Radiative Transfer in the Earth System

178 Pages·2010·2.781 MB·English

by  Charlie Zender

Online: http://dust.ess.uci.edu/facts Updated: Thu18th Nov,2010,14:22
Radiative Transfer in the Earth System
byCharlieZender
UniversityofCalifornia,Irvine
DepartmentofEarthSystemScience [email protected]
UniversityofCalifornia Voice: [phone]
Irvine,CA 92697-3100 Fax: [phone]
Copyright(cid:13)c 1998–2010,CharlesS.Zender
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Facts about FACTs: This document is part of the Freely Available Community Text (FACT)
project. FACTs are created, reviewed, and continuously maintained and updated by members of
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project website. FACTs 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 FACTs and
theirURLsarelistedinTable1. Becauseofitsinternationalscopeandavailabilitytostudentsofall
Table1: FreelyAvailableCommunityTexts
Format URL Location
RadiativeTransferintheEarthSystem
DVI http://dust.ess.uci.edu/facts/rt/rt.dvi
PDF http://dust.ess.uci.edu/facts/rt/rt.pdf
Postscript http://dust.ess.uci.edu/facts/rt/rt.ps
ParticleSizeDistributions: TheoryandApplicationtoAerosols,Clouds,andSoils
DVI http://dust.ess.uci.edu/facts/psd/psd.dvi
PDF http://dust.ess.uci.edu/facts/psd/psd.pdf
Postscript http://dust.ess.uci.edu/facts/psd/psd.ps
NaturalAerosolsintheClimateSystem
DVI http://dust.ess.uci.edu/facts/aer/aer.dvi
PDF http://dust.ess.uci.edu/facts/aer/aer.pdf
Postscript http://dust.ess.uci.edu/facts/aer/aer.ps
incomelevels,theFACTprojectmayimpactmorestudents,andtoagreaterdepth,thanimaginable
to before the advent of the Internet. If you are interested in learning more about FACTs and how
youmightcontributetoorbenefi[email protected].
ii
Notes for Students of ESS 223,
Earth System Physics:
This monograph on Radiative Transfer provides some core and some supplementary reading ma-
terial for ESS 223. We will discuss much of the material in the first twenty pages, and the figures
attheend. TheIndexbeginningonpage168isalsohelpful.
Notes for Students of ESS 236,
Radiative Transfer and Remote Sensing:
Yadayadayada.
CONTENTS iii
Contents
Contents iii
ListofFigures vi
ListofTables 1
1 Introduction 1
1.1 PlanetaryRadiativeEquilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 RadiativeTransferEquation 4
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 MeanIntensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4 ActinicFlux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.5 ActinicFluxEnhancement . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.6 EnergyDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.7 Spectralvs.Broadband . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.8 ThermodynamicEquilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.9 PlanckFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.10 HemisphericQuantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.11 Stefan-BoltzmannLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.12 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.13 ExtinctionandEmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.1.14 OpticalDepth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.15 GeometricDerivationofOpticalDepth . . . . . . . . . . . . . . . . . . . 26
2.1.16 StratifiedAtmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 IntegralEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 FormalSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 ThermalRadiationInAStratifiedAtmosphere . . . . . . . . . . . . . . . 31
2.2.3 AngularIntegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 ThermalIrradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.5 GreyAtmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.6 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.7 PhaseFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.8 LegendreBasisFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.9 Henyey-GreensteinApproximation . . . . . . . . . . . . . . . . . . . . . 40
2.2.10 DirectandDiffuseComponents . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.11 SourceFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.12 RadiativeTransferEquationinSlabGeometry . . . . . . . . . . . . . . . 42
2.2.13 AzimuthalMeanRadiationField . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.14 AnisotropicScattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
CONTENTS iv
2.2.15 DiffusivityApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.16 Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Reflection,Transmission,Absorption . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 BRDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.2 LambertianSurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.3 Albedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.4 FluxTransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 Two-StreamApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.1 Two-StreamEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.2 LayerOpticalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4.3 ConservativeScatteringLimit . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5 SolarHeating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 ChapterExercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 RemoteSensing 62
3.1 RayleighLimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2 AnomalousDiffractionTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 GeometricOpticsApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 SingleScatteredIntensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 SatelliteOrbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.6 AerosolCharacterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6.1 MeasuringAerosolOpticalDepth . . . . . . . . . . . . . . . . . . . . . . 69
3.6.2 AerosolIndirectEffectsonClimate . . . . . . . . . . . . . . . . . . . . . 69
3.6.3 AerosolEffectsonSnowandIceAlbedo . . . . . . . . . . . . . . . . . . 69
3.6.4 A˚ngstro¨mExponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 GaseousAbsorption 70
4.1 LineShape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.1 LineShapeFactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.1.2 NaturalLineShape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.3 PressureBroadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.4 DopplerBroadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.1.5 VoigtLineShape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 MolecularAbsorption 81
5.1 MechanicalAnalogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1.1 VibrationalTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.2 IsotopicLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.3 CombinationBands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 PartitionFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 DipoleRadiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 TwoLevelAtom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 LineStrengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5.1 HITRAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.6 Line-By-LineModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
CONTENTS v
5.6.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 BandModels 96
6.1 Generic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.1 BeamTransmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.2 BeamAbsorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1.3 EquivalentWidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1.4 MeanAbsorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2 LineDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.1 LineStrengthDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.3 MeanLineIntensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.4 MeanAbsorptanceofLineDistribution . . . . . . . . . . . . . . . . . . . 106
6.2.5 Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.6 MultiplicationProperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3 TransmissioninInhomogeneousAtmospheres . . . . . . . . . . . . . . . . . . . . 116
6.3.1 Constantmixingratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.2 H-C-GApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 TemperatureDependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5 TransmissioninSphericalAtmospheres . . . . . . . . . . . . . . . . . . . . . . . 120
6.5.1 ChapmanFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 RadiativeEffectsofAerosolsandClouds 121
7.1 SingleScatteringProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1.1 MaxwellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 SeparationofVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.1 AzimuthalSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2.2 PolarSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2.3 RadialSolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2.4 PlaneWaveExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.2.5 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2.6 MieTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2.7 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2.8 OpticalEfficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.9 OpticalCrossSections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.10 OpticalDepths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2.11 SingleScatteringAlbedo . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2.12 AsymmetryParameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2.13 MassAbsorptionCoefficient . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.3 EffectiveSingleScatteringProperties . . . . . . . . . . . . . . . . . . . . . . . . 134
7.3.1 EffectiveEfficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3.2 EffectiveCrossSections . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.3.3 EffectiveSpecificExtinctionCoefficients . . . . . . . . . . . . . . . . . . 136
7.3.4 EffectiveOpticalDepths . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.3.5 EffectiveSingleScatteringAlbedo . . . . . . . . . . . . . . . . . . . . . . 136
LISTOFFIGURES vi
7.3.6 EffectiveAsymmetryParameter . . . . . . . . . . . . . . . . . . . . . . . 136
7.4 MeanEffectiveSingleScatteringProperties . . . . . . . . . . . . . . . . . . . . . 136
7.4.1 MeanEffectiveEfficiencies . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4.2 MeanEffectiveCrossSections . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4.3 MeanEffectiveSpecificExtinctionCoefficients . . . . . . . . . . . . . . . 138
7.4.4 MeanEffectiveOpticalDepths . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4.5 MeanEffectiveSingleScatteringAlbedo . . . . . . . . . . . . . . . . . . 138
7.4.6 MeanEffectiveAsymmetryParameter . . . . . . . . . . . . . . . . . . . . 139
7.5 BulkLayerSingleScatteringProperties . . . . . . . . . . . . . . . . . . . . . . . 139
7.5.1 AdditionofOpticalProperties . . . . . . . . . . . . . . . . . . . . . . . . 139
7.5.2 BulkOpticalDepths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.5.3 BulkSingleScatteringAlbedo . . . . . . . . . . . . . . . . . . . . . . . . 140
7.5.4 BulkAsymmetryParameter . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.5.5 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8 GlobalRadiativeForcing 140
9 ImplementationinNCARmodels 150
10 Appendix 151
10.1 VectorIdentities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.2 LegendrePolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.3 SphericalHarmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
10.4 BesselFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
10.4.1 SphericalBesselFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.4.2 RecurrenceRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.4.3 PowerSeriesRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.4.4 AsymptoticValues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.5 GaussianQuadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.6 Gauss-LobattoQuadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.7 ExponentialIntegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Bibliography 161
Index 168
List of Figures
1 CrossSectionandQuantumYieldofNitrogenDioxide . . . . . . . . . . . . . . . 11
2 VerticalDistributionofPhotodissociationRates . . . . . . . . . . . . . . . . . . . 12
3 ClimatologicalMeanAbsorbedSolarRadiation . . . . . . . . . . . . . . . . . . . 141
4 ClimatologicalMeanEmittedLongwaveRadiation . . . . . . . . . . . . . . . . . 142
5 ENSOTemperatureandOLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6 SeasonalShortwaveCloudForcing . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7 ZonalMeanShortwaveCloudForcing . . . . . . . . . . . . . . . . . . . . . . . . 145
8 SeasonalLongwaveCloudForcing . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9 ZonalMeanLongwaveCloudForcing . . . . . . . . . . . . . . . . . . . . . . . . 147
10 ClimatologicalMeanNetCloudForcing . . . . . . . . . . . . . . . . . . . . . . . 148
11 ENSOCloudForcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
List of Tables
1 FACTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
2 WaveParameterConversionTable . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 ActinicFluxEnhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 SurfaceAlbedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 TemperatureDependenceofα . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
p
6 Pressure-BroadenedHalfWidths . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 MechanicalAnaloguesofImportantGases . . . . . . . . . . . . . . . . . . . . . . 85
8 HITRAN database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9 Full-rangeGaussianquadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10 Half-rangeGaussianquadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
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
sensinginstruments. OurapproachowesmuchtoBohrenandHuffman[1983](particlescattering),
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
theflavoroftheresearcherswhocontributetoit.
1.1 Planetary Radiative Equilibrium
Theimportantrolethatradiationplaysintheclimatesystemisperhapsbestillustratedbyasimple
example showing that without atmospheric radiative feedbacks (especially, ironically, the green-
house effect), our planet’s mean temperature would be well below the freezing point of water.
Earthissurroundedbythenearvacuumofspacesotheonlywaytotransportenergytoorfromthe
planet is via radiative processes. If E is the thermal energy of the planet, and FASR and FOLR are
theabsorbedsolarradiationandemittedlongwaveradiation,respectively,then
∂E
= FASR −FOLR (1)
∂t
On timescales longer than about a year the Earth as a whole is thought to be in planetary radiative
equilibrium. That,is,theglobalannualmeanplanetarytemperatureisnearlyconstantbecausethe
1 INTRODUCTION 2
absorbedsolarenergyisexactlycompensatedbythermalradiationlosttospaceoverthecourseof
ayear. Thus
FASR = FOLR (2)
The total amount of solar energy available for the Earth to absorb is the incoming solar flux (or
irradiance)atthetopofEarth’satmosphere,F (akathesolarconstant),timestheinterceptingarea
(cid:12)
of Earth’s disk which is πr2. Since Earth rotates, the total mean incident flux πr2F is actually
⊕ ⊕ (cid:12)
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 /4. The fraction of
(cid:12)
incident solar flux which is reflected back to space, and thus unable to heat the planet, is called
the aplanetary albedo or spherical albedo, R. Satellite observations show that R ≈ 0.3. Thus only
(1−R)ofthemeanincidentsolarfluxcontributestowarmingtheplanetandwehave
FASR = (1−R)F /4 (3)
(cid:12)
Earthdoesnotcooltospaceasaperfectblackbody(41a)ofasingletemperatureandemissivity.
NeverthelessthespectrumofthermalradiationFOLR whichescapestospaceandthuscoolsEarth
does resemble blackbody emission with a characteristic temperature. The effective temperature
T of an object is the temperature of the blackbody which would produce the same irradiance.
E
InvertingtheStefan-BoltzmannLaw(73)yields
T ≡ (FOLR/σ)1/4 (4)
E
For a perfect blackbody, T = T . For a planet, the difference between T and the mean surface
E E
temperatureT is dueto theradiative effectsof theoverlying atmosphere. The insulatingbehavior
s
oftheatmosphereismorecommonlyknownasthegreenhouseeffect.
Substituting(3)and(4)into(2)
(1−R)F /4 = σT4 (5)
(cid:12) E
(cid:18)(1−R)F (cid:19)1/4
(cid:12)
T = (6)
E
4σ
For Earth, R ≈ 0.3 and F ≈ 1367Wm−2. Using these values in (6) yields T = 255K.
(cid:12) E
ObservationsshowthemeansurfacetemperatureT = 288K.
s
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
itsfrequency
E = hν (7)
1 INTRODUCTION 3
where h is Planck’s constant. Regrettably, almost no radiative transfer literature expresses quanti-
tiesinfrequency.
Arelatedquantity,theangularfrequencyω measurestherateofchangeofwavephaseinradi-
ans per second. Wave phase proceeds through 2π radians in a complete cycle. Thus the frequency
andangularfrequencyaresimplyrelated
ω = 2πν (8)
Since radians are considered dimensionless, the units of ω are s−1. However, angular frequency is
also rarely used in radiative transfer. Thus some authors use the symbol ω to denote the element
ofsolidangle,asindω. Thereadershouldbecarefulnottomisconstruethetwomeanings. Inthis
textweuseω onlyinfrequently.
Most radiative transfer literature use wavelength or wavenumber. Wavelength, λ (m), mea-
sures the distance between two adjacent peaks or troughs in the wavefield. The universal relation
betweenwavelengthandfrequencyis
λν = c (9)
wherecisthespeedoflight. Sincecdependsonthemedium,λalsodependsonthemedium.
Thewavenumber ν˜m−1,isexactlytheinverseofwavelength
1 ν
ν˜ ≡ = (10)
λ c
Thus ν˜ measures the number of oscillations per unit distance, i.e., the number of wavecrests per
meter. Using (9) in (10) we find ν˜ = ν/c so wavenumber ν˜ is indeed proportional to frequency
(andthustoenergy). Historicallyspectroscopistshavefavoredν˜ratherthanλorν. Becauseofthis
history, it is much more common in the literature to find ν˜ expressed in CGS units of cm−1 than
in SI units of m−1. 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 ν˜ expressed in CGS wavenumber units (cm−1) and
energyinSIunits(J)isobtainedbyusing(10)in(7)
E = 100hcν (11)
Thereisanother,distinctquantityalsocalledwavenumber. Thissecondaryusageofwavenum-
berinthistextisthetraditionalmeasureofspatialwavepropagationandisdenotedbyk.
k ≡ 2πν˜ (12)
The wavenumber k is set in Roman typeface as an additional distinction between it and other
symbols1.
Table 2 summarizes the relationships between the fundamental parameters which describe
wave-likephenomena.
1ThescriptkisalreadyusedforBoltzmann’sconstant,absorptioncoefficients,andvibrationalmodes
2 RADIATIVETRANSFEREQUATION 4
Table2: WaveParameterConversionTableab
Variable ν λ ν˜ ω k τ
Units s−1 m cm−1 s−1 m−1 s
c ν 2πν 1
ν − 2πν
ν 100c c ν
c 1 2πc 2π λ
λ −
λ 100λ λ λ c
1 ν˜ 1
ν˜ 100cν˜ − 200πν˜
100ν˜ 200πc 100cν˜
ω 2πc ω ω 2π
ω −
2π ω 200πc c ω
kc 2π k 2π
k ck −
2π k 200π ck
1 1 2π 2π
τ cτ −
τ 100cτ τ cτ
aThespeedoflightiscms−1.
bTableentriesexpressthecolumnintermsoftherow.
2 Radiative Transfer Equation
2.1 Definitions
2.1.1 Intensity
The fundamental quantity defining the radiation field is the specific intensity of radiation. Spe-
cific 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 I are Joule meter−2 second−1
λ
steradian−1 meter−1. InSIdimensionalnotation,theunitscondensetoJm−2s−1sr−1m−1. TheSI
unitofpower(1Watt≡1Joulepersecond)ispreferred,leadingtounitsofWm−2sr−1m−1. Often
the specific intensity is expressed in terms of spectral frequency I with units Wm−2sr−1Hz−1 or
ν
spectralwavenumber(alsoI )withunitsWm−2sr−1(cm−1)−1.
ν˜
Consider light travelling in the direction Ωˆ through the point r. Construct an infinitesimal
elementofsurfaceareadS intersectingrandorthogonaltoΩˆ. TheradiantenergydE crossingdS
intimedtinthesolidangledΩinthefrequencyrange[ν,ν +dν]isrelatedtoI (r,Ωˆ)by
ν
dE = I (r,Ωˆ,t,ν)dSdtdΩdν (13)
ν
It is not convenient to measure the radiant flux across surface orthogonal to Ωˆ, 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 nˆ, then we