% $Header$

% Purpose: Theory and Practice of Radiative Transfer in the Earth System

% Copyright (c) 1998--2008, Charles S. Zender
% Permission is granted to copy, distribute and/or modify this document
% under the terms of the GNU Free Documentation License (GFDL), Version 1.2
% or any later version published by the Free Software Foundation;
% with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
% GFDL: http://www.gnu.ai.mit.edu/copyleft/fdl.html

% The original author of this software, Charlie Zender, wants to improve it
% with the help of your suggestions, improvements, bug-reports, and patches.
% Charlie Zender <zender at uci dot edu>
% Department of Earth System Science
% University of California, Irvine
% Irvine, CA 92697-3100

% URL: http://dust.ess.uci.edu/facts/rt/rt.pdf

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% Plausible nomenclatural changes: 
% Ste94 discriminates between optical depth and optical thickness
% Optical depth = tau(z) = vertical optical depth
% Optical thickness = tau(s) = slant optical thickness
% Absorption/absorptivity/absorptance/absorbance:
% Absorption is process or absolute quantity, e.g., 100 W m-2
% Absorptance is fractional measure in [0.0,1.0]
% Absorptivity is as defined by Ramanathan
% Absorbance is defined by chemists as log10(I_o/I) (according to SGW)
% Same for reflection, transmission

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\begin{document} % End preamble

\hypersetup{ % A command provided by \hyperref
pdftitle={Radiative Transfer in the Earth System},
pdfsubject={Radiative Transfer in the Earth System},
pdfauthor={Charlie Zender},
pdfkeywords={rt ess236 ess}
} % end \hypersetup

\begin{center}
Online: \url{http://dust.ess.uci.edu/facts} \hfill Updated: \shortdate\today, \xxivtime\\
\bigskip
{\Large \textbf{Radiative Transfer in the Earth System}}\\
\bigskip
by Charlie Zender\\
University of California, Irvine\\
\end{center}
Department of Earth System Science \hfill \url{zender@uci.edu}\\
University of California \hfill Voice: (949)\thinspace 824-2987\\
Irvine, CA~~92697-3100 \hfill Fax: (949)\thinspace 824-3256

% GFDL legalities: http://www.gnu.ai.mit.edu/copyleft/fdl.html
\bigskip\noindent
Copyright \copyright\ 1998--2008, Charles S. Zender\\
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version~1.2
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
Texts.
The license is available online at
\url{http://www.gnu.ai.mit.edu/copyleft/fdl.html}.
\\
\\
\textbf{Facts about FACTs:}
This document is part of the 
\href{http://dust.ess.uci.edu/facts}{Freely Available Community Text (\acr{fact}) project}.  
\acr{fact}s are created, reviewed, and continuously maintained and updated
by members of the international academic community communicating with
eachother through a well-organized project website. 
\acr{fact}s are intended to standardize and disseminate our fundamental
knowledge of Earth System Sciences in a flexible, adaptive,
distributed framework which can evolve to fit the changing needs and
technology of the geosciences community. 
Currently available \acr{fact}s and their \acr{url}s are listed in
Table~\ref{tbl:fact}. 
\begin{table}[h] % h = ``here'' = position of table
\begin{minipage}{\hsize} % Minipage necessary for footnotes KoD95 p. 110 (4.10.4)
\renewcommand{\footnoterule}{\rule{\hsize}{0.0cm}\vspace{-0.0cm}} % KoD95 p. 111
\begin{center}
\caption[\acr{fact}s]{\textbf{Freely Available Community Texts}%
\label{tbl:fact}}
\vspace{\cpthdrhlnskp}
\begin{tabular}{ r l }
\hline \rule{0.0ex}{\hlntblhdrskp}% 
Format & \acr{url} Location \\[0.0ex]
\hline \rule{0.0ex}{\hlntblntrskp}%
& \\[-1.0ex]
\multicolumn{2}{c}{\centering{Radiative Transfer in the Earth System}} \\[-0.5ex]
DVI & \url{http://dust.ess.uci.edu/facts/rt/rt.dvi} \\
PDF & \url{http://dust.ess.uci.edu/facts/rt/rt.pdf} \\
Postscript & \url{http://dust.ess.uci.edu/facts/rt/rt.ps} \\[1.0ex]
\multicolumn{2}{c}{Particle Size Distributions: Theory and Application to Aerosols,
Clouds, and Soils \hfill} \\[-0.5ex]
DVI & \url{http://dust.ess.uci.edu/facts/psd/psd.dvi} \\
PDF & \url{http://dust.ess.uci.edu/facts/psd/psd.pdf} \\
Postscript & \url{http://dust.ess.uci.edu/facts/psd/psd.ps} \\
\multicolumn{2}{c}{Natural Aerosols in the Climate System \hfill} \\[-0.5ex]
DVI & \url{http://dust.ess.uci.edu/facts/aer/aer.dvi} \\
PDF & \url{http://dust.ess.uci.edu/facts/aer/aer.pdf} \\
Postscript & \url{http://dust.ess.uci.edu/facts/aer/aer.ps} \\
%& \\[1.0ex]
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}
Because of its international scope and availability to students
of all income levels, the \acr{fact} project may impact more students,
and to a greater depth, than imaginable to before the advent of the
Internet. 
If you are interested in learning more about \acr{fact}s and how you
might contribute to or benefit from the project please contact
\url{zender@uci.edu}. 

\pagenumbering{roman}
\setcounter{page}{1}
\pagestyle{headings}
\thispagestyle{empty}
%\onecolumn

\clearpage
\begin{center}
\bigskip
{\Large \textbf{Notes for Students of ESS~200B,\\ Earth System Physics:}}\\
\bigskip
\end{center}
This monograph on Radiative Transfer provides some core and some
supplementary reading material for ESS~200B.
We will discuss much of the material in the first twenty pages,
and the figures at the end.
\csznote{
% Equivalence to Hou02
Much of the material in \cite{Hou02}, Chapters~2 and~4, is described
at a more advanced level in the rest of this monograph.
\citeauthor{Hou02}~\S\,2.2 is our \S\,\ref{sxn:plk_fnc}, \ref{sxn:ext},
and \ref{sxn:str_atm};
\citeauthor{Hou02}~\S\,2.3 is our \S\,\ref{sxn:rte_plk_sln};
Part of \citeauthor{Hou02}~\S\,4.1 is our \S\,\ref{sxn:sct}; 
Part of \citeauthor{Hou02}~\S\,4.2 is our \S\,\ref{sxn:two_srm}; 
Part of \citeauthor{Hou02}~\S\,4.3 is our \S\,\ref{sxn:eqv_wdt} and \S\,\ref{sxn:lnshp}; 
\citeauthor{Hou02}~\S\,4.4 is our \S\,\ref{sxn:HCG} and \S\,\ref{sxn:lsd}; 
\citeauthor{Hou02}~\S\,4.5 is our \S\,\ref{sxn:rte_sln_frm}--\ref{sxn:rte_plk_sln}; 
This FACT has no equivalent to \citeauthor{Hou02}~\S\,4.6--4.8.
\citeauthor{Hou02}~\S\,4.9 is our \S\,\ref{sxn:nbm}; 
and \citeauthor{Hou02}~\S\,4.10 is our \S\,\ref{sxn:rad_frc}. 
% \stepcounter{idxpg} % [nbr] Page number of Index
} % end csznote
The Index beginning on page~\pageref{sxn:idx} is also helpful.
\begin{center}
\bigskip
{\Large \textbf{Notes for Students of ESS~236,\\ Radiative Transfer and Remote Sensing:}}\\
\bigskip
\end{center}
Yada yada yada.
\clearpage
\tableofcontents
\listoffigures
\listoftables
\pagenumbering{arabic}
\setcounter{page}{1}
%\markleft{Radiative Transfer}
%\markright{}
\thispagestyle{empty}

\section{Introduction}\label{sxn:ntr}

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 \cite{BoH83} (particle scattering), 
\cite{GoY89} (band models), and \cite{ThS99} (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. 

\subsection{Planetary Radiative Equilibrium}\label{sxn:nrg_bln}
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
\begin{eqnarray}
\frac{\partial \nrg}{\partial \tm} & = & \flxabssw - \flxolr
\label{eqn:nrg_bln_dfn}
\end{eqnarray}
On timescales longer than about a year the Earth as a whole is thought
to be in \trmidx{planetary radiative equilibrium}.
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
\begin{eqnarray}
\flxabssw & = & \flxolr
\label{eqn:rdn_eqm_dfn}
\end{eqnarray}
The total amount of solar energy available for the Earth to absorb is
the incoming solar flux (or \trmidx{irradiance}) at the top of Earth's
atmosphere, $\flxslrtoa$ (aka the \trmidx{solar constant}), times the
intercepting area of Earth's disk which is $\mpi \rdsrth^{2}$.
Since Earth rotates, the total mean incident flux $\mpi \rdsrth^{2}
\flxslrtoa$ 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\trmidx{planetary albedo} or \trmidx{spherical albedo}, $\rfl$.
Satellite observations show that $\rfl \approx 0.3$.
Thus only $(1 - \rfl)$ of the mean incident solar flux contributes
to warming the planet and we have
\begin{eqnarray}
\flxabssw & = & (1 - \rfl) \flxslrtoa / 4
\label{eqn:flx_abs_sw_dfn}
\end{eqnarray}

Earth does not cool to space as a perfect blackbody
(\ref{eqn:plk_frq_dfn}) 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 \trmdfn{effective temperature} $\tptffc$ of an object is the
temperature of the blackbody which would produce the same irradiance.
Inverting the \trmidx{Stefan-Boltzmann Law} (\ref{eqn:stf_blt_dfn})
yields
\begin{eqnarray}
\tptffc & \equiv & ( \flxolr / \cststfblt )^{1/4}
\label{eqn:tpt_ffc_dfn}
\end{eqnarray}
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 \trmdfn{greenhouse effect}.

Substituting (\ref{eqn:flx_abs_sw_dfn}) and (\ref{eqn:tpt_ffc_dfn})
into (\ref{eqn:rdn_eqm_dfn})
\begin{eqnarray}
(1 - \rfl) \flxslrtoa / 4 & = & \cststfblt \tptffc^{4} \\
\tptffc & = & \left( \frac{(1 - \rfl) \flxslrtoa}
{4 \cststfblt} \right)^{1/4}
\label{eqn:tpt_ffc_rth}
\end{eqnarray}
For Earth, $\rfl \approx 0.3$ and $\flxslrtoa \approx 1367$\,\wxmS.
Using these values in (\ref{eqn:tpt_ffc_rth}) yields $\tptffc =
255$\,K\@.
Observations show the mean surface temperature $\tptsfc = 288$\,K\@.

\subsection[Fundamentals]{Fundamentals}\label{sxn:fnd}
The fundamental quantity describing the electromagnetic spectrum is
\trmdfn{frequency}, $\frq$.
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, \xs.
Frequency is intrinsic to the oscillator and does not depend on the
medium in which the waves are travelling.
The \trmdfn{energy} carried by a photon is proportional to its
frequency 
\begin{eqnarray}
\nrg & = & \cstplk \frq
\label{eqn:nrg_dfn}
\end{eqnarray}
where $\cstplk$ is \trmdfn{Planck's constant}.
Regrettably, almost no radiative transfer literature expresses
quantities in frequency.

A related quantity, the \trmdfn{angular frequency} $\frqngl$ 
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
\begin{eqnarray}
\frqngl & = & 2\mpi \frq
\label{eqn:frq_ngl_dfn}
\end{eqnarray}
Since radians are considered dimensionless, the units of $\frqngl$ are
\xs. 
However, angular frequency is also rarely used in radiative transfer. 
Thus some authors use the symbol $\omega$ to denote the element 
of \trmidx{solid angle}, as in $\dfr\omega$.
The reader should be careful not to misconstrue the two meanings.
In this text we use $\omega$ only infrequently.

Most radiative transfer literature use \trmdfn{wavelength} or
\trmdfn{wavenumber}.
Wavelength, $\wvl$ (m), measures the distance between two adjacent
peaks or troughs in the wavefield.
The universal relation between wavelength and frequency is
\begin{eqnarray}
\wvl \frq & = & \cstspdlgt
\label{eqn:wvl_dfn}
\end{eqnarray}
where $\cstspdlgt$ is the \trmdfn{speed of light}.
Since $\cstspdlgt$ depends on the medium, $\wvl$ also depends on the
medium. 

The \trmdfn{wavenumber} $\wvn$\,\xm, is exactly the inverse of
wavelength 
\begin{eqnarray}
\wvn & \equiv & \frac{1}{\wvl} = \frac{\frq}{\cstspdlgt}
\label{eqn:wvn_dfn}
\end{eqnarray}
Thus $\wvn$ measures the number of oscillations per unit distance,
i.e., the number of wavecrests per meter.
Using (\ref{eqn:wvl_dfn}) in (\ref{eqn:wvn_dfn}) 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 \xcm\ than in SI units of
\xm. 
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 (\xcm)
and energy in SI units (J) is obtained by using (\ref{eqn:wvn_dfn}) in 
(\ref{eqn:nrg_dfn})
\begin{eqnarray}
\nrg & = & 100 \cstplk \cstspdlgt \frq
\label{eqn:nrg_wvn_dfn}
\end{eqnarray}

There is another, distinct quantity also called \trmidx{wavenumber}. 
This secondary usage of wavenumber in this text is the traditional
measure of spatial wave propagation and is denoted by $\wvnbr$. 
\begin{eqnarray}
\wvnbr & \equiv & 2\mpi \tilde{\frq}
\label{eqn:wvnbr_dfn}
\end{eqnarray}
The wavenumber $\wvnbr$ is set in Roman typeface as an additional
distinction between it and other symbols
\footnote{The script $\kkk$ is already used for Boltzmann's constant, 
absorption coefficients, and vibrational modes}.

Table~\ref{tbl:wv_cnv} summarizes the relationships between the
fundamental parameters which describe wave-like phenomena.
\begin{table}
\begin{minipage}{\hsize} % Minipage necessary for footnotes KoD95 p. 110 (4.10.4)
\renewcommand{\footnoterule}{\rule{\hsize}{0.0cm}\vspace{-0.0cm}} % KoD95 p. 111
\begin{center}
\caption[Wave Parameter Conversion Table]{\textbf{Wave Parameter Conversion Table}%
% fxm: 20050513 latex2html appears to choke when math mode is used in footnotes
\footnote{The speed of light is $\cstspdlgt$\,\mxs.}%
\footnote{Table entries express the column in terms of the row.}%
\label{tbl:wv_cnv}}
\vspace{\cpthdrhlnskp}
\begin{tabular}{ >{$\displaystyle}r<{$} *{6}{>{$\displaystyle}c<{$}} } % KoD95 p. 94 describes '*' notation
\hline \rule{0.0ex}{\hlntblhdrskp}% 
\mbox{Variable} & \frq & \wvl & \wvn & \frqngl & \wvnbr & \tau \\[0.0ex]
\mbox{Units} & \mbox{\xs} & \mbox{m} & \mbox{\xcm} & \mbox{\xs} & \mbox{\xm} & \mbox{s} \\[0.0ex]
\hline \rule{0.0ex}{\hlntblntrskp}%
\frq & - & \frac{\cstspdlgt}{\frq} & \frac{\frq}{100\cstspdlgt} & 2\mpi\frq & \frac{2\mpi\frq}{\cstspdlgt} & \frac{1}{\frq} \\[3.0ex]
\wvl & \frac{\cstspdlgt}{\wvl} & - & \frac{1}{100\wvl} & \frac{2\mpi\cstspdlgt}{\wvl} & \frac{2\mpi}{\wvl} & \frac{\wvl}{\cstspdlgt} \\[3.0ex]
\wvn & 100\cstspdlgt\wvn & \frac{1}{100\wvn} & - & \frac{\wvn}{200\mpi\cstspdlgt} & 200\mpi\wvn & \frac{1}{100\cstspdlgt\wvn} \\[3.0ex]
\frqngl & \frac{\frqngl}{2\mpi} & \frac{2\mpi\cstspdlgt}{\frqngl} & \frac{\frqngl}{200\mpi\cstspdlgt} & - & \frac{\frqngl}{\cstspdlgt} & \frac{2\mpi}{\frqngl} \\[3.0ex]
\wvnbr & \frac{\wvnbr\cstspdlgt}{2\mpi} & \frac{2\mpi}{\wvnbr} & \frac{\wvnbr}{200\mpi} & \cstspdlgt\wvnbr & - & \frac{2\mpi}{\cstspdlgt\wvnbr} \\[3.0ex]
\tau & \frac{1}{\tau} & \cstspdlgt\tau & \frac{1}{100\cstspdlgt\tau} & \frac{2\mpi}{\tau} & \frac{2\mpi}{\cstspdlgt\tau} & - \\[3.0ex]
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}

\section{Radiative Transfer Equation}\label{sxn:rte}

\subsection{Definitions}

\subsubsection[Intensity]{Intensity}\label{sxn:ntn}
The fundamental quantity defining the radiation field is the
\trmdfn{specific intensity} of radiation.
Specific intensity, also known as \trmdfn{radiance}, 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 \jxmSssrm.
The SI unit of power (1\,Watt $\equiv$ 1\,Joule~per~second) is
preferred, leading to units of \wxmSsrm.
Often the specific intensity is expressed in terms of spectral
frequency $\ntnfrq$ with units \wxmSsrhz\ or
spectral wavenumber (also $\ntnwvn$) with units \wxmSsrwvn.

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
\begin{equation}
\dfr\nrg = \ntnfrq(\psnvct,\nglhat,\tm,\frq) \,\dfr\sfc \,\dfr\tm \,\dfr\ngl \,\dfr\frq
\label{eqn:ntn_dfn1}
\end{equation}
It is not convenient to measure the radiant flux across surface
orthogonal to $\nglhat$, as in (\ref{eqn:ntn_dfn1}), 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 (\ref{eqn:ntn_dfn1}) to account for projection
of $\dfr\sfc$ onto~$\dfr\xsx$.
If the angle between $\nrmhat$ and $\nglhat$ is $\plr$ then
\begin{equation}
\cos \plr = \nrmhat \cdot \nglhat
\label{eqn:plr_dfn}
\end{equation}
and the projection of $\dfr\sfc$ onto $\dfr\xsx$ yields
\begin{equation}
\dfr\xsx = \cos \plr \,\dfr\sfc
\label{eqn:dxsx_dfn}
\end{equation}
so that
\begin{equation}
\dfr\nrg = \ntnfrq(\psnvct,\nglhat,\tm,\frq) 
\cos \plr \,\dfr\xsx \,\dfr\tm \,\dfr\ngl \,\dfr\frq
\label{eqn:ntn_dfn}
\end{equation}
The conceptual advantage that (\ref{eqn:ntn_dfn}) has over
(\ref{eqn:ntn_dfn1}) is that (\ref{eqn:ntn_dfn}) 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 \trmdfn{stratified atmosphere} 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 \trmdfn{plane-parallel}
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 \trmidx{optical depth} $\tau$ (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 $\tau$ in a
plane parallel atmosphere is denoted by $\ntnfrq(\tau,\plr,\azi)$.
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 ($\tau$), then the field is
\trmdfn{homogeneous}. 
If $\ntnfrq$ is not a function of direction ($\nglhat$), then the
field is \trmdfn{isotropic}. 

\subsubsection[Mean Intensity]{Mean Intensity}\label{sxn:ntn_bar}
The \trmdfn{mean intensity} 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.
\begin{eqnarray}
\ntnmnfrq = \frac{\int_{\ngl} \ntnfrq \,\dfr\ngl}{\int_{\ngl} \dfr\ngl}
\label{eqn:ntn_bar_dfn1}
\end{eqnarray}
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 $4\mpi\rds^{2}$, there must be $4\mpi$
steradians in a sphere.
It is straightforward to demonstrate that the differential element of
area in \trmidx{spherical polar coordinates} 
is $\rds^{2} \sin \plr \,\dfr\plr \,\dfr\azi$. 
Thus the element of solid angle is
\begin{eqnarray}
\ngl & = & \AAA / \rds^{2} \nonumber \\
\dfr\ngl & = & \rds^{-2} \,\dfr\AAA \nonumber \\ 
& = & \rds^{-2} \, \rds^{2} \sin \plr \,\dfr\plr \,\dfr\azi \nonumber \\
& = & \sin \plr \,\dfr\plr \,\dfr\azi 
\label{eqn:ngl}
\end{eqnarray}
The \trmdfn{field of view} of an instrument, e.g., a telescope, is
most naturally measured by a solid angle.

The definition of $\ntnmnfrq$ (\ref{eqn:ntn_bar_dfn1}) 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 (\ref{eqn:ntn_bar_dfn1}) is
\begin{eqnarray}
\int_{\ngl} \dfr\ngl & = & \int_{\plr=0}^{\plr=\mpi} \int_{\azi=0}^{\azi=2\mpi} 
\sin \plr \,\dfr\plr \,\dfr\azi \nonumber \\
& = & \left[ \azi \right]_{0}^{2\mpi} \int_{\plr=0}^{\plr=\mpi} \sin
\plr \,\dfr\plr \nonumber \\
& = & 2 \mpi \int_{\plr=0}^{\plr=\mpi} \sin \plr \,\dfr\plr \nonumber \\
& = & 2 \mpi \left[ - \cos \plr \right]_{0}^{\mpi} \nonumber \\
& = & 2 \mpi [-(-1) - (-1)] \nonumber \\
& = & 4 \mpi
\label{eqn:srd_sph_dfn}
\end{eqnarray}
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 
\trmdfn{isotropic radiation}.
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 (\ref{eqn:srd_sph_dfn}) to (\ref{eqn:ntn_bar_dfn1}) yields
\begin{eqnarray}
\ntnmnfrq & = & \frac{1}{4 \mpi} \int_{\ngl} \ntnfrq \,\dfr\ngl
\label{eqn:ntn_bar_dfn}
\end{eqnarray}
$\ntnmnfrq$ has units of radiance, \wxmSsrhz.
If the radiation field is azimuthally independent (i.e., $\ntnfrq$
does not depend on $\azi$), then 
\begin{eqnarray}
\ntnmnfrq & = & \frac{1}{2} \int_{0}^{\mpi} \ntnfrq \sin \plr \,\dfr\plr
\label{eqn:ntn_bar_plr_dfn}
\end{eqnarray}

Let us simplify (\ref{eqn:ntn_bar_plr_dfn}) by introducing the change
of variables 
\begin{eqnarray}
\plru & = & \cos \plr \\
\dfr\plru & = & -\sin \plr \,\dfr\plr
\label{eqn:plru_dfn}
\label{eqn:cov_plru}
\end{eqnarray}
This maps $\plr \in [0, \mpi]$ into $\plru \in [1, -1]$ so that
(\ref{eqn:ntn_bar_plr_dfn}) becomes
\begin{eqnarray}
\ntnmnfrq & = & -\frac{1}{2} \int_{1}^{-1} \ntnfrq \,\dfr\plru \nonumber \\
\ntnmnfrq & = &  \frac{1}{2} \int_{-1}^{1} \ntnfrq \,\dfr\plru
\label{eqn:ntn_bar_plru_dfn}
\end{eqnarray}

The \trmdfn{hemispheric intensities} or \trmdfn{half-range
intensities} are simply the up- and downwelling components 
of which the full intensity is composed
\begin{subequations}
\label{eqn:ntn_hms_dfn}
\begin{align}
\ntnfrq(\tau,\nglhat) = \ntnfrq(\tau,\plr,\azi) & = \left\{ 
\begin{array}{r@{\quad:\quad}ll}
\ntnupfrq(\tau,\plr,\azi) & 0 < \plr < \mpi/2 \\
\ntndwnfrq(\tau,\plr,\azi) & \mpi/2 < \plr < \mpi
\end{array} \right. \\
\ntnfrq(\tau,\nglhat) = \ntnfrq(\tau,\plru,\azi) & = \left\{ 
\begin{array}{r@{\quad:\quad}l}
\ntnupfrq(\tau,\plru,\azi) & 0 \le u < 1 \\
\ntndwnfrq(\tau,\plru,\azi) & -1 < u < 0
\end{array} \right.
\end{align}
\end{subequations}

\begin{subequations}
\begin{align}
\label{eqn:ntn_hms_up}
\ntnupfrq(\tau,\plrmu,\azi) & = \ntnfrq(\tau,+\plrmu,\azi) 
= \ntnfrq(\tau,0 < \plr < \mpi/2,\azi) = \ntnfrq(\tau,0 < \plru <
1,\azi) \\
\label{eqn:ntn_hms_dwn}
\ntndwnfrq(\tau,\plrmu,\azi) & = \ntnfrq(\tau,+\plrmu,\azi) 
= \ntnfrq(\tau,\mpi/2 < \plr < \mpi,\azi) = \ntnfrq(\tau,-1 < \plru <
0,\azi) 
\end{align}
\end{subequations} 

\subsubsection[Irradiance]{Irradiance}\label{sxn:flx}
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
\cite[]{Mad87}.  
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 $\cos \Theta = \nglhat \cdot \nglhatprm$, thus
\begin{eqnarray}
\flxfrq & = & \int_{\ngl} \ntnfrq \cos \plr \,\dfr\ngl \nonumber \\
& = & \int_{\plr=0}^{\plr=\mpi} \int_{\azi=0}^{\azi=2\mpi} 
\ntnfrq \cos \plr \, \sin \plr \,\dfr\plr \,\dfr\azi 
\label{eqn:flx_dfn}
\end{eqnarray}
In a plane-parallel medium, this defines the net specific irradiance
passing through a given vertical level. 
Note the similarity between (\ref{eqn:ntn_bar_dfn}) and
(\ref{eqn:flx_dfn}).  
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 (\ref{eqn:flx_dfn}) 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 (\ref{eqn:flx_dfn}) by introducing the change of
variables $u =\cos \plr$, $du = -\sin \plr \,\dfr\plr$.
This maps $\plr \in [0, \mpi]$ into $\plru \in [1, -1]$: 
\begin{eqnarray}
\flxfrq & = & \int_{\plru=1}^{\plru=-1} \int_{\azi=0}^{\azi=2\mpi} 
\ntnfrq \plru \, (- \dfr\plru) \,\dfr\azi \nonumber \\
& = & \int_{\plru=-1}^{\plru=1} \int_{\azi=0}^{\azi=2\mpi} 
\ntnfrq \plru \,\dfr\plru \,\dfr\azi
\label{eqn:flx_udfn}
\end{eqnarray}

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
\begin{eqnarray}
\flxfrq \,\dfr\frq & = & - \flxwvl \,\dfr\wvl \nonumber \\
\flxfrq & = & - \flxwvl \, \frac{\dfr\wvl}{\dfr\frq} \nonumber \\
& = & - \flxwvl \, \frac{d}{\dfr\frq} \left( \frac{\cstspdlgt}{\frq} \right) \nonumber \\
& = & - \flxwvl \, \left( - \frac{\cstspdlgt}{\frq^{2}} \right) \nonumber \\
\flxfrq & = & \frac{\cstspdlgt}{\frq^{2}} \flxwvl =
\frac{\wvl^{2}}{\cstspdlgt} \flxwvl \\
\label{eqn:flx_frq_wvl}
\flxwvl & = & \frac{\cstspdlgt}{\wvl^{2}} \flxfrq =
\frac{\frq^{2}}{\cstspdlgt} \flxfrq 
\label{eqn:flx_wvl_frq}
\end{eqnarray}
Thus $\flxfrq$ and $\flxwvl$ are always of the same sign.

\subsubsection[Actinic Flux]{Actinic Flux}\label{sxn:flx_act}
A quantity of great importance in photochemistry is the total
convergence of radiation at a point.
This quantity, called the \trmdfn{actinic flux}, $\flxact$, 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
\begin{eqnarray}
\flxactfrq & = & \int_{4\mpi} \ntnfrq \,\dfr\ngl \nonumber \\
& = & 4\mpi \ntnmnfrq
\label{eqn:flx_act_dfn}
\end{eqnarray}
Thus the actinic flux is simply $4\mpi$ times the mean
intensity~$\ntnmnfrq$ (\ref{eqn:ntn_bar_dfn}). 
$\flxactfrq$~has units of \wxmShz\ which are identical to the units of
irradiance~$\flxfrq$ (\ref{eqn:flx_dfn}). 
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 \trmdfn{photo-absorption}.
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 $\cncA$\,[\xmC], the
actinic radiation field~$\flxactfrq$, and the efficiency with with
each molecule absorbs photons.
This efficiency is called the \trmdfn{absorption cross-section},
\trmdfn{molecular cross section}, or simply \trmdfn{cross-section}.
The absorption cross-section is denoted by $\xsxabs$ and has units of 
[\xmS].
In the literature, however, values of $\xsxabs$ usually appear in CGS
units [\xcmS].
To make the frequency-dependence of $\xsxabs$ explicit 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~\A\ will absorb a photon with frequency in
$[\frq,\frq+\dfr\frq]$ is proportional to $\flxactfrq(\frq) \xsxabsoffrq$%
\footnote{The proportionality constant is $(\cstplk\frq)^{-1}$, i.e., 
when the actinic radiation field $\flxactfrq(\frq)$ is converted
from energy (\jxmSshz) to photons (\phtxmSshz) then 
$\flxactfrq(\frq) \xsxabsoffrq$ is the probability of absorption per
second per unit frequency}. 
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 $\flxabsfrq$\,[\wxmC] be the energy absorbed per unit time, per
unit frequency, per unit volume of air.
Then
\begin{eqnarray}
\flxabsfrq(\frq) & = & \cncA \flxactfrq(\frq) \xsxabsoffrq
\label{eqn:flx_abs_dfn}
\end{eqnarray}
where $\cncA$\,\xmC\ is the number concentration of~\A.

Photochemists are interested in the probability of absorbed radiation  
severing molecular bonds, and thus decomposing species \AB\ into
constituent species \A\ and \B. 
Notationally this process may be written in any of the equivalent
forms 
\begin{rxnarray}
\AB + \hnu & \yields & \AAA + \BBB \nonumber \\
\AB + \hnu & \yields^{\frq > \frqnot} & \AAA + \BBB \nonumber \\
\AB + \hnu & \yields^{\wvl < \wvlnot} & \AAA + \BBB
\label{rxn:pch_AB}
\end{rxnarray}
Both forms indicate that the efficiency with which reaction
(\ref{rxn:pch_AB}) proceeds is a function of photon energy.
The second form makes explicit that the photodissociation reaction
does not proceed unless $\frq < \frqnot$, where $\frqnot$ is 
the \trmdfn{photolysis cutoff frequency}.
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 \AB\ will result in the 
photodissociation of \AB, and the completion of (\ref{rxn:pch_AB}), is 
called the \trmdfn{quantum yield} or \trmdfn{quantum efficiency} and
is represented by $\qntyld$.
As a probability, $\qntyld$ is dimensionless%
\footnote{Azimuthal angle and quantum yield are both
dimensionless quantities denoted by $\qntyld$.
The meaning of $\qntyld$ should be clear from the context.}. 
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 \trmdfn{photolysis rate coefficient} for the
photodissociation of a species \A\ is the number of photodissociations
of \A\ occuring per unit time, per unit volume of air, per unit
frequency, per molecule of \A. 
In accord with convention we denote the specific photolysis rate
coefficient by $\prcfrq$.
The units of $\prcfrq$ are \xshz.
\begin{eqnarray}
\prcfrq & = & 
\frac{\flxactfrq(\frq) \xsxabs(\frq) \qntyld(\frq)}{\cstplk \frq}
\nonumber \\
& = & 
\frac{4 \mpi \ntnmnfrq(\frq) \xsxabs(\frq) \qntyld(\frq)}{\cstplk \frq}
\label{eqn:prc_frq_dfn}
\end{eqnarray}
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 (\ref{eqn:prc_frq_dfn}) 
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 \trmdfn{photolysis rate coefficient} $\prc$ is obtained
by integrating (\ref{eqn:prc_frq_dfn}) over all frequencies which
may contribute to photodissociation
\begin{eqnarray}
% ThS99 p. 164 (5.80)
\prc & = & \int_{\frq > \frqnot} \prcfrq(\frq) \,\dfr\frq
\nonumber \\
& = & \int_{\frq > \frqnot} 
\frac{\flxactfrq(\frq) \xsxabs(\frq) \qntyld(\frq)}
{\cstplk \frq} \,\dfr\frq \nonumber \\
& = & \frac{4\mpi}{\cstplk} 
\int_{\frq > \frqnot} 
\frac{\ntnmnfrq(\frq) \xsxabs(\frq) \qntyld(\frq)}{\frq} 
\,\dfr\frq
\label{eqn:prc_dfn}
\end{eqnarray}
As mentioned above, evaluation of (\ref{eqn:prc_dfn}) requires
essentially a complete knowledge of the radiative and photochemical
properties of the environment and species of interest.
Moreover, $\prc$ is notoriously difficult to compute where there 
is any uncertainty in the input quantities.

Integration errors due to the discretization of (\ref{eqn:prc_dfn})
are quite common. 
To compute $\prc$ with high accuracy, regular grids must have
resolotion of $\sim$1\,nm in the ultraviolet, \cite[]{Mad89}.
Much of the difficulty is due to the steep but opposite gradients 
of $\flxactfrq$ and $\qntyld$ which occur in the ultraviolet.
High frequency features in $\xsxabs$ worsens 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
\cite[][]{CBI87,DaS91,TMA89,StT901,Pet95,LaC981,WZP00}. 
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 
\begin{rxnarray}
\NOd + \hnu & \yields^{\wvl < 420\mathrm{\,nm}} & \NO + \Ou
\label{rxn:NO2+hv_NO+O}
\end{rxnarray}
If [\NOd] denotes the number concentration of \NOd\ 
in a closed system where photolysis is the only sink of \NOd, 
then
\begin{eqnarray}
\frac{\dfr[\NOd]}{\dfr\tm} & = & - \prcNOd [\NOd] + \SSS_{\NOd}
\label{eqn:prc_NO2_dfn}
\end{eqnarray}
where $\SSS_{\NOd}$ represents all sources of \NOd.
The terms in (\ref{eqn:prc_NO2_dfn}) all have dimensions of \xmCs.
The first term on the RHS is the \trmdfn{photolysis rate} of \NOd\
in the system. 

Figure~\ref{fgr:j_NO2_spc} shows the spectral distribution of actinic flux
in a clear mid-latitude summer atmosphere, and the absorption
cross-section and quantum yield of \NOd.
\begin{figure*}
\centering
\includegraphics[width=0.8\hsize]{/data/zender/fgr/rt/j_NO2_act_flx}\vfill
\includegraphics[width=0.8\hsize]{/data/zender/fgr/rt/j_NO2_abs_xsx}\vfill
\includegraphics[width=0.8\hsize]{/data/zender/fgr/rt/j_NO2_qnt_yld}\vfill
\caption[Cross Section and Quantum Yield of Nitrogen Dioxide]{
(a)~Spectral distribution of actinic flux $\flxact$\,[\phtxmSsum] at TOA
and at the surface for a \trmidx{mid-latitude summer} (MLS) atmosphere
with a unit optical depth of dust or sulfate in the lowest kilometer.
(b)~Absorption cross section of \NOd, $\xsxabsNOd$\,[\mSxmlc].
(c)~Quantum yield of \NOd, $\qntyldNOd$ (\ref{rxn:NO2+hv_NO+O}).
\label{fgr:j_NO2_spc}}
\end{figure*}

Figure~\ref{fgr:j_NO2} shows the vertical distribution of
$\prcNOd$\,[\xs] for the conditions shown in
Figure~(\ref{fgr:j_NO2_spc}). 
\begin{figure*}
\centering
\includegraphics[width=0.8\hsize]{/data/zender/fgr/rt/j_NO2_arese_19951011}\vfill
\includegraphics[width=0.8\hsize]{/data/zender/fgr/rt/j_NO2_rlt_arese_19951011}\vfill
\caption[Vertical Distribution of Photodissociation Rates]{
Vertical distribution of $\prcNOd$\,[\xs] (\ref{rxn:NO2+hv_NO+O})
for the conditions shown in Figure~(\ref{fgr:j_NO2_spc}). 
(a)~Absolute rates. 
(b)~Rates normalized by clean sky rates.
\label{fgr:j_NO2}}
\end{figure*}

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 (\ref{eqn:prc_dfn}).
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.

\subsubsection[Actinic Flux Enhancement]{Actinic Flux Enhancement}\label{sxn:flx_act_cld}
The actinic flux~$\flxactfrq$ (\ref{eqn:flx_act_dfn}) 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 \trmdfn{actinic flux enhancement}.
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~\ref{tbl:flxact}. 
\begin{table}
\begin{minipage}{\hsize} % Minipage necessary for footnotes KoD95 p. 110 (4.10.4)
\renewcommand{\footnoterule}{\rule{\hsize}{0.0cm}\vspace{-0.0cm}} % KoD95 p. 111
\begin{center}
\caption[Actinic Flux Enhancement]{\textbf{Actinic Flux Enhancement by
Scattering}%
\footnote{Terminology: $\flxdwnfrq$ is downwelling irradiance of
source, $\rfllmboffrq$ is Lambertian reflectance of surface (or
cloud), $\ntnfrqofngl$ is resulting intensity field, $\flxupfrq$ is
upwelling irradiance, $\ntnmnfrq$ is mean intensity, $\flxactfrq$ is
actinic flux.}%
\label{tbl:flxact}}
\vspace{\cpthdrhlnskp}
\begin{tabular}{ >{\raggedright}p{8em}<{} *{6}{>{$\displaystyle}c<{$}} } % KoD95 p. 94 describes '*' notation
\hline \rule{0.0ex}{\hlntblhdrskp}% 
Description & \flxdwnfrq & \rfllmboffrq & \ntnfrqofngl & \flxupfrq & \ntnmnfrq & \flxactfrq \\[0.0ex]
\hline \rule{0.0ex}{\hlntblntrskp}%
Collimated, non-reflecting & \flxslrfrq & 0 & \flxslrfrq\dltfncofnglhatmnglhatnot & 0 & \frac{\flxslrfrq}{4\mpi} & \flxslrfrq \\[1.0ex]
Isotropic, non-reflecting & \flxslrfrq & 0 & 
\begin{array}{rcl}
\ntndwnfrq & = & \flxslrfrq/\mpi \\
\ntnupfrq & = & 0
\end{array}
& 0 & \frac{\flxslrfrq}{2\mpi} & 2 \flxslrfrq \\[1.0ex]
Collimated, reflecting & \flxslrfrq & 1 & 
\begin{array}{rcl}
\ntndwnfrq & = & \flxslrfrq\dltfncofnglhatmnglhatnot \\
\ntnupfrq & = & \flxslrfrq/\mpi
\end{array}
& \flxslrfrq & \frac{3\flxslrfrq}{4\mpi} & 3\flxslrfrq \\[1.0ex]
Isotropic, reflecting & \flxslrfrq & 1 & \flxslrfrq/\mpi & \flxslrfrq & \frac{\flxslrfrq}{\mpi} & 4 \flxslrfrq \\[1.0ex]
\hline
\end{tabular}
\end{center}
\end{minipage}
\end{table}
The scenarios differ in the isotropicity of the downwelling radiance
(\trmidx{collimated} or \trmidx{isotropic}) and the
\trmidx{reflectance} $\rfllmboffrq$ (0 or~1) of the lower boundary,
which is taken to be a \trmidx{Lambertian surface}.
All scenarios are driven by the same downwelling irradiance, which 
is taken to be the direct solar beam.
The scenarios are arranged in order of increasing actinic flux
$\flxactfrq$, shown in the final column. 
$\flxactfrq$ is seen to increase with both the number and the
brightness of reflecting surfaces.
The reflectivity of natural surfaces is no more than 90\% nor less than
5\%\footnote{Glaciers are the most reflective surfaces in nature, with
$\rfllmboffrq \lesssim 0.9$. 
Maximum cloud reflectance is $\lesssim 0.7$.
Dark forests and ocean have $\rfllmboffrq \gtrsim 0.05$.
See discussion in \S\ref{sxn:lmb} and Table~\ref{tbl:alb_sfc}.}, so that 
the $\flxactfrq$ in Table~\ref{tbl:flxact} 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
(\ref{eqn:flx_act_dfn}).
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
\begin{eqnarray}
\flxactfct & \equiv & \flxactfrq/\flxdwnfrq
\label{eqn:flx_act_fct_dfn}
\end{eqnarray}
Table~\ref{tbl:flxact} 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 ($\tau \gtrsim 3$) 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 $\flxdwnfrq \approx \flxslrfrq$.
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 
\begin{eqnarray}
\flxactfsh & \equiv & \frac{\flxactfrq}{4 \flxdwnfrq}
\label{eqn:flx_act_fsh_dfn}
\end{eqnarray}
Clearly $0 \le \flxactfsh \le 1$.
Table~\ref{tbl:flxact} 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~\ref{tbl:flxact}.  

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.

\subsubsection[Energy Density]{Energy Density}\label{sxn:nrg_dns}
Another quantity of interest is the density of radiant energy per unit
volume of space.
We call this quantity the \trmdfn{energy density} $\nrgdnsfrq$.
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 \trmidx{actinic flux}
$\flxactfrq$ (\ref{eqn:flx_act_dfn}) and thus to the mean intensity
$\ntnmnfrq$.
\begin{eqnarray}
% ThS99 p. 40 (2.9)
\nrgdnsfrq & = & \int_{4\mpi} \,\dfr\nrgdnsfrq \nonumber \\
& = & \frac{4\mpi}{\cstspdlgt} \ntnmnfrq
\label{eqn:nrg_dns_dfn}
\end{eqnarray}
The units of $\nrgdnsfrq$ are \jxmChz.

\subsubsection[Spectral vs.\ Broadband]{Spectral vs.\ Broadband}\label{sxn:spc_bb}
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 \trmdfn{spectral} 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 \trmdfn{band-integrated} radiant
quantity.
Band-integrated radiant fields are often called \trmdfn{narrowband} 
or \trmdfn{broadband} Depending on the size of the frequency range, 
\trmdfn{broadband} radiant
are obtained by integrating over all frequencies:
\begin{eqnarray}
\ntn & = & \int_{0}^{\infty} \ntnfrq(\frq) \,\dfr\frq \nonumber \\
\flx & = & \int_{0}^{\infty} \flxfrq(\frq) \,\dfr\frq \nonumber \\
\flxact & = & \int_{0}^{\infty} \flxactfrq(\frq) \,\dfr\frq \nonumber \\
\flxabs & = & \int_{0}^{\infty} \flxabsfrq(\frq) \,\dfr\frq \nonumber \\
\prc & = & \int_{0}^{\infty} \prcfrq(\frq) \,\dfr\frq \nonumber \\
\nrgdns & = & \int_{0}^{\infty} \nrgdnsfrq(\frq) \,\dfr\frq \nonumber \\
\htr & = & \int_{0}^{\infty} \htrfrq(\frq) \,\dfr\frq \nonumber
\label{eqn:bb_dfn}
\end{eqnarray}

\subsubsection[Thermodynamic Equilibria]{Thermodynamic Equilibria}\label{sxn:tdy_eqm}
Temperature plays a fundamental role in radiative transfer because 
$\tpt$ determines the population of excited atomic states, which
in turn determines the potential for \trmdfn{thermal emission}.
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 \trmdfn{planetary radiative equilibrium}
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 \trmdfn{thermodynamic equilibrium}, or~TE\@.

Radiation in thermodynamic equilibrium with matter plays a fundamental 
role in radiation transfer. 
Such radiation is most commonly known as \trmdfn{blackbody radiation}.  
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 \trmdfn{local thermodynamic  
equilibrium}, or LTE\@.  

\subsubsection[Planck Function]{Planck Function}\label{sxn:plk_fnc}
The Planck function $\plkfrq$ describes the intensity of blackbody
radiation as a function of temperature and wavelength
% 19991011: subequation/align environment is more cramped than equivalent eqnarray
\begin{subequations}
\label{eqn:plk_dfn}
\begin{align}
\label{eqn:plk_frq_dfn}
\plkfrq(\tpt,\frq) & = \frac{2 \cstplk \frq^{3}}{\cstspdlgt^{2} 
( \me^{\cstplk \frq / \cstblt \tpt} - 1)} \\
\label{eqn:plk_wvl_dfn}
\plkwvl(\tpt,\wvl) & = \frac{2 \cstplk \cstspdlgt^{2}}{\wvl^{5} 
( \me^{\cstplk \cstspdlgt / \wvl \cstblt \tpt} - 1)}
\end{align}
\end{subequations}
Blackbodies emit isotropically---this considerably simplifies 
thermal radiative transfer. 
The correct predictions (\ref{eqn:plk_frq_dfn}) 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 (\ref{eqn:plk_frq_dfn}) and (\ref{eqn:plk_wvl_dfn})
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
\begin{eqnarray}
\plkfrq \,\dfr\frq & = & - \plkwvl \,\dfr\wvl
\label{eqn:plk_frq_wvl_eqv}
\end{eqnarray}
Once again, the negative sign arises as a result of the opposite
senses of increasing frequency versus increasing wavelength.
We may derive (\ref{eqn:plk_wvl_dfn}) from (\ref{eqn:plk_frq_dfn}) 
by using (\ref{eqn:wvl_dfn}) in (\ref{eqn:plk_frq_wvl_eqv}) 
\begin{eqnarray}
\plkfrq & = & - \plkwvl \, \frac{\dfr\wvl}{\dfr\frq} \nonumber \\
& = & \frac{\plkwvl \cstspdlgt}{\frq^{2}} \nonumber \\
& = & \plkwvl \cstspdlgt \left( \frac{\wvl^{2}}{\cstspdlgt^{2}} \right) \nonumber \\
\plkfrq & = & \frac{\wvl^{2}}{\cstspdlgt} \plkwvl = 
\frac{\cstspdlgt}{\frq^{2}} \plkwvl \\
\label{eqn:plk_frq_wvl_dfn}
\plkwvl & = & \frac{\frq^{2}}{\cstspdlgt} \plkfrq =
\frac{\cstspdlgt}{\wvl^{2}} \plkfrq
\label{eqn:plk_wvl_frq_dfn}
\end{eqnarray}
These relations are analogous to (\ref{eqn:flx_frq_wvl}).

The Planck function (\ref{eqn:plk_dfn}) has interesting behavior in
both the high and the low energy photon limits. 
In the high energy limit, known as \trmdfn{Wien's limit},
the photon energy greatly exceeds the ambient thermal energy
\begin{eqnarray}
\cstplk \frqmshmax \gg \cstblt \tpt
\label{eqn:wien_dfn}
\end{eqnarray}
In Wien's limit, (\ref{eqn:plk_dfn}) becomes
\begin{subequations}
\label{eqn:plk_wien_dfn}
% cf ThS99 p. 94, KiK80 p. 94, GoY89 p. 30 (2.41)
\begin{align}
\label{eqn:plk_frq_wien_dfn}
\plkfrq(\tpt,\frq) = & \frac{2 \cstplk \frq^{3}}{\cstspdlgt^{2}}
\me^{-\cstplk \frq / \cstblt \tpt} \\
\label{eqn:plk_wvl_wien_dfn}
\plkwvl(\tpt,\wvl) = & \frac{2 \cstplk \cstspdlgt^{2}}{\wvl^{5}} 
\me^{-\cstplk \cstspdlgt / \wvl \cstblt \tpt}
\end{align}
\end{subequations}

In the very low energy limit, known as the \trmdfn{Rayleigh-Jeans
limit}, the photon energy is much less than the ambient thermal energy 
\begin{eqnarray}
\cstplk \frqmshmax & \ll & \cstblt \tpt
\label{eqn:ryl_jean_dfn}
\end{eqnarray}
Thus in the Rayleigh-Jeans limit the arguments to the exponential in
(\ref{eqn:plk_dfn}) are less than~$1$ so the exponentials may be
expanded in Taylor series.
Starting from (\ref{eqn:plk_frq_dfn}) 
\begin{eqnarray}
% cf ThS99 p. 94, KiK80 p. 94
\plkfrq(\tpt,\frq) & \approx & 
\frac{2 \cstplk \frq^{3}}{\cstspdlgt^{2}}
\left(1 + \frac{\cstplk \frq}{\cstblt \tpt} -1 \right)^{-1}
\nonumber \\
& = & 
\frac{2 \cstplk \frq^{3} \cstblt \tpt}{\cstspdlgt^{2} \cstplk \frq}
\nonumber \\
& = & 
\frac{2 \frq^{2} \cstblt \tpt}{\cstspdlgt^{2}}
\end{eqnarray}
Similar manipulation of (\ref{eqn:plk_wvl_dfn}) may be performed 
and we obtain
\begin{subequations}
\label{eqn:plk_ryl_jean_dfn}
\begin{align}
\label{eqn:plk_frq_ryl_jean_dfn}
\plkfrq(\tpt,\frq) & \approx  
\frac{2 \frq^{2} \cstblt \tpt}{\cstspdlgt^{2} } \\
\label{eqn:plk_wvl_ryl_jean_dfn}
\plkwvl(\tpt,\wvl) & \approx  
\frac{2 \cstspdlgt \cstblt \tpt}{\wvl^{4}}
\end{align}
\end{subequations}

The frequency of extreme emission is obtained by taking the partial
derivative of (\ref{eqn:plk_frq_dfn}) with respect to frequency with
the temperature held constant
\begin{eqnarray}
\frac{\partial \plkfrq}{\partial \frq} 
\bigg|_{\tpt}
& = &
\frac{2 \cstplk}{\cstspdlgt^{2}} \times
\frac{1}{( \me^{\cstplk \frq / \cstblt \tpt}- 1)^{2}} \times
\left( 3 \frq^{2} ( \me^{\cstplk \frq / \cstblt \tpt}- 1) -
\frq^{3} \frac{\cstplk}{\cstblt \tpt} \me^{\cstplk \frq / \cstblt \tpt} \right)
\nonumber
\end{eqnarray}
To solve for the \trmdfn{frequency of maximum emission}, $\frqmshmax$, 
we set the RHS equal to zero so that one or more of the LHS factors
must equal zero 
\begin{eqnarray}
\frac{2 \cstplk \frqmshmax^{2}
\left[ 3 ( \me^{\cstplk \frqmshmax / \cstblt \tpt}- 1 ) -
\frac{\cstplk \frqmshmax}{\cstblt \tpt} \me^{\cstplk \frqmshmax / \cstblt \tpt} \right] }
{\cstspdlgt^{2} ( \me^{\cstplk \frqmshmax / \cstblt \tpt}- 1)^{2}}
& = & 0
\nonumber \\
3 ( \me^{\cstplk \frqmshmax / \cstblt \tpt}- 1 ) -
\frac{\cstplk \frqmshmax}{\cstblt \tpt} \me^{\cstplk \frqmshmax / \cstblt \tpt}
& = & 0
\nonumber \\
\frac{\cstplk \frqmshmax}{\cstblt \tpt} \me^{\cstplk \frqmshmax / \cstblt \tpt}
& = &
3 ( \me^{\cstplk \frqmshmax / \cstblt \tpt}- 1 )
\nonumber \\
\frac{\cstplk \frqmshmax}{\cstblt \tpt}
& = &
3 ( 1 - \me^{-\cstplk \frqmshmax / \cstblt \tpt})
% \frqmshmax & = &
% \frac{3 \cstblt \tpt ( 1 - \me^{-\cstplk \frqmshmax / \cstblt \tpt})}{\cstplk}
\label{eqn:frq_max_dfn}
\end{eqnarray}
An analytic solution to (\ref{eqn:frq_max_dfn}) 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 $\xxx \approx 2.8215$ so that 
\begin{eqnarray}
% KiK80 p. 98 (27)
\frac{\cstplk \frqmshmax}{\cstblt \tpt} & \approx & 2.82 \nonumber \\
\frqmshmax & \approx & 2.82 \cstblt \tpt / \cstplk \nonumber \\
& \approx & 5.88 \times 10^{10} \, \tpt \qquad\mbox{\hz}
\label{eqn:frq_max_apx}
\end{eqnarray}
where the units of the numerical factor are \hzxk.
Thus the frequency of peak blackbody emission is directly
proportional to temperature.
This is known as \trmdfn{Wien's Displacement Law}.

A separate relation may be derived for the wavelength of maximum
emission $\wvlmshmax$ by an analogous procedure starting from
(\ref{eqn:plk_wvl_dfn}). 
The result is
\begin{eqnarray}
% Ste94 p. 68 (2.45), ThS99 p. 94 (4.5)
\wvlmshmax & \approx & 2897.8 / \tpt \qquad\mbox{\um}
\label{eqn:wvl_max_apx}
\end{eqnarray}
where the units of the numerical factor are \umk.
Note that (\ref{eqn:frq_max_apx}) and (\ref{eqn:wvl_max_apx})
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 (\ref{eqn:frq_max_apx}) 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 $\wvlmshmax$\,\um.
Then a first approximation is that the planetary temperature is close
to $2897.8 / \wvlmshmax$.

\subsubsection[Hemispheric Quantities]{Hemispheric Quantities}\label{sxn:flx_hms}
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 \trmdfn{hemispheric} irradiances measure the radiant energy
transport in the vertical direction.
These hemispheric irradiances depend only on the
corresponding \trmdfn{hemispheric intensities}.
Let us assume the intensity field is azimuthally independent, i.e., 
$\ntnfrq = \ntnfrq(\plr)$ only.
Then the azimuthal contribution to (\ref{eqn:flx_udfn}) is $2\mpi$ and
\begin{eqnarray}
\flxfrq & = & 2 \mpi \int_{\plru=-1}^{\plru=1} \ntnfrq \plru \,\dfr\plru \nonumber \\
& = & 2 \mpi \left( 
\int_{\plru=-1}^{\plru=0} \ntnfrq \plru \,\dfr\plru +
\int_{\plru=0}^{\plru=1} \ntnfrq \plru \,\dfr\plru \right)
\label{eqn:hms_u}
\end{eqnarray}
We now introduce the change of variables $\plrmu = |\plru| = |\cos
\plr|$. 
Referring to (\ref{eqn:plru_dfn}) we find
\begin{subequations}
\label{eqn:plrmu_dfn}
\begin{align}
\plrmu =  | \cos \plr | & = \left\{ 
\begin{array}{r@{\quad:\quad}ll}
\cos \plr & 0 < \plr < \mpi/2 \\
-\cos \plr & \mpi/2 < \plr <  \mpi
\end{array} \right. \\
\plrmu =  |u| & = \left\{ 
\begin{array}{r@{\quad:\quad}l}
 u & 0 \le u < 1 \\
-u & -1 < u < 0
\end{array} \right.
\end{align}
\end{subequations}
Most formal work on radiative transfer is written in terms of
$\plrmu$ rather than $\plru$ or $\plr$.
Substituting (\ref{eqn:plrmu_dfn}) into (\ref{eqn:hms_u})
\begin{eqnarray}
\flxfrq & = & 2 \mpi \left( 
\int_{\plrmu=1}^{\plrmu=0} \ntnfrq (-\plrmu) \, (-\dfr\plrmu) +
\int_{\plrmu=0}^{\plrmu=1} \ntnfrq \plrmu \,\dfr\plrmu \right) \nonumber \\
& = & 2 \mpi \left( 
\int_{\plrmu=1}^{\plrmu=0} \ntnfrq \plrmu \,\dfr\plrmu +
\int_{\plrmu=0}^{\plrmu=1} \ntnfrq \plrmu \,\dfr\plrmu \right) \nonumber \\
& = & 2 \mpi \left( 
-\int_{\plrmu=0}^{\plrmu=1} \ntnfrq \plrmu \,\dfr\plrmu +
\int_{\plrmu=0}^{\plrmu=1} \ntnfrq \plrmu \,\dfr\plrmu \right) \nonumber \\
& = & -\flxdwnfrq + \flxupfrq \nonumber \\
& = & \flxupfrq - \flxdwnfrq
\end{eqnarray}
where we have defined the \trmdfn{hemispheric fluxes} or
\trmdfn{half-range fluxes}
\begin{subequations}
\label{eqn:flx_hms}
\begin{align}
\label{eqn:flx_up_frq}
\flxupfrq & = 2 \mpi \int_{0}^{1} \ntnfrq (+\plrmu) \plrmu \,
\dfr\plrmu \\
\label{eqn:flx_dwn_frq}
\flxdwnfrq & = 2 \mpi \int_{0}^{1} \ntnfrq (-\plrmu) \plrmu \,
\dfr\plrmu
\end{align}
\end{subequations} 
The hemispheric fluxes are positive definite, and their difference is
the net flux.
\setlength{\fboxsep}{6pt} % KoD95 p. 92
\newline\fbox{\parbox{\hsize}{  % KoD95 p. 138
\begin{equation}
\flxfrq = \flxupfrq - \flxdwnfrq
\label{eqn:flx_hms_2}
\end{equation} 
}} % end \fbox
\begin{equation} 
\fbox{$ \displaystyle % KoD95 p. 147
\flxfrq = \flxupfrq - \flxdwnfrq $} % end \fbox
\label{eqn:flx_hms_3}
\end{equation} 
The superscripts $^+$ and $^-$ denote \trmdfn{upwelling}
(towards the upper hemisphere) and \trmdfn{downwelling} (towards the
lower hemisphere) quantities, respectively.
The net flux $\flxfrq$ is the difference between the upwelling and
downwelling hemispheric fluxes, $\flxupfrq$ 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 $\flxupfrq = \flxdwnfrq$ (\ref{eqn:flx_hms_3}).
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 $\plkfrqtpt$
(\ref{eqn:plk_frq_dfn}) 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 (\ref{eqn:flx_up_frq}) 
and we obtain
\begin{eqnarray} 
\flxplkfrq & = & 2 \mpi \int_{0}^{1} \plkfrq \plrmu \,\dfr\plrmu \nonumber \\
& = & 2 \mpi \plkfrq \int_{0}^{1} \plrmu \,\dfr\plrmu \nonumber \\
& = & 2 \mpi \plkfrq \frac{\plrmu^{2}}{2} \bigg|_{\plrmu = 0}^{\plrmu = 1} \nonumber \\
% fxm: 20040124 tth breaks here on use of \textstyle unless enclosed with braces
& = & 2 \mpi \plkfrq \, ({\textstyle\frac{1}{2}} - 0) \nonumber \\
& = & \mpi \plkfrq
\label{eqn:plk_hms_dfn}
\end{eqnarray} 
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 
$\frac{1}{2}$ difference between the naive and the correct solutions.

\subsubsection[Stefan-Boltzmann Law]{Stefan-Boltzmann Law}\label{sxn:stf_blt}
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 $\plkfrq$ (\ref{eqn:plk_frq_dfn})
directly to obtain the broadband intensity 
$\plkfnc \equiv \int_{0}^{\infty} \plkfrq(\frq) \,\dfr\frq$,
it is traditional to integrate $\plkfrq$ first over the hemisphere 
(\ref{eqn:plk_hms_dfn}). 
By proceeding in this order, we shall obtain the total hemispheric
blackbody irradiance $\flxupplk$ in terms fundamental physical
constants and the temperature of the body.
\begin{eqnarray}
\flxupplk & = & \mpi \int_{0}^{\infty} \plkfrq \,\dfr\frq 
\nonumber \\
& = & \mpi \int_{0}^{\infty} 
\frac{2 \cstplk \frq^{3}}{\cstspdlgt^{2} 
( \me^{\cstplk \frq / \cstblt \tpt} - 1)} \,\dfr\frq
\nonumber \\
& = & \frac{2 \mpi \cstplk}{\cstspdlgt^{2}} 
\int_{0}^{\infty} 
\frac{\frq^{3}}{\me^{\cstplk \frq / \cstblt \tpt} - 1} \,\dfr\frq
\label{eqn:stf_blt_1}
\end{eqnarray}
To simplify (\ref{eqn:stf_blt_1}) we make the change of variables 
\begin{eqnarray}
\xxx & = & \frac{\cstplk \frq}{\cstblt \tpt} \nonumber \\
\frq & = & \frac{\cstblt \tpt \xxx}{\cstplk} \nonumber \\
\dfr\frq & = & \frac{\cstblt \tpt}{\cstplk} \,\dfr\xxx \nonumber \\
\dfr\xxx & = & \frac{\cstplk}{\cstblt \tpt} \,\dfr\frq \nonumber
\label{eqn:cov_x}
\end{eqnarray}
This change of variables maps $\frq \in [0,\infty)$ to
$\xxx \in [0,\infty)$.
Substituting this into (\ref{eqn:stf_blt_1}) we obtain
\begin{eqnarray}
\flxupplk 
& = & \frac{2 \mpi \cstplk}{\cstspdlgt^{2}} 
\int_{0}^{\infty} 
\left( \frac{\cstblt \tpt}{\cstplk} \right)^{3}
\frac{\xxx^{3}}{\me^{\xxx} - 1} \, \frac{\cstblt \tpt}{\cstplk} \,\dfr\xxx
\nonumber \\
& = & \frac{2 \mpi \cstblt^{4} \tpt^{4} }{\cstspdlgt^{2} \cstplk^{3}} 
\int_{0}^{\infty} 
\frac{\xxx^{3}}{\me^{\xxx} - 1} \,\dfr\xxx
\label{eqn:stf_blt_2}
\end{eqnarray}

The definite integral in (\ref{eqn:stf_blt_2}) is $\mpi^{4}/15$. 
Proving this is a classic problem in mathematical physics which
involves the \trmidx{Riemann zeta function} (and thus prime number
theory), the \trmidx{Gamma function}, and contour integration.
The procedure used to obtain this result is interesting so we briefly
summarize it here.
The \trmdfn{Riemann zeta function} $\rmnztafnc(\xxx)$ for real $\xxx > 1$
may be defined as
\begin{eqnarray}
\rmnztafnc(\xxx) & \equiv & \frac{1}{\gmmfnc(\xxx)}
\int_{0}^{\infty} \frac{\uuu^{\xxx-1}}{\me^{\uuu}-1} \,\dfr\uuu
\label{eqn:rmn_zta_dfn}
\end{eqnarray}
Comparing (\ref{eqn:stf_blt_2}) with the Riemann zeta function
definition (\ref{eqn:rmn_zta_dfn}), we see that $\xxx=4$, i.e, 
\begin{eqnarray}
\flxupplk 
& = & \frac{2 \mpi \cstblt^{4} \tpt^{4} }{\cstspdlgt^{2} \cstplk^{3}} 
\gmmfnc(4)\rmnztafnc(4)
\label{eqn:stf_blt_rmn}
\end{eqnarray}

The integral (\ref{eqn:rmn_zta_dfn}) is analytically solvable for 
the special case of integers $\xxx = \nnn$.
We may transform the integrand from a rational fraction into a
\trmidx{power series} using algebraic manipulation:
\begin{eqnarray}
\frac{\uuu^{\xxx-1}}{\me^{\uuu}-1} 
& = & \frac{\uuu^{\xxx-1}}{\me^{\uuu}-1} \times \frac{\me^{-\uuu}}{\me^{-\uuu}} \nonumber \\
& = & \frac{\uuu^{\xxx-1}\me^{-\uuu}}{1-\me^{-\uuu}} \nonumber \\
& = & \uuu^{\xxx-1}\me^{-\uuu} \times \frac{1}{1-\me^{-\uuu}}
\label{eqn:rmn_ntg}
\end{eqnarray}
The integration limits in (\ref{eqn:rmn_zta_dfn}) are $[0,\infty)$
so it is always true that $\me^{-\uuu} < 1$ in (\ref{eqn:rmn_ntg}),
and thus in the integrand of (\ref{eqn:rmn_zta_dfn}).

Recall that the sum of an infinite \trmdfn{power series} with 
initial term $\aaa_{0}$ and ratio $\rrr$ is
\begin{eqnarray}
\sum_{\kkk=0}^{\infty} \aaa_{0}\rrr^{\kkk} & = & 
\lim_{\kkk \to \infty} \aaa_{0} + \aaa_{0}\rrr + \aaa_{0}\rrr^{2} + \cdots
+ \aaa_{0}\rrr^{\kkk-1} + \aaa_{0}\rrr^{\kkk} \nonumber \\
& = & \aaa_{0}/(1-\rrr) 
\quad \mbox{for\ } |\rrr| < 1
\label{eqn:pwr_srs}
\end{eqnarray}
Hence the last term in (\ref{eqn:rmn_ntg}) is the sum of a power
series (\ref{eqn:pwr_srs}) with initial term $\aaa_{0} = 1$ and 
ratio $\rrr = \me^{-\uuu}$.
\begin{eqnarray}
\frac{1}{1-\me^{-\uuu}}
& = & \sum_{\kkk=0}^{\infty} \me^{-\kkk\uuu}
\label{eqn:xpn_pwr_srs}
\end{eqnarray}
Substituting (\ref{eqn:xpn_pwr_srs}) into (\ref{eqn:rmn_ntg}) 
we obtain
\begin{eqnarray}
% http://mathworld.wolfram.com/RiemannZetaFunction.html
\frac{\uuu^{\xxx-1}}{\me^{\uuu}-1} 
& = & \uuu^{\xxx-1}\me^{-\uuu} \sum_{\kkk=0}^{\infty} \me^{-\kkk\uuu} \nonumber \\
& = & \sum_{\kkk=0}^{\infty} \uuu^{\xxx-1} \me^{-(\kkk+1)\uuu} \nonumber \\
& = & \sum_{\kkk=1}^{\infty} \uuu^{\xxx-1} \me^{-\kkk\uuu}
\label{eqn:rmn_ntg_pwr_srs}
\end{eqnarray}
where the last step shifts the initial index to from zero to one.

Using the inifinite series representation (\ref{eqn:rmn_ntg_pwr_srs}) for
the integrand of (\ref{eqn:rmn_zta_dfn}) yields
\begin{eqnarray}
\rmnztafnc(\xxx) & = & \frac{1}{\gmmfnc(\xxx)}
\int_{0}^{\infty} \left[ \sum_{\kkk=1}^{\infty} \uuu^{\xxx-1}
  \me^{-\kkk\uuu} \right] \,\dfr\uuu \nonumber
\label{eqn:rmn_pwr_srs_dfn_1}
\end{eqnarray}
Integration and addition are commutative operations.
Interchanging their order yields
\begin{eqnarray}
\rmnztafnc(\xxx) & = & \frac{1}{\gmmfnc(\xxx)}
\sum_{\kkk=1}^{\infty} 
\left[ \int_{0}^{\infty} \uuu^{\xxx-1} \me^{-\kkk\uuu}  \,\dfr\uuu \right]
\label{eqn:rmn_pwr_srs_dfn_2}
\end{eqnarray}
We change variables from 
$\uuu \in [0,+\infty]$ to $\yyy \in [0,+\infty]$ with
\begin{eqnarray}
\yyy & = & \kkk\uuu \nonumber \\
\dfr\yyy & = & \kkk \,\dfr\uuu \nonumber \\
\uuu & = & \yyy/\kkk \nonumber \\
\dfr\uuu & = & \kkk^{-1} \,\dfr\yyy
\label{eqn:rmn_cov}
\end{eqnarray}
so that (\ref{eqn:rmn_pwr_srs_dfn_2}) becomes
\begin{eqnarray}
\rmnztafnc(\xxx) & = & \frac{1}{\gmmfnc(\xxx)}
\sum_{\kkk=1}^{\infty} 
\left[ \int_{0}^{\infty} \left( \frac{\yyy}{\kkk} \right)^{\xxx-1} 
\me^{-\yyy} \kkk^{-1} \,\dfr\yyy \right] \nonumber \\
& = & \frac{1}{\gmmfnc(\xxx)}
\sum_{\kkk=1}^{\infty} \kkk^{-(\xxx-1)} \kkk^{-1}
\left[ \int_{0}^{\infty} \yyy^{\xxx-1} \me^{-\yyy} \,\dfr\yyy \right] \nonumber \\
& = & \frac{1}{\gmmfnc(\xxx)}
\sum_{\kkk=1}^{\infty} \kkk^{-\xxx} [ \gmmfnc(\xxx) ] \nonumber \\
& = & \sum_{\kkk=1}^{\infty} \kkk^{-\xxx}
% fxm: Exactly where must we have assumed \xxx is integer \nnn?
\label{eqn:rmn_pwr_srs_dfn_3}
\end{eqnarray}
where we replaced the integral in brackets with the 
\trmidx{Gamma function} 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 $\Sigma_{1}^{\infty} \kkk^{-\nnn}$
(\ref{eqn:rmn_pwr_srs_dfn_3}) for positive, even integers~$\nnn$.
Our immediate concern is $\nnn = 4$.
Consider the complex function \cite[][p.~97]{CCP83}
\begin{eqnarray}
% CCP83 p. 97
\fnc(\zzz) & = & \frac{\mpi\cot\mpi\zzz}{\zzz^{4}}
\label{eqn:cnt_ntg_fnc}
\end{eqnarray}
This function 
\begin{enumerate*}
\item Is \trmdfn{analytic} throughout the complex plane
\item Has first order poles at all integer values on the real axis
  (except the origin)
\item Has a fifth order pole at the origin
\item Satisfies $\lim_{|\RRR| \to \infty} \fnc(\zzz) = 0$
  where $\zzz = \RRR\me^{\mi\plr}$
\end{enumerate*}
Therefore (\ref{eqn:cnt_ntg_fnc}) obeys the \trmidx{residue theorem}
for suitably chosen contours.
In other words, fxm.

Having shown 
\begin{eqnarray}
% NB: Contour integration of Riemann zeta function in AM201 notes p. 117
% Full solution is now in end of AM201 notes
\rmnztafnc(4) & = & \mpi^{4}/90
\label{eqn:rmn_zta_four}
\end{eqnarray}

Using $\gmmfnc(4) = 3! = 3 \times 2 \times 1 = 6$
and $\rmnztafnc(4) = \mpi^{4}/90$ from (\ref{eqn:rmn_zta_four}),
Equation~(\ref{eqn:stf_blt_rmn}) becomes
\begin{eqnarray}
\flxupplk 
& = & \frac{2 \mpi \cstblt^{4}}{\cstspdlgt^{2} \cstplk^{3}} 
\times 6 \times \frac{\mpi^{4}}{90} \times \tpt^{4}
\nonumber \\
& = & \frac{2 \mpi^{5} \cstblt^{4}}{15 \cstspdlgt^{2} \cstplk^{3}} \tpt^{4}
\nonumber
\label{eqn:stf_blt_3}
\end{eqnarray}
This is known as the \trmdfn{Stefan-Boltzmann Law} of radiation,
and is usually written as 
\begin{eqnarray}
\label{eqn:stf_blt_dfn}
\flxupplk & = & \cststfblt \tpt^{4} \qquad \mbox{where} \\
\label{eqn:cst_stf_blt_dfn}
\cststfblt & \equiv & 
\frac{2 \mpi^{5} \cstblt^{4}}{15 \cstspdlgt^{2} \cstplk^{3}}
\end{eqnarray}
$\cststfblt$ is known as the \trmdfn{Stefan-Boltzmann constant}
and depends only on fundamental physical constants.
The value of $\cststfblt$ is $5.67032 \times 10^{-8}$\,\wxmSkQ.
The thermal emission of matter depends very strongly (quartically) on 
$\tpt$ (\ref{eqn:stf_blt_dfn}). 
This rather surprising result has profound implications for Earth's
climate. 

We derived $\flxupplk$ (\ref{eqn:stf_blt_dfn}) directly so that
the Stefan-Boltzmann constant would fall naturally from the
derivation. 
For completeness we now present the broadband blackbody intensity
$\plkfnc$
\begin{eqnarray}
\plkfnc & = & \int_{0}^{\infty} \plkfrq(\frq) \,\dfr\frq \nonumber \\
& = & \frac{2 \mpi^{4} \cstblt^{4}}{15 \cstspdlgt^{2} \cstplk^{3}} \tpt^{4} \nonumber \\
& = & \flxupplk / \mpi = \cststfblt \tpt^{4} / \mpi 
\label{eqn:plk_fnc_dfn}
\end{eqnarray}
The factor of $\mpi$ difference between $\plkfnc$ and $\flxupplk$ 
is at first confusing.
One must remember that $\plkfnc$ is the broadband
(spectrally-integrated) intensity and that $\flxupplk$ is the 
broadband hemispheric irradiance
(spectrally-and-angularly-integrated).  

\subsubsection[Luminosity]{Luminosity}\label{sxn:lmn}
The total thermal emission of a body (e.g., star or planet) is called  
its \trmdfn{luminosity}, $\lmn$.
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.
\begin{eqnarray}
\lmnslr & = & 4\mpi\dstslr^{2} \flxslrtoa
\label{eqn:lmn_dfn_1}
\end{eqnarray}
The meaning of the \trmdfn{solar constant} $\flxslrtoa$ is made clear
by (\ref{eqn:lmn_dfn_1}).
$\flxslrtoa$ 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 $\sim 1.5 \times 10^{11}$\,m\ and 
$\flxslrtoa \approx 1367$\,\wxmS\ so 
$\lmnslr = 3.9 \times 10^{26}$\,W.
Note that $\lmn$ has dimensions of power, i.e., \jxs.

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 \trmidx{effective temperature} (\ref{eqn:tpt_ffc_dfn}) of
our Sun is $\tptffc$, the radius $\rdsslr$ at which this emission must
originate is defined by combining (\ref{eqn:stf_blt_dfn}) with
(\ref{eqn:lmn_dfn_1})
\begin{eqnarray}
4\mpi\rdsslr^{2} \cststfblt \tptslr^{4} & = & 4\mpi\dstslr^{2} \flxslrtoa 
\nonumber \\
\rdsslr^{2} & = & \frac{\dstslr^{2} \flxslrtoa}{\cststfblt \tptslr^{4}} 
\nonumber \\
\rdsslr & = &
\frac{\dstslr}{\tptslr^{2}}\sqrt{\frac{\flxslrtoa}{\cststfblt}} 
\label{eqn:rds_slr_dfn}
\end{eqnarray}
For the Sun-Earth system, $\tptslr \approx 5800$\,K so
$\rdsslr = 6.9 \times 10^{8}$\,m. 
Solar radiation received by Earth appears to originate from a portion
of the solar atmosphere known as the \trmdfn{photosphere}. 

\subsubsection[Extinction and Emission]{Extinction and Emission}\label{sxn:ext}
Radiation and matter have only two forms of interactions,
\trmdfn{extinction} and \trmdfn{emission}.
Lambert\footnote{Beer is usually credited with first formulating this
law, but actually the similarly-name Bougher deserves the credit.},
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
\begin{equation}
\frac{\dfr\ntnfrq}{\dfr\pth} = -\extcffoffrq \ntnfrq \qquad\mathrm{Extinction\ only}
\label{eqn:ext_dfn}
\end{equation}
Here $\extcffoffrq$ is the \trmdfn{extinction coefficient}, 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 \trmdfn{extinction coefficient} 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 $\mpc$\,[\kgxmS],  
number path of molecules $\nbr$\,[\nbrxmS],
and geometric distance $\pth$\,[\m]. 
Each path measure has a commensurate extinction coefficient: 
\trmidx{the mass extinction coefficient} $\extcffmss$\,[\mSxkg] 
(i.e., optical cross-section per unit mass), 
\trmidx{the number extinction coefficient} $\extcffnbr$\,[\mSxmlc] 
(i.e., optical cross-section per molecule), 
\trmidx{the volume extinction coefficient} $\extcffvlm$\,[\mSxmC]$=$[\xm]
(i.e., optical cross-section per unit concentration).
These coefficients are inter-related by 
\begin{eqnarray}
\extcffmss\,\mbox{[\mSxkg]} & = & \frac{\extcffnbr\,\mbox{[\mSxmlc]}}{\dns\,\mbox{[\kgxmC]}} \nonumber \\
\extcffnbr\,\mbox{[\mSxmlc]} & = & \frac{\extcffmss\,\mbox{[\mSxkg]} \times \mmw\,\mbox{[\kgxmol]}}{\cstAvagadro\,\mbox{[\mlcxmol]}} \nonumber \\
\extcffvlm\,\mbox{[\xm]} & = & \extcffnbr\,\mbox{[\mSxmlc]} \times \nbrcnc\,\mbox{[\mlcxmC]}
\label{eqn:np}
\end{eqnarray}

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~\ref{sxn:lnshp_fct} we state our preference for working in 
mass 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
\begin{equation}
\frac{\dfr\ntnfrq}{\dfr\pth} = \extcffoffrq \srcfrq \qquad\mathrm{Emission\ only}
\label{eqn:msn_dfn}
\end{equation}
where $\srcfrq$ is known as the \trmdfn{source function}.
The source function plays an important role in radiative transfer
theory.
We show in \S\ref{sxn:rte_sln_frm} 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,
\begin{eqnarray}
\frac{\dfr\ntnfrq}{\dfr\pth} & = & -\extcffvlm \ntnfrq + \extcffvlm \srcfrq \nonumber \\
\frac{1}{\extcffvlm} \frac{\dfr\ntnfrq}{\dfr\pth} & = & -\ntnfrq + \srcfrq
\label{eqn:rte_dfn_smp}
\end{eqnarray}
Equation~(\ref{eqn:rte_dfn_smp}) is the \trmdfn{equation of radiative
transfer} in its simplest differential form.

\subsubsection[Optical Depth]{Optical Depth}\label{sxn:tau}
We define the \trmdfn{optical path} $\tautld$ between
points $\pnt_{1}$ and $\pnt_{2}$ as
\begin{eqnarray}
\tautld(\pnt_{1},\pnt_{2}) & = & \int_{\pnt_{1}}^{\pnt_{2}} \extcffvlm \,\dfr\pth
\nonumber \\ 
\dfr\tautld & = & \extcffvlm \,\dfr\pth
\label{eqn:tau_tld_dfn}
\end{eqnarray}
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 \trmidx{optical
depth}. 
The optical depth $\tau$ is the vertical component of the optical path
$\tautld$, i.e., $\tau$ measures extinction between vertical levels.
For historical reasons, the optical depth in planetary atmospheres is
defined $\tau = 0$ at the top of the atmosphere and $\tau = \taustr$
at the surface. 
This convention reflects the astrophysical origin of much of radiative
transfer theory.
Much like pressure, $\tau$ 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 $\zzz_{2} > \zzz_{1}$, and
then allow $\zzz_{2} \rightarrow \infty$
\begin{eqnarray}
\tau(\zzz_{1},\zzz_{2}) & = & \int_{\zzzprm=\zzz_{1}}^{\zzzprm=\zzz_{2}}
\extcffvlm \,\dfr\zzzprm \nonumber \\
\tau(\zzz,\infty) & = & 
\int_{\zzzprm=\zzz}^{\zzzprm=\infty} \extcffvlm \,\dfr\zzzprm \\
 & = & 
\int_{\zzzprm=\zzz}^{\zzzprm=\infty} \extcffvlm \,\dfr\zzzprm
\label{eqn:tau_int_dfn}
\end{eqnarray}
Equation~(\ref{eqn:tau_int_dfn}) is the integral definition of optical
depth. 
The differential definition of optical depth is obtained by
differentiating (\ref{eqn:tau_int_dfn}) with respect to the lower
limit of integration and using the fundamental theorem of
differential calculus  
\begin{eqnarray}
\dfr\tau & = & [\extcffvlm(\infty) - \extcffvlm(\zzz)] \,\dfr\zzz \nonumber
\\
& = & -\extcffvlm(\zzz) \,\dfr\zzz
\label{eqn:tau_dfn}
\end{eqnarray}
where the second step uses the convention that $\extcffvlm(\infty) =
0$.  
By convention, $\tau$ is positive definite, but
(\ref{eqn:tau_dfn}) shows that $\dfr\tau$ 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.

\subsubsection[Geometric Derivation of Optical Depth]{Geometric Derivation of Optical Depth}\label{sxn:tau_geo}
The optical depth of a column containing spherical particles may be
derived by appealing to intuitive geometric arguments. 
Consider a concentration of $\cnc$\,\xmC\ 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 \trmdfn{collimated} beam
of sunlight from one side, how much energy reaches the opposite side?

For this thought experiment, we will neglect the effects of
scattering\footnote{The effects of diffraction will not be explicitly
included either, but are implicit in the assumption of the horizontal
homogeneity of the radiative flux.}.
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 $\mpi \rds^{2} \fshext$, where $\fshext = 1$
for perfectly absorbing particles.
If $\fshext < 1$, then each particle removes fewer photons than
suggested by its geometric size\footnote{Although intuition suggests a
spherical particle should not remove more energy from a collimated
light beam than it can geometrically intercept, this is not the case. 
As discussed later, diffraction around the particle is important. 
In fact, for $\rds \gg \wvl$, $\fshext \rightarrow 2$. 
However, most of the extinction due to diffraction occurs as
scattering, not absorption.}. 
Conversely, if $\fshext > 1$ each particle removes more photons than 
suggested by its geometric size.
Each particle encountered removes $\mpi \rds^{2} \fshext
\flx(\hgt)$\,\wxmS\ 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
\begin{eqnarray}
\flxdlt & = & - \mpi \rds^{2} \fshext \flx \cnc \hgtdlt \nonumber \\
\frac{\flxdlt}{\flx} & = & - \mpi \rds^{2} \fshext \cnc \hgtdlt
\label{eqn:flx_slr_toa_dlt_xmp_1}
\end{eqnarray}
If we take the limit as $\hgtdlt \rightarrow 0$ then
(\ref{eqn:flx_slr_toa_dlt_xmp_1}) becomes 
\begin{eqnarray}
\frac{1}{\flx} \,\dfr\flx & = & -  \mpi \rds^{2} \fshext \cnc \,\dfr\hgt
\nonumber \\
{\dfr(\ln \flx)} & = & - \mpi \rds^{2} \fshext \cnc \,\dfr\hgt
\label{eqn:flx_slr_toa_dlt_xmp_2}
\end{eqnarray}
Let us define the \trmdfn{volume extinction coefficient} $\kkk$ as
\begin{equation}
\kkk \equiv \mpi \rds^{2} \fshext \cnc
\label{eqn:ext_xmp_2}
\end{equation}
Finally we define the optical depth $\tau$ in terms of the extinction
\begin{eqnarray}
\tau & \equiv & \kkk \hgtdlt \\
\tau & = & \mpi \rds^{2} \fshext \cnc \hgtdlt
\label{eqn:tau_xmp_2}
\end{eqnarray}
The dimensions of $\kkk$ are \xm\ and therefore $\tau$ is
dimensionless. 
All quantities which compose $\tau$ are positive by convention,
therefore $\tau$ itself is positive definite.

We are now prepared to solve (\ref{eqn:flx_slr_toa_dlt_xmp_2}) for
$\flx(\tau)$. 
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 $\tau$ 
\begin{equation}
\flx(\tau) = \flxslrtoa \me^{-\tau}
\label{eqn:flx_slr_toa_dlt_xmp_4}
\end{equation}
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 \trmdfn{extinction law}.

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. 
\begin{equation}
\lim_{\rds \gg \lambda} \fshext = 2
\label{eqn:fsh_ext_xmp}
\end{equation}
Thus particles larger than 5--10\,\um\ 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 $\tau$ (\ref{eqn:tau_xmp_2}) in terms of the aerosol mass
$\mss$, rather than number concentration $\cnc$.
For a monodisperse aerosol of density $\dns$, the mass concentration is
\begin{equation}
\mss = \frac{4}{3} \mpi \rds^{3} \dns \cnc
\label{eqn:mss_xmp}
\end{equation}
If we substitute 
\begin{equation}
\mpi \rds^{2} \cnc = \frac{3 \mss}{4 \rds \dns} \nonumber
\end{equation}
into (\ref{eqn:tau_xmp_2}) we obtain
\begin{equation}
\tau = \frac{3 \mss \fshext \hgtdlt}{4 \rds \dns}
\label{eqn:tau_xmp_3}
\end{equation}
Typical cloud particles have $\rds \sim 10$\,\um\ so that for visible
solar radiation with $\wvl \sim 0.5$\,\um\ we may employ
(\ref{eqn:fsh_ext_xmp}) to obtain 
\begin{equation}
\tau = \frac{3 \mss \hgtdlt}{2 \rds \dns}
\label{eqn:tau_xmp_4}
\end{equation}
Thus $\tau$ 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!

\subsubsection[Stratified Atmosphere]{Stratified Atmosphere}\label{sxn:str_atm}
We obtain the radiative transfer equation in terms of optical path
by substituting (\ref{eqn:tau_tld_dfn}) into (\ref{eqn:rte_dfn_smp})
\begin{eqnarray}
% ThS99 p. 151 (5.42)
\frac{\dfr\ntnfrq}{\dfr\tautld} & = & -\ntnfrq + \srcfrq
\label{eqn:rte_tau_tld_dfn}
\end{eqnarray}
A \trmidx{stratified atmosphere} 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.
\begin{subequations}
% ThS99 p. 154
\label{eqn:dfr_tau_tld_tau_dfn}
\begin{align}
\label{eqn:tau_tld_tau_dfn}
\dfr\tautld & = \plru^{-1} \,\dfr\tau \\
\label{eqn:tau_tau_tld_dfn}
\dfr\tau & = \plru \,\dfr\tautld
\end{align}
\end{subequations}
Substituting 
(\ref{eqn:tau_tld_tau_dfn}) into (\ref{eqn:rte_tau_tld_dfn})  
we obtain
\begin{eqnarray}
% ThS99 p. 159 (5.64)
\plru \frac{\dfr\ntnfrq}{\dfr\tau} & = & -\ntnfrq + \srcfrq
\label{eqn:rte_tau_dfn}
\end{eqnarray}
This is the differential form of the radiative transfer equation in a
plane parallel atmosphere and is valid for all angles.
The solution of (\ref{eqn:rte_tau_dfn}) is made difficult because
$\srcfrq$ \textit{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\index{surface reflectance}.
When combined, these two hemispheric boundary conditions span a
complete range of polar angle, and are thus sufficient to solve 
(\ref{eqn:rte_tau_dfn}).
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 (\ref{eqn:rte_tau_dfn}) into
its constituent upwelling and downwelling radiation components.

Using the definitions of the \trmidx{half-range intensities}
(\ref{eqn:ntn_hms_dfn}) in (\ref{eqn:rte_tau_dfn}) we obtain 
\begin{subequations}
% ThS99 p. 155 (5.51, 5.52)
\label{eqn:rte_plru_dfn}
\begin{align}
\label{eqn:rte_plru_dwn_dfn}
-\plru \frac{\dfr\ntndwnfrq}{\dfr\tau} & = \ntndwnfrq - \srcdwnfrq 
\qquad 0 < \plr < \mpi/2, \plru = \cos \plr > 0 \\
\label{eqn:rte_plru_up_dfn}
-\plru \frac{\dfr\ntnupfrq}{\dfr\tau} & = \ntnupfrq - \srcupfrq
\qquad \mpi/2 < \plr < \mpi, \plru = \cos \plr < 0
\end{align}
\end{subequations}
where we have simply multiplied (\ref{eqn:rte_tau_dfn}) by $-1$
in order to place the negative sign on the LHS for reasons that
will be explained shortly.
The definitions of $\srcdwnfrq$ and $\srcupfrq$ are exactly analogous
to (\ref{eqn:ntn_hms_dfn}).

We now change variables from $\plru$ to $\plrmu$
(\ref{eqn:plrmu_dfn}). 
Replacing $\plru$ by $\plrmu$ in (\ref{eqn:rte_plru_dwn_dfn})
is allowed since $\plrmu = \plru > 0$ in this hemisphere.
In the upwelling hemisphere where $\mpi/2 < \plr < \mpi$, 
$\plru < 0$ so that $\plrmu = -\plru$ (\ref{eqn:plrmu_dfn}).
This negative sign cancels the negative sign on the LHS of 
(\ref{eqn:rte_plru_up_dfn}), resulting in 
\begin{subequations}
% ThS99 p. 155 (5.51, 5.52)
\label{eqn:rte_hlf_dfn}
\begin{align}
\label{eqn:rte_hlf_dwn_dfn}
-\plrmu \frac{\dfr\ntndwnfrq}{\dfr\tau} & = \ntndwnfrq - \srcdwnfrq \\
\label{eqn:rte_hlf_up_dfn}
\plrmu \frac{\dfr\ntnupfrq}{\dfr\tau} & = \ntnupfrq - \srcupfrq
\end{align}
\end{subequations}
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 (\ref{eqn:rte_hlf_dwn_dfn})
and (\ref{eqn:rte_hlf_up_dfn}) 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 $\ntnupfrq$.
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~(\ref{eqn:rte_plru_dfn}) states that $\ntnupdwn$ depends
explicitly only on $\ntnupdwn$ and on $\srcupdwn$, but has no
\textit{explicit} dependence on $\ntndwnup$ or on $\srcdwnup$.
Thus it may appear that $\ntnup$ and $\ntndwn$ are completely
decoupled from eachother.
However, we shall see that in problems involving scattering, 
$\srcupdwn$ depends explicitly on $\ntndwnup$ because scattering
may change the trajectory of photons from upwelling to downwelling and
visa versa.
By coupling $\ntnup$ 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 (\ref{eqn:rte_hlf_dfn}) from 
being locally-dependent to depending on the global radiation field.

As a special case of (\ref{eqn:rte_tau_dfn}), consider a stratified,
non-scattering atmosphere in thermodynamic equilibrium.
Then the source function equals the Planck function $\srcfrq =
\plkfrq = \plkfrqtpt$ and we have
\begin{equation}
% ThS99 p. 151 (5.42)
\plru \frac{\dfr\ntnfrq}{\dfr\tau} = -\ntnfrq + \plkfrq
\label{eqn:rte_plk}
\end{equation}
Equation~(\ref{eqn:rte_plk}) is the basis of radiative transfer in the
thermal infrared, where scattering effects are often negligible.
The solution to (\ref{eqn:rte_plk}) is described in
\S\ref{sxn:rte_plk_sln}. 

\subsection{Integral Equations}\label{sxn:rte_ntg}

\subsubsection[Formal Solutions]{Formal Solutions}\label{sxn:rte_sln_frm}
It is useful to write down the formal solution to
(\ref{eqn:rte_tau_dfn}) before making additional assumption about
the form of the source function $\srcfrq$.
\begin{eqnarray}
\plru \frac{\dfr\ntnfrq}{\dfr\tau} & = & - \ntnfrq + \srcfrq \nonumber \\
\plru \,\dfr\ntnfrq & = & - \ntnfrq \,\dfr\tau + \srcfrq \,\dfr\tau \nonumber \\
\plru \,\dfr\ntnfrq + \ntnfrq \,\dfr\tau & = & \srcfrq \,\dfr\tau \nonumber \\
\dfr\ntnfrq + \frac{\ntnfrq}{\plru} \,\dfr\tau & = & \plrurcp \srcfrq \,
\dfr\tau
\label{eqn:rte_src_dff}
\end{eqnarray}
Multiplying (\ref{eqn:rte_src_dff}) by the \trmdfn{integrating factor}
$\exptauou$ 
\begin{eqnarray}
\exptauou \,\dfr\ntnfrq + \frac{\exptauou \ntnfrq}{\plru} \,\dfr\tau & =
& \plrurcp \exptauou \srcfrq \,\dfr\tau \nonumber \\
\dfr( \exptauou \, \ntnfrq ) & = & \plrurcp \exptauou \srcfrq
\,\dfr\tau \nonumber \\
\frac{\dfr( \exptauou \, \ntnfrq )}{\dfr\tau} & = & \plrurcp
\exptauou \srcfrq 
\label{eqn:rte_sln_dff}
\end{eqnarray}
The LHS side of (\ref{eqn:rte_sln_dff}) 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 (\ref{eqn:rte_sln_dff}) 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 (\ref{eqn:plrmu_dfn}), $+\plrmu$ and 
$-\plrmu$ uniquely specify the angles for which these boundary
conditions apply. 

The solution for upwelling radiance is obtained by replacing $\plru$
in (\ref{eqn:rte_sln_dff}) by $-\plrmu$ because $\plru < 0$ for
upwelling intensities.
\begin{eqnarray}
\frac{\dfr( \expmtauomu \, \ntnupfrq )}{\dfr\tau} & = & -\plrmurcp
\expmtauomu \srcupfrq 
\label{eqn:rte_hlf_up_sln_dff}
\end{eqnarray}
We could have arrived at (\ref{eqn:rte_hlf_up_sln_dff}) by starting
from (\ref{eqn:rte_hlf_up_dfn}), and proceeding as above except using 
$\expmtauomu$ as the integrating factor.
We now integrate from the surface to level $\tau$ (i.e., along
a path of decreasing $\tauprm$) and apply the boundary
condition at the surface
\begin{eqnarray}
\left.
\expmtaupomu \ntnfrq(\tauprm,+\plrmu) 
\right|_{\tauprm = \taustr}^{\tauprm=\tau} 
& = & - \plrmurcp \int_{\tauprm = \taustr}^{\tauprm=\tau} 
\expmtaupomu \srcupfrq \,\dfr\tauprm \nonumber \\
\expmtauomu \ntnfrqoftaupmu -
\expmtausomu \ntnfrq(\taustr,+\plrmu)
& = & \plrmurcp \int_{\tau}^{\taustr}
\expmtaupomu \srcupfrq \,\dfr\tauprm \nonumber \\
\expmtauomu \ntnfrqoftaupmu  
& = & \expmtausomu \ntnfrq(\taustr,+\plrmu) + 
\plrmurcp \int_{\tau}^{\taustr} 
\expmtaupomu \srcupfrq \,\dfr\tauprm \nonumber \\
\ntnfrqoftaupmu  
& = & \me^{(\tau - \taustr)/\plrmu} \ntnfrq(\taustr,+\plrmu) + 
\plrmurcp \int_{\tau}^{\taustr} 
\me^{(\tau - \tauprm)/\plrmu} \srcupfrq \,\dfr\tauprm \nonumber
\end{eqnarray}
Note that $\taustr > \tau$ and $\tauprm > \tau$ 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 $\tau$).
The physical meaning of the transmission factors is more clear if we
write all transmission factors as negative exponentials.
\begin{equation}
% ThS99 p. 157 (5.56)
\ntnfrqoftaupmu  
 = \me^{-(\taustr - \tau)/\plrmu} \ntnfrq(\taustr,+\plrmu) + 
\plrmurcp \int_{\tau}^{\taustr} 
\me^{-(\tauprm - \tau)/\plrmu} \srcfrq(\tauprm,+\plrmu) \,\dfr\tauprm
\label{eqn:rte_sln_up}
\end{equation}
The first term on the RHS is the contribution of the boundary (e.g.,
Earth's surface) to the upwelling intensity at level $\tau$.
This contribution is attenuated by the optical path of the radiation
between the ground and level $\tau$.
The second term on the RHS is the contribution of the atmosphere to
the upwelling intensity at level $\tau$.
The net upward emission of each parcel of air between the surface and
level $\tau$ is $\srcupfrq(\tauprm)$, but this internally emitted
radiation is attenuated along the slant path between $\tauprm$ and
$\tau$. 
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 (\ref{eqn:rte_sln_dff}) by $\plrmu$ because $\plrmu = \plru$ in the
downwelling hemisphere. 
The resulting expression must be integrated from the upper boundary 
down to level $\tau$, and a boundary condition applied at the top.
\begin{eqnarray}
\frac{\dfr( \exptauomu \, \ntndwnfrq )}{\dfr\tau} & = & \plrmurcp
\exptauomu \srcdwnfrq \nonumber \\
\left. 
\exptaupomu \ntnfrq(\tauprm,-\plrmu) 
\right|_{\tauprm = 0}^{\tauprm=\tau} 
& = & \plrmurcp \int_{\tauprm = 0}^{\tauprm=\tau} 
\exptaupomu \srcdwnfrq \,\dfr\tauprm \nonumber \\
\exptauomu \ntnfrqoftaummu - 
\ntnfrq(0,-\plrmu)
& = & \plrmurcp \int_{0}^{\tau} 
\exptaupomu \srcdwnfrq \,\dfr\tauprm \nonumber \\
\exptauomu \ntnfrqoftaummu  
& = & \ntnfrq(0,-\plrmu) +
\plrmurcp \int_{0}^{\tau} 
\exptaupomu \srcdwnfrq \,\dfr\tauprm \nonumber \\
\ntnfrqoftaummu  
& = & \expmtauomu \ntnfrq(0,-\plrmu) +
\plrmurcp \int_{0}^{\tau} 
\me^{(-\tau + \tauprm)/\plrmu} \srcdwnfrq \,\dfr\tauprm  \nonumber \\
\ntnfrqoftaummu  
& = & \expmtauomu \ntnfrq(0,-\plrmu) +
\plrmurcp \int_{0}^{\tau} 
\me^{-(\tau - \tauprm)/\plrmu} \srcfrq(\tauprm,-\plrmu) \,\dfr\tauprm
\label{eqn:rte_sln_dwn}
\end{eqnarray}

The upwelling and downwelling intensities in a stratified atmosphere
are fully described by (\ref{eqn:rte_sln_up}) and
(\ref{eqn:rte_sln_dwn}). 
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
(\ref{eqn:rte_sln_up}) and (\ref{eqn:rte_sln_dwn}) does not 
rely on this assumption. 
It is straightforward to relax this assumption and replace
$\ntnfrqoftaumu$ by $\ntnfrq(\tau,\plrmu,\azi)$ and 
$\srcfrqoftaumu$ by $\srcfrq(\tau,\plrmu,\azi)$ in the above.  

\subsubsection[Thermal Radiation In A Stratified Atmosphere]{Thermal
Radiation In A Stratified Atmosphere}\label{sxn:rte_plk_sln}
Consider a purely absorbing, stratified atmosphere in thermodynamic 
equilibrium.
Then the source function equals the Planck function $\srcfrq =
\plkfrq = \plkfrqtpt$ (\ref{eqn:plk_dfn}) and the radiative transfer
equation is given by (\ref{eqn:rte_plk}). 
It is important to remember that $\plkfrq$ is the \textit{complete}
source function only because we are explicitly neglecting all
scattering processes. 
Thus we need only define the boundary conditions in order to use
(\ref{eqn:rte_sln_up}) and (\ref{eqn:rte_sln_dwn}) to fully
specify $\ntnfrq$. 
We assume that the surface emits blackbody radiation into the upper
hemisphere 
\begin{equation}
\ntnfrq(\taustr,+\plrmu) = \plkfrq[\tpt(\taustr)]
\label{eqn:plk_bc_btm}
\end{equation}
For brevity we shall define $\plkfrqstr = \plkfrq[\tpt(\taustr)]$.
At the top of the atmosphere, we assume there is no downwelling
thermal radiation.
\begin{equation}
\ntnfrq(0,-\plrmu) = 0
\label{eqn:plk_bc_top}
\end{equation}

The solutions for upwelling and downwelling intensities are then
obtained by using $\srcfrq(\tauprm,\plrmu) = \plkfrq(\tauprm)$ (the
Planck function is isotropic) and substituting (\ref{eqn:plk_bc_btm})
and (\ref{eqn:plk_bc_top}) into (\ref{eqn:rte_sln_up}) and
(\ref{eqn:rte_sln_dwn}), respectively
\begin{eqnarray}
\label{eqn:plk_sln_up}
\ntnfrqoftaupmu & = & 
\me^{-(\taustr - \tau)/\plrmu} \plkfrq(\taustr) + 
\plrmurcp \int_{\tau}^{\taustr} 
\me^{-(\tauprm - \tau)/\plrmu} \plkfrq(\tauprm) \,\dfr\tauprm \\
\label{eqn:plk_sln_dwn}
\ntnfrqoftaummu & = & 
\plrmurcp \int_{0}^{\tau} 
\me^{-(\tau - \tauprm)/\plrmu} \plkfrq(\tauprm) \,\dfr\tauprm
\end{eqnarray}
The first term on the RHS of (\ref{eqn:plk_sln_up}) is the thermal
radiation emitted by the surface, attenuated by absorption in the
atmosphere until it contributes to the upwelling intensity at level
$\tau$.  
The second term on the RHS contains the upwelling intensity arriving
at $\tau$ contributed from the attenuated atmospheric thermal emission
from each parcel between the surface and $\tau$. 
The $\plrmu^{-1}$ factor in front of the integral accounts for the
slant path of the thermally emitting atmosphere.
The RHS of (\ref{eqn:plk_sln_dwn}) 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 (\ref{eqn:plk_sln_up}) and
(\ref{eqn:plk_sln_dwn}).

\subsubsection[Angular Integration]{Angular Integration}\label{sxn:ngl}
Once the solutions for the hemispheric intensities are known, it is
straightforward to obtain the hemispheric fluxes by performing the
angular integrations (\ref{eqn:flx_up_frq})--(\ref{eqn:flx_dwn_frq}).
\begin{equation}
\flxdwnfrqoftau = 2 \mpi \int_{\plrmu=0}^{\plrmu=1} \ntnfrq
(\tau,-\plrmu) \plrmu \,\dfr\plrmu
\label{eqn:flx_dwn_frq_2}
\end{equation}
Consider first the downwelling flux in a non-scattering, thermal,
isotropic, stratified atmosphere obtained by substituting
(\ref{eqn:plk_sln_dwn}) into (\ref{eqn:flx_dwn_frq_2})
and interchanging the order of integration
\begin{eqnarray}
\flxdwnfrqoftau & = & 2 \mpi \int_{\plrmu=0}^{\plrmu=1}
\left( \plrmurcp \int_{\tauprm = 0}^{\tauprm = \tau} 
\me^{-(\tau - \tauprm)/\plrmu} \plkfrq(\tauprm) \,\dfr\tauprm 
\right) \plrmu \,\dfr\plrmu \nonumber \\
& = & 2 \mpi \int_{\tauprm = 0}^{\tauprm = \tau} 
\int_{\plrmu=0}^{\plrmu=1}
\me^{-(\tau - \tauprm)/\plrmu} \plkfrq(\tauprm) \,\dfr\plrmu \,\dfr\tauprm
\nonumber \\ 
& = & 2 \mpi \int_{\tauprm = 0}^{\tauprm = \tau} 
\plkfrq(\tauprm) \left(
\int_{\plrmu = 0}^{\plrmu = 1}
\me^{-(\tau - \tauprm)/\plrmu} \,\dfr\plrmu \, \right) \,\dfr\tauprm
\label{eqn:flx_dwn_frq_tau}
\end{eqnarray}
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 \trmidx{exponential integrals} defined in
Appendix~\ref{sxn:xpn}, the inner integral in parentheses in
(\ref{eqn:flx_dwn_frq_tau}) is $\xpn_{2}(\tau - \tauprm)$
(\ref{eqn:xpn_2_dfn}).  
\begin{equation}
\flxdwnfrqoftau = 2 \mpi \int_{\tauprm = 0}^{\tauprm = \tau} 
\plkfrq(\tauprm) \xpn_{2}(\tau - \tauprm) \,\dfr\tauprm
\label{eqn:flx_dwn_xpn}
\end{equation}
Similar terms arise when we consider the horizontal upwelling flux
obtained by substituting (\ref{eqn:plk_sln_up}) into
(\ref{eqn:flx_up_frq}) and we obtain
\begin{eqnarray}
\flxupfrqoftau & = & 2 \mpi \int_{\plrmu=0}^{\plrmu=1} \ntnfrq
(\tau,+\plrmu) \plrmu \,\dfr\plrmu
\nonumber \\
& = & 2 \mpi \int_{\plrmu=0}^{\plrmu=1} \left( 
\me^{-(\taustr - \tau)/\plrmu} \plkfrq(\taustr) + 
\plrmurcp \int_{\tau}^{\taustr} 
\me^{-(\tauprm - \tau)/\plrmu} \plkfrq(\tauprm) \,\dfr\tauprm
\right) \plrmu \,\dfr\plrmu 
\nonumber \\
& = & 2 \mpi \plkfrq(\taustr) 
\int_{\plrmu=0}^{\plrmu=1} 
\me^{-(\taustr - \tau)/\plrmu} \plrmu \,\dfr\plrmu
+ 2 \mpi \int_{\tau}^{\taustr} 
\plkfrq(\tauprm) \left(
\int_{\plrmu=0}^{\plrmu=1} 
\me^{-(\tauprm - \tau)/\plrmu} 
\,\dfr\plrmu \right) \,\dfr\tauprm
\nonumber \\
& = & 2 \mpi \plkfrq(\taustr) \xpn_{3}(\taustr - \tau) + 
2 \mpi \int_{\tauprm = \taustr}^{\tauprm = \tau}
\plkfrq(\tauprm) \xpn_{2}(\tauprm-\tau) \,\dfr\tauprm
\label{eqn:flx_up_xpn}
\end{eqnarray}
Subtracting (\ref{eqn:flx_dwn_xpn}) from (\ref{eqn:flx_up_xpn}) we
obtain the net flux at any layer in a non-scattering, thermal,
stratified atmosphere
\begin{eqnarray}
\flxfrqoftau & = & \flxupfrqoftau - \flxdwnfrqoftau \nonumber \\
& = & 2 \mpi \left[
\plkfrq(\taustr) \xpn_{3}(\taustr - \tau) + 
\int_{\tau}^{\taustr}
\plkfrq(\tauprm) \xpn_{2}(\tauprm - \tau) \,\dfr\tauprm -
\int_{0}^{\tau} 
\plkfrq(\tauprm) \xpn_{2}(\tau - \tauprm) \,\dfr\tauprm
\right]
\label{eqn:flx_net_xpn}
\end{eqnarray}
Equations~(\ref{eqn:flx_dwn_xpn}) and (\ref{eqn:flx_up_xpn}) may not
seem useful at this point but their utility becomes apparent in
\S\ref{sxn:flx_trn} where we define the \trmdfn{flux
transmissivity}. 

\subsubsection[Thermal Irradiance]{Thermal Irradiance}\label{sxn:thr}
Assume a non-scattering planetary surface at temperature $\tpt$ emits
blackbody radiation such that $\ntnfrq(\taustr,+\plrmu) =
\plkfrq(\tpt)$ (\ref{eqn:plk_bc_btm}).   
What is the total upwelling thermal irradiance from the surface? 
From (\ref{eqn:flx_up_frq}) we have
\begin{eqnarray}
\flxupfrq 
& = & 2 \mpi \int_{0}^{1} \plkfrqtpt \plrmu \, \dfr\plrmu \nonumber \\
& = & 2 \mpi \plkfrqtpt \int_{0}^{1} \plrmu \,\dfr\plrmu \nonumber \\
& = & 2 \mpi \plkfrqtpt \left[ \frac{\plrmu^{2}}{2} \right]_{0}^{1} \nonumber \\
& = & 2 \mpi \plkfrqtpt \left( \frac{1}{2} - 0 \right) \nonumber \\
\flxupfrq & = & \mpi \plkfrqtpt \nonumber \\
\frac{1}{\mpi} \flxupfrq & = & \plkfrqtpt 
\label{eqn:flx_frq_sfc_up}
\end{eqnarray} 
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 (\ref{eqn:flx_frq_sfc_up}) over frequency to obtain the
total upwelling thermal irradiance
\begin{eqnarray}
\frac{1}{\mpi} \int_{0}^{\infty} \flxupfrq  \,\dfr\frq & = &
\int_{0}^{\infty} \plkfrqtpt \,\dfr\frq \nonumber \\ 
\frac{1}{\mpi} \flxup & = & \int_{0}^{\infty} \plkfrqtpt \,\dfr\frq
\nonumber \\ 
\frac{1}{\mpi} \flxup & = & \frac{\cststfblt \tpt^{4}}{\mpi} \nonumber \\ 
\flxup & = & \cststfblt \tpt^{4}
\label{eqn:flx_sfc_up}
\end{eqnarray} 
Thus the factor of $\mpi$ from the azimuthal integration nicely cancels
the factor of $\mpi$ from the \trmidx{Stefan-Boltzmann Law}.
Equation~(\ref{eqn:flx_sfc_up}) 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 $\flxdwn(\zzz_{\btmsbs}) = \cststfblt \tpt_{\btmsbs}^{4}$
and $\flxup(\zzz_{\topsbs}) = \cststfblt \tpt_{\topsbs}^{4}$, respectively.

\subsubsection[Grey Atmosphere]{Grey Atmosphere}\label{sxn:gry}
Consider an atmosphere transparent to solar radiation and partially
opaque to thermal radiation governed by (\ref{eqn:rte_plk}).
The exact solution, including angular dependence, is given in
\S\ref{sxn:rte_plk_sln}.
The hemispheric fluxes and net flux may only be obtained exactly by
accounting for the angular dependence as in \S\ref{sxn:ngl}.
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 
\trmidx{diffusivity factor} (e.g., \S\ref{sxn:dff_prx}).
These methods are all related to the \trmidx{two-stream approximation}.

% Method used in Sal96 p. 232
One such method \cite[][p.~232]{Sal96} 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 \ref{eqn:flx_net_xpn}), is
\begin{eqnarray}
2 \xpn_{3}(\taustr - \tau) & = & 
\exp \left( -\frac{\taustr - \tau}{\plrmubar} \right)
\label{eqn:dfs_fct_apx}
\end{eqnarray}
Inspection (or differentiation) shows that the atmosphere within one
optical depth makes the dominant contribution to~(\ref{eqn:dfs_fct_apx}).
For $\dlt\tau = \taustr - \tau = 1$, the diffusivity factor 
\begin{eqnarray}
\plrmubar^{-1} & \approx & \frac{5}{3}
\label{eqn:plr_mu_bar_dfn}
\end{eqnarray}
The hypothetical collimated beam of radiation is inclined to the zenith
by $\arccos(3/5) \approx 53.13\dgr$.
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 (\ref{eqn:dfs_fct_apx}), the upwelling
hemispheric irradiance (\ref{eqn:flx_up_frq}) for blackbody radiation is
\begin{eqnarray}
\flxupfrq & = & 
2 \mpi \int_{0}^{1} \ntnfrq (+\plrmu) \plrmu \, \dfr\plrmu \nonumber \\
& \approx & 2 \mpi \int_{0}^{1} \ntnfrq (+\plrmubar) \plrmu \, \dfr\plrmu \nonumber \\
& = & 2 \mpi \plrmubar \ntnfrq(+\plrmubar) \int_{0}^{1} \plrmu \, \dfr\plrmu \nonumber \\
& = & 2 \mpi \ntnfrq (+\plrmubar) \left[ \frac{\plrmu^{2}}{2} \right]_{0}^{1} \nonumber \\
& = & 2 \mpi \ntnfrq (+\plrmubar) \left[ \frac{1}{2} - 0 \right] \nonumber \\
& = & \mpi \ntnfrq (+\plrmubar) \nonumber \\
\label{eqn:flx_ntn_apx}
\end{eqnarray}
% Hou02 p. 12
An analogous relationship holds for the downwelling irradiance.

Based on (\ref{eqn:plr_mu_bar_dfn}) and (\ref{eqn:flx_ntn_apx}),
the irradiance structure of the thermal atmosphere may approximated
by performing a direct angular integration of (\ref{eqn:rte_hlf_dfn}).
With our approximation, radiances $\ntn$ integrate directly to
irradiances $\flx$ modulo the diffusivity factor $\plrmubar$. 
(\ref{eqn:rte_plk})
\begin{subequations}
% Hou02 p. 12 (2.6)
\label{eqn:dff_hlf_dfn}
\begin{align}
\label{eqn:dff_hlf_dwn_dfn}
-\plrmubar \frac{\dfr\flxdwnfrq}{\dfr\tau} & = \flxdwnfrq - \mpi\flxplkfrq \\
\label{eqn:dff_hlf_up_dfn}
\plrmubar \frac{\dfr\flxupfrq}{\dfr\tau} & = \flxupfrq - \mpi\flxplkfrq
\end{align}
\end{subequations}

It is instructive to examine an idealized \trmdfn{grey atmosphere},
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 
\trmidx{radiative equilibrium temperature profile} and the 
\trmidx{greenhouse effect}. 
We simplify (\ref{eqn:dff_hlf_dfn}) in two ways.
First, we introduce a scaled optical depth
$\dfr\tautld = \plrmubar^{-1}\dfr\tau = \frac{5}{3}\dfr\tau$.
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 $\lambda = 5$--$20$\,\um\ range where most of
Earth's terrestrial radiative energy resides.
\begin{subequations}
% Hou02 p. 12 (2.6)
\label{eqn:gry_hlf_dfn}
\begin{align}
\label{eqn:gry_hlf_dwn_dfn}
-\frac{\dfr\flxdwn}{\dfr\tautld} & = \flxdwn - \mpi\plkfnc \\
\label{eqn:gry_hlf_up_dfn}
\frac{\dfr\flxup}{\dfr\tautld} & = \flxup - \mpi\plkfnc
\end{align}
\end{subequations}

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
(\ref{eqn:gry_hlf_dfn}) will adjust to a temperature profile
determined by \trmidx{radiative equilibrium}.
Let the time rate of change of temperature $\tpt$ of a parcel be
denoted by $\htr$\,[\kxs], the parcel \trmidx{warming rate}\footnote{% 
Conventionally $\htr$ is called the \trmidx{heating rate}, which is a
misnomer since the dimensions of $\dfr\tpt/\dfr\tm$ are \kxs.
Thus we use the less common but more accurate terminology ``warming rate''.
We reserve ``heating rate'' for a measure power dissipation, i.e.,
energy per unit time, in \jxs, \jxmCs, or \jxkgs.} or \trmidx{cooling rate}. 
The warming rate is the rate of net energy deposition divided by the
\trmidx{specific heat capacity at constant pressure} 
$\heatcpcspcprs$\,[\jxkgK] times the density $\dnsatm$\,[\kgxmC]
\begin{eqnarray}
%\frac{\dfr\flx}{\dfr\hgt} & = & \dnsatm \heatcpcspcprs 
\htr \equiv \frac{\dfr\tpt}{\dfr\tm} & = & 
\frac{1}{\dnsatm \heatcpcspcprs} \frac{\dfr\flx}{\dfr\hgt} \\
\frac{\mbox{K}}{\mbox{s}} & = & 
\left( \frac{\mbox{kg}}{\mbox{\mC}} \times \frac{\mbox{J}}{\mbox{kg\,K}} \right)^{-1} 
\times \frac{\mbox{J}}{\mbox{\mSs}} \times \frac{1}{\mbox{m}} \nonumber
\label{eqn:flx_dvr_hgt_dfn}
\end{eqnarray}
The forcing term on the RHS, $\dfr\flx/\dfr\hgt$ is the 
\trmidx{radiative flux divergence}, 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,
\begin{eqnarray}
\frac{\dfr\flx}{\dfr\hgt} & = & \frac{\dfr}{\dfr\hgt} (\flxdwn - \flxup)
\label{eqn:flx_dvr_dfn}
\end{eqnarray}

By definition, the net radiative heating vanishes 
($\dfr\tpt/\dfr\hgt = 0$) at all levels of an atmosphere in
radiative equilibrium.
Setting (\ref{eqn:flx_dvr_dfn}) equal to zero and integrating, we obtain
\begin{eqnarray}
\flxdwn - \flxup & = & \flxnot
\label{eqn:flx_dvr_dfn}
\end{eqnarray}
The net radiative flux $\flx(\hgt) = \flxnot$ is constant in 
radiative equilibrium.

Adding and subtracting (\ref{eqn:gry_hlf_dfn}), we obtain
\begin{subequations}
% Hou02 p. 12 (2.6)
\label{eqn:gry_hlf_rdc}
\begin{align}
\label{eqn:gry_hlf_dwn_rdc}
\frac{\dfr}{\dfr\tautld}(\flxup - \flxdwn) & = \flxup + \flxdwn - 2\mpi\plkfnc \\
\label{eqn:gry_hlf_up_rdc}
\frac{\dfr}{\dfr\tautld}(\flxup + \flxdwn) & = \flxup - \flxdwn = \flxnot
\end{align}
\end{subequations}
Defining $\psi = \flxup - \flxdwn$ and $\phi = \flx = \flxup - \flxdwn$,
\begin{subequations}
% Hou02 p. 12 (2.6)
\label{eqn:gry_hlf_rdc_grk}
\begin{align}
\label{eqn:gry_hlf_dwn_rdc_grk}
\frac{\dfr\phi}{\dfr\tautld} & = \psi - 2\mpi\plkfnc \\
\label{eqn:gry_hlf_up_rdc_grk}
\frac{\dfr\psi}{\dfr\tautld} & = \phi = \flxnot
\end{align}
\end{subequations}
Since $\phi = \flxup - \flxdwn$ is constant,
(\ref{eqn:gry_hlf_up_rdc_grk}) shows that $\dfr\phi/\dfr\tautld = 0$
and thus
\begin{eqnarray}
\psi = 2\mpi\plkfnc
\label{eqn:gry_psi_dfn}
\end{eqnarray}
Substituting this into (\ref{eqn:gry_hlf_up_rdc_grk}),
\begin{eqnarray}
\frac{\dfr}{\dfr\tautld}(2\mpi\plkfnc) & = & \flxnot \nonumber \\
\frac{\dfr\plkfnc}{\dfr\tautld} & = & \dpysty \frac{\flxnot}{2\mpi} \nonumber \\
\plkfnc(\tautld) & = & \dpysty \frac{\flxnot\tautld}{2\mpi} + \cstone
\label{eqn:flx_dvr_prs_dfn}
\end{eqnarray}
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 \trmidx{TOA}.
This implies that $\phi = \psi(0) = \flxup(0) = \flxnot$.
We must therefore have $\plkfnc(0) = \phi/2\mpi$ by
(\ref{eqn:gry_psi_dfn}).
Using this result in (\ref{eqn:flx_dvr_prs_dfn}),
\begin{eqnarray}
\plkfnc(0) & = & \dpysty \frac{\phi}{2\mpi} = 
\dpysty \frac{\flxnot(0)}{2\mpi} + \cstone \nonumber \\
\cstone & = & \dpysty \frac{\phi}{2\mpi} = \frac{\flxnot}{2\mpi} \nonumber \\
\label{eqn:cst_one_dfn}
\end{eqnarray}
Substituting (\ref{eqn:cst_one_dfn}) back into
(\ref{eqn:flx_dvr_prs_dfn}) shows  
\begin{eqnarray}
\plkfnc(\tautld) & = & \dpysty \frac{\flxnot\tautld}{2\mpi} + \frac{\flxnot}{2\mpi} \nonumber \\
& = & \dpysty \frac{\flxnot}{2\mpi}(\tautld + 1)
\label{eqn:gry_plk_flx_dfn}
\end{eqnarray}
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 
$\flxup(\tautldstr) = \mpi\plkfnc(\tptsfc)$ where $\tptsfc$\,[\K] is the
surface skin temperature.
The atmospheric temperature just above the surface is given by
(\ref{eqn:gry_plk_flx_dfn}) as
$\plkfnc(\tautldstr) = \frac{\flxnot}{2\mpi}(\tautldstr + 1)$.
Thus there is a temperature discontinuity between the near-surface air
and the ground.

Climate models typically express $\htr$ (\ref{eqn:flx_dvr_hgt_dfn}) in 
terms of the flux gradient with respect to pressure by invoking the
hydrostatic equilibrium condition (\ref{eqn:hyd_eqm}) 
\begin{eqnarray}
\htr \equiv \frac{\dfr\tpt}{\dfr\tm} & = & 
\frac{1}{\dnsatm \heatcpcspcprs} \frac{\dfr\flx}{[-(\dnsatm\grv)^{-1}\dfr\prs]} \nonumber \\
& = & -\frac{\grv}{\heatcpcspcprs} \frac{\dfr\flx}{\dfr\prs} \\
& = & -\frac{\grv}{\heatcpcspcprs} 
\frac{(\flxdwn_{\kkk} - \flxup_{\kkk})-(\flxdwn_{\kkk+1} - \flxup_{\kkk+1})}
{\prs_{\kkk}-\prs_{\kkk+1}} \nonumber \\
& = & \frac{\grv}{\heatcpcspcprs} 
\frac{(\flxdwn_{\kkk} - \flxdwn_{\kkk+1}) + (\flxup_{\kkk+1} - \flxup_{\kkk})}
{\prs_{\kkk+1}-\prs_{\kkk}}
\label{eqn:flx_dvr_prs_dfn}
\end{eqnarray}
where the subscript denotes the $\kkk$th vertical interface level in
the atmosphere. 

\subsubsection[Scattering]{Scattering}\label{sxn:sct}
Energy interacting with matter undergoes one of two processes,
\trmdfn{scattering} 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 \trmdfn{isotropic
scattering}. 
In general, the angular dependence of the scattering is described by
the \trmdfn{phase function} 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
\trmdfn{scattering angle}
$\nglsct$ between incident and emergent directions.
\begin{eqnarray}
\cos \nglsct & = & \nglhatprm \cdot \nglhat
\label{eqn:ngl_sct_dfn}
\end{eqnarray}
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
\trmdfn{forward scattering}, and corresponds to $\nglsct < \mpi/2$.
When scattered radiation has been reflected back into the
hemisphere from whence it arrived, it is called \trmdfn{back
scattering}, 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 \trmidx{spherical polar coordinates}.
\begin{eqnarray}
\nglhat & = & \sin \plr \cos \azi \, \ihat + \sin \plr \sin \azi \, \jhat +
\cos \plr \, \khat \\
\nglhatprm & = & \sin \plrprm \cos \aziprm \, \ihat + \sin \plrprm \sin
\aziprm \, \jhat + \cos \plrprm \, \khat
\label{eqn:ngl_crt_dfn}
\end{eqnarray}

The scattering angle $\nglsct$ is simply related to the inner product 
of $\nglhatprm$ and $\nglhat$ by the cosine law
\begin{eqnarray}
\cos \nglsct & = & \nglhatprm \cdot \nglhat \nonumber \\
& = & \sin \plrprm \cos \aziprm \sin \plr \cos \azi + \sin \plrprm
\sin \aziprm \sin \plr \sin \azi + \cos \plrprm + \cos \plr \nonumber \\
& = & \sin \plrprm \sin \plr \, ( \cos \aziprm \cos \azi + \sin \aziprm
\sin \azi ) + \cos \plrprm \cos \plr \nonumber \\
& = & \sin \plrprm \sin \plr \cos (\aziprm - \azi) + \cos \plrprm \cos \plr 
\label{eqn:ngl_sct_dfn2}
\end{eqnarray}

\subsubsection[Phase Function]{Phase Function}\label{sxn:phz_fnc}
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 \citet{Van57}, \citet{JWW76},
\citet{WiG76}, \citet{Wis771}, \citet{Wis79}, and \citet{Bou98}.

The \trmidx{phase function} $\phzfnc(\cos \nglsct)$ is normalized so
that the total probability of scattering is unity
\begin{subequations}
% ThS99 p. 75 (3.24)
\label{eqn:phz_fnc_nrm}
\begin{align}
\label{eqn:phz_fnc_nrm_sht}
\frac{1}{4\mpi} \int_{\ngl} \phzfnc(\cos \nglsct) \,\dfr\ngl & = 1 \\
\label{eqn:phz_fnc_nrm_lng}
\frac{1}{4\mpi} \int_{\azi = 0}^{\azi = 2\mpi} 
\int_{\plr = 0}^{\plr = \mpi} 
\phzfnc(\plrprm,\aziprm;\plr,\azi) \, \sin \plr \,\dfr\plr
 \,\dfr\azi
& = 1
\end{align}
\end{subequations}
The dimensions of the phase function are somewhat ambiguous.
If the $(4\mpi)^{-1}$ factor in (\ref{eqn:phz_fnc_nrm}) 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, \xsr.
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., (\ref{eqn:src_sct_frq_dfn}).
Furthermore, the units of $\phzfnc$ are probability per steradian,
\xsr.  

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 $\phi$, and depend only on $\plr$.
In this case the phase function normalization
(\ref{eqn:phz_fnc_nrm_lng}) simplifies to 
\begin{eqnarray}
\label{eqn:phz_fnc_nrm_plr}
\frac{1}{2}
\int_{\plr = 0}^{\plr = \mpi} 
\phzfnc(\plrprm;\plr) \, \sin \plr \,\dfr\plr
& = & 1
\end{eqnarray}
In accord with the discussion above, the factor of $1/2$ is
dimensionless, as is