Motivations, successes, and problems, with the Lagrangian ...
Transcript of Motivations, successes, and problems, with the Lagrangian ...
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Motivations, successes, and problems, with the Lagrangian-remap dynamical core in MOM6
AlistairAdcroftPrincetonUniversity/NOAA-GFDL
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• Wintonetal.,1998:z-coordinatemodelsexcessmixinginoverflows– Numerouscoordinate/mixingstudies:Griffies etal.,2000;Chassignetetal.,2003;Leggetal.,2006;Burchard&Rennau 2008;Megann etal.,2010;…
• Dunneetal.,2012:comparedESM2MandESM2G,both1° resolution– ESM2M:z-coordinate,shallowAMOC– ESM2G:isopycnal layermodel,deepAMOCwithoverflows
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Z-coordinatesv.isopycnal coordinates
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
Dunneetal.,2012
ESM2M
ESM2G
Talley,2003Ganachaud &Wunsch, 2003
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• Delworth etal.,2012,coupledmodelseries(CM2.1,CM2.5,CM2.6):– 50kmatmosphere– 1°,¼° and0.1° ocean– 1° alonehasSGSeddyparameterization(Gent-McWilliams)
• Griffies etal.,2015,diagnosedhowtransienteddiesina0.1° oceantransportheatupwards– CM2.6leastheatuptakeofCM2.xseries– ArguethatCM2.5hasmostheatuptakeduetoweakeddiesandnoGM
Roleofmesoscaleeddies
Evolutionofhorizontallyaveragedpotential temperature,θ (°C).
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CM2.5 CM2.6¼° ocean 1/10° oceanCM2-1deg
1.1°C 1.3°C 0.8°C
1° ocean
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• Measureworkdonebymixinginspindownexperiments– Turningoffexplicitmixingleavesonlynumericalmixing*
• Ilicak etal.,2012,argued¼° z-coordinatemodelhadnumericalmixingaslargeas“real”mixing
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Measuringspuriousmixing
[Wm-2]
Ilicak etal.,2012
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• ReducingspuriousmixingmotivatedMOM6development– Generalcoordinatetoavoidlimitationsofpotentialdensity(usedbyisopycnal layermodels)
• MOM6broadlyreproducedpriorsolutionsusingz-coordinates
• Adoptionofhybridz-densitycoordinatesreducedheatuptake– success?
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MOM6andOM4
OM41/4° z-coordinates
OM41/4° hybrid-coordinates
Adcroftetal.,2019
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• HydrostaticPrimitiveEquations(forocean)ingeneralcoordinates𝑟 = 𝑟(𝑥, 𝑦, 𝑧, 𝑡) :
Generalcoordinates(Boussinesq)
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
𝑧+ =𝜕𝑧𝜕𝑟
Starr,1945;Kasahara,1974;…
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• Integratingbetweensurfacesofconstant𝑟 converts𝑧+ → ℎ = ∫ 𝑧+𝑑𝑟 and𝜕+ → 𝛿+
• Choosing�̇� = 0 followstheverticalmotion→VerticallyLagrangian– removesexplicitverticaltransports→noverticalCFL
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Layerintegratedgeneralcoordinateequations
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
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Lagrangianremapalgorithm
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Reconstruction
𝜃†
𝑧5
𝜃5
Dynamics+“physics”
ℎ†ℎ5
�̇� = 𝟎
Average
𝑧589
𝜃589
Gridgeneration
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Eulerian,ALEandLRMalgorithmsside-by-side
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𝑣;589 = 𝑣;5 + Δ𝑡 − 9?@𝛻B𝑝 +⋯
𝜕E𝑤 = −𝛻 G 𝑣;589
𝜕E𝑝 = −𝑔𝜌 𝑧, 𝑆5, 𝜃5
𝜃589 = 𝜃5 − Δ𝑡 𝛻 G 𝑣;589𝜃5 +𝜕E 𝑤𝜃5 + ⋯
𝑣;589 = 𝑣;5 + Δ𝑡 − 9?@𝛻B𝑝 +⋯
𝛿K(𝑤∗ + 𝑤M) = −𝛻 G ℎ5𝑣;589
𝜕E𝑝 = −𝑔𝜌 𝑧, 𝑆5, 𝜃5
ℎ589𝜃589 = ℎ5𝜃5
−Δ𝑡 𝛻 G ℎ5𝑣;589𝜃5
+𝛿K 𝑤∗𝜃5 + ⋯
ℎ† = ℎ5 − Δ𝑡𝛻 G (ℎ5𝑣;†)
𝜕E𝑝 = −𝑔𝜌 𝑧, 𝑆5, 𝜃5
ℎ†𝜃† = ℎ5𝜃5 − Δ𝑡 𝛻 G ℎ5𝑣;†𝜃5+⋯
ℎ589 ← 𝛿K𝑍 𝑧†
𝜃589 = 𝜃† 𝑍(𝑧†)
𝑣;† = 𝑣;5 + Δ𝑡 − 9?@𝛻B𝑝 +⋯
Eulerian A.L.E. Langrangian-remap
wΔ𝑡Δ𝑧 < 1
𝑤∗Δ𝑡Δ𝑧 < 1
ℎ589 = ℎ5 + Δ𝑡𝛿K(𝑤M)
𝑤∗ = 𝑤− 𝑤M
introduces grid motion 𝑤M
Grid generation
Remap
SeeGriffies etal.,JAMES2020
Assumingexplicit-in-timetransport
Assumingexplicit-in-timetransport
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Sub-cyclingwithLagrangianverticaldynamics
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
𝑣;S89 = 𝑣;S + 9TUV?@−𝛻W𝑝 − 𝜌𝛻WΦ +⋯
ℎS89 = ℎS − 9TΔ𝑡𝛻W G (ℎ
S𝑣;S89)
𝛿K𝑝 = −𝜌 𝑧, 𝑆5, 𝜃5 𝛿KΦ
ℎ∗𝐶∗ = ℎ5𝐶5 −𝑀Δ𝑡 𝛻 G [ ℎS𝑣;S89𝐶5T
S\9
ℎ589 ← 𝛿K𝑍 ℎ∗ ; 𝐶589 = 𝐶∗ 𝑍(ℎ∗) ; …
×𝑀
×𝑁
Internal gravity waves
𝚫𝒕𝒄𝒊𝒈𝚫𝒙 < 𝟏
𝐌𝚫𝒕𝒖𝒉𝚫𝒙 < 𝟏
𝑈l89 = 𝑈l + 9mΔ𝑡 −𝛻𝜂
l + ⋯𝜂l89 = 𝜂S − 9
mΔ𝑡𝛻W G (𝐻𝑈l89)
Barotropic gravity waves
×𝐿 𝚫𝒕√𝒈𝑯𝑳𝚫𝒙 < 𝟏
Tracers
Vertical remap
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Interpolationforgridgeneration• Accurate• Continuous (singlevalued)– resultingprofilesaregenerallynotconservative
Reconstructionforremapping• Accurate• Conservative– profilesoftenbecomediscontinuoustobeconservative
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Consistentgrid-generationandremapping
Areas not equal
Discontinuity between layers
Second order interpolation (i.e. linear)
Third order reconstruction
Piecewise Parabolic (PPM)
Discrepancies reduce with higher order
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Remappingimplementations
𝑧589
𝜃K589
𝜃†
𝑧† 𝑧589
𝜃K589
𝜃†
𝑧†
Remappingvia“advectiveflux” Remappingby“projection”
ℎK589𝜃K589 = ℎKt𝜃K
t + u 𝜃t𝑑𝑧Evwxy
zwx
Evwxy
{−u 𝜃t𝑑𝑧
Ev|xy
zwx
Ev|xy
{ℎK589𝜃K589 = u 𝜃t𝑑𝑧
Ev|xy
zwx
Evwxy
zwx
AccurateconservationbutlocallyinaccurateforCFL>1 Inaccuratetotalconservation butaccuratelocally
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Remapby“sublayers”• Dividesupersetofold/newgridsintosublayers(2N-1)
• Integrateprofileforeachsub-layer– Bydefinitionusesonlyonesourcelayerprofile
• Replacelargest sub-layerwith
• Sumsub-layerstonewlayer
Accurateconservationwhenremapping
𝑧589
𝜃K589
𝜃†
𝑧†
ℎ}~𝜃}~ = u 𝜃t𝑑𝑧E�|xy
�
E�wxy
�ℎK589𝜃K589 = [ℎ}~𝜃}~
S
}\l
ℎ}~𝜃}~ = ℎKt𝜃K
t − [ 1− 𝛿}�} ℎ}�~ 𝜃}�~S
}�\l(insight fromHallberg)
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• PLM:twodegreesoffreedom– Cellmean+slope
• PPM:threedegreesoffreedom– Verywidelyused– Cellmean+twoedgevalues
• PQM:fivedegreesoffreedom– Cellmean+twoedgevalues+twoedgeslopes
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Piecewise*Methods (*=C,L,PorQ)PLM
PPM
PQM
InspiredbyDaru &Tenaud, JCP2004– introduced OSi i=1..7
Successiveschemesprovidemoreflexibilitytorepresentstructures→moreaccurate
(White&Adcroft, JCP2008)
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IllustratingmethodsandspuriousdiffusionLayeredIsopycnal
PCM
PPMih4
PQMih6
zPLM-PCM
PPMih4-PPMih4
PQMih4-PQMih4
ρ
Inappropriate interpolator leads to collapse of stratification
Diffusive behavior reduces with increasing order of remapping but
does not vanishMinimal smoothing
of interface
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• Denseflowdownaslope– Hydrostaticandadiabatic
• Layeredisopycnal solutionistrulyadiabatic
• z*andterrain-followingbothhavemodifiedwatermasses
• LRM+densitycoordinates(foundbyinterpolation)behavesverysimilarlytolayeredmodel
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft 16
Overflows
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• Lateralprocesses(incl.transport)actingoninclinedisopycnals
• Alongisopycnal processesinnon-isopycnal coordinates
• Dia-surfacetransportinvertical– Minimizingdia-surfacemotionreducesnumericaldiffusion
• Choosingacoordinateclosetoisopycnal isdesirabletominimizespuriousdiffusion
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft 17
Sourcesofspuriousmixing
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• Alreadyknowthatpureisopycnal coordinateshavelimitations
• Bleck,2002,introducedhybridz-rhocoordinateinHYCOM
• z-tilde:Leclair &Madec,2011;Petersenetal.,2015
• adaptive:Hofmeisteretal.,2010;Gräwe etal.,2015;Gibson(thesis)2019
• OM4used“HYCOM1”whichisasimpleinterpretationofactualHYCOMhybridcoordinate– OngoingworkwithWallcraft andChassignet toimprovegridgeneration(c.f.AMOCresults)
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft 18
Hybridandothercoordinates
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• Comparingz*tohybridcoordinateprovidesmagnitudeofspuriousnumericalmixing
• Reducingresolutionfrom¼° to½° (eddyingtonon-eddying)inhibitsresolvedre-stratification– ½° requireseddyparameterization
• Refiningresolutionfrom¼° to⅛°reducesheatuptake:– moreefficienteddies– and/orlessnumericalmixing?
• Thisreallytellsuswe(inadvertently)managedtoperfectlycompensateforweakeddiesat¼° resolution
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft 19
Balancingheattransferbyeddiesandmixing
More efficient eddies?
Parameterized eddies
Spurious mixing
Zanna et al. 2019
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• MainconcernisOM4withhybridcoordinatestillhasashallowAMOC– Isthismixinginoverflows?
– ItturnsoutwedidhavetoomuchparameterizedmixingbutreducingthathashadnoaffectondepthofAMOC.Infactnothingwe’vetriedsofarseemsto!
20
OM4AMOC
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
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• Motivatedbyresultswithisopycnal layeredmodelswebuiltMOM6,capableofusingarbitrarilygeneralcoordinatesfollowingHYCOM’spioneeringhybridcoordinate
• LagrangianRemapAlgorithmhassignificantadvantagesforEarthSystemModelswithmanyconstituents
• High-ordernumericalmethods+LRMdeliverlowlevelsofspuriousmixing
• InOM4,puttingeverythingtogether,amysteryremainsastowhyAMOCisshallowerthaninESM2G
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Wrapup
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
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• Potentialenergy
• Availablepotentialenergy(APE)
• ρ*istheadiabaticallyre-arrangedstatewithminimalpotentialenergy
• RPEcanonlybechangedbydiapycnalmixing– Mixingraisescenterofmass
Quantifyingspuriousmixingusingenergetics
Wintersetal.,JFM1995Ilicak etal.,OM2012
ρ+
ρ-
ρ+
ρ-
½(ρ- +ρ+)
APE
½(ρ- +ρ+)ρ+
ρ-½(ρ- +ρ+)
APE=PE−RPE
RPE=g�ρ∗zdV
PE=g�ρzdV
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft
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• OM4istheice-oceancomponentofGFDL’slatestcoupledmodelCM4(Heldetal.,2019)
• OM4configuration– Identicalsetup/parameterstoCM4– Developedalmostexclusivelyincoupledmode• UncoupledOM4
– Nominallyeddy-permitting¼°horizontalresolution• Non-eddying½° witheddyparameterization(GM+EKEscheme)
Ingredients:• MOM6
– usinghybridz-ρ verticalcoordinates• ePBL
– Reichl andHallberg,2019• Scale-awareMLErestratification
– Fox-Kemperetal.,2011• Shear-dependentmixing
– Jacksonetal.,2008• Internal-wavedrivenmixing
– Melet etal.,2012• BBL
– Leggetal.,2006
Adcroftetal.,JAMES 2019
ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft 23
OM4
c.f.HYCOM,Bleck 2002