T. Satomura

April1988

26r

Dynamic Normal Mode Initialization
for a Limited-area Model

By TakehikoSatomura(*)
CentreNational de RecherchesMit1orotogiquesIEERM
31057 ToulouseCEDEX,France(Manuscriptreceivedl4 November1987. in revisedform 29 Januarv 1988)

Abstract
The dynamic normal mode initialization scheme (DNI) proposed by Sugi is applied to alimited-area model. Performance of two integration schemesfor DNI is examined. and it is shown that
Okamura's time integration scheme is inefficient and a backward implicit scheme is appropriate for
DNI applied to the limited-area model. A1l physical processescan be inciuded in the integration cycleof DNL
Horizontal wind and precipitation rate are compared usrng a control forecast, a forecast with thenonlinear normal mode initialization (NMI). and with DNL The deformation of horizontal wind neart h e m o u n t a i n s i s r e c o n s t r u c t e d b y D N I b e t t e r t h a n b y N M I . T h e a r e a - a v e r a g ep
dr e c i p i t a t i o n i s n o tlargely improved by DNI, but the rms error of the precipitation from the control forecast is smaller in
DNI than in NMI in the first three hours.

l.

lntroductiontion, and normal mode initialization(NMI)(- I ).
Limited-areanumerical models have been The NMI method is efficient in eliminating
developedto study and to forecastmeso-scale high-frequency gravity waves from the lnitialw e a t h e rp h e n o m e n as i n c e t h e m i d 1 9 7 0 ' s .T h e field and it is appiied to limited-areamodels
primitive equationsare used as the governing ( B r i d r e , 1 9 8 2 , B o u r k e a n d M c G r e g o r ,1 9 8 3 ,equations of the model, and they allow theC r a p i e t ,1 9 8 5 ; B i j l s m aa n d H a f k e n s c h e i d1 .9 8 6 )propagationof gravitywaves.In the atmosphere, as weil as to global or hemisphericmodelsfor
fast propagating gravity waves typicaliy have which NMI was originaliy formuiated.It is wellsmall amplitudescomparedwith the ampiitudes known, however. that NMI has one seriousof the more siowly propagatingsynoptic scale defect: the iterativemethod used in NMI does
weather systems. In numerical atmospheric not convergeif one inciudesa1lverticalmodes
models,however,large-amplitudehigh-frequency and/or diabatic processes (Williamson andgravity wavesare observedunlessspecialcareis
T e m p e r t o n 1. 9 8 1 ;B a l l i s ha n d B a e r ,1 9 8 5 :R a s c h ,taken in specifyingthe initial conditions.The
1 9 8 5 a ) .A l t h o u g h a n u n d e r r e l a x a t i o nm e t h o dspecificationof suitablybalancedinitial fieldsis(Kitade, 1983) and a secant approximation
calledthe initialization.
method (Rasch. 1985b) were proposedto
Three types of the initialization are so far
improve this defect, they could not achieve
proposed:staticinitialization,dynamicinitializa- c o m p i e t e s u c c e s sa s p o i n t e d o u t b y R a s c h(*)Permanentaffiliation:

Meteorologicai

I n s t i t u t e , N a g a m i n e ,T s u k u b a . 3 0 5 J A P A N
O 1 9 8 8 , M e t e o r o l o g i c aSl o c i e t yo f J a p a n

Research(+1) For comprehensive reviews. the reader who isinterested in the former two methods is referred toB e n g t s s o n( i 9 7 5 ) a n d w h o i s i n t e r e s t e di n t h e N M I i s t o
D a l e y( 1 9 8 1 ) .

262

Journal of the Meteorological Society of Japan( 1 9 8 5 a , ba
) n d b y B a l l i s ha n d S e l a( 1 9 8 5 ) .K u d o(1984) proposeda new iterationmethod which
was similarto the method of Rasch(1985b).It
appearsto be more robust than Rasch'smethod
but. unfortunately. its characteristicshave notbeen describedin detail. Another initialization
method cailed the bounded derivativemethod(Browninget a1.,1980) also fails to convergein
thesecases(Ballishand Baer,1985).
In addition, it is difficult in limited-areamodels to define normal modes which satisfy
time-dependent
boundaryconditions.Progressrr
relieving this difficulty is being made (Bourke
and McGregor.1983;Juvanondu Vachat,1985).
Recently, a method. simpler than NMI inconcept,was proposedby Bratseth(1982) and
was applied to a global forecast model
successfuliy by Sugi ( 1986). All physicalprocessesused in the model were taken into
account through the initialization but noproblem of convergence was found. The
procedure of this method is summarizedas
follows: l) integrate the model equationsin a
forward/backwardcycle with a selectivedamping
schemesuchas the Euler-backwardor Okamura's
scheme (for these schemes,see Haltiner andWilliams, 1982), 2) during the integrationcycle,
keep the nonllnearterms constant,and 3) after
integration cycles,update the nonlinearterms
and repeat the integration cycle and thenonlinearupdating.This method is, therefore.avariation of dynamic initialization.Sugi (1986)
a l s o p r o v e dt h a t t h i s m e t h o d i s e q u i v a l e n tt o
N M I u ' i t h a f r e q u e n c yd e p e n d e nut n d e r r e l a x a t i o n f a c t o r , a n d n a m e d i t " d y n a m i c a ln o r m a l
m o d e i n i t i a l i z a t i o n "( h e r e a f t e r ,w e a b b r e v i a t e
i t t o D N I f o l l o w i n g S u g i ) . T h e o l d v e r s i o no f
dynamic initialization was also capableof
el i m in a t i n g h i g h- f r e quenc y g r av i t l ' \ \ , av e s
f r o m t h e i n i t i a l f i e l d o f a h e m i s p h e r i cm o d e l( T e m p e r t o n .1 9 7 3 , 1 9 1 6 ) . I t u s e d , h o w e v e r ,m u c h m o r e c o m p u t a t i o n a lt i m e t h a n D N I
b e c a u s et h e n o n l i n e a rt e r m s r , i ' e r eu p d a t e d a t
e a c hf o r w a r d , , ' b a c k u , airndt eg r a t i o n .
In this paper, DNI will be applied to alimited-areamodel. As Anthes (1983) pointed
out. physical processesare important in alimited-areamodel.In somelimited-areamodels.
NMI rs alreadyused (as remarkedeariy in thisV o l .6 6 , N o . 2

section) and has succeededin eliminatinggravity
oscillations. Only few vertical modes, however,are initialized without irreversible processes
because of the convergenceproblem. Higtrly
diabatic phenomena such as heavy rainfall
associatedwith a cold front may be suppressedby the adiabaticNMI as the Hadiey circulationand tropical depressions
were suppressedby it rngiobalmodeis(e.9.Kitade, 1983;Rasch,1985b).
Therefore. DNI, in which the problem of
convergencewas not found in the examination
by Sugi (1986), is an attractive method. In
addition,the normal modesarenot calculatedin
DNI and, therefore. the boundary pioblems
associatedwith this areabsent.
The choiceof time integrationschemegoverns
the efficiency and the stability of DNL Fromtheseview points, we sha11
discusstwo schemes:
Okamura's scheme and a backward implicit
scheme. In the case of the backward implicit
scheme,it will be shownthat we haveto evaluate
implicitly even the nonlinear terms to performthe integration cycie stably. By estimatingthe
noniinear terms approximately, we will show
some numerical resultsand comparewith theresultsof NMI.
2.

Time integrationschemefor DNI

As mentionedin section1, it is importantfor
DNI to choose an efficient time integration
scheme.In this section,we discussthe efficiency
and the stabiiity of two seiectivedamping
schemes:Okamura's scheme and a backward
implicit scheme.First. their linear characteristics
are examinedand. next, the effectsof nonlinear
termsareevaiuated.
2.I Lineqr characterisfics(a) Okamura'sscheme
This schemeproposedby Okamura (Haltiner
and Wiliiams,1980),which wasusedto initialize
the FSU globalmodelby Sugi(1986).consistsof
a forward step. a backward step. and an
extrapoiation.
u'**/':utr*/t(Ag)t,u t x r o : u t { t t - / 1 t ( AL u * ' ) t + / luL".t:JYtr-2ut**,(2.1)

April 1988

T. Satomura

where lt is the time step, At is the time
derivative, n is the number of the integration
cycle. The linear responsefunction of one cycie
forward-backwardintesrationis

R y ( a ) : + - -2 (to /t)" ,(2.2)

where only the linear terms are consideredand &.)is the frequencyof the normal mode.
A necessary condition to complete theintegrationcycle stablyis givenby(2.3a)lRr(rr.r-"")l<1,

where rd-u* is the maximum frequency of themodei normal modes. The condition (2.3a) isrewritten as

gn

16"

4n
Perlod2hJhFig. 1. Damping characteristics of Okamura's scheme(dashed line) and of the impiicit scheme
withn:740
withn:32(solid line).

rewritten as
l o gp ( a " )l o g { l l - 2 ( r o l"a ^ ^ ) ' t l(27)( 2 . 3 b ) where
ro5,is the frequency to be reduced.The0(ot^^*/t 1l .

curve (2.7) is graphedin Fig. 2 and the numberof cyclesneededfor the globalmodel used by
Sugi (1986) and for the limited-areamodel weuse are indicated in the figure by FSUGCM andPERIDOT, respectiveiy.Using (2.4), we can(2.4)
estimatethe number approximatelyas

For the model used (for the brief descriptionof the mode1,see the next section), zrr-^* isestimatedas. /--.L!

0J-". =

LL

p6vys/ l^v m r n,*ftfiffi

where C7,-u* is the maximum phasevelocity of, nd
t h e m o d e l n o r m a l m o d e s( - 3 0 0 m / s e c )alx^rn is the minimumgrid sizeof the model(30km).Thus(2.3) becomes(2.8)

Thus, the number of integrationcyclesis very
sensitiveto the grid size.Usually, /x-1^ in the( 2 . 5 ) limited area model is smaller than that in the0</t{35 sec.
global model and, therefore, the efficiency of
The damping factor after n integration cyclesis
Okamura'sschemeis much reduced.
givenasp(a):R ' ( r , r )L " : i l - 2 Q t l t ) ' ) "

.

(2.6)

Therefore,in order to reduce the amplitudesof
fast oscillationswhose periods are less than 2hours, for example,to be lessthan one fourth of
the initial value,we must compieteat least740
cycles.The damping factor p(<o) in this caseis
shownin Fig. I by the dashedline.
The fact that many integrationcyclesmust be
performed to reduce the fast osciilations
indicatesan inefficiencyin Okamura'sscheme
for iimited-area models, which was aiso
mentionedin Sugi (i986). The inefficiencycan
be describedanotherway as follows. Choosingt h e n u m b e r o f c y c l e sn e e d e di s
Jt:a^'^*,

400
n

300

200

100

t09p
'' joq( 1 1 - u2 z r t u z ^ " , 1 1
p = 00 5C

c o5orRtoor
dc /0.0,

FSU 0 15
GCI/

Fig. 2. Number of necessary integration cycles asfunction of the ratio o)cru^^,.

Journal of the Meteorological Society of Japan

In the model we used, the computation time
on a CRAY-I is 0.6 sec for one cycle of
Okamura's scheme; 440sec to integrate 740
c y c l e s . T h i s i s l o n g e rt h a n t h e c o m p u t a t i o n a ltime for a l2 hour forecast by the model.
(b) Backward implicit scheme
To overcome the probiem of Okamura's
schemedescribedabove. consider the backward
implicit scheme:
I uti'l

At : u l , , +/ t ( A , u t * / t ,*)ot\ u ' , *r : u L - - / t ( A , u , *r ) ' .

(a) Okamura'sscheme
With (2.11)<2.13), the integration cycle(2.1) is rewrittenas
f[fI/':X'"11* / t (io * a)l,o' 1X'"*:X'"*(l ial t) / t \i (o a) alx'",I
lXt"*t:3Xt"-2/.\* ,(2.14)

where the nonlinear term is not changedin thecycie.The responsefunction isiq-',
R ; ( r r r :) l l * ( a l t ) " |wherethe subscriptTis dropped.
We may now derive responsefunctionsand
the conditionsfor /t.

(2 .e )The linear responsefunction of one
infagretion

Vol. 66, No. 2

R " ( a r )- - 1 - 2 a ( o * i a ) / r ' .

(2.10)and the scheme(2.9) is unconditionally stable.
As far as linear terms are considered, nocondition for /t is imposed,and it permitsus todecrease
the number of integrationcyclesn. lnfact for lt:240 sec,n is oniy 32 and almostthesame damping characteristicsas in the previous
subsectionare obtained.The damping factor for
n=32 and lt:240 secis alsoshownin Fig. I by
the solid line.
2.2 Effects of nonlinear terms
By consideringthe effects of nonlinear terms
in a simpie manner, we can derivemore precise
responsefunctions and necessaryconditions forthe time step /t. As a first step, we rewrite thegoverningequationsof atmosphericmotion as(2.15)The stability condition I R (ar) < I is written as(/

n2 .t -t\O'/J/ t ' < l|a. o ( l + + ) f(2.16)

Further,we can rewrite(2.16) asao)-j,(2.t7a)alt <l ,(2.r7b)

and

where
_

|

/ , , a ' \oo:!oo\l_rA),

is a frequencymodified by the nonlinearity.
The condition (2.17a) indicates thar the( 2 . i 1 ) schemewill be unstableif the fiequency o has theAra,:i<o1a1*ri(ar, ar,... , an) ,opposite sign to the frequency of the correswhere a, and a1 are the amplitude and the
ponding normal mode @. A breakdown of
frequencyof the 7 th normalmode respectively, (2.11a) would occur if wind is strong and thea n d r i i s t h e n o n l i n e a rt e r m . E q a t i o n( 2 . 1 1 )i s
Doppler-shiftedfrequencyhas the oppositesign.
derived by a proper linear transformation (e.g. On the other hand, (2.11b) is the sameformasDaley. 1981). If we presumethat a, is of the(2.3b) except e)^n* is replacedby rrr.Sinceo,rform
could be larger than rrr-u* by, for example,the(2.t2) Doppler shift, (2.17b) restrictsthe time step toai:XiexPl(iot-a)t).
shorterthan (2.3b).
where o, and ai are the frequency and damping
index when the nonlinearity is included. then

(2. I 1) canbe rewrittenas
7:li(o-a)-ala,(b) Backward implicit scheme
Explicit evaluationof the nonlineor teftn.
Using(2.11) and (2.12), the integrationcycle(2.l3)(2.9) is rewrittenas

665

T. Satomura

April i988

a'
( X ' i/ ' : / " t * i a l t X ' * * + / t \ i ( o - a ) - d l T i ,1
ot-io/ tx.L,**4t- lt li (o -1!,)- a\x'" ,l.x;*r- 7'o"function is
The response
R1.(ar):( 2 .l 8 )where the nonlinear term is evaiuatedin an
cwnlinifm2nnerfunctionis
The response-u\o-aIial/t'n /_.\_ |,Rru(<rr): ------T-t-;-i,i--,t-Tu) ut

- l l a ) / t " 1 2 +a ' o f / t n - ,ll-a(oxN-:@=''(2.20)
To know the characteristicsof the scheme(2.18), examine two extreme cases: zero
nonlinear damping(a:0) and zero moduiation
to the frequercy (o--a).
In the case of the zero damping, (2.20)
becomeso(o-Zot\/t'<"/ n(2.2r)

aand.in the caseof the zeromodulation,(a'-a') lt !2 .

(2.22)

The condition (2.21) is rather complicatedandexaminationof it in detail would not be useful.
T h e c o n d i t i o n( 2 . 2 2 ) i s , o tnh e o t h e r h a n d ,s i m p l e
and can be interpretedclearly:if the damping(orthe ampiifying) effect of the noniinearity rs
strongenough(i.e. a' ) r"), a conditionon /t
is imposed. In fact, it occurs in two cases,
0000UTC and 0600UTC 4 June 1986. In thesecases, the term which parameterizestheturbulent vertical flux is large and a time step
much smaller than 240 sec must be taken. It
reducesthe efficiencyof the scheme.

-o) /t(1*alt -iolt) (lliolt)(2.24)

We againexaminethe two extremesa:0 ando:.a. In the caseof zero damping a:0, thenecessarycondition for the stability, lR(ar) I S
1.becomes(2.2s)

oa7-0.

/ 1 t^9)l1,.

condition for the stability isand the necessary

l* alt -i(oT h i s c o n d i t i o ni s t h e s a m ea s ( 2 . 1 7 a ) a n di t a l s omeans that the schemewill be unstable if. forexample,wind is strong enoughto reversethe
signof the frequencydue to Doppler shift.
ln the caseof zero modulation. o:tri, thestabilityconditionbecomes(l , - - *r" \ ' + , 2 9,,+' q , ' >, ,0 ,< o " t d -/\u'+d- )'\2.26)

which is always satisfiedinsofar as all variabies
It, a, anda are real.
In the following sections,we shall use thebackward implicit scheme (2.9). The implicit
evaluationof the nonlinearterms is best sinceno
condition for the time step was fcund in theaboveanalysis;it is difficult in practice,however.
An approximate schemeto estimate the terms
q,iil be used. and the detailsof this schemewili
be presentedin Appendix A. We shall use thev a i u e sn = 3 2 a n d / t : 2 4 0 s e c w i t h n o n l i n e a rterms updating once every 8 integration cycles.
The reasonwhy thesevaluesare chosenwlll be
explainedlater.
3.

The meso-scale
model

The three-dimensional
forecastmodel usedis
a 15 o-level, hydrostatic,limited-areamodel
developedby the French WeatherService(seeI m b a r de t a l . , 1 9 8 6 , 1 9 8 7 ;B o u g e a u l t1. 9 8 7 ;f o r
the detailed description of the model). The
forecastvariablesare surfacepressure,horizontal
Implicit evaluation of nonlinear term.
wind, temperature,and specifichumidity. The
The integrationcycle (2.9) is rewrittenin this model includesthe followingphysicaiprocesses:
caseassurface flux. vertical turbulent flux, radiation,
clouds, and precipitation by deep cumulus
Jt :/-'^+ 1t (io - a)xt*-tt ,lxL'
Iconvectionand by large-scalecondensation.lno' - ia / t/.t,*, - / t I i (o - a) - dl r'.,:t'.
\xr,, :r';addition,land-surface
processes
are describedby(2.23) soil temperatureand molstureusing two levels.

V o l .6 6 ,N o . 2

Journal of the Meteorological Society of Japan

Time integration uses a semiimplicit methodwith a leap-frog scheme and a low-passfilter(Asselin, 1972). The laterai boundary conditionis the Davies(191O relaxationmethod.
A 51 x 51 horizontalgrid matrix, centeredat
46N 3T, is usedwith a time stepof 240 secanda grid incrementof 35km at the center.
4.

Numerical experiment starting with
data.

real

In this section we will describe theof DNI with the backwardimplicit
characteristlcs
schemeand comparethem with those of NMI.
The initial data are the fine-mesh analysisof
0000UTC 4 June 1986. The preparationof the
input data for the fine-mesh anaiysisand thedetaiisof the method of the analysisis describedin Durand (1985), Imbard et al. (1986, 1987)
and Durand and Bougeault(1987).The analyzed
data have been prepared in the frame of the
Hapex-Mobilhyproject (Andr6 "t o1., 19$$) andare describedby Mercusotet al. (1986). NMI ofthe model includesonly adlabatictermsand uses
Machenhauer's
iterationscheme.It initializesthelargest two vertical modes.(*2) The surfacepressureand the temperatureat the lowestlayerof the model are not altered during initialization(seeBridre,19821Craplet,1985;for the detailed
descriptionof NMI of the model).
On the 4 June 1986. a cold front withaccompanyingprecipitationwas passingacrosscentral France as shown in Fig. 3a and strong
wind was reveaied,as shown in Fig. 3b. Thissituation would be severeon both NMI and DNI.
becauseNMI in the model does not include
diabaticprocesses
at all and because
DNI may beunstablein this strongwind condition.
4.1 Convergenceof integration cycle
Figure 4 shows the mean square rate ofsurfacepressurechange

B(p):(?m7J",(4.1)

wherep, rs surfacepressure.versusthe number(*2) The reason why NMI in the model initializes only
two vertical modes is to avoid large modification ofobserved surface data. If it is needed. the number of thevertical modes can be increased to four, at least(Juvanon du. Vachat. personal communication).

2 O 6 l

t

+

?/,.

'..s\\\s'.: ''-

:).'

,

1'

4,:':':':iiitr;:rr:i *::2fi ir
1::::ttiii

j j j l j I . (. \ \ \ " \ N N i ^ - i

BRSE 9/ 5.f'.

q

.

;: ;\

0k'U I ECH. 0 aX(b)
Fig. 3. The initial atmospheric state at 0000UTC 4
June 1986. (a) Synoptic chart, (b) horizontal windat the lowest layer of the model produced by thefine-mesh analysis.

of integration cycles. Although we use anapproximationto the exact implicit evaiuationof
the nonlineartermsas describedin theAppendix
A, the shocksby the nonlinear-term
updatingare
small and ,B(p,) decreases
rather smoottrll,by
two orders.

26'7

I. Satomura

BAL
104
10"
102

100
8

16
Step

24

32

F i g . 4 . M e a s u r e o f b a l a n c e- B ( p " ) a s a f u n c t i o n o f t h e
n u m b e r o f i n t e g r a t i o nc - v c l e s .
xt0-6

l0-l

1

2

3
V e r l i c oMl o d e

45

Fig.6. Measure of balance BAL as a function ofvertical mode. The vaiue before the initialization isindicated bv the solid line.

a good balanceat the secondverticalmode,DNI
and NMI have aimost the sameBAL at the firstmode. At the higher modes, DNI shows goodit initializesall verticalmodes,inbalancebecause
contrast to NMI which initiaiizes onlv the firsttwo modes.

0
i
2
3
4
numberlerm updolingof nonlheor
Fig. 5. Diabatic heating rate as a function
number of nonlinear term updatings.

of thediabatic
Figure 5 shows volume-averaged
heatingrate for each nonlinear term updating.It
is nearly constant in this caseand indicatesthat
updating 4 times is sufficient for obtaining thestate.
convergedby an
The degreeof balanceis represented
index BAL:BAL:ZIA,Zil,,

4.2 Surfacepreswre
Figure 7 shows the time variation of surfacepressureat three points:Paris,Toulouse,and thecoastof the CorsicaIsiand,which are markedin
Fig. 3b by P, T, and C, respectiveiy.
The fast
osciliationswhich haveamplitudes4-5 hPa andperiods0.7-2 hours are eliminatedby both DNI
and NMI. The initial surfacepressure
is modified
by 1-2 hPa by DNI, since DNI does not fix
surface pressure nor the iowest-layer air
temperature.Thesequantitiesare,however,notalteredduringNMI of the model.
After 6 forecasthours, the fast oscillafionsare
dampedalso by the model itself even if we do
not initialize the data,and the surfacepressureis
almostthe sameasthat of DNI and NMI.

(4.2)

J

whereZ, is the amplitudefor a gravity mode andthe summationis taken over al1 gravity modes.
Figure 6 shows BAL after NMI with four
iterations and BAL after DNI. Whi-leNMI shows

5.

Data-simulationexperiment.

In the previoussection,it was shownthat DNI
eliminated high-frequencygravity waves.It was
not shown. however,that the physicalquantities
after DNI were the desired ones, that is,Journal of the Meteorological Society of Japan-no iniliolizolion

hPo

| 004
I002
I 000

3

6
Time

9

12UTC

I

6
Time

9

!2UTC

3

6
Time

9

|2UTC

Fig.7. Time variation of surface pressure (a) at Paris,(b) at Toulouse, and (c) at the coast of the Corsica
Island, starting with real data.

comparableto real atmospherevalues.Unlesswe
observe the atmosphereperfectly, it is notpossibleto show it strictiy. A method of avoiding
the problem of the lack of observeddata is"identical-twin" method
so-calledor "datasimuiation" experiment.This method uses thecontrol data simulated by the model insteadof
the observeddata. We shall compare the data
after DNI and NMI with the control atmosphere
given by the simulation in the following
sub-sections.
An advantageof the method is thatail data are givenat the sametime-spaceposition,
and, therefore,there is no problemin comparing
them. But the results will be sensitiveto theforecasting model itself, as well as to theinitializationprocess.
A prerequisiteto a meaningfulexperimentis
that the control data are not affected by the
initial conditions but are determined by theV o l . 6 6 ,N o . 2

forecastmodel itself. As discussedin the previous
section, differences due to the initialization
schemesnearly disappearedafter 6 forecast
hours.The prerequisite,
therefore,appearsto besatisfiedand one may perform the experimentas
follows: as a control exneriment,usethe 6 to 12
forecast hours based on the initial data from
0000UTC 4 June i986 with NMI. A1so.regard
the forecastfor 0600UTC asthe "perfect" initial
data and the forecastby the control experiment
then represents
the "perfect" forecast.
To provide a smoothingeffect by an objective
analysisandto imitate imbalancesamongphysicalvariables, we take 9 x 9 points averagefor
horizontal wind for the perfect initial data, and.
then, we initialize the averageddata by DNI or
NMI. Finally. we compare the 6-hours forecast(i.e. from 0600 to 1200UTC),starting from theaveraged-initiaiized
data. with the control forecast. By this averaging,the wind data lose
structuressmaller than 300km; this is greaterthan the resolution of the fine-meshanalysisof
the model (-l00km). Therefore,for initialization schemes.this experimentmay be a more
severetest than in reaisituation.
5.I Surfacepressure
Figure 8 shows time variation of surfacepressure at three points. Both initiaiization
methods eliminate fast oscillations,as in theprevious section. Although the initial surfacepressuregiven by the DNI differs 0.2^O.5hPa
from that by the control and the NMI. theforecast of the surfacepressurewith DNI is notworsethan NMI.
5.2 Hoizontal wind
Figure 9 shows the initial horizontai wind in
the lowest layer, where the effects of mountains
are most significant. Using the 9 x 9 points
average, smail structures represented in theperfect data (Fig. 9a) have disappeared
asshown
in Fig. 9b; this is most evident near themountainsin the Alps and the Pyrenees.
We findthat the lowest-iayerwind is improvedlittle by
NMI (Fig. 9c) becauseNMI initializesonly the
first two vertical modes.By DNI. on the otherhand. the deformationof the wind due to themountainsis reconstructedto a certainextent

A p r i l1 9 8 8

T. Satomura

for NMI.
5.3 Precipitation
precipitaTime variation of the area-averaged
tion rate is shown in Fig. 10a. The averaged
precipitation rate in the first hour after DNI isintermediate between the perfect forecast andthe adiabatic NMI. Subsequently,the mean
precipitation rate recovers quickly in bothexperiments.The underestimationof precipitation by DNI is mainly due to the averaging
processused,becauseit almostdisappears
in thecasewhere the perfect data is initialized by DNI(closed circle in Fig. 10a). The result that theforecastwith NMI, which inciudesoniy adiabatic
terms, for the perfect data underestimaestheprecipitation(closedsquarein Fig. 10a)indicatesthe importance of the inclusion of physicalprocesses
in initialization process.
Figure 10b shows the rms error from thecontroi experiment for the precipitation. The
forecast of the first three hours after DNI is
9
4
2UTC
Time
better than that after NMI, while the difference.--- control
after the 4 hours forecast.
disappears
no iniliolizotion
Figures1lb and 1lc show differencesof the---- NMI
precipitation in the first one hour after DNI andafter NMI, respectively, from the control(Fig. 11a). Both forecastswith DNIexperimentFig. 8. As in Fig. 7. except for the simulation-data
andNMI
underestimate
the maximum precipitaexperiment.
tion, but the difference from the controi
experiment is smaller in the forecastwith DNI(Fig. 9d). The improvementis stiil inadequate,
than that with NMI.
but complete reconstructionis not possible
uniess the wind field is forced to adjust the
5.4 Computation time and stability of DIttI
pressurefield. for example.The result, however,The computation time (CPU time) of thelndicatesthat, in order to reconstructthe wind implicit DNI for the operationalmodel of theover mountains.higher vertical modesshould be FrenchWeatherServiceis around30 secon the
inciuded in the initialization. In the problem of
CRAY-2 for the case of nonlinearupdating 4
the inclusionof the surfacefriction, the necessity times in 32 integrationcycles.It is about 3 timesof higher vertical modes is also reported byiarger than that of the adiabatic NMI with 4Baliishand Baer(1985).
iterationsof 2 verticalmodes(-10 sec).WhireAt higher levels,the differencebetweenDNI
the differenceon CPU time is large, this is not aand NMI is much smaller,and the rms error ER
gravedefect of DNI if we weigh the differenceof
from the perfect data for the whole domain is the two methods:NMI includesoniy adiabatic
nearlythe same:processes
while DNI includesall processes
in themode1.
E R u : 1 . l 3 l m / s e c , E R v : l . 1 1 7m / s e c ,A defect rather more seriousthan the CPU
for DNl. andtime consumptionis the instability suggestedby(3.25). Figure 12 shows the difference in
E R u : I . 1 8 5m / s e c , E R y : I . 1 7 4 m / s e c ,Journal of the Meteorological Society of JapanV o l . 6 6 .N o . 2'.|, x,i.N€u1( b )Fig. 9. Horizontal wind in the lowest layer of the m o d e l a t t h e i n i t i a l t i m e ( 0 6 0 0 U T C ) . ( a ) T h e"perfect"i n i r i a l d a t a . ( b ) a f t e r 9 x 9 p o i n t s a v e r a g i n g , (c) after averaging and NMI, and (d) after averaging
and DNI.

MediterraneanSea. Although the amplitudeof
the instabilityis largestfor I,r, it is oniy about^-o^following the averageof the horizontal wind:0.25-0.5-C. It contaminates
the initializedfield
only
a
as
Fig.
iittle,
shown
in
13.
'r
/l'r -Tt t5.DN I .
u t r S - , r S .p e r f e c t If we increasethe time step /t of the
We observesignsof the instabilitycharacterized integrationcycles.we can decreasethe numberof the cyclesand the CPU time. The amplitude
by a pattern like a checkerboardover thelowest-layer air temperature lT * between the"nerfeer" initizl detr
and the data after DNI

2 1|

T. Satomura

April 1988
mm/h

mm/n

0 .t 5015.--. NMI*'- DNII2 UTC

12UTC

9
(a)

Fig. 10. (a) Time variation of the area-averaged
precipitation. Closed circle and square indicate thearea-averagedprecipitation for the first on hour after
DNI and NMI with the perfect data. respectively.
(b) Time variation of the rms error of
precipitation from the conrrol experiment.

?

("

;lr

of the checkerboardpattern,however,increases(the dependenceof the responsefunction on z4l
is evaiuated in Appendix B). The region
contaminated by the instability aiso extends'
Therefore.we choose the valueslt=240 sec.
n=32 at the sacrificeof the CPU time.
6.

Summary and discussion
We have applied the dynamic normal mode

ff N
-a

t

-f--'-i

.)

6/86

R

0fTU

Qt

(i]r

4 ,)'

t"

!1

(?

ECH. ? tX 60-90

(b)

OpTC

uH
E
R5SLE 9 /

6/8
U66 F

OHHTI U
0U

t cCi l i.l . t
Et X 6 0 -_6655

lr

OPTC

t( cc ))(a) Precipitation for the first one hour by the control experiment: (b) difference of

Fig- 11.
the precipitation of the first one hour between the control experiment and the experimentwith DNI; (c) with NMI. The contour interval rs 0.5mm and the dashedlines indicate
negative values.

Vol. 66. No. 2

J o u r n a l o f t h e M e t e o r o l o g i c a lS o c i e t y o f J a p a n-)

Fig. 1 2. Difference of the initial air temperature at the"perfect"
initial data andlowest layer between thethe data after DNI following the averaging. The
contour interval is 0.2511 and the dashed lines
indicate negative values.

initializationto the French limited-areamodel.
Because of the inefficiency of Okamura's
scheme,we used the backwardimpiicit scheme
for the time integrationcycle. After 32 cyclesof
integrationwith the noniinearterms. updating 4
times,fast oscillations
weresuccessfuliydamped.
The computation time of DNI (-30 sec)was
almost 3 times iargerthan NMI. This might not
be a grave disadvantage
becauseonly 2 verticalmodesare initiaiizedby adiabaticNMI. If morevertical modes, for example5 or 6 modes,areinitializedby NMl(-3), the CPU time would benearlythe same.
"identical twin" method,
By thethe horrzontal wind and the precipitation rate werecompared among the control experiment,theexperimentstartingwith NMI, and the experiment with DNI. The deformatlon of thelowest-layerwind due to the mountainswas
reconstructedthroughDNI better than NMI,but
the improvementwas not sufficient.It was also
shown that the rms error of the precipitationfrom the control experimentwas smallerin theexperimentwith DNI than that with NMI for the
first 3 hours.The area-averaged
precipitationwas("3) In order to resolve an Ekman type circulation
through NMI. all vertical modes must be initialized(Ballish and Baer.1985). It is 15 for the Frenchl i m i t e d - a r e am o d e i .

EnSE S/ 6/86 A 0fTU ECfl. 6 EX

5

T K=lS(b)(a) The air temperatureat the lowest iayer tn
Fig. i3.
"perfect" data, and (b) after DNl.
theaisoimprovedby DNI.
In spite of the aboveimprovcmentusingDNI,
there is a discrepancybetween the perfect
forecastand the forecastafter DNI. Two possible
reasonsfor the differencebetweenthe perfect
forecastand the experimentwith DNI are: 1)
processused.the structure
due to the averaging
which accounts for a certain amount ofprecipitation was lost, and the initialization
processcould not reconstructit;2) since thetime integrationschemein DNI (implicit scheme)
is different from that of the forecast model(semi-impiicitleap-frog scheme),the balanced
state after DNI would be different from that of
the forecastmodel. The former problem is less

April 1988

T. Satomura

severein a real situationbecausethe fine mesh DNI, however.could be performedstably.The
analysisfor this model hasbetter resolutionthan
applicationof DNI to a modelwith ,y'x:1Okmis
the averageddata. The latter is, however, now beingtested.
inevitable for DNI becauseDNI has to use the
Acknowledgments
selective damping scheme whose cut-off frequency is low enoughfor the time integration
The author would like to thank P. Bougeault
schemesof the forecastmodel to permit thefor the encouragementthrough this work aswell
wavepropagatingneutraiiy.NMI or \TvlI (Bourke as for many valuablecommentsand discussions.
and McGregor, 1983) with physical processes C. Mercusotprovidedthe fine-meshanalysisdata
would be better if the iteration converges. and supported the programming.P. Lacarrere
becausethey can use the same time difference prepared graphic programs. The author isscheme as the forecast model. The balance indebted to R. Juvanondu Vachat.R. Daley,J.
condition of the iterationin NMI, however.has Noonan, and the anonymousreviewersfor their
been suggested
asbeing different from the model constructivecriticismsand suggestions.
balancecondition(Rasch,1985b),especiallyfor
The computations were carried out at
low frequencymodes(Errico, 1984).Therefore. Direction de ia M6t6oroiogieNationale,Paris.
as long as a good iteration method for \rI\{I orand at Centre de Calcul Vectoriel pour la
NMI is not found. DNI hasthe advantagethat we
Recherche.
Palaiseau,
France.
can include any physicalprocesswith small riskThis research was conducted during theof divergence.
author'svisit to Centre National de Recherches
We havecomparedDNI and NMI only in rainy
M6t6orologiques
under the auspicesof Science
weather. Certainly, it is not sufficient toand TechnologyAgency,Tokyo. Japan.
conciudewhich methodis practicallybestfor the
initialization of short-range(from a few hours toAppendix A
one day) forecastsby meso-scalenumerical Detail of the implicit
integration cycle
modeis.We alsohaveto examineother synoptic
To obtain the desiredcharacteristics
showninconditions of fair weather.In addition, we havesection 2.2. we have to calculatethe nonlinear
to compare the numerical resultsnot only withterms by the implicit method. It is difficult,the data by the identical twin method but also
however,to calculatethem strictly. Therefore,
with the fine-meshanalysisof observeddata. In
we estimate the terms roughlv as described
the caseof rainy weather,the precipitationor
below.
other variablesrelatedto latent heat reieasewill
The governingequationsarerewrittenas
be good indicesto examinethe performanceof
the initialization method, since then the
06ui0'P:N,,
differencesamong initialization method will be
0pl3"P:Nu,
most significant,as reported in Salmon andWarner(1986) and in Zhangand Fritsch(1986).
r .. l )u , r n 1 " 7 \ \ " ( a" , , I ; . u t . t d o - - N( A
cpo d.lo'
In the caseof fair weather,other variablessuch^ -, f,,^as the surfaceheat flux or the shearvector in the+ d ur)d o : N,,
A'Z -t\,(0,umixing layer are possibiygood indices.Thoseareleft to further study.
Another candidatefor which DNI is suitable where n.and o are horizontalwind, P:6+R"TZ,
wouid be modelsfor smali-scale
phenomena.
For@ the geopotentialof dry air, Ro and cpo arethesuch phenomena, the diabatic processes gasconstant and the specificheat respectiveiy.7(condensation,
surfaceflux. etc.) would be most the horizontaliyaveraged
temperature.
Z=ln(p"),important, and all vertical modes should bep" the surface pressure, T the temperature,
initiaiizedwith diabatic processes.
The current o =p/ p,, and the subscribedquantitiesly' denote
iteration method in NMI might not converge nonlinear terms and other residualterms notunder such condition. The inteerationcvcle ininciudedin the left-hand-sides.
The separationofV o l .6 6 .N o . 2

J o u r n a l o i t h e M e t e o r o l o g i c a lS o c i e t l ' o f J a p a n

11 .1

the linear and the nonllnear(and the residual)
The responsefunction of this estimation
terms ls exactly the same as that of theschemecan be derived straightforwardlyby thesemi-implicitschemeof the mode1.-Itshouldbe
same method as that in the section 2.2. The
noted that the Corioiisterms are taken onto theresultis not shownbecauseof its complexitv.
right-handsides.As far as this separationis used
Appendix B
in the time integration of the forecastmodel. no
probiemhas been experienced.
We do not prove Dependenceof the responsefunction on lt
analytically that it works well in the forward/
By including the effect of the finite time step
backwardcycle. but, in the caseexamined.we
lt in the estimation of the nonlinear termagainexperienced
no problem.
(2.13), we can estimatethe dependenceof the
For simplicity,we rewrite(A.1) as
nonlinearresponsefunction on ,4t in the caseaJtl,'land iolt
1.
(A.2)
a,X+ L(X):^n/(X)Equation(2.1I ) is rewrittenas
where X is the physical quantity. I and l/
representthe iinear and the nonlinear terms:i@at*ttlr,(B.l)/trespectively.
Then. the impllcit methodin (2.23)
i s w r i tt e n a s
where the subscript 7 is dropped and thex'lx';"Ixi*,:xl-- tl t L( x'i / ) + / t N(x',J/ L),(A.3a)

t ' + / t L ( X ' * * , )- / t N ( x ' i / ' ) .
(A.3b)As mentioned at the beginning of thts
appendix, it is difficuit to perform theintegration(A.3a) owing to the implicit nonlinear term. We avoid this difficulty by
estimatingthe valueusingan approximatevalueof Xf /' asfollows:(i) Integratethe linearterm impiicitly and thenonlinearterm explicitly for one step as
f r t+ , t _ y t _ I t L ( * , ,where the implicit calcula'tionfor the linear term
is easiiy performed by modifying the semiimplicit schemeof the model.
(ii) Regardingf t-tt' as an approximatevalueof Xtgrt, we compietethe integrationcycle as,,- /t L(x'*'r') +y'/,\I(i'-r'). rsin(o/t/2) iot,ir_.rlo,_s,,-t.\-e
_)
.
Jt 2

,

(8.2)

which correspondsto the finite differenceformof (2.13). Assuming \o,4tl<<i, the righthand-sideof (B.2) is approximatedas.f r,r:i{o(

io/tt

I

(B.l)

,';o'-".

U s i n g ( B . 3 ) , t h e r e s p o n s ef u n c t i o n ( 2 . 2 4 )becomes^ - f l -f( oi ' l {r ' e2 )t- i t( oRt , ) + , 1t l { ( x ' " ) ,(A.4)

f rln'':-l)backwardimplicit schemeis used. By assumrng( 2 . 1 2 )a n d a 1 = 0 .( 8 . 1 ) i s r e w r i t t e na su)Jt

and it is approximatedas
R:l-Jt' o(o*a) ,(B.5)

w h e r e l a l t i z i I a n d 1 o / t " 4 l a r ea s s u m e d .
This equationindicatesthat the scheme(2.23) is
Italsou n s t a b i e i fo a . 1 0 a n d l o l { l a i .
i n d i c a t e tsh a t t h e a m p l i f i c a t i o nf a c t o rd e f i n e db 1 ' R - l i s p r o p o r t i o n atlo J l ' .

.
I x i , *,: r l - r ' + I t L (x',,,)- r/r^v1 i '*r';(A.5)

Since we perform the step (i) only when the
nonlinear term is updated. the additional
computationaltime is not large.

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216Journal of the Meteorological Society of Japan

EWTi )r{;td('6 h+*lffi#EF
-

+
,t'l+

*
uE

Vol. 66, No. 2

F?nffi|t)*+*)-17-t7.Eix*Bl+je>rSugiarffift L Et Yffi#.#+ * r' &tSH.f
t i* (DNI) ",trft + t /r,r: ff ffi L, + D H Hr,lt ? ffi .r /:.
DNIA.,EIU+itN:ffiffii, 4tr6, ffi';.r4 2 tvy:gy,b^+ -/^ t L{OkamuraO^+_ aitir)*rl
B<, 4./ 7'r > v l- x+-/.riftt'-&l:1m3fu6.
{ F f r R p f . l l .>o J4r ) v r / v l . i . L / : ^ + * a
X fil[uaircqffi< : ! r. I D, + i tvN.6 * i"La k-. D ryzWjEE':+nSE{LjEEl:eo b : t nr{. * /:.
.b, lk(D.Lrt]"hct:. ( 1) LIIBI(N.r.6^+l4lo*bD[&rtz'v v*> tirDNIo;5r\NMI l1,,
. R < F 4 , 3 h 6 " ( 2 ) D N rt . i z , f f i r 4 . , * h w l t € o 4 * : r r , l . 3 < r r r . " ( 3 ) T # a t r 3 f f i F i o K ^ €q+rz 2 **;F#rr DNI i: l g HAb h't.&'&* i-l 6 "(.*) nFfiE: A*-ffinFfr