the control of gravity waves in data assimilation 27 …...the control of gravity waves in data...

Meteorological Training Course Lecture Series

ECMWF, 2002 1

The control of gravity waves in data assimilation27 April 1999

By Adrian Simmons

European Centre for Medium-Range Weather Forecasts

The needto control gravity waves in dataassimilationand the generalconceptof initialization are discussed.The basicmethodof non-linearnormal-modeinitialization is introduced,and aspectsof its implementationin modelsare presented,covering the vertical and horizontalmodal decomposition,implicit initialization and diabatic initialization. The control ofgravity wavesin theECMWF variationaldataassimilationsystemis described.An introductionto thealternative methodofdigital filtering is also given.

Table of contents

1 . Introduction

2 . Non-linear normal-mode initialization

2.1 Basic method

2.2 Vertical decomposition

2.3 Horizontal decomposition

2.4 Implicit initialization

2.5 Diabatic initialization

3 . Control of gravity waves in the ECMWF variational data assimilation system

4 . Digital filtering

4.1 Adiabatic, non-recursive filtering

4.2 Diabatic, recursive filtering

4.3 Diabatic, non-recursive filtering

APPENDIX A . Definition of operators for the ECMWF vertical finite-difference scheme

APPENDIX B . The Lamb Wave

APPENDIX C . The non-recursive implementation of the recursive filter

1. INTRODUCTION

Theexistenceof balanceis fundamentalto thedynamicsandpredictabilityof theatmosphere.Atmosphericmotion

is dominatednot by rapid fluctuationsassociatedwith soundandgravity waves,but ratherby the moreslowly

changingweathersystems.Over muchof theglobethesearecloseto thefamiliar stateof geostrophicbalancein

which winds blow parallel to contours of constant geopotential height on an isobaric surface (Fig. 1).

The control of gravity waves in data assimilation

2 Meteorological Training Course Lecture Series

ECMWF, 2002

Figure 1. The operational ECMWF analysis of height and wind at 500hPa for 12 UTC 13 April 1997.

Fasterwavemotionsare,however, excitedif theatmosphereis suddenlydisturbed.Themassivevolcaniceruption

of Krakatau(betweenJava andSumatra)in1883exciteda wave whosepropagationcouldbetracedin thesurface

pressurefield aroundto theantipodalpointandbackto thesource(Fig.2 ). Thesignaltookabout35hoursto com-

pleteits outwardandreturntrip, moving atabout320ms-1, aroundthespeedof sound.Thiswaswhathascometo

beknown asa Lambwave,or in numericalweatherpredictionasanexternalgravity wave.More commonly, and

on a muchsmallerscale,propagatinginternalgravity wavesmaybeexcitedby intenseconvective systems.Fig. 3

presentsobservationsof wavestructurein mid-tropospherictemperatureandwind fieldsoverJapanapparentlyas-

sociated with the occurrence of exceptionally heavy rainfall.



ECMWF, 2002 3

Figure 2. Position at two-hourly intervals of the surface-pressure wave excited by the volcanic eruption of

Krakatau, fromTaylor(1929).



ECMWF, 2002

Figure 3. Fine-scalestructurein temperatureat500hPa(left) andwind at550hPa(right) overJapanat12UTCon

1 July 1971, fromNinomiya (1983)

Numericalweatherpredictionmodelsmostlyassumehydrostaticbalance,andthusdonotsupporttheexistenceof

soundwaves.They aregenerallybased,however, ontheprimitiveequationsof motion,andLambandinternalgrav-

ity wavescanbeexcitedin thesemodels.In particular, they will beexcitedin dataassimilationif thesystemuses

observationsin suchawayasto produceanalysesin whichthebalancepresentin thebackgroundstateis disturbed.

Thewavesin thiscasecontaminatetheshort-rangeforecastthatprovidesthebackgroundstatefor thenext analysis,

adding to the difficulty of extracting useful information from the observations.

Theprimarycontrolof gravity wavesin amoderndataassimilationsystemis throughthemultivariateformulation

of theanalysis.In suchasystem,anobservationof surfacepressureor temperaturethatdiffersfromthebackground

will generallyresultin ananalysiswhichdiffersfrom thebackgroundnotonly in surfacepressureandtemperature,

but alsoin wind, thuspreventinganunrealisticdeviation from geostrophicbalance.This aspectof thecontrolof

gravity wavesis notthemainconcernof theselecturenotes,althoughweshallreturnto thetopicbriefly later. Rath-

er, we shall be concerned mostly with what is known as the process ofinitialization.

Theterminitialization is usuallyusedto referto theadjustmentof analysedinitial modelconditionsto preventex-

cessivegravity-waveactivity in thesubsequentforecast.Thisdynamicalinitialization is neededif themultivariate

formulationof theanalysisis unableto provideanadequatebalancethroughoutthemodeldomain.This is thetype

of initialization thatwill bediscussedhere.More widely, initialization maybedefinedasanadjustmentof thein-

itial conditionseither to maintaindynamicalor physical balanceor to satisfysomephysical constraint.Thusit

couldincludetheadjustmentof fieldsto ensurethatrelativehumiditieslie in therangefrom 0 to 100%,or thead-

justmentof fieldsto ensurethatinitial rainfall ratesmatchestimatesfrom observations.This is referredto asphys-

ical initialization.

Althoughtheprincipaluseof dynamicalinitialization is to remove imbalanceintroducedduringtheanalysisstep

in dataassimilation,it hasadditionalapplicationsin numericalweatherprediction.It is oftenusedto remove im-

balance in an initial state in which the model resolution or orography has been changed, for example in:

• Incremental variational data assimilation (as discussed later);

• Creatinginitial conditionsfor ensemblepredictionbasedonamodelwith resolutionlower thanthat

of the model used in the data assimilation;



ECMWF, 2002 5

• Testing of a new higher resolution version of a model or new specification of orography.

Initializationmayalsobeusedto removeimbalancewhenrunningforecastsfrom analysesproducedby otherfore-

castingcentres.Thisis doneoccasionallyto investigateforecastingfailuresandis underinvestigationin thecontext

of ensemble prediction.

Somefurthercharacteristicsof atmosphericmotionmustbetakeninto accountin consideringthetopicof initiali-

zation.A rich structureof largely stationarygravity-wave motionmaybeexcitedby flow over orography. Fig. 4

shows a classicalexample.Thesewavescanhave an importantdirecteffect on the troposphericcirculation,and

togetherwith theconvectively-excitedwavesplayacritical rolein aspectsof thecirculationof thestratosphereand

mesosphere.Their effectsarein generalparametrizedin large-scalemodels,but they arecomingincreasinglyto

beresolveddirectly in limited-areaandfiner-resolutionglobalmodels,andarea targetfor short-rangemesoscale

prediction. They should not be unduly suppressed by initialization.

Figure 4. Section of potential temperature crossing the Rocky Mountains in Colorado, for 17 February 1970,

from Lilly and Kennedy (1973).

Tidal motion in theatmosphereis a form of forcedgravity-wave motionwhich alsoshouldnot besuppressedby

initialization.Thestrongsemi-diurnalcomponentat thesurfaceis forcedby thedaily variationin solarheatingof

thestratosphere,dominatingthemorestrongly-forceddiurnalcomponentdueto differencesin downwardpropa-

gation.It is seenin quitepronouncedoscillationsin surfacepressurein thetropicsandsubtropics(Fig. 5 andFig.

17 ) andneedsto behandledwell if othersignalsin theobservationsareto be interpretedeffectively. Moreover,

thequasi-steadytropicalcirculationsystems(asillustratedschematicallyin Fig.6 ) characteristicallyinvolveabal-

ancebetweendiabaticheating(or cooling)andtheadiabaticcooling(or heating)associatedwith verticalmotion.



ECMWF, 2002

Care is needed to ensure that initialization does not produce analyses in which these circulations are weakened.

Figure 5. The semi-diurnal tide. The upper plot shows surface pressure (hPa) and the lower plot the tendency of

pressure(hPa/hour)at thelowestmodellevel. Thequantityplottedis where

denotes the mean ECMWF analysis atUTC for January 1997, truncated spectrally at T10 to remove local

orographic and station-specific features. Solid contours denote positive values and dashed contours negative

values. The nature of atmospheric tides is described inChapman and Lindzen (1970).

Figure 6. Schematic view of the mean tropospheric circulation in the equatorial plane, fromWebster(1983).

Themethodof non-linearnormal-modeinitializationthathasbeenusedextensively atECMWFto initialize its glo-

bal modelsis describedin thefollowing section.Theway gravity wavesarecontrolledin theCentre’s variational

dataassimilationsystemis describedandillustratedin section3. An alternative approachto initialization, thatof

digital filtering, hasbeendevelopedin recentyears,particularlyfor applicationin limited-areamodels.An intro-

duction to this method is given in section 4.

1 4⁄( )

00

06–

12

18–+( )



ECMWF, 2002 7

2. NON-LINEAR NORMAL-MODE INITIALIZATION

2.1 Basic method

Considera forecastmodellinearizedaboutastateof restwhich is staticallystable.Thegoverningequationsof the

model can be written in the following general form:

(1)

where is a columnvectorof suitably-scaledmodelvariables,of dimension say, and is a symmetric,real

matrix.Explicit examplesof sub-elementsof will begivenin thefollowing subsections.Theeigenval-

uesof arethefrequenciesof thesmall-amplitudewave motionsthattherestingstatecansupport,andthecor-

respondingeigenfunctionsdescribethestructuresof thewaves.In limiting casesthesewavescanbeseparatedinto

a setof “meteorological”or “Rossby”waveswhich move relatively slowly westward,andsetsof eastward-and

westward-moving gravity modes,thegraver of which move muchfasterthentheRossbywaves.More generally,

thecategorizationis complicatedin thetropicsby theexistenceof theso-calledmixedRossby-gravity andKelvin

waves.Comprehensive studiesof themodesof theshallow-waterequations(known asthe“Hough modes”)have

been made byLonguet-Higgins(1968) andKasahara(1976).

The general non-linear model equations can be written in the form:

(2)

wheretheterm representsboththenon-lineartermsandtheresiduallineartermsthatarisefrom differences

between the actual state of rest and that used to compute.

Considera subsetcomprising orthonormaleigenvectors, , of with large eigenvalues

. Theserepresentthegravity wavesthatareto beinitialized.A generalmodelstate canbewrit-

ten in the form:

(3)

Underidealizedconditionsin which the eigenvectorsrepresentacompletesetof gravity waves, represents

thecomponentof theatmosphericstatecomprisingsolelyRossbywaves.Moregenerally, representstheresid-

ual stateorthogonalto thesetof gravity modes,which comprisesRossbywaves,theslower gravity wavesthat

will not be initialized, and mixed waves. We denote the component that will be initialized by:

(4)

The aredeterminedby takingthescalarproductsof thetwo sidesof (3) with the , andusingorthogonality,

, and orthonormality, , to obtain

(5)

or

dd----- ( ) =

×

dd----- ( ) ( )+=

( )

1

2 …

, , ,( )

υ1 υ2 … υ

, , ,( )

α 1=

∑+=

α 1=

∑=

α ⋅ 0=

⋅ δ =

α ⋅=



ECMWF, 2002

(6)

where is the matrix whose columns are the , for .

Substituting(3) into (2) and projecting onto the modes to be initialized gives:

(7)

where .

2.1 (a) Linear normal-mode initialization:Linear normal-modeinitialization comprisessimply setting to

zerotheamplitudes of the modesto beinitialized. It thusinvolvesreplacingtheuninitializedstate by

, where

(8)

and is the identity matrix.

Fig. 7 presentsanexampleof theimpactof linearnormal-modeinitializationon theevolutionof surfacepressure

at two points,onein theGreatPlainsof North Americaandonein theHimalayas.It is takenfrom thefirst study

of normal-modeinitializationfor theoriginalECMWFfinite-differencemodel(TempertonandWilliamson,1981;

Williamson andTemperton,1981),andpresentsan extremecasein that the initial datahadbeentaken from an

analysisproducedusingadifferentmodel.Imbalancein theinterpolatedinitial conditionsfor theECMWFmodel

resultedin large-amplitudeoscillationsin thesurfacepressure,asindicatedby thesolid linesin thefigure.Theam-

plitudeof theseoscillationswasclearlyreducedby applyinglinearnormal-modeinitialization (dashedlines),but

significantfluctuationsremained.Also worthy of noteis thelarge,rapidfall in surfacepressureat theHimalayan

point at the start of the integration from the initialized analysis.

Figure 7. Surface pressure (hPa) as a function of time before (solid) and after (dashed) linear normal-mode

initialization at 40oN 90oW (left) and 30oN 90oE (right), fromTemperton and Williamson(1981).

! =

×

" 1 2 …

, , ,=

dd----- α ( ) υ α $#% α1 α2 … α

, , , ,( )+=

# ! ( ) =

α '&)(

)( * !–( ) '&=

*



ECMWF, 2002 9

2.1 (b) Non-linear normal-mode initialization:Linear normal-modedoesnot prevent gravity waves from

growing to significantamplitudebecausethenon-linearor residuallineartermsrepresentedby arenotneg-

ligible. In non-linearnormal-modeinitialization(Machenhauer, 1977),theinitializedgravity-waveamplitudes

aresetto non-zerovalueswhich arechosento make thegravity-wave tendenciessmall.An approximatesolution

of the equation

(9)

is sought. Equation(7) is used to define an iterative solution of(9):

. (10)

Two iterationsof theprocedureareusuallyfoundto besufficient.Thestartingvaluescouldbetakenfrom a linear

normal-modeinitialization ( ), but in practiceit is found to besufficient to startfrom theuninitialized

analysis( ). This is themoststraightforwardstartingpointwhentheprocedureis implementedin

termsof updatesapplieddirectly to modelvariables,asoutlinedbelow in equations(12) to (15). It is, moreover,

thenecessarystartingpoint for aneffective full-field diabaticinitializationusingtheapproachdescribedin 2.5(a).

An exampleof theworking of non-linearnormal-modeinitialization is givenin Fig. 8 . This is for thesamecase

aspresentedin Fig. 7 for linearnormal-modeinitialization.Fig. 8 alsocontainsinformationrelatedto thevertical

decompositionof normalmodeswhich is discussedin thefollowing subsection.For themoment,attentionshould

bedirectedfirst towardstheuppertwo panelsandthecurveslabelled“0” which show theforecastfrom uninitial-

izedinitial conditionsand“5” whichshow theforecastfrom theconditionsmostfully initializedby thenon-linear

normal-modetechnique.Thesetwo panelscanbecompareddirectly with Fig. 7 which shows the forecastfrom

linearnormal-modeinitialization. It is clearthatthenon-lineartechniqueis muchmoresuccessfulthanthelinear

technique.It notonly preventshigh-frequency oscillationsin surfacepressureover the24-hourintegrationperiod,

but alsoavoidsthelargechangeto theinitial surfacepressureat theHimalayanpoint,a changethatis rejectedby

themodelin thefirst few time stepsof the forecastfrom linear initialization. Comparisonof the lower-right and

upper-left panelsof Fig.8 showshow thefirst iterationis sufficient to removemostof theshort-periodoscillations

presentwhenrunningfrom theuninitializedanalysis,with theseconditerationremoving mostof whatis left after

the first iteration.

( )α

dd----- α ( ) 0=

α ,+( ) 1 υ --------

– # - α1+ 1–( ) α2+ 1–( ) … α + 1–( ), , , ,( )=

α 0( ) 0=

α 0( ) &⋅=



ECMWF, 2002

Figure 8. Surface pressure (hPa) as a function of time at 40oN 90oW (upper left) and 30oN 90oE (upper right)

with no initialization (solid) and after non-linear normal-mode initialization of the first three and five vertical

modeswith two iterations.Surfacepressureis alsoshown at40oN 90oW with one,two andfour modesinitialized

with two iterations (lower left), and with five modes initialized with one and two iterations (lower right). From

Williamson and Temperton(1981).

Equation(10) is notnormallysolveddirectly. (7) and(10)arecombinedto expresstheiterativenon-linearnormal-

modeinitializationprocedurein termsof themodificationsmadeto thegravity-waveamplitudesateachiteration:

(11)

Equation(11) is thenconvertedinto the initialization incrementappliedto modelvariables.Startingthe iteration

∆α ( ) + 1+( ) α ,+ 1+( ) α .+( )–1 υ --------

d

dα +( )

–= =



ECMWF, 2002 11

from the uninitialized analysis, the non-linear initialization procedure becomes:

(12)

(13)

where is the diagonalmatrix of eigenvalues for . Thenon-linearmodeltendency

is computed using the latest approximation, , to the initialized state:

(14)

The final initialized state, , is given by:

(15)

where is the number of iterations.

This“explicit” formof non-linearnormal-modeinitializationthusinvolvesarepeatedsetof operationscomprising:

• Onetimestepof the model to computesthe tendenciesin physical space,startingfrom the latest

approximation to the linearized state;

• A projection of these tendencies onto the gravity modes to be initialized;

• Division of the gravity-mode tendencies by the frequencies of the modes;

• Projection back to physical space;

• Addition of the increment to form a new approximation to the initialized state.

Equation(13) may be written in the alternative form:

(16)

Equation(16)is usedasthestartingpointfor thederivationof “implicit” non-linearnormal-modeinitialization.As

discussedin subsection2.4, thisapproachusesamodifiedform of thegoverning,linearizedequationswhichena-

blesincrementsto be calculatedin physical spacewithout useof explicit projectionsto andfrom gravity-wave

space.

2.2 Vertical decomposition

Normal-modeinitialization is a practicalmethodbecauseit is possibleto separatetheverticalandhorizontalde-

pendenceof thegravity-wave modes.Furthermore,it hasto beappliedonly to a limited numberof modesin the

vertical.

We considersmall-amplitudeperturbationsabouta stateof restwith temperature andsurfacepressure ,

with eastwardandnorthwardwind perturbations and , andtemperatureperturbation . Theequationsare

setout in theform appropriatefor amodel(suchasthatof ECMWF) in which thelogarithmof surfacepressureis

a basicprognosticvariable,with perturbationgiven by . We restrict the presentationhereto the casein

0( ) &=

∆ /+ 1+( ) Λ 1– ! dd----- ( )

+( )=

Λ × υ " 1 2 …

, , ,=

d 0⁄( ) +( ) /+( )

1+( ) 0( ) ∆

( )1=

+∑+=

)(

)( 0( ) ∆ /+( )

+ 1=

2∑+=

#

∆ 1+ 1+( ) ! dd------

+( ) dd------

+( )

= =

354 67849 ' : ' 3

'

6ln7

( )'



ECMWF, 2002

whichthereferencetemperature is isothermal,ratherthanvaryingwith pressure.Thissimplifiestheequations,

andis not a seriousrestrictionin practice.Effectsof unrepresentivity of thereferencestateof restaretaken into

accountin non-linearnormal-modeinitialization throughthe term . Moreover, gravity-wave phasespeeds

andverticalmodestructuresarenotstronglydependentonthechoiceof referencestate,asillustratedin Appendix

A.

The dry, linearized, primitive equations for a terrain-following vertical coordinate are then:

(17)

(18)

(19)

(20)

Here is the divergence:

(21)

is the pressure-coordinate vertical velocity:

(22)

and is the temperature-dependent part of the perturbation geopotential:

(23)

is theplanetaryrotationrate, is theplanetaryradius, is longitude, is latitudeand is thegasconstant

of dry air. , where is thespecificheatof dry air at constantpressure.In evaluatingthepressureon

a coordinatesurfacein equations(19), (20), (22) and(23), thesurfacepressureis takento bethereferencevalue

.

Weconsideramodelwith levels,define to bethecolumnvectorcomprisingthevaluesof at the levels,

anddefine , and similarly. Thecolumnvectorwith eachelementequalto is denotedby , andwe

use matrices and , and vector to represent a general finite-difference scheme:

354

( )

9 ′∂∂-------- 2Ω θsin : ′ 1; θcos

---------------λ∂

∂ φ′ < 34 67ln( )′+( )–=

: ′∂∂------- 2– Ω θ 9 ′sin

1;--- θ∂∂ φ′ < 34 67

ln( )′+( )–=

3′∂∂--------

κ346----------ω′=

∂∂ 6

ln7

( )′ 167 4-------- = ′6>

0

?A@CB∫–=

= ′

= ′ 1; θcos---------------

9 ′∂λ∂

--------θ∂

∂ : ′ θcos( )+ =

ω′

ω′ = ′6>

0

?∫–=

φ′

φ′ < 3 ′6----------- 6d??A@CB∫=

Ω ; λ θ <κ <ED ?⁄= D ?

6784F G 9 ′

FH I J < 34 µ

γ τ δ

16784-------- = 6d

0

?K@CB∫ δ

! I→



ECMWF, 2002 13

Explicit expressionsfor , and aregivenin AppendixA for theverticaldiscretizationschemedevelopedfor

the ECMWF model bySimmons and Burridge(1981).

We further define:

(24)

Then, using(17), (18) and the combination of(19) and(20), the linearized equations become:

(25)

(26)

(27)

where

(28)

Thenormal-modeinitializationproceduredeterminesanincrementto the“mass”variable . Toobtainincrements

in temperatureandthe logarithmof surfacepressure,equation(27) is usedtogetherwith thevectorformsof the

equations for and :

(29)

(30)

Giveninitializationincrement , thecorrespondingincrementsto temperatureandthelogarithmof surfacepres-

sure, and , are given by

(31)

and

(32)

κ346----------ω τ– I→

φ' γ J→

< 34( )67

ln( )' µ67

ln( )'→

γ τ δ

Lγ J µ

67ln( )'+=

∂∂G

2Ω θsin H 1; θcos---------------

λ∂∂L

–=

∂∂ H 2– Ω θ

Gsin

1;--- θ∂∂L

–=

∂∂L M I–=

Mγτ µδ

!+=

L

J 67ln( )'

∂∂ J τ I–=

∂∂ 67

ln( )' δ! I–=

∆L

∆ J ∆67

ln( )'

∆ J τM

1– ∆L

=

∆67

ln( )' δ! M

1– ∆L

=



ECMWF, 2002

Now let denotethediagonalmatrixof eigenvalues of , for , andlet denotethematrix

whose columns are the eigenvectors of . can be written: .

If the prognostic variables are transformed in the following way:

and

equations(25), (26) and(27) becomea setof uncoupledshallow waterequationswith “equivalent” depths

:

(33)

(34)

(35)

Here is the divergence associated with velocities and as defined by(21):

(36)

Thegoverningequationfor puregravity wavesis obtainedby settingtherotationrate, , to zero,takingthetime

derivative of (35), and using(33) and(34) to eliminate the rate of change of divergence . This gives:

(37)

For scalessmallenoughfor a localplanegeometryto bevalid, thephasespeedof thepuregravity wavesis .

Onthespherethepuregravity-wavemodesarethesphericalharmonicfunctions , where is an

associatedLegendrefunction.Thesphericalharmonicfunctionsareeigenfunctionsof theLaplacianoperatorwith

eigenvalues . The pure gravity-wave frequencies are thus .

Examplesof modesarepresentedherefor thecurrentoperational31-level ECMWFmodelandfor a50-level ver-

sionof themodelplannedfor operationalimplementation.Thelocationof themodellevelsfor thesetwo vertical

resolutionsisshown in Fig.9 . Table1showsphasespeedsandequivalentdepthsof thefastermodesfor areference

temperature, , of 300Kandareferencesurfacepressure, , of 1000hPa.Correspondingverticalstructuresof

the divergence fields (weighted by ) are shown in Fig. 10.

Φ Φ NM O

1 2 …F

, , ,= M M M Φ 1–=

9 ˆ N 1– G( )N=

:ˆ N 1– H( )N=

P Nˆ 1– L( ) N=

FΦNRQ⁄

∂∂9 ˆ N

2Ω θsin : ˆ N 1; θcos---------------

λ∂∂P ˆ N

–=

∂∂: ˆ N

2– Ω θ 9 ˆ Nsin1;--- θ∂

∂P ˆ N

–=

∂∂P ˆ N

ΦN = Nˆ–=

= Nˆ 9 ˆ N : ˆ N

= Nˆ 1; θcos---------------

λ∂∂9 ˆ N

θ∂∂ : ˆ N θcos( )+

=

Ω= Nˆ

22

∂∂P ˆ N

Φ N P ˆ N∇2– 0=

ΦNP+S

θsin( ) T S λ P

+S

UVU 1+( ) ; 2⁄–1;--- UWU 1+( )ΦN

34 67 46



ECMWF, 2002 15

Figure 9. The distribution of full model levels for 31-level (left) and 50-level (right) vertical resolutions, plotted

for a distribution of surface pressure which varies from 1013.25 to 500 hPa.

TABLE 1. GRAVITY-WAVE PHASESPEEDSAND EQUIVALENT DEPTHSFOR THE VERTICAL MODES OF THE 31-LEVEL

AND 50-LEVEL VERSIONSOF THE ECMWF MODEL FORA REFERENCETEMPERATURE OF 300K AND A REFERENCE

SURFACE PRESSUREOF 1000hPa. ONLY MODES WITH GRAVITY-WAVE PHASESPEEDSFASTER THAN 50ms-1ARE

INCLUDED.

Mode numberPhase speed

(ms-1)for 31-level model

Phase speed (ms-1)

for 50-level model

Equivalent depth (km)

for 31-level model


for 50-level model

1 343 347 12.0 12.3

2 203 262 4.2 7.0

3 119 194 1.4 3.8

4 78 145 0.6 2.1

5 56 112 0.3 1.3

OΦ N ΦN Φ NRQ⁄ ΦNRQ⁄



ECMWF, 2002

Figure 10. Structures of the vertical modes of the 31-level and 50-level models with gravity-wave phase speeds

faster than 50 ms-1, for a reference temperature of 300K and a reference surface pressure of 1000 hPa.

Thegravestmodemovesthefastest,anddiffersin structurefrom theothermodesin thatit is theonly modewhose

-weightedamplitude(or energy density)decreaseswith height.It is generallyreferredto astheexternalgravity

wavein thecontext of numericalweatherprediction,but is known in geophysicalfluid dynamicsastheLambwave.

It is shown in AppendixB that thereis a correspondinganalyticalsolutionof thecontinuousequationsin which

thedivergenceandtemperaturevaryin theverticalas , andthephasespeedis givenby . This

givesaphasespeedof 347ms-1 for areferencetemperatureof 300K,with =287m2K-1s-2 and =0.286.Thisval-

ue is reproduced to the nearest ms-1 by the 50-level resolution.

Theremaining(“internal”) modesprovideamathematically-acceptablebasisfor representingtheverticalstructure

of modelvariables,but donotcorrespondto modesof thecontinuousequations.They haveanoscillatorystructure

6 90 0.8

7 75 0.6

8 63 0.4

9 54 0.3

TABLE 1. GRAVITY-WAVE PHASESPEEDSAND EQUIVALENT DEPTHSFOR THE VERTICAL MODES OF THE 31-LEVEL

AND 50-LEVEL VERSIONSOF THE ECMWF MODEL FORA REFERENCETEMPERATURE OF 300K AND A REFERENCE

SURFACE PRESSUREOF 1000hPa. ONLY MODES WITH GRAVITY-WAVE PHASESPEEDSFASTER THAN 50ms-1ARE

INCLUDED.

Mode numberPhase speed

(ms-1)for 31-level model

Phase speed (ms-1)

for 50-level model


for 31-level model


for 50-level model

OΦN ΦN ΦNRQ⁄ ΦNRQ⁄

6

6 κ– < 34( ) 1 κ–( )⁄< κ



ECMWF, 2002 17

with amplitudeapproximatelyproportionalto , andrepresentstandingwavesthatcanexist becauseof the

reflective nature of the upper boundary condition applied in the model (Lindzenet al., 1968).

It canbeseenfrom Table1 thatonly a limited numberof modesareassociatedwith gravity-wavephasespeedsof

theorderof 50ms-1 or more.It is foundnecessaryto initialize only thesemodes.Indeed,applyingthenon-linear

normal-modetechniqueto theslower, higher-ordermodesis counter-productive, asthe iterative solutionfails to

convergefor thesemodes.Theexamplepresentedin Fig. 8 illustratestheextentto which high frequency oscilla-

tionsaresuppressedby initializing up to thefive gravestmodes.Initializing four or five modesappearsadequate

in this case.Theseresultswerefor a nine-level verticalresolutionfor which thegravity-wave phasespeedsof the

fourth and fifth modes were 70ms-1 and 39ms-1 respectively, for a 300K reference temperature.

2.3 Horizontal decomposition

Thehorizontaldecompositionis illustratedmostconvenientlyfor thecaseof aglobalspectralmodelsuchasused

atECMWF. Firstly, theshallow-watermomentumequationsarerewrittenin termsof thedivergence, (givenby

(36)), and relative vorticity, :

(38)

The non-linear shallow-water equations are:

(39)

(40)

and

(41)

Theseequationsarerecastin termsof thespectralcoefficients , and in spherical-harmonicexpansions:

(42)

(43)

and

(44)

As theprognosticvariablesarerealvalued, , and arethecomplex conjugatesof and ,

andwe needconsideronly theequationsfor . Theseequationscanbeseparatedinto sets,onefor

16

( )⁄ω 0=

= NˆξN

ξN 1; θcos---------------

λ∂∂: ˆ N

θ∂∂ 9 ˆ N θcos( )–

=

∂∂ ξN 2Ω θcos;-------------------- :XN– 2Ω θsin = Nˆ– # ˆ ξ N+=

∂∂ = Nˆ 2Ω θsin ξN 2Ω θcos;-------------------- 9 N ∇2P Nˆ–– # ˆ Y N+=

∂∂ P Nˆ ΦN– = Nˆ # ˆ Z N+=

ξ+S = +

S P+S

ξ N ξ+S ( )

P+S

θsin( ) T S λ

+ S=

[∑S [

–=

[∑=

= ˆ N = +S ( )

P+S

θsin( ) T S λ

+ S=

[∑S [

–=

[∑=

P ˆ N φ+S ( )

P+S

θsin( ) T S λ

+ S=

[∑S [

–=

[∑=

ξ +S– = +

S– φ +S– ξ+

S = +S

φ +S

\ 0≥ ] 1+



ECMWF, 2002

each zonal wavenumber . We introduce scaled variables:

(45)

(46)

and

(47)

For each , weuse , and to denotecolumnvectorsof dimension with elements , and

, for .

The shallow water equations(39), (40) and(41) may then be written in the form:

(48)

where and are diagonal matrices, with diagonal elements and , where

(49)

and

(50)

is asymmetrictridiagonalmatrixwith diagonalelementszero,andoff-diagonalelements for , with

(51)

In theabsenceof rotation , andthenon-trivial eigenvectorsof aregravity waveswith frequencies

, as noted earlier.

With rotation,thebasicCorioliseffect(asoccursin anf-planegeometry)is representedby thematrix , while the

matrix representsthevariationof theCoriolis effect with latitude,theso-called“beta” effect. Purebarotropic

Rossby waves are obtained by imposing and have frequencies .

Equation(48) provides an explicit form for :

\

ξ+S ;

UWU 1+( )--------------------------ξ+

S=

= ˜ +S ;

UWU 1+( )-------------------------- = +

S=

φ+S 1

ΦN----------φ+S

=

\ ξ = ˜ φ ] \– 1+ ξ+S = ˜ +

Sφ +S U \^\ 1+ … ], , ,=

dd-----

ξ

= ˜φ

β _ 0_ β `0 ` 0

ξ

= ˜φ

# ˜ ξ

# ˜ Y# ˜ φ

+ ξ

= ˜φ

# ˜ ξ

# ˜ Y# ˜ φ

+= =

β ` β+S D +

β+S 2Ω \UVU 1+( )

---------------------=

D +1;--- UVU 1+( )Φ N=

_ a +S U \>

a +S 2ΩU------- U 2 1–( ) U 2 \ 2–( ) 4U 2 1–( )⁄=

β _ 0= = D +±

_β

= ˜ 0= β+S



ECMWF, 2002 19

(52)

Eigenvaluesandeigenfunctionsof canreadilybeobtainedusingstandardsoftwarefor matrixanalysis.Separa-

tion into Rossbyandgravity modescanin practicebeachievedsimply on thebasisof wave frequencies;theeast-

ward moving waves and faster westward moving waves are the modes to be initialized.

2.4 Implicit initialization

Althoughformally straightforwardfor globalspectralmodels,explicit normal-modeinitializationhasademanding

requirementfor computerstorageof thenormalmodesin thecaseof highhorizontalresolution.Similarconsider-

ationsapply to globalmodelsbasedon alternative horizontaldiscretizations.Two additionalproblemsmayarise

for a limited-areamodel.Firstly, themapprojectionmaycausehorizontalseparabilityto belost.Secondly, there

maybea problemin defininglateralboundaryconditionsfor thenormalmodes.This led to thedevelopmentof

implicit normalmodeinitialization.Thisis presentedherefor thecaseof aglobalspectralmodel;Temperton(1988)

introducedthemethodfor a regionalmodelusingapolarstereographicprojectionandfinite elementswith anon-

uniform grid.

We start from the formula for initialization increments given by equation(16):

Theapproachreliesonbeingableto determinethe“f ast”gravity-wavecomponent from thetotal ten-

dency andonbeingableto solve thelinearsystemfor theincrements . Thiscanbeachieved

by approximatingtheoperator . Theapproximationis to neglectthe“beta-effect” in thevorticity equation.Equa-

tion (52) is replaced by:

(53)

Equation(49) indicatesthattheapproximationwill bea goodonefor smallhorizontalscales(large ) for which

the non-zero elements of the diagonal matrix are small.

With this approximation, the linearized form of(48) gives:

(54)

(55)

and

(56)

β _ 0_ β `0 ` 0

=

∆ + 1+( ) ! dd------

+( ) dd------

+( )

= =

d⁄d( )b+( )

d⁄d( ) +( ) ∆ /+ 1+( )

0 _ 0_ β `0 ` 0

=

Uβ+S

β

ξ>>------ _ = ˜=

= ˜dd------- β = ˜ -_ ξ ` φ+( )+=

φdd------ c` = ˜=



ECMWF, 2002

The Rossby modes are stationary and non-divergent:

(57)

and, from(55), satisfy the balance equation:

(58)

From(54) and(56), a gravity mode (which has ) must have:

(59)

The expanded form of(16) is

(60)

Temperton(1989)showedthat(57), (58)and(59)canbeusedto derive formulaefor initialization incrementsthat

do not require explicit computation of gravity modes:

(61)

and

(62)

with

(63)

Thematrix is pentadiagonal,but theequationsto besolved((61)and(62)) canin facteachbesplit into

two separateequationsfor oddandevenspectralcomponentswith diagonally-dominanttridiagonalmatricesonthe

left-hand sides. Their solution is thus straightforward.

In practicethis implicit methodhasbeenfoundto give resultsthatarenegligibly differentfrom explicit normal-

modeinitializationfor mostof thewavenumberrangeof highresolutionmodels.A mixedschemehasbeenadopted

for theECMWFspectralmodel.Theimplicit methodis usedby default to initialize totalwavenumbersin therange

, with theexplicit methodusedfor , astheneglectof the termbecomesincreasinglyinaccurate

ξdd----- = ˜ 0= =

` φ _ ξ–=

= ˜ 0≠

ξ _d` 1– φ=

0 _ 0_ β `0 ` 0

∆ξ + 1+( )

∆ = ˜ + 1+( )

∆φ + 1+( )

=

ddξ

+

( )

dd=

+

( )

ddφ

+

( )

_ 2 ` 2+( )∆ = ˜ + 1+( ) _ dξd------

+( ) c` dd----- φ

+( )+=

_ 2 ` 2+( ) ` 1– ∆φ + 1+( )( ) d

d----- = ˜ +( )

β∆= ˜ +( )–=

∆ξ + 1+( ) _e` 1– ∆φ +( )( )=

_ 2 ` 2+( )

U 21> U 21≤ β



ECMWF, 2002 21

as the total wavenumber decreases.

2.5 Diabatic initialization

The formalismof non-linearnormal-modeinitialization presentedin 2.1 is general,but in practiceit wasfound

necessaryto suppressthediabatic(parametrized)componentsof theterm in earlyapplicationsof themeth-

od (WilliamsonandTemperton,1981).In particular, thelargeandrapidly varyingtemperaturetendenciesarising

from theparametrizationof convectionwerefoundto inhibit convergenceof themethod.Moreover, thebasicini-

tialization condition expressed by(9):

is inappropriatefor thewestward-moving thermaltidal wavesforcedby thedaily westwardprogressionof thesolar

heating of the atmosphere.

Theform of initialization in whichonly theadiabatictendenciesfrom themodelarecomputedat eachiterationis

referredto asadiabaticnon-linearnormal-modeinitialization. Although successfulin preventingthe growth of

high-frequency oscillationsin subsequentforecasts,adiabaticinitialization also suppressesthe slowly-varying

large-scalecirculationsthatareafeatureof thetropicalatmosphere.Two of theapproachesthathavebeenadopted

to circumvent the problem are outlined below.

2.5 (a) Full-field initialization. A diagnosisand discussionof the problemsof adiabatic initialization has

beengivenby Wergen(1987),who alsodescribedthediabaticnon-linearnormal-modeinitialization schemein-

troducedoperationallyat ECMWF in 1982.This schemewasusedin conjunctionwith theCentre’s OptimumIn-

terpolationanalysisuntil the latter was replacedby a variationaldataassimilationschemeearly in 1996.The

diabatic initialization scheme was modified in 1986 to include the treatment of tides specified below.

Theapproachinvolvesestimatingasteadylarge-scalediabaticforcing.This is appliedasatendency termthatdoes

notchangeduringtheiterationsof thenon-linearnormal-modeinitialization,therebyavoidingtheproblemof non-

convergence.The tidal problemis dealtwith by subtractinganestimateof the tidal tendency from theadiabatic

tendency. The steps involved are:

• Performinga short forecast(~2h) from the uninitialized analysis,using all parametrizationsbut

suppressing the diurnal cycle in the radiation scheme;

• Time-averaging the diabatic tendencies computed at each timestep of this forecast;

• Projecting the time-averaged tendenciesonto gravity modes, keeping only low-frequency

components (with periods>11h);

• Subtracting tidal tendencies computed from a (10-day) time series of the most recent analyses;

• Adding the resulting fixed filtered tendenciesto the adiabatic tendenciescomputedfor each

iteration of the initialization procedure;

• Computingthe initialization incrementin the usualway, using the modified tendenciesfrom the

preceding step.

This approach, though rather cumbersome, worked well in practice.

2.5 (b) Incremental initialization. Thesimpleralternative approachof incrementalinitialization (Puri et al.,

1982;Ballish et al., 1992)wasadoptedwhenECMWF moved to thevariationaldataassimilationscheme.This

approachdoesnot necessarilyrely on usingthenormal-modetechniquefor initialization. In particular, it canbe

used in conjunction with the digital filtering technique described later.

( )

dd-----α 0=



ECMWF, 2002

We illustratetheapproachherein its basicform. Let denotea backgroundstate,andasbefore theunini-

tializedstateand the initialized state.Let denotethe resultof anadiabaticinitialization of . The

incremental initialization scheme is then:

(64)

Provided the differencesbetween and aresmall, the schemepreservesthe large-scalediabaticbalance

presentin thebackgroundfield. However, if a componentis handledpoorly by theinitialization procedureandis

improvedin comparedwith , thethermaltide for example,theimprovementin maynotcarrythrough

to the initialized analysis .

As appliedin a conventionaldataassimilationsystem,the state is the short-rangeforecastthat providesthe

backgroundfor theanalysis, is theresultof theanalysisand is thestartingpoint for thenext forecast.The

approachhas,however, wider applicability. (64) couldalsobeused,for example,to provide initial conditionsfor

aforecastfrom ananalysisproducedusingadifferentdataassimilationsystem,suchasthatof adifferentforecast-

ing centre.In thiscase, wouldbethe(uninitialized)analysisproducedby astandarddataassimilationusingthe

forecastmodelfor which initialization is required,and would betheanalysisproducedby thedifferentdata

assimilation system.

3. CONTROL OF GRAVITY WAVES IN THE ECMWF VARIATIONAL DATA ASSIMILATION SYS-

TEM

Thebasicvariationaldataassimilationproblemis to determinethemodelstate thatminimizesascalarcostfunc-

tion . comprises three elements:

(65)

where is thebackgroundcostfunctiondefinedin termsof thedeviation of theanalysisfrom thebackground

state and is theobservationcostfunctiondefinedin termsof thedeviation of theanalysisfrom theobser-

vationsin thecaseof three-dimensionalvariationalassimilation(3D-Var). In thecaseof four-dimensionalassimi-

lation (4D-Var), is theobservationcostfunctiondefinedin termsof thedeviation from theobservationsof a

forecastfrom thestate . Thetwo elements and arediscussedin thecompanionlecturenotesonDataas-

similationconceptsandmethods. Theprimarycontrolof gravity wavesin thedataassimilationcomesin general

throughamultivariateformulationof the term,althoughin 4D-Varacontributioncancomethroughthemodel

integration that is implicit in the term .

Theterm representsadditionalconstraintsontheanalysis.Thesecouldincludephysicalconstraints,for exam-

plethattherelativehumiditybebetween0 and100%,or aconstraintonhighfrequenciesbasedondigital filtering.

In the current implementation at ECMWF, is given by

(66)

where denotestheprojectionof the tendency ontogravity wavesand denotesa simple

energy-basednormdefinedby a weightedsumof squaresof spectralcoefficients.Theoverall weightingfactor

wasadjustedduringthedevelopmentof ECMWF’s3D-Varsystemto removeoscillationsin surfacepressure(typ-

ically a fraction of an hPa in magnitude) that were found to occur in the absence of (Courtier et al., 1998).

f '&)( g U ( )

( fhg U ij&( ) g U f( )–+=

'& f'& f '&

(f

'& )(

f'&

f klm

+ +=

ff k

k f k

fkm

m

mε d

d------

dfd---------

–2

=

d d⁄( ) d d⁄( ) 2

ε

m



ECMWF, 2002 23

Examplesof theevolutionof thecostfunction andits threecomponentsover70 iterationsof a3D-Varminimi-

zationarepresentedin Fig. 11 . Theupperpanelshows how in absoluteterms makesonly a tiny contribution

to theoverall costfunction.Thebackgroundstate is usedasthestartingvaluefor theminimization,and

and arethusinitially zero. is subsequentlydecreasedsubstantially1, at theexpenseof someincreasein

andaslight increasein . Theplotsin thelowerpanelof Fig.11 indicateanincreasein small-scalegravity-wave

activity when is excludedfrom theminimization(but computedasadiagnostic).In thiscase growsto about

twice the valuethat developswhenit providespart of the constrainton the analysis.It shouldbe notedthat the

elementsof thecostfunctionaredefinedglobally;despiteitssmalloverallvalue mayprovideanimportantlocal

constraint, close to steep orography in particular.

Thevariationaldataassimilationschemeis implementedin anincrementalform in which theminimizationis car-

ried out at a lower resolutionthanthat of the backgroundforecast.Until March1999,initialization wasapplied

twice in theprocedure.Thelow-resolutionanalysisrequiresaninterpolatedlow-resolutionbackgroundfield, and

thiswasinitializedapplyingadiabaticnon-linearnormal-modeinitializationfor scaleswith . Then,oncom-

pletionof thelow-resolutionanalysis,incrementaladiabaticnon-linearnormal-modeinitialization wasappliedin

forming the high-resolution analysis :

(67)

Here is the incrementof the low-resolutionanalysis(the differencebetweenthe low-resolution

analysisandthelow-resolutionbackground)interpolatedto thehighresolution.Thisincrementalinitializationwas

appliedoperationallyonly for scaleswith from May 1997onwards,for theprimarypurposeof adjusting

the low-resolutionanalysisto thehigh-resolutionorography. It was,however, appliedto all scalesin theoriginal

operational implementation of 3D-Var, as discussed further below.

1. The sharp fall near iteration number 30 is due to the initiation of variational quality control.

m

f fm k f5mm m

m

U 20≥

nn fog U f nδ( ) p q)→+( ) g U irf( )–+=

nδ( ) p q)→

U 20≥



ECMWF, 2002

Figure 11. Evolution of the cost function and its components , and during the minimization in a

standard cycle of 3D-Var (upper panel) and plotted on a logarithmic scale (lower panel) both for the standard

cycle (solid) and for a modified cycle (dashed) in which was calculated for diagnostic purposes but not

activated in the minimization.

Someexamplesof thetimeevolutionof thesurfacepressurearepresentedin Figs.12and13 . Plotsareshown for

thetwo points40oN 90oW and30oN 90oE usedto illustrateresultsfrom earlierstudiesof initialization in Figs.7

and8 . Theforecastswerecarriedoutafterthreecyclesof 3D-Varusingtheinitializationconfigurationin question.

Fig. 12 shows theimpactof excludingtheconstraint . Both forecastsshown werefrom assimilationsin which

the incrementalinitialization wasappliedon all scales,andthis is evidently sufficient to preventhigh-frequency

gravity-wave oscillationswhether is activatedor not. Suppressing giveslittle changeat 40oN 90oW over

theGreatPlains,but hasmodifiedthestartingvalueof surfacepressureby about0.5hPaat theHimalayanpoint.A

pronounced semi-diurnal tidal oscillation can be seen at the latter point.

k f mm

mm m



ECMWF, 2002 25

Figure 12.Surfacepressure(hPa)asafunctionof timefor acontrolforecast(solid)andaforecastfollowing three

cycles of 3D-Var in which was not activated (dashed), at 40oN 90oW (upper) and 30oN 90oE (lower).

Fig. 13 shows theeffectof removing theincrementalinitializationof thelargerscales( ), andof removing

initialization (andthe constraint)completely. Removing the larger-scaleinitialization allows somehigh- fre-

quency oscillationsto develop,althoughevenat theHimalayanpoint theamplitudeis barelyover 0.1hPa. Com-

pletelyremoving initializationhasa largereffect,especiallyin thefirst few stepsat theHimalayanpoint.Gravity-

wave oscillationsareneverthelessmuchsmallerthanin theforecastfrom theuninitializedanalysisshown in Fig.

7 , presumablybecausethepresentforecastscomefrom aconsistentandmuchmoremoderndataassimilationsys-

tem.

Furtherexaminationof theseissueswascarriedoutaspartof thedevelopmentof the50-level versionof themodel

which becameoperationalin March1999.It wasfoundthat continuesto play a smallbut usefulrole,but that

theinitializationstepscouldbeeliminatedwithoutsignificantdeteriorationof analysisandforecastquality. As the

amplitudesof theinternalmodesvaryapproximatelyas for small , eliminationof theinitializationsteps

mU 20<m

m

16

( )⁄6



ECMWF, 2002

avoided a problem of large initialization increments close to the top of the 50-level model.

Figure 13. Surface pressure (hPa) as a function of time for a control forecast (solid) and for forecasts following

three cycles of 3D-Var with no large-scale initialization (dashed) and no initialization at all (dotted), at 40oN

90oW (upper) and 30oN 90oE (lower).

Theremoval of theincrementalinitializationof thescaleswith wasimplementedoperationallyatECMWF

in May 1997at thesametimeasachangeto thebackgroundterm , moving from theformulationdescribedby

Courtieret al.(1998)(referredto as“old” ) to thatreportedby Bouttieret al.(1997)(the“new” ). Thenew

wasusedfor theforecastsshown in Figs.12and13 . Someidealizedtestscarriedoutprior to thechangepro-

vide examples of the working of the initialization and of the background constraint.

U 20< f f f f



ECMWF, 2002 27

Fig. 14 shows incrementsin 850hPa heightdueto several idealizedisolatedobservationsof this field, specified

suchthat theobserveddeviation from thebackgroundfield wasthesameat eachpoint. Resultsareshown for a

singlecycle of 3D-Var. Theold reducesto a univariateformulationin thetropics,andproduceslocalizedin-

crementsof similar magnitudeat all locations(upper-left panel).However, theincrementalinitialization (applied

toall scales)removesmostof theincrementin thetropics(lower-left panel).Thenew imposesasemi-empirical

(closeto linear)balance.It producesmoreof a large-scaleincrement,andsmallerlocal incrementsin thetropics

(upper-right panel).More of eachlocal incrementsurvives initialization. Incrementalinitialization thusplaysa

smallerrole in imposingbalanceon theanalysisin thecaseof thenew . This wasan importantfactorin the

decision to remove initialization for scales in the operational system.

Figure 14.Analysisincrementsin 850hPaheightfor asetof idealizedheightobservationsat850hPa,for the

operationalprior to May 1997(left) andthatoperationalafterMay 1997(right), with no initialization(upper)and

after incremental non-linear normal-mode initialization (left).

Close-upsof theheightincrementsat onelocationandtheassociatedwind incrementsarepresentedin Fig. 15 .

Themultivariateformulationsof theold andnew bothproduceincrementswhich arecloseto beingin geos-

trophicbalance,andinitializationchangesaremuchsmallerthanin thetropics.They areslightly smallerwith the

new than the old . The height increments are reduced by initialization and the wind increments are increased.

Thenew producesa divergentcomponentto thewind incrementat theground,ascanbeseenin theplotsfor

1000hPashown in Fig. 16 . Theincrementin divergenceis shallow, andsurvivesinitializationbecausethelatteris

applied only to the first five, relatively deep, vertical modes.

f f

fU 20<

f

f f

f



ECMWF, 2002

Figure 15. Analysis increments in 850hPa height and wind for a set of idealized height observations at 850hPa,

for the operationalprior to May 1997(left) andthatoperationalafterMay 1997(right), with no initialization

(upper) and after incremental non-linear normal-mode initialization (left).

Figure 16.Analysisincrementsin 1000hPaheightandwind for asetof idealizedheightobservationsat850hPa,

for the operationalafterMay 1997,with no initialization(left) andafterincrementalnon-linearnormal-mode

initialization (right).

Thethermaltide providesa final example.Thevariationalanalysisis ableto “draw” to thetidal signalpresentin

thesurfacepressureobservations,but thesignalis not fully retainedin theensuingforecast.Theanalysisthuspro-

ducesincrementswhich improve thedescriptionof thetides.A fractionof theimprovementis lost,however, if in-

crementalinitialization is appliedto large scales.Fig. 17 illustrateshow the analysisgenerallyfits betterthe

f

f



ECMWF, 2002 29

surface-pressureobservationsfrom afrequentlyreportingtropicalislandstationwhentheinitializationis restricted

to scales .

Figure 17. Surface pressure (hPa) from 00UTC 8 February to 18UTC 14 February 1997, as observed at

SeychellesInternationalAirport (5oS,56oE; dashedline) andasanalysedat this location(solid)with (upper)and

without (lower) large-scale initialization.

4. DIGITAL FILTERING

Themethodof digital filtering providesanapproachto initialization that is conceptuallysimpleandeasierto im-

plementthannon-linearnormal-modeinitialization.It involvesgeneratingasequenceof modelfieldsandthenap-

plying afilter to theresultingtimeseriesfor eachmodelgrid-pointandvariableor eachmodelspectralcoefficient.

Thefilter is chosento reducetheamplitudesof high-frequency componentsof thetimeseriesto acceptablelevels.

The forecast is then run from an appropriate point within the filtered time series.

U 20≥



ECMWF, 2002

4.1 Adiabatic, non-recursive filtering

We considerfirst the simplestcaseof adiabatic,non-recursivefiltering. We denotethe uninitializedanalysisby

. A forward adiabatic integration is carried out for timesteps to generate the set of values:

where denotesthemodelstateafter timesteps.Also, a backwardadiabaticintegrationof thesamelength

is carried out to generate values:

The filtered initial state, or initialized analysis, is then given generally by:

(68)

where

(69)

Theoriginal applicationof digital filtering for initialization by Lynch andHuang(1992)useda modificationof a

basic filter defined by:

(70)

This representsthediscreteequivalentof thefilter of acontinuousfunctionthatleaveslow-frequency components

unchangedbut removeshigh-frequency componentscompletely, multiplying a fourier component by

, where

(71)

The modified “Lanczos” filter was defined by:

(72)

The term

0( ) #

1( ) 2( ) … 2( ), , ,

1+( ) U

1–( ) 2–( ) … 2–( ), , ,

s 0( ) g U ir 0( )( ) 1t--- t+ 1+

( )

+ 2–=

2∑= =

t t++ 2

–=

2∑=

t+

U υm∆ ( )sin U π

----------------------------=

υ ( )expuυ( )

uυ( )

1 υ υm

≤

0 υ υm

>

=

t+

U π # 1+( )⁄[ ]sinU π # 1+( )⁄-------------------------------------------

U υ

m∆ ( )sin U π

----------------------------

=

U π # 1+( )⁄[ ]sinU π # 1+( )⁄-------------------------------------------



ECMWF, 2002 31

in (72) provides what is known as a Lanczos window.

Thefilterswhosecoefficientsaregivenby (70)and(72)havethepropertyof leaving thephaseof asinusoidalwave

unchanged(apartfrom apossible180o shift) while reducingtheamplitudeof thewave.Thereductionin amplitude

is shown asa functionof wave periodin Fig. 18 . Thecalculationis for a cut-off period( ) of six hours,a

span( ) alsoof six hoursanda timestep( ) of 15 minutes.Thesolid line denotestheamplituderesponse

for the continuousfilter . The basicfilter (dashedline) exhibits the familiar Gibbsoscillations,which are

greatlyreducedbyapplicationof theLanczosfilter (dottedline).TheLanczosfilter giveslessattenuationof longer-

period waves, but weaker filtering of waves with periods just shorter than six hours.

Theimpactonwavesof unit inputamplitudeandperiodsof 3, 6, 12and24hoursis shown in Fig. 19 . Theampli-

tudesof the6-, 12- and24-hourperiodwavesarereducedlesswhentheLanczoswindow is included.Thephase

of the3-hourwave is reversedby thebasicfilter. It hassmalleramplitudeandno phasereversalwith theLanczos

window.

An example of the noise reduction found when

andHuang(1992)appliedthemethodto a versionof theHIRLAM modelis presentedin Fig. 20 . TheLanczos

filter wasusedwith six-hourcutoff andspan,andthemodelwasrun with a six-minutetimestep.Fig. 20 shows

thatdigital filtering initializationreducednoisemoreeffectively thanthenormal-modeinitializationschemedevel-

oped for HIRLAM. Other diagnostics confirmed the success of the digital filtering approach.

Figure 18. Amplitude of the filtered wave as a function of wave period for an input sinusoidal wave of unit

amplitude.Theidealizedcontinuousfilter hasa6-hourcutoff, andthecorrespondingbasicdiscretefilter andbasic

filter modified by a Lanczos window are shown for a 15-minute timestep and 6-hour span, following Lynch and

Huang (1992).

2π υm

⁄2# ∆ ∆ u

υ( )



ECMWF, 2002

Figure 19.Theresponseof wavesof periods3,6,12and24hoursto thebasicfilter (upper)andthefilter modified

by the Lanczos window (lower).



ECMWF, 2002 33

Figure 20.Evolutionof themeanabsolutesurfacepressuretendency (hPa /3h) in 24-hourforecastsstartingfrom

an uninitialized analysis (solid line) and from analyses initialized using non-linear normal-mode initialization

(dotted) and non-recursive digital filtering (dashed), from Lynch and Huang(1992).

4.2 Diabatic, recursive filtering

Thebackward integrationusedto generatethevalues in the approachdescribedin the

precedingsubsectioncannotbecarriedout usingparametrizationsof irreversiblephysicalprocesses.Useof a re-

cursivefilter offersonewayto usethedigital filtering methodfor diabaticinitialization(LynchandHuang,1994).

Considerthesequenceof values of modelvariablesfrom consecutivetimestepsof afore-

caststartingfrom theuninitializedanalysis , possiblyincludingdiabaticandfrictional processes.A recursive

filter of order is defined in general by the values and the values

in the following expression for the filtered value at step :

(73)

Theprocessis startedby applyinglower-orderfilters to computevaluesof for . A non-recursive im-

plementation of this filter is set out inAppendix C.

We illustrate the caseof the Second-orderQuick-Startfilter presentedby Lynch and Huang(1994).This has

and:

(74)

with

(75)

and

1–( ) 2–( ) … 2–( ), , ,

0( ) 1( ) … +( ) …, , , , 0( )

# # 1+ ; 0( ) ; 1( ) … ; 2( ), , , #v 1( ) v 2( ) …v 2( ), , , s +( ) U

s +( ) ; ( ) 1+ –( )

0=

2∑

v ( )s + –( )

1=

2∑+=

s +( ) U #<

# 2=

w + 1+( ) ; 0( ) + 1+( ) ; 1( ) +( ) ; 2( ) + 1–( )+ +( )v 1( ) w +( ) v 2( ) w + 1–( )+( )+=

w 0( ) 0( )=



ECMWF, 2002

(76)

If we define:

(77)

and

(78)

the filter coefficients are given by:

(79)

(80)

(81)

and

(82)

The upperpanelof Fig. 21 shows the input andfiltered waves for a 12-hourinput period,a cutoff frequency

( ) of threehoursanda timestepof 15 minutes.Thefilter causeslittle reductionin amplitudeof this wave,

but introducesadelayor phase-lagof a little morethanhalf anhour. A verysimilardelaycanbeseenin thelower

panel ofFig. 21 for waves with longer periods.

w 1( ) ;x 1( ) 1 ;–( ) 0( )+=

µm υ

m∆ 2

------------ tan=

y µm

1 2+=

; 11 µ

m+

--------------=

; 0( ) 12--- ; 1( ) ; 2( )

y1 y–-----------

2= = =

v 1( ) 21 y–1 y+------------

=

v 2( ) 1 y–1 y+------------

2–=

2π υm

⁄



ECMWF, 2002 35

Figure 21. The input wave and filtered output wave for 12-hour input period and a second-order recursive filter

(upper) and the filtered output for 12-, 24- and 48-hour input periods (lower).

Thefilteredresponseto input wavesof periods12,3, 1 and0.5hoursareshown in Fig. 22 , for 15- and1-minute

timesteps.Theamplitudeof thewavewith 3-hourperiodis reducedby about30%,andit too is delayedby a little

over half anhour. Waveswith shorterperiodsaredampedconsiderablyover thefirst houror so.Thatwith half-

hour period is soon damped completely in the case of the 15-minute timestep.

Therapidinitial dampingof short-periodwavesandtheuniformity of thephase-lagsfor thelonger-periodwaves

meansthataninitializedforecastmaybesuccessfullylaunchedby applyingthesecond-orderfilter for aspanof an

houror two,andthensettingthemodel’sclockbackby half anhouror soto accountfor thedelay, beforeextending

theforecastfrom theend-pointof thefilteredsequence.Theeffectof applyingthisprocedureon thelevel of noise

in a forecastusingtheHIRLAM systemcanbeseenby comparingthedashedandsolid curvesin Fig. 23 . The

procedureis evidentlysuccessful,andis sufficient for mostforecastingpurposes.It doesnot,however, provide in-

itialized conditionsat theanalysistime or for thefollowing houror so,suchasmaybeusefulfor diagnosticpur-

posesor physical initialization. The alternative of simply assumingthat the filtered valueat the endof the span

appliesat the time of theuninitializedanalysisis shown by thedottedcurve in Fig. 23 . This introducesa phase

shift of about an hour, the effect of which can be reduced by adopting the incremental approach presented in2.5.



ECMWF, 2002

Figure 22. The filtered output wave for 12-, 3-, 1- and 0.5-hour input periods and a second-order recursive filter

with 15-minute timestep (upper) and 1-minute timestep (lower).



ECMWF, 2002 37

Figure 23. Evolution of the mean absolute surface pressure tendency (hPa /3h) over six-hour forecasts starting

from an uninitialized analysis (solid line) and from using two types of recursive digital filtering schemes, from

Lynch and Huang(1994).

4.3 Diabatic, non-recursive filtering

Anotherapproachto diabaticinitialization is to carryout aninitial backwardadiabaticintegrationover a time in-

terval followedby a forwarddiabaticintegrationoveraninterval , andthento applynon-recursivefiltering

to thesequenceof valuesfrom thediabaticintegration.Lynchetal.(1997)describeaparticularlyefficientvariation

of thisapproachthathasbeenusedto initializebothlimited-areaand(in incrementalform)globalmodelsatMétéo-

France.

Theprincipalcostof digital filtering initializationis thatof themodelintegration.Improvedefficiency arisespartly

from usingaDolph-Chebyshev filter (Lynch, 1997)thatrequiresashorterspanto achievethesamedegreeof noise

reductionastheLanczosfilter. A furthergaincomesfrom filtering theresultsof thebackwardadiabaticintegration.

This yieldsa filteredvalueat a time prior to theanalysistime,andthis providesinitial conditionsfor a dia-

baticforecastover aninterval . Theinitialized analysisis derivedfrom a non-recursive filtering of this diabatic

forecast.

Fig. 24 shows thatsimilar levelsof noisereductionareobtainedby usingtheLanczosand(shorter-span)Dolph-

Chebyshev filters without filtering the backward adiabaticforecast,andby usingthe Dolph-Chebyshev filter on

both the backward adiabatic and (half-length) forward diabatic integrations.

A disadvantageof thisgeneralapproachto diabaticinitializationis thattheinitializedfieldsaresubjectto errordue

to changesbroughtaboutby diabaticprocessesover thefirst half of the forward integrationthat is filtered.This

error is reducedby usingfilters thatneeda shorterspanto beeffective,andby thefiltering of thebackwardinte-

gration, which enables the length of the diabatic forecast to be halved.

323

32⁄3



ECMWF, 2002

Figure 24.Evolutionof themeanabsolutesurfacepressuretendency (hPa/3h) for thefirst six hoursof a forecast

startingfrom anuninitializedanalysis(solid line) andfrom analysesinitializedusingthreenon-recursivediabatic

digital filtering schemes, fromLynchet al.(1997).



ECMWF, 2002 39

APPENDIX A . DEFINITION OF OPERATORS FOR THE ECMWF VERTICAL FINITE-DIFFER-

ENCE SCHEME

Specificationof theformsfor thematrices and andthevectors and hasbeengivenby SimmonsandBurr-

idge(1981),althoughwith anotationdifferentto thatusedhere.Wepresentthegeneralcaseof areferencetemper-

aturethatvarieswith pressure,with values at the“full” levelsof themodel,for . We assume

ahybridverticalcoordinatein whichthepressureatthe“half” levels, , isdefinedasafunctionof thesurface

pressure, , with and .

The forms are:

(A.1)

(A.2)

(A.3)

and

(A.4)

Here

(A.5)

and

(A.6)

All expressions involving a pressure are evaluated for the reference surface pressure .

If the reference temperature profile is isothermal, with , (A.2) reduces to

(A.7)

γ τ δ µ

34 NO

1 2 …F

, , ,=6 N 1 2⁄+67 61 2⁄ 0=

6 p 1 2⁄+

67=

γ zN0

Ol|<

< αNO |

=

<6 N 1 2⁄+6 N 1 2⁄–-----------------

lnO^|

>

=

τ zN6 N∆6 z∆

----------κ354 z γ Nbz 1

26 z∆

------------- ~ zN 354 z 1+( )34 z–( ) ~ z 1–( ) N 354 z 354 z 1–( )–( )+ +=

δN6 N∆67 4---------=

µN < 354 N6 N 1 2⁄+-----------------

67 67∂∂6

N 1 2⁄+

< 354 z 16 z 1 2⁄+------------------

67 67∂∂6

z 1 2⁄+

16 z 1 2⁄–-----------------

67 67∂∂6

z 1 2⁄–

–

zN 1+=

p∑+=

αN 1

6 N 1 2⁄–6 N∆----------------

6 N 1 2⁄+6 N 1 2⁄–-----------------

ln–O

1>

2lnO

1=

=

~ zN 6 N∆( ) 67∂∂6

z 1 2⁄+

0Ol|

>6 N∆Ol|

≤

–=

678434 N 354

=

τzN6 N∆6 z∆

----------κ354 z γ Nz=



ECMWF, 2002

and noting that

(A.4) becomes simply

(A.8)

We illustrateresultsby comparingmodescomputedfor the idealizedtemperatureprofile shown in Fig. 25 with

modescomputedfor isothermalreferenceprofileswith temperaturesof 245K and300K. The idealizedprofile is

constructedto belinear in in regionsrepresentative of thetroposphereandstratosphere.This distribution of

temperature has a mean value of approximately 245K

Figure 25. An idealized vertical profile of temperature.

Gravity-wave phasespeedsfor thefirst nine modesarepresentedin Table2 andmodestructuresfor the profile

TABLE 2. GRAVITY-WAVE PHASESPEEDSFOR THE FIRST NINE MODES OF A 50-LEVEL VERSION OF THE ECMWFMODEL FOR THE REFERENCETEMPERATURE PROFILE SHOWN IN Fig. 25 AND FOR TWO UNIFORM REFERENCE

TEMPERATURES. THE REFERENCESURFACE PRESSUREIS 1000HPA.

Mode numberPhase speed (ms-1)

for referencetemperatureprofile shown in Fig. 25

Phase speed (ms-1)for 245K reference

temperature


temperature

1 316 313 347

16 p 1 2⁄+------------------

67 67∂∂6

p 1 2⁄+

1=

µ N < 354=

6ln

O



ECMWF, 2002 41

shown in Fig.25 andfor the245Kreferencetemperatureareshown in Fig.26. Thecorrespondingmodestructures

for the300Kreferencetemperaturecanbeseenin theright-handpanelof Fig. 10 . Thegravity-wavephasespeeds

computedfor themeantemperatureof 245K arequitesimilar (within about5%) to thosecomputedfor therefer-

encestatewith varyingtroposphericandstratospherictemperatures.Modestructuresarequalitatively similar, but

evidently reflectthedifferencesin staticstability in thetroposphereandstratosphere.Phasespeedsfor the245K

referencetemperaturearesmallerthanthosefor the300Kreferenceby afactorequalto thesquarerootof theratio

of thetemperatures,andmodestructuresareidenticalfor thetwo isothermalreferencestates.This is becausethe

uniform referencetemperature appearsonly asa simplefactormultiplying eachelementof thematrix de-

fined by equation(28)

2 249 237 262

3 180 175 194

4 131 131 145

5 100 102 112

6 78 82 90

7 64 67 75

8 54 57 63

9 47 49 54

TABLE 2. GRAVITY-WAVE PHASESPEEDSFOR THE FIRST NINE MODES OF A 50-LEVEL VERSION OF THE ECMWFMODEL FOR THE REFERENCETEMPERATURE PROFILE SHOWN IN Fig. 25 AND FOR TWO UNIFORM REFERENCE

TEMPERATURES. THE REFERENCESURFACE PRESSUREIS 1000HPA.

Mode numberPhase speed (ms-1)

for referencetemperatureprofile shown in Fig. 25


temperature


temperature

O

354 M



ECMWF, 2002

Figure 26. Structures of the first nine vertical modes of the 50-level model for the reference temperature profile

shown in Fig. 25 and for a uniform reference temperature of 245K. The reference surface pressure is 1000hPa.

APPENDIX B . THE LAMB WAVE

Theverticalstructureandequivalentdepthof theLambwave canbespecifiedanalytically. A temperatureprofile

of the form:

(B.9)

enables the term in (17) and(18) to be written:

(B.10)

With alsovaryingin theverticalas , thetemperatureandsurface-pressureequations,(19)and(20),

can both be satisfied only if . Thus, if we write:

(B.11)

(B.12)

3' α

34 667 4-------- α 67

ln( )'–=

φ' < 34 67ln( )'+

φ' < 354 67ln( )'+ < 354

667 4--------

α 67ln( )'=

= '66 7 4

⁄( )α

α κ–=

< 3 4 6 7ln( )'

P ˆ=

9 '66 784--------

κ– 9 ˆ=



ECMWF, 2002 43

(B.13)

and

(B.14)

equations(17) to (20) reduce to the shallow water equations:

(B.15)

(B.16)

(B.17)

The phase speed of the plane wave in the absence of rotation is thus .

Since and , where is thespecificheatat constantvolume,thephasespeedmaybe

written . This may be recognized as the speed of sound in a gas of temperature .

With height, , astheverticalcoordinate,theverticalstructure becomes . Wave

energy densityvariesas . Horizontalwindsandtemperaturethusincreaseexponential-

ly with increasingheight,but waveenergy densitydecreasesawayfrom theground.Theverticalvelocityvanishes

identically. This may be seen by writing, to first order in wave amplitude:

(B.18)

The vertical velocity, , is then given to this order by:

(B.19)

and are given by:

(B.20)

and

(B.21)

Using(B.9) and(B.10), (B.21) becomes:

: '667 4--------

κ– : ˆ=

= '667 4--------

κ– = ˆ=

9 ˆ∂∂------ 2Ω θsin : ˆ 1; θcos

---------------λ∂

∂ P ˆ–=

:ˆ∂∂------ 2– Ω θ 9 ˆsin

1;--- θ∂∂ P ˆ–=

P ˆ∂∂-------

< 3541 κ–------------ = ˆ–=

< 354( ) 1 κ–( )⁄

κ <D ?⁄= D ? D<+= DD ? D⁄( ) < 34 34 66784

⁄( ) κ– Q κ( ) < 354( )⁄( )exp

Q 1 2κ–( ) < 354( )⁄–( )exp

0

6( ) '+=

DD -------

′∂∂------- ω+ ′

d 0

d6--------==

0

'

0

< 34Q-----------

67 46--------

ln=

'1Q--- φ' < 354 67

ln( )'+( )=



ECMWF, 2002

(B.22)

and substituting(B.20) and(B.22) into (B.19):

(B.23)

The right-hand side of(B.23) is equal to zero by virtue of equation(19).

APPENDIX C . THE NON-RECURSIVE IMPLEMENTATION OF THE RECURSIVE FILTER

Consider the recursive filter defined in general by equation(73):

Define a vector of dimension by

(C.24)

and define similarly.

Let and denote matrices whose rows are given by and for .

The non-recursive form of the recursive filter is then:

(C.25)

and can be constructed inductively:

(C.26)

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'< 3 'Q κ----------=

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3'∂∂--------

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+( ) _+( ) +( )=

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the control of gravity waves in data assimilation 27 …...the control of gravity waves in data...

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