules per unit volume in the state with \nrgone, and \ppltwo in the state with \nrgtwo. The frequency of transition between the two levels is \hnunot = \nrgtwo − \nrgone.
The five processes by which an idealized two-level molecule may change state are ^ +
+ ^
+ ^ + 2
+ + ^ + + +
+ + ^ + + where  is a collision partner and  and  represent the kinetic energy of the two molecule system before and after the collision, respectively. The most frequent collision partners are, naturally, nitrogen and oxygen molecules.
The Einstein coefficient \atwoone  is the transition probability per unit time for . Spontaneous emission, as its name implies, requires no external stimulus. Thus decay of excited molecules occurs even in the absence of a radiation field. Each emission reduces the population of excited molecules and increases the population of ground state molecules by one. Hence
\dfr \pplone

\dfr\tm


 
=
\atwoone \ppltwo
(316)
The Einstein coefficient \bonetwo determines the rate of radiative absorption. \bonetwo is also called the Einstein coefficient for . More specifically, \bonetwo is the proportionality constant between the mean intensity of the radiation field \ntnmnfrq and the probability per unit time of an absorption occuring. The transition probability per unit time for radiative absorption is \bonetwo \ntnmnfrq. \bonetwo has units of [()−1]. The rate of radiative absorptions per unit time per unit volume is
\dfr \ppltwo

\dfr\tm


 
=
\pplone


 



 
\xsxabsoffrq \ntnfrq

\hnu


 
 \dfr\ngl  \dfr\frq
=
4\mpi \pplone


 
\xsxabsoffrq \ntnmnfrq

\hnu


 
 \dfr\frq
(317)
where \xsxabsoffrq refers to absorption cross-section of the single transition available in the two-level atom. Comparing the final two expressions we see that
\bonetwo
=
4\mpi \xsxabsoffrq

\hnu \lnshpoffrq


 
(318)
\bonetwo is intimately related to the \lnstr of the molecular transition . In particular, \bonetwo may be expressed in terms of the weighted transition-moment squared.
The Einstein coefficient \btwoone determines the rate of or . \btwoone \ntnmnfrq is transition probability per unit time for stimulated emission. \btwoone has the same dimensions as \bonetwo.
\dfr \pplone

\dfr\tm


 
=
\ppltwo


 



 
\xsxabsstmoffrq \ntnfrq

\hnu


 
 \dfr\ngl  \dfr\frq
=
4\mpi \ppltwo


 
\xsxabsstmoffrq \ntnmnfrq

\hnu


 
 \dfr\frq
(319)
where \xsxabsstmoffrq is the absorption cross-section for stimulated emission.
Gaseous absorption cross-sections \xsxabsoffrq are extremely narrow and so the integrals over frequency domain in (317) and (319) are determined by a small range of frequencies about \frqnot, the line center frequency. Exact descriptions of line broadening about \frqnot are discussed in §0.0.10.0.1. It is instructive to ignore the details of the finite shape of line transitions for now and to assume the line absorption cross section behaves as a delta-function with integrated cross section \aaaonetwo
\xsxabsoffrq
\aaaonetwo \dltfncoffrqfrqnot
(320)
\aaaonetwo
=
\mpi \chglct2

\msslct \cstspdlgt


 
(321)
where \msslct is the , \chglct is the electron charge, and \oscstr is the of the transition. With these definitions.
In thermodynamic equilibrium the number of transitions from state 1 to state 2 per unit time per unit volume equals the number of transitions from state 2 to state 1 per unit time per unit volume. Combining (316), (317), and (319) we obtain
\dfr \pplone

\dfr\tm


 
=
\dfr \ppltwo

\dfr\tm


 
\pplone \bonetwo \ntnmnfrq
=
\ppltwo \atwoone + \ppltwo \btwoone \ntnmnfrq
(322)

0.0  Line Strengths

We repeat the definition of (230) for convenience
\lnstr =


 
\xsxabsoffrq  \dfr\frq
(323)
In practice, (323) is either measured in the laboratory or predicted from ab initio methods.

0.0.0  

The ("high-resolution transmission") database provides the parameters required to computed absorption coefficients for all atmospheric transitions of interest. The database is defined by ], and its usage is defined in their Appendix A.
The idiosyncratic units of line parameters can be confusing. Most data are provided in CGS units. For reference, Table  presents the equivalence between the symbols used in ], their Appendix A, and our symbols.
Table 8: database2324
This Text Description
S_^ Line strength
_air Pressure-broadened HWHM
_self Self-broadened HWHM
_^ Transition frequency
E_ Lower state energy
n Exponent in temperature-dependence of pressure-broadened halfwidth
Air-broadened pressure shift of transition frequency
Frequency in wavenumbers
The \lnstrlnnot (230) is tabulated in in CGS units of wavenumbers times centimeter squared per molecule, , often written . Line strengths for weak lines can be less than 10−36 , i.e., unrepresentable as IEEE single precision (4 byte) floating point numbers. Thus it is more robust to work with \lnstrlnnot units of per mole (rather than per molecule) in applications where using IEEE double precision (8 byte) floating point numbers is not an option either due to memory limitations or computational overhead. Following are the conversions from the tabulated values of \lnstrlnnot in  to more useful units, including , , and to fully SI units of  and . [] = [] 10^-4
[] = [] 10^-4
[] = [] 10^-4 100
[] = [] 10^-4 100 ^2 where \cstAvagadro is . Equation (0.0.1) is uncertain.
The energy of the lower state of each transition, \nrglwrln, is required to determine the relative population of that state available for transitions. supplies this energy in wavenumber units in the tabulated parameter \wvnlwrln which is simply related to \nrglwrln by
\nrglwrln
=
\cstplk \cstspdlgt \wvnlwrln
(324)
Of course, consistent with and spectroscopic conventions, \wvnlwrln is archived in CGS, .
All data have been scaled from the conditions of the experiment to a temperature and pressure of 296 K and 1013.25 mb, respectively. Thus line strengths must typically be scaled from -standard conditions, \lnstrlnnot(\tptnot), to the atmospheric conditions of interest, \lnstrln(\tpt).
\lnstrln(\tpt)
=
\lnstrlnnot(\tptnot) \prtfnc(\tptnot)

\prtfnc(\tpt)


 
\me−\cstplk \cstspdlgt \wvn\iii / \cstblt \tpt

\me−\cstplk \cstspdlgt \wvn\iii / \cstblt \tptnot


 
1 − \me−\cstplk \cstspdlgt \wvn\iii \jjj / \cstblt \tpt

1 − \me−\cstplk \cstspdlgt \wvn\iii \jjj / \cstblt \tptnot


 
(325)
where \prtfnc is the total internal partition function. The second, third and fourth factors on the RHS of (325) are the temperature-dependent scalings of, respectively, the total partition function, the Boltzmann factor for the lower state, and the stimulated emission factor. The scaling presented in (325) is general and contains no approximation.
The classical approximation ,] assumes that \prtfnc is the product of the rotational and the vibrational partition functions
\prtfnc(\tpt)
\prtvbr(\tpt) \prtrtt(\tpt)
(326)
In this approximation, \prtvbr (311) and \prtrtt (307) are treated as independent components of \prtfnc. This allows \prtfnc to be computed as discussed in §0.1. Most high precision calculation improve this approach by using more exact methods described in ] and ].
For many applications, a simplified version of (325) is employed
\lnstrln(\tpt)
=
\lnstrlnnot(\tptnot)
\tptnot

\tpt


 



 
exp
\nrglwrln

\cstblt


 

1

\tpt


 
1

\tptnot


 



 



 
=
\lnstrlnnot(\tptnot)
\tptnot

\tpt


 



 
exp
\nrglwrln

\cstblt


 

1

\tptnot


 
1

\tpt


 



 



 
(327)
where \mmm is an empirical parameter of order unity and The parameterization (327) incorporates the temperature dependence of both partition functions and the stimulated emission factor into the single factor (\tptnot/\tpt)\mmm. The exact temperature dependence of Boltzmann factors is retained. If the parameter \mmm is available, use of (327) offers decreased computational overhead relative to (325).
Pressure-broadened halfwidths \hwhmprsnot are also provided at the reference temperature and pressure, \hwhmprsnot ≡ \hwhmprs(\prsnot,\tptnot). literature refers to broadening features are as (rather than pressure-broadened) because all parameters in the database are normalized to Earth's natural isotopic mixing ratios. Thus while the broadening process is collision-broadening, the tabulated line width refers to collision-broadening by air of Earth's isotopic composition, hence the label. actually provides two halfwidths, the self-broadened halfwidth \hwhmslfnot and the air-broadened or halfwidth \hwhmfrnnot. Air-broadening accounts for collision-broadening by foreign molecules, i.e., all molecules except the species undergoing radiative transition. Thus air-broadening is, to a good approximation, due nitrogen and oxygen molecules. The other form of broadening, refers to collision-broadening by molecules of the species undergoing the radiative transition. Because reasonances may develop between members of identical species, in general \hwhmslfnot ≠ \hwhmfrnnot. The pressure-broadened halfwidth of transitions of radiatively active species  at arbitrary pressure and temperature scales as
\hwhmprs(\prs,\tpt)
=

\tptnot

\tpt


 



 

\prs − \prsprtA

\prsnot


 
\hwhmfrnnot + \prsprtA

\prsnot


 
\hwhmslfnot


 
(328)
where \prsbrdtptdpnxpn is an empirical fitting parameter supplied by and \prsprtA is the partial pressure of species . Clearly self-broadening effects are appreciable only for species with signifcant partial pressures, i.e.,  and . For minor trace gases, \prsprtA << \prs and \prsprtA << \prsnot so \prs − \prsprtA ≈ \prs and \prsprtA/\prsnot ≈ 0. For such gases, (328) simplifies to
\hwhm(\prs,\tpt)
\hwhmnot \prs

\prsnot


 

\tptnot

\tpt


 



 
(329)
which has no dependence on self-broadening.
Collision-broadening may also cause a of the line transition frequency away from the tabulated line center frequency \wvnnot = \wvn(\prsnot) to a shifted frequency \wvnshf = \wvn(\prs). supplies a parameter \prsshf(\prsnot) in CGS wavenumbers per atmosphere () with which to calculate the line center frequency change due to pressure-shifting.
\wvnshf(\prs)
=
\wvnnot + \prsshf(\prsnot) \prs

\prsnot


 
(330)
The prescriptions for adjusting line centers (330) and half-widths (328) from (\prsnot,\tptnot) to to arbitrary \prs and \tpt should be used to determine the corrected line shape profile of each transition considered. In the lower atmosphere, application of (330) and (328) leads to a corrected Lorentzian profile (237)
\lnshplrn(\wvn,\wvnnot,\prs,\tpt)
=
1

\mpi


 
  \hwhmlrn(\prs,\tpt)

 { \wvn − [\wvnnot + \prs \prsshf(\prsnot)/\prsnot] }2 + \hwhmlrn(\prs,\tpt)2


 
\lnshplrn(\frq,\frqnot,\prs,\tpt)
=
1

\mpi


 
  \hwhmlrn(\prs,\tpt)

 (\frq − \frqshf)2 + \hwhmlrn(\prs,\tpt)2


 
(331)
Of course this section has only discussed application of database parameters to atmospheres in conditions. Application of to conditions is discussed in ].
It is necessary to compute statistics of \lnstrln for use in (§). There is no reason not to apply the full approach of (325) when computing these statistics. Certain well-documented narrow band models use more approximate forms appropriate for specific applications. ] uses
\lnstrln(\tpt)
=
\lnstrlnnot(\tptnot)
\tptnot

\tpt


 



 
\me−\cstplk \cstspdlgt \wvn\iii / \cstblt \tpt

\me−\cstplk \cstspdlgt \wvn\iii / \cstblt \tptnot


 
1 − \me−\cstplk \cstspdlgt \wvn\iii \jjj / \cstblt \tpt

1 − \me−\cstplk \cstspdlgt \wvn\iii \jjj / \cstblt \tptnot


 
(332)
which is a hybrid of (325) and (327) with \mmm = 1.5.

0.0  Line-By-Line Models

0.0.0  Literature

The classic paper on the water vapor continuum is ]. ] compare a correlated-\kkk method to line-by-line model results. ] describe the a field experiment to directly compared measured and modeled spectral radiances to both band and line-by-line models. ] uses a line-by-line model to determine atmospheric solar absorption. ] presents an economical algorithm for selecting variable wavelength grid resolution such that absorption coefficients may be computed to a given level of accuracy. ] compare a correlated-\kkk model () to the well-known line-by-line model . ] describe a fully scattering line-by-line model. ] presents test-case spectra for evaluating line-by-line radiative transfer models in cold and dry atmospheres. ] use observations to constrain magnitude of any non-Lorentzian continuum in the near-infrared. ] discuss the . ] present an algorithm that computes absorption coefficients to a specified error tolerance by using a pre-computed lookup table of where interpolation is appropriate.

0  Band Models

Band models, also called narrow band models, discretize the radiative transfer equation intervals for which the line statistics and the Planck function are relatively constant. The literature describing these models extends back to Goody (1957). Band models have traditionally been applied to thermal source functions (
string :autorefequation187a
)-(
string :autorefequation187b
), but they may also be applied to the solar spectral region ] (MODTRAN3) if additional assumptions are made.

0.0  Generic

The following presentation assumes that the absorption path is homogeneous, i.e., at constant temperature and pressure. Important corrections to these assumptions are necessary in inhomogeneous atmospheres. These corrections are discussed in §.
This discussion makes use of an arbitrary frequency interval \dltfrq which represents the discretization interval of narrow band approximation. It is important to remember that \dltfrq is best determined empirically by comparison of the narrow band approximation to approximations. Typically, \dltfrq is 5-10 . ] showed \dltfrq = 5 \xcm bands are optimal for . ] uses \dltfrq = 5 \xcm for , , , , but \dltfrq = 10 \xcm for . ] recommends \dltfrq = 5 \xcm for , 10  for , 5-10  for , and 5  for all other trace gases.

0.0.0  Beam Transmittance

The gaseous absorption optical depth \tauabsoffrq is the product of the spectrally resolved molecular cross-section \xsxabsoffrq () and the absorber path \abspth (227)
\tauabsoffrq
=
\xsxabsoffrq \abspth
(333)
The transmittance \trnbmoffrq between two points in a homogeneous atmosphere is the negative exponential of \tauabsoffrq (
string :autorefequation182
),
\trnbmoffrq
=
\me−\tauabsoffrq
=
\me−\xsxabsoffrq \abspth
(334)
Note that \trnbmoffrq is the spectrally and directionally resolved transmittance because (a) it pertains to a monochromatic frequency interval and (b) it applies to radiances, not to irradiances (which require an additional angular integration). Narrow band models are based on the \trndltfrq of a narrow but finite frequency interval \dltfrq between \frqnot−\dltfrq/2 and \frqnot+\dltfrq/2.
\trndltfrq
=
1

\dltfrq


 



 
\trnbm  \dfr\frq
=
1

\dltfrq


 



 
\me−\tauabs  \dfr\frq
(335)
where \frqnot is the central frequency of the interval in question
Using (229), we rewrite (335) as
\trndltfrq = 1

\dltfrq


 



 
exp[−\xsxabsoffrq \abspth]  \dfr\frq
(336) If the lines are Lorentzian (i.e., pressure-broadened) then the transmittance may be written in terms of the line strength and absorber amount using (0.0.1) and (262)
\xsxabsoffrq \abspth = \lnstr \hwhmlrn \abspth

\mpi(\frq2 + \hwhmlrn2)


 
(337)
Thus (335) becomes
\trndltfrq = 1

\dltfrq


 



 
exp
\lnstr \hwhmlrn \abspth

\mpi(\frq2 + \hwhmlrn2)


 



 
 \dfr\frq
(338)

0.0.0  Beam Absorptance

In analogy to \trnbm (
string :autorefequation181
) we define the \absbm
\absbm
=
1 − \trnbm
=
1 − exp[−\xsxabsoffrq \abspth]
(339)
where \xsxabsoffrq is the absorption cross-section and \abspth is the absorber path. As described in §0.0.1, we refer to \absbm as the beam absorptance becuase it pertains to a monochromatic frequency, and refers to absorption of radiance, not irradiance.
The \absdltfrq is the complement of the band transmittance
\absdltfrq
=
1 − \trndltfrq
=
1

\dltfrq


 



 
\absbm  \dfr\frq
=
1

\dltfrq


 



 
1 − \me−\tauabs  \dfr\frq
=
1

\dltfrq


 



 
1 − exp[−\xsxabsoffrq \abspth] \dfr\frq
(340)
The relation between the band absorptance and band transmittance is the same as the relation between the monochromatic beam absorptance and the monochromatic beam transmittance (339). \absdltfrq measures the mean absorptance within a finite frequency (or wavelength) range comprising many lines. \absdltfrq does not include any contribution from outside the \dltfrq range. Thus \absdltfrq may neglect contribution from the far wings of some lines in the band. Moreover, it is important to remember that energy absorption follows the band absorptance only if the energy distribution within the band is close to linear.
If the line shape is Lorentzian (237), then (338) applies so that
\absdltfrq = 1

\dltfrq


 



 
1 − exp
\lnstr \hwhmlrn \abspth

\mpi(\frq2 + \hwhmlrn2)


 



 
 \dfr\frq
(341)

0.0.0  Equivalent Width

The spectrally integrated monochromatic beam absorptance of a line is called its
\eqvwth
=



 
\absbm  \dfr\frq
\eqvwthabspth
=



 
1 − exp[−\xsxabsoffrq \abspth]  \dfr\frq
(342)
Thus the absorber amount must be specified in order to determine the equivalent width. The lower limit of integration, −∞, is actually an approximation which is mathematically convenient to retain so that (342) is analytically integrable for the important line shape functions. Recall that we are working with a frequency coordinate \frq which is defined to be \frq = 0 at line center \frqnot (263). In practice the error caused by using the analytic expressions resulting from ∫−∞+∞ rather than those resulting from ∫−\frqnot+∞ is negligible except for lines in the microwave.
The name "equivalent width" reminds us that \eqvwth is the width of a completely saturated rectangular line profile that has the same total absorptance. The relation between \abspth and \eqvwthabspth is called the . The curve of growth of a gas can be measured in the laboratory. The interpretation of the curve of growth played a very important role in advancing our understanding and representation of gaseous absorption.
If the line shape is Lorentzian (237), then (341) applies
\eqvwth =


 
1 − exp
\lnstr \hwhmlrn \abspth

\mpi(\frq2 + \hwhmlrn2)


 



 
 \dfr\frq
(343)

0.0.0  Mean Absorptance

Although the spectrally resolved absorptance (339) and the band absorptance (340) are dimensionless, the equivalent width (342) has units of frequency (or wavelength). It is convenient to define a dimensionless mean absorptance for a transition line \absavg
\absavg
=
1

\mls


 



 
\absbm  \dfr\frq
=
1

\mls


 



 
1 − exp[−\xsxabsoffrq \abspth]  \dfr\frq
=
\eqvwth

\mls


 
(344)
where \mls is the mean line spacing between adjacent lines in a given band. Note that \absdltfrq (340) differs from \absavg (344). \absavg accounts for the absorptance due to a single line over the entire spectrum and renormalizes that to a mean absorptance within a specified frequency (or wavelength) range (\mls). In contrast to \absdltfrq, \absavg does not neglect any absorption in the far wings of a line. Therefore the sum of the average absorptances of all the lines centered within \dltfrq may exceed, by a small amount, the band absorptance computed from (340)


 
\absavgi ≥ \absdltfrq
(345)
This difference should make it clear that \absavg is much more closely related to \eqvwth than to \absdltfrq.
For lines with Lorentzian profiles (237), \eqvwthavg and \absavg are
\eqvwth = \absavg \mls
=



 
1 − exp
\lnstr \hwhmlrn \abspth

\mpi(\frq2 + \hwhmlrn2)


 



 
 \dfr\frq
(346)
We shall rewrite (346) in terms of three dimensionless variables, \xxx, \yyy, and \uuu = /
=
= ^-1  
=  
= /
=
=
= 2 This change of variables maps \frq ∈ [−∞,+∞] to \xxx ∈ [−∞,+∞]. We obtain
\absavg
=
1

\mls


 



 
1 − exp
−(2 \hwhmlrn \uuu)
\hwhmlrn

\mls2 \xxx2 + \mls2 \yyy2


 



 



 
 (\mls  \dfr\xxx)
=



 
1 − exp
2 \hwhmlrn2 \uuu

\mls2 (\xxx2 + \yyy2)


 



 
 \dfr\xxx
=



 
1 − exp
2 \mls2 \yyy2 \uuu

\mls2 (\xxx2 + \yyy2)


 



 
 \dfr\xxx
=



 
1 − exp
2 \uuu \yyy2

\xxx2 + \yyy2


 



 
 \dfr\xxx
(347)
The solution to this definite integral may be written in terms of modified of the first kind \bslIcpx(\uuu) (see §)
\absavg
=
2 \mpi \yyy \uuu \me−\uuu [ \bslIfnc0(\uuu) + \bslIfnc1(\uuu) ]
2 \mpi \yyy \ldnfnc(\uuu)
=
2 \mpi \hwhmlrn \mls−1 \ldnfnc(\uuu)
\eqvwth
=
2 \mpi \hwhmlrn \ldnfnc(\uuu)
(348)
where \ldnfnc(\uuu) is the ]. The dimensionless optical path \uuu (0.0.1) is therefore a key parameter in determining the gaseous absorptance. Two important limiting cases of (348) are \uuu << 1 and \uuu >> 1.
For small optical paths, \lnstr \abspth << 1 (227), or, equivalently, \uuu << 1 (0.0.1). This is the . In this limit the exponential in (346) or (347) may be replaced by the first two terms in its Taylor series expansion. Thus all line shapes have the same weak-line limit
\eqvwth = \absavg \mls



 
\xsxabsoffrq \abspth  \dfr\frq
\lnstr \abspth


 
\lnshplrnoffrq  \dfr\frq
\lnstr \abspth
(349)
where we have use the generic normalization property of the line shape profile (240) in the final step.
For large optical paths, \lnstr \abspth >> 1 (227), or, equivalently, \uuu >> 1 (0.0.1). This is the . To examine this limit we first note that the line HWHM is much smaller than the frequencies where line absorptance is strong, i.e., \hwhmlrn << \frq in (346). Equivalently, \yyy << \xxx (0.0.1)-(0.0.1) so that \yyy2 may be neglected relative to \xxx2 in the denominator of (347)
\absavg



 
1 − exp( −2 \uuu \yyy2/\xxx2 )  \dfr\xxx
One further simplification is possible. Both terms in the integrand are symmetric about the origin so we may consider only positive \xxx if we double the value of the integral.
\absavg
=
2


 
1 − exp( −2 \uuu \yyy2/\xxx2 ) \dfr\xxx
(350)
The change of variables \zzz = 2 \uuu \yyy2 \xxx−2 maps \xxx ∈ (0,+∞) to \zzz ∈ (+∞,0) = 2 ^2 ^-2
= (2 )^1/2 ^1/2
= (2 )^1/2 (-)^-3/2  
= - 2^-1/2 ^1/2 ^-3/2   Substituting this into (350) leads to
\absavg
2


 
(1 − \me−\zzz) ( − 2−1/2 \uuu1/2 \yyy ) \zzz−3/2  \dfr\zzz
21/2 \uuu1/2 \yyy


 
(1 − \me−\zzz) \zzz−3/2  \dfr\zzz
\yyy

 

2 \uuu
 

 



 
\zzz−3/2 − \zzz−3/2 \me−\zzz  \dfr\zzz
The first term in the integrand of (351) is directly integrable ∫\zzz−3/2  \dfr\zzz = −2 \zzz−1/2. The second term in the integrand of (351) is the complete \gmmfnc(−1/2) = . Section  describes the properties of gamma functions25.
Combining these results we find that the mean absorptance of an isolated Lorentz line in the (351) reduces to
\absavg
\yyy

 

2 \uuu
 

 

−2 \zzz−1/2


 
− (− 2

 

\mpi
 

 
)


 
=
\yyy

 

2 \uuu
 

 
( ∞+ 2

 

\mpi
 

 
)
=
2 \yyy

 

2 \mpi \uuu
 

 
(351)
fxm: The first term must vanish, but how? Substituting (0.0.1)-(0.0.1) into (351)
\absavg
2
\hwhmlrn

\mls


 



 


 

2 \mpi
 

 
  ⎛


\lnstr \abspth

2 \mpi \hwhmlrn


 
 


 
\eqvwth = \absavg \mls
=
2

 

\lnstr \abspth \hwhmlrn
 

 
(352)
In the we see that the mean absorptance \absavg of an isolated line increases only as the square-root of the mass path. Physically, the strong line limit is approached as the line core becomes saturated and any additional absorption must occur in the line wings. Put another way, transition lines do not obey the exponential on which our solutions to the radiative transfer equation are based. One important consequence of this result is that complicated gaseous spectra must either be decomposed into a multitude of monochromatic intervals, each narrow enough to resolve a small portion of a transition line, or some new statistical means must be developed which correctly represents line absorption in both the weak line and strong line limits.
In summary, the equivalent width \eqvwth (342) of an isolated spectral line behaves distinctly differently in the two limits (349) and (352)
\eqvwth = \absavg \mls
=



\lnstr \abspth
   :    Weakline limit
2

 

\lnstr \abspth \hwhmlrn
 

 
   :    Strongline limit
(353)

0.0  Line Distributions

Inspection of realistic gaseous absorption spectra reveals that line strengths in complex bands occupy a seemingly continuous distribution space, with line strengths \lnstr varying over many orders of magnitude in a single band. A key point discussed further in ] is that the variabilities in line spacing and in line widths within a band are negligible (and usually order of magnitude smaller) in comparison to the dynamic range of \lnstr. A band containing a suitably large number of lines, therefore, may be amenable to the approximation that the line strength distribution may be represented by a continuous function of \lnstr. The \pdfoflnstr is the probability that a line in a given spectral region will have a line strength between \lnstr and \lnstr + \dfr\lnstr. The function \pdfoflnstr must be correctly normalized so that probability of a line having a finite, positive strength is unity



 
\pdfoflnstr  \dfr\lnstr
=
1
(354)
A number of functional forms for \pdfoflnstr have been proposed.

0.0.0  Line Strength Distributions

] proposed the , now also known as the
\pdfoflnstr
=
\lnstravg−1  \me−\lnstr/\lnstravg
(355)
where \lnstravg is a constant which, in the next section, we show to be equal to the mean line intensity. A prime advantage of (355) is its arithmetic tractability. The zeroth and first moments of (355) are both solvable analytically. The following sections use this property to demonstrate the absorptive characteristics of line distributions. However, more complex distributions can improve upon the exponential distribution (355) by better representing the observed line shape distribution of many important species in Earth's atmosphere, such as .
] proposed an improved, albeit more complex, analytic form for the line strength distribution function. First we present a simplified, approximate version of this PDF which illustrates the essential differences between the Malkmus distribution and simpler line strength distributions.
\pdfoflnstr
=
\lnstr−1  \me−\lnstr/\lnstravg
(356)
The difference with (355) is the prefactor has changed from \lnstravg−1 to \lnstr−1. The \lnstr−1 dependence increases the number of weaker lines relative to stronger lines, which is an improvement over (355) and simpler distributions (such as the Elsasser or Godson distributions). The Malkmus distribution (356) is now perhaps the most commonly used line distribution function.
The astute reader will note that (356) is not normalizable on the interval [0,+∞) (cf. §). A key point of the Malkmus distribution, therefore, is the truncation or tapering of \pdfoflnstr so that statistics of the resulting \pdfoflnstr are well-behaved. A straightforward line strength distribution function with most of the desired properties is the "truncated" distribution which is non-zero only between a fixed maximum and minimum line strength, \lnstrmax and \lnstrmin, respectively. By convention, \lnstrmin is deprecated in favor of the ratio \lnstrrat between the maximum and minimum line strengths considered in the band
\lnstrmin
=
\lnstrmax / \lnstrrat
\lnstrrat
\lnstrmax / \lnstrmin
(357)
The line strength ratio \lnstrrat measures the dynamic range of line strengths considered in a band, which can be quite large, e.g., greater than 106 for  bands. With this convention,
\pdfoflnstr
=




0
   :    \lnstr < \lnstrmax/\lnstrrat
(\lnstr ln\lnstrrat)−1
   :    \lnstrmax/\lnstrrat < \lnstr < \lnstrmax
0
   :    \lnstr > \lnstrmax
(358)
] appears to have been the first to examine (358), although it is closely related to the full Malkmus distribution. The normalization of (358) is demonstrated in §. Clearly (358) contains no contributions from lines weaker than \lnstrmax/\lnstrrat or stronger than \lnstrmax. The main disadvantage to (358) is that it is discontinuous, and thus more difficult to treat computationally.
] had the insight to identify the following continuous function
\pdfoflnstr
=
\me−\lnstr/\lnstrmax − \me−\lnstrrat \lnstr / \lnstrmax

\lnstr ln\lnstrrat


 
(359)
We shall call (359) the full Malkmus line strength distribution. In practice, (359) is only applied in the limit \lnstrrat → ∞. Rather than simply truncating (356) so that \pdfoflnstr is non-zero only between some lower and upper bound (358), ] developed (359) because it has many useful properties:



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