TotestandstudythebehaviorofourIITMimplementation,westudyspheroidswithanoutershellofmeltwater.Weconsiderasolidicecoreaswellas“fluffy”mixedair-icecoreswithdensities0.1,0.2,and0.3g/cm3. Numericalcalculationsarecarriedforbothprolate andoblatespheroidswithasemi-minortosemi-majorratiosequalto1/2(2)forfrequencies13.6GHz,35.6GHz,and94GHz.Thecorrespondingsizeparametersare~0.25,~0.64,and1.7.
WewouldliketothanktheNASACenterforClimateSimulation(NCCS)forpartialsupportofthisongoingeffort.
CraigPelissier,PhDScienceSystemsandApplicationsIncorporationNASAGoddardSpaceFlightCenter,Code606,Greenbelt,[email protected]
Acknowledgements
ContactInformation
Invariant Imbedded T-Matrix Method for Axial Symmetric Hydrometeors with Extreme Aspect Ratios
CraigPelissier1,3, Kwo-Sen Kuo1,2, Tom Clune1, Ian Adams1, S. Joseph Munchak1
1NASA Goddard Space Flight Center, Maryland, USA; 2Earth System Science Interdisciplinary Center, University of Maryland-College Park, Maryland, USA3Science Systems and Applications Incorporation, Maryland, USA
IntroductionOneofthemostfundamentalandcrucialquantitiesforaccurateprecipitationretrievalsisthesingle-scatteringproperty(SSP)ofeachindividualhydrometeorintheprecipitatingvolume. Thus,animportantstepinminimizingretrievaluncertaintyistoensurethattheparticlesusedforsingle-scatteringcalculationsresemblethosethatoccurinnature,allowingtheSSPstobecomputedpreciselywithnumericaltechniques.
Hydrometeorswithrotationalsymmetry(axialsymmetry)aremostefficientlycomputedusingtheT-Matrixmethods.Forhydrometeorswithauniformdielectric,theExtendedBoundaryConditionMethod(EBCM)isthemostefficient.However,formixed-phasehydrometeorswithnon-uniformdielectricsEBCMisnotapplicable.Inthesecases,wecanusetheInvariantImbeddedT-Matrix(IITM),whichwedevelopedforthispurpose.ItisalsonotedthathydrometeorswithmoreextremeaspectratioscanbecomputedwithIITMthanEBCMandthatsuchhydrometeorsarefoundinprecipitatingsystems.Forhydrometeorswithoutanyrotationalsymmetry,however,volumeintegralmethodslikeDiscreteDipoleApproximation(DDA)andtheMethodofMoments(MoM)mustbeemployed.
AlgorithmFlow-ChartforSSPs
AnefficientC++parallelIIITMcodehasbeendeveloped.ThiscodeaccuratelycomputesSSPsofnon-uniformhydrometeorswithrotationalsymmetry.Thecodeparallelizesbytakingadvantageoftheblockstructureinducedbyrotationalsymmetryaboutthez-axisbydistributingblocksamongMPIprocesses.Longdoubleprecisionswasrequiredtoachievesatisfactoryaccuracy.ThisIITMcodewillallowtheefficientcomputationofmixed-phaserotationally-symmetrichydrometeorsnecessaryformodelingprecipitatingsystems.
ItischallengingtouseT-MatrixMethodsforirregularhydrometeors,anditisnotclearthatnumericalprecisionandcomputeconstraintswillallowaccurateSSPstobecomputedusingthismethod.Currently,DDAapproachesareemployed.
35.6GHz
=Prolate spheroid =Oblatespheroid
%Massmeltwater
13.6GHz
94GHz
%Massmeltwater
%Massmeltwater
=0.1g/cm3
=0.3g/cm3
=0.2g/cm3
=0.916g/cm3
T-MatrixmethodsarecomputationallymoreefficientthanDiscreteDipoleApproximations(DDAs)forscatteringtargetsthatpossessahighdegreeofsymmetry.Forthecaseofspheroidsshown,itonlytakesafewminuteson32corestocomputetheorientationaveragedscatteringproperties.ThereasonforthisisthattheT-Matrixbecomessparseandstructured,reducingtheproblemsizesignificantly.Irregularparticlesincurasubstantiallylargercostasthemaximumallowedangularmomentum,Lmax,increases,sincetheblockdiagonalstructureislost.Specifically,thecomputationalcostforaxialandnosymmetry:
Afurthercomplicationisthatsurfaceintegralscarriedoutoverthespherebecomehighlydiscontinuousforirregularparticles.Inthiscase,the“nice”propertiesofGauss-Legendrequadraturearenolongertrue,andpartitioningtheintegralswouldrequireanumericalroutinetofindtheboundaries.
Axialsymmetry No symmetry
FLOPS ~ Lmax4 × # shells ~ Lmax
6 × # shellsHydrometeorsSSPs
NoSymmetry(non-uniform)
DDA,MoM
RotationalSymmetry
UniformDielectric
ExtremeAspectRatios
IITM
ModerateAspectRatios
EBCM
Non-UniformDielectric
IITM
SphericalSymmetry
(non-uniform)
Multi-layeredLorenz-Mie
ImplementationTheIITMistypicallyanorderofmagnitudemorecomputationallyexpensivethanEBCM.TomakeIITMpracticalrequiresanefficientimplementationthatisoptimizedandtakesadvantageofparallelismintheIITMalgorithm.
AlgorithmSchematic
SOV
rk
rk-1
1. InitializewithSeparationofVariables(SOV).
2. Performquadratureoversphericalshellrk-1 --- Gauss-Legendrequadrature.
3. UsingrecursionrelationstogettheT-Matrixatrk,T(rk).---matrixinversion+linearalgebra.Rombergintegrationused.
BlockStructure/Parallelism
M=0
M=1
M=2
Rotationalsymmetrydecouples“m”
Tl’l,m’m=Tl’lm dm’m
AngularQuadrature—Gauss-Legendre
NumericalConvergence/Precision
Mixed-Phase Hydrometeor Spheroids
M=0 M=2 M=3
Process:0123456
M=1 M=-1 M=-3M=-2
Naïvescaling—nocommunication.Scaleswell~Lmax, thepartialwavecutoff.
z
z
S2
S11. Surfacediscontinuityleadsto
significantnumericalerror.2. Integrationmustbebrokenup:
I=∫S1+∫S2
3. Gauss-quadraturedominatestheruntime.
4. Functionsarecachedforeachshelltospeedupperformance.
Irregular Hydrometeors
Conclusions
OblateSpheroid Extinctionefficiency ScatteringEfficiencyIITM+SOV(this) 3.28513(2)(2) 2.29014(2)(2)
EBCM[1] 3.2854 2.2903
IITM+SOV[1] 3.2850 2.2901
[1]Bi,L.,P.Yang,G.W.Kattawar,and M.I.Mishchenko,2013:EfficientimplementationoftheinvariantimbeddingT-matrixmethodandtheseparationofvariablesmethodappliedtolargenonspherical inhomogeneousparticles. J.Quant.Spectrosc.Radiat.Transfer, 116,169-183,doi:10.1016/j.jqsrt.2012.11.014
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Doubleprecisionfails.
LongdoubleprecisionfailsforL>96.
M=-1
References
Comparisonwithresultsfrom[1]witherrorestimatesassociatedwithpartialwavetruncationandthenumberofradialshells(dl)(dr),respectively.
2017FallMeetingNewOrleans,LA,USA11-15December2017
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