Energy balance climate models

374 Pages·2017·22.314 MB·English

by  Kim, Kwang-Yul, North, Gerald R

EnergyBalanceClimateModels
WileySeriesinAtmosphericPhysicsandRemoteSensing
SeriesEditor:AlexanderKokhanovsky
Wendisch,M./Brenguier,J.-L.(eds.) North,G.R./Kim,K.-Y.
AirborneMeasurementsfor EnergyBalanceClimate
EnvironmentalResearch Models
MethodsandInstruments
2013 Forthcoming:
Kokhanovsky,A./Natraj,V.
CoakleyJr.,J.A./Yang,P. AnalyticalMethodsin
AtmosphericRadiation AtmosphericRadiative
Transfer
APrimerwithIllustrativeSolutions
2014
Huang,X./Yang,P.
Stamnes,K./Stamnes,J.J.
RadiativeTransferProcessesin
RadiativeTransferinCoupled WeatherandClimateModels
EnvironmentalSystems
AnIntroductiontoForwardandInverse
Modeling
Davis,A.B./Marshak,A.
2015
Multi-dimensionalRadiative
Transfer
Tomasi,C./Fuzzi,S./Kokhanovsky,A. Theory,Observation,andComputation
AtmosphericAerosols
LifeCyclesandEffectsonAirQualityand
Climate
Minnis,P.etal.
2016
SatelliteRemoteSensingof
Clouds
Weng,F.
PassiveMicrowaveRemote
SensingoftheEarth Zhang,Z.etal.
forMeteorologicalApplications PolarimetricRemoteSensing
AerosolsandClouds
2017
Energy Balance Climate Models
GeraldR.NorthandKwang-YulKim
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WewishtodedicatethisbooktothememoryofthosewhoworkedwithusonEBMsbut
weretakenawaymuchtoosoon,ThomasJ.Crowley,JohnMengel,andWan-HoLee.
vii
Contents
Preface xiii
1 ClimateandClimateModels 1
1.1 DefiningClimate 3
1.2 ElementaryClimateSystemAnatomy 7
1.3 RadiationandClimate 9
1.3.1 SolarRadiation 9
1.3.2 AlbedooftheEarth–AtmosphereSystem 13
1.3.3 TerrestrialInfraredRadiationintoSpace(TheIRorLongwave
Radiation) 14
1.4 HierarchyofClimateModels 15
1.4.1 GeneralCirculationModels(GCMs) 16
1.4.2 EnergyBalanceClimateModels 17
1.4.3 AdjustableParametersinPhenomenologicalModels 19
1.5 GreenhouseEffectandModernClimateChange 20
1.6 ReadingThisBook 20
1.7 CautionaryNoteandDisclaimer 22
NotesonFurtherReading 23
Exercises 23
2 GlobalAverageModels 27
2.1 TemperatureandHeatBalance 27
2.1.1 BlackbodyEarth 28
2.1.2 Budyko’sEmpiricalIRFormula 29
2.1.3 ClimateSensitivity 30
2.1.4 ClimateSensitivityandCarbonDioxide 31
2.2 TimeDependence 31
2.2.1 FrequencyResponseofGlobalClimate 32
2.2.2 ForcingwithNoise 35
2.2.3 PredictabilityfromInitialConditions 37
2.2.4 ProbabilityDensityoftheTemperature 39
2.3 SpectralAnalysis 40
2.3.1 WhiteNoiseSpectralDensity 41
2.3.2 SpectralDensityandLaggedCorrelation 41
2.3.3 AR1ClimateModelSpectralDensity 42
viii Contents
2.3.4 ContinuousTimeCase 42
2.4 NonlinearGlobalModel 44
2.4.1 Ice-AlbedoFeedback 44
2.4.2 LinearStabilityAnalysis:ASlope/StabilityTheorem 46
2.4.3 RelaxationTimeandSensitivity 47
2.4.4 FiniteAmplitudeStabilityAnalysis 48
2.4.5 PotentialFunctionandNoiseForcing 49
2.4.6 RelationtoCriticalOpalescence 52
2.5 Summary 52
SuggestionsforFurtherReading 53
Exercises 53
3 RadiationandVerticalStructure 57
3.1 RadianceandRadiationFluxDensity 58
3.2 EquationofTransfer 61
3.2.1 ExtinctionandEmission 61
3.2.2 TerrestrialRadiation 62
3.3 GrayAtmosphere 63
3.4 Plane-ParallelAtmosphere 64
3.5 RadiativeEquilibrium 65
3.6 SimplifiedModelforWaterVaporAbsorber 68
3.7 CoolingRates 72
3.8 SolutionsforUniform-SlabAbsorbers 73
3.9 VerticalHeatConduction 75
3.9.1 K>0 77
3.10 ConvectiveAdjustmentModels 77
3.11 LessonsfromSimpleRadiationModels 79
3.12 CriticismoftheGraySpectrum 80
3.13 AerosolParticles 82
NotesforFurtherReading 83
Exercises 83
4 GreenhouseEffectandClimateFeedbacks 85
4.1 GreenhouseEffectwithoutFeedbacks 85
4.2 InfraredSpectraofOutgoingRadiation 85
4.2.1 GreenhouseGasesandtheRecord 92
4.2.2 GreenhouseGasComputerExperiments 92
4.3 SummaryofAssumptionsandSimplifications 99
4.4 LogDependenceoftheCO Forcing 101
2
4.5 RunawayGreenhouseEffect 102
4.6 ClimateFeedbacksandClimateSensitivity 105
4.6.1 EquilibriumFeedbackFormalism 107
4.7 WaterVaporFeedback 108
4.8 IceFeedbackfortheGlobalModel 109
4.9 ProbabilityDensityofClimateSensitivity 110
4.10 MiddleAtmosphereTemperatureProfile 112
4.10.1 MiddleAtmosphereResponsestoForcings 113
Contents ix
4.11 Conclusion 115
NotesforFurtherReading 116
Exercises 116
5 LatitudeDependence 119
5.1 SphericalCoordinates 120
5.2 IncomingSolarRadiation 121
5.3 ExtremeHeatTransportCases 122
5.4 HeatTransportAcrossLatitudeCircles 122
5.5 DiffusiveHeatTransport 123
5.6 DerivingtheLegendrePolynomials 125
5.6.1 PropertiesofLegendrePolynomials 127
5.6.2 Fourier–LegendreSeries 128
5.6.3 IrregularSolutions 128
5.7 SolutionoftheLinearModelwithConstantCoefficients 129
5.8 TheTwo-ModeApproximation 129
5.9 PolewardTransportofHeat 133
5.10 Budyko’sTransportModel 134
5.11 RingHeatSource 136
5.12 AdvancedTopic:FormalSolutionforMoreGeneralTransports 137
5.13 IceFeedbackintheTwo-ModeModel 138
5.14 PolarAmplificationthroughIceCapFeedback 140
5.15 ChapterSummary 141
5.15.1 ParameterCount 142
NotesforFurtherReading 142
Exercises 142
6 TimeDependenceinthe1-DModels 145
6.1 DifferentialEquationforTimeDependence 146
6.2 DecayofAnomalies 146
6.2.1 DecayofanArbitraryAnomaly 147
6.3 SeasonalCycleonaHomogeneousPlanet 148
6.4 SpreadofDiffusedHeat 153
6.4.1 EvolutiononaPlane 155
6.5 RandomWindsandDiffusion 157
6.6 NumericalMethods 159
6.6.1 ExplicitFiniteDifferenceMethod 159
6.6.2 Semi-ImplicitMethod 162
6.7 SpectralMethods 163
6.7.1 GalerkinorSpectralMethod 163
6.7.2 PseudospectralMethod 164
6.8 Summary 166
6.8.1 ParameterCount 166
NotesforFurtherReading 167
Exercises 167
6.9 AppendixtoChapter6:SolarHeatingDistribution 169
6.9.1 TheEllipticalOrbitoftheEarth 171
x Contents
6.9.2 RelationBetweenDeclinationandObliquity 172
6.9.3 ExpansionofS(𝜇,t) 172
7 NonlinearPhenomenainEBMs 175
7.1 FormulationoftheNonlinearFeedbackModel 176
7.2 Stürm–LiouvilleModes 178
7.2.1 OrthogonalityofSLModes 179
7.3 LinearStabilityAnalysis 180
7.4 FinitePerturbationAnalysisandPotentialFunction 184
7.4.1 NeighborhoodofanExtremum 185
7.4.2 RelationtoGibbsEnergyorEntropy 187
7.4.3 AttractorBasins—NumericalExample 187
7.5 SmallIceCapInstability 187
7.5.1 PerturbationofanExactIce-FreeSolution 190
7.5.2 FrequencyDependenceoftheLengthScale 191
7.6 SnowCapsandtheSeasonalCycle 193
7.7 Mengel’sLand-CapModel 193
7.8 ChapterSummary 196
NotesforFurtherReading 199
Exercises 199
8 TwoHorizontalDimensionsandSeasonality 203
8.1 BeachBallSeasonalCycle 203
8.2 EigenfunctionsintheBoundedPlane 205
8.3 EigenfunctionsontheSphere 208
8.3.1 LaplacianOperatorontheSphere 208
8.3.2 LongitudeFunctions 209
8.3.3 LatitudeFunctions 209
8.4 SphericalHarmonics 211
8.4.1 Orthogonality 211
8.4.2 Truncation 212
8.5 SolutionoftheEBMwithConstantCoefficients 212
8.6 IntroducingGeography 214
8.7 GlobalSinusoidalForcing 216
8.8 Two-DimensionalLinearSeasonalModel 217
8.8.1 AdjustmentofFreeParameters 219
8.9 PresentSeasonalCycleComparison 220
8.9.1 AnnualCycle 220
8.9.2 SemiannualCycle 220
8.10 ChapterSummary 220
NotesforFurtherReading 224
Exercises 224
9 PerturbationbyNoise 229
9.1 Time-IndependentCaseforaUniformPlanet 230
9.2 Time-DependentNoiseForcingforaUniformPlanet 234
9.3 Green’sFunctionontheSphere: f =0 235
Contents xi
9.4 ApportionmentofVarianceataPoint 237
9.5 StochasticModelwithRealisticGeography 238
9.6 ThermalDecayModeswithGeography 243
9.6.1 StatisticalPropertiesofTDMs 246
NotesforFurtherReading 248
Exercises 249
10 Time-DependentResponseandtheOcean 253
10.1 Single-SlabOcean 254
10.1.1 ExampleswithaSingleSlab 255
10.1.2 EventualLevelingoftheForcing 258
10.2 PenetrationofaPeriodicHeatingattheSurface 259
10.3 Two-SlabOcean 262
10.3.1 DecayofanAnomalywithTwoSlabs 266
10.3.2 ResponsetoRampForcingwithTwoSlabs 268
10.4 Box-DiffusionOceanModel 269
10.5 SteadyStateofUpwelling-DiffusionOcean 271
10.5.1 All-OceanPlanetaryResponses 273
10.5.2 RampForcing 274
10.6 UpwellingDiffusionwith(andwithout)Geography 274
10.7 InfluenceofInitialConditions 276
10.8 ResponsetoPeriodicForcingwithUpwellingDiffusionOcean 277
10.9 SummaryandConclusions 280
Exercises 282
11 ApplicationsofEBMs:OptimalEstimation 287
11.1 Introduction 287
11.2 IndependentEstimators 288
11.3 EstimatingGlobalAverageTemperature 290
11.3.1 Karhunen–LoèveFunctionsandEmpiricalOrthogonalFunctions 292
11.3.2 RelationshipwithEBMs 296
11.4 DeterministicSignalsintheClimateSystem 298
11.4.1 SignalandNoise 299
11.4.2 FingerprintEstimatorofSignalAmplitude 299
11.4.3 OptimalWeighting 299
11.4.4 InterferingSignals 302
11.4.5 AllFourSignalsSimultaneously 303
11.4.6 EBM-GeneratedSignals 306
11.4.7 CharacterizingNaturalVariability 310
11.4.8 DetectionResults 311
11.4.9 DiscussionoftheDetectionResults 314
NotesforFurtherReading 317
Exercises 317
12 ApplicationsofEBMs:Paleoclimate 321
12.1 Paleoclimatology 321
12.1.1 InterestingProblemsforEBMs 322