Institute for Meteorology of the University M�unchen, Germany

On Normal Mode Aided Modelling: Application to Asian Monsoon

Frank Schmidt

With 13 Figures

Received July 19, 1999Revised December 29, 1999

Summary

A new concept of utilizing generalized normal mode initi-alization (NMI) for modelling is investigated. This NMI-balancing adapts real data to model solutions. For eigen-modes of typical (climatological) structure and a simple 1-or more-layer shallow water model, anomalous ¯ow ofspecial episodes or events can be identi®ed with modegroups giving distinctive signals. If these signals are partof the model solution or of the complement of the data to themodel solution and how this is changed by introducingadditional physical mechanisms to the model, gives hintsfor improvement.

In applying this to multi-scale interactions of Asianmonsoon, in particular a seventeen-years backgroundknowledge and the anomalous ¯ow of years 1991, 1977,1978 we have a challenging test frame.

Particular topics are:

(1) Mei Yu rains in China, the representation of particularaspects of them by certain mode groups, the degree ofindependence of the corresponding signals on othermodes, and the role of dissipation;

(2) detection of El Ni~no-Southern Oscillation (ENSO) inatmospheric (reanalysis) data minus the NMI-balancedmodel solution;

(3) Madden-Julian Oscillations (MJO) and the introductionof diabatic processes, and ®nally

(4) aspects of dynamic stability.

From among the results which are new or we simply havebecome aware of we pull out that atmospheric processesgenerally can be represented by modes or mode groupsneither straight forward nor uniquely. Introduction of somedissipation schemes to the model was not helpful but ratherdestroyed the realistic structure of atmospheric dynamics. Aparameterization of diabatic processes passes signalspresent in the data but not in the NMI-balanced model

solution through to such being part of the latter, in case ofMJO but not yet in case of ENSO. Ocean dynamics provesto be indispensable. Nevertheless there are imprints ofessential atmospheric processes hidden on certain (e.g.,most unstable) modes and partially recovered by NMI-balancing.

1. Introduction

Let us consider meteorological modelling asorientated at reality, i.e., to be veri®ed with realdata and, nevertheless, remaining transparent. Achallenging touchstone is modelling of monsoonand interacting mechanisms. While this is amulti-scale task, let us select the response on thelarge scale dynamics, the region of greater SE-Asia and periods much larger than three days (fora wave to go once around a latitude circle, i.e., ofspeeds much slower than 100 m/s) for ourexperimentation ®eld. And let us identify typicalpatterns and their relations and interactions.

We do not intend to state a closed theory butrather give a chain of examples. The mainpurpose will be to raise ideas. Moreover,considering patterns is the way thinking may goin its most vivid and obvious manner.

So in dealing with Asian monsoon and itsmodi®cations, we will be aware of the Indian andtropical part of monsoon equatorward of theintertropical convergence zone (ITCZ) and thesubtropical part beyond of ITCZ associated withMei Yu rains in China (Krishnamurti, 1985; Tao

Meteorol. Atmos. Phys. 73, 189±210 (2000)

and Chen, 1987; Ropelewski and Halpert, 1989;Johnson, 1992), of Eurasian snow cover and itsmemory (Barnett et al., 1988; He et al., 1995;Webster and Yang, 1992), of trade winds andeasterly jets, of Madden-Julian oscillations(MJOs; Gill, 1980; Lau and Chan, 1986a; Lauand Peng, 1987), of El Ni~no with SouthernOscillation (ENSO; Lau and Chan, 1986b;Ropelewski and Halpert, 1989; Webster andYang, 1992), of the East Asian wave Pattern(EAP; Webster, 1983; He et al., 1997) etc. andlast but not least of the presence of theHimalayas (Yanai et al., 1992; Schmidt, 1987;He et al., 1995). Moreover, we realize the hugenumber of interacting mechanisms (Yasunari andSeki, 1992; Webster, 1983; Tao and Chen, 1987;Lau et al., 1995; Lau and Chan, 1986b; Lau andPeng, 1990; He et al., 1996; Frederiksen andFrederiksen, 1993). Literature mentioned in thissection constitutes only a typical sample.

How shall we constitute typical patterns?There is a whole spectrum of techniques toconstruct patterns which enlighten physicalstructures, reaching from the purely statisticalones with data as the primary basis to thosewhere a model is the primary tool. Starting fromdata we may mention empirical orthogonalfunctions (EOFs; e.g., Lau and Chan, 1986a, inconnexion with MJO; Workshop ECMWF, 1977,where also their capability in supporting modeldesign is mentioned) and principal oscillationpatterns (POPs; Hasselmann, 1988; Schnur et al.,1992) among others. If promoted by modelswhich emphasize dynamics and interactions, letus mention normal modes (NMs). In between we®nd extended EOFs (EEOFs; Lau and Chan,1986a), cyclostationary POPs and complex POPs(v. Storch et al., 1995), principal interactionpatterns (PIPs; Hasselmann, 1988) and nonlinearnormal mode initialization (NMI; Baer, 1977;Machenhauer et al., 1977), where some dynamicsand/or variational methods have been introduced.

Due to the close relationship to models andmodelling plans we favour to utilize NMs. Ifposed ingeneously they not only give an directinsight to speci®c oscillations of the model withrespect to a basic state, but also to the atmo-sphere. Hence, if the basic state is typical and themodel complete and correct then the oscillationsindicate plausible structures and relations. And ifthe model does not function the oscillations will,

as we hope, specify contexts where modelling isto be improved.

Normal modes of a general basic ¯ow(depending on all space coordinates) have beenused by Frederiksen (1978) and many others toidentify typical and unstable patterns of stormtracks, and also by Frederiksen and Frederiksen(1993, 1997) to uncover mechanisms of Aus-tralian monsoon. Use of these modes in case of ageneral basic ¯ow is an expensive matter suchthat only low-order (rhomboidally truncatedR15) quasi-geostrophic or large-scale primitive2-layer models have been employed. As to theauthor's knowledge, no NMI has been workedwith yet for this kind of general basic states.Corresponding experiments with PIPs had to quitfor truncations equivalent to and larger thantriangular T3.5 (v. Storch et al., 1995).

In order to emphasize main features ofmethods based on normal modes, we give a briefoverview in the next section. In Sect. 3 modeland data are discussed and their treatment to givea basic state and corresponding normal modes.We also study the anomalous signal of July 1991,i.e., the sum of all modes of July 1991 minus 17-years-mean of July, as control case. In Sect. 4 wepresent certain aspects of 1991 mode groups ascorrelated with physical signals and discuss theirsensitivity to changes of model and of basic ¯ow.We ®nally conclude by summarizing and alsodiscussing future applications and possibleimprovements.

2. Review of Normal Modes (NMs)and De®nition of Normal ModeInitialization (NMI)

Due to the more general use of NMI put forwardin the sections to follow and not suggested inliterature so far we review the de®nition of NMsand give a preliminary de®nition of NMI-balanc-ing in this section. For simplicity and withoutdanger of confusion we abandon the usual kindof notation of vectors etc. within Sect. 2.

(a) NMs

Let

@a

@t� F�a� � b �2:1�

190 F. Schmidt

be a system of equations, where a is a vector ofall components of all prognostic variablesexpanded in, say, spherical harmonics, F is theoperator of all (linear and nonlinear) termsdepending on a, and b speci®es the forcingvector. Let furthermore �a(�,') be a typicalsolution dependent on longitude and latitude,where `̀ typical'' means

@�a

@t� �bÿ F��a� � 0; �2:2�

hence a slow solution of the model F. Generally`̀ typical'' is identi®ed with (large scale) `̀ cli-mate''. If then F(a) is expanded with respect to�a� aÿ a0,

F�a� �F��a��rF��a� �a0 �rrF��a� � � a0a0 � � � �we get

@a0

@t� A � a0 � n�a0� �2:3�

as a system of prognostic equations for theanomaly a0. Here

A � rF��a� �2:4�is the linear operator, and n(a0) includes bilinearterms (the regular dynamic terms like advectionetc.), higher-order terms (generally coming fromparameterization processes like diabatic heating)and anomalous forcing b0 � bÿ �b. In Sect. 3 Awill be constituted by the expansion coef®cientsof linear combinations of operators J� ; . . .�,J��; . . .�, r � r . . . , � etc. as given by themodel and evaluated for a climatological (17years, time-independent) state �a� i; �i; pi�, and Ais to be applied to the vector of coef®cients of theanomaly a0� 0i; �0i; p0i�.

Solving the eigenvalue problem for A,

AT � T� �2:5�with T built by the eigenvectors or normal modesti as columns and � by the eigenvalues �i in themain diagonal and zeros otherwise, then

a0 � Tx �X

tixi �2:6�is an expansion of a0 with respect to theeigenvectors. In order to have a sound platformof discussion, we here simply assume that thenumber of eigenvalues is identical to themaximum number possible, i.e., equal to thenumber of lines of A. By this statement we do

admit that some of the eigenvalues be zero(stationary and of vanishing growth rate).

Then, generally two aspects of this presenta-tion are not quite precise.

First, the climate �a as a mean of (slow)solutions (e.g., average of 17 years of mean July)needs not be a (slow) solution again, since, e.g.,

F�a1 � a2� : � �a1 � a2�2 � a21 � a2

2 � 2a1� a2

� F�a1� � F�a2� � I�a1; a2��2:7�

and the interactions I(a1, a2) vanish only (in astatistical sense) in case of (statistical) indepen-dence.

This remains a question, since subsequentyears are typically coupled by a biannualoscillation etc., but may be of minor signi®cancefor a 17-year mean as used in this paper.

Secondly,

�a is the climate of the atmosphere

and not of the given model�2:8�

Hence besides other possible de®ciencies, �amay be a non-slow solution of the given model(2.1) also for this reason.

We may reduce this problem, if necessary, byslightly adjusting �a with respect to the model aswill become obvious.

Generally, let us emphasize that due to (2.4)NMs do not only depend on the basic state �a butalso on model F and, while furthermore data areinformation of one monthly mean, i.e., at one ®xedpoint of time, that due to (2.3) or the followingdiagonalized form (2.9) oscillations owe theirfrequencies and growth rates only to the model.

(b) NMI

So, if data and their resolution to NM oscillationpatterns shall characterize the atmosphere, thenthe model should be as realistic as possible. Or,posed in another way, NM-answers differentfrom known signals of monsoon etc. indicateurgency of additional modelling or parameter-ization.

As given below in (3.1), the standard versionof the model used in this study is very simple, atleast much simpler than the atmosphere, namelywithout dynamics of an ocean, without humidity(snow), without any diabatic processes, but with

On Normal Mode Aided Modelling: Application to Asian Monsoon 191

some care put on representation of orography orthe Himalayas, in particular. Hence there areplenty of occasions to stress differences betweenmodel and atmosphere and thus call for addi-tional modelling.

Which structures of the atmosphere can beidenti®ed by the model? Hence, which dynamicswill be `̀ cut out'' by the model, i.e., can be madetransparent? And how can the rest of dynamicsnot being understood by the model yet, beopened to model control by parameterization ±as indicated by NMs?

In order to resolve knotting or linkage betweenmodel and atmosphere a generalized NMI willprove helpful, i.e., a balancing of initially givenatmospheric data in terms of something likemodel prognosis as speci®ed below. So, multi-plying (2.3) through by Tÿ1 from left yields thediagonalized form

@x

@t� � � x � Tÿ1 � n�a0� �: m�x� �2:9�

due to (2.5). Split into two parts with, on the oneside, modes x1 we are not especially interested inor which are especially authentic or whichbelong to a certain well established process(and including stationary modes with vanishinggrowth rates because of numerics) and, on theother side, the rest of the modes x2 includingthose we are interested in, (2.9) evolves into

@

@t

x1

x2

� �� �1

�2

� �x1

x2

� �� m1�x1; x2�

m2�x1; x2�� �

:

�2:10�By eliminating the tendencies of the latter

modes,

@x2

@t� 0; �2:11�

we may ®nd an interesting (balanced) solution ofthe model if we proceed as follows. A wellde®ned iteration

a�n�1� � G�a�n�� �2:12�is set up if dimension(x1) additional conditionsare speci®ed like `̀ x1 ®xed'' (`̀ free'' NMI) or`̀ geopotential ®xed'' (`̀ forced'' NMI) etc. For anexplicit formulation see Appendix. Trivially,for free NMI only modes x2 can have changed,while for forced NMI the above meaning of therole of x1 and x2 is to be modi®ed. In case of

convergence a(n�1)� a(n)�: M(a0) for all ngreater than some n0, �a�M(a0) is a candidateof a model solution wanted. The balanced state iscalled a `̀ typical'' model solution if j�1j� j�2jor only 0�j�2j for any members �i of thediagonal matrices �i and i� 1, 2, respectively. Incase of no convergence of iteration we maymodify the selection of x2 or of anomaly a0 oreven of basic state �a. And, moreover, a ®rstiteration step already gives a hint of a tendency tothe model solution and the quali®cation ofparameterization.

For reference of historical use (Baer, 1977;Machenhauer et al., 1977; Daley, 1980; referencesin the latter), NMI is generally introduced for basicstate �a at rest and a slow third of modes x1, theRossby modes then, while the remaining modes x2

are the gravity modes, and ®nally the balancedsolution is a point of the slow manifold in phasespace (Leith, 1980). The expression `̀ forced'' NMIis for historical reasons, presumably since some-thing like `̀ ®xed geopotential'' grabs into therelation of `̀ slow'' Rossby and `̀ fast'' gravitymodes. For the following we will adopt the graphicrepresentation suggested in Leith (1980), but nowgeneralized for these �- and '-dependent basic¯ows �a and arbitrary subdivisions x1, x2 of gravity-Rossby mixed-up modes (Fig. 1).

Fig. 1. Phase space split into modes x1 ®xed during NMIand the rest x2

192 F. Schmidt

NMs themselves (without one eye on NMI) havebeen de®ned for general basic states like climate(e.g., Frederiksen and Frederiksen, 1997). But thenonly instability properties of single modes havebeen studied and no mode groups of actualanomalies on the basis of these normal modesgiven by the climate �a and (2.4), and no NMI hasbeen set up for this general basic state.

For further discussion of Fig. 1 let us mentionthat generally the balance equation without anyspecial parameterizations is a quadratic equation.And if x1 is a set-up of slow modes, there is sensein x2� 0 if x1� 0, since fast modes may bethought of as being forced by slow ones and soondissipated if forcing is no longer present. So onlyone of the two branches of the solution is beingselected as physically meaningful. For the moregeneral NMI procedure with a particular selec-tion of modes (including the trivial modes) kept®xed and the rest including those processes weare interested in for balance, we cannot rule outone of the two branches for physical reasons nearat hand. Moreover, if special parameterizationslike for diabatic processes with condensation orfor turbulence etc. are included, the order ofbalance equation may increase to n� 2. Corre-spondingly multi-branched may be the solution.Compare also Tribbia (1981).

Let us emphasize another aspect of Fig. 1: If theinitial state of NMI-iteration is a projection of a0 onthe abscissa x1, then the result M(a0) of NMI mayrecover some of the signal a0, especially if it is`close' to M. This emphasizes some kind of stabi-lity of which we will give examples in section 4d.

Concluding this chapter on NMs we noticethat changing the basic state �a changes A, hencethe modes and thus the phase space. Changingthe model will also change the linear operator Aetc. and the interesting manifold M. Changingonly the nonlinear part of the model or itsforcing, so A and modes and phase space arepreserved but we have a different balance M dueto NMI. Finally, changing only actual data forgiven basic state and model and hence givenphase space, this will change the data point a0and x, respectively.

3. The Actual Constitution of NMs

Constitution of NMs (their structure and ampli-tude) means constitution of the model F, that of a

basic state �a, and moreover as a new aspect(since we are not only interested in single modesbut also in actual mode groups), de®nition ofactual anomalies a0.

We select a spherical harmonic shallow water(SWk) of k (k� 1, 2) dynamically active layers ofconstant densities �i for i� 1, . . . , k underneath ageostrophically balanced passive layer. There arearrangements of ¯ux correction at orography(Schmidt, 1990) corresponding with the follow-ing formulation of the continuity equation. Thisis an essential point with respect to theHimalayas and their dominant role.

Tuned to Fig. 2, the explicit version of system(2.1) in case of k� 2 active layers i� 1, . . . , k(model SW2) reads

@�i

@t� dr � ��a;i

~Vi� ÿ dr ��

f�0

�i

~V0

�� Zi;

@�i

@t� dr � �~kx�a;i

~Vi�

��

�d

2~V

2

i � gXi

v�1

�v ÿ �vÿ1

�i

hv

�ÿ dr �

�~kx�0

�i

~V0

�� Di;

@�i

@t� d�~Vi � r�i �r � ~Vi� � Pi (3.1)

with horizontal coordinates longitude � andlatitude ' and corresponding directions ~i and ~jpointing to increasing coordinates, moreover

Fig. 2. Layers of the baroclinic model SW2

On Normal Mode Aided Modelling: Application to Asian Monsoon 193

with ~k �~ix~j directed to increasing height, withmodi®ed nabla operator r :�~i�@=@�� �~j cos'�@=@'� (equal to the classical one multiplied bycos'), and with Laplacean

� � 1

cos2'

�@

@�

@

@�� cos'

@

@'cos'

@

@'

�� 1

cos2'r2 �: dr2;

with d :� (1/cos2'), furthermore with vorticity�i�� i, absolute vorticity �a,i� �i� f, diver-gence �i���i, stream function i, velocitypotential �i, angular momentum

~Vi :�~kxr i �r�i ��ÿ cos'

@ i

@'� @�i

@�

�~i

��@ i

@�� cos'

@�i

d'

�~j; (3.2)

hi height of the layer's upper boundary, hB� h3

orography and

�i � ln�hi ÿ hi�1�: �3:3�All the dependent variables are normalized by

the radius of earth, the period of one day androughly the maximum resolved height of orogra-phy.

Within the standard version of the model thereis neither moisture nor any diabatic effect present,no dissipation and also no dynamic ocean layerunderneath. Hence there is also no forcing,Zi�Di�Pi� 0 in (3.1).

Prognostic variables, say i, �i, �i, and orogra-phy hB are expanded in spherical harmonics.Operations map spherical harmonics on sphericalharmonics except for the exponential: Spectralhi� exp(�i)� hi�1 is received by Fourier trans-form. Moreover, spectral handling is employed asput forward by Robert (1966). Formally for SW1,we have only i� 1� k and hB� h2.

Different horizontal resolutions Tn have beenselected, i.e., with triangular truncation at n� 20for barotropic SW1 and n� 15 for baroclinicSW2, respectively. As a consequence theadmitted maximum dimension N of phase spaceis 3(20� 1)2� 1323 and 6(15� 1)2� 1536,respectively. Triangular truncation is preferredto rhomboidal since we are interested in wavetrains along great circles rather than in stormtracks in middle latitudes.

For construction of basic state we useEuropean Center (ECMWF) reanalysis monthlymean data of years 1979±1995 supplied by theGerman Climate Computing Center (DKRZ).Delivered on 17 pressure levels, they wereinterpolated, respectively extrapolated to theearth' surface, by cubic splines and transformedto vertical �-coordinate. Separation into threelayers was done by introduction of interfaces at�/�max� 0.6 and 0.22, and each layer wasrepresented by its mean density. Where upperboundaries of layers cut orography, a thin foil isput on orography and spectral representationexpanded globally.

We concentrate on July and de®ne a (pre-liminary with respect to (2.7), (2.8)) basic state �aas the mean of 17 years (i.e., of years 1979 to1995) of July.

Figure 3 portrays this basic state within theAsian monsoon region [0, �]x[ÿ(�/6), (�/3)], i.e.,for latitudes north of 30S and south of 60N, forthe eastern hemisphere and consistent with thelower troposphere SW1. The layer's thicknessshows minima due to the volumina occupied byorography: The Himalayas and East Africa showtheir physiognomies. Equator crossing ¯ow ismost emphasized east of Africa (Somalia).Out¯ow from the Himalayas as a consequenceof ¯ow correction is not removed here since itdoes not show up in actual anomalies.

As suggested by Hoskins and Karoly (1971)and generally used, the linear operator A asde®ned in (2.4) and commented upon subse-quently is evaluated by taking the linear part ofthe model (3.1) with respect to �a, lettinganomalies a0 run through the standard basisvectors ei, i� 1, . . . , N of phase space, where ei isthe unit vector with value 1 at the i-th place andzero otherwise, and gathering tendencies of the®rst step of forecast as the column vectors of ±Aconsistent with (2.3). Without loss of generalityA is real and hence its rank is rk(A)�N.

Inspite of the fact that the general basic ¯ow isnot the state of rest (but a �- and '-dependentclimatic state at a ®xed time), so we do not haveone third of slow Rossby modes and two thirds offast gravity modes, the spectrum (Fig. 4) againsuggests subdivision into a slower third of modesand a signi®cantly faster rest of modes. This istrue for all resolutions N, as well as forbarotropic and baroclinic domain. The subdivi-

194 F. Schmidt

Fig. 3. Basic state for the `̀ Asian monsoon region'' north of 30S and south of 60N of the eastern hemisphere showinglogarithm of layer thickness � and wind~v triangularly truncated at N� 20 for the lower troposphere. All variables are scaledby earth radius, maximum model height of the Himalayas and 1 day. The range of � there is [ÿ 2.0,ÿ 0.15] and maximum~v is 0.23. In dimensional units, here marked by asterisks, h� ÿ h�B � e� � 6:4 km with range [0.87, 5.5] km andj~v �j � 0:23 � 73:7m/s � 17m/s

Fig. 4. Spectrum of normal modes of SW1 and basic state of Fig. 3 with frequencies of modes given by full line of range[ÿ 500, 500] and growth rates dashed of range [ÿ 0.14, 0.13]. In dimensional units, 2� over frequency is duration in days forone lap along the latitude circle. So for mode 62 non-dimensional frequency 0.1 corresponds to 60 days, and for mode 109frequency 0.2 to 30 days, hence the MJO signal should be searched for among modes 62 to 109. For further information seeFig. 3

On Normal Mode Aided Modelling: Application to Asian Monsoon 195

sion mark is close to a period of 3 days whichcorresponds, for N� 1323 e.g., to mode 426.Evidently the slower modes are the moreunstable ones. They are supposed to carry theinteresting information.

In order to resolve monsoon-relevant signals attheir best we could also select clusters of yearswhich have been very wet in China or El Ni~noyears for basic state. Here we do not do so. Wealso do not present (as is generally done, e.g.,Frederiksen and Frederiksen, 1993) single modesof the given basic ¯ow with striking structure orgrowth rate as representing monsoon (`̀ themonsoon mode'') etc. Indeed, single modes aretaken to represent aspects of monsoon. Butgenerally we increase sensitivity by consideringgroups of modes of anomalies of special years, soof July 1991 known for its heavy rain falls overthe most populated eastern part of China.

Figure 5 shows the sum of all modes multi-plied by the principal coef®cients of the anomaly,(2.6), hence the anomaly itself (a well de®nedpoint in phase space of Fig. 1) with its logarithmof relative topography, �, of the lower tropo-spheric layer SW1 having a greater than normaldepression at the Himalayas and with greaterthan normal equator crossing ¯ow off the Somalicoast, between 45 and 60E, and at 90E, i.e., fromwest of Australia. Furthermore there is a strong

anomalous subtropical-latitude high pressureregion pumping warm and moist air into the eastof China and furthermore at the same time asystem steering in polar air from north. Moregenerally, we can identify a convergence line andfollow it from about Madagaskar in ENEdirection through the Indian Ocean to SE Asianear (100E, 25N) and furthermore, after a jumpto north east of the Himalayas, pronounced againin 40N from 100E until the paci®c coast andJapan at about 160E.

And as mentioned above, the anomaly ®eldsdo indeed no longer show any out¯ow from thethinned layer above the Himalayas related to ¯uxcorrection.

In the following chapter we will highlightcertain ideas, structures and results as theyemerge from a series of experiments.

4. Experiments

(a) On Mei Yu and Dissipation

Inspite of the critical or uncertain relationshipbetween atmosphere and model we search forMei Yu type signals (as de®ned subsequently) atmodes with realistic periods of more than onemonth and of striking growth rate. And workingwith lower tropospheric SW1 at maximum

Fig. 5. Like Fig. 3, but for anomaly a0 of 1991 with [ÿ 0.064, 0.0081] range of � and maximum wind 0.087

196 F. Schmidt

dimension N� 1323, indeed, we ®nd modes 88±89 (more precisely

P89i�88 tixi, of period 48 days)

and 96±97 (44 days) which show aspects of MeiYu, namely polar air meeting tropical air along a

zonal convergence line at about 30 to 40N aboveeast China, the ®rst mode couple with tropical aircoming from SW (Fig. 6a) and the other withtropical air coming rather from S (Fig. 6b).

Fig. 6. Maps of components of 1991 anomaly a0 above Asian monsoon region showing in¯ow to Mei Yu from differentdirections for SW1 truncated at N� 20, namely (a) modes 88±89 with in¯ow from SW with range [ÿ 0.0025, 0.0030] for �and maximum wind 0.025, and (b) modes 96±97 with ¯ow from S and range [ÿ 0.0067, 0.00087] for � and maximum wind0.038; (c) ®rst EOF pattern of precipitation anomaly for June, July, August with 11% of total variance based on data of years1951 to 1996 of China and normalized by the mean, due to Zhu, 1998

On Normal Mode Aided Modelling: Application to Asian Monsoon 197

These structures appear inspite of the lack ofconvection and humidity in the model and merelyunder the presence of a realistic hard orography atT20 and a simple passive geostrophic uppertroposphere. Obviously these processes, thoughmissing in the model did nevertheless have their

impact on atmospheric dynamics which now waspicked up by the modes presented. Records ofprecipitation do roughly support these dynamicalstructures (Fig. 6c; associated ®elds, i.e., projec-tions of precipitation on these modes have not beencomputed).

Fig. 6 (continued)

Fig. 7. Map like in Fig. 6 but due to modes 1±109 of periods longer than 30 days with range [ÿ 0.027, 0.00033] for � andmaximum wind 0.064

198 F. Schmidt

These signatures of Mei Yu are not typicalenough, however, in the following respect: If weconsider modes 1±109 (of periods longer than 30days) and their joint structure (Fig. 7), the signalspartly (e.g., the convergence line above China,¯ow crossing equator near 90E) disappear ormove further to E although they are present in thetotal anomaly at the right place (Fig. 5) and alsoin the slower part of it (with periods longer thanthree days, not shown). Obviously much of thevariance of the modes is cancelled. Informationis not orthogonal as for EOFs. So there is no(unique) `̀ Mei Yu mode'' and we also cannot say`̀ mode xx is the Mei Yu mode with in¯ow fromSW''.

Is there the possibility to `̀ drive down'' theamplitudes of modes or to change their structuresuch that redundancy is reduced? A symmetriza-tion of the linear operator A would cause NMs tobecome orthogonal (and also eigenvalues tobecome real and hence modes to becomestationary). Dissipation accentuates the maindiagonal of the operator and so increasessymmetry. Will there be a measurable effect oforthogonalization by introduction of well knowndissipation schemes?

We experiment with the following schemeswhere the terms at the right-hand side ofequations like (3.1), but written for and �,are de®ned by

�ÿ1 Zi

Di

� �:� ��

�

� �; �4:1�

or

� �

hÿ hB

dr � �hÿ hB�r

�

� �� �dr� � r

�

� �� ��

�

� �;

�4:2�

or

� �

hÿ hB

dr � �hÿ hB�~kxr r�

( )

� ��

dr� �~kxr r�

( )� 0

��

� ��;

�4:3�

or ®nally

� 0; �4:4�

respectively. (4.2) is due to a suggestion by Gent(1992) for the plane velocity equation,

@~v

@t� � � � � �

hÿ hB

����dpr� � �hÿ hB��

���dpr�~v;�4:5�

with corresponding classical cartesian nabla

operator � ���dp r�, but now changed to matchspherical geometry.

However, for decreasing i� 4, 3, 2, 1 in (4.1)we ®nd strongly increasing growth rates of thefast modes (Fig. 8) and, moreover, even true forboth the baroclinic and the barotropic version ofthe model. This lets us simply conclude that theatmosphere does not work with this type ofdamping appended to the given model. Data andthis type of modelling are not consistent. Anddissipation is no solution to thinning down thespectrum to only one or a few Mei Yu modes.More sophisticated physics (like humidity andconvection) may play a role, though not ofobvious impact on a symmetri®cation of thelinear operator A.

In order to make things quite clear we mentionat this point that only NMs of the standardversion of SW1 have been utilized so far to studypatterns of Mei Yu, and only the spectrum ofeigenvalues of dissipation-changed SW1 wasglimpsed at to ®nd the inconsistency of para-meterization and data. No NMI has beenintroduced yet.

(b) On ENSO

The atmospheric dynamic part of ENSO may beportrayed by the global scale tropical divergence®eld of the lower troposphere with negativevalues (convergence) in the western Paci®c andpositive values in its eastern part, more empha-sized in La Ni~na years and reduced in El Ni~noyears (like 1991 or 1987), respectively. Can thissignal be found in single modes or in compositsof only a few modes although there are no oceanin the model, no snow cover and no diabaticprocesses and not their joint memory? Or can wedetect a wave train like the East Asian Paci®cPattern (EAP) as a consequence of ENSO?

Obviously the lack in physics is too serioussuch that there is no mechanism strong enough tocommunicate with model dynamics in an

On Normal Mode Aided Modelling: Application to Asian Monsoon 199

obvious way. But let us remember that in case ofa linear operator A of maximum rank theensemble of all normal modes covers the wholeinformation of the data a0. Furthermore, that NMIprojects actual atmospheric data a0 (e.g., ofanomalies) on an (interesting or slow etc.) modelsolution M(a0). So we can identify a non-modelpart of the data (or `̀ model complement'' withinthe whole body of data) with a0 ÿM(a0) and mayreach a signi®cant answer within this comple-ment if basic state and NMI are speci®edingeniously.

Now in order to assess ENSO-related struc-tures, let us reduce dependence of the linearoperator A on basic state solution �a (which is stillthe 17 years' mean of July) to dependence ononly the zonal mean of this basic state and let usrealize zonal mean of interactions of non-zonalcomponents as additional forcing (Schmidt andUnzicker, 1995) on the right hand side of (2.3).

Does it surprise that we ®nd well de®nedstructures for the model complement? No! Data�a� a0 are `̀ reanalysis data'', so balanced already.So are the model data M(a0) due to NMI. Takingfor a0 the non-zonal part of the basic state �a thenwe ®nd for the model complement somethinglike a Walker circulation with convergence closeto the Phillippines and divergence in the easternPaci®c of the lower troposphere (Fig. 9a) and a

reversed situation in the upper troposphere (notshown). This is true as well for (1) the baroclinicSW2 as for (2) the barotropic SW1 if iterativelyscanned through the lower and the upper layer ofthe troposphere. Taking the difference of the ElNi~no year 1987 to the zonal mean of the basicstate �a for a0 instead, i.e., the 1987 anomaly ofthe model complement, then we ®nd about thispattern, but of smaller amplitude and of oppositesign (for the lower layer Fig. 9b), thus for 1987 areduced Walker circulation. For 1988, a non-ElNi~no year, this circulation is intensi®ed. Theseresults in this paragraph are due to T. L. Zhao(1999).

While the wind ®eld (in Fig. 9) is mainlysupported by the divergent component (due tothe diabatic processes of model complement),there is vorticity associated with already a smalldistance off the tropical region and there arewave trains forced to spread out (Hoskins andKaroly, 1981; Gill, 1980). And indeed, we ®ndan anomalous wave train (i.e., a wave trainforced by the anomaly) along East Asia (EAP)consistent with the Walker divergence, respec-tively high pressure near the Phillippines andshowing a depression adjacent in the north aboveeastern China (not shown, from T. L. Zhao, 1999)

While these signals are generally brought inrelation to ENSO, they here are outside of model

Fig. 8. Spectrum like in Fig. 4 but for model SW1 with damping with ranges [ÿ 500, 500] and [ÿ 0.14, 0.14] for full anddashed line, respectively

200 F. Schmidt

control: They are part of the model complement,i.e., residual, and cannot be correlated with acertain realistic frequency nor a speci®c mechan-ism within the model. So the invitation orchallenge is to construct a correspondingmechanism within the model and verify itsfeatures by NMs. This modelling would have to®rst of all set up and deal with diabatic processes,but (as to ENSO also) secondly with additionalocean dynamics among others.

For clarity let us emphasize at this point againthat getting the model solution M(a0) or themodel complement a0 ÿM(a0) indeed is based onNMI.

(c) On Madden Julian Oscillations (MJO)and Introduction of Diabatic Processes

We here will only take up modelling of diabaticprocesses and remind of the correspondingphysical environment. The de®nition of whatwe will understand by MJO will be worked out inthe following.

There are the well known results due to Gill(1980 etc.) on thermal forcing in low latitudeswith signals as qualitatively summarized inFig. 10a for symmetric forcing about the equatorand in Fig. 10b for asymmetric forcing, respec-tively. For SW1, i.e., the barotropic standard

Fig. 9. For zonal mean of basic state in Fig. 3 taken as �a, the map of wind of (a) the model complement a0 ÿM(a0) with a0 thenon-zonal part of basic state, and (b) the 1987 anomaly of a0 ÿM(a0) with a0 the difference of 1987 data to �a

On Normal Mode Aided Modelling: Application to Asian Monsoon 201

model without any diabatic heating, we are solucky to ®nd about this pattern (Fig. 10c) inmode 156±157 (19 days), which however is atperiods much shorter than 30d and which werather must attribute to a Kelvin wave. Generally,thermal forcing ®elds are much more involvedlike those drawn from outgoing longwaveradiation (OLR) by Lau and Chan (1986a).When maximum OLR is identi®ed with antic-yclonic sinking etc. and then this pattern of day 5EEOF (Lau and Chan, 1986a, their Fig. 9) istransposed into a ¯ow pattern, we may sketchthis as in Fig. 10d. We will keep this pattern inmind as a possible candidate representing MJOand return to it later in this section.

Moreover, from modelling of conditionalinstability of the second kind (CISK) we knowthe problem of retarding a Kelvin wave basedoscillation to the observed and much slowerMJO. Nonlinear moist convection (that is con-vection restricted to that area with only low-levelconvergent ¯ow) and relatively moderate to lowcondensation levels seem to be necessary ingre-dients for locking modes to a slowly migratingcirculation pattern (e.g., Lau and Peng, 1987). Sowithout these mechanisms present in the modelwe ®nd an ersatz read off (like in Fig. 10c) fromatmospheric data of modes at wrong speedand wrong (smaller) amplitude and no signal atthe observed speed. We might also call this`̀ aliasing''.

A layered model does not know unstablestrati®cation. Hence condensational heat is onlyrealized as a result of the joint action ofcondensation and convective reshuf¯e: Localheating of a layer appears as a local mass lossof this layer in favour of the adjacent higher one,and upper layer condensational heating wouldappear as a mass ¯ow to a still higher layer ofsmaller density. Similarly infrared radiationalcooling mainly appears at the ground, inducing amass loss to a layer blowing up still underneathbut above the ground and of higher density. Wehere will have to simplify this and ®t it in SW2model concept. Namely condensational heatingthen happens locally as a consequence ofconvergent ¯ow close to the ground andcorresponding updraft. The condensation ratewill be proportional to moisture availability orspeci®c humidity q (introduced by climatologyor additional dynamics) and appears as mass

Fig. 10. Flow patterns according to Gill (1980) due tothermal forcing (a) symmetric about the equator, and (b)asymmetric in the northern hemisphere. (c) Flow patternwith some kind of Kelvin structure of mode 156±157 ofperiod 19 days. (d) Sketch of ¯ow pattern corresponding toan EEOF of OLR in Fig. 9 by Lau and Chan (1986a)

202 F. Schmidt

exchange from the lower to the upper layer. Longwave radiative loss appears (e.g.) as globallyhomogeneous mass loss of the upper to the lowerlayer and such that the overall mass of each layeris conserved as the result of both condensationalheating and radiative cooling.

On the level of prognostic variables �1 and �2

we may write

@�2

@t� � � � � P1 :� ����2�ÿ ÿ �

�2�h2 ÿ hB�@�1

@t� � � � � P2 :� ÿ��2

�1

h2 ÿ hB

h1 ÿ h2

���2�ÿ

� �

�1�h1 ÿ h2� ; �4:6�

where

� :� �

4�

��S2

�2�h2 ÿ hB����2�ÿ cos'd�d'

and

f ��; '�ÿ :� 0

f ��; '��

if f ��; '� >�

� �0:

This parameterization allows for a true mate-rial circulation with local convergence in thelower layer due to dynamics, with local ascent,with advection or dispersion in the upper layerand with global subsidence. Here � represents ahumidity availability parameter.

Together with exchange of mass also momen-tum is exchanged. It is conserved, if integratedover both layers but not layerwise. On the levelof angular momentum equations this reads

@~V2

@t� � � � �

~V2� ÿ "�2�h2 ÿ hB� ;

@~V1

@t� � � � � �2

�1

h2 ÿ hB

h1 ÿ h2

�~V1 ÿ ~V2�����2�ÿ

� "ÿ ~V1�

�1�h1 ÿ h2� �4:7�

with

" :� �

4�

��S2

�2�h2 ÿ hB�~V2���2�ÿ cos'd�d':

Computation of mass as well as of momentumterms for any of the layers is possible spectrally

only by means of Fourier transform. But due to�i :� ln(hiÿ hi�1) and

h2 ÿ hB

h1 ÿ h2

� exp��2 ÿ �1� �4:8�

computation is non-critical.This modelling of diabatics in SW2 has its foot

prints on a barotropic part SW1. Considering alower tropospheric layer SW1, e.g., mass balanceis straight forward. Momentum balance as toradiative cooling with its momentum transportfrom the layer above needs an additional closure,where upper winds' climatology may be helpful.

As a result we may consider a whole hierarchyof parameterizations of different quality andcompleteness. All of them are generally non-linear, however, since for example for �f :�sin'and f 0 �ÿ sin' we have

�fÿ� f 0ÿ � ÿj sin'j 6� 0 � ��f� f 0�ÿ: �4:9�

So these parameterizations of diabatic processeswill not enter the linear operator A, so will notchange the normal modes or dependence of data onthem. So is there any advantage for detecting theMJO signal at realistic speed? If so, which one?

Modelling of diabatic processes as givenabove was stimulated by and followed theestablished CISK parameterization (see Lau andPeng, 1987, or Hendon, 1988), so there is achance to also ®nd a MJO signal in these layer-concept models (as opposed to models withlevels at constant values of height or pressure).Hence, taking

MCISK�a0� ÿMstandard�a0�; �4:10�the difference of solutions of a model with andwithout diabatic parameterization, could give theclue to identify the MJO signal as a group ofNMs at realistic frequencies (see Fig. 11).

With regard to the model solutions, which typeof NMI should be selected? If on the one hand wechoose a forced initialization by preserving mass(i.e., relative topography �i) as the most reliablevariable (as to data inquiry, though alreadyreanalysed data) and if, moreover, we restrictdiabatic modelling to a minimum disposition ofonly mass exchange, the signal can only beimprinted on the wind ®eld by the nonlinearinteractions of NMI (see Appendix). It can havedeveloped at realistically slow frequencies, how-ever. If on the other hand we choose free NMI

On Normal Mode Aided Modelling: Application to Asian Monsoon 203

(where projection through a0 in Fig. 11 would beparallel to x2) in its conventional form, then slowmodes will be kept constant, so just the modeswithin that area of phase spacewherewe would liketo ®nd a realistic signal, do not change. Then (4.10)would address only faster modes, i.e., too fastmodes. So instead we would like to keep ®xed onlythose (slow or less slow) modes for which the MJOsignal is not expected. So select

x1 � fgroup � of �modes � not � included � in � the

� process � under � discussiongand

x2 � frest � of �modes � including � those � of

� discussed � processg�4:11�

Will then the NMI be stable and yield an MJOsignal? Or will there be different answers (Fig. 1,dashed part) depending on numerics of iterationwithin NMI or will iteration be divergent? (Ifstable, we will speak of dynamical stability ofthese modes. For further discussion, see Sect. d.)Moreover, how will the signal depend on thechoice of preserved modes?

In order to state more precisely the type ofnonlinearity within parameterization of diabaticprocesses let us still mention that the function� 7! �ÿ de®nes a spray over modes of all

frequencies due to the edges of �ÿ alongmanifolds of vanishing � and diagonalization.However within `̀ free NMI'', contributions tomodes x1 which are to be ®xed are abandoned.

Both, for forced (case 1) and for free NMI, forthe latter with modes of periods longer than 60days (case 2) and 1 year (case 3) kept ®xed,respectively, the signal (4.10) of diabatic terms ofbaroclinic SW2 was computed and also wasemphasized by scanning through the atmospherewith the higher resolved barotropic SW1 (seeTable 1). It was found to be astonishingly uniqueand relatively slow, i.e., the amplitude of thegroup of fastest two thirds of modes is smaller bymore than one order of magnitude than the groupof slowest third of modes (< 3 days).

Moreover, in all cases considered the structure[but not quite the amplitude] of the signal behavesstable in the following respect: Giving up all claimto angular momentum exchange (4.7) and leavingproduction of corresponding angular momentumeffects of the signal entirely to nonlinear interac-tions of NMI (type b) is not so different from resultsby employing the full diabatic scheme (4.6), (4.7)(type a). Furthermore we can see this signal in aselected layer if working with only the barotropicSW1 (and necessarily reduced diabatic scheme,type c). By the way, anomalies a0 are also notcoated much with the globally averaged long waveradiational part of diabatic scheme (see type d). Wegenerally can skip this part of the diabatic scheme(4.6), (4.7). A review of all NMI cases and types ofexchange is given in Table 1. Due to the Appendixall combinations 1a, 1b, . . . , 3d are possible. Anontrivial selection of experiments has beenconducted.

Fig. 11. Sketch of two model solutions in phase space dueto standard and to CISK parameterization, respectively

Table 1. View of NMI-cases and Exchange-Types Discussed

Cases of NMI

1. forced (� ®xed)2. free (modes 1±61 ®xed, i.e., 1 to 60d)3. free (modes 1±19 ®xed, i.e., 1 to 1a)

Types of exchange

(a) exchange of mass and momentum within SW2(b) exchange of only mass within SW2(c) mass exchange within SW1, i.e.,

reorganization of mass due to condensationand radiative cooling

(d) exchange of mass in SW1 due to onlycondensation

204 F. Schmidt

Now considering the total signal for case 3, itshows convergence lines in the Indian andChinese monsoon regions. Moreover, breakingdown the signal to slowest modes of 1 year to 60days which were just not kept ®xed (modes 20±

61, Fig. 12a), to the MJO periods of 30 to 60days (modes 62±109), to periods of 30 to 12 days(modes 110±223, Fig. 12b), and to the restincluding the fastest two thirds of the spectrum,we ®nd some kind of a red spectrum with

Fig. 12. Balanced results of 1991 anomalies of the lower layer of the baroclinic model SW2 due to free NMI with modes ®xedof periods longer than 1 year. Presented are (a) modes 19 to 61 of periods 1 year until 60 days with �-range [ÿ 0.0021, 0.0021]and maximum wind 0.010, and (b) modes 110 to 223 of periods 30 to 12 days, �-range [ÿ 0.0011, 0.0012] and maximum wind0.0037

On Normal Mode Aided Modelling: Application to Asian Monsoon 205

decreasing amplitudes, but where all parts are ofsimilar structure characterized by these conver-gence lines. This is for one iteration within NMI.For three iterations this is even more emphasized.The pattern reminds of the quadrupole-EOFpattern presented by Lau and Chan (1986a) intheir Fig. 9 and sketched here in Fig. 10d, wherestructures are cyclically reset from or with in¯owfrom SW to India and from SE to China andblocked by the Himalayas. In particular, turningback some pages to section (a), we see the closerelationship to Mei Yu dynamics with a severezonally oriented convergence line indicatingcollision of [cold dry] polar air and [warm moist]air of lower latitude origin (Fig. 6).

In case 3 we started from a pretty rough ®eldwith some of the structure of Fig. 5, while in case2 modes 1±61 are a better approximation to the®elds of Fig. 5. Consequently the iteration withinNMI is smoother in the latter case.

Why cannot we ®nd a MJO signal becomingclearer for a moderate group of modes ofappropriate range of frequencies instead of thisred structure of the signal? Is this a specialfeature of July 1991 since this month presents amaximum of monsoon activity? We do not provethis here by studying, e.g., the situation for Juneor August of the same year. We simply let theidea stand, open for future inquiry.

Furthermore, we do not introduce an addi-tional ocean layer in order to intensify or clear upnormal modes of ENSO as correlated tomonsoon. We also do not try to design somedynamical counterpart of Eurasian snow coverand its memory. We are going to conclude withonly highlighting some aspects of what we called`̀ dynamical stability'' before.

(d) Dynamical Stability

Let us select a certain group of modes. They will becalled dynamically stable if (free) NMI is stablewith only these modes (taken to be x1) kept ®xed inthe iteration. If we identify these modes with thecoordinate space drawn as abscissa in Fig. 1 and doassume dynamical stability, then there is ade®nitive intersection M(a0) of the complementsubspace x2 drawn through the data point and thedynamical manifold M given by the model.

Obviously, as mentioned before, a dynamicallystable group of modes must include the stationary

modes with vanishing growth rates (`̀ trivialmodes'') in order to avoid division by vanishingeigenvalue within the iteration procedure. More-over, there may be more than one differentdynamically stable solutions (Fig. 1, especiallythe dashed part).

Let us take the barotropic SW1 at resolutionT20 (with N� 1323 the degree of freedom of theeigenvalue problem) as the basis of our discus-sion, and let us select the group of most unstableand slow modes (with growth rates > 0.031 andaugmented by the trivial modes) as the ®xed ones.These are 43 modes with periods longer than 3days (Fig. 13a). Let us also select a subset of thesemodes with only periods longer than 30 days.Surprisingly, structures are nearly identical (dif-ferences as to height and to velocity are smallerthan the signal by about one order of magnitude;not shown). Both groups of modes show a certainsignal of convergence in northern China reachingfurther into the paci®c ocean. The faster unstablemodes appear to carry super¯uous variance.

The corresponding rests of less unstablemodes slower than 3 days (up to mode number425) exhibit most of the total signal with equatorcrossing ¯ow near Somali, turning to becomewesterly and staying so all the way to 120E,diluted in particular by southerly ¯ows near 100Eand 120E, the former de¯ected by the Himalayasand the latter merged by tropical southeasterlywinds and then hit by northerly winds near 45N.In contrast, the full slow signal (all modes slowerthan 3 days) display merged southerly winds andno supply by tropical easterlies east of 120E.

Now, running through the NMI, the signal ofmost unstable modes is updated (Fig. 13b) byequator crossing ¯ow near Somali, by southerliesnear 120E fed by tropical easterlies and meetingnortherlies near 45N, which recovers an essentialpart of the total signal. Much of the slow andessential information seems to be intimate part ofthe model and those unstable modes! A visualexplanation in phase space has been given closeto the end of Sect. 2. This is also some kind ofstability. Moreover, there is a third meaning inwhich this pattern may be called dynamicallystable: The slower subset of most unstable modesand the larger group as de®ned above causenearly identical results. Both mode groups arenot dynamically stable with respect to thede®nition given at the beginning of this chapter:

206 F. Schmidt

More than one iterations of the balancing processwithin NMI will diverge vehemently. By con-trast, if keeping ®xed all 425 modes slower than3 days, NMI will hardly change this slow signal(only with a difference 1.5 orders of magnitudesmaller than the initial or ®nal signal).

5. Discussion

How can normal modes of a given model withrespect to typical (real 17 years mean) data forbasic state and checked for actual data aid animprovement of the model? This topic of the

Fig. 13. Map of 1991 anomalies of (a) the 43 most unstable modes (i.e., with periods longer than 3 days and growth rates> 0.031 and) with �-range [ÿ 0.010, 0.0065] and maximum speed 0.12, and (b) the full answer due to NMI with �-range[ÿ 0.17, 0.029] and maximum wind 0.92

On Normal Mode Aided Modelling: Application to Asian Monsoon 207

paper is studied by drawing the actual data (oractual anomaly or actual signal) to a balancestate (or slow solution) of the model by aparticular form of normal mode initialization(NMI) and taking the differences from real dataas indications for additional model design.

Generally, NMI will not be convergent forarbitrary basic ¯ows. We even know aboutproblems for basic state of rest. However, it isnot necessary to ®nd solutions of NMI oper-ationally. It is suf®cient to ®nd solutions at all foringenious selection of basic state and data inorder to get hints for improvements of modeling.

Moreover, we do not stand up to improve awell known and operationally used model, butinstead to show problems of NMI-aided design inprinciple, indeed for a couple of very simple 1- to2-layer shallow water models with lack of evensimplest physical processes like diabatic heatingor dissipation etc. such that there is ample needfor additional parameterization.

We study structures like Mei Yu rains in China,effects of the Himalayas, El-Ni~no-Southern-Oscil-lation (ENSO), Madden-Julian oscillations (MJO)etc. and look at the degree of independence ofcorresponding patterns and mode groups fromother modes (for comparison an optimum ofindependence can be found for empirical orthogo-nal functions), look on possibilities of orthogona-lization by dissipation, design diabatic heating forthe given model and check wave CISK in thiscontext, and ®nally investigate kinds of dynamicstability of balanced states.

Generally the physical signals are characterizedby typical time scales, the data, however, we use aremonthly means, namely 17 years of July or July1991, hence only points of time. So it is the givenmodel which borrows time dependence to thesignals. The linear operator de®ning the modesdepends as well of the model as of the basic state.Within the balancing procedure (NMI) a certaingroup of modes is kept ®xed and the rest of themodes taken to adjust to the model. The ®xedmodes may be the slowest modes or the mostunstable or those with a special interest on etc. Sowe have a multidimensional problem with mixeddependencies on model and data.

Most signals studied can be expanded in asometimes more and generally less restrictednumber of modes. In particular this is true for thebalancing effect. Introducing either changes to

the model or to the basic state (like balancingbecause of (2.7), (2.8)), do both change themodes. Never, however, has there been found onesingle typical `̀ monsoon mode'' or `̀ Mei Yumode'' etc.!

Several types of dissipation have been studied.As indicated by growth rates of modes, the trivialcase of no dissipation is the most stable or theone most compatible with atmospheric data. Aneffect of reducing dependence between modes orincreasing orthogonality could not be found.ENSO is a signal not present in the model, but inits complement of the data. Introduction of onlydiabatic processes did not make ENSO a signalcontrolled by the model. Experiments with anadditional ocean layer have not been conductedyet.

However, the introduction of diabatic pro-cesses (or wave CISK) has produced a typicalsignal of MJO type (Fig. 12). It is neither thatone found for forcing symmetric about theequator like in Gill (1980) nor a similar structurefor asymmetric heating since forcing is muchmore involved in reality. The pattern instead isclosely related to a certain phase of ®rst orsecond EEOF of Lau and Chan (1986a). Again, itis swept over many time scales.

Future inquiry is necessary for further proof ofnormal mode aided modeling. While one point ofthis paper was mainly to communicate the ideaof using NMI for model design, this has to besubstantiated systematically by introducing dataof different years, months etc. and by introducingphysical processes and additional ocean layersinto the model etc. Another part of this paperdealt with the physical interpretation of thenormal modes and their eigenvalues. Proceduresto get modes or mode groups more independentof each other (like EOFs are) is highly desirable,not only for sake of stable convergence but alsofor better interpretation.

The value of NMI for analysis and interpreta-tion alone, however, is beyond any doubt.

Appendix

The central part of iteration (2.12) is given as aconsequence of the tendency of x2 due to (2.10) togetherwith (2.11) and Tÿ1n(a0)�m(x) from (2.9), namely�2x2�m2� (Tÿ1)2n(a0) or

x2 � �ÿ12 �Tÿ1�2n�a 0�: �A1�

208 F. Schmidt

Then for free NMI this x2 together with the ®xed x1 givesa new guess of a0 due to (2.6).

For forced NMI, say, a 01 is ®xed, so rewritten (2.6), i.e.,

a 01a 02

� �� T11 T12

T21 T22

� �x1

x2

� �; �A2�

is to be solved with known a 01, x2 in order to complete oneiteration step of (2.12), so x1 � Tÿ1

11 �a 01 ÿ T12x2� andconsequently

a 02 � T21Tÿ111 �a 01 ÿ T12x2� � T22x2: �A3�

This is without much limitation of generality: Assumingto have a regular basis T of eigenvectors. Necessary for setup, however, is the existence of �ÿ1

2 , hence a consistentselection of x2 not containing stationary modes withvanishing growth rates.

If diabatic processes are parameterized there is anothernonlinear vector p(a0) to be added to the right-hand side of(2.3), giving

@a 0

@t� A � a0 � n�a0� � p���; ��

p ;���; ; ��� �

: �A4�

In case of only mass exchange, p ,�� 0 and onlycomponents p� belonging to mass (continuity) equations arenonzero. Introducing (unchanged) diagonalization (2.5), aterm q(x) has to be added on the right hand side of (2.9) and(2.10) such that generally in¯uence of mass exchange isalso found in the x2-component of (2.10).

In case of forced NMI with � ®xed and � de®ning knowna 01, this signal is passed to a 02 due to (A2) or (A3) whichthen represents , �.

Acknowledgements

Thanks go to Dr. K. Arpe of DKRZ for supplying withreanalysis data and to Ms. Lotz-Seidel for drawing andpreparing some of the figures. Furthermore, the congenialunderstanding and constructive remarks of two anonymousreviewers are highly appreciated.

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Author's address: Frank Schmidt, MeteorologischesInstitut der Universit�at, Theresienstraûe 37, D-80333M�unchen, Germany.

210 F. Schmidt: On Normal Mode Aided Modelling: Application to Asian Monsoon