Turbulence and Dispersion in the Planetary Boundary Layer

252 Pages·2017·6.105 MB·English

by  Francesco Tampieri (auth.)

Physics of Earth and Space Environments
Francesco Tampieri
Turbulence and
Dispersion in
the Planetary
Boundary Layer
123
Turbulence and Dispersion in the Planetary
Boundary Layer
Physics of Earth and Space Environments
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Francesco Tampieri
Turbulence and Dispersion
in the Planetary Boundary
Layer
123
FrancescoTampieri
CNRISAC
Bologna,Italy
ISSN1610-1677 ISSN1865-0678 (electronic)
PhysicsofEarthandSpaceEnvironments
ISBN978-3-319-43602-9 ISBN978-3-319-43604-3 (eBook)
DOI10.1007/978-3-319-43604-3
LibraryofCongressControlNumber:2016950206
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Preface
The purpose of this book is to give an updated and problematic description of
theatmosphericturbulenceandthe dispersionin the planetaryboundarylayer,for
students, and perhapsto researchers,interested in the atmosphericsciences. Well-
settled argumentscoexists with topics investigatedin the last years or even under
discussion:thus, the readerwill find differencesin the treatmentof the arguments
thatreflecttheseaspects.Thehopeistogiveacomprehensiveviewofthephysics
of the planetaryboundarylayer,with the certaintiesandthe uncertainties,to raise
interestandtostimulatefutureresearch.
Acoupleofobservations(whichshouldbecomesuggestionstothereader).The
firstone:manyboringcomputationsareputintotheexercises.Mostofthemrefer
to analytic solutions of simplified problems. Although numerical modelling is of
increasinguse,theexactsolutionsmaybeabenchmarkformodelsandareusefulto
betterevidencethebasicmechanismsandapproximations.Thestudentshouldnot
overlookthisaspect.
Thesecondone:intheAppendixtotheIntroduction,somedatasetsarecited,and
referenceis given to downloadthe data. Note that the list is not exhaustive;other
data sets are freely available. The data typically refer to variables averaged over
some time interval. Most of the figures of the book are made using such data, so
thateveryfigurecanbeunderstoodasanexerciseforthestudent,whocanremake,
modifyandpossiblyimproveit.(Asageneralruleforthefigures,thedataareplotted
directly, with various symbols, or binned in intervals of the independent variable
andreportedintermsofthemedianandthe10thandthe90thpercentiles,witherror
bars.)
Iwouldliketoacknowledgehereallwhocontributedtothegenesisofthebook:
first, all my students who pushed me to build (as far as possible for me) a clear
andunifyingpictureofthetopicandthecolleaguesfordiscussionsandcriticism,in
particularthosewhogavemedataandspecialimages.SpecialthankstoDomenico
Anfossi, Alessandra Lanotte, Silvia Trini Castelli and Sergej Zilitinkevich, who
suggested specific argumentsand supplied the proper material. Last but not least,
v
vi Preface
I must remember two people who addressed my research and thus are in part
responsibleforthegenesisofthisbook:OttavioVittori,whotaughtmenottocross
theroadonzebracrossing,andJulianHunt,whointroducedmetothemysteriesof
turbulence.
Contents
1 Introduction .................................................................. 1
1.1 TheBasicDefinitionofthePlanetaryBoundaryLayer............... 1
1.2 AFewWordsAboutTurbulence....................................... 2
1.3 TheStructureandEvolutionofthePBL............................... 3
1.3.1 LocalEquilibrium .............................................. 4
1.3.2 HeterogeneitiesandUnsteadiness ............................. 6
1.3.3 TheBoundaryLayerDepth .................................... 8
1.4 TheTransportProblemandtheTurbulentDispersion ................ 9
1.5 Observations............................................................. 10
1.6 NumericalExperimentsandSimulations.............................. 11
Appendix ...................................................................... 11
References..................................................................... 14
2 ASummaryofMathematicsandPhysicsforPBL....................... 17
2.1 EulerianandLagrangianDescription.................................. 17
2.2 TheEquationsforVelocityandPassiveScalars....................... 18
2.2.1 TheNavier-StokesEquations(NSE)inaRotating
ReferenceFrame................................................ 18
2.2.2 TheHydrostaticPressureandthePBLFormofNSE......... 19
2.2.3 TheContinuityEquation ....................................... 20
2.2.4 TheEquationforaPassiveScalar.............................. 20
2.2.5 ALittleThermodynamics...................................... 21
2.2.6 The Equations for the Temperature
andforthePotentialTemperature.............................. 23
2.2.7 TheNondimensionalFormoftheEquations.................. 24
2.3 StochasticVariables..................................................... 25
2.3.1 ProbabilityDensityFunctionandMoments................... 25
2.3.2 Averaging ....................................................... 26
2.3.3 CovariancesandSpectra........................................ 28
vii
viii Contents
2.4 ReynoldsAveragedEquations.......................................... 34
2.4.1 TheEquationsfortheFirst-OrderMoments .................. 34
2.4.2 TheEquationsfortheFluctuations ............................ 36
2.4.3 TheEquationsfortheSecond-OrderMoments
ofVelocity....................................................... 36
2.4.4 TheEquationfortheTemperatureVariance................... 37
2.4.5 TheEquationsfortheHeatFluxes............................. 38
2.4.6 TheInterpretationoftheFluctuationCovariances
andtheEddyDiffusionModel................................. 39
2.5 UniversalFeaturesofShear-DominatedTurbulence.................. 41
2.5.1 VelocityCovariancesandSpectra.............................. 42
2.5.2 TheSpectraofthePassiveTracerVariances .................. 46
2.5.3 SomeConsequencesofIsotropy............................... 46
2.5.4 FinalRemarks................................................... 47
Exercises....................................................................... 48
References..................................................................... 48
3 TheBasicParadigm:HorizontalHomogeneityOverFlatTerrain..... 51
3.1 TheGoverningEquations............................................... 51
3.2 InnerandOuterScalingfromtheWindProfile........................ 53
3.3 Similarity,ObukhovLengthandBeyond.............................. 54
3.4 TheSurfaceLayerinNeutralandUnstableConditions............... 56
3.4.1 TheQuasi-NeutralConditionsandtheMeanWindProfile... 56
3.4.2 UnstableConditions............................................ 60
3.4.3 The Higher-OrderMomentsof the Velocity
ComponentsandoftheTemperatureFluctuations............ 69
3.5 TheOuterRegioninNeutralConditions .............................. 77
3.5.1 TheMeanVelocityintheEkmanLayer....................... 77
3.5.2 Truly-Neutraland Conventionally-Neutral
BoundaryLayers................................................ 78
3.5.3 ResistanceLaws ................................................ 79
3.6 SomeFeaturesoftheConvectiveBoundaryLayer.................... 80
3.6.1 Second-andThird-OrderMomentsofFluctuations .......... 82
3.6.2 TheMorningGrowthoftheCBL.............................. 83
3.6.3 TheDay-NightTransitionandtheResidualLayer(RL)...... 87
3.7 StableBoundaryLayers ................................................ 88
3.7.1 LocalSimilarity................................................. 93
3.7.2 TheSecond-OrderMoments................................... 98
3.7.3 TheNieuwstadt(1984)Model................................. 101
3.7.4 TheNeutralandStableBoundaryLayerDepth............... 105
3.8 SomeRemarksAbouttheSpectra...................................... 106
Exercises....................................................................... 111
References..................................................................... 112
4 HorizontalHeterogeneities.................................................. 117
4.1 ExplicitTreatmentvs.Parameterization............................... 117
4.1.1 ACriterionforHorizontalHomogeneity...................... 117
Contents ix
4.2 InternalBoundaryLayers............................................... 118
4.2.1 RoughnessLengthChanges.................................... 119
4.2.2 TheThermalIBLattheSea-LandTransition ................. 121
4.3 TheBoundaryLayerOverHillsandValleys .......................... 123
4.3.1 TheLinearizedEquations...................................... 123
4.3.2 TheInnerandOuterLayerConceptintheNeutralFlow..... 124
4.3.3 TheOuterLayerandtheStratificationEffects................ 127
4.3.4 ADiscussionAbouttheInnerLayer........................... 130
4.3.5 TheTurbulentWakeandtheSeparation....................... 132
4.3.6 SpectraModifications .......................................... 134
4.4 SubgridEffectsoftheHeterogeneousSurfaceFeatures .............. 134
4.4.1 DistributionsofRoughnessElementsonaFlatSurface...... 134
4.4.2 TheEffectiveRoughnessofTopography...................... 135
4.5 LowWind,SmallVerticalFluxes...................................... 137
4.6 CanopyFlowandtheUrbanPBL...................................... 140
4.6.1 SomeScalesandtheDragDuetotheBuildings.............. 141
4.6.2 TheFlowAbovetheCanopy................................... 144
4.6.3 TheAverageFlowinaVolumewithanArray
ofSolidObstacles:TheUrbanCanopyLayer................. 145
4.6.4 HeterogeneousUrbanCanopy ................................. 148
Exercises....................................................................... 150
References..................................................................... 150
5 TurbulentDispersion ........................................................ 155
5.1 TheTransportProblemforFluidParcels .............................. 155
5.1.1 ProbabilityDensityFunctions,Concentration
andWellMixedCondition(WMC)............................ 156
5.2 AbsoluteDispersionofTracerParcels................................. 157
5.2.1 SteadyHomogeneousConditions:Taylor(1921)............. 157
5.2.2 ExtensiontoInhomogeneousConditions...................... 160
5.3 Two-ParcelDispersion.................................................. 162
5.3.1 TheDispersionintheInertialSubrange....................... 163
5.3.2 TheDiffusivePhase ............................................ 164
5.4 Meandering.............................................................. 165
5.5 ObservationsofDispersion............................................. 167
5.5.1 Mikkelsenetal.(1987):HorizontalMeandering
andRelativeDispersionintheSurfaceLayer................. 167
5.5.2 VerticalandLateralDispersioninaLaboratoryCBL ........ 167
5.6 TheStochasticApproachtotheAbsoluteDispersion
ofTracerParcels ........................................................ 169
5.6.1 TheLinkBetweentheEulerianandLagrangian
Descriptions..................................................... 169
5.6.2 TheModelwithUncorrelatedVelocities:N D3 ............. 170
5.6.3 TheModelwithUncorrelatedAccelerations:N D6 ......... 173