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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. The Index beginning on page 361 is also helpful.

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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

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 (73) yields

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

1.2 Fundamentals

The fundamental quantity describing the electromagnetic spectrum is frequency, ν. Frequency measures the oscillatory speed of a system, counting the number of oscillations (waves) per unit time. Usually ν 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

A related quantity, the angular frequency ω measures the rate of change of wave phase in radians per second. Wave phase proceeds through 2π radians in a complete cycle. Thus the frequency and angular frequency are simply related

Most radiative transfer literature use wavelength or wavenumber. Wavelength, λ (m), measures the distance between two adjacent peaks or troughs in the wavefield. The universal relation between wavelength and frequency is

The wavenumber m-1, is exactly the inverse of wavelength

There is another, distinct quantity also called wavenumber. This secondary usage of wavenumber in this text is the traditional measure of spatial wave propagation and is denoted by k.

Table 2 summarizes the relationships between the fundamental parameters which describe wave-like phenomena.

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Table 2: Wave Parameter Conversion Tablea b Variable ν λ ω k τ Units s-1 m cm-1 s-1 m-1 s ν - 2πν               λ -               100c - 200π               ω -               k ck -               τ cτ -

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2 Radiative Transfer Equation

2.1 Definitions

2.1.1 Intensity

The fundamental quantity defining the radiation field is the specific intensity of radiation. Specific intensity, also known as 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 Iλ are Joule meter^-2 second^-1 steradian^-1 meter^-1. In SI dimensional notation, the units condense to J m-2 s-1 sr-1 m-1. The SI unit of power (1 Watt ≡ 1 Joule per second) is preferred, leading to units of W m-2 sr-1 m-1. Often the specific intensity is expressed in terms of spectral frequency Iν with units W m-2 sr-1 Hz-1 or spectral wavenumber (also I) with units W m-2 sr-1 (cm-1)-1.

Consider light travelling in the direction through the point r. Construct an infinitesimal element of surface area dS intersecting r and orthogonal to . The radiant energy dE crossing dS in time dt in the solid angle dΩ in the frequency range [ν,ν + dν] is related to Iν(r,) by

(13)

It is not convenient to measure the radiant flux across surface orthogonal to , as in (13), when we consider properties of radiation fields with preferred directions. If instead, we measure the intensity orthogonal to an arbitrarily oriented surface element dA with surface normal , then we must alter (13) to account for projection of dS onto dA. If the angle between and is θ then

(14)

and the projection of dS onto dA yields

(15)

so that

(16)

The conceptual advantage that (16) has over (13) is that (16) builds in the geometric factor required to convert to any preferred coordinate system defined by dA and its normal . In practice dA 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 ν as a subscript, as in Iν, in favor of the more explicit, but lengthier, notation I(ν). Three of the dimensions are superfluous to climate models and will be discarded: The time dependence of Iν 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 t. We reduce the number of spatial dimensions from three to one by assuming a stratified atmosphere which is horizontally homogeneous and in which physical quantities may vary only in the vertical dimension z. Thus we replace r by z. This approximation is also known as a 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, Iν(z,). Often the optical depth τ (defined below), which increases monotonically with z, is used for the vertical coordinate instead of z. The angular direction of the radiation is specified in terms of the polar angle θ and the azithumal angle φ. The polar angle θ is the angle between and the normal surface that defines the coordinate system. The specific intensity of radiation traveling at polar angle θ and azimuthal angle φ at optical depth level τ in a plane parallel atmosphere is denoted by Iν(τ,θ,φ). Specific intensity is also referred to as intensity.

Further simplification of the intensity field is possible if it meets certain criteria. If Iν is not a function of position (τ), then the field is homogeneous. If Iν is not a function of direction (), then the field is isotropic.

2.1.2 Mean Intensity

The mean intensity is an integrated measure of the radiation field at any point r. Mean intensity ν is defined as the mean value of the intensity field integrated over all angles.

The definition of ν (17) demands the radiation field be integrated over all angles, i.e., over all 4π steradians. Evaluating the denominator demonstrates the properties of angular integrals. The denominator of (17) is

It is convenient to return briefly to the definition of isotropic radiation. Isotropic radiation is, by definition, equal intensity in all directions so that the total emitted radiation is simply 4π times the intensity of emission in any direction.

Applying (19) to (17) yields

Let us simplify (21) by introducing the change of variables

The hemispheric intensities or half-range intensities are simply the up- and downwelling components of which the full intensity is composed

Iν(τ,) = Iν(τ,θ,φ)	=	(25a)
Iν(τ,) = Iν(τ,u,φ)	=	(25b)

2.1.3 Irradiance

The spectral irradiance Fν 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 [Madronich, 1987]. Consider a surface orthogonal to the ′ direction. All radiant energy travelling parallel to ′ crosses this surface and thus contributes to the irradiance with 100% efficiency. Energy travelling orthogonal to ′ (and thus parallel to the surface), however, never crosses the surface and does not contribute to the irradiance. In general, the intensity Iν() projects onto the surface with an efficiency cos Θ = ⋅′ , thus

Let us simplify (27) by introducing the change of variables u = cos θ, du = - sin θ dθ. This maps θ ∈ [0, π] into u ∈ [1,-1]:

The irradiance per unit frequency, Fν, is simply related to the irradiance per unit wavelength, Fλ. The total irradiance over any given frequency range, [ν,ν + dν], say, is Fν dν. The irradiance over the same physical range when expressed in wavelength, [λ,λ - dλ], say, is Fλ dλ. The negative sign is introduced since -dλ increases in the same direction as +dν. Equating the total irradiance over the same region of frequency/wavelength, we obtain

2.1.4 Actinic Flux

A quantity of great importance in photochemistry is the total convergence of radiation at a point. This quantity, called the actinic flux, FJ, determines the availability of photons for photochemical reactions. By definition, the intensity passing through a point P in the direction within the solid angle dΩ is Iν dΩ. We have not multiplied by cos θ since we are interested in the energy passing along (i.e., θ = 0). The energy from all directions passing through P is thus

The usefulness of actinic flux FJ ν becomes apparent only in conjunction with additional, species-dependent data describing the probability of photon absorption, or 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 NA [m-3], the actinic radiation field FJ ν , and the efficiency with with each molecule absorbs photons. This efficiency is called the absorption cross-section, molecular cross section, or simply cross-section. The absorption cross-section is denoted by α and has units of [m-2]. In the literature, however, values of α usually appear in CGS units [cm-2]. To make the frequency-dependence of α explicit we write α(ν). If α depends significantly on temperature, too (as is true for ozone), we must consider α(ν,T).

The probability, per unit time, per unit frequency that a single molecule of species A will absorb a photon with frequency in [ν,ν + dν] is proportional to FJ ν (ν)α(ν)² . Thus α(ν) 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 Fα ν [W m-3] be the energy absorbed per unit time, per unit frequency, per unit volume of air. Then

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

The probability that a photon absorbed by AB will result in the photodissociation of AB, and the completion of (33), is called the quantum yield or quantum efficiency and is represented by φ. As a probability, φ is dimensionless³ . In addition to its dependence on ν, φ depends on temperature T for some important atmospheric reactions (such as ozone photolysis). We explicitly annotate the T -dependence of φ only for pertinent reactions. Measurement of φ(ν) for all conditions and reactions of atmospheric interest is an ongoing and important laboratory task.

The specific 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 Jν. The units of Jν are s-1 Hz-1.

The total photolysis rate coefficient J is obtained by integrating (34) over all frequencies which may contribute to photodissociation

Integration errors due to the discretization of (35) are quite common. To compute J with high accuracy, regular grids must have resolotion of ~1 nm in the ultraviolet, [Madronich, 1989]. Much of the difficulty is due to the steep but opposite gradients of FJ ν and φ which occur in the ultraviolet. High frequency features in α worsens this problem for some molecules.

The utility of J has motivated researchers to overcome these computational difficulties by brute force techniques and by clever parameterizations and numerical techniques [Chang et al., 1987; Dahlback and Stamnes, 1991; Toon et al., 1989; Stamnes and Tsay, 1990; Petropavlovskikh, 1995; Landgraf and Crutzen, 1998; Wild et al., 2000]. 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 J is the first order rate coefficient in photochemical reactions. For example, JNO2 is the first order rate coefficient for

Figure 1 shows the spectral distribution of actinic flux in a clear mid-latitude summer atmosphere, and the absorption cross-section and quantum yield of NO2.

Figure 1: (a) Spectral distribution of actinic flux FJ [# m-2 s-1 μm-1] at TOA and at the surface for a mid-latitude summer (MLS) atmosphere with a unit optical depth of dust or sulfate in the lowest kilometer. (b) Absorption cross section of NO2, αNO2 [m2 molecule-1]. (c) Quantum yield of NO2, φNO2 (36).

Figure 2 shows the vertical distribution of JNO2 [s-1] for the conditions shown in Figure (1).

Figure 2: Vertical distribution of JNO2 [s-1] (36) for the conditions shown in Figure (1). (a) Absolute rates. (b) Rates normalized by clean sky rates.

In this section we have assumed the quantities FJ ν , α, and φ are somehow known and therefore available to use to compute J. Typically, α and φ 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 (35). The essence of forward radiative problems is to determine I so that quantities such as Jν and Fα may be determined. In inverse radiative transfer problems which are encountered in much of remote sensing, both J and the species concentration are initially unknown and must be determined. We shall continue describing the methods of forward radiative transfer until we have tool