properties of equations - ecmwf · 2 . observed behaviour 3 . toy problems 3.1 introductory remarks...

Meteorological Training Course Lecture Series

ECMWF, 2002 1

Properties of the equations of motion

By Mik e Cullen

European Centre for Medium-Range Weather Forecasts

Abstract

Somegenericpropertiesof thenonlinearequationsof fluid flow aredemonstratedwith simpleillustrativeproblems.Propertiesof the shallow watermodelarethendescribed,andthe solutionsshown to be closeto that of a ‘balanced’approximationtothem.

In threedimensions,thegeneralisationof theconceptof ‘balance’leadsto modelsfrom whichsoundwaveshavebeenfiltered,in particular the incompressibleand anelasticmodels.Propertiesof their solutionsare described.In particular, analyticsolutionof theequationsrequiressolutionof anelliptic problem,which thusalsohasto besolvedin numericalmodelsusingtheequations.Thehydrostaticequationscanbesolvedwithoutsolvinganelliptic problem,but it is shown thatthismeansthatthesolutionsbreakdown for weakstratification.Useof thehydrostaticapproximationin numericalmodelsrequiresuseof anumerical equivalent of a non-hydrostatic pressure to ensure stability.

Operationalmodelsaremorecorrectlyviewedassolvingspace-timeaveragesof theequations.Both EulerianandLagrangianaveragingproceduresareillustrated.In particular, both suggestthat the averagedvariablerepresentingthe fluid trajectoryisbest treated as different from that representing the momentum.

Averagedequationscanberelatedto filteredmodelsin whichall inertia-gravity wavesareremoved.While suchmodelsdonotgive a completedescriptionof the atmosphere,sincethey excludereal waves,they candescribethe motionsthat arewell-resolvedandpredictableby operationalmodels.Theirpropertiesarethususefulin designingmodels,particularlytheway thatthe computation of the resolved flow is related to the sub-grid models which parametrise the unresolved motions.

Keywords: Nonlinear EquationsAveraging Balance

Table of contents

1 . Introduction

2 . Observed behaviour

3 . Toy problems

3.1 Introductory remarks

3.2 Examples

3.3 Two-dimensional incompressible Euler equations

4 . Shallow water equations

4.1 Basic properties

4.2 Properties of ‘slow’ solutions

5 . Three dimensional equations

5.1 Basic equations, and filtering of fast waves

5.2 Solution of the inviscid incompressible equations

5.3 Practical implications of solution procedure

5.4 The hydrostatic approximation


2 Meteorological Training Course Lecture Series

ECMWF, 2002

6 . Averaged equations

6.1 Averaged equations, and their approximation by balanced models

6.2 Computations of averaged solutions.

6.3 Balanced models

6.4 Structure of the solution of the semi-geostrophic model

7 . Summary

REFERENCES

1. INTRODUCTION

In operationalweatherforecastandclimatemodelswe have to solve theequationsof fluid motionandthermody-

namics,subjectto boundaryconditionsat the Earth’s surfaceandprescribedexternal forcing due to radiation.

While the basiclaws governingdynamicsandthermodynamicsarewell-known, andthe laws governingphase

changesbetweenwatervapour, liquid water, andice arewell established,many compromiseshave to bemadein

applyingtheselawsin anoperationalmodel.Thecurrentoperationalversionof theECMWFmodelhasaresolution

of about40kmin thehorizontaland60 levels in thevertical.Realprocessesoccuron scalesdown to millimetres

or less,andregionsof theatmospherewheresmallscaleprocessesareimportantareoftenhighly concentrated,for

instancein convectiveupdraughts.Thusacoarsescaleaverageof theflow maybequitemisleading,anaverageof

a localised updraught looks like a smooth wave.

Theselecturesconcentrateonrepresentingtheequationsof motionandthermodynamics.Observationsshow that,

qualitatively, theatmospherebehavesin asimilarwayall thetime.Weathersystemsmovearound,andtheoverall

level of activity varieswith theseason.A modelhasto representtheevolutionof individualweathersystemsaccu-

rately, andalsoto maintainthestatistical‘steadystate’which ensuresthat it alwaysrepresentsa possiblestateof

the atmosphere after the limits of deterministic predictability are passed.

No modelcanbeaccuratein all respects,andcompromisesarenecessary. In theselectures,we identify theprop-

ertiesof theequationswhich control thelong-termbehaviour of thesolutions.Thesewill beimportantaspectsto

treat accurately in numerical models.

2. OBSERVED BEHAVIOUR

Themostusefulway to gain informationaboutthesolutionsof theequationsof motionandthermodynamicsis to

studyobservationsof theatmosphere.Satellitepicturesshow largescalecloudpatterns,Figure1, whichoncloser

examinationcontainmoreandmorefine-scaledetail,Figure2. Thus,asnumericalmodelsreachhigherandhigher

resolution,new phenomenawill appearall thetime.Therewill alsoalwaysbefeaturesthatarenot well resolved,

andthusrepresentedinaccurately. It maywell bebeneficialto excludethemfrom themodeluntil they canbetreat-

edaccurately. Thusthefundamentalproblemis to maintainaccuratetreatmentof well-resolvedfeatures,while ex-

cluding those not treated accurately and representing their bulk effect by a parametrisation.



ECMWF, 2002 3

Figure 1. METEOSAT visible picture covering North Atlantic and Europe.

Figure 2. AVHRR visible picture covering the UK at a similar time toFigure 1.

The separationproblemwould be easyif therewasa cleardifferencein spaceandtime-scalebetweendifferent



ECMWF, 2002

phenomena.However, observationsshow clearlythatthis is not thecase.Figure3 is aclassicalsetof observations

from commercialaircraft,dueto GageandNastrom(1986,Fig.2).Theenergy spectrumis essentiallyacontinuous

functionof horizontalscale,thereis no ‘spectralgap’ which would make modellingeasier. Thesituationis sum-

marisedin Figure4, dueto Smagorinsky(1974),which shows typical spaceandtime scalesof atmosphericphe-

nomena.Whateverresolutionis chosenwill cutacrosstheactivescalesof somephenomena,resultingin inaccurate

treatment.

Figure 3. Wavenumber spectra of zonal and meridional velocity and potential temperature computed from 3

groups of flight segments. The meridional wind spectra are shifted one decade to the right and the potential

temperature spectra two decades. FromGageand Nastrom (1986, Fig.2)

Observationsalsodemonstratethattheequationshaveverynon-smoothsolutions.Figure5 is aballoonobservation

of Brunt-Vaisalafrequency, essentiallystaticstability, which shows very largevariationson thesmallestscalere-

solvableby theballoon.This illustratesthatdifferentiatedquantities,in particular, will notberobustdiagnosticsof

atmosphericbehaviour, andnot besafequantitiesto usein a numericalmodel.Anotherdifficulty is thatthesmall

scalevariationsarenot universal,socannotberegardedashomogeneousturbulencein spectralspace.Figure6 is

anexampleof thewind tracefromananemometer. Thereareintensefastfluctuationsall thetime,muchlargerwhen

thewind isstronger. Superposedonthesearelargerscalechanges.Ideally, wewouldwishtopredictthelargerscale

changes,togetherwith a measureof theamplitudeof thesmall-scalefluctuations.We would not expect,or need,

to predict the small scale fluctuations deterministically.



ECMWF, 2002 5

Figure 4. Characteristic horizontal and time-scales of atmospheric motions. AfterSmagorinsky (1974).

Figure 5. Plot of Brunt-Vaisala frequency against height from a balloon observation.



ECMWF, 2002

Figure 6.Anemographtracefor BellambiPointon26December1996(wind speedin knots),takenfrom Battand

Leslie (1998), Fig. 7.

Wecanseefrom Figure1 andFigure2 thatthereis coherentlargescaleorganisationin theflow, suchasthecloud

bandwith wavesextendingSWtoNEacrosstheAtlantic.Thereis alsoaregionof regularcellularconvectionsouth

of Iceland,and,in Figure2, a regulargravity wave-trainextendingacrossIrelandandScotland.It is well-known

thatmany differenttypesof organisationarepossible,dependingon theatmosphericstateandthespaceandtime-

scales examined.

Figure 7. Daily sea-level pressure maps for December 1999, from Weather Log (Royal Meteorological Society).

In operationalmedium-rangeforecasting,theprimejob is to predictthecoherentmotionsassociatedwith weather



ECMWF, 2002 7

systems.A sequenceof weathermaps,suchasFigure7, shows thattheseevolve in anirregularnon-periodicway,

but thatstatisticallythey arein steadystate.Qualitatively, weathermapslook similar from dayto day. Thegeneral

level of activity only varieswith season,which is onamuchlongertime-scalethanthatof theindividualsystems.

3. TOY PROBLEMS

3.1 Introductory remarks

Very elementarymodelsof realphenomenaareoftenlinear, sothatthey canbesolvedanalytically. Thesolutions

canbeusedto validatenumericalcalculations.Suchtestswereusedto establishthebasicruleswhich numerical

methodshave to obey to approximatesolutionsover largetime periods.Essentiallya schemewhich is consistent

with the analyticequationwill converge to the solutionprovided that it is stable(RichtmyerandMorton, 1967,

p.45).However, realisticmodelshave to benonlinear, andcannotbesolvedanalyticallyexceptin specialcases.

Thusnumericalmethodshave to beused,andtendto beassumedcorrectif thesolutionsarecorrectin thespecial

caseswhereanalyticsolutionis possible,anddonot ‘blow up’ for generaldata.However, theproofthatthenumer-

ical methodsaregiving thecorrectanswerrequiresknowledgethat theoriginal equationhassolutions,andeven

thenit maybeverydifficult to provethatanumericalmethodis stable.In particular, sayingthatamethodhas th

orderaccuracy requirestheassumptionthatthesolutionhas continuousderivatives,allowing theTaylorex-

pansion to be constructed. Spectral methods can only converge to solutions with infinitely many derivatives.

Equationswhichcorrectlydescriberealfluid flowsmusthavesolutions,sincefluid propertiesdonotdivergeto in-

finity. We canthusexpectthecompressibleNavier-Stokesequationsto besoluble(thoughthis hasnot yet been

proved),sincethey area very well validatedmodelof realfluids. However, problemsstartwhensimplifying as-

sumptionsaremade.For instance,thereis no suchthing asaninviscid fluid, but realviscosityin theatmosphere

actsonscaleslessthanmillimetresexceptin theveryhighatmosphere.Thusit is negligible onpracticalmodelling

scales,andit is naturalto neglect theviscosity, giving theEulerequations.However, it is not at all clearthat the

Eulerequationshavesolutions.It maywell bethatany realflow generatessmallenoughscalesfor viscosityto be

important,andthusequationswhich omit it will ‘blow up’. This will leadto a lossof predictability, asdefinedby

Lorenz(1969),sincethe solutionwill dependcritically on unresolvableprocesses.The distinction is easiestto

makeatrigid boundaries.TheNavier-Stokesequationswill besolvedwith ano-slipboundarycondition.However,

theonly boundaryconditionthatcanbesetfor theEulerequationsis oneof nonormalflow, with freeslip allowed

alongtheboundary. In practice,therewill alwaysbeathin boundarylayerwhereviscosityis important.It is there-

fore very dangerous to make deductions from the inviscid equations.

Anotherexampleis theuseof thehydrostaticapproximation.This is accurateenoughfor numericalmodelswith

resolutionlessthanabout10km,andis oftenusedathigherresolutions.However, wewill seethatit is veryunlikely

that theseequationshave solutions.Whenareasof weakstratificationdevelop,we will show that thenon-hydro-

static pressure is critical. Thus results deduced from hydrostatic equations will always be suspect.

We now illustratetheseissueswith somevery simpleproblemswhich containelementsfoundin thegeneralfluid

equations.

3.2 Examples

The first example is

(1)

1–

-------

2=



ECMWF, 2002

with at . The solution is

(2)

whichblowsupat . Thisexampleillustratesthegenerictendency of thesolutionsof nonlinearequationsto

collapse to singularities, as is believed to happen with vorticity in three-dimensional incompressible flow.

The second example is the one-dimensional Burger’s equation

(3)

This is written in, successively

(i) Lagrangian form

(ii) Eulerian advective form

(iii) Eulerian flux form

Equation(3) can be solved by direct integration by characteristics, giving

(4)

Givenperiodicboundaryconditions,andnon-constantinitial data,thesolutionalwaysbecomesmultivaluedasil-

lustrated inFig. 8

This is because,given , thesolutionhasvaluesboth and

at .

Oncethesolutionbecomesmultivalued,thederivativeswith respectto in eq.(3) cannotbecalculated.However,

theLagrangianequationstill makessense,asit simply statesthatparticlesmove at a constantspeed.In orderto

determinehow to continuethesolution,it is necessaryto returnto thephysicsof theproblemfrom whichtheequa-

tion was derived. There are then three options:

(i) Say that the equation breaks down, and do not attempt to continue the solution.

(ii) If the equationwas derived from the physics of particles,and there is no reasonwhy particles

shouldnot overtake eachother, acceptthesolutionof theLagrangianform of eq.(3) andabandon

the Eulerian forms. We will see that this is a very useful interpretationin two- and three-

dimensional fluid mechanics. It is not natural for a genuinely one-dimensional problem.

(iii) UsetheEulerianflux form, interpretedasdefiningmomentumbalancewhenintegratedoverafinite

interval in . Thus we have

1=

0=

11

–-----------=

1=

,( )

-------- 0

∂∂

∂∂

+ 0

∂∂

12---

∂∂ 2+ 0

=

=

=

=

,( )

,( ) 0,–( )=

1 0,( )

1

2 0,( ),

2

2

1

1

2>,>,= =

1

2

1

1

;+

2

1–( )

1

2–( )⁄= =



ECMWF, 2002 9

(5)

Figure 8. Solutions of eq.(3) for at t=0, 0.1, 0.2, 0.3

Then,afterthesolutionfirst becomesmultivalued,thesolutioncanbecontinuedasadiscontinuitypropagatingat

a speed

(6)

SufficesL andR referto valuesto left andright of thediscontinuity. Thissolutioncanbeprovedto bethelimit of

thatof aviscousversionof theequationastheviscositytendsto zero.Thusif theoriginalphysicalproblemwasa

slightly viscousflow, theinviscidequationcanbesolvedwith adiscontinuity, andall largescaleaspectsof theflow

predictedwithoutknowledgeof theviscoussub-layer. Undertheseconditionsdeterministicpredictabilityis main-

tained. This situation occurs in three-dimensional supersonic flow, but is not typical of subsonic flow.

Thenumericalmethodto beusedalsodependson thephysical interpretationof the problem.In case(ii) , semi-

Lagrangianmethodswill be accurateandEulerianmethodswill not be useable.In case(iii) , only Eulerianflux

formmethodswill givethecorrectsolution.Furthermore,sincethesolutionisdiscontinuous,thenumericalmethod

cannotbemorethanfirst orderaccurate,asastepdiscontinuitycannotbeapproximatedto morethanfirst orderby

asmoothfunction.Theskill comesin blendingmethodswhicharefirst orderaccurateatdiscontinuitieswith ones

which are of higher order accuracy elsewhere.

While this examplesuggeststhatadvectionalwaysproducesdiscontinuities,this is anartefactof theone-dimen-

sional geometry. The next example shows that in two dimensions it is possible to maintain smoothness.

∂∂

1

2

∫ 1

2---

2[ ] 1

2+ 0=

0,( ) 2π

( )sin=

2

2–

2

–-----------------------



ECMWF, 2002

3.3 Two-dimensional incompressible Euler equations

These equations, for a uniformly rotating fluid, can be written

(7)

Here, arevelocitycomponents, is thepressureand theCoriolisparameter. Wecanwrite eq.(7) in vorti-

city form as

(8)

Typical boundary conditions would be that on the boundary of a closed domain.

Given smoothinitial data,it canbe proved that thereis a uniquesolutionwhich stayssmoothfor infinite time.

Thereforeit canbeapproximatedby stablenumericalmethodsof arbitrarilyhigh ordersof accuracy, in particular

spectralmethods.It is importantto understandwhy theresultis true,becausethis will determinewhatproperties

anumericalmethodwill haveto satisfyif it is to benonlinearlystableandaccurateoverlongperiodsof integration.

A detailedaccountof theproof,with references,is setout in Gerard(1992).A moreextensive text on thesubject

is Lions(1996).Thebasicreasonthatit is possibleis becausethevorticity is conservedby advection,soany solu-

tion mustbearearrangementof theinitial data.Thekey stepis to write thevelocity in termsof thevorticity using

the Biot-Savart integral:

(9)

is thetermfrom theboundaryconditions.Thisallowsthevelocitygradientsto beestimatedin termsof thevor-

ticity gradients by

(10)

The rate of growth of the vorticity gradient can be written

(11)

--------

∂∂

–+ 0 -------- ∂∂

+ + 0

∇ u⋅ 0

=

=

=

,

ζ

+( ) ---------------------- 0

u ∂∂ψ

∂∂ψ,–

∇2ψ ζ

=

=

=

u n⋅ 0= Ω

2πψ ζ x'( ) x x'–( )ln

'

u

Ω∫+

∂∂ψ

∂∂ψ,–

=

=

∇

ζ + 1 ∇ ζ +( )

ζ +

--------------------------+ ln≤

-------∇ ζ +( ) ψ – ψ

ψ– ψ ∇ ζ +( )+ 0=



ECMWF, 2002 11

Combining this with(10) gives an estimate of the form

(12)

where growsexponentiallyin time.Thisgrowth is thesymptomof thecharacteristicenstrophy cascadeof two-

dimensionalturbulence(Leith 1983)andis associatedwith aninversecascadeof kineticenergy, andhencestream-

function, to large scales. This has been illustrated in many computations, e.g.Farge and Sadourny (1989)

Therearetwo mainimplicationsof this proof for numericalmethods.Firstly, thevaluesof thevorticity mustnot

bechangedby advection.Thisfavourstheuseof quasi-monotoneschemes.Secondly, therateof growth of thevor-

ticity gradientsmustbe restrictedto the analytic valuewhile they arewell-resolved. This is aidedin Eulerian

schemesby theuseof energy andenstrophy-conservingschemes,which control themeanscaleof theflow (Ar-

akawa, 1966),andin semi-Lagrangianschemesby usingarea-preservingtrajectorymappings.Thelatterareanac-

tive researcharea,seefor instanceLeslieandPurser(1995).Sincetheexact solutiongeneratesarbitrarily small

scalesin partsof theflow, a dissipationmechanismis requiredto remove enstrophy at thesmallestscales,even

though the analytic solution is actually smooth without requiring viscosity.

4. SHALLO W WATER EQUATIONS

4.1 Basic properties

These equations are normally written as

(13)

is thedepthof thefluid, typically chosento haveonly smallperturbationsaboutameanvalue . If theseequa-

tionsarederivedastheverticalmeanof theflow of anisentropicatmospherewith a freeupperboundary, hasto

bereplacedby in thepressuregradientterm.If they arederivedastheverticalmeanof theflow

of anisentropicatmospherebetweenrigid upperandlower boundaries, mustbereplacedby in thepres-

suregradientterm.Theequationscanalsobeappliedto a singlelayerwith differentdensityfrom therestof the

fluid. In this caseg becomesa ‘reduced’gravity, andit is likely that canbeof thesamesizeas , andeven

become zero over part of the domain (as in outcropping density layers in the ocean), seeFigure 9.

∇u2

( ) u2≤

-------- ∂

∂ –+ 0 --------

∂∂

+ + 0

∂∂ ∇ u( )⋅+ 0

=

=

=

0 κ κ, γ 1–( ) γ⁄=

γ 1–

0



ECMWF, 2002

Figure 9. Schematicdiagramof solutionof shallow waterequationswith zerodepth oversomeof thedomain.

If we linearisetheequationsabouta stateof rest,andassumesolutionsproportionalto , we find so-

lutions . The propagating solutions are inertio-gravity waves, with speed

(14)

They arepuregravity wavesif andpureinertialwavesif . In thegeneralnonlinearcase,thesolution

corresponding to the zero value of becomes potential vorticity advection.

The natureof the solutionsin any specificcasedependson the Froudenumberdefinedby ,

where is a velocity scale;theRossbynumber , where is a horizontallengthscale,andtheRossby

radius . If is suchthat , thenthegravity wavesarefastcomparedwith theadvection

speed,irrespective of horizontalscale.If the inertio-gravity waves are fast comparedwith advection

speeds, but this condition is only satisfied on large scales (much larger than 100km forU=10 ms-1).

The equations conserve the energy integral

(15)

and the potential vorticity following fluid particles, where .

The inertio-gravity waveswill alwaysbreakaftera time of order , where is a typical flow speed.When

they break,momentumbalanceis preservedwhile energy is dissipated.AccuratenumericalsolutionrequiresEul-

erianflux formschemeswhichconservemomentum.Theequationsconservethepotentialvorticity in aLagrangian

sense, until the waves break. Conservation is then lost.

If and is a mean value of , then Rossby waves can propagate. Their frequency is

(16)

where are horizontalwavenumbers.The wave speedthus increaseswith wavelength.On large horizontal

scales the last term of the denominator dominates, and the wave speed becomes independent of wavelength.

ρ! #" %$ ω &+ +( )

ω 0 ω2, 0 ' 2 ( 2+( ) 2+= =

) 2

' 2 ( 2+----------------+=

0= 0=

ω*,+

2 - 2 0( )⁄=- - /.( )⁄ .. 0( )( ) ⁄= 0

*,+1«021

1«

12--- 2 2+( ) 2+

∫

3 ζ +( )= ⁄ ζ ∂∂

∂∂

–=

) -⁄ -

β ∂∂

0≠= 0

β '–

' 2 ( 2 02 0⁄+ +

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

' (,



ECMWF, 2002 13

4.2 Properties of ‘slow’ solutions

If is constant, the solution of the linearised equations corresponding to satisfies

(17)

Thusthevelocity is geostrophicandnon-divergent.In caseswhere is small,wecanthusapproximateequa-

tions(13) by the quasi-geostrophic equations, which can be written

(18)

wherethesuffix g representsadvectionby thegeostrophicwind and is thequasi-geostrophicpotentialvorticity.

BourgeoisandBeale(1994)provedthatthereis asolutionof theshallow waterequations(13)closeto thesolution

of (18) if is small.In orderto studythebehaviour of (18), weconsidertwo limiting cases.If

thenwe canapproximate in (18) by . Theevolution equationis thenexactly equation(7) for two-

dimensionalincompressibleflow with astreamfunction . Thepropertiesof (18)arethusdetermined

by vorticity conservation. If , then(18) can be approximated by

(19)

This is called the ‘equivalent barotropic’ model.The advection term in the secondequationis small because

is only asmallpartof thepotentialvorticity. Theadvectionof thelargerpart is identicallyzero.

Thus we expect the dynamics in this regime to be much less active.

We illustratethesetwo casesby computationswith a shallow watermodelon a spherewith a grid of 288x193

points.Thevalueof is takenasuniform,sothatthetwo limiting regimes canbereproducedby

takingdifferentvaluesof , referredtoas‘deep’and‘shallow’. Themodelusedasemi-Lagrangian,semi-implicit

schemewith asvariables,sono conservationpropertieswereexactly enforced.theinitial datahaswave-

numbersbetween3 and19.Thedeformationradius correspondsto wavenumber3 in the‘deep’caseandwav-

enumber13 in the ‘shallow’ case.Thusthe regimescomputedbothhave fairly closeto . Thediagnostics

illustratedarethepotentialandkineticenergies,thevelocitygradientnorm , thepo-

tential enstrophy and the potential enstrophy gradient norm . The result for the

‘deep’ case is shown in Figure 10.

ω 0=

∂∂ ,

∂∂ ∇ u⋅,– 0= = =

- )⁄

3 –( ) 0

54----------∇2 ,6 3 -----------

–

0

=

=

3- )⁄ *,+

1. .

«,«3 –

---∇2 ψ ( ) ⁄=021

1. .

»,«

3 –( ) 0

–

0-----

∂∂

– u ∇ ---∇2 ⋅+ 0

=

=

---∇2 7 0⁄–

. . . . »,«

0 , , . . . ∇

( )2 ∇ ( )2+( )

∫ 3 2( )

∫ ∇ 3( )2

∫



ECMWF, 2002

Figure 10. Diagnostics from shallow water integration with .

Theresultsshow almostexactconservationof potentialenstrophy, indicatingthat thenumericalmethods,which

donotusepotentialvorticity asavariable,arenotintroducingdamping.They show nearconservationof thekinetic

energy (16%lossin 20days)andpotentialenergy. Sincegeostrophicbalanceis quiteaccurateon thescalesmod-

elled,thekinetic energy densityis essentially andthepotentialenergy is . Thusindividual

conservationof kineticandpotentialenergy indicatesthatthemeanscaleof theheightfield will beconserved.The

velocity gradientnorm decreasesby 58% over 20 days,showing that enstrophy conservation is not sufficient to

controlall thevelocitygradients.Theenstrophy gradientnormincreasesrapidlyover thefirst two days.After this

time,thecomputationscannotresolvethefilamentationof thepotentialvorticity field.Contourdynamicsmethods,

Dritschel and Ambaum (1997), have to be used to follow the evolution further.

Thebehaviour of thesecomputationsis exactly thatexpectedfrom thetheoryof two-dimensionalincompressible

flow exceptfor thelackof aninversecascadeof energy to largescales.This is becausetherateof changeof poten-

tial energy is given by

(20)

andthis vanishesfor non-divergentflow if thereis no flow acrosstheboundaries.Thedivergenceis very small in

shallow waterintegrationswith small,sothekineticenergy will bealmostconservedandthemeanscaleof

theheightfield conserved.In puretwo-dimensionalturbulenceasdescribedby (7), thereis nopotentialenergy; so

this constraintdoesnot operate.Theinversecascadecanonly operatein shallow waterflow with sothat

the geostrophic constraint does not operate.

0 105=

0.5 1– ∇ ( )2 0.5 2

∂∂ 2∫

8 u ∇ 1

2--- 2

2∇ u⋅+⋅

∫–=

- )⁄. .

«



ECMWF, 2002 15

Figure 11. Diagnostics from shallow water integration with .

Figure11 illustratesthebehaviour of theshallow waterequationswith . As expectedfrom equation(19),

theeffectsof advectionareweakenedandtherateof increaseof theenstrophy gradientnormis muchless.Thus

thereis nolongerastrongcascadeof enstrophy to smallscales.Thepotentialvorticity distributionnow looksmuch

like theheightdistribution,asshown in (19). Bothchangeonly veryslowly in time.Thekineticenergy decreaseis

12%, so the mean scale of the height field hardly changes.

Theseresultsillustratetheimportanceof thedeformationradius in determiningthebehaviour of ‘slow’ solu-

tionsof theshallow waterequationswith . On scalessmallerthan , thedynamicsis essentiallyvortex

dynamics.Onscaleslargerthan , thesolutionsarealmoststationary. Furtherdetailsof thesecomputationsare

givenin Cullen(2002).More detailedstudyof theseissuesis givenby FargeandSadourny (1989)andLarichev

andMcWilliams (1991).It shouldalsobe notedthat the ‘slow’ solutionsof the shallow waterequationscanbe

approximatedmuchmoreaccuratelythanby thequasi-geostrophicequations.However, thequalitativedistinction

between flows with greater than or less than remains.

5. THREE DIMENSION AL EQUATIONS

5.1 Basic equations, and filtering of fast waves

Westartwith thethree-dimensionalcompressibleNavier-Stokesequations.Thesearevalid up to heightsin theat-

mosphere where the continuum hypothesis breaks down. They can be written as follows:

0.0 5.0 10.09 15.0: 20.09Days;

0.4

0.6

0.8

1.0

1.2

1.4

Rat

io to

initi

al v

alue

Potential energyKinetic energyPotential enstrophyVelocity gradient normPot. enstrophy gradient norm

0 5000=

. . >

. - )«

. .

. .



ECMWF, 2002

(21)

Here, is theExnerpressure, thepotentialtemperatureand , where is theratio of specific

heats.Therestof thenotationis standard.Thetypicalboundaryconditionswouldbe on thelateraland

lower boundaries of a domain , and as .

This systemof equationshas5 independentevolution equations.Two of theseare soundwaves, with speed

, two areinertio-gravity waves,andonehasthetime-scaleof theadvectingvelocity. The inertio-

gravity wave frequency is approximately

(22)

where is theBrunt-Vaisalafrequency , arehorizontalwavenumbers,and is averticalscale.Fig-

ure4 shows that thereis a largescaleseparationbetweensoundwavesandmeteorologicallysignificantmotions,

which tendto have theadvective time-scale.Thereis alsoa largescaleseparationbetweentheviscousdissipation

scalesandmeteorologicalmotions.However, thereis notmuchscaleseparationbetweeninertio-gravity wavesand

meteorological motions.

In all circumstancesrelevant to weatherforecastingwe have , so that it is appropriateto studyequations

from which soundwaveshave beenfiltered.Sinceviscouseffectsarealsowell-separatedin general,we initially

studyinviscidequations.However, viscosityisalwaysimportantnearrigid boundaries.It isalsolikely thatthelocal

generationof small scaleswill make viscosityimportantin the interior of theatmosphereon occasions.We will

see how this possibility appears naturally in the analysis of the inviscid system.

Thesimplestmethodof filtering is to imposeincompressibility, asin thetwo-dimensionalcase.If viscosityis also

omitted, this gives

(23)

is determined implicitly by

(24)

Theseequationsareonly accurateif . Onlargerscales,thefinite propagationspeedof soundwavesmatters.

u --------

=<θ∇Π >? ,,–( )+ + ν∇2u

∂∂ρ ∇ ρu( )⋅+ 0

θ -------- κ∇2θ

0ρθΠ

Π @ 0⁄( )κ

=

=

=

=

=

Π θ κ γ 1–( ) γ⁄= γu n⋅ 0=

Ω ρ 0→, A ∞→

) γ ρ⁄( )=

2 B 2 C 2 ' 2 ( 2+( )+

' 2 ( 2 B 2+ +-------------------------------------------------

C 2 θ--- A∂

∂θ ' 2 ( 2, B

u )«

u --------

=<θ∇Π >? ,,–( )+ + 0

∇ u⋅ 0θ -------- 0

=

=

=

Π

∇ <

θ∇Π( )⋅ ∇ > –,–,( ) u ∇u⋅( )–( )⋅=

#. )«



ECMWF, 2002 17

However, they arenotaccuratein thepresenceof thelargebasicstateverticaldensitygradientof theatmosphere,

as is alargeterm.Usefulequationswith thesamemathematicalstructureastheincompressibleequations

are obtained by making, instead, the anelastic approximationLipps and Hemler (1982), which replaces(23) by:

(25)

where are prescribed time-independent functions satisfying the equation of

state: andthehydrostaticequation . Primesindicatepertur-

bations from the reference profiles. Eq.(24) becomes

(26)

Theseequationsare‘balanced’in thesensethatthepressureis determinedimplicitly from thevelocityfield.There

is a conservedpotentialvorticity . However, unlike thetwo-dimensionalcase,therearestill 3

independentevolution equationsleft. Theequationsarethusnot ‘balanced’in thesenseof Hoskins et al. (1985),

becausethepotentialvorticity alonedoesnot determineall theothervariables.In particular, internalgravity and

inertial waves are supported.

The limitation canbe avoidedin numericalmodelsby usinga semi-implicit approximationto thecom-

pressibleequations,asusedin theUK Met Office‘new dynamics’,Cullenet al. (1997).Thiseffectively eliminates

the soundwaves.However, thereis still a difficulty in representingthe true upperboundarycondition.Thebest

methodis acurrentresearchtopic.Inappropriatechoicesmayre-introducethecondition , whichwouldnot

be acceptable in global models, though adequate for local mesoscale models.

We will seein section6 that three-dimensionalequationswhich canbe describedin termsof the evolution of a

singlescalarcanbederivedby assumingthattypical frequenciesareslow comparedwith theinertial frequency, so

theRossbynumber is small,or thattheinternalFroudenumber , where is a heightscale,

is small.While typicalvaluesof theseratiosfor weathersystemsareabout0.1,therearesignificantregionswhere

they areO(1)or greater. Thusany systemof equationsbasedonpotentialvorticity alonewill notbeveryaccurate,

asFigure 4 suggests.

5.2 Solution of the inviscid incompressible equations

Theresultswhich wereprovedby thevorticity methodof 4.2no longerhold,sincevorticity is not conserved.We

thereforeusean alternative methodwhich is basedon approximating(25) by a semi-Lagrangian,semi-implicit

schemeasusedin theECMWFandmany othermodels,andproving thatthesequenceof approximationsconverg-

es.For simplicity, we describethemethodfor thespecialcasewhere areindependentof .

A similar method can be used for the general case, but is more complicated to explain.

Weassumethat(25)is to besolvedoverafinite timeinterval . Wewill dividethisinterval into time-steps,

andproveconvergenceas . In orderto constructanapproximationto thesolutionoverasinglestep , we

initially ignorethepressuregradienttermsin (25)andintegrateby characteristics.With constant,thisgivesafirst

guess solution for the positions of particles initially at :

∂ρ ∂A⁄

u --------

=<θ0∇Π'

>? θ' θ0⁄–,,–( )+ + 0

∇ ρ0u( )⋅ 0θ -------- 0

=

=

=

ρ0 A( ) θ0 A( ) Π0 A( ),, 0 A( )

0ρ0 A( )θ0 A( )Π0 A( )=

=<θ0∂Π0 ∂A⁄ + 0=

∇=<

ρ0θ∇Π'( )⋅ ∇ ρ0D θ' θ0⁄,–,( ) ρ0u ∇u⋅–( )⋅=

ζ +( ) ∇θ⋅( ) ρ0⁄

E. )«

E. )«

- /.( )⁄ - C2F( )⁄ F

ρ0 A( ) θ0 A( ) Π0 A( ),, A

0 G,[ ] ∞→ δ

HJILK, ,

0( )

0( ) A 0( ),,



ECMWF, 2002

(27)

Themapping to doesnot satisfythecontinuityequationor theboundary

conditions. Therefore we correct this first guess by finding satisfying

(28)

where is aconvex function.Theconvexity is becausethemapfrom to is closeto theidentity:

(29)

Thedifference representstheeffectof thepressuregradienttermintegratedover thetime-step.

Theprocedureis illustratedin Figure12 for aone-dimensionalcross-sectionwithout rotation.Thesolid linesrep-

resentfirst guesstrajectories,whicharestraightlines,andthedashedlinesrepresentthequadraticcurvesobtained

by solving (28). Thesecurvesforce themappingbackto the identity, which is theonly possiblesolutionin one

dimension with fixed boundaries.

Figure 12. Illustration of the projection of a first guess trajectory onto one that satisfies the continuity equation.

8M M,( )

0( )

0( ) ⁄ 0( )

0( ) ⁄–,+( )

Hx 0( ) δ

,( )

8M 0( )

8M–( ) δ ( )

0( )

M–( ) δ ( )

Ix 0( ) δ

,( )

sin–cos+

M 0( )

M–( ) δ ( )

0( )

8M–( ) δ ( )

Kx 0( ) δ

,( )

sin+cos+

A 0( ) N 0( )δ 1

2--- θ'

θ0-----δ

2+ +

=

=

=

=

0( )

0( ) A 0( ),,H

δ

( )I

δ

( )K

δ

( ),,x δ

( )

X x δ

( )( ) ∇O x δ

( )( )=

O x δ

( ) X x δ

( )( )

X x δ

( )( ) x δ

( ) 12--- < θ0∇Π' x δ

( )( )δ

2+ ∇ 1

2--- x δ

( )2 1

2--- < θ0Π'δ

2+

= =

x δ

( ) X x δ

( )( )–



ECMWF, 2002 19

The‘polar factorisation’theoremof Alexandrov, seePogorelov(1973,p.475),andBrenier(1991)provesthatthis

constructionis alwayspossibleunderappropriateconditions.It is equivalentto solvingaMonge-Ampereequation

(a nonlinearelliptic problem)for the trajectories.The right-handside of the Monge-Ampereequationwill be

smoothif the velocity gradientsarebounded.The result is a measure-preservingmapping

which satisfies the boundary conditions.

If this argumentis usedin thetwo-dimensionalcase,theboundon thevelocity gradientsfollows from (12). Caf-

farelli (1996)hasproved in two-dimensionsthat, undersuitableconditions,the solutionof the MongeAmpere

equationis smoothfor a smoothright-handside.It is thenpossibleto prove that . It

shouldthenbepossibleto breakany finite time interval into intervalsof length , andconstructa se-

quence of measure-preservingmappings . Provided that

, thevelocity will evolve continuouslyin time as andthediscretesequence

of mappingswill convergeto ameasure-preservingmappingsatisfying(25). Thustheresultsobtainedby thevor-

ticity argument in3.3 should be reproduced.

In threedimensions,thenecessaryboundon thevelocity gradientscannotbeproved,andit is possiblethat they

increase according to the equation

(30)

As we saw in 3.2, this hasinfinite solutionsin finite time.At presentit is a majorresearchquestionasto whether

singularitiescanbegeneratedfrom smoothinitial datain three-dimensionalincompressibleor anelasticflows.The

majority opinion is that they can.If so, it may still be possibleto follow this constructionprovided that we can

prove . However, the limit solutionmay involve particleslosing their identity and

‘mixing’, thus implying the effect of a small but finite viscosity.

A numericalmodelthatfollowsthisprocedureatfinite resolutioncanbeexpectedto bestableandconvergeasres-

olutionincreases.Thekey propertiesthatmustbesatisfiedaretheaccuratesolutionalongtrajectories,in particular

theuseof quasi-monotoneschemesto ensurethatnew valuescannotbegeneratedby advection;andtheimplicit

modificationof thetrajectoriesto enforcethecontinuityequationandboundaryconditions.Thepossiblemixing

of trajectoriesastheresolutionincreaseswill beautomaticallyhandledby theneedto averageall quantitiesto grid-

points,sono extra viscosityshouldberequired.Therequirementthat themappingfrom time-stepto time-stepis

measurepreservingis just asappropriatefor a modelsolving (25) averagedover a finite region asfor the exact

solution.Thedifferencefrom thevorticity methodof section3.3is thatthemonotonicityrequirementnow applies

to the velocity components, rather than the vorticity.

Notethat,if thesolutionsarenotsmooth,andparticularlyif mixing of trajectoriesoccurs,derivativesof theveloc-

ity field will becomemeaningless.However, quantitiessuchasvorticity anddivergencemaystill bemeaningfulin

a volume-integratedsense.ThusKelvin’s circulationtheoremandlocal massbalancerelationswould still hold.

However, thepresenceof singularitieswould preventany proof thatsolutionswereunique,sotherewill bea fun-

damentallossof predictabilityin thesenseof Lorenz(1969).Any resultsdeducedby manipulatingor differentiat-

ing the vorticity or divergence field will also be invalid.

This methodof solutioncanin principlebeappliedto thefully compressibleequations(21) provided . This

is relatedto theoreticalresults,seeLions (1996),whichshow that,if bothsolutionsexist, thereis asolutionof the

compressibleequationscloseto that of the incompressibleequations,whereclosenessis measuredin termsof

. Thecontinuityequationwhichis enforcedby theprojectionnow expressesconservationof mass,ratherthan

non-divergence.

We canusethesolutionprocedureto illustratesomeotherpropertiesof thesolution.If thefirst guesspressureis

x δ

( ) X x δ

( )( )→

x δ

( ) X x δ

( )( )– P δ

2( )=

0 G,( ) δ

x 0( ) x δ

( ) … x δ

( ) … x G( )→ →→ →→x δ

( ) X x δ

( )( )– P δ

2( )= δ

0→

dd ∇u

∇u 2=

x δ

( ) X x δ

( )( )– P δ

2( )=

)«

)⁄



ECMWF, 2002

chosen to be the hydrostatic pressure , then(27) becomes

(31)

The largestterm that contributesto is thusremoved.However, the horizontalderivativesof

may blow up. The non-hydrostaticpressureis still critical in orderto enforcethe continuity equation,even

though the total pressure may be dominated by the hydrostatic part.

If the rotationis rapid, then will be small,andthe projectionof onto a measure-

preservingmappingwill bemuchsimpler. Thuswe canexpectrapidrotationto improve thebehaviour of theso-

lution.

5.3 Practical implications of solution procedure

We summarise the implications of the solution procedure set out in the previous subsection:

(i) Quasi-monotoneschemesshould be used,as good behaviour of the equationsintegratedalong

particle paths (without the pressure gradient term) is required.

(ii) In supportof this, notethat ECMWF only obtainedsatisfactorybehaviour of their modelat T213

resolutionafterquasi-monotoneschemeswereintroduced,(Tempertonetal. 2001).Now they arein

use,satisfactoryresultsat resolutionsup to T799havebeenobtainedwithout furtherchangesto the

schemes.Experiencewith very high resolutionsimulations,suchasof bubbleconvection,is also

that monotone schemes are needed.

(iii) An elliptic problemhasto besolved to force the trajectoriesto satisfythecontinuityequationand

boundaryconditions. Implicit recalculationof the trajectoriesin semi-Lagrangianmethodsis

needed.

(iv) Eulerianmethodsshouldbedesignedto beequivalentto semi-Lagrangianmethods,i.e. to upwind

quasi-monotoneinterpolation. Note that the derivatives nominally approximatedby Eulerian

methodsmay not exist, but derivationsbasedon characteristicswill be valid. Flux form schemes

have the advantage of enforcing the continuity and boundary conditions exactly.

Wedemonstratepoint (iii) by computationsusingananelasticmodeldevelopedby P. Smolarkiewicz. Theexplicit

versionof this modelis describedin SmolarkiewiczandMargolin (1997)andtheimplicit versionby Smolarkie-

wicz et al. (1999).Theareaof integrationwas2000kmsquare.All integrationsshown used91 levelswith a300m

verticalspacingandasemi-Lagrangiansemi-implicitschemewith a5 minutetime-step.Thelowerboundarycon-

dition wasfree-slip,andno viscosityor turbulencemodelwasused.Theexampleis of flow at 10 ms-1 impinging

on the Scandinavian orography.

Figure13showstheflow atthebottomlevel of themodelandtheflow Jacobianat thebottomlevel.Theflow Jaco-

bianis theratioof thevolumeof fluid at thebeginningandendof a time-stepasinferredfrom thedeparturepoint

calculationin thesemi-Lagrangianscheme.Thevaluesshouldbe1. Theerrorsrangefrom +19%to -28%,andare

largestin theregion whereexperimentswith a shortertime-stepandmoreaccurateboundaryconditionshow the

numerical errors in the flow are greatest.

RQSM M

,( )

0( ) 0( ) ⁄

0( )

0( ) ⁄–,+( )H

x 0( ) δ

,( )8M 1

2---

∂∂ RQ( )δ

2–

0( )

SM–( ) δ ( )

0( )

M–( ) δ ( )

Ix 0( ) δ

,( )

sin–cos+

M 12---

∂∂ TQ( )δ

2–

0( )

M–( ) δ ( )

0( )

SM–( ) δ ( )

Kx 0( ) δ

,( )

sin+cos+

A 0( ) N 0( )δ

+

=

=

=

=

X x 0( ) δ

,( ) x 0( )–

RQ

xM

x 0( )– X x 0( ) δ

,( ) x 0( )–



ECMWF, 2002 21

Figure 13. Low-level flow and flow Jacobian. Contour interval 3%.

Figure14showsacross-sectionof thepotentialtemperature,includingbreakinggravity wavesforcedby theorog-

raphy. Theerrorsin theflow Jacobianrangefrom+12%to -19%,butarelessthan3%overmostof theregionwhere

the flow is smooth.

Figure 14. Cross-section of potential temperature and flow Jacobian. Contour interval 3%.

Theseexamplesshow thatthelargesterrorsarewherethetrajectoriesaremostdistorted,or whereapproximations



ECMWF, 2002

madeat thelower boundarylimit theaccuracy of thescheme.It shouldbepossibleto improve thesimulationsby

correcting these errors.

5.4 The hydrostatic approximation

Thoughthehydrostaticequationsarenormallywrittenin pressurecoordinates,for uniformity wewrite theapprox-

imation directly into(21) as

(32)

Theseequationsalmostcertainlyblow upin anirrecoverableway. Suppose is uniform,thenit will remainsofor

all time.Then is independentof and , and areindependentof . If, initially, and dependon

, thepressuregradientterm,which is independentof , will beunableto preventfluid trajectoriescolliding, as

in theexampleof 3.2. Theequationof stateconstrains , in particular, if aresmooth, is alsosmooth.

Thecontinuityequationthenrequires to tendto infinity. Thereis no known way of resolvingthedifficulty in

two dimensionsanalogousto theone-dimensionalconstructionof 3.2. Eitherviscosityor non-hydrostaticpressure

hasto be introduced.However, if and areinitially independentof , they remainso for all time andtwo-

dimensional theory can be used to show that the problem can be solved.

Scaleanalysisshows thatthehydrostaticapproximationis only valid on time-scalesgreaterthanthereciprocalof

thebuoyancy frequency, . Typical valuesare100s(aperiodof 600s)in thetroposphereand

10s(aperiodof 60s)in thestratosphere.However, in moistsaturatedneutralregionsof theatmosphere, ,

assumingphasechangesareinstantaneous.ThecurrentECMWFmodeltimestepis 900sfor ahorizontalresolution

of 40km.Thuswe canexpectto resolve thetypical periodwhenthehydrostaticequationsbreakdown at abouta

10km resolution. However, it will break down locally in weakly stratified regions before this.

In theaboveexample, . Thusthehydrostaticapproximationis invalid,andthebreakdown of thesolution

shouldberesolvedby includinganon-hydrostaticpressureratherthanviscosity. In thecurrentECMWFsemi-im-

plicit scheme,areferencebasicstatewith a largestaticstability is usedin theimplicit step.This introducesacom-

putationalnon-hydrostaticpressureof , where is thebuoyancy frequency of thereference

state.Thisis sufficientto stabilisethemodel.However, it mayintroduceerrorsinto thesimulation.Figure15shows

theeffectof changingthereferenceprofileaveragedover12forecastswith theECMWFmodel(CY23R4)atT511

resolution.

Thehydrostaticequations,unlike theanelasticequations,canbesolvedexplicitly. However, evenin theUK Met

Office explicit Unified Model (CullenandDavies(1991),a referenceprofile is usedin theshorttime-stepswhich

computethepressuregradientandverticaladvection,sothatthesamestabilisationmechanismis present.It is im-

portantto rememberthata hydrostaticmodelstabilisedthis way will not give correctsolutionswherethestratifi-

,( ) -------------------=<

θ ∂∂Π

∂∂Π,

5U ,–( )+ + 0

=<θ A∂

∂Π 0

∂∂ρ ∇+

=+

ρu( )⋅ 0

θ -------- 0

0ρθΠ

Π @ 0⁄( )κ

=

=

=

=

=

θ

A∂∂Π

∂∂Π

∂∂Π, A

A Aρ

∂∂Π

∂∂Π, ρ

A∂∂N

A

C 1– θ ∂θ ∂A⁄( )⁄= C 1– ∞→

C 1– ∞=

P C C0–( )δ

2( ) C0



ECMWF, 2002 23

cation is weak, the solutionsin theseregions will be a computationalartefact. It is thus important that the

parametrisations maintain an adequate level of stability.

Figure 15. Anomaly correlations for 12 forecasts for periods between August 1998 and September 1999 using

different reference profiles of temperature in the semi-implicit scheme.

6. AVERAGED EQUATIONS

6.1 Averaged equations, and their approximation by balanced models

Theprecedingsectionshave discussedexactequationsandapproximationsto themwhich arevalid for subsonic

flow. However, it is clearthatoperationalmodelsgreatlyunder-resolve thesolutionsof theseequations,asillus-

tratedby theobservationsreviewedin Section2. In practice,it is morerealisticto view themassolvingequations

thathavebeenaveragedin spaceandtime.Thisis well-known to large-eddymodellers,whoarealwaysdisciplined

in defining averaging scales in the problems they solve.

Figure4 illustratesthataveragingwill involve compromises.For instance,a horizontalaveragingscaleof 80km,

compatiblewith theT511ECMWF modelresolutionof 40km,cutsthroughthedominantscaleof frontal zones

andof internalandexternalgravity waves.Theplannedresolutionof 15kmachievablelaterin thedecadeis com-

patiblewith a30kmaveragingscale,whichis still in themostactivescalefor mesoscaleweathersystemsandgrav-

ity waves. Thus accurate representation of these phenomena will not be possible.

Theoryalsoshows thataveraginginvolvescompromises.Averagedequationsincludetermsrepresentingcorrela-

168 HOUR FORECASTS

AREA=N.HEM TIME=12 DATE=19980825/...VANOMALY CORRELATION FORECAST

1000 hPa GEOPOTENTIAL (%)

100W 90 80 70 60 50 40 30 20 10

T330KX100

90

80

70

60

50

40

30

20

10

T35

0KY

T330K is WORSE than T350K at the 2.0% level (t test)T330K is WORSE than T350K at the 2.0% level (sign test)Z

12 CASES

MEAN

MAGICS 6.3 styx - nac Tue Oct 30 17:13:23 2001 Verify SCOCAT

168 HOUR FORECASTS

AREA=N.HEM TIME=12 DATE=19980825/...

ANOMALY CORRELATION FORECASTV 500 hPa GEOPOTENTIAL (%)

[

100W 90 80 70 60 50 40 30 20 10

T330KX100

90

80

70

60

50

40

30

20

10

T35

0KY

T330K is WORSE than T350K at the 10.0% level (t test)\

12 CASES

MEAN

MAGICS 6.3 styx - nac Tue Oct 30 17:13:27 2001 Verify SCOCAT



ECMWF, 2002

tionsbetweenresolvedandunresolvedmotions,whichcanonly bemodelledempirically. They thusneverhavethe

fundamentalvalidity of theoriginalequations.Accuratedeterministicpredictionwith averagedequationscanonly

beexpectedif thereis aspectralgap,whichmeansthatinteractionsbetweenresolvedandunresolvedflowswill be

weak.However, Figure3 shows thatthereis nosuchgapin atmosphericdata.Accuratepredictionis alsopossible

if theaveragingcanselectresolvedandunresolvedprocesseswhichinteractweaklyfor otherreasons.For instance,

flows which arealmostentirelyrotationalinteractweaklywith flows thatarealmostentirelydivergent.This situ-

ationis only likely if theaveragingreflectsaphysically importantscaleof theflow. However, in weatherforecast-

ing, typical resolutionsarefiner thantheRossbyradius,above which therotationalconstraintdominates,but not

fine enough to resolve all the key scales associated with stratification.

If theequationsareaveragedin spaceandtimein aEuleriansense,andthesub-gridmodelis correct,thesolutions

mustbesmoothon theaveragingscalebecausetherealflow is smoothwhenaveragedto this scale.Theanalysis

of section5.2 suggeststhat this will beachieved in an incompressiblenon-hydrostaticincompressiblemodelby

usingmonotoneadvectionschemesandimplicit calculationof trajectories.Thishasbeenverifiedby computations

usingthesamemodelasusedin section5.3. However, it is importantthatsub-gridmodellingof motionswhichare

sub-gridscalebut have a majorlarge-scaleimpact,suchasconvection,needto assumethattheir input datarepre-

sentsaveragedvaluesandbe designedso that the solutionof the resolved andsub-gridmodelstogetherstays

smooth on the averaging scale.

Thereis no reasonwhy theform of theunaveragedNavier-Stokesequationshasto beretainedoncetheequations

areaveraged.For instance,thehydrostaticequationsbecomeappropriatewhentheaveragingscaleis greaterthan

about15km,astheregionsof weakstability wherethehydrostaticequationsblow up areno longerresolved.An-

otherpossibilityis discussedby GentandMcWilliams (1996),andis widely usedin oceanmodels.Averagingthe

advection term gives

(33)

assumingthatperturbationquantitiesaverageto zero.Thefinal termcanbemodelledin termsof meanquantities

as , where is theantisymmetricpartand thesymmetricpart. canbemodelledasadvectionby

a ‘bolus’ velocity, e.g.Stokesdrift or ‘entrainmentvelocity’. Thisprovidesanon-diffusiveelementto thesub-grid

model, and allows smoothness to be achieved without relying entirely on diffusion.

Ideally, we would like to averagewithout imposingsmoothness,soasto capturesharpchangessuchasthatillus-

tratedin Figure6. Onepossiblemethodis to useLagrangianaveraging,sothatweaverageoverair parcelsasthey

move in spaceandtime, but allow differentair massesto beseparatedby sharpboundaries.An exampleof this

approachissetoutbyAndrewsandMcIntyre(1978)andBuhlerandMcIntyre(1998).They show thattheLagrang-

ian mean momentum equation takes the from

(34)

is the ‘pseudo-momentum’associatedwith thewavesremovedby averaging.The trajectoryis definedby the

‘Lagrangianmean’velocity. is aperturbationpressure.Onceagainweseethatthetrajectoryis definedby adif-

ferent velocity from the momentum.

A naturalaveragingscaleis givenby theinertial frequency . Thesemi-geostrophicmodelis anaccurateapprox-

imationto (34) if . It is obtainedby neglecting andassuming is in ge-

ostrophicbalance.This model can be proved to have solutions,e.g. Cullen and Purser(1989),Benamouand

Brenier(1998),but thecondition restrictsits accuracy to scalesgreaterthan

u ∇θ⋅

u ∇θ⋅ u ∇θ u' ∇θ'⋅+⋅=

]θ θ+

] ]

u p+( ) -----------------------

=<θ∇ Π π+( ) D? ,,–( )+ + 0=

pπ

^

⁄( ) wind di rection–( ) « π u p+

^ ⁄( ) wind di rection–( )

«



ECMWF, 2002 25

the Rossby radius (Cullen, 2000). Accuracy of semi-Lagrangian advection schemes requires that

, which is a conditionof thesameform but muchlessrestrictive at opera-

tional resolutions.Thecurrentoperationalresolution,T511,is integratedwith a15minutetime-step.Theaccuracy

conditionis thus12 timeslessrestrictive thanthesemi-geostrophicconditionin middlelatitudes.It thusallows a

wider variety of solutions, such as inertial waves.

Anothernaturalscaleis definedby theconditionthat theFroudenumber is small.This requiresinternal

gravity wavesto propagatefasterthantheadvectionvelocity. In theECMWFmodelwith 60 levels,only 5 vertical

modeshave a phasespeedgreaterthan100ms-1, andonly 15 greaterthan30ms-1. Thusbalancedmodelsbasedon

fast gravity wave speeds are only likely to work with limited vertical resolution.

A finer naturalaveragingscalewould begivenby thebuoyancy frequency. As discussedin section5.4, this is not

resolvedin operationalmodels.Thuswe eitherhave to accepta mixtureof well- andpartly-resolvedmotions,or

impose a rather restrictive averaging scale.

6.2 Computations of averaged solutions.

Sincetheaveragingscalesusedin operationalmodelsdo not correspondto a physicalscaleseparation,eventhe

correctaveragedbehaviour maynot look like what is expected.We illustratewith anexampleof flow over Scan-

dinavia shown in Figure16. Thereis anorth-westerlyflow roughlyat right anglesto theScandinavianorography.

A satellitepictureat thesametime,Figure17, showspossiblewaveactivity over themountains,but skiesareclear

over theBaltic. Thusweexpectthatthesolutionof theequationswill containlargeverticalmotionsover theorog-

raphy, but only smallverticalmotionsover theBaltic. A cross-sectionfrom a forecastusingthethenoperational

modelat T319 resolutionis shown in Figure18. The cross-sectionline extendsacrossthe Baltic. However, the

modelorography, shown in thecross-section,hasadownslopewhichalsoextendshalf-wayacrosswheretheBaltic

shouldbe.Thecross-sectionshowsverticalmotionsextendingright acrosstheBaltic. However, they maywell be

arealisticrepresentationof how theatmospherewouldhaverespondedto themodelorography. Otherexperiments

(not shown) show that the model waves are not numerical artefacts.

Figure 16. Mean sea-level pressure over NW Europe for 18 March 1998.

^ ⁄( ) wind di rection–( ) δ

1–«

-_C`F⁄



ECMWF, 2002

Figure 17. METEOSAT visible picture for 18 March 1998, 1200 UTC.

Figure 18. Cross-section of vertical velocity from operational formulation at T319 and a 20 minute time-step.

Weshow thattheforecastsfrom theoperationalmodelmaywell beaccurateforecastsof theaveragedstateof the

atmosphereby usingtheresearchmodelusedin section5.3. Theproblemsolvedis exactlyasin 5.3. Wecompare

integrationswith a 10kmgrid wherethe resultsareaveragedto a 80kmgrid with resultsfrom themodelwith a

40kmgrid. Becauseof numericalerrors,wecanonly expectresultsfrom a40kmgrid to beverifiableat80kmres-



ECMWF, 2002 27

olution.Theagreementis quitegood,bothfor thelow-level flow, Figure19, andfor theverticalmotion,Figure20.

Theoperationalmodelusedto produceFigure18 hasa resolutionof about60km.Theresultsshown in Figure20

aresufficiently similar to this to suggestthat theoperationalmodelmaybegiving a reasonablepredictionof the

averagedflow. A similar comparisonwasmadebetweena 20kmmodelandthe10kmresultsaveragedto 40km.

Again there was reasonable agreement.

Figure 19. Comparison of low-level flow produced by 40km model (left) and 10km model averaged to 80km

(right)

Figure 20. Comparison of vertical velocity cross-section produced by 40km model (left) and 10km model

averaged to 80km (right).



ECMWF, 2002

6.3 Balanced models

As discussedin section5.1, theexactequationsgoverningtheatmospheresupport5 time-like solutions.Two of

thesecanbefilteredwill little lossin accuracy becausetheflow is very subsonic.However, we cannotfurtherre-

ducetheequationsto giveasystemwith onetime-likevariableby eliminatinginertio-gravity wavesandretainac-

curacy in all circumstances.This is becausethe inertio-gravity wave frequency, (22), is similar to the advection

frequency undermany circumstances.In orderto obtaina ‘balancedmodel’ with a singletime-like solution,we

mustassumethat either the Rossbynumber , the Froudenumber , or the aspectratio is

small.Theresultingequationswill havemuchsimplersolutionsthaneitherthefull equations,(21), or theanelastic

approximation(25). Typically they canbedescribedby advectionof potentialvorticity, togetherwith thesolution

of anelliptic problemto obtaintheothervariables,Hoskinsetal. (1985).However, wecanexpectthesolutionsto

be accurate approximations to those of the full equations on large scales.

An exampleis shown in Figure21. This is asequenceof ECMWFanalysesof potentialvorticity in theuppertrop-

osphereandaccumulatedprecipitation.We canseethecloserelationbetweenthemovementanddevelopmentof

thepotentialvorticity anomalywith themainareasof precipitation.It is likely thatthis correspondencewould be

retained in a balanced model, though the small scale detail would be lost.

Figure 21. Analyses of potential vorticity on the 315K surface and 12 hour accumulated precipitation ending at

analysis time.

Therearemany examplesof balancedmodels,basedon differentexpansionsin Rossbyor Froudenumbersor as-

- /.⁄ -_C`F⁄ F .⁄



ECMWF, 2002 29

pectratio.Two examplesarethebalanceequations,andthesemi-geostrophicmodel.Theformeris a muchmore

accurateapproximationin the context of the shallow waterequations,Cullen (2000),but in threedimensionsis

liableto spontaneousinstabilityMcWilliams et al. (1998).Thesemi-geostrophicmodelcanalwaysbesolved,Cul-

lenandPurser(1989),BenamouandBrenier(1998),giving a ‘slow manifold’, but thesolutionswill notbeasac-

curate.This reflectsthecomplexity of thetruesolution,forcingachoicebetweenanaccuratelocalapproximation

by asimplemodelor a lessaccurateglobalapproximation.Figure22is aschematicillustrationof thischoice.The

thin line representstheexactsolutionof theequationsof motion.The thick solid line representstheLagrangian

mean,with occasionalsharpchanges.The short-dashedline representsthe semi-geostrophicsolution,which is

muchsimplerthanthesolutionof theequationsof motion,asmany typesof motionarefiltered.Thesolutional-

waysexistsbut slowly losesaccuracy. Thelong-dashedline representsthesolutionof themoreaccuratebalance

equations,which cannotbesolvedunderall conditions.Thebreakdownswill usuallycorrespondto placeswhere

theexactsolutiondoesnotbehavesmoothly, socannotbecloselyapproximatedby asimplersolutionMcWilliams

et al., 1998).

Figure 22. Schematic graph of solution curves of exact, averaged and approximate models.

A practicaldifficulty with theuseof balancetechniquesin operationalmodelsis thesolutionof theelliptic problem

to derive othervariablesfrom thepotentialvorticity. In orderto derive thedivergence,or verticalmotion,or age-

ostrophic wind; we have to solve an ‘omega’ equation. This takes the generic form

(35)

Theaccuracy of thebalancedmodelassumesthattheeigenvaluesof arelarge,sothata realisticestimateof the

verticalmotion is producedfrom theforcing . However, asdiscussedin section6.1, balanceassumptionsare

notapplicableon thesmallerhorizontalandverticalscalesresolvedby theECMWF model,andthiswill beasso-

ciatedwith smalleigenvaluesof . For instance,thereare11 verticalmodesin the60 level modelwith internal

. ω*

=

.ω

*.



ECMWF, 2002

gravity wavespeedslessthan3ms-1. Straightforwardapplicationof (35)will thusleadto very largeandunrealistic

results.Thisis why, for instance,nonlinearnormalmodeinitialisationcannotbeusedfor morethanabout5 vertical

modes.

6.4 Structure of the solution of the semi-geostrophic model

As shown in section6.1, thesemi-geostrophicmodelis mostusefullyconsideredasanapproximationto theaver-

agedequations,sowemustincludethesub-gridmodelterms.In thesenoteswecannotconsidermorethanavery

limited partof thesub-gridmodel.A fuller discussionof this materialis includedin CullenandSalmond(2002).

The semi-geostrophic equations can then be written as:

(36)

Theboundarylayerfriction is representedby verticaldiffusion,and is a thermodynamicsourceterm.Themix-

ing coefficient is computedonly from thebalancedvariables.Thelastequationof (36) hasbeenmodifiedto

includethe friction terms,so thatgeostrophicbalancehasbeenreplacedby Ekmanbalance,hencethesuffix .

Write thelastequationof (36) in theform , where is amatrixoperator. Theequationscan

then be written, following Schubert (1985) as

(37)

where

(38)

and

ua -----------

=<θ∇Q Π >U ,–( )+ + A∂

∂ b a A∂∂ 2

a a,( ) ,( )–( )

∂∂ρ ∇ ρu( )⋅+ 0

θ -------- c

0ρθΠ

Π @ 0⁄( )κ

> a A∂∂ b a A∂

∂ a

– a A∂

∂ b a A∂∂ a

–,+,+

=<θ∇Π=

=

=

=

=

=

cb a d∇Π F a a θ,–,( )= F

Q

a– a–

N

∂∂

F 1– ∇Π+ H=

Q

a + a A∂

∂ b a A∂∂

+

afe a –

– A∂

∂ b a A∂∂

a –

age–

Θ Θ Θ e

Q h+=



ECMWF, 2002 31

(39)

Wehaveallowedadditionallyfor theeffectsof moisture,notincludedin (36), by using to representthepotential

temperaturein unsaturatedregionsandthemoistequivalentpotentialtemperature in saturatedregions,and

to represent a non-local matrix describing the convective plumes.

Equation(37)showsthattherateof changeof thebalancedpressureis drivenby theforcingterm . Theresponse

to theforcing,expressedasthetotalwind , is determinedby thepotentialvorticity matrix . contains

geostrophicadvection,radiation,andverticaldiffusion,all of whichhaveslow time-scales.If containsfasttime

scales(comparedto ), thisdecompositionis inappropriate.It is equallyimportantnotto forceoperationalmod-

els on fast time-scales, since the response is unlikely to be modelled realistically.

ThePV matrix includesthestabilisingeffectof boundarylayerfriction, andthedestabilisingeffectof moisture.

If theatmosphereis stableto moist(slantwise)ascent,thefirst matrix termin (38) hasall positive eigenvalues.If

aneigenvaluebecomeszero,thesmoothtransportvelocity is replacedby mixing over theregionwhere

thezeroeigenvalueexists.If thereis anegativeeigenvalue,thesecondterm,whichrepresentsthenon-localeffects

of convection,comesinto play. A ‘correct’ convectiveparametrisationfor thefull equations(21)shouldensurethat

the total has no negative eigenvalues, or else the model will still support explicit convective instability.

It canbeproved,CullenandPurser(1989),thatthereis anessentiallyuniquechoiceof trajectorythatsatisfies(37),

andmaintainsapressurewhoseimpliedgeostrophicwind andtemperatureareinertially andstaticallystable.If the

actualstateis potentiallyunstableto moistconvection,this solutionwill includeconvective plumes,asillustrated

in Figure23, anddiscussedin moredetailby Shuttset al. (1988).Realstateswhich arestaticallyor inertially un-

stableevolve on time-scalesfasterthan , sothatthesemi-geostrophicapproximationis inappropriate.Theef-

fectof theapproximationis to forcesuchfastmotionsto happeninstantaneously, sothatthereis asharpboundary

between‘slow’ and‘f ast’motions.As shown earlierin Figure3, thisis asimplificationof therealcase,wherethere

areno sharpdistinctions.In operationalmodelsof resolutionsimilar to ECMWF, Figure4 shows thatthereis not

enoughresolutionto treatmany ‘f ast’motionsaccurately, sothey have to beparametrised.Thepracticalsituation

is thus not that different from the simplified picture of equation(37).

Sincethetrajectoryis determinedimplicitly by theforcing,implicit calculationof thetrajectoryis preferredin op-

erationalmodels.Theboundarylayerfriction, whichappearsin thelowestorderbalance,andtheconvectivemass

transport, should also be determined implicitly.

Figure24andFigure25show theeffectof usingapredictor-correctorschemeto integratethedynamicsandphysics

in CY23R4of theECMWF model.Furtherdetailsaregiven in CullenandSalmond(2002).In this schemeeach

time-stepis repeated,allowing implicit adjustmentof the trajectoriesandtheparametrisedtransports.Figure25

shows thattheeffect is largerthanthatobtainedby simplyhalvingthetimestep.Figure24shows thatit is compa-

rable,exceptat theendof the forecastrange,to thedifferencemadeby changingthehorizontalresolutionfrom

T319 to T511.

H

ua ∇ a⋅–

u a ∇ a⋅

u a ∇Θ⋅ c+

=

Θθ h Q h

H N, ,( ) Q HH 1–

Q

N, ,( )

Q

1–



ECMWF, 2002

Figure 23. Left: Fluid element picture showing a vertical cross-section of a frontal zone with moisture (hatched

elements). Right: the striped elements have been cooled by precipitation falling from the convecting elements.

Figure 24.Left: comparisonof predictor-correctorphysicswith controlforecast,bothatT511resolution.Right:

comparison of control forecasts at T511 and T319 resolutions.

0i 1j 2k 3l 4m 5n 6o 7p 8q 9r 10

Forecast Days0

10

20

30

40

50

60

70

80

90

100%t DATE1=19980825/... DATE2=19980825/...

AREA=N.HEM TIME=12 MEAN OVER 14 CASESuANOMALY CORRELATION FORECASTu 500 hPa GEOPOTENTIAL

FORECAST VERIFICATION

CY23R4 IMPPHYS

CY23R4 CTL

MAGICS 6.3 styx - nac Fri Dec 7 14:33:59 2001 Verify SCOCOM

0i 1j 2k 3l 4m 5n 6o 7p 8q 9r 10

Forecast Days0

20

40

60

80

100

120

140

160

180Mv DATE1=19980825/... DATE2=19980825/...

AREA=N.HEM TIME=12 MEAN OVER 14 CASES

ROOT MEAN SQUARE ERROR FORECAST

500 hPa GEOPOTENTIAL


CY23R4 IMPPHYS

CY23R4 CTL

MAGICS 6.3 styx - nac Fri Dec 7 14:33:59 2001 Verify SCOCOM

0i 1j 2k 3l 4m 5n 6o 7p 8q 9r 10

Forecast Days0

10

20

30

40

50

60

70

80

90

100%t DATE1=19980825/... DATE2=19980825/...


ANOMALY CORRELATION FORECAST



CY23R4 T319 CTL

CY23R4 T511 CTL

MAGICS 6.3 styx - nac Fri Jan 11 09:12:08 2002 Verify SCOCOM

0i 1j 2k 3l 4m 5n 6o 7p 8q 9r 10

Forecast Days0

20

40

60

80

100

120

140

160

180Mv DATE1=19980825/... DATE2=19980825/...


ROOT MEAN SQUARE ERROR FORECAST



CY23R4 T319 CTL

CY23R4 T511 CTL

MAGICS 6.3 styx - nac Fri Jan 11 09:12:08 2002 Verify SCOCOM



ECMWF, 2002 33

Figure 25. Left: comparison of predictor-corrector physics against control forecasts, both with 20 minute

timesteps. Right: comparison between control forecasts with 10 and 20 minute timesteps.

7. SUMMAR Y

We have discussedthepropertiesof somevery simplemodelswhich arerelevant to aspectsof atmosphericflow,

andthenillustratedhow they give informationaboutthesolutionof thecompleteproblem.Themainpointswhich

are relevant to the design of operational models are summarised below.

(i) The full solutionsof the equationsof motion have well-behaved solutions,but they are far too

detailed to compute correctly in operational models.

(ii) Simplificationsof the full equationsmayor maynot have well-behavedsolutions.Thehydrostatic

equationsdo not have solutionsfor weak stratification.The semi-geostrophicequationsalways

have solutions, but these only describe relatively large scale flow where rotation is dominant.

(iii) Accuratesolutionof thecompleteequationsfor high Reynoldsnumbersubsonicflow (therelevant

case)requiresimplicit determinationof the trajectoryandsolutionof an elliptic problem.Proper

treatment of rigid boundaries is particularly important.

(iv) Operationalmodelsare most realistically thought of as solutionsof averagedequations.Only

certainaveragingscalescorrespondto physicalscaleseparations,otherchoicesof scalewill not be

soeffective in practice.Theaverageof a realatmosphericstatemaynot look like whatonewould

expect.

168 HOUR FORECASTS

AREA=N.HEM TIME=12 DATE=19980825/...wANOMALY CORRELATION FORECASTw 500 hPa GEOPOTENTIAL (%)

100x 90 80 70 60 50 40 30 20 10

Imp physy100

90

80

70

60

50

40

30

20

10

CT

L 2

0m

Imp phys is BETTER than CTL 20m at the 2.0% level (t test)Imp phys is BETTER than CTL 20m at the 10.0% level (sign test)

14 CASES

MEAN

MAGICS 6.3 styx - nac Tue Jan 8 15:43:14 2002 Verify SCOCAT

168 HOUR FORECASTS

AREA=N.HEM TIME=12 DATE=19980825/...zANOMALY CORRELATION FORECASTz 500 hPa GEOPOTENTIAL (%)

100 90 80 70 60 50 40 30 20 10

CTL 10m100

90

80

70

60

50

40

30

20

10

CT

L 2

0m

CTL 10m is BETTER than CTL 20m at the 5.0% level (t test)|

14 CASES

MEAN

MAGICS 6.3 styx - nac Tue Jan 8 15:16:33 2002 Verify SCOCAT



ECMWF, 2002

(v) Balancedmodelsarea usefulguide to the large-scalebehaviour of the atmosphere,in particular

how to combine resolved and parametrised processes.

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properties of equations - ecmwf · 2 . observed behaviour 3 . toy problems 3.1 introductory remarks...

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