motivations, successes, and problems, with the lagrangian

Motivations, successes, and problems, with the Lagrangian-remap dynamical core in MOM6

AlistairAdcroftPrincetonUniversity/NOAA-GFDL

• 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

• 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).

ECMWFAnnualSeminarLagrangian-remapdynamiccoreofMOM6- Adcroft 3

CM2.5 CM2.6¼° ocean 1/10° oceanCM2-1deg

1.1°C 1.3°C 0.8°C

1° ocean

• Measureworkdonebymixinginspindownexperiments– Turningoffexplicitmixingleavesonlynumericalmixing*

• Ilicak etal.,2012,argued¼° z-coordinatemodelhadnumericalmixingaslargeas“real”mixing


Measuringspuriousmixing

[Wm-2]

Ilicak etal.,2012

• ReducingspuriousmixingmotivatedMOM6development– Generalcoordinatetoavoidlimitationsofpotentialdensity(usedbyisopycnal layermodels)

• MOM6broadlyreproducedpriorsolutionsusingz-coordinates

• Adoptionofhybridz-densitycoordinatesreducedheatuptake– success?


MOM6andOM4

OM41/4° z-coordinates

OM41/4° hybrid-coordinates

Adcroftetal.,2019

• HydrostaticPrimitiveEquations(forocean)ingeneralcoordinates𝑟 = 𝑟(𝑥, 𝑦, 𝑧, 𝑡) :

Generalcoordinates(Boussinesq)


𝑧+ =𝜕𝑧𝜕𝑟

Starr,1945;Kasahara,1974;…

6

• Integratingbetweensurfacesofconstant𝑟 converts𝑧+ → ℎ = ∫ 𝑧+𝑑𝑟 and𝜕+ → 𝛿+

• Choosing�̇� = 0 followstheverticalmotion→VerticallyLagrangian– removesexplicitverticaltransports→noverticalCFL

7

Layerintegratedgeneralcoordinateequations


Lagrangianremapalgorithm


Reconstruction

𝜃†

𝑧5

𝜃5

Dynamics+“physics”

ℎ†ℎ5

�̇� = 𝟎

Average

𝑧589

𝜃589

Gridgeneration

Eulerian,ALEandLRMalgorithmsside-by-side


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

Sub-cyclingwithLagrangianverticaldynamics


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

10

Interpolationforgridgeneration• Accurate• Continuous (singlevalued)– resultingprofilesaregenerallynotconservative

Reconstructionforremapping• Accurate• Conservative– profilesoftenbecomediscontinuoustobeconservative


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

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


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)


• PLM:twodegreesoffreedom– Cellmean+slope

• PPM:threedegreesoffreedom– Verywidelyused– Cellmean+twoedgevalues

• PQM:fivedegreesoffreedom– Cellmean+twoedgevalues+twoedgeslopes


Piecewise*Methods (*=C,L,PorQ)PLM

PPM

PQM

InspiredbyDaru &Tenaud, JCP2004– introduced OSi i=1..7

Successiveschemesprovidemoreflexibilitytorepresentstructures→moreaccurate

(White&Adcroft, JCP2008)

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


• Denseflowdownaslope– Hydrostaticandadiabatic

• Layeredisopycnal solutionistrulyadiabatic

• z*andterrain-followingbothhavemodifiedwatermasses

• LRM+densitycoordinates(foundbyinterpolation)behavesverysimilarlytolayeredmodel


Overflows

• Lateralprocesses(incl.transport)actingoninclinedisopycnals

• Alongisopycnal processesinnon-isopycnal coordinates

• Dia-surfacetransportinvertical– Minimizingdia-surfacemotionreducesnumericaldiffusion

• Choosingacoordinateclosetoisopycnal isdesirabletominimizespuriousdiffusion


Sourcesofspuriousmixing

• 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)


Hybridandothercoordinates

• Comparingz*tohybridcoordinateprovidesmagnitudeofspuriousnumericalmixing

• Reducingresolutionfrom¼° to½° (eddyingtonon-eddying)inhibitsresolvedre-stratification– ½° requireseddyparameterization

• Refiningresolutionfrom¼° to⅛°reducesheatuptake:– moreefficienteddies– and/orlessnumericalmixing?

• Thisreallytellsuswe(inadvertently)managedtoperfectlycompensateforweakeddiesat¼° resolution


Balancingheattransferbyeddiesandmixing

More efficient eddies?

Parameterized eddies

Spurious mixing

Zanna et al. 2019

• MainconcernisOM4withhybridcoordinatestillhasashallowAMOC– Isthismixinginoverflows?

– ItturnsoutwedidhavetoomuchparameterizedmixingbutreducingthathashadnoaffectondepthofAMOC.Infactnothingwe’vetriedsofarseemsto!

20

OM4AMOC


• Motivatedbyresultswithisopycnal layeredmodelswebuiltMOM6,capableofusingarbitrarilygeneralcoordinatesfollowingHYCOM’spioneeringhybridcoordinate

• LagrangianRemapAlgorithmhassignificantadvantagesforEarthSystemModelswithmanyconstituents

• High-ordernumericalmethods+LRMdeliverlowlevelsofspuriousmixing

• InOM4,puttingeverythingtogether,amysteryremainsastowhyAMOCisshallowerthaninESM2G

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Wrapup


• 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


• 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


OM4

c.f.HYCOM,Bleck 2002

motivations, successes, and problems, with the lagrangian

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