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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. Much of the material in Houghton [2002], Chapters 2 and 4, is described at a more advanced level in the rest of this monograph. Houghton § 2.2 is our § 2.1.9, 2.1.13, and 2.1.16; Houghton § 2.3 is our § 2.2.2; Part of Houghton § 4.1 is our § 2.2.6; Part of Houghton § 4.2 is our § 2.4; Part of Houghton § 4.3 is our § 6.1.3 and § 4.1; Houghton § 4.4 is our § 6.3.2 and § 6.2.1; Houghton § 4.5 is our § 2.2.1– 2.2.2; This FACT has no equivalent to Houghton § 4.6–4.8. Houghton § 4.9 is our § 6; and Houghton § 4.10 is our § 8. The Index beginning on page  413 is also helpful.

Yada yada yada.

Contents

List of Figures

List of Tables

1 Introduction

This document describes mathematical and computational considerations pertaining to radiative transfer processes and radiative transfer models of the Earth system. Our approach is to present a detailed derivation of the tools of radiative transfer needed to predict the radiative quantities (irradiance, mean intensity, and heating rates) which drive climate. In so doing we begin with discussion of the intensity field which is the quantity most often measured by satellite remote sensing instruments. Our approach owes much to Bohren and Huffman [1983] (particle scattering), Goody and Yung [1989] (band models), and Thomas and Stamnes [1999] (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.

1.1 Planetary Radiative Equilibrium

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

On timescales longer than about a year the Earth as a whole is thought to be in 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

The total amount of solar energy available for the Earth to absorb is the incoming solar flux (or irradiance) at the top of Earth’s atmosphere, $F\odot$ (aka the solar constant), times the intercepting area of Earth’s disk which is $❐r{\oplus }^2$. Since Earth rotates, the total mean incident flux $❐r{\oplus }^{2}F\odot$ 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 $F\odot ∕4$. The fraction of incident solar flux which is reflected back to space, and thus unable to heat the planet, is called the planetary albedo or spherical albedo, $\mathcal{R}$. Satellite observations show that $\mathcal{R}\approx 0.3$. Thus only $\left(1-\mathcal{R}\right)$ of the mean incident solar flux contributes to warming the planet and we have

Earth does not cool to space as a perfect blackbody ( 41a) of a single temperature and emissivity. Nevertheless the spectrum of thermal radiation $FOLR$ which escapes to space and thus cools Earth does resemble blackbody emission with a characteristic temperature. The effective temperature $TE$ of an object is the temperature of the blackbody which would produce the same irradiance. Inverting the Stefan-Boltzmann Law ( 62) yields

For a perfect blackbody, $T=TE$. For a planet, the difference between $TE$ and the mean surface temperature $Ts$ is due to the radiative effects of the overlying atmosphere. The insulating behavior of the atmosphere is more commonly known as the greenhouse effect.

Substituting ( 3) and ( 4) into ( 2)

For Earth, $\mathcal{R}\approx 0.3$ and $F\odot \approx 1367$W m$-2$. Using these values in ( 6) yields $TE=255$ K. Observations show the mean surface temperature $Ts=288$K.

1.2 Fundamentals

The fundamental quantity describing the electromagnetic spectrum is frequency, $\nu$. Frequency measures the oscillatory speed of a system, counting the number of oscillations (waves) per unit time. Usually $\nu$ is expressed in cycles-per-second, or Hertz. Units of Hertz may be abbreviated Hz, hz, cps, or, as we prefer, s$-1$. Frequency is intrinsic to the oscillator and does not depend on the medium in which the waves are travelling. The energy carried by a photon is proportional to its frequency

where $h$ is Planck’s constant. Regrettably, almost no radiative transfer literature expresses quantities in frequency.

A related quantity, the angular frequency $\omega$ measures the rate of change of wave phase in radians per second. Wave phase proceeds through $2❐$ radians in a complete cy