a convection scheme for data assimilation: description and initial tests

411

A convection scheme for dataassimilation: Description and initial

tests

Philippe Lopez and Emmanuel Moreau

Research Department

To be submitted to Q. J. Roy. Meteor. Soc.

March 2004

Series: ECMWF Technical Memoranda

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Copyright 2004

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A convectionschemefor dataassimilation:Descriptionandinitial tests

Abstract

A new simplified parametrizationof subgridscaleconvective processeshasbeendevelopedandtestedintheframework of theECMWF IntegratedForecastingSystemfor thepurposeof variationaldataassimila-tion, singularvectorcalculationsandadjointsensitivity experiments.Its formulationis basedon Tiedtke’s(1989)nonlinearconvectionschemeusedat ECMWF, but a setof simplificationshasbeenappliedto sub-stantiallyimproveits linearbehaviour. Theseincludethespecificationof asingleclosureassumptionbasedon convective availablepotentialenergy, theuncouplingof theequationsfor theconvective massflux andupdraughtcharacteristicsanda unified formulationof the entrainmentanddetrainmentrates. Simplifiedrepresentationsof downdraughtsandmomentumtransportarealsoincludedin thenew scheme.

Despitethesesimplifications,the forecastingability of thenew convective parametrizationis shown to re-mainsatisfactoryevenin seasonalintegrations.A detailedstudyof its Jacobiansandthevalidity of thelinearhypothesisis presented.Thenew schemeis alsotestedin combinationwith thenew simplifiedparametriza-tion of large-scalecloudsandprecipitationdevelopedby TompkinsandJaniskova (2004). In contrastwithMahfouf’s (1999)simplifiedconvectiveparametrization,its tangent-linearandadjointversionsaccountforperturbationsof all convectivequantitiesincludingconvectivemassflux, updraughtcharacteristicsandpre-cipitation fluxes. Thereforethe new schemeis expectedto be beneficialwhen combinedwith radiativecalculationsthataredirectlyaffectedby condensationandprecipitation.

Examplesof applicationsof thenew moistphysicsin 1D-Varretrievalsusingmicrowavebrightnesstemper-aturemeasurementsaswell asin adjointsensitivity experimentsarepresented.

1 Introduction

In Meteorology, variationaldataassimilationis usedto obtaina statisticallyoptimal representationof theat-mosphericstatefor a given date(the analysis) throughthe combinationof backgroundinformation from apreviousshort-rangeforecastwith usefulinformationcontainedin a setof availableobservations.Practically,variationalmethodsarebasedon theminimizationof a costfunction thatmeasurestheglobaldistanceof thenew atmosphericstateto both the backgroundstateandthe observations. The major underlyingassumptionin thesetechniquesis thatall processesinvolvedcanbeconsideredaslinearor at leastonly weaklynonlinear.In four-dimensionalvariationaldataassimilation(4D-Var), theminimizationof thecostfunctionis performedover a time window of severalhours,which permitstheassimilationof observationsat their actualtime. Forinstance,at the EuropeanCentrefor Medium-rangeWeatherForecasts(ECMWF), the twice-daily 4D-Varanalyses(Courtieret al. 1994;Rabieret al. 2000)arecurrentlyobtainedby assimilatingobservationsover a12-hourwindow.

The propagationof the 4D-Var incrementsthroughthe assimilationwindow via adjoint integration and theconversionof the modelstateinto equivalentobserved quantitiesrequirethe useof a setof parametrizationsthatdescribetheatmosphericdynamicandphysicalprocesses.In particular, adetailedenoughrepresentationofmoistprocesses(convectionandlarge-scalecondensation)becomesessentialwhenobservationsdirectlyrelatedto cloudsandprecipitationareto be assimilated.However, moist diabaticprocessesareby naturenonlinearandnon-differentiableor discontinuousin time (e.g. saturationthreshold,precipitationformation).A carelessinclusionof raw parametrizationsof thesephenomenacanquickly resultin seriousviolationof thebasiclinearassumptionmadein variationaldataassimilation.This is particularlytrue in thecaseof convectionasshownin Zupanski(1993)andVukicevic andErrico (1993)for instance.In recentyears,many effortshave beended-icatedto thedesignof simplifiedparametrizationsof atmosphericmoist processesfor thespecificpurposeofoperationaldataassimilation(e.g.Janiskova etal. 1999;Mahfoufetal. 1996;Mahfouf1999[MAH99]; Mah-fouf andRabier2000,Fillion andMahfouf 2000).Zou (1993)successfullyperformed4D-Var assimilationofpseudo-observationsincludingtheadjointof simplelarge-scalecondensationandconvectionparametrizationsin the NCEPmodel,with no specialtreatmentof ”on-off” processes.Nonetheless,ZupanskiandMesinger

TechnicalMemorandumNo. 411 1


(1995)underlinedthat the importanceof discontinuitiesin moist processesis likely to dependon the modelusedandon the chosenmeteorologicalsituation. Theseauthorsindicatedthat their convectionschemehadfirst to be modified in order to eliminateits discontinuousbehaviour. Only thenwas it possibleto obtainabetterprecipitationforecastusingthe4D-Varassimilationof 24-houraccumulatedsurfacerainfall observationsprior to the beginning of the forecast. In other respects,MarecalandMahfouf (2003)demonstratedthat anindirect ’1D-Var + 4D-Var’ assimilationof precipitationobservationswasmorerobust thana direct 4D-Varassimilation,partly becauseof thepresenceof strongnonlinearitiesin Tiedtke’s (1989)mass-fluxconvectionscheme.

Thispaperintroducestheformulation,theimplementationandtheapplicationof anew simplifiedparametriza-tion of convection(calledSIMPCV2 hereafter)suitablewherever linearizedcomputationsareneeded.SIM-PCV2 is muchmore detailedthan the MAH99 simplified convection scheme(SIMPCV hereafter). In fact,thenew parametrizationof convectionhasbeendesignedsuchasto retainsomefundamentalsimilaritieswithTiedtke’s (1989)nonlinearscheme(operationallyusedin forecastmode;OPNLCV hereafter)andat thesametime to reducethehigh degreeof complexity of the latter that is a sourceof strongnonlinearities,detrimentalto 4D-Varassimilation.Table2 summarizesall usefulacronymsdefinedandusedthroughoutthepaper.

Section2 describesSIMPCV2,while section3 presentsits validationin season-longintegrations.In section4SIMPCV2 is first comparedto Tiedtke’s parametrizationin termsof Jacobiansandin termsof linearity overa single timestepfor initial perturbationsof variousamplitudes.Then, the linear behaviour of SIMPCV2 isassessedin 12h-longintegrationsfor initial perturbationsthataretypical of analysiserrors.This set-upcorre-spondsto theoneusedin theECMWF 4D-Var assimilationsystem.Finally, somepreliminaryapplicationsofthenew simplifiedconvectionschemearepresentedin section5.

2 Description of the new parametrization

Previousstudiesunderlinedtheneedfor an improvementof the linearizedversionof ECMWF’s parametriza-tions of moist processesto be usedin a wide rangeof applicationsincluding 4D-Var assimilation,singularvectorcomputationsand1D-Var retrievals from rain observations(Marecal andMahfouf 2000). TompkinsandJaniskova (2004)recentlydevelopeda new simplifiedparametrizationof large-scalecloudsandprecipita-tion. A new simplified versionof theoperationalconvectionschemewasalsorequired.This new simplifiedmass-fluxscheme(SIMPCV2) wasdesignedsuchasto retainsomebasicsimilaritieswith OPNLCV andatthesametime to reduceits degreeof complexity thatcanbea sourceof unwantedstrongnonlinearities.Notethatmodificationsdescribedby Bechtoldetal (2004)havebeenrecentlyappliedto Tiedtke’soriginalnonlinearconvectionschemeto improve forecastscores.

In the new simplified scheme,all typesof convection (shallow, mid-level, anddeep)aretreatedin the sameway. In particular, thelink betweenthemodelcontrolvariablesandthecloudbasemassflux (theso-calledclo-sure assumption) is expressedthrougha singleformulationthatdependson thereleaseof convective availablepotentialenergy (CAPE) in time. In contrastwith OPNLCV, theequationsthatdescribetheverticalevolutionof theupdraughtmassflux, Mu (in kg m� 2 s� 1), andof theupdraughtthermodynamiccharacteristics,Φu (drystaticenergy, s, andtotal waterqt � q � qc whereq denotesspecifichumidity andqc cloud condensate),areuncoupled:

∂Mu

∂z � � εu � δu � Mu (1)

∂Φu

∂z � � εu � Φu � Φ � (2)

2 TechnicalMemorandumNo. 411


whereεu and δu are the fractional entrainmentand detrainmentrates,respectively. Here, as well as in therestof thepaper, overbarsdenotefield valuesin theenvironment,that is averagedover a modelgridbox,andsubscriptor superscript”u” relateto theupdraughtcharacteristics.As explainedbelow, theuncouplingallowsthe removal of the iterative calculationsinvolved in OPNLCV for updatingthecloud basemassflux therebyleadingto aneasierdevelopmentof theadjoint.BasedonSiebesmaandJakob(personalcommunication),εu isparametrizedascε � � z � zst � � 10� 5 m� 1 wherecε=0.4andzst is thestartingheightof theupdraught(in m). Asfor detrainment,observations(e.g. Cohen2000)suggestthatδu is usuallysmallerthanεu in thelower partoftheupdraughtandcomparableto εu furtherup,which is takeninto accountby writing

δu � εu � 1 � cδ exp � � z � zst

2000 � (3)

wheretheconstantcδ is setto 0.7.Closeto cloudtopwhereupdraughtbuoyancy becomesnegative,aconstantorganizeddetrainmentrateof 2 � 10� 4 m� 1 is addedto δu computedfrom Eq.(3).

Convectionis assumedto beactivatedonly if thebulk convective updraughtverticalvelocity remainspositiveat cloudbase.Theupdraughtis assumedto originatefrom thesurfaceonly if its initial verticalvelocity, ws, ispositive. ws is calculatedfrom thesurfaceheatfluxesusingHoltslagandMoeng(1991)

ws � 1 � 2 u3� � 1 � 5 gzκ FV

T � 13

(4)

whereκ=0.4 (von Karman’s constant)andFV � � w� Tv � is thesurfaceflux of virtual temperature,Tv (primesdenoteperturbations).For simplicity, thefriction velocityu� is arbitrarilysetequalto 0.1m s� 1 asin OPNLCV.Theinitial temperatureandspecifichumidity departuresof theupdraughtfrom theenvironment,δTu andδqu,areassumedto be dependenton the surfacesensibleand latentheatfluxes,FS andFL (in W m� 2; positivedownward)respectively, accordingto

δTu � � 1 � 5 FS

ρCpws(5)

δqu � � 1 � 5 FL

ρLvws(6)

whereρ is thedensityandCp thespecificheatcapacityof theair at constantpressure,andLv is thelatentheatof vaporisationof water. If convectioncannotbeinitiatedfrom thesurface,theconvective ascentmayoriginatefrom higherlevelsprovidedrelative humidityexceeds80%.In thiscase,theinitial verticalvelocityof thebulkupdraughtis setequalto 1 m s� 1.

Regardlessof whetherthe updraughtoriginatesfrom the surfaceor from higherup, the vertical evolution ofits kinetic energy is computedfollowing SimpsonandWiggert (1969),which involvesthebuoyancy, � � z� , aswell astheentrainmentof environmentalair into theupdraught,

∂w2u

∂z � � αεuw2u � β � � z� (7)

whereα � 23, β=1 andwu is theupdraughtverticalvelocity. Thebuoyancy in theupdraughtis definedas

� � z� � g

Tv� Tu

v � Tv �� gquc (8)

which includesthecloudwaterloadinsidetheupdraught,quc.

Using a similar approach,a simple parametrizationof convective downdraughtsthat are usually associatedwith the strongcooling dueto the evaporationof precipitationhasbeenimplemented.Uncoupledequations



describetheevolutionwith heightof thedowndraughtmassflux, Md, andof thedowndraughtthermodynamicalcharacteristics,Φd:

∂Md

∂z � � � εd � δd � Md (9)

∂Φd

∂z � εd � Φd � Φ � (10)

wherethefractionalentrainmentanddetrainmentratesin thedowndraughtsarerespectively εd � 2 � 10� 4 m� 1

andδd � cδ � z, with cδ =0.3.

In practice,Eq. (1) andEq. (9) aresolvedin termsof theratiosµu � Mu � Mclbu andµd � Md � Mclb

u respectively,whereMclb

u denotestheupdraughtmassflux at cloudbase(clb). µu andµd arerespectively initialized to 1 atcloudbaseandto � 0.3at thelevel of freesinkingfollowing Tiedtke (1989).Oncetheverticalprofilesof µu � z�andµd � z� areknown, Mclb

u is calculatedin sucha way that theCAPE in theatmosphericcolumnis removedover a characteristicresolutiondependenttimescale,τ (3600s for resolutionscoarserthanT319,1200s fromT319to T511,600sbeyondT511),accordingto Gregoryetal. (2000)

Mclbu � CAPE

τ1�

cloud 1� αqCpTv

∂s∂z � α ∂q

∂z � � µu � z� � µd � z� gdzρ

(11)

whereα � RvRd� 1 andCAPE � �

cloud � � z� dz. Sinceearlierexperimentssuggestedthatthevaluesof thecloud-

basemassflux weretoo low in thecaseof shallow convection,Mclbu is multiplied by 3 in suchsituations.

SIMPCV2alsofeaturesa simplerepresentationof convective momentumtransport.It is basedon OPNLCVbut in contrastwith thelatter thesameentrainmentanddetrainmentratesasfor heatandmoistureareappliedto thetwo wind componentsfor thesake of simplicity.

The computationof precipitationformation, � (in kg kg� 1 m� 1), from the updraughtcloud condensateisinspiredby Tiedtke (1989)

� � K � z� Cf quc

wu(12)

where

K � z� �� 1 � 2 � 10� 3 s� 1 if z - zclb � 1500m0 otherwise

(13)

� representsautoconversionandcollection/aggregation processes,while thecoefficient Cf crudelyaccountsfor the dependenceof precipitationformation efficiency on temperature.Practically, Cf is set equal to �1 � 4

15 � 260 � Tu � , with the restriction1 � Cf � 5. It shouldbe notedthat in OPNLCV Tiedtke’s originalformulationof precipitationformationhasbeenreplacedby Sundqvist’s (1988).

The tangent-linearand adjoint versionsof SIMPCV2 are available and have beentestedin different typesof experiments,someof which will be presentedin section5. Simplificationsand modificationsthat wereneededto improve the scheme’s linear behaviour will be summarizedin section4.4. It is worth noting thatunlikeSIMPCV, SIMPCV2handlesperturbationsof all convectivequantitiesincludingmassflux, all updraughtcharacteristicsaswell asprecipitationfluxes.This is expectedto bebeneficialwhencombiningSIMPCV2witha radiative transfermodelfor frequenciesthataresensitive to cloudcondensateand/orhydrometeors.



3 Season-long integrations

AlthoughSIMPCV2wasnot designedfor thepurposeof mediumor long-rangeweatherforecasting,it is stillimportantto checkthatits resultsin aforecastmodedonotdeparttoomuchfrom thoseobtainedwith OPNLCV.As anexample,season-longintegrationswith aT95 spectralresolution(approx.200km) in thehorizontaland60verticallevelsarepresentedherefor thewinter1987-1988usingeitherOPNLCVor SIMPCV2in cycle26r3of theECMWF forecastmodel.Therespective experimentswill bereferredto asOPNLCVandSIMPCV2.

For eachexperiment,an ensembleof three4-monthlong runswerestartedfrom 1, 2 and3 November1987respectively to ensuremorerobust statistics.The actualvalidationfocuseson the last threemonthsof theseintegrations,namelyDecember, JanuaryandFebruary. Three-monthaveragesareshown for four differentfieldsrelatedto cloudsandprecipitation:thetotal surfaceprecipitation(Fig. 1), thetotal cloudcover (Fig. 2), thenetshortwave radiationat the top of the atmosphere(SWR-TOA; Fig. 3), and the outgoinglongwave radiationat the top of the atmosphere(OLR-TOA; Fig. 4). Correspondingobservationsfrom the Global PrecipitationClimatologyProject(GPCP),from the InternationalSatelliteCloud ClimatologyProject(ISCCP),andfromtheEarthRadiative BudgetExperiment(ERBE),areusedto validatethemodelfields.

Eachfiguredisplays(a) theobserved field, (b) thefield from OPNLCV, (c) thefield from SIMPCV2,(d) thedifferencebetweenOPNLCV andobservations,and(e) the differencebetweenSIMPCV2 andobservations.Theglobalmeanof eachfield appearsin thetitle of eachpanelalongwith theroot meansquare(RMS) errorsat thetopof panels(d) and(e).

In termsof precipitation,Fig.1.(d)and(e) indicatethatOPNLCVandSIMPCV2exhibit rathersimilardiscrep-ancieswith respectto theGPCPobservations,exceptover southernAfrica. In bothexperiments,theprecipita-tion simulatedin theInter-TropicalConvergenceZone(ITCZ) tendsto besubstantiallyoverestimatedover thecentralandeasternPacific,over theAtlantic andover theIndianOcean.Local departuresreach15 mm day� 1

in OPNLCV and8 mm day� 1 in SIMPCV2 slightly eastof the dateline. On the otherhand,precipitationisclearlyunderestimatedover theIndonesianwarmpool by up to 12 mm day� 1 with bothconvectionschemes.Also notethesimilardoublestructureof theITCZ in theIndianOceanin Fig. 1.(b) and(c), whichdoesnotap-pearin theobservations.Overall, theaverageratio of convective precipitation(not shown) on thetotal amountreaches56%in OPNLCVversus49%in SIMPCV2.Themeandeparturefrom theGPCPobservationsis lowerin SIMPCV2 (0.12 mm day� 1) thanin OPNLCV (0.22 mm day� 1), but the RMS error is slightly higher inSIMPCV2(1.91versus1.86mmday� 1).

Figure2.(d) and(e) indicatethatthedeparturesfrom observedcloudcoverexhibit similarpatternsin OPNLCVandSIMPCV2: too low cloudinessover the oceanictrade-windregionsoften affectedby stratocumuli(westof SouthAmerica,SouthAfrica, California andMorroco), excessive cloud cover over CentralAmerica, theAndes,southwestof India andover SoutheastAsia. Over the extra-tropicaloceans,cloudsaremorewidelyunderestimatedin SIMPCV2 than in OPNLCV. In fact this correspondsto a lack of low-level clouds(notshown) that partly resultsfrom the useof the CAPE closurefor shallow convection in SIMPCV2 insteadofthemoistureconvergenceclosureappliedin OPNLCV. Globally, cloud amountin SIMPCV2 shows a deficitof roughly5.8%while OPNLCV overestimatesit by only 1.1%.TheRMS erroris largerin SIMPCV2thaninOPNLCV(20.80versus17.73%).

Figure3.(d)and(e)andFig.4.(d)and(e)exhibit SWR-TOA andOLR-TOA deficienciesthatareconsistentwiththedeparturespreviouslyoutlinedfor cloudcover. Theexcessof SWR-TOA in thetrade-windregionsaffectedby stratocumuli(upto 80W m� 2 off thePeruviancoast)resultsfrom thelackof suchcloudsin thesimulations.Both figuressuggestthatclouddepthfrom OPNLCV or SIMPCV2is too low over theIndonesianwarmpool,theAmazonandWestAfrica. On theotherhand,thesimulatedcloudverticalextentseemsto beexaggeratedover the westernIndian Oceanas well as off the coastof Nordestein SIMPCV2. On the planetaryscale,



60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

120°W�

120°W� 60°W

60°W 0°

0° 60°E

60°E 120°E�

120°E�

Total Precipitation DJF 87/88, GPCP (Mean=2.70)mm/day

0.5�1�2�3�5�7 10�20�30�a

60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

120°W�

120°W� 60°W

60°W 0°

0° 60°E

60°E 120°E�

120°E�

Total Precipitation DJF 87/88, OPNLCV (Mean=2.91)mm/day

0.5�1�2�3�5�7 10�20�30�b

60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

120°W�

120°W� 60°W

60°W 0°

0° 60°E

60°E 120°E�

120°E�

Total Precipitation DJF 87/88, SIMPCV2 (Mean=2.82)mm/day

0.5�1�2�3�5�7 10�20�30�c

60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

120°W�

120°W� 60°W

60°W 0°

0° 60°E

60°E 120°E�

120°E�

Total Precipitation DJF 87/88, OPNLCV - GPCP (Mean=0.22, RMS=1.86)!

mm/day

-20

-15

-10

-5

-2

-0.50.5�2�5�10�15�20�d

60°S60°S

30°S 30°S

0°0°

30°N 30°N

60°N60°N

120°W�

120°W� 60°W

60°W 0°

0° 60°E

60°E 120°E�

120°E�

Total Precipitation DJF 87/88, SIMPCV2 - GPCP (Mean=0.12, RMS=1.91)"

mm/day

-20

-15

-10

-5

-2

-0.50.5�2�5�10�15�20�e

Figure 1: Total precipitationover winter 1987-88: (a) as from GPCPobservations,(b) from OPNLCV and (c) fromSIMPCV2experiments(seetext for moredetails). Dif ferencesOPNLCV-observationsandSIMPCV2-observationsareshown in panels(d) and(e) respectively. Meanandroot meansquaredifference(whenrelevant)arementionedat thetop

of eachpanel.Unitsarein mm day# 1.



30°S 30°S$0°%

0°

30°N 30°N$120°W

%

120°W%

60°W

60°W 0°

0° 60°E

60°E 120°E%

120°E%

Total Cloud Cover DJF 87/88, ISCCP (Mean=63.53)

20

40

60

80

100

110a

30°S 30°S$0°%

0°

30°N 30°N$120°W

%

120°W%

60°W

60°W 0°

0° 60°E

60°E 120°E%

120°E%

Total Cloud Cover DJF 87/88, OPNLCV (Mean=64.61)

20

40

60

80

100

110b

30°S 30°S$0°%

0°

30°N 30°N$120°W

%

120°W%

60°W

60°W 0°

0° 60°E

60°E 120°E%

120°E%

Total Cloud Cover DJF 87/88, SIMPCV2 (Mean=57.75)

20

40

60

80

100

110c

30°S 30°S$0°%

0°

30°N 30°N$120°W

%

120°W%

60°W

60°W 0°

0° 60°E

60°E 120°E%

120°E%

Total Cloud Cover DJF 87/88, OPNLCV - ISCCP (Mean=1.08, RMS=17.73)&

-60

-40

-20

10

30

50d

30°S 30°S$0°%

0°

30°N 30°N$120°W

%

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E%

120°E

Total Cloud Cover DJF 87/88, SIMPCV2 - ISCCP (Mean=-5.79, RMS=20.80)&

-60

-40

-20

10

30

50e

Figure2: Sameasin Fig. 1, but for total cloudcoverusingISCCPobservations.Unitsarein %.



30°S'

30°S

0° 0°(30°N'

30°N

120°W(

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E(

120°E

Top Shortwave Radiation DJF 87/88, ERBE (Mean=269.04))

W/m2*

50100150200250300350400450

a

30°S'

30°S

0° 0°(30°N'

30°N

120°W(

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E(

120°E

Top Shortwave Radiation DJF 87/88, OPNLCV (Mean=269.20)W/m2

50100150200250300350400450

b

30°S'

30°S

0° 0°(30°N'

30°N

120°W(

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E(

120°E

Top Shortwave Radiation DJF 87/88, SIMPCV2 (Mean=271.94)W/m2

50100150200250300350400450

c

30°S'

30°S

0° 0°(30°N'

30°N

120°W(

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E(

120°E

Top Shortwave Radiation DJF 87/88, OPNLCV - ERBE (Mean=0.17, RMS=17.12)

W/m2

-100-70-50-30-20-101020305070100

d

30°S'

30°S

0° 0°(30°N'

30°N

120°W(

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E(

120°E

Top Shortwave Radiation DJF 87/88, SIMPCV2 - ERBE (Mean=2.91, RMS=19.86)

W/m2

-100-70-50-30-20-101020305070100

e

Figure3: Sameasin Fig. 1, but for top netshortwaveradiationusingERBEobservations.Units arein W m+ 2.



30°S,

30°S

0° 0°-30°N,

30°N

120°W-

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E-

120°E

Outgoing Longwave Radiation DJF 87/88, ERBE (Mean=-237.06)W/m2

-360-330-300-270-240-210-180-150-120

a

30°S,

30°S

0° 0°-30°N,

30°N

120°W-

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E-

120°E

Outgoing Longwave Radiation DJF 87/88, OPNLCV (Mean=-247.05)W/m2.

-360-330-300-270-240-210-180-150-120

b

30°S,

30°S

0° 0°-30°N,

30°N

120°W-

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E-

120°E

Outgoing Longwave Radiation DJF 87/88, SIMPCV2 (Mean=-245.43)W/m2.

-360-330-300-270-240-210-180-150-120

c

30°S,

30°S

0° 0°-30°N,

30°N

120°W-

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E-

120°E

Outgoing Longwave Radiation DJF 87/88, OPNLCV - ERBE (Mean=-9.99, RMS=14.38)/

W/m2

-90-70-50-30-20-1010200301502703904d

30°S,

30°S

0° 0°-30°N,

30°N

120°W-

120°W 60°W

60°W 0°

0° 60°E

60°E 120°E-

120°E

Outgoing Longwave Radiation DJF 87/88, SIMPCV2 - ERBE (Mean=-8.38, RMS=14.00)/

W/m2.

-90-70-50-30-20-1010200301502703904e

Figure 4: Sameasin Fig. 1, but for outgoinglongwaveradiationat thetop of theatmosphereusingERBEobservations.Units arein W m + 2.



SIMPCV2showsaSWR-TOA positivebiasof 2.91W m5 2, while OPNLCVis almostperfectlybalanced(biasof 0.17W m5 2). In termsof OLR-TOA, both experimentsleadto comparablemeandeviationsrespectivelyequalto 6 9.99W m5 2 and 6 8.38W m5 2.

Theresultspresentedin thissectiondemonstratethat,althoughnotdesignedfor thepurposeof longsimulations,SIMPCV2behavesreasonablywell with respectto observationsandresemblesthefull operationalconvectionscheme.

4 Linear behaviour

4.1 Jacobians

Thesensitivities or Jacobiansof theoutputsof SIMPCV2to theinput temperatureandspecifichumidity havebeencomputedfor severalhundredsof initial atmosphericprofilesusingthefinite-differenceapproachandbyperturbingspecifichumidityandtemperatureoneachverticallevel, separately. Thestudyof theseJacobianshaspermittedefficient isolationandmodificationof portionsof thecodethatcould leadto excessive sensitivitieslocally (seesubsection4.4). Sharpvertical gradientsof sensitivities are not desirablesincethey can leadto irregular profilesof analysisincrements(despitethe subsequentsmoothingeffect of the backgrounderrorcovariancematrix).

Figures5 and6 display typical Jacobiansof variousoutputquantitiesfrom OPNLCV with respectto inputspecifichumidityandtemperatureperturbations.Figure5 focusesontheJacobiansof theconvective tendenciesof temperatureandspecifichumidity andon thesensitivities of theconvective updraughtmassflux. Figure6displaysthe Jacobiansof convective precipitationprofilesandCAPE. Similar plots areshown in Fig. 7 and8 for SIMPCV2. Note that thedynamicrangeof thecolourscaleis determinedby theextremevaluesof thefield plottedin eachpanel. For theselectedgrid point (4.397 N, 159.967 E), deepconvectionextendsbetweenmodellevels55 (resp.56) and30 (resp.28) with OPNLCV (resp.SIMPCV2),andsurfaceconvective rainfallis equalto 53.54(resp. 85.82)mm day5 1. Table1 in theappendixgivesthecorrespondencebetweenmodelandpressurelevels.

Several conclusionscan be drawn from theseplots. First, the Jacobiansof 8 ∂T∂ t 9 conv with respectto T and

of 8 ∂q∂ t 9 conv with respectto q are confinedaroundthe main diagonal,which indicatesthat the effect of the

perturbationremainslocalized. Secondly, the structureof the Jacobiansof all otherfields is almostentirelydictatedby the shapeof the CAPE Jacobians(bottompanelsof Fig. 6 and8). This is not surprisingsincethe convective massflux is directly relatedto CAPE throughEq. (11). In contrast,it shouldbe notedthatthe denominatorof Eq. (11) hasa negligible influenceon the Jacobians.An additionalsourceof sensitivityof surfacerainfall to temperatureonly appearsaroundmodellevel 40 with OPNLCV. Thirdly, thesignof theJacobianscaneasilybe explainedby the changesof vertical stability inducedby temperatureperturbationsandby thechangesin thewaterphasecausedby specifichumidity perturbations.Focusingon SIMPCV2forinstance,aheatingappliedbelow modellevel 46meanswarmerair in theconvective updraught,hencestrongerpositivebuoyancy in theascentvia Eq.(8), enhancedCAPE(Fig.8.f) andconvectivemassflux (Fig.7.f), whichfinally resultsin moreconvective precipitation(Fig. 8.b andd). With OPNLCV suchresponseto low-leveltemperatureperturbationsis confinedto thelowestmodellevel. Similarly andfor bothconvectionschemes,amoisteningof theenvironmentbelow level 40 implies morewatervapouravailablein theupdraughtthroughentrainment,that is a strongerlatentheatingthroughcondensationin the ascent,which eventually leadstohigherCAPE values(Fig. 8.e) andenhancedprecipitation(Fig. 8.a andc). Note that the peakof sensitivityseenat level 57 and58 in Fig. 6.(e) resultsfrom therecentinclusionof a mixing of updraughtcharacteristicsbelow cloud basein OPNLCV (Bechtoldet al. 2004). Overall, it shouldbe emphasizedthat the Jacobians



(a) Jacobian of dT/dt w.r.t. Qv :

60 50 40 30Qv Model levels;60

50

40

30

dT/d

t Mod

el le

vels<

-1.130 / 4.534

(b) Jacobian of dQv/dt w.r.t. Qv =


50

40

30

dQv/

dt M

odel

leve

ls

> -16.009 / 14.822

(c) Jacobian of dT/dt w.r.t. T :

60 50 40 30T Model levels;60

50

40

30

dT/d

t Mod

el le

vels<

-22.293 / 22.106

(d) Jacobian of dQv/dt w.r.t. T :


50

40

30

dQv/

dt M

odel

leve

ls

> -1.976 / 2.146

(e) Jacobian of Mu w.r.t. Qv (*1.e+04)?


50

40

30

Mu

Mod

el le

vels

-10.846 / 85.909

(f) Jacobian of Mu w.r.t. T (*1.e+04)?


50

40

30

Mu

Mod

el le

vels

-70.746 / 76.316

Profile number 11@LCUM=1, LMFLIM=0, KSTUP=19, KCBOT=55, KCTOP=30, KTYPE=1

LON= 159.961, LAT= -4.390, RRcv= 53.54, RRst= 0.00A

Min MaxB



Min MaxBFigure 5: Jacobiansof output temperature(a,c)andspecifichumidity (b,d) tendenciesandconvective updraughtmassflux (e,f) with respectto inputspecifichumidityandtemperaturefor OPNLCV. Thex-axis(resp.y-axis)showsthemodellevelsat which the input (resp.output)variableis considered.Jacobiansof themassflux (e,f) have beenscaledby 104.Extremevaluesof the Jacobiansaregiven at the top left of eachpanel(SI units). Solid (resp. dashed)white isolines

correspondto positive(resp.negative)sensitivities.



(a) Jacobian of RRconv w.r.t. Qv


50

40

30

RR

conv

Mod

el le

vels

-0.945 / 4.469

(b) Jacobian of RRconv w.r.t. T =


50

40

30

RR

conv

Mod

el le

vels

-6.127 / 4.909



Min MaxB

(c) Jacobian of RRconv w.r.t. Qv=

-1.04 0.45C 1.94D 3.43C 4.92DJacobian (mm dayE -1 (g/kg)-1)

60

50

40

30

Mod

el le

vels

-0.945 / 4.469

(d) Jacobian of RRconv w.r.t. T:

-6.74 -3.70 -0.67 2.36F 5.40CJacobian (mm dayE -1 K-1)

60

50

40

30

Mod

el le

vels

-6.127 / 4.909



(e) Jacobian of CAPE w.r.t. Qv:

-5.1 6.1G 17.4H 28.6I 39.9DJacobian (mE 2 s-2 (g/kg)-1)

60

50

40

30

Mod

el le

vels

-4.646 / 36.235

(f) Jacobian of CAPE w.r.t. T:

-31.46 -15.41 0.63J 16.68G 32.73GJacobian (m2 s-2 K-1)

60

50

40

30

Mod

el le

vels

-28.603 / 29.757



Figure 6: Jacobiansof convectiveprecipitationprofiles(a,b),surfaceconvectiveprecipitation(c,d) andCAPE(e,f) withrespectto input specifichumidity andtemperaturefor OPNLCV. First row: samelayout asin Fig. 5; otherpanels:thex-axisshowsthevaluesof theJacobianswhile they-axisshowsthemodellevelsatwhich theinputvariableis perturbed.



(a) Jacobian of dT/dt w.r.t. Qv :


50

40

30

dT/d

t Mod

el le

vels<

-8.975 / 6.985

(b) Jacobian of dQv/dt w.r.t. Qv


50

40

30

dQv/

dt M

odel

leve

ls

> -42.240 / 37.682

(c) Jacobian of dT/dt w.r.t. T :


50

40

30

dT/d

t Mod

el le

vels<

-39.506 / 39.885

(d) Jacobian of dQv/dt w.r.t. T :


50

40

30

dQv/

dt M

odel

leve

ls

> -2.394 / 5.067

(e) Jacobian of Mu w.r.t. Qv (*1.e+04)?


50

40

30

Mu

Mod

el le

vels

-17.367 / 140.230

(f) Jacobian of Mu w.r.t. T (*1.e+04)?


50

40

30

Mu

Mod

el le

vels

-180.350 / 85.215



Min MaxB



Min MaxBFigure7: Sameasin Fig. 5 but for SIMPCV2.



(a) Jacobian of RRconv w.r.t. Qv


50

40

30

RR

conv

Mod

el le

vels

-1.736 / 14.558

(b) Jacobian of RRconv w.r.t. T


50

40

30

RR

conv

Mod

el le

vels

-18.622 / 9.110



Min MaxB

(c) Jacobian of RRconv w.r.t. Qv

-1.9 2.4G 6.8G 11.1 15.5KJacobian (mm dayE -1 (g/kg)-1)

60

50

40

30

Mod

el le

vels

-1.736 / 14.073

(d) Jacobian of RRconv w.r.t. T:

-20.5L -13.2F -5.8 1.5G 8.8GJacobian (mm dayE -1 K-1)

60

50

40

30

Mod

el le

vels

-18.622 / 8.032



(e) Jacobian of CAPE w.r.t. Qv:

-3.7 8.2G 20.0L 31.9 43.7FJacobian (mE 2 s-2 (g/kg)-1)

60

50

40

30

Mod

el le

vels

-3.355 / 39.729

(f) Jacobian of CAPE w.r.t. T:

-42.3M -25.1K -7.9 9.3G 26.6JJacobian (m2 s-2 K-1)

60

50

40

30

Mod

el le

vels

-38.481 / 24.142



Figure8: Sameasin Fig. 6 but for SIMPCV2.



exhibit asmootherstructureanda largeramplitudewith SIMPCV2thanwith OPNLCV.

4.2 Linearity over a single time-step

Reasonablywell-behaved physicalparametrizationsin termsof linearity arenecessaryto ensurea successfulconvergenceof theminimizationin variationaldataassimilation.Therefore,thelinearbehaviour of themodi-fied convectionschemehasbeenassessedthroughthecalculationof nonlinearresidualsandcomparedto thatof OPNLCV. For a givennonlinearoperatorM andits tangent-linear(TL) versionM N , thenonlinearresidualsaredefinedas

RESNL O M 8 x P λδx 9 6 M 8 x 9 6 M NRQ x S λδx (14)

wherex denotesthe input temperatureand specifichumidity in the presentcase. The residualshave beencomputedfor theconvective tendencies∂T T ∂ t and∂qT ∂ t thatareproducedby theconvective parametrizationM. Thecalculationsof Eq. (14) have beenrepeatedfor valuesof thescalingfactorλ rangingfrom 105 5 to 1andfrom -1 to -105 5. Thereferencevectorof perturbationsδx wassetequalto typical valuesof thestandarddeviationof thebackgroundmodelerrors.An exampleof verticalprofilesof sucherrorsis shown in Fig. 9.

0.0U 0.5U 1.0U 1.5U 2.0U 2.5UBackground error on T and qV 60

50

40

30

20

Mod

el le

velW

q

T

Figure 9: Vertical profilesof typical valuesof the standarddeviation of the modelbackgrounderrorson temperature(dashedline; in K) andspecifichumidity (solid line; in g kg + 1).

Figures10and11show nonlinearresiduals,averagedover100atmosphericprofileswith deepconvection,thatareobtainedwhenrespectively temperatureandspecifichumidity perturbationsof varyingsizeareappliedatthelowestmodellevel (60). Resultsareshown for this level becauseinputperturbationsimposedthereusuallybring large modificationsof convective activity but similar conclusionscanbe drawn for other levels. Thefield actuallyplottedin Fig. 10 and11 is log10 8 RESNL T RESmax

NL 9 whereRESmaxNL denotesthe maximumvalue

of the residualfor a given plot andfor both OPNLCV andSIMPCV2. This normalizationpermitsthe com-parisonof theresidualsobtainedwith thetwo parametrizationsandthesmallerthevaluesof theplottedfield,



T tendency OPNLCVX

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on T60 perturbation

60

50

40

30

Mod

el le

vels

a

log10(RES/RESZ max)

-30[ -10 -7\ -4] -2 -1_ 0Y

T tendency SIMPCV2`

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on T60 perturbation

60

50

40

30

Mod

el le

vels

b

log10(RES/RESZ max)

-30[ -10 -7\ -4] -2 -1_ 0Y

q tendency OPNLCVX

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on Ta 60 perturbation

60

50

40

30

Mod

el le

vels

c

log10(RES/RESZ max)

-30[ -10 -7\ -4] -2 -1_ 0Y

q tendency SIMPCV2`

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on Ta 60 perturbation

60

50

40

30

Mod

el le

vels

d

log10(RES/RESZ max)

-30[ -10 -7\ -4] -2 -1_ 0YFigure 10: Nonlinearresidualsof convective temperatureandspecifichumidity tendenciesasfunctionsof parameterλthatdeterminesthesizeof theinput temperatureperturbationsimposedon modellevel 60. Thedisplayedresidualshavebeenaveragedover onehundredconvective profiles. Thefield actuallyplottedis log10 b RESNL c RESmax

NL d whereRESmaxNL

denotesthe maximumvalueof the residualfor a givenpair of input/outputparametersandfor both studiedconvectionschemes(i.e. for a givenrow of thefigure).Left: OPNLCV, right: SIMPCV2.



T tendency OPNLCVX

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on qa 60 perturbation

60

50

40

30

Mod

el le

vels

a

log10(RES/RESZ max)

-30[ -10 -7 -4 -2 -1 0Y

T tendency SIMPCV2`

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on qa 60 perturbation

60

50

40

30

Mod

el le

vels

b

log10(RES/RESZ max)

-30[ -10 -7 -4 -2 -1 0Y

q tendency OPNLCVX

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on q60 perturbation

60

50

40

30

Mod

el le

vels

c

log10(RES/RESZ max)

-30[ -10 -7 -4 -2 -1 0Y

q tendency SIMPCV2`

-100 -10-2 -10-4 10Y 4 10Y 2 10Y 0

Scaling factor on q60 perturbation

60

50

40

30

Mod

el le

vels

d

log10(RES/RESZ max)

-30[ -10 -7 -4 -2 -1 0YFigure11: Sameasin Fig. 10, but wheninput specifichumidity perturbationsareimposedon modellevel 60.



the morevalid the linear assumption.Figures10 and11 demonstratethat the nonlinearresidualsarealmostsystematicallyoneorderof magnitudesmallerwith thenew schemethanwith theoperationalone. Theweakasymmetriesaboutthey-axis resultfrom thefact that large negative temperatureor moistureperturbationsatthesurfacecanturnoff convectiontherebyleadingto a lesslinearbehaviour of thescheme.

4.3 Validity of the linear assumption in time

In additionto thepreviouslinearity testsperformedonconvectionalone,it is crucialto verify thattheinclusionof SIMPCV2 in TL integrationscan leadto a betterapproximationof the finite differenceof two nonlinearintegrationswhenotherprocessessuchasvertical diffusion, large-scalecondensationandradiationarealsoactivatedandwhentheperturbationsinvolvedarenotnecessarilylimited in size.Indeed,in 4D-Varassimilationfor instance,the magnitudeof suchperturbationsmay reachseveral K for temperature,several g kg5 1 forspecifichumidityandaseveralm s5 1 for wind.

Themeanabsoluteerror, E, betweena TL run andthecorrespondingpair of nonlinearintegrationsis definedas

E O eM 8 x P δx 9 6 M 8 x 9 6 M N Q x S δx e (15)

wherex is equalto a given backgroundmodelstateandthe perturbationδx is setto the differencebetweenthis backgroundandtheoperationalverifying analysis.Thesizeof theinitial perturbationsimposedto theTLmodelis thereforethatof analysis6 background departures.

In the2004versionof theECMWF4D-Varsystem,themodeltrajectoryis calculatedusingOPNLCV, while theminimizationandtheevolutionof analysisincrementsinvolve thesimplifiedlinearizedconvectiveparametriza-tion SIMPCV. SinceSIMPCV2is animprovedapproximationto OPNLCV, its usein theTL integrationshouldlead to a reductionof the linearity error. It is also importantto checkhow SIMPCV2 behaves when com-binedwith the new simplified parametrizationof large-scalecloudsandprecipitation(CLOUDST, hereafter;TompkinsandJaniskova 2004).

Theintegrationspresentedherearerun over 12 hourswith a spectralresolutionof T159and60 vertical levelsin order to matchthe currentassimilationwindow and inner-loop horizontalandvertical resolutionsof theECMWF 4D-Varsystem.Thefull simplifiedphysicsis activatedin all TL integrationsbut one(adiabaticrun).

Figures12and13respectively displayzonalmeanerrordifferencesfor temperature(in K) andspecifichumidity(in g kg5 1) whentheTL modelis integrated:

(a) with the operationalsimplified physicalpackage(SIMPOP:vertical diffusion, gravity wave drag,deepconvection,large-scalemoistprocessesfrom MA99 andradiationfrom Janiskova etal. (2002))insteadofnosimplifiedphysicsatall (ADIAB),

(b) with SIMPCV2asasubstitutefor SIMPCV, and

(c) with bothSIMPCV2andCLOUDSTinsteadof SIMPOPmoistphysics.

Theglobalrelative changein thelinearityerroris mentionedat thetopof eachpanelandnegativevaluesof theplottedfield indicateimprovementsof thelinearity.

First of all, panel(a) shows thattheinclusionof theoperationalsimplifiedphysicsimprovesthelinearity errorby 13.96%for temperatureand18.4%for specifichumidityrelative to theadiabaticTL run. Statisticscomputedfor panel(b) of Fig. 12and13 indicatethatSIMPCV2furtherreducestheglobalmeanTL errorby 0.49%and



80f

OgN 60O

gN 40O

gN 20O

gN 0

hOg

20OgS 40O

gS 60O

gSi

80OgSi60

50

40

30

20

10

Error difference SIMPOP - ADIAB: -13.96 % - Temperature - 2000121512+12 (T159 L60)j

K

-0.5

-0.2

-0.15

-0.1

-0.05

-0.025

-0.01

0.01

0.025

0.05

0.1

0.15

0.2

0.5

80f OgN 60O

gN 40O

gN 20O

gN 0

h Og

20OgS 40O

gS 60O

gSi

80OgSi60

50

40

30

20

10

Error difference SIMPCV2 - SIMPOP: -0.49 % - Temperature - 2000121512+12 (T159 L60)

K

-0.5

-0.2

-0.15

-0.1

-0.05

-0.025

-0.01

0.01

0.025

0.05

0.1

0.15

0.2

0.5

80f ON 60ON 40ON 20

k ON 0h O 20OS 40OS 60OS

i80OSi60

50

40

30

20

10

Error difference (SIMPCV2 + CLOUDST) - SIMPOP: -2.1 % - Temperature - 2000121512+12 (T159 L60)

K

-0.5

-0.2

-0.15

-0.1

-0.05

-0.025

-0.01

0.01

0.025

0.05

0.1

0.15

0.2

0.5

a

b

c

Figure 12: Impactof the new moist physicson the tangent-linearapproximationerror asdefinedin the text. Panel(a)shows the temperatureerror difference(in K) when SIMPOPis usedin the tangent-linearcalculationsinsteadof nophysicsat all (adiabaticrun). Thetwo otherpanelsshow thefurtherchangesin theerrordueto theuse(b) of SIMPCV2and(c) of SIMPCV2combinedwith CLOUDST. Errorchangesaredisplayedaszonalmeansandmodellevelsareshown

on theverticalaxis.Theglobalrelativeerrorreductionappearsat thetop of eachpanel.



80f

ON 60ON 40ON 20k

ON 0h

O 20OS 40OS 60OSi

80OSi60

50

40

30

20

10

Error difference SIMPOP - ADIAB: -18.4 % - Specific humidity - 2000121512+12 (T159 L60)

g/kgl

-0.5

-0.2

-0.15

-0.1

-0.05

-0.025

-0.01

0.01

0.025

0.05

0.1

0.15

0.2

0.5

80f ON 60ON 40ON 20ON 0

h O 20OS 40OS 60OSi

80OSi60

50

40

30

20

10

Error difference SIMPCV2 - SIMPOP: -0.4 % - Specific humidity - 2000121512+12 (T159 L60)

g/kgl

-0.5

-0.2

-0.15

-0.1

-0.05

-0.025

-0.01

0.01

0.025

0.05

0.1

0.15

0.2

0.5

80f OgN 60O

gN 40O

gN 20O

gN 0

h Og

20OgS 40O

gS 60O

gSi

80OgSi60

50

40

30

20

10

Error difference (SIMPCV2 + CLOUDST) - SIMPOP: -1.45 % - Specific humidity - 2000121512+12 (T159 L60)

g/kgl

-0.5

-0.2

-0.15

-0.1

-0.05

-0.025

-0.01

0.01

0.025

0.05

0.1

0.15

0.2

0.5

a

b

c

Figure13: Sameasin Fig. 12 but for specifichumidity.



0.40%respectively. The strongestreductionof the error occursbetween607 S and707 N and is confinedtomodellevels39 to 50. Betweenthesurfaceandmodellevel 50 theerror is slightly increased.Above level 40,error changesremainsmall and limited to the Tropics. Figures12.(c) and13.(c) show that the combinationof SIMPCV2with CLOUDSTbringsanoverall reductionof E equalto 2.10%for temperatureand1.45%forspecifichumiditywith respectto theexperimentwith thecurrentoperationalsimplifiedmoistphysics.

4.4 Elimination of instability and nonlinearity sources

In someearlierJacobiancomputations,somesourcesof excessive sensitivity were isolatedandremoved byadequatemodificationsof thecode.Thesecodeadjustmentsconsistedof

- theremoval of thedependenceof organizeddetrainmenton theupdraughtverticalvelocity,

- the specificationof a constantupdraughtvertical velocity (3 m s5 1) in the calculationof precipitationformation(Eq. (12)) whenever thebuoyancy becomesnegative at levelsbelow 7000m.

It wasalsofound that the activation of the massflux limiter that is requiredto prevent numericalinstabilitytriggeredby the violation of the Courant-Friedrich-Levy criterion leadsto ratherunrealisticJacobians.Forinstance,thesensitivity of thesurfacerainfall ratewith respectto temperatureat lower levelsbecomesat leastoneorderof magnitudesmallerandoppositeto thatshown in thebottomright panelof Fig. 7. However, onlyabout5%of thetotalnumberof convectivegrid pointsareaffectedby themassflux limiter for timestepssmallerthan1800s andfor thecurrentmodelresolution(T511L60).

Furthermore,in someTL evolution experimentsrun with SIMPCV2a spuriousamplificationof TL perturba-tionsoccurredata few geographicalpointsaftera few hoursof integration,which thencontaminatedtheentireGlobe.In fact,suchproblemscanarisewhentheperturbationof a givenfield canbecomeaslargeasthefielditself, whichwasfoundto bethecasefor

- thecloudbasemassflux, Mclbu from Eq. (11),

- thebuoyancy, mn8 z9 , asdefinedin Eq. (8),

- theinitial verticalvelocity of theupdraught,ws, from Eq. (4).

Toavoid theseinstabilities,thecorrespondingperturbationsarereducedby afactorof 0.25for Mclbu and0.35formo8 z9 andws. For thelatterfield, thelimitation is appliedonly whenws is lower than0.5m s5 1. Thesefactors

weretunedsoasto provideagoodcompromisebetweenstabilityandvalidity of thelinearassumption,andarerequiredwhenrunningSIMPCV2in all tangent-linearandadjointcalculationsinvolving time integration.

5 Applications

5.1 1D-Var in rainy areas

SIMPCV2hasbeenextensively testedin 1D-Var retrievalsof temperatureandspecifichumidity profilesfromobserved rainfall rate(RRs)andmicrowave brightnesstemperatures(TBs). The two correspondingmethodswill be abbreviated as1D/RR and1D/TB hereafter. In suchexperiments,vertical profilesof incrementsoftemperatureandspecifichumidityareobtainedby minimizingacostfunctionthatmeasuresthefit of themodelstatebothto theobservationsandto a backgroundstate.Error statisticson modelandobservationsneedto beproperlyspecified.The new moist physicalpackage(SIMPCV2+CLOUDST)is calledto convert the modelstate(T,q) into precipitationfields that can be usedin turn to simulatemicrowave brightnesstemperatures



1000p30°S 30°S

25°S25°S

175°W

175°Wq 170°Wr

170°W

as -1

0.1t0.5t12u3v5w10

20u50w100

Surface RR guessx

mm/hy

-1

0.1z0.5z1{2|3}5~10{20|50~100{

1000p30°S 30°S

25°S25°S

175°W

175°Wq 170°Wr

170°W

b -1

0.1

0.5

1

2

3

5

10

20

50

100

Surface RR TMI obsx

mm/h

-1

0.1z0.5z1{2|3}5~10{20|50~100{

1000�30°S 30°S

25°S25°S

175°W

175°Wq 170°Wr

170°W

c� -1

0.1

0.5

1

2

3

5

10

20

50

100

Surface RR after 1D/RR�

mm/hy

-1

0.1z0.5z1{2|3}5~10{20|50~100{

1000�30°S 30°S

25°S25°S

175°W

175°Wq 170°Wr

170°W

d�

-1

0.1

0.5

1

2

3

5

10

20

50

100

Surface RR after 1D/TB�

mm/h

-1

0.1z0.5z1{2|3}5~10{20|50~100{

1000p30°S 30°S

25°S25°S

175°W

175°Wq 170°Wr

170°W

e� -30

-20

-10

-5

-3

-1

-0.5

0.5

1

3

5

10

20

30

1D/RR TCWV increments

kg/m2

-30

-20

-10

-5

-3

-1

-0.5

0.5�1�3�5�10�20�30�

1000p30°S 30°S

25°S25°S

175°W

175°Wq 170°Wr

170°W

f -30

-20

-10

-5

-3

-1

-0.5

0.5

1

3

5

10

20

30

1D/TB TCWV increments

kg/m2

-30

-20

-10

-5

-3

-1

-0.5

0.5�1�3�5�10�20�30�

Figure14: 1D-VarexperimentsusingSIMPCV2andCLOUDST: (a)surfaceRRsassimulatedfrom thebackgroundstateof (T,q), (b) asderivedfrom TRMM multi-channelmicrowaveTBsusingthePATER algorithm,andasrecomputedfromthemodifiedmodel(T,q) profilesafter (c) 1D/RRand(d) 1D/TB (usingthe10, 19, 22 and37 GHz channelsin verticalandhorizontalpolarizations).1D-Var Total ColumnWaterVapour(TCWV) incrementsfrom (e) 1D/RRand(f) 1D/TB

arealsoplotted.In all panelsbackgroundmean-sea-level isobarsaredrawn every2 hPa.



usingthe radiative transfermodelof Bauer(2002). The iterative minimizationof the cost function involvestheadjointversionof themoistphysics.Theadjointof theradiative transfermodelis alsorequiredin 1D/TB.More technicaldetailson 1D/RRand1D/TB in precipitatingareascanbefoundin Moreauetal. (2004).

An exampleof 1D-Var retrieval usingobservationsfrom the Tropical Rainfall MeasuringMission (TRMM)polar-orbitingsatelliteis shown in Fig.14for tropicalcycloneAmi at1800UTC 14January2003in thewesternPacific. Notethatthesurfacerainfall ratesusedasobservationsin 1D/RRareobtainedby applyingtheretrievalalgorithmPATER developedby Baueret al. (2001).Figure14 demonstratesthatboth1D-Var approachesareableto produceadjustmentsof themodel’s temperatureandspecifichumidity thatimprovesurfacerainfall rateswith respectto thePATER observations. This is evenmoreremarked in thecaseof 1D/TB sincethis methodworksdirectlyon theTBs andhasthereforeno explicit informationonprecipitation.In theproposedexample,1D/TB hasbeenappliedto the10,19,22and37GHzchannelsin verticalandhorizontalpolarizations.A rathersystematicandcommonfeaturein 1D/RR and1D/TB usingSIMPCV2 andCLOUDST is thepredominanceof specifichumidity incrementsrelative to thosein temperature(afterconversioninto equivalentincrementsofsaturationspecifichumidity). Thissupportsthechoicemadeby MarecalandMahfouf (2003)to defineTCWVpseudo-observationsthatareobtainedby verticalintegrationof the1D-Varspecifichumidity increments.TheseTCWV pseudo-observationscan in turn be fed into the 4D-Var assimilationsystem. In other respects,oneshouldalso note that the incrementsare usually systematicallytwice as large with 1D/TB as with 1D/RR.This is the resultof the sensitivity of the selectedmicrowave frequenciesnot only to rain but also to cloudliquid watercontentandwatervapour, but alsoto a lesserextent to thespecificationof theobservation errorstatistics.1D/TB retrievalsonly usingthe22GHzchannelleadto TCWV incrementsthataremorecomparablein magnitudeto thosefrom 1D/RR(notshown).

In thenearfuture,SIMPCV2andCLOUDSTwill beusedoperationallyin the1D-Varcalculationsinvolvedinthe ’1D-Var + 4D-Var’ methodproposedby MarecalandMahfouf (2003)to assimilateTBs from theSpecialSensorMicrowave Imager(SSM/I).Eventually, thenew simplifiedmoistphysicswill betestedwithin 4D-Varitself.

5.2 Sensitivity experiments

The new moist physicshasalsobeencomparedwith the operationaloneto computethe sensitivity to initialconditionsof a givencharacteristic,J, of theforecastat thefinal time overa selectedgeographicaldomain,� .Thegradientof J with respectto themodel’s initial state,∇0J, is obtainedfrom thegradientof J at final time,∇ f J, throughtheintegrationof theadjointmodelwith simplifiedmoistphysics,M � . Formally this relationcanbewritten

∇0J � B � 12 M � P� P ∇ f J (16)

wherethe norm basedon theECMWF 4D-Var backgrounderror covariancematrix, B, (DerberandBouttier1999)is usedat theinitial time,P is thegeographicalprojectorandtheasteriskdenotestheadjointoperator.

In theexampledescribedhere,the forecastcharacteristicof interest,J, is chosento be thequadraticerror ofspecifichumidity

J � ��P � qf c � P�� qan � P�� 2 dP

gd � (17)

whereqf c is thespecifichumidity from a 48-hourforecastvalid at 1200UTC 12 November1999andqan thesamefield from theverifying analysis.J involvesboth vertical integrationandhorizontalaccumulationoverdomain � with longitudesrangingfrom 0� to 5� E andlatitudesbetween40� N and45� N. Themainpatternsof



mean-sea-level pressure(MSLP) and500 hPa geopotentialheightat thebeginning andat theendof 48-hourwindow aresummarizedin Fig. 15. By 1200UTC 12November1999,a low haddevelopedoverSpain,whichled to torrentialrainfall over southernFrance(up to 620 mm in 24 hourslocally; BechtoldandBazile2001)dueto theenhancingeffectof theintensemoisturesupplythroughsurfaceevaporationover theGulf of Lyons.

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40°N

50°N

60°N

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40°W

30°W

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20°W

10°W

10°W

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0°

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10°E

20°E

20°E 30°E

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b

Figure 15: MSLP (black solid line; every 5 hPa) and500 hPa geopotentialheight(red dashedline: every 40 m) fieldsfrom theoperationalanalysisat1200UTC (a)10Novemberand(b) 12 November1999.

Sensitivity fieldsarecomparedusingeithertheSIMPOPmoistphysicspackageor therevisedone(SIMPCV2andCLOUDST) in theadjointcomputations.Figure16 displaysthesensitivities of J computedat 1200UTC12 November1999 to specifichumidity on model levels 46 (left) and 55 (right): panels(a) and (b) showthesensitivities at thebeginningof theadjoint integrationwhile thefour otherpanelsdisplaythesensitivitiesat the endof the 48-hourbackward adjoint integrationusingeither the SIMPOPmoist physics(c andd) orSIMPCV2+CLOUDST(e andf). Notethatsensitivities have beennormalizedby thepressurethicknessof thecorrespondingmodellayer.

Figure16.(a)and(b) exhibit expectedpatternsof ∂J∂q at thebeginningof theadjointintegration.Cross-sections

(not shown) suggestthat the largestsensitivities at the end of the adjoint integration are confinedbetweenmodellevel 43 andthesurface,with maximumvaluesat level 46. Furthermore,sensitivity patternsobtainedwith SIMPCV2+CLOUDSTareusuallysimilar to thoseobtainedwith theSIMPOPmoistphysicsasillustratedby panels(c) and(e). Somesignificantdifferencesonly appearbelow modellevel 50 over theGulf of Lyonsasindicatedby panels(d) and(f), which couldbeattributedto thedirect link betweensurfaceheatfluxesandconvective activity in SIMPCV2 throughEq. (4)-(6). Thesepreliminarysensitivity experimentsdemonstratethat the new simplified moist physicsis ableto remainstableduring the 48 hoursof adjoint integration(i.e.no spuriousamplificationof perturbations)and that it producessensitivity patternsthat areconsistentwiththeoperationalones.Thereseemsto bea potentialadditionalimpactof physicalprocessesthatareexplicitlyrepresentedin SIMPCV2andCLOUDSTadjointversionsbut not in SIMPOP. Whetherthis impactis actuallybeneficialwill be determinedin the context of forecastsensitivity experimentsaswell as in singularvectorcalculations(usedfor instanceto generateinitial perturbationsin anensemblepredictionsystem)andof coursein 4D-Vardataassimilation.



0.051660Ö-0.708300

30°N

40°N

50°N

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0°

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10°E

20°E

20°E 30°E

30°E

Diagnostic: 7 Iteration: 0Wednesday 10 November 1999 12UTC ECMWF Sensitivity Gradients t+48 VT: Friday 12 November 1999 12UTC Model Level 46 **specific humidity

a

× ∂J∂q46 Ø 0

0.385052Ù-0.028088

30°N

40°N

50°N

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10°W

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20°E 30°E

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Diagnostic: 7 Iteration: 0Wednesday 10 November 1999 12UTC ECMWF Sensitivity Gradients t+48 VT: Friday 12 November 1999 12UTC Model Level 55 **specific humidity

b

× ∂J∂q55 Ø 0

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Diagnostic: 7 Iteration: 0Wednesday 10 November 1999 12UTC ECMWF Sensitivity Gradients t+0 VT: Wednesday 10 November 1999 12UTC Model Level 46 **specific humidity

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f

× ∂J∂q55 Ø � 48

Figure 16: Sensitivities of specifichumidity forecasterror at 1200 UTC 12 November1999 to specifichumidity onmodellevels 46 (left panels)and55 (right panels)at the sametime (a andb) and48 hoursbeforehandwhenusingtheSIMPOPmoistphysics(c andd) or SIMPCV2+CLOUDST(e andf) in adjointcalculations.Positiveandnegativevaluesof sensitivity areshown with redsolid andbluedashedisolinesrespectively. Sensitivities arerendereddimensionlessby

thenormalizationapplied(seetext).



6 Conclusions

A new simplifiedparametrizationof subgridscaleconvective processeshasbeendevelopedandtestedin theframework of theECMWF IntegratedForecastingSystemfor thepurposeof variationaldataassimilation,sin-gularvectorcalculationsandadjointsensitivity experiments.It isbasedonthefull nonlinearschemethatis usedin theECMWF operationalforecasts,but it wasdemonstratedthata setof simplificationscould leadto a sub-stantialimprovementof its linearbehaviour without significantlydegradingits forecastingability evenin sea-sonalintegrations.Thenew schemehasalsobeensuccessfullycombinedwith thenew simplifiedparametriza-tion of large-scalecloudsandprecipitationdevelopedby TompkinsandJaniskova (2004). In contrastwith theMAH99 simplifiedconvective parametrization,its tangent-linearandadjointversionsaccountfor perturbationsof all convective quantitiesincludingconvective massflux, updraughtcharacteristicsandprecipitationfluxes.Thereforethe new schemeis expectedto be beneficialin the 4D-Var context whencombinedwith radiativecalculationsthat aredirectly affectedby condensationandprecipitation. In particular, this is the reasonwhySIMPCV2hasbeensuccessfullyusedin 1D-Var retrievalsof TCWV from microwave brightnesstemperaturesin precipitatingareas(Moreauet al. 2004). In thenearfuture,theimpactof thenew simplifiedmoistphysicsin 4D-Vardataassimilationitself will betested.

Acknowledgements

Wewould like to thankJean-Franc¸oisMahfouf for hiskey-role in initiating thiswork andhelpingusin its earlystage. Marta Janiskova, Adrian Tompkins,Martin Miller, Anton Beljaars,PeterBechtold,PeterBauerandJean-Noel Thepautshouldbeacknowlegdedfor their usefulreviews of this paper. We aregratefulto INRIA(Institut National de Rechercheen Informatiqueet en Automatique)for providing the M1QN3 mimizationcode. This studywassupportedby theEuropeanSpaceAgency projectsEuroTRMM-2 (#12651/97/NL/NB)andEGPM(#3-10600/02/NL/GS).



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Model level number Pressure(hPa)

5 110 415 1220 3625 9630 20235 35340 53945 72850 88455 97960 1012

Table 1: Pressure(in hPa) on every five model levels betweenlevels 5 and60. Valuesare given assuminga surfacepressureof 1013.25hPa.

Acronym DescriptionOPNLCV Nonlinearconvectionschemeasusedin operationalforecasts

and4D-Var trajectory(cycle26r3,Tiedtke 1989)SIMPCV Simplifiedconvectionschemeasusedin operationaltangent-linear

andadjointcalculations(cycle 26r3;Mahfouf1999)SIMPOP All simplifiedphysicalparametrizationsasusedin operational

tangent-linearandadjointcalculations(cycle 26r3;Mahfouf1999;Janiskova et al. 2002)

CLOUDST New simplifiedparametrizationof cloudsandprecipitation(TompkinsandJaniskova 2004)

SIMPCV2 New simplifiedconvectionscheme(topic of thispaper)

Table2: List of acronymsusedin thetext.


a convection scheme for data assimilation: description and initial tests

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