Adiabatic formulation of models - ECMWF · 2.2 The shallow water equations 2.3 Map projections 3 . Horizontal discretization 3.1 Finite differences 3.2 Spectral technique 3.3 Finite-element

Meteorological Training Course Lecture Series

ECMWF, 2002 1

Adiabatic formulation of models

By Terry Davies

Table of contents

1 . Introduction

2 . Governing equations

2.1 Horizontal coordinate systems

2.2 The shallow water equations

2.3 Map projections

3 . Horizontal discretization

3.1 Finite differences

3.2 Spectral technique

3.3 Finite-element techniques

4 . Vertical coordinates

4.1 Pressure coordinate

4.2 Sigma coordinates

4.3 Hybrid coordinates

4.4 Isentropic coordinates

5 . Vertical discretization

5.1 Vertical differencing schemes for baroclinic primitive equation models

REFERENCES

1. INTRODUCTION

Thesenotesarenot a verbatimrecordof the lectureseries,but rathera condensationof thetopicscovered.They

describesomeof thetechniquesusedin theadiabaticformulationof primitive-equationmodelsof theatmosphere.

Somesectionshavebeentakendirectly from earliernotesby D. Burridgeandfrom apaperby A. SimmonsandD.

Dent. These notes are not complete and need to be supplemented by the following sources:

(i) ‘Numerical predictionanddynamicmeteorology’by Haltiner andWilliams. This is a very useful

textbook and reference book describing many of the basic principles and techniques used in NWP.

(ii) ‘Numerical methodsfor weatherprediction’, ECMWF Seminarnotes1983 and, in particular,

chaptersby Mesinger, JanjicandArakawa.Unfortunatelythisvolumeis outof print but youmaybe

able to find a copy in your home institution.


2 Meteorological Training Course Lecture Series

ECMWF, 2002

In additionto theabove,usefulmaterialcanalsobefoundin Philips’s articlein ‘Planetaryfluid dynamics’edited

by P. Morel and‘Numericalmethodsusedin atmosphericmodels’by MesingerandArakawa,GarpPublications

Series No. 17, Volume 1, (1976).

2. GOVERNING EQUATIONS

Taking an inertial (or fixed) frame of reference, Newton's second law can be expressed as

(1)

whereF is thesumof all theforcesperunit mass. is thevelocityasobservedin theinertial frame.Weneedtoexpress this equation in a reference frame on earth and which rotates with the earth.

Let be the angular velocity of the earth, then

(2)

whereV is thevelocity relative to theearthandr representsthepositionvectorof theparticleasmeasurefrom theorigin at the earth's centre.

Now consider the time derivative of a vector in the inertial frame and

in therotatingframewhere and areunit vectorsalongtheorthogonalaxes

in the two frames. Thus

Now , etc. so that we can write

(3)

Using the expression in(1) and using(2) we can show that

Thesecondtermof theright-handsideis theCoriolis forceandthethird termthecentrifugal force.In atmospheric

motions,theforcesweareconcernedwith arethoseof gravitation,friction andpressure.Normally, thecentrifugal

force is included in the gravitational force which is taken as constant .

The equation of motion can be written as

(4)

DVaD�----------- F=

Va

Ω

Va V Ω r×+=

V V′ � i′ V′� j′ V′ � k′+ +=V V � i V� j V � k+ += i′ j′ k′, , i j k, ,

DVD�--------- dV′ �

d�------------i′ dV′�

d�------------ j′ dV′ �

d�-----------k′+ +=

dV �d�----------i dV�

d�---------- j dV �

d�----------k V � di

d�----- V� dj

d�----- V � dk

d�------+ + + + +=

di d�⁄ Ω i×=

DVD�--------- dV

d�------- Ω V×+=

DVaD�----------- dV

d�------- 2Ω V Ω Ω r×( )×+×+=

� k=

d�d�-------- α∇�– 2Ω V � k– F+×–=



ECMWF, 2002 3

whereF denotesthefrictional forceand is thespecificvolume, is thedensity. Theequationof masscontinuity expresses the fact that local density changes are brought about by mass divergence.

(5)

or, alternatively, .

The first law of thermodynamics can be written

(6)

where is the heatingor cooling from diabaticeffects, is the rate of changeof internal energy and

represents the work done by expansion.

For a perfect gas, and we have the equation state of

(7)

where is thegasconstantand and arethespecificheatsat constantpressureandvolume,

respectively.

Using we can rewrite (6) as

(8)

Alternatively, we can expand and rearranging(8) we have

, (9)

where .

In termsof potentialtemperature (or where ) where ,

hPa.

We have and, on substituting in(9), and rearranging we find

(10)

2.1 Horizontal coordinate systems

Applicationof theequationsof motionrequirestheuseof anappropriatecoordinatesystem.Theobviouschoice

for globalmodelsis to choosesphericalpolarcoordinateswith longitude , latitude andradialdistance . The

map factors in this system are , , and the velocity components are

α 1 ρ⁄= ρ

∂ρ∂ �------ ∇ ρV⋅–=

dρ d�⁄ ρ∇ V⋅–=

� d�d�----- � dα

d�-------+=

�d� d�⁄

� dα d�⁄( )� v =

� � ρ =� � v–= � v

� v =� vdd�------- �

dαd�-------+=

� dαd�------- � d

d�----- � �--------- �

d

d�------- � �---------

d�d�------–= =

� � d

d�------- � �---------ω–=

ω d� d�⁄=Θ � 0 �⁄( )κ = ΠΘ= Π �� 0⁄( )κ= κ ��⁄=� 0 1000=

� dΘd�------- � ΠdΘ

d�------- � Θ κ�----Π

d�d�------+=

� dΘd�------- Θ----

�=

λ ϕ ��λ � ϕcos= � ϕ �= �� 1=



ECMWF, 2002

t

Theradialdistance where is theradiusof theearthand is theverticaldistanceabove thesurface

of the earth.

The components of the momentum equation become

(11)

(12)

(13)

where

From equations(11), (12) and(13), it can be shown that

(14)

i.e. therateof changeof kinetic energy is equalto therateat which theforcesof pressurefriction andgravity do

work.

We can also show that

(15)

which is the angular momentum principle.

For long integrations e.g. climate runs, it is important that these principles are maintained.

In thestudyof larger-scaleatmosphericmotions,at leastfor motionswe areusuallyconcernedwith in numerical

modelling, the following approximations can be justified.

(i) Since , we replace by the constant value .

(ii) The Coriolis and metric terms proportional to are ignored.

(iii) We make the hydrostatic approximation to equation(13) so that

� �λdλd�------ � ϕdλ

d�------cos= = , the eastward component

� �ϕdϕd�------ � dϕ

d�------= = , the northward component

� �� d�d�------ d�

d�------= = , the radial component, t

� � �+= � �

d�d�------- 2Ω � θ 2Ω � ϕ �� ϕtan�-------------------

��--------–

1ρ � ϕcos-------------------

∂�∂λ------– � λ+ +cos–sin=

d�d�------ 2Ω � ϕ � 2 ϕtan�------------------–

��--------–

1ρ �------

∂�∂ϕ------– � ϕ+sin–=

d�d�-------- 2Ω � ϕ � 2 � 2+�-----------------

1ρ---∂�

∂ �------– �– ��+ +cos=

dd�----- ∂∂ �-----

�ϕacos

--------------- ∂∂λ------

��---

∂∂ϕ------ � ∂

∂ �-----+ + +=

dd�----- 1

2---V2

V 1ρ---∇�– F � k–+ ⋅=

dd�----- � Ω � ϕcos+( ) � ϕcos[ ] � ϕ 1ρ � ϕcos-------------------

∂�∂λ------– � λ+ cos=

� �« � �ϕcos



ECMWF, 2002 5

(16)

i.e. the gravity force balances the vertical pressure gradient force.

Under these assumptions,

(17)

(18)

(19)

(20)

where is the Coriolis parameter.

Equations(18), (19), (20)togetherwith thecontinuityequation(5), theequationof state(7)andthethermodynamic

equation((8), (9) or (10)) give ustherequirednumberof equationsfor theunknowns ; theseare

known collectively as the primitive equations.

However, thereis no predictionequationfor sincewe ignoreverticalaccelerations( ) by our useof the

hydrostatic approximation. is usually diagnosed from a form of the continuity equation.

2.2 The shallow water equations

Theprimitive equationsarenonlinear, andit is only possibleto solve themapproximatelyusingnumericalmeth-

ods.Someof thepropertiesof theprimitive equations,andalsoof thenumericalschemesusedin their solution,

can be examined by studying simplified forms of the equations.

One such set of equations are the shallow water equations.

If weassumeconstantdensity, thehorizontalpressureforcein theequationsof motionis independentof heightso

that

(21)

where is the height of the free surface.

If wefurtherassumethatthevelocityfield is initially independentof height,thenit will remainsoandconsequently

we can omit the vertical advection terns.

We also assume that the fluid is incompressible so that the continuity equation is

(22)

which can be integrated from to (the bottom to the free top surface) to give

∂�∂�------ ρ�–=

� ϕdλd�------ �,acos � dϕ

d�------ �, d�

d�------= = =

d�d�------- 1ρ ϕacos------------------

∂�∂λ------– � � �� ϕtan�----------- � λ+ + +=

� �� ------- 1ρ �-------∂�∂ϕ------– � �– � 2 ϕtan�-----------– � ϕ+

∂�∂�------ ρ�–=

� 2Ω ϕsin=�� ρ , , , , ,

� d� d�⁄�

∇� ρ� ∇ �=�

∇ V ∂�

∂�-------+⋅ 0=

� 0= � �=



ECMWF, 2002

(23)

At the bottom whilst at the top, , the rate at which the free surface rises or falls.

Thus

(24)

or

The set of equations is completed by the momentum equations which are

(25)

Theequationsof motionabovearein aform known astheadvective form dueto thepresenceof the term.

The equations of motion can easily be rearranged into their vector invariant form viz:

(26)

where

is the vorticity in spherical coordinates (cf. with in a Cartesian system) and

is the kinetic energy.

Theequationsof motioncanbecombinedandrearrangedinto theirvorticity/divergenceform.Considertheshallow

water equations in a Cartesian system where the vorticity is .

(27)

Operating on the -equation by , the -equation by and subtracting, we obtain

(28)

Thedivergencecanbeobtainedby operatingonthe -equationby andthe -equationby andadding

to obtain

�∇ V �"!#� 0–+⋅ 0=

�0 0=

�$! d� d�⁄=

�$! ∂ �∂ �------ V ∇

�⋅+

�∇ V⋅–= =

∂�

∂ �------ V ∇� �

∇ V⋅+⋅+ 0=

∂V∂ �------- V ∇V � k V � ∇

�+×+⋅+ 0=

V ∇V⋅

∂V∂ �------- ζ %+( )k V ∇ �

� &+( )+×+ 0=

ζ 1 ϕacos--------------- ∂

�∂λ------

∂∂ϕ------ � ϕcos( )– =

ζ ∂ � ∂'⁄ ∂ � ∂(⁄–=&0.5 � 2 � 2+( )=

ζ ∂ � ∂'⁄ ∂ � ∂(⁄–=∂ �∂ �------ �

∂ �∂'------ �

∂ �∂(------ � �– �

∂�

∂'------+ + + 0=∂ �∂ �------ �

∂ �∂'------ �

∂ �∂(------ � � �

∂�

∂(------+ + + + 0=� ∂ ∂'⁄ � ∂ ∂(⁄

∂ζ∂ �------ ∇ ζ �+( )V[ ]⋅+ 0=

� ∂ ∂'⁄ � ∂ ∂(⁄



ECMWF, 2002 7

(29)

2.3 Map projections

Sofar, we have useda sphericalpolarcoordinatesystemin our derivationof theprimitive equations.However, it

is sometimesnecessaryto mapontoa plane.This is oftendonefor limited-areamodellingand,of course,is nec-

essary when producing charts.

Theadvectiveform of theshallow waterequationsin ageneralizedlinearcoordinatesystem with correspond-

ing map factors and are

(30)

(31)

(32)

In this system, the vorticity and divergence are given by

(33)

(34)

When , the angles between intersecting curves are preserved and the system is conformal.

We have already shown that a spherical polar coordinate system has .

2.3 (a) Polar stereographic projection. Thisprojectionis usedfor mapsandwaswidely usedin modellingin

limited area models and the hemispheric domain. If is the radius of a latitude circle on the map then

(35)

Thelongitudinalangleis definedas sothatwith axeswith origin at thepole, positiveeastwards,

positivenorthwardsfromourreferencelongitude,then and . In termsof and these

become

(36)

with , where is the map factor ( ). The wind

components( ) in the ( ) directionsare andarerelatedto thevelocities( ) in

the spherical system by and .

∂)∂ �------- ∇ ζ �+( ) * �×[ ] ∇

2 � � � 2 � 2+2

-----------------+ +⋅+ 0=

'+(,� � � �

∂ �∂ �------

�� ------∂ �∂'------

�� ------∂ �∂(------ � �

1� � � �------------ �∂� �

∂'--------- �∂� �

∂(---------– +–�� ------

∂�

∂'------+ + + 0=

∂ �∂ �------

�� ------∂ �∂'------

�� ------∂ �∂(------ � �

1� � � �------------ �∂� �

∂'--------- �∂� �

∂(---------– +�� ------

∂�

∂(------+ + + + 0=

∂�

∂ �------1� � � �------------

∂∂'------� � � �( ) ∂∂(------

� � � �( )++ 0=

ζ )

ζ 1� � � �------------∂

∂'------� � �( ) ∂∂(------

� � �( )–=

) 1� � � �------------∂

∂'------� � �( ) ∂∂(------

� � �( )+=� � � �=

�λ ϕ

�ϕ,acos �= =

�

� 2 Φ2---- whereΦ,atan π

2--- ϕ–= =

Θ λ= ',(,( ) '( ' � Θsin= ( � Θcos= ϕ λ

' 2 ϕ λsinacos1 ϕsin+( )

------------------------------ (, 2 ϕ λcosacos1 ϕsin+( )

-------------------------------–= =

� �.- ϕ( ) ϕcos= - ϕ( ) 2 1 ϕsin+( )⁄= � � � � 1 - ϕ( )⁄= =��, '+(, � � �/' ˙ �, � �0( ˙= = � s � s,� �s λ

�s λsin–cos=

� �s λ

�s λcos+sin=



ECMWF, 2002

2.3 (b) Mercator (cylindrical) projection. This is commonlyusedfor mapsor for modelling in tropical re-

gions. The (east–west) and (north–south) coordinates are defined by

(37)

where is the latitude at which the projection is true (usually , the equator).The map factor

.

3. HORIZONTAL DISCRETIZATION

So far, we have derived the basichydrodynamicalequationsthat govern atmosphericmotions.Given the initial

fieldsof massandvelocity it is possibleto treatthesolutionof theseequationsasaninitial valueproblemandthus

obtainthemassandvelocity distribution at somefuturetime.However, thepartialdifferentialequationsarenon-

linearandcanonly besolvedby usingnumericaltechniques.Furthermore,thenonlinearityof theequationsmeans

thatany smallerrorsin theinitial conditionsarelikely to grow, makingthesolutionuselessin adeterministicsense

after a number of days.

Thereareavarietyof methodsthatcanbeemployedin thesolutionof theequations.Weshallput theminto three

broad categories—finite difference, spectral and finite-element techniques.

3.1 Finite differences

Thefirst numericaltechniquesusedin thesolutionof theprimitive equationswerebasedon finite differences.In

thistechniquevaluesandderivativesof thedependentvariablesarerepresentedatdiscretepointsonagrid.Usually

the grid is regular in the coordinatesystemused,at leastin the horizontal.In the vertical mostcentreshave a

non-uniformgrid asit seemsdesirableto havefinerresolutionin theboundarylayerandaroundthejet stream.Fur-

therenhancementsto theperformance(andto someextenttheaccuracy) of finite differenceschemescanbemade

by staggeringthe dependentvariables,i.e. by holding variablesat different grid-points. Normally staggered

schemesgiveabettersimulationof geostrophicadjustmentandthey areeithermoreaccurateor moreeconomical

than equivalent non-staggered schemes.

In thedesignof any numericalschemecertainconditionsneedto bemetto maintainstability. Sincetheprimitive

equationsarenonlinear, they areproneto nonlinearcomputationalinstability undercertainconditions.Nonlinear

instability canbeavoidedby designingschemesin sucha way asto conserve particularproperties,normallyen-

strophy or kinetic energy. Suchconservationpropertiesareusuallymetby satisfyingintegral constraintsthatare

alsosatisfiedby thecontinuoussystemof equations.By maintainingtheseconstraintsin longerintegrations,the

statistical structure of the atmosphere is maintained and it may help to reduce systematic errors.

Thedesignof conservativefinite-differenceschemeswaspioneeredbyArakawa(1966),Lilly (1964)andSadourny

(1975),andareview canbefoundin Arakawa(1984).Failureto conserveenstrophy, in particular, eventuallyleads

to a spuriouscascadeof energy to smallerscales.It is possibleto limit or control this falsecascadeby removing

energy from thesesmallerscalesby theuseof lateraldiffusion,but excessive lateralsmoothingleadsto enhanced

energy dissipation.

To achieveconservationin thediscretizedsystem,successive termsneedto cancelsothatsummingoveradomain

(equivalent to integrating the continuous equations) leaves only boundary values.

Conservationpropertiesdo not necessarilyendow a particularschemewith increasedlocal accuracy, which may

bemoreadvantageousfor theforecastingof small-scalephenomena.Thus,thechoiceof aschemewith or without

' (

' ϕ0λ (,acos ϕ0 1 ϕsin+ ϕcos-------------------- lnacos= =

ϕ0 ϕ0 0=- ϕ( ) ϕ0cosϕcos

-------------- 1� �------1� �------= = =



ECMWF, 2002 9

conservation is likely to be dictated by the problem being solved.

For globalmodels,themoststraightforwardgrid to useis basedonlinesof constantincrementsin latitudeandlon-

gitude(Fig. 1 ). However, on sucha grid thelongitudinalpointsconverge,andeast–westresolutionconsequently

increasesasthepolesareapproached.Unlessspecialmeasuresaretaken,this increasein resolutionresultsin the

violationof theCFL conditionwhenexplicit time-steppingschemesareused.In someformulations,thepolesare

singular points and usually have to be treated separately.

Figure 1Illustration of a regular latitude/longitude grid near a pole.

We candemonstratetheeffectsof converging meridiansby consideringa linearizedversionof theshallow-water

equations

(38)

(39)

(40)

wherewe have inserted which will act to highlight the effects of longitudinaldifferencingand, if

necessary, asa smoothingoperator. We will examinethebehaviour of two differentschemeson anunstaggered

grid, i.e. are held at the samepoints.First, with a leapfrogscheme,the differentiation is

replacedby . We usecentraldifferencesfor the spacedifferencingand,after substituting

solutionsof the form , etcwe arrive at a systemof equations,thedeterminantof

which is

∂ �∂ �------

��1� ϕcos----------------

∂�

∂λ------+ 0=

∂ �∂ �------

��---

∂�

∂ϕ------+ 0=

∂�

∂ �------2� ϕcos---------------- 1

∂ �∂λ------ ∂

∂ϕ------ � ϕcos( )++ 0=

1 1 ϕ( )=�� and �, ∂ � ∂ �⁄��3 1+ ��3 1––( ) 2∆ �⁄��3 λ34� ˆ i * ∆λ 5 ∆ϕ+( )[ ]exp=



ECMWF, 2002

(41)

(Note we have also used in the difference equation for ).

Evaluating the determinant, we obtain roots and the solutions of

(42)

where (43)

i.e. (44)

or

(45)

Revertingbackto onedimension,this is simply or , theusualCFL sta-

bility condition. For a regular Cartesian, two-dimensional grid we would have .

Now considerour latitude–longitudegrid whereasthepolesareapproached , becomessmallanddom-

inatestheCFL condition.Thereareanumberof waysto circumventthisproblem.Oneof thesimplestis to let the

timestepdecreasetowardsthepoletherebymaintainingtheratio , but this is prohibitively expensive. It is

just aseasyto restrictthenumberof wavesin thesolution,i.e. not letting the termapproach1. Again,

therearea varietyof waysof doingthis in practice.Essentiallywe obtaintheFouriercomponentsin thelongitu-

dinaldirectionandeithertruncateor smooththosemodes(thehigherwavenumbers)thatwouldbeunstableby our

stability criterionequation(45). We thenperformaninversetransformto obtainour desiredsolution.Essentially

wearetrying to controltheeffectof in thefirst ternof (45) . For aparticular

latitude, we only retain those waves so that , or

, givesus thevalueof thatcanberetainedfor a particularlatitude.If we chooseto

truncatethesolutionataparticularwavenumber, theprocessis known asFourierdropping.Alternatively, wecan

smooth the amplitude of a wave that would be unstable by using

Wealsohavesomefreedomonhow weapplythechoppingor smoothing.Wecanapplythefilter to thedependent

variables,the tendenciesor the increments.ArakawaandLamb(1981)apply thesmoothingoperators within

the differencing as in equations(38) and(40).

It is instructive to repeattheabove stability analysisusinga forward–backwardschemefor time differencing,i.e.

thetimedifferencingusesaforwardschemeandtheheighttermin thefirst two equationsis at thelatesttimelevel.

The same procedure as before gives.

λ2 1– 0 λ�61 ∆�

� ϕcos( )∆λ----------------------------2i * ∆λ( )sin0 λ2 1– λ� ∆

�� ∆ϕ-----------2i 5 ∆ϕ( )sin

λ2 1 ∆ �� ϕcos( )∆λ----------------------------2i * ∆λ( )sin λ

2 ∆ �� ∆ϕ-----------2i 5 ∆ϕ( )sin λ

2 1–

0=

ϕ 7 1+cos ϕ 7 1–cos ϕ 7cos= = �λ 1±=

λ4 2λ2– 1 4 8 λ2+ + 0=

8 � 2 ∆ �( )2 1 2 * ∆λ( )2sin� 2 ∆λ( )2 ϕ2cos-----------------------------------5 ∆ϕ( )2sin

� 2 ∆ϕ( )2------------------------+=

λ2 1 2 8– 48 2 4 8–( )±=λ 1 provided that A 1



ECMWF, 2002 11

(46)

Evaluatingthedeterminantasbeforewefind werequire for i.e. in onedimension

and,for a regularCartesiantwo-dimensional . Themaximumtime stepfor theforward–back-

ward scheme is twice that for the leapfrog scheme and it has the further advantage of being a 2-level scheme.

As with theleapfrogscheme,thesmoothingfactor playsthesamerole in determiningthosewave numberswe

can retain by requiring .

Anothersolutionto theproblemof converging meridiansis to usea grid developedby Kuriharawherepointsare

skippedasthe polesareapproached,therebyroughly maintainingthe sameresolution.In its simplestform this

leadsto overlappingof grid boxesin thenorth–southdirection,which complicatesa little thedifferencingalgo-

rithms. A more serious drawback of such a grid appears to be associated with larger errors near the poles.

Theabove examplesinvolvedtheuseof unstaggeredgrids.However, thespacedifferencingin equations(38) to

(40) only involvesvariablesat pointsadjacentto thoserequiredin thesolutionof theequations.Therefore,it is

possibleto carry the velocity componentsandheightfield at alternatepoints,therebyreducingthe computation

time andstorageby half. Thetruncationerrorremainsunaltered.By staggeringthevariablesin this way we have

effectively made theshortestresolvablewavelength.If we want to include wavesthenwe canreduce

thegrid lengthby half and,in this case,for comparablecomputationaleffort astheunstaggeredgrid we increase

the accuracy.

In two-dimensions,therearemoreoptionsavailablewhenstaggeringthevariables.Oneof themostrecentreviews

canbefoundin JanicandMesinger(1984).Thevariouschoicesof gridsaredenotedby thelettersA to E. TheA

grid is the unstaggered arrangement. The other grids are as shown below.

Theimportancein thestaggeringof thevariablesis not just for computationalefficiency, but is alsorelatedto the

behaviour in thegeostrophicadjustmentprocess.By examiningthedispersionpropertiesfor eachof thegrids,the

relativemeritsof eachof thegridscanbeassessed.JanicandMesinger(1984)suggestthatthereis little to choose

between the B and C grids whereas the unstaggered A grid is poor.

Wehaveconsideredsofar theadjustmentor gravity-wavepartof oursystemof equations.If necessary, theadvec-

tive termscanbeincludedin our simplifiedsystem.If we thencarryout a stability analysis,theCFL criterion is

usuallymodifiedby theadditionof avelocity termsuchas comparedwith . Now

typically hasa valueof about300m/swhereasmaximumvaluesof arearound100m/s.Hence,thesta-

bility conditionis dominatedby thegravity-wavespeed.Theadditionof thenonlinearadvectiontermscomplicates

thesolutionandmakesit muchmoreexpensive to solve. However, sincethestability conditionfor thewinds is

aroundathird asrestrictiveasthegravity-waveconditionthisexpensecouldbereducedby somehow allowing the

advectionto setthe time stepratherthanthe gravity-wave speed.Therearebroadlytwo approaches.Oneis the

λ 1– 0 ��1 ∆�

� ϕcos( )∆λ----------------------------i * ∆λ( )sin0 1 � ∆

�� ∆ϕ-----------i 5 ∆ϕ( )sin

2 1 ∆ �� ϕcos( )∆λ----------------------------i * ∆λ( )sin λ2 ∆ �� ∆ϕ-----------i 5 ∆ϕ( )sin λ 1–

0=

8 4≤ λ 1≤ � 2 ∆ � ∆'⁄ 2<2� 2 ∆ � ∆'⁄ 2<1

1 * ϕ,( ) * ∆λ ϕcos ϕ0cos⁄



ECMWF, 2002

semi-implicit approach,wherethegravity-wavetermsaretreatedimplicitly andtheadvectiontermsaretreatedex-

plicitly. Alternatively, asplit–explicit approachcanbeadopted,wherebothpartsaretreatedexplicitly but separate-

ly sothatdifferencingschemessuitedto therelevantcharacteristicsof eachproblemcanbeused.For example,a

forward–backwardschemefor thegravity-waveadjustmentandaLax–Wendroff or aHeunschemefor theadvec-

tion step.If this approachis used,severaladjustmentstepscanbetakenfor every advectionstep(typically 3 ad-

justment steps for each advection step).

3.2 Spectral technique

TheoperationalECMWF forecastmodelusesa spectraltechniquefor its horizontaldiscretization.Over thepast

decadeor so this techniquehasbecomethe mostwidely usedmethodof integratingthe governingequationsof

numericalweatherpredictionover hemisphericor globaldomains.Following thedevelopmentof efficient trans-

form methodsby Eliasenet al. (1970)andOrszag(1970),andtheconstructionandtestingof multi-level primi-

tive-equationmodels(e.g.Bourke, 1974;HoskinsandSimmons,1975;Daleyet al., 1976),spectralmodelswere

introducedfor operationalforecastingin AustraliaandCanadaduring1976.Thetechniquehasbeenutilizedoper-

ationally in theUSA since1980,in Francesince1982,andin Japanandat ECMWF since1983.Implementation

of aspectralmodelis plannedin theFederalRepublicof Germany. Themethodis alsoextensively usedby groups

involved in climate modelling.

A comprehensiveaccountof thetechniquehasbeengivenby Machenhauer(1979),andafurtherdescriptionof the

methodhasbeengivenby JarraudandSimmons(1984).Referenceshouldbemadeto theseor otherreviews for

further discussion of most of the points covered below.

In theusualapplicationof themethod,thebasicprognosticvariablesarevorticity, divergence,temperature,a hu-

midity variable,andthelogarithmof surfacepressure.Theirhorizontalrepresentationis in termsof truncatedseries

of sphericalharmonicfunctions,whosevariationis describedby sinesandcosinesin theeast–westandby associ-

ated Legendre functions in the north–south. The horizontal variation of a variable is thus given by

(47)

where is the latitude and is the sine of the longitude. For the particular choice of normalization adopted at

and

.

As is a realvariable, is equalto thecomplex conjugateof , enablingthemodelto dealexplicitly

with the for . The sum

defines the "Fourier coefficients", of the variable .

<

<λ µ,( )

< 3=?>3=

µ( ) i - λ[ ]exp3 ==@∑= A–=

A∑=

λ µ

>3=

µ( ) 2B 1+ B -–( )!B -+( )!----------------------1

23 B !------------ 1 µ2–( )= 2⁄ d3 =+

dµ3 =+----------------- µ2 1–( ) - 0≥,=

>3=– µ( ) > 3

=µ( )=

< < 3=– < 3

=< 3= - 0≥

< 3=9>3=

µ( )3 ==@∑

< =µ( )



ECMWF, 2002 13

It is becomingincreasinglycommonfor theso-called"triangular"truncationof theexpansionto beused.Thistrun-

cationis definedby , andgivesuniform resolutionover thesphere.Thesymbol“TN” is the

usualway of definingtheresolutionof sucha truncation; beingthesmallesttotal wave numberretainedin the

expansion.Thesmallestresolvedhalf-wavelengthin any particulardirectionis then (320km for T63,190

km for T106), although the corresponding lateral variation is of larger scale.

Derivatives of a spectrally represented variable are known analytically:

(48)

and

(49)

where

with

Theorthogonalityof thesphericalharmonicsenablesthe to bedeterminedfrom accordingto theformula

(50)

Thecomputationalefficiency of theevaluationof theFouriercoefficientsis enhancedfor globalmodelsby writing

the summation over (known as theinverse Legendre transform) in the form

(51)

Evaluatingthesetwo sumsseparatelyenablesa simpleevaluationof the inversetransformfor thecorresponding

latitude in the opposite hemisphere:

(52)

C Dconstant= = D

π � D⁄<

∂<

∂λ------- i - < 3

=9>3=

µ( ) i - λ[ ]exp3 ==@∑= A–=

A∑=

∂<

∂µ-------

< 3= d> 3

=dµ

-------------- i - λ[ ]exp3 ==

@∑= A–=

A∑=

1 µ2–( )d>3=

dµ-------------- B ε3 1+

= >3 1+= B 1+( )ε 3

= >3 1–=

+–=

ε3= B 2 - 2–

4B 2 1–-------------------

12---

=

< 3= <

< 3= 1

4π------

<λ µ,( )

>3=

µ( ) i– - λ[ ]exp λd µd0

2π

∫1–

1

∫=

B

< 3=9>3=

µ( )3 ==@∑

< 3=E>3=

µ( )< 3=9>3=

µ( )3 = 1+=3 =+( ) odd

@∑+3 ==3 =+( ) even

@∑=

< 3= >3=

µ–( )3 ==

@∑

< 3= >3=

µ( )< 3= >3=

µ( )3 = 1+=3 =+( ) odd

@∑–3 ==3 =+( ) even

@∑=



ECMWF, 2002

Similarly, the latitudinal integral in(50), the direct Legendre transform, need be taken only over one hemisphere:

(53)

The spherical harmonics are eigenfunctions of the Laplacian operator on the sphere:

(54)

where is theradiusof theearth.This is utilized in deriving wind componentsfrom vorticity anddivergence.If

and aretheeastwardandnorthwardvelocity componentsmultiplied by thecosineof latitude,the(relative)

vorticity and divergence are defined by

(55)

and

(56)

In terms of a stream function and velocity potential :

Thus

(57)

and

(58)

< =µ( )>3=

µ( ) µd1–

1

∫< =

µ( )< =

µ–( )+[ ]>3=

µ( ) µ for - B even+d0

1

∫=

< =µ( )< =

µ–( )–[ ]>3=

µ( ) µ for - B odd+d0

1

∫=

∇2>3=

µ( ) i - λ[ ]exp{ } BEB 1+( )� 2--------------------->3=

µ( ) i - λ[ ]exp–=

�; �)

ξ 1�---1

1 µ2–( )-------------------∂ �

∂λ------- ∂

;∂µ--------–

=

) 1�---1

1 µ2–( )-------------------∂

;∂λ-------- ∂ �

∂µ-------+

=

ψ χ

; 1�--- 1 µ

2–( )∂ψ∂µ-------– ∂χ

∂λ------+

=

� 1�---∂ψ∂λ------- 1 µ2–( )∂χ∂µ

------+

=

ξ ∇2ψ and ) ∇2χ= =

; � B 1– B 1+( ) 1– 1 µ2–( )ξ3= d>F=3

dµ-------------- i -G) 3

=H>3=

– I - λ[ ]exp

3 ==

@∑= A–=

A∑=

� �– B 1– B 1+( ) 1– i - ξ3=H>3=

1 µ2–( ) ) 3= d> = 3

dµ--------------+

I - λ[ ]exp

3 ==@∑= A–=

A∑=



ECMWF, 2002 15

Thesequenceof calculationsfor eachtime stepof a spectralmodelcannow bebroadlysummarizedasfollows.

Theanalyticalrepresentationof basicfields(Eqs.(47)) is usedto evaluatethesefields,andderivedfieldssuchas

horizontalvelocities(Eqs.(48)andEqs.(49)) and(in someimplementations)temperatureandmoisturegradients,

atanarrayof grid points.Thesegrid-pointvaluesareusedto computecontributionsto thetendenciesof thefields

at thegrid pointsfrom bothresolveddynamicalprocessesandfrom theparametrizedprocesses.Thesetendencies

areprojectedback to give tendenciesof the spectralcoefficients,enablingtime-steppingto be completed.The

transformationto andfrom grid-pointvaluesis particularlyefficient in theeast–west,for whichFastFourierTrans-

formationcanbeused.Gaussianquadratureis employedfor thelatitudinalintegrationusedin theprojectionback

to spectralspaceEqs.(53); this requiresthatthegrid in thenorth–southhasaparticulardistribution (theso-called

"Gaussian"latitudes)for eachresolution,althoughthis canberegardedasa regulargrid for all practicalapplica-

tionsof outputgrid-pointproductsfrom spectralmodelsattoday'soperationalresolutions.An exacttransformation

of quadratictermsis achievedby usinga grid which hasa finer resolutionthanthatof thespectralrepresentation,

with at least pointsin theeast–westand from poleto polefor TN truncation(Fig. 2 ). If,

asis generally(but not necessarily)thecase,thephysicalparametrizationsarecomputedon this grid, thenprog-

nosticfieldssuchassurfacetemperatureor snow cover, andoutputfieldssuchasprecipitation,aredefinedandup-

dated on this grid.

Figure 2. Illustrationof variousspectraltruncations:Triangulartruncation andRhomboidal

truncation

In theformulationadoptedatECMWF, completetendenciesof thetemperature,moistureandlogarithmof surface

pressurearecomputedin grid-point space,andtransformeddirectly into thecorrespondingtendenciesof spectral

coefficients.For vorticity anddivergence,integrationby partsis used.Thewind tendenciescanbewritten in the

form

(59)

and

(60)

3D

1+( ) 3D

1+( ) 2⁄

TC 8 J )+ +=

RC 8 +=

∂;

∂ �-------- �$K1�---

∂ L∂λ-------–=

∂ �∂ �------- �$M

1 µ2–( )�-------------------

∂ L∂µ-------–=



ECMWF, 2002

where , and are quantities that can readily be computed in grid-point space.

Using equations(50), (55) and(56), (59) and(60) yield

(61)

and

(62)

Thefactthatthesphericalharmonicsareeigenfunctionsof theLaplacianoperatormakesit relatively simpleto im-

plementanefficient semi-implicit treatmentof gravity-wave terms.It is alsostraightforwardto implementlinear

formsof "horizontal"diffusionalongthecoordinatesurfacesdefinedby theverticaldiscretization.Suchdiffusion

is widely usedin spectralmodels,with largelysatisfactoryresults.Therecan,however, asin finite-differencemod-

els,be problemsassociatedwith diffusion alongcoordinatesurfaceswhich slopesignificantlyin the vicinity of

steepterrain,andtheintroductionof otherformsof diffusionmayrequiresignificanteffort andcomputationalcost.

Recentdevelopmentsof thespectraltechniquehave beenalonga numberof lines.Apart from computationalre-

finementsof the type discussedin the following section,the techniquehaslent itself to somefurther gains in

time-steppingefficiency in high-resolutionapplications(Simmonset al., 1988),andthereis promiseof furtherin-

creasesof efficiency with theuseof a semi-Lagrangiantreatmentof advection(Ritchie, 1988).Theapplicability

of thetechniquehasbeenwidenedby thedevelopmentof limited-areaspectralmodels(e.g.Tatsumi,1986),and

of astretchedcoordinatewithin theframework of aglobalmodel(CourtierandGeleyn, 1988,following Schmidt,

1977).

3.3 Finite-element techniques

Finite-elementsaremuchlesswidespreadin operationalNWP whencomparedwith eitherspectralor grid-point

models.RPNin Canadahavethemostpracticalexperiencein usingfinite-elementsfor bothhorizontalandvertical

discretization. A review article byStaniforth (1987) gives an introduction and contains useful references.

Thereis anexperimentalversionof theCentre'smodelthatusesfinite elementsin theverticaland,apartfrom noise

problemsattheupperboundary, thisversionappearsto givegreateraccuracy morecheaplythanincreasingvertical

resolution using finite differences.

�$K �$M L

∂ξ3=

∂ �-------------1

4π �----------1

1 µ2–--------------

∂ �$M∂λ

----------∂ �$K∂µ

----------– > 3

=µ( ) i - λ–[ ]exp λd µd

0

2π

∫1–

1

∫=

14π �----------

1

1 µ2–-------------- i -G�$M > 3

=µ( ) 1 µ2–( ) �$K d

>3=

dµ--------------+

i - λ–[ ] λd µdexp0

2π

∫1–

1

∫=

∂ ) 3=

∂ �--------------1

4π �----------1

1 µ2–--------------

∂�$K∂λ

----------∂ �$M∂µ

---------- � ∇2 L–+ > 3

=µ( ) i - λ–[ ]exp λd µd

0

2π

∫1–

1

∫=

14π �----------

1

1 µ2–-------------- i -G�"K > 3

=µ( ) 1 µ2–( )– �$M d

>3=

dµ--------------

i - λ–[ ] λd µdexp0

2π

∫1–

1

∫=

BEB 1+( )4π � 2--------------------- L

>3=

µ( ) i - λ–[ ] λd µdexp0

2π

∫1–

1

∫+



ECMWF, 2002 17

4. VERTICAL COORDINATES

Sofar, heightabovemeansealevel hasbeenusedastheverticalcoordinatein oursystemof equations.For mid-lat-

itudesynoptic-scaledisturbancestheCoriolis forceandpressure-gradientforcearein approximatebalance—the

geostrophicrelationshipwhichis , adiagnosticrelationshiprelatingthevelocityfield to the

pressure gradient.

However, thedensityvarieswith latitudeandheightwhichmake theaboveequationlesseasyto usethananalter-

nativesystemwhichusespressureastheverticalcoordinate.Othercoordinatesaremoreusefulfor numericaltech-

niquesin solvingthecompleteequationsof motion.Wecanderivedasystemof equationsfor ageneralizedvertical

coordinatewhichis assumedto berelatedto theheight by asingle-valuedmonotonicfunction.Thiswasderived

by Kasahara (1974).

We have

(63)

and

(64)

Thus

(65)

Using the above, we have

(66)

(67)

(68)

The total derivative can be transformed into the system using(66) and(67), thus

(69)

However, by definition, the total derivative in the system is

� k V× 1 ρ⁄( ) �∇–=

�

ζ ζ '+(N� �, , ,( ) or � �O'+( ζ �, , ,( )= =

∂8∂ P------- ζ

∂8∂ P------- �

∂8∂�-------

∂�∂ P------ ζ where P+ '+( or

�, ,= =

∂8∂�-------

∂8∂ζ-------∂ζ

∂�------=

∂8∂ P------- ζ

∂8∂ P------- �

∂8∂ζ-------∂ζ

∂�------∂�∂ P------ ζ+=

∂8∂ �------- ζ

∂8∂ �------- �

∂8∂ζ-------∂ζ

∂�------∂�∂ �------ ζ+=

∇ζ 8 ∇ �Q8 ∂8∂ζ-------∂ζ∂�------∇ζ�+=

∇ζ V⋅ ∇ � V⋅ ∂V∂ζ-------∂ζ∂�------ ∇ζ �⋅+=

d d�⁄ ζ

dd�----- d

d�----- ζ V ∇ζ

� ∂�∂ �------ ζ– V ∇ζ⋅–

∂ζ∂�------

∂∂ζ------+⋅+=

ζ



ECMWF, 2002

(70)

where is the vertical velocity in the system.

From(69) and(70) we have

(71)

Using(67). We can write the horizontal pressure forces as

(72)

where is the geopotential. The hydrostatic equation becomes

(73)

The horizontal equation of motion (neglecting friction) becomes

(74)

To derive the continuity equation, from(71) we have

(75)

Thus

(76)

Using(66) and(67) and substituting in the continuity equation

we have

i.e.

dd�----- ∂∂ �----- ζ V ∇ζ ζ̇

∂∂ζ------+⋅+=

ζ̇ dζ d�⁄= ζ

ζ̇ ∂ζ∂�------ �∂�∂ �------ ζ– V ∇ζ �⋅( )–=

1ρ---∇ �R�– 1ρ---∇ζ�–

1ρ---∂�

∂�------∇ζ �+1ρ---∇ζ�– ∇ζΦ–= =

Φ � �=∂�∂ζ------ ρ∂Φ∂ζ

-------–=

d�d�-------- 1ρ---∇ζ�– ∇ζΦ– � k V×–=

� ∂�∂ �------ ζ � ∇ζ � ζ̇

∂�∂ζ------+⋅+=

∂ �∂�-------

∂ �∂ζ-------∂ζ

∂�------∂ζ∂�------

∂∂ζ------ ∂�

∂ �------ ζ∂V∂ζ------- ∇ζ� V ∇∂�∂ζ------⋅ ζ̇

∂∂ζ------ ∂�

∂ζ------

∂�∂ζ------∂ζ̇

∂ζ------+ + +⋅+= =

∂ζ∂�------

dd�----- ∂�∂ζ------

∂V∂ζ------- ∇ζ�⋅+ ∂ζ̇∂ζ------+=

dd�----- ρln( ) ∇ � ∂ �∂�-------+⋅+ 0=

dd�----- ρln( ) ∇ζ V ∂V∂ζ-------

∂ζ∂�------ ∇ζ�

∂ζ∂�------

dd�----- ∂�∂ζ------

∂V∂ζ------- ∇ζ�⋅+ ∂ζ̇∂ζ------+ +⋅–⋅+



ECMWF, 2002 19

(77)

or

(78)

Alternatively, by expanding(77) and using the hydrostatic equation, we can obtain

(79)

Theaboveequationsfor ageneralizedverticalcoordinatemaynow beusedto examinealternativecoordinatesys-

tems.

4.1 Pressure coordinate

and so that the equation of motion(74) becomes

(80)

and the hydrostatic equation is

(81)

The first term in the continuity equation(77) vanishes, so that

(82)

where is the vertical velocity. Integrating (80) from where , to an arbitrary

level, , yields

(83)

At the surface we assume no normal flux so that

(84)

In the pressure system, the geostrophic approximation gives

(85)

dd�----- ρ∂�∂ζ------ ∇ζ V

∂ζ̇∂ζ------+⋅+ln 0=

dd�----- ∂�∂ζ------ ∇ζ V

∂ζ̇∂ζ------+⋅+ln 0=

∂∂ζ------ ∂�

∂ �------ ∇ζ V∂�∂ζ------

∂∂ζ------ ζ̇∂�∂ζ------ +⋅+ 0=

ζ �=∇ζ� ∇�4� 0= = ∂� ∂�⁄ 1=

d�d�-------- ∇� Φ– � k V×–=

∂Φ∂�-------

1ρ---–=

∇�S� ∂ω∂�-------+⋅ 0=

ω d� d�⁄ � ˙= = � 0= ω 0=�

ω ∇� V⋅ �d0

�∫=

ωsd� sd�--------- ∂� s∂ �--------- Vs ∇� s⋅+ ∇� V �d⋅( )

0

�UT∫–= = =

� k V× ∇� Φ–=



ECMWF, 2002

This relatesthe horizontalvelocity to the geopotentialgradientand,sincethereis no longera dependenceon

density, the same gradient implies the same wind at any height.

4.2 Sigma coordinates

Thelowerboundarycondition(82) is noteasyto applyin practice.To circumventthisproblem,Phillipssuggested

the use of as the vertical coordinate. The hydrostatic equation becomes

(86)

and the equation of motion becomes

(87)

The continuity becomes

Expanding the first term and rearranging we have

(88)

The boundary conditions are at and 1, so that integrating(86) from to 1 gives

(89)

i.e. the surface pressuretendency is given by the massdivergence integrated over the entire atmosphere.

Integrating(86) from to an arbitrary level gives our equation for , viz

(90)

Althoughthelower boundaryconditionin thesigmacoordinatesystemis easierto applythanin pressurecoordi-

nates,thesystemdoeshave a drawbackin thetermsfor thehorizontalpressureforce.In pressurecoordinatesthe

pressureforceis simply thegradientof geopotentialwhereasin sigmacoordinatesit is thedifferenceof two terms

which are large in magnitude when compared with the result, i.e.

(91)

In designinga differenceschemefor theabove,caremustbetakenin selectinga schemethatminimizestheerror

in thedifferencebetweentheterms.Evenso,wherethegradientin orography is steep,it is almostimpossibleto

devise a consistent scheme that is free from appreciable error.

σ �V� *⁄=

� * ρ∂Φ∂σ-------–=

∂V∂ �------- V ∇V σ̇

∂V∂σ-------+⋅+ ∇σΦ–

� � *---------∇� *– � k V×–=

d � *ln( )d�-------------------- ∇σ V ∂σ̇∂σ------+⋅+ 0=

∂� *∂ �--------- ∇ � * �( )⋅–

∂ � σ̇( )∂σ

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

σ̇ 0= σ 0= σ 0=

∂� *∂ �--------- ∇ � * �( ) σd⋅

0

1

∫–=

σ 0= σ̇

� * σ̇– σ∂� *∂ �--------- ∇ � * �( ) σd⋅0

σ

∫+=

∇Φ– � � *---------∇� *–



ECMWF, 2002 21

Over steeporography, thesigmalevelsslopeappreciablyeven in thestratosphere(seeFig. 3 ) andtheproblems

with thehorizontalpressureforceremain.It seems,therefore,thataschemewhichcombinestheadvantageof sig-

macoordinatesnearthesurfaceandpressurecoordinatein thestratospheremayproveadvantageous.Suchschemes

exist and are known as hybrid coordinates.

-coordinates

-coordinates

-coordinates

Figure 3Illustration of the three vertical coordinate systems.

4.3 Hybrid coordinates

Therearevariouswaysof defininga coordinatesystemthathasthepropertiesof sigmacoordinatesin the lower

atmosphere and pressure in the stratosphere.Arakawa and Lamb (1977) use the definition

(92)

where

�

�

σ

ηW �XWY� I–( )π

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



ECMWF, 2002

(93)

where is an interfacialpressure(e.g.nearthe tropopause)and is thepressureat the top of themodel.In

fact, setting gives the standard -system.In this hybrid systemthe boundaryconditionsare

applied at .

SimmonsandBurridge(1981)showed that an alternative way of defininghybrid coordinateswasto definethe

half-levels by the relation

(94)

where is a constantpressure.This form of coordinateallows a direct control over the flattening of the

coordinatesurfacesasthepressuredecreases.Notethatwe candefinesigmacoordinatesor hybrid coordinatesas

in (89) in the same manner as(91), so that for sigma coordinates

(95)

An alternative form of hybrid coordinates equivalent to the Arakawa–Lamb system is

(96)

Thereasonfor definingthehalf-levels in thisway is dueto theappearanceof termsinvolving in

theequationswhich,whendiscretized,become (see,for example,thecontinuityequa-

tion (78)).

4.4 Isentropic coordinates

In coordinates(isentropic),for dry adiabaticmotions,potentialtemperature is conserved,sothattheenergy

equation becomes and the isentropic surfaces become material surfaces.

From(74), the horizontal pressure force is

Using where , we have

i.e.

Hence, using equation of state, we have

(97)

π� I � T for � T � � I≤ ≤–� * � I for � I � � *≤ ≤–

=

� I � T� I � T 0= = ση 1±=

�XW 1 2⁄+ 8ZW 1 2⁄+ � 0 J�W 1 2⁄+ � *+=� 0

�FW 1 2⁄+ σW 1 2⁄+ � *=

�XW 1 2⁄+ � I 1 2⁄+ ηW 1 2⁄+ηI 1 2⁄+----------------- for η W 1 2⁄+ ηI 1 2⁄+≤=

�XW 1 2⁄+ � I 1 2⁄+ η W 1 2⁄+ ηI 1 2⁄+–( ) � * � I 1 2⁄+–( )1 ηI 1 2⁄+–( )-------------------------------------------------------------------------------+=

�FW 1 2⁄+ ∂� ∂η⁄∆�XW �FW 1 2⁄+ �XW 1 2⁄––=

Θ ΘdΘ d�⁄ 0=

1 ρ⁄( )– ∇Θ� ∇ΘΦ–Θ Π⁄ Π �V� 0⁄( )κ=

∇Θ Θln( ) 0 ∇Θ ln( ) κ∇Θ �ln( )–= =

1�----∇Θ�–

�� ---------∇Θ =

1ρ---∇Θ�– ∇Θ � =



ECMWF, 2002 23

Thehorizontalpressureforcecanthusbewrittenas or where is called

the Montgomery stream function. The hydrostatic equation(73) can be written .

Again, using the definition for above we have

Differentiating w.r.t. gives

Substituting for in the hydrostatic equation, using the equation of state and rearranging, we obtain

(98)

or alternatively

For the continuity equation it is convenient to use one of the forms(78) or (79) with replacing

The boundary conditions are, at the top and at the earth’s surface .

The isentropiccoordinateshows (aswith the pressurecoordinate)the advantageof a simplerpressuregradient

force,but alsothedisadvantageat thesurfacewhich is intersectedby -surfaces.It is possibleto introduceyet

anotherhybrid schemeusing -coordinatesnearthesurfaceand -coordinatesaway from thesurface.However,

how to join or overlapthesesystemsis not asstraightforwardasin thesigma–pressuretypeof hybrid coordinate.

For further details seeBleck (1978) and references therein.

5. VERTICAL DISCRETIZATION

Eventhoughcomputerpowerhasincreasedenormouslyin thelast10to 15yearstheverticalresolutionin primitive

equationmodelsis quitelow, 10 to 20 levelsin theverticalbeingtypical andtheselevelsaregenerallyirregularly

spacedto takeaccountof thestrongvariationof atmosphericquantitiesin thevertical.For example,highresolution

neartheearth'ssurfaceis requiredwith many formulationsto representthetransitionbetweentheturbulenceregion

close to the surface and the free atmosphere.

However, thereareanumberof problemsarisingfrom thechoiceof verticalcoordinatesystemandthedistribution

of discretemodellevelsin thevertical.Firstly, theinevitablepresenceof anartificial upperboundarycaneventually

contaminateforecasts.Theconclusionof several theoreticalstudiesis thathaving themodeltop at zeropressure

is, in somerespects,asa resultof inevitablefinite-differenceerrors,equivalentto having themodeltop at some

smallfinite non-zeropressure.On theotherhand,it hasbeenpointedout thattheinfluenceof loweringtheupper

boundaryis not damagingto thestructureof theplanetarywavesin thetroposphere,if it is placedno lower than

themiddlestratosphereandif themodelhasseverallayersin thestratospherewheredampingeffectscanattenuate

waves reflected from the top.

∇Θ � Φ+( )– ∇ΘC– C � Φ+=∂� ∂Θ⁄ ρ ∂Φ ∂Θ⁄( )–=

Θ

Θln κ �� 0------ln–ln=

Θ

1Θ----

1

----∂

∂Θ------- ��0�-----------

∂�∂Θ-------–=

∂� ∂Θ⁄∂

∂Θ------- � Φ+( ) � Θ------------=

∂C

∂Θ--------- Π=

Θ ζ

Θ̇ 0= Θ̇ ∂Θ ∂ �⁄ V* ∇Θ*⋅+=

Θσ Θ



ECMWF, 2002

Experimentally(Mechosoet al., 1984),it hasbeenshown that the impactof theupperboundaryon forecastsis

considerableat theuppermostmodellevelsin all cases,andthatloweringtheupperboundaryto thelower strato-

spherecanaffect significantlymiddle andlower troposphericforecastsin somecases.The comparisonbetween

integrations with 16 and 19-level models described below supports these results.

The detrimentaleffects of using terrain-following coordinateshave beenthe subjectof study for many years.

Small-scaleerrorsandcomputationalnoisewherethecoordinatesurfacesaresteepor rapidlychangingin thehor-

izontalaregeneratedasa resultof thediscreteapproximationsmadefor thetermsin thepartialdifferentialequa-

tions.Theparticularproblemof thepressuregradientforceis well known. Thehorizontalpressuregradientforce

(see equation(91)) in a -coordinate model is given by

andnearsteeporography the nearcancellationof the two termsin this expressionis a potentialsourceof large

spatial truncation errors.

Theseerrorscanbedecreasedto someextentby formulatingcarefullythedifferencingapproximationsfor thehy-

drostaticequationandpressuregradientforce,but cannotbetotally eliminatedaslongasaterrain-following coor-

dinateis used.Thereis alsosomeconcernabouttheaccuraterepresentationof theadvectionof quantitiessuchas

moisturenearsignificantirregularitiesin thecoordinatesurface,andtherearealsoproblemsin the treatmentof

horizontaldiffusionterms.Overall,theundesirableeffectscanbepartlyremovedby usingahybrid coordinatesys-

temwhichchangessmoothlyfrom aterrain-following specificationnearthelowerboundary, to ahorizontalor iso-

baricdefinition in theuppertroposphereandstratosphere.An alternative methodhasalsobeendeveloped,using

quasi-horizonalcoordinatesandrepresentingmountainsassteps,althoughthishasnotbeentestedexhaustively in

practiceasyet.A full discussionof thenumericalproblemsassociatedwith terrainfollowing coordinatesystems

is given inMesinger and Janic (1984) andSimmons and Burridge (1981).

In comparisonwith theschemeusedfor horizontaldiscretization,theverticaldiscretizationschemein usein the

majorityof large-scalemodelsof theatmospherecanbedescribedasrudimentarybased,asthey generallyare,on

simplefinite differencesthatprovide second-orderaccuracy, at best,with the irregularity of the level spacingre-

ducing the local accuracy to first order.

Very few attemptshave beenmadeto employ theuseof analyticalfunctionsthatarecontinuouslydifferentiable

anddefinedglobally; thetwo reallypotentiallyusefulapproaches—byFrancis(1972)usingLaguerrepolynomials

andby MachenhauerandDaley (1972)who usedLegendrefunctions—have not led to practicalschemesfor nu-

mericalmodelsof theatmosphere.Theuseof Laguerrepolynomialsleadsto severestability criteria,whereasthe

use of Legendre polynomials gives rise to computational problems at the upper boundary.

In contrastto theglobal functionalapproachesof FrancisandMachenhaurandDaley, theuseof local functional

approachesbasedon Galerkinimplementationsof finite elementshasbeenfairly successful.A successfulfinite

elementformulationfor the vertical discretizationin -coordinateprimitive equationmodelswasdescribedby

StaniforthandDaley (1977),anda 36-hourforecastfrom realdatawaspresentedto demonstratetheviability of

the method.

Theremainderof thissectionis devotedto adescriptionof averticalfinite differencingschemefor a -coordinate

model.

5.1 Vertical differencing schemes for baroclinic primitive equation models

In this subsectionI shall discussthemethodologyunderlyingthedesignof many verticaldifferencingschemes.

σ

∇Φ– � � *---------∇� *–

σ

σ



ECMWF, 2002 25

The approach is based on the imposition of some important conservation laws.

The equations introduced inSection 1 may be rewritten as

(99)

(100)

(101)

(102)

with

(103)

For completeness we again state the upper and lower "no flux" boundary conditions

(104)

For ourdiscussionweshallrequiremassandenergy conservation.In particular, wewantthework doneby pressure

forces to be balanced by the changes in the total potential energy arising from (the -term).

The integrated continuity equation is

(105)

The divergence term in(105) integrates to zero over the globe, which means that the total mass is conserved.

A "kinetic energy" equation can be derived from the momentum equations in the form

(106)

∂� s∂ �--------- ∇σ � sv( )

∂∂σ------ � sσ̇( )+⋅+ 0 continuity of mass=

∂v∂ �------ [ k v× σ̇

∂v∂σ------ ∇σ Φ

&+( ) � ∇σ � sln( )+ ++ + 0 momentum=

∂∂ �----- � s � ( ) ∇σ � sv � ( )

∂∂σ------ � sσ � ˙( ) � ωσ-------------–+⋅+ 0 thermodynamics=

∂Φ∂ σln( )----------------- � hydrostatic equation–=

[ � 1� ϕcos----------------∂ �∂λ------

∂∂ϕ------ � ϕcos( )–

+=

& 12--- � 2 � 2+( ) 1

2--- v v⋅( )= =

ω � sσ̇ σ ∂� s∂ �--------- v ∇σ� s⋅+ +=

� sσ̇( )σ 0= 0=� sσ̇( )σ 1= 0=

� ω( ) σ⁄ ω

∂� s∂ �--------- ∇σ � sv σd

0

1

∫⋅–=

� sv ∂v∂ �------ � sσ̇v∂v∂σ------ � sv∇σ & � sv ∇σΦ � ∇σ � sln+( )⋅+ +⋅+⋅

or

0=

� s∂&

∂ �------- � sσ̇∂&

∂σ------- � sv∇σ & � sv ∇σΦ � ∇σ � sln+( )⋅+ ++ 0=



ECMWF, 2002

Using the continuity equation,(99) and(106) may be rewritten as

(107)

In orderto constructthetotalenergy equationweneedto expressthepressuregradienttermsin analternativeform.

Using the continuity and hydrostatic equations we have

(108)

Combining(107) and(108) we have

Now

Addition of thekineticenergy equationandtheflux form of thethermodynamicequation(equation(101)) andin-

tegration from to 1 gives

(109)

Equation(109)is anexpressionof theenergy conservationlaw for the -system(99) to (103). Thehorizontaldi-

vergencetern in (109) integratesto zeroover thewholeglobe,which meansthat the total energy, potentialplus

kinetic, is conserved.

∂∂ �----- � s

&( ) ∇σ � sv&( ) ∂∂σ------ � sσ̇

&( ) � sv ∇σΦ � ∇σ � sln+( )⋅+ +⋅+ 0=

� sv ∇σΦ⋅ ∇σ � svΦ( ) Φ ∇σ � sv⋅( )+⋅=∇σ � svΦ( ) Φ ∂� s∂ �---------

∂∂σ------ � sσ̇( )+

+⋅=

∇σ � svΦ( ) ∂∂σ------ Φ σ∂� s∂ �--------- � sσ̇+

∂Φ∂σ------- σ

∂� s∂ �--------- � sσ̇+ –+⋅=

∂ � s&( )∂ �------------------ ∇σ � sv

&( ) ∂∂σ

------ � sσ&̇( ) ∇σ � svΦ( ) ∂∂σ------ Φ σ∂� s∂ �--------- � sσ̇+

+⋅+ +⋅+

∂Φ∂ �------- σ

∂� s∂ �--------- � sσ̇+ – � � sv ∇σ � sln⋅+ 0=

∂Φ∂ �------- σ

∂� s∂ �--------- � sσ̇+ – � � sv ∇σ � sln⋅+

�

σ

--------- σ∂� s∂ �--------- � sσ̇+ � v ∇σ� s⋅+

� ωσ

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

σ 0=

∂∂ �----- � sΦs � s

& � +( ) σd0

1

∫+

∇σ � sv & � Φ+ +( ) σd0

1

∫⋅+ 0=

σ



ECMWF, 2002 27

Figure 4Vertical disposition of variables

Thedispositionof variablesandthegrid structurein thevertical for the -coordinatemodelwe shallbeusingis

illustratedin Figs.4 and5 . Theprimaryvariables, and , arekeptat thefull levelsandtheverticalvelocity

and the geopotential are kept at half levels. The variable grid spacing is defined by

(110)

σv σ̇

∆σ W∆σW σW 1 2⁄+ σW 1 2⁄––=



ECMWF, 2002

Figure 5Vertical distribution of levels for the 16-level (top) and 19-level (bottom) models.

(i) The equation of continuity. The continuity equation is taken in the form

(111)

Multiplying (111) by and summing (integration) from gives

(112)

For we have

(113)

∂� s∂ �--------- ∇σ � svW( )

∆ � sσ̇( )W∆σ W---------------------+⋅+ 0=

∆σ W * 1 … \, ,=

σW 1 2⁄+ ∂� s∂ �--------- ∇σ � sv W( )∆σW]� sσ̇ W 1 2⁄++⋅[ ]W 1=

^∑+ 0=

� sσ̇1 2⁄ 0=( )\ D=

∂� s∂ �--------- ∇σ � svW( )∆σW⋅W 1=

^∑– ∇σ � svW( )∆σ W3 1=

@∑⋅–= =



ECMWF, 2002 29

Equation(113) is a finite differenceanalogueof (105), which obviously conservesthe total (global)massof the

model atmosphere.

(ii) Energy conservation. The finite difference momentum equation is

(114)

where with .

The finite difference kinetic energy equation is

and this may be rewritten as

(115)

Using the finite difference continuity equation we may rewrite equation(115) in the flux form

(116)

where .

In order to derive the analogue of the identity(108) we need a hydrostatic equation; we use

(117)

For the finite difference model we have

∂vW∂ �-------- [ k v

12---

σ̇ W 1 2⁄+ v W 1+ vW–( ) σ̇ W 1 2⁄– vW vW 1––( )+∇σσW---------------------------------------------------------------------------------------------------

+×+

∇σ ΦW & W+{ } � W ∇σ � sln+ + 0=ΦW αW ΦW 1 2⁄+ βW ΦW 1 2⁄–+= αW βW+ 1=

� svW ∂vW∂ �--------12--- � sv W σ̇W 1 2⁄+ vW 1+ vW–( ) σ̇W 1 2⁄– v W vW 1––( )+∆σW---------------------------------------------------------------------------------------------------

⋅+⋅

� svW ∇σ& W_� svW ∇σΦW`� ∇σ � sln+{ }⋅+⋅+ 0=

� s∂& W∂ �----------

12---� sσ̇W 1 2⁄+ v W 1+ vW⋅ � sσ̇W 1 2⁄–– vW vW 1–⋅

∆σW------------------------------------------------------------------------------------------------ 1

2---vW v W ∆ � sσ̇( ) W∆σW---------------------⋅–+

� sv W ∇σ & W]� svW ∇σΦ W`� ∇σ � sln+{ }⋅+⋅+ 0=

∂∂ �----- � s

& W( ) ∇σ � sv W & W( ) ∆σ � sσ̇v2

σ

2-------------------

W� sv W ∇σΦW`� ∇σ � sln+{ }⋅+ +⋅+ 0=

)

v2σ

W 1 2⁄+ v W 1+ v W (geometric mean)⋅=

)

∆σΦ∆ σln-------------

W � –=



ECMWF, 2002

(118)

Note:

If we requiretotal energy conservationthentheform of thelasttermin (118)is a constrainton our choicefor the

term, , in the thermodynamic equation.

The thermodynamic equation for our model is

(119)

If the term is chosen to maintain energy conservation we have

(120)

The term can be interpreted as a definition of for the model.

From(120)wenotethatwestill haveadegreeof freedomremaining,namelythechoiceof and , subjectto

. Theseweightscanbe chosenin sucha way that no spuriousangularmomentumis generatedby

pressure forces,Simmons and Burridge (1981).

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� svW ∇σΦW⋅ ∇σ � svW ΦW( ) Φ W ∇σ � svW( )⋅–⋅=∇σ � svW ΦW( ) ΦW ∂� s∂ �---------

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αW βW+ 1=



ECMWF, 2002 31

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Documents

Adiabatic formulation of models - ECMWF · 2.2 The shallow water equations 2.3 Map projections 3 . Horizontal discretization 3.1 Finite differences 3.2 Spectral technique 3.3 Finite-element