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Low-Order Stochastic Mode Reduction for a Realistic Barotropic Model Climate

CHRISTIAN FRANZKE, ANDREW J. MAJDA, AND ERIC VANDEN-EIJNDEN

Courant Institute of Mathematical Sciences, New York University, New York, New York

(Manuscript received 24 March 2004, in final form 12 October 2004)

ABSTRACT

This study applies a systematic strategy for stochastic modeling of atmospheric low-frequency variabilityto a realistic barotropic model climate. This barotropic model climate has reasonable approximations of theArctic Oscillation (AO) and Pacific/North America (PNA) teleconnections as its two leading principalpatterns of low-frequency variability. The systematic strategy consists first of the identification of slowlyevolving climate modes and faster evolving nonclimate modes by use of an empirical orthogonal function(EOF) decomposition. The low-order stochastic climate model predicts the evolution of these climatemodes a priori without any regression fitting of the resolved modes. The systematic stochastic modereduction strategy determines all correction terms and noises with minimal regression fitting of the vari-ances and correlation times of the unresolved modes. These correction terms and noises account for theneglected interactions between the resolved climate modes and the unresolved nonclimate modes. Low-order stochastic models with only four resolved modes capture the statistics of the original barotropic modelmodes quite well. A budget analysis establishes that the low-order stochastic models are dominated bylinear dynamics and additive noise. The linear correction terms and the additive noise stem from the linearcoupling between resolved and unresolved modes, and not from nonlinear interactions between resolvedand unresolved modes as assumed in previous studies.

1. Introduction

The understanding of low-frequency variability ontime scales of more than 10 days to one season hasattracted a lot of attention. Most of the midlatitudelow-frequency variability can be described by a few re-curring persistent teleconnection patterns (Wallace andGutzler 1981; Barnston and Livezey 1987). These pat-terns have a strong impact on surface climate and arerelated to global warming (Marshall et al. 2001; Thomp-son and Wallace 2001). The atmospheric circulationshows a large range of variability both on spatial andtemporal scales. This variability ranges from these per-sistent large-scale teleconnection patterns, to synopticscale waves with a time scale of a few days, and rathersmall scale boundary layer processes acting on anhourly time scale. Due to the nonlinear character of theequations of motion, all of these processes are dynami-cally coupled. It is quite interesting for climate studiesto develop systematic low-order models for the large-scale teleconnection patterns of the atmosphere. Oneneeds to properly account for the coupling with shortertime scale unresolved processes and this is known as the

closure problem. The issue of whether the large-scalelow-frequency atmospheric variability can be modeledsystematically by a low-dimensional stochastic dynami-cal system is the main topic of this paper.

In numerical atmospheric models the atmosphericfields are represented on a particular grid or are ex-panded in spherical harmonics, in so-called spectralmodels. All features smaller than the grid size orsmaller than the smallest wavelength are neglected. Pa-rameterizations are used to account for these neglectedprocesses and their interaction with the resolved scales.Highly truncated spectral models (Charney and De-Vore 1979; Reinhold and Pierrehumbert 1982; Legrasand Ghil 1985) have been developed to give insight intothe dynamics of low-frequency variability and atmo-spheric regime behavior, but they exhibit only very ide-alized flow patterns.

This raises the question if the descriptions used innumerical models are optimal choices in describinglarge-scale atmospheric features. Decomposing the at-mospheric fields more directly in terms of the recurringteleconnection patterns could have advantages. Pos-sible choices of better basis functions are empirical or-thogonal functions (EOF) (Schubert 1985; Selten 1995;Achatz and Branstator 1999) and principal interactionpatterns (PIP) (Hasselmann 1988; Kwasniok 1996,2004). Truncated EOF models show a climate drift dueto the neglected interactions with the unresolved

Corresponding author address: Dr. Christian Franzke, CourantInstitute of Mathematical Sciences, New York University, 251Mercer Street, New York, NY 10012.E-mail: [email protected]

1722 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 62

© 2005 American Meteorological Society

JAS3438

modes. Selten (1995) and Achatz and Branstator (1999)parameterize these neglected interactions by a lineardamping, whose strength is determined empirically. Amore powerful tool to extract the dynamics of a systemare PIPs (Kwasniok 1996, 2004). The calculation ofPIPs takes into account the dynamics of the model forwhich one tries to find an optimal basis and also ofteninvolves ad hoc closure through linear damping and anansatz for nonlinear interactions. Crommelin and Ma-jda (2004) compare different optimal bases. They findthat models based on PIPs are superior to models basedon EOFs and optimal persistence patterns (OPP; Del-sole 2001). On the other hand, they also point out thatthe determination of PIPs can show sensitivities regard-ing the calculation procedure, at least for some low-order atmospheric dynamical systems with regime tran-sitions. This feature makes PIPs possibly a less attrac-tive basis.

The low-order models described above are based onphysical principles combined with an ad hoc closureprinciple involving damping. They take the governingsystem of equations and project them on more or lesssuitable basis functions. Another method is to deter-mine the model by empirical fitting procedures. Thisapproach linearizes the equations of motion aroundsome basic state, determines the linear operator em-pirically, and adds some kind of forcing: this forcing canbe random and is considered as taking account of theneglected nonlinear processes (Newman et al. 1997;Whitaker and Sardeshmukh 1998; Zhang and Held1999; Winkler et al. 2001) or represents external forcinglike tropical heating (Branstator and Haupt 1998). Toensure stability of these linear models these studies addlinear damping according to various ad hoc principles.There is a recent survey of such strategies (Delsole2004).

Another approach is to fit simple stochastic modelsto scalar teleconnection indices. Wunsch (1999) arguesthat the North Atlantic Oscillation (NAO) time seriescannot be easily distinguished from a random station-ary process. This is confirmed by Feldstein (2000), whoconcludes that the NAO is a Markov process with ane-folding time scale of about 10 days by fitting a first-order autoregressive process. Stephenson et al. (2000)find that interannual variations of the NAO have abroadband spectrum, which is close to being whitenoise by analyzing monthly mean data. But they alsopresent evidence for long-range dependencies by fittinga fractional differenced model. Percival et al. (2001) fita first-order autoregressive and a fractionally differ-enced model to the North Pacific (NP) index. Bothmodels show that the NP index exhibits significant cor-relations and also have a long tail of small but positivecorrelations for long lags. But their statistical tests can-not distinguish the superiority of one model over theother.

All these previous studies show encouraging resultsbut have also their disadvantages in other respects.

Majda et al. (1999, 2001, 2002, 2003; collectively here-after MTV) provide a systematic framework for how toaccount for the effect of the unresolved degrees of free-dom on the resolved modes. The stochastic mode re-duction strategy put forward in that work predicts alldeterministic and stochastic correction terms. It hasbeen applied and tested on a wide variety of simplifiedmodels and examples. The idealized models where theprocedure has been tested, include those with trivialclimates (MTV02), periodic orbits or multiple equilib-ria (MTV03), and heteroclinic chaotic orbits coupled toa deterministic bath of modes satisfying the truncatedBurgers equation (Majda and Timofeyev 2000, 2004);the MTV procedure has been validated in these ex-amples even when there is little separation of timescales between resolved and unresolved modes. To in-vestigate the influence of topographic stress on the an-gular momentum budget, they applied this strategy toidealized flow over topography and derived a nonlinearreduced stochastic model with multiplicative noises(MTV03). This stochastic model turns out to be supe-rior compared to a standard linear model with dampingand white noise forcing. Also the explicit assumptionsof varying the correlation times between the resolvedmodes and the unresolved modes in the MTV proce-dure for the climate model was checked explicitly(MTV03) by varying the topographic height. Here aseamless framework of the systematic stochastic modereduction strategy from the MTV papers is developedfor direct implementation in complex geophysical mod-els. This seamless framework makes the practicalimplementation of the MTV procedure simpler withthe same reduced stochastic equations for the unre-solved modes. The mode reduction procedure is ap-plied to a realistic barotropic model climate. Wepresent results for the resulting low-order stochastic cli-mate model with as little as only two resolved modes.The basis functions for this model are the dominantteleconnection patterns of the barotropic model cli-mate. In section 2, we present the barotropic model andits climate. Section 3 describes the seamless MTVframework with details in an appendix, and section 4gives the results of the reduced stochastic model. Thispaper closes with a summary and conclusions.

2. Barotropic model climate

a. Barotropic model

In this study a standard spectral barotropic model onthe sphere is used with realistic orography. This modelis forced with a forcing that was calculated from obser-vations in order to get a realistic climatological meanstate and low-frequency variability. The model equa-tion is given by the barotropic vorticity equation

!"

!t! "J##, " $ f $ h% "

"

$$ D%3" $ F, #1%

JUNE 2005 F R A N Z K E E T A L . 1723

where & is the relative vorticity, f ! 2' sin ( stands forthe Coriolis parameter, ' is the angular velocity of thesphere, h is the topography; ) is the Ekman dampingcoefficient, D the coefficient of the scale-selectivedamping, and F is an external time-independent forcing(see appendix A). The equation has been nondimen-sionalized using the radius of the earth as unit of lengthand the inverse of the angular velocity of the earth asunit of time. The Jacobi operator J(a, b) is defined by

J#a, b% ! !!a!&

!b!'

"!a!'

!b!&",

where * denotes the sine of the latitude (, and + thelongitude. The nondimensional orography h is relatedto the realistic orography h, by h ! A0h,/H, whereA0 ! 0.2 is a factor that determines the strength of thesurface wind that blows across the orography, and H !10 km, a scale height (see Selten 1995). The model istruncated at T21. By restricting the spectral model tomodes with zonal wavenumber plus total wavenumberbeing even, a model of hemispheric flow is obtainedwith a total number of 231 variables. The model hasbeen integrated for 105 days after a spinup period of1000 days, using a fourth-order Adams–Bashforth timestep scheme with a 45-min time step. The Ekman damp-ing time scale is set to 15 days and the strength of thescale selective damping is such that wavenumber 21 isdamped at a time scale of 3 days.

b. Climatology

The time mean streamfunction and the standard de-viation are shown in Fig. 1. The simulated climate isdifferent from the observations in some respects. The

simulated ridges on the west coasts of North Americaand Europe are more pronounced than in the observa-tions (Fig. 1a). The magnitude of the simulated vari-ability is comparable with the observations (Fig. 1b).The simulated variability captures the maxima over theNorth Atlantic in the right location and amplitude. Theobserved maxima over the North Pacific is split in two:one is shifted southwestward and the other northwest-ward. But by considering all processes that the modellacks due to its barotropic nature (e.g., baroclinic insta-bility, tropical convection, ENSO) and its coarse reso-lution (T21), the model climate is reasonably close tothe observations. For the purpose here of testing thesystematic stochastic mode reduction strategy themodel climate is close enough to the observations.

EOFs of the streamfunction data are calculated. Forthe calculation, the time mean is subtracted from thedata; the norm chosen is the kinetic energy norm inorder to preserve nonlinear symmetries in the dynamics(see appendix B; Kwasniok 1996, 2004). The first EOFis characterized by a center of action over the Arcticthat is surrounded by a zonal symmetric structure inmidlatitudes (Fig. 2a). This pattern bears resemblanceto the Arctic Oscillation/Northern Hemisphere Annu-lar Mode (AO/NAM) (Thompson and Wallace 1998).This pattern explains about 14% of the total variance.The second EOF pattern is displayed in Fig. 2b andresembles qualitatively the Pacific/North America(PNA) pattern (Wallace and Gutzler 1981). It consistsof a meridional dipole over the Pacific Ocean at the endof the Pacific subtropical jet and two further centers ofaction over Canada and Florida. But it misses the cen-ter of action over the North Atlantic (Wallace andThompson 2002). This pattern explains about 10% of

FIG. 1. (a) Climatological mean of T21 barotropic model (contour interval is 1.5 - 107 m2 s"1). (b) Standarddeviation of low-pass filtered streamfunction of T21 barotropic model (contour interval is 2 - 106 m2 s"1).


the total variance. The successive EOF patterns be-come increasingly small scale and do not correspond towell-known teleconnection patterns found in observa-tional data. The cumulative explained variance of thefirst seven EOFs is more than 50% of the total variance(Fig. 3).

The autocorrelation functions for the first seven ki-netic energy norm EOFs are displayed in Fig. 4. Some

of the autocorrelation functions, such as modes 1, 4, 6,and 7, show only weak oscillations or no oscillations atall. While others, such as modes 2, 3, and 5, have stron-ger oscillations. The time scale set by the autocorrela-tion function is an important measure in this study be-cause for the development of a stochastic climate modelwe have to make the truncation between slow and fastmodes. The autocorrelation time scale, defined as the

FIG. 2. Kinetic energy norm EOF: (a) EOF1 and (b) EOF2.

FIG. 3. Spectrum of the explained variances.


integral over the absolute value of the autocorrelationfunction is

Tk#s% !.ak#t%ak#t $ s%/

.ak2/

, #2%

where k indicates the kth EOF mode, s the time lag,and .·/ denotes a time average; Tk(s) indicates how slowor fast a variable loses its memory. In Fig. 5 we showthe autocorrelation time scale for all EOF modes. Thefirst remarkable feature of this plot is that the decay ofthe autocorrelation time scale is not a monotonic func-tion of the index of the EOF modes. Mode 1 has theslowest decay with an integral timescale of about 12days, but modes 3, 4, and 6 decay slower than mode 2.The first seven modes decay more slowly than 6 days,and all remaining modes decay faster.

The whole idea behind the stochastic climate modelis the separation between slowly evolving low-fre-quency (resolved) modes and considerably faster evolv-ing unresolved modes. The stochastic mode elimination

procedure (MTV01) is rigorously valid in the limit thatthe ratio of the autocorrelation time scale of the slowestunresolved mode to the fastest resolved modes goes tozero. Several examples from MTV02 and MTV03 showthat for ratios of up to 0.5, and sometimes even 1.0, thisapproach is still applicable. For the cases presentedhere this ratio is relatively large; in particular, by con-sidering only the first two modes as resolved this ratiois 1.5, for the first four modes as climate modes thisratio becomes 1.1, and in the case of the first sevenmodes 0.67. As shown below, despite these large ratiosthe stochastic low-order models perform reasonablywell.

The third-order moment as defined by

!k#s% !.ak

2#t $ s%ak#t%/

.ak2#t%/3(2 #3%

measures the deviations from Gaussianity. This quan-tity is normalized so that !k(s) ! 0 for Gaussian vari-ables. The leading EOF modes of the barotropic model

FIG. 4. Autocorrelation function of the amplitude of the kinetic energy norm EOFs of the barotropic model: (a) EOF1(solid line) and EOF2 (dashed line), (b) EOF3 (solid line) and EOF4 (dashed line), (c) EOF5 (solid line) and EOF6(dashed line), and (d) EOF7 (solid line).


show only modest departures from Gaussianity (Fig. 6).The first three modes show the largest departures forsmall lags of about 15 days, which decay quickly to zero.These modest departures from Gaussianity indicatethat the EOF modes of the barotropic model are likelyto be dominated by a linear Gaussian process. This isfurther confirmed by the one-dimensional probabilitydensity functions (PDFs) (Fig. 7). All EOF modes ofthe barotropic model have a quasi-Gaussian structure.To highlight this feature, we also display the corre-sponding theoretical Gaussian distribution togetherwith the PDFs in Fig. 7. The barotropic model PDFs areclose to their corresponding Gaussian distributions.The largest deviations from Gaussianity have EOFs 1,3, and 5.

It is worth pointing out that these leading EOF pat-terns in the present barotropic model have a similarstructure and steady-state statistics as those from a re-cent perpetual January simulation of the National Cen-ter for Atmospheric Research Community ClimateModel (NCAR CCM0) general circulation model(Berner 2003).

3. Systematic stochastic mode reduction

We illustrate the ideas for stochastic climate model-ing for the barotropic model in (1), which we write inthe following symbolic form:

!#

!t! L# $ B##, #% $ F, #4%

where 0 denotes the streamfunction vector, L is a linearoperator, B is a quadratic nonlinear operator, and Fdenotes a constant forcing. By defining u ! 0 " 0,where 0 is the time mean streamfunction, Eq. (4) be-comes

!u!t

! Lu $ L# $ B##, #% $ B#u, #% $ B##, u%

$ B#u, u% $ F. #5%

By defining a new linear operator, Lu ! Lu $ B(u, 0)$ B(0, u) and F ! F $ L0 $ B(0, 0), we can writethis as

!u!t

! Lu $ B#u, u% $ F. #6%

In stochastic climate modeling, the variable u is decom-posed into an orthogonal decomposition through thevariables u and u,, which are characterized by stronglydiffering time scales. The variable u denotes a low-fre-quency mode of the system, which evolves slowly intime compared to the u, variables. By decomposing u !u $ u, in terms of kinetic energy norm EOFs (appendixB) we can write them as

u ! 1i!1

N

aiei ! 1i!1

R

)iei $ 1j!R$1

N

*jej, #7%

with u ! 1Ri!12iei and u, ! 1N

j!R$13jej, where R is thenumber of resolved modes, ai denote the EOF expan-

FIG. 5. Spectrum of the autocorrelation time scale.


sion coefficients, 2i (3j) are the expansion coefficientsof the resolved (unresolved) modes, and N ! 231 forthe present application. The use of the kinetic energynorm ensures the conservation of kinetic energy by the

nonlinear operator (Selten 1995; Kwasniok 1996). Byproperly projecting the EOF expansions, derived fromthe barotropic model, onto Eq. (6), we get two sets ofequations for resolved 2i and unresolved 3i modes:

)i#t% ! +Hi) $ 1

jLij

)))j#t% $1+ 1

jLij

)**j#t% $ 1jk

Bijk))))j#t%)k#t% $

2+ 1

jkBijk

))*)j#t%*k#t% $1+ 1

jkBijk

)***j#t%*k#t%, #8%

*i#t% ! +Hi* $

1+ 1

jLij

*))j#t% $1+ 1

jLij

***j#t% $1+ 1

jkBijk

*)))j#t%)k#t% $2+ 1

jkBijk

*)*)j#t%*k#t% $1

+2 1jk

Bijk****j#t%*k#t%,

#9%

where the interaction coefficients are defined in appen-dix C and the nonlinear operators have been symme-trized; that is, Bijk ! Bikj in (8) and (9). The upperindices 2 and 3 indicate the respective subsets of thefull operators in (6). Here 4 is a small positive param-eter that controls the separation of time scale between

slow and fast modes and measures the ratio of the cor-relation time of the slowest unresolved mode u, to thefastest resolved mode u. In placing the parameter infront of particular terms we tacitly assume that theyevolve on a faster time scale then the terms involvingthe resolved modes alone. Ultimately, 4 is set to the

FIG. 6. Third-order moment of the amplitude of the kinetic energy norm EOFs of the barotropic model: (a) EOF1 (solidline) and EOF2 (dashed line), (b) EOF3 (solid line) and EOF4 (dashed line), (c) EOF5 (solid line) and EOF6 (dashedline), and (d) EOF7 (solid line).


FIG. 7. PDF of the amplitude of the kineticenergy norm EOFs of the barotropic model.The solid lines are the barotropic modelmodes, and the dashed lines are correspondingGaussian PDFs with the same variance as thebarotropic model modes. The means are notsubtracted.


value 4 ! 1 in developing all final results (MTV02,MTV03), that is, introducing 4 is only a technical step inorder to carry out the MTV mode reduction strategy.Such a use of 4 has been checked on a wide variety ofidealized examples where the actual value of 4 rangesfrom quite small to order 1 (MTV02, MTV03; Majdaand Timofeyev 2004). In particular, here in (8) and (9)we make the following assumptions:

• The external forcings H2 and H3 act on a slow timescale of order 4. This assumption leads to their van-ishing in the effective equations.

• L22 and B222 act on a time scale of order one.• The linear operators L23, L32, and L33 act at most on

a faster time scale of order 4"1.• The self-interaction of the unresolved modes B333

acts on the fastest time scale of order 4"2, whereasB223, B233, B322, and B323 act on a time scale oforder 4"1.

Furthermore, it is assumed that the dynamical systemgiven only by unresolved modes with interaction B333 isergodic and mixing, and can be represented by a sto-chastic process, and that all unresolved modes arequasi-Gaussian distributed (based on the model dy-namics this assumption is valid). With these assump-tions, the seamless MTV procedure (see appendix D)predicts the following effective stochastic equations forthe resolved variables alone:

d)i#t% ! 1j

Lij)))j#t%dt $ 1

jkBijk

))))j#t%)k#t%dt

$ Hidt $ 1j

Lij)j#t%dt $ 1jk

Bijk)j#t%)k#t%dt

$ 1jkl

Mijkl)j#t%)k#t%)l#t%dt

$ 52 1j

, ij#1%#)#t%%dWi

#1% $ 52 1j

, ij#2%dWi

#2%.

#10%

This stochastic differential equation system is in Itôform (Gardiner 1985). This system consists of the baretruncation (minus the bare forcing); additional newforcing; linear, quadratic, and cubic nonlinear interac-tion terms; as well as additive and multiplicative noises.All of these correction terms and noises are predictedby the systematic stochastic mode reduction strategyand account for the interaction between resolved andunresolved modes, as well as for the self-interactionbetween the unresolved modes. As described belowthese correction terms stem from certain physical pro-cesses. The explicit values of the coefficients in (10) aredetermined by the seamless MTV procedure in appen-dix D with the only inputs from the unresolved modesinvolving their variances and the integrated autocorre-lation components (the unresolved correlation times).

To see which of these correction terms play a vital

role in the integrations of the low-order stochasticmodel, that is, (10), we group the interaction terms be-tween resolved and unresolved modes according totheir physical origin and set a parameter +i in front ofthe corresponding interaction coefficient. The interac-tion between the triads B233 and B323 give rise to ad-ditive noise and a linear correction term; we will namethese triads additive triads and set a +A in front of them(MTV99, MTV01, MTV02, MTV03). The second typeof triad interaction is between B223 and B322. Theseinteractions create multiplicative noises and cubic non-linear correction terms (MTV99, MTV01, MTV02,MTV03); we will call them multiplicative triads in thefollowing and indicate them by a +M. The linear cou-pling between the resolved and unresolved modes L23

and L32 give rise to additive noise and a linear correc-tion term (MTV01), which are called the augmentedlinearity here, indicated by a +L. We set a +F in front ofthe last remaining interaction term L33, the linear cou-pling of the unresolved modes. The quadratic nonlinearcorrections, a forcing term, and a further multiplicativenoise contribution are caused by the interaction be-tween the linear coupling terms and the multiplicativetriads. Another forcing correction term comes from theinteraction between additive triads and the linear cou-pling of the fast modes.

After regrouping the terms the stochastic climatemodel from (10) can be written according to thesephysical processes as

d)i#t% ! 1j

Lij)))j#t%dt $ 1

jkBijk

))))j#t% )k#t%dt

$ &A2 1

jLij

#2%)j#t%dt $ &A521j

, ij#2%dWj

#2%

$ &M2 !1

jLij

#3%)j#t%dt $ 1jkl

Mijkl)j#t%)k#t%)l#t%dt"$ &L

2 !1j

Lij#1%)j#t%dt"

$ &M&L#Hj#1%dt $ 1

jkBijk)j#t% )k#t%dt%

$ &A&FHj#2%dt

$ 521j

,ij#1%#)#t%% dWj

#1%, #11%

where the nonlinear noise matrix !(1) satisfies

&L2 Qij

#1% $ &L&M1k

Uijk)k#t% $ &M2 1

klVijkl)k#t%)l#t%

! 1k

, ik#1%#)#t%%, jk

#1%#)#t%%. #12%

It is guaranteed (MTV01) that the operator on the left-hand side of (12) is always positive definite, ensuringthe existence of the nonlinear noise matrix on the right-hand side. All coefficients are defined explicitly in ap-pendix D.


The first line in Eq. (11) is the bare truncation (minusbare forcing), the second line denotes the additive triadcontributions. The third line indicates the multiplicativetriad contributions, the fourth line is the augmentedlinearity, the fifth line denotes the interaction betweenmultiplicative triads and the linear coupling, and thesixth line comes from the interaction between additivetriads and the linear coupling of the fast modes. The lastline and (12) determine a multiplicative noise operator,whose contributions stem from the linear coupling, themultiplicative triads, and the interaction between mul-tiplicative triads and the linear coupling. By consideringonly contributions from the linear coupling, in otherwords setting +M ! 0 in (12), this multiplicative noiseoperator reduces to an additive noise operator.

For integrating this stochastic differential equation asplit step numerical procedure is utilized with a fourth-order Runge–Kutta scheme for the deterministic partand an Euler-forward scheme for the stochastic part. Atime step size of 0.01 days has been used for all inte-grations. Statistics of the stochastic model are calcu-lated by time averaging an individual solution inte-grated over a long time of the order of 106 days.

4. Stochastic climate model results

a. Bare truncation

First, we test the performance of the low-order modelwithout any closure to account for the neglected inter-actions with the unresolved EOFs, the so-called baretruncation (this contains the bare forcing). This is doneto see how well the bare truncation dynamics capturethe projected dynamics of the EOFs of the originalbarotropic model. To analyze the flow characteristicsof these low-order systems we integrate the modelstarting from 10 000 normally distributed initial condi-tions for 1000 days and save only the last model state.Integrations go to a fixed point for low-order modelsconsisting of only two or three resolved modes. Thefixed point is always the same for the respective low-order model. Low-order models (those with 4 to 7modes) go into a stable periodic orbit (not shown). Theperiod orbit is the same for every realization. This in-dicates that the bare truncation does not capture thedynamics of the leading EOF modes of the barotropicmodel very well.

Long integrations for single realizations show thatthe bare truncation low-order models have large means(the barotropic model mode means are zero) and toomuch variance. The autocorrelation function of the7-mode system is dominated by a stable oscillation (Fig.14). The corresponding PDFs show a double peakstructure and also too much variance (Fig. 15). Theseresults emphasize the importance of properly account-ing for the neglected unresolved interactions. Climatedrifts and too-large variances of truncated EOF modelshave also been reported by Selten (1995), Achatz and

Branstator (1999), and for PIP models (Kwasniok1996). These studies add linear damping to their modelsin order to eliminate the climate drift and to reduce thevariances.

b. Stochastic climate model with only diagonalcorrelations for unresolved modes

Now we apply the systematic stochastic mode reduc-tion strategy that properly accounts for the neglectedinteractions with the unresolved modes. In appendix Dwe develop a general framework for stochastic modereduction, which was summarized in the last section.First we will present results from a simplified frame-work, which requires the least input of informationfrom the unresolved modes where we assume that allcross correlations are zero between the unresolvedmodes. This means that we set Bij(t) ! Bi(t)6ij in Eqs.(D16)–(D31). Results for the general strategy will bepresented in the next section.

First we describe the results for the 4-mode reducedstochastic system. The means of the individual modesare small; therefore, the climatological streamfunctionis very close to the barotropic model mean (not shown).The amplitude is well captured and the pattern corre-lation between both model means is 0.99 (Table 1).Figure 8 shows the horizontal distribution of the stan-dard deviation of the streamfunction and the climato-logical transient eddy forcing " 7"2" · (u!&,). To allowfor a fair comparison, the corresponding barotropicmodel standard deviation and transient eddy forcinghave been calculated from the first four EOFs only.The stochastic model reproduces the key features of thestandard deviation quite well. Only the amplitude isunderestimated by a factor of about 2. The patterncorrelation between both is 0.98. Also the key featuresof the climatological transient eddy forcing are repro-duced fairly well, although the amplitude is againunderestimated. The pattern correlation for the tran-sient eddy forcing is 0.91. Figure 9 shows the autocor-relation functions of the four leading modes in the re-duced model. It captures roughly most of the oscilla-tions of modes 2 and 4. The stochastic climate modeldoes not reproduce the oscillations of modes 1 and 3.The stochastic climate model reproduces reasonablywell the low-frequency envelop of the autocorrelationfunction of the original EOFs of the barotropic model,

TABLE 1. Pattern correlation between barotropic model andstochastic model for various truncations. To allow for a fair com-parison, the corresponding barotropic model fields have been cal-culated with the same number of EOFs as for the stochasticmodel.

Numberof modes Mean Std dev

Transienteddy forcing

2 0.99 0.98 0.864 0.99 0.98 0.917 0.99 0.99 0.97


as can be seen in a semilogarithmic plot of the absolutevalue of the autocorrelation function (Fig. 10). The low-frequency envelop of the barotropic model is wellcaptured by modes 1, 3, and 4 and slightly overesti-mated for mode 2. The largest discrepancies occurfor lags larger than 60 days. At these large lags theautocorrelation functions have already decayed tosmall values; therefore, these differences are morelikely due to numerical errors than any systematicdifferences of the models. The third-order momentshows deviations from Gaussianity that are roughlyof the same magnitude as for the modes of the baro-tropic model (not shown). The PDFs (Fig. 11) have a

nearly Gaussian structure. They also show that thestochastic climate model underestimates the variancesby a factor of about 1.5 to 2. In comparison with thebare truncation integration, the predicted correctionterms and noises improve the results considerably.It has to be noted that, by assuming that the resolvedforcing H2

i in (9) acts on a time scale of the order of1 instead of 4 (as assumed above), the stochastic modelexperiences a climate drift. These means are still con-siderably smaller than the means of the bare trunca-tion integrations. The assumption that the resolvedforcing H2

i acts on a time scale of the order of 1 hasonly a negligible effect on the autocorrelation func-

FIG. 8. Horizontal distribution of streamfunction standard deviation: (a) barotropic model (contour interval is5 - 105 m2 s"1), and (b) 4-mode stochastic model (contour interval is 2 - 105 m2 s"1). Transient eddy forcing:(c) barotropic model (contour interval is 5 m2 s"2), and (c) 4-mode stochastic model (contour interval is 2 m2 s"2).For this comparison the barotropic model streamfunction has been reconstructed from the first four EOFs.


tions, the third-order moments, and the variances (notshown).

Also for the 2-mode system with only diagonal cor-relations the geographical distribution of the means,standard deviations, and transient eddy forcing agreereasonably well with the barotropic model (Table 1).Figure 12 shows the autocorrelation function for the2-mode system. Mode 1 has not changed in comparisonwith the 4-mode system (Fig. 9a) while mode 2 has lostits oscillation; but again the 2-mode low-order modelcaptures roughly the low-frequency envelop of thebarotropic model modes reasonably well (Figs. 12c,d).Also in this case we get quasi-Gaussian PDFs, and thestochastic climate model underestimates the variancesby a factor of 1.5 to 2 (Fig. 13), but less than for the4-mode system. The third-order moments show devia-tions from Gaussianity that are of the right strength(not shown). Recall that this 2-mode stochastic model isa systematically derived numerical model for the twoleading principal teleconnection patterns with AO andPNA behavior.

By increasing the number of resolved modes to 7, the

geographical distribution of the means, standard devia-tions, and transient eddy forcing are reasonably wellreproduced again (Table 1). The pattern correlation forthe mean streamfunction and the standard deviation isstill high and the pattern correlation for the transienteddy forcing has improved to 0.97. Only the amplitudeof the standard deviation and transient eddy forcing areunderestimated (not shown). The autocorrelation func-tions of the first three modes are further improved (Fig.14). Modes 1 and 2 follow more closely the barotropicmodel modes and mode 1 also exhibits a dip around lag25 days. Mode 3 now also shows an oscillation, which ismuch slower than the corresponding barotropic modelmode oscillation (Fig. 14c). The full autocorrelationfunction is reproduced well for modes 6 and 7. Overall,the autocorrelation functions of the stochastic climatemodel captures the low-frequency envelope of thebarotropic model modes. The corresponding PDFs arenearly Gaussian (Fig. 15). The stochastic climate modelunderestimates the variances by a factor of about 2 to 3for the first 5 modes and has the right variances formode 6 and 7. Also this stochastic climate model has

FIG. 9. Autocorrelation function for the 4-mode system; solid line is stochastic model, and dashed line is barotropicmodel. Only diagonal correlations are used.


vanishing means and the deviations from Gaussianityare of the right order (not shown).

The low-order stochastic models reproduce the geo-graphical distribution of the means, standard devia-tions, and transient eddy forcing well. Furthermore,they capture reasonably well the low-frequency os-cillations of the modes of the barotropic model buthave some difficulties simulating the high-frequencyoscillations. All low-order stochastic models presentedabove have modes that are more persistent than thecorresponding barotropic model modes. They all un-derestimate the variances but possess self-consistentvanishing means. By comparing these results with thecorresponding integrations of the bare truncation, thepredicted correction terms and noises reduce the meansand variances considerably. Now they have roughlythe right sizes and are no longer orders of magnitudeoff as for the bare truncation integrations. Anotherproperty is that the variances are roughly equally dis-tributed between the various modes; the variance doesnot decrease as rapidly with increasing mode number

as for the barotropic model modes. However, by con-sidering that we developed very low-order modelswith as little as only two resolved modes, the stochasticclimate model reproduces reasonably well the keystatistical features of the leading EOF modes for thebarotropic model. Furthermore, this was achieved with-out ad hoc additions of dissipation and white noiseforcing but such terms were produced automaticallyfrom the systematic procedure itself. As indicated bythe autocorrelation time spectrum in Fig. 5, as in manyclosure procedures, one should not expect conver-gence of the stochastic mode reduction procedureas the number of resolved modes R increases sincethe ratio of correlation times remains about 1 forlarge R.

c. Stochastic climate model with full crosscorrelations for unresolved modes

Now we present results from the general stochasticmode reduction strategy as developed in appendix D

FIG. 10. Autocorrelation function for the 4-mode system using a semilogarithmic axis. Solid line is stochastic model,and dashed line is barotropic model. Only diagonal correlations are used.


with the full matrix of unresolved cross-correlationtimes. We will focus on the 4-mode system; the sameconclusions apply to the 2- and 7-mode systems.

As for the simplified diagonal strategy discussed ear-lier the low-order stochastic model captures the low-frequency envelope of the autocorrelation function ofthe barotropic model modes (Fig. 16). Now mode 1decays faster and the oscillation of mode 2 is weakerand is a better fit for the low-frequency envelope of thebarotropic model mode, and mode 4 is again very closeto the barotropic model mode (Fig. 9). Including thecross correlations leads to only minor changes of thevariances and means (not shown). This indicates thatthe cross correlations of the unresolved modes have aneffect on the resolved modes. They lead to improve-ments for certain modes, but do not affect the meansand variances.

d. Budget analysis

The above sections show that the systematic low-order stochastic climate model is performing reason-

ably well. This raises the question regarding which ofthe various correction terms in (10) are important. Wewill now investigate this question. In Eq. (11) the cor-rection terms are decomposed corresponding to theirphysical origin. A close investigation of the various cor-rection terms and noises reveals that the augmentedlinearity, +L together with its associated additive noiseterm, produce the largest tendencies. The tendenciescoming from the remaining correction terms and noisesare at least an order of magnitude smaller during theintegration of the model. By including only the baretruncation terms (minus bare forcing), the augmentedlinearity and its associated noise term, we are able toreproduce the results of the full stochastic climatemodel integrations as described above (not shown).This amounts to setting +L to 1 and all other +i to zeroin (10).

Now the question arises how well a purely linearmodel with additive noise could perform. This is aninteresting question since most of the previous studieson stochastic climate modeling use a linear model with

FIG. 11. PDF for the 4-mode system. Solid line is stochastic model, and dashed line is barotropic model. Themeans are not subtracted.


additive noise. To test this hypothesis we use only thelinear part of the bare truncation (minus bare forcing),the augmented linearity correction terms, and its asso-ciated additive noise term to integrate the low-orderstochastic climate model. This integration shows thatsuch a linear stochastic model reproduces the full sto-chastic climate model results nearly perfectly (Fig. 17).There are only some minor differences in the autocor-relation functions of mode 3 and 4 (cf. Figs. 16 and 17).The differences in the variances and means are onlymarginal between the full stochastic model and the lin-ear stochastic model (not shown).

It is important to point out that the linear correctionterms and the additive noise stem from the linear cou-pling between the resolved and unresolved modes asshown in (11). In most previous studies about stochasticclimate modeling it is argued that the added additivenoise accounts for the neglected nonlinear interactionsand the forcing (Newman et al. 1997; Winkler et al.2001). In terms of the present context these studies onlyconsider the additive triads, which are leading to a lin-ear correction term and additive noise. These correc-

tion terms have a negligible role in the present study.Also the corrections terms stemming from multiplica-tive triads, which lead to the other nonlinear correctionterms, are negligible in this study.

5. Summary and discussion

This study applies a systematic strategy for stochasticmode reduction to a realistic barotropic model climate.The barotropic model climate has realistic versions ofthe AO and PNA teleconnection patterns as the lead-ing EOFs. The systematic strategy requires first theidentification of slowly evolving climate modes and fastevolving nonclimate modes. The low-order stochasticclimate model consists of the climate modes as resolvedmodes and the stochastic mode reduction procedurepredicts all forcing, linear, quadratic, and cubic correc-tion terms as well as additive and multiplicative noises;these correction terms and noises account for the inter-action of the resolved climate modes with the neglectednonclimate modes and for the self-interaction of the

FIG. 12. Autocorrelation function for the 2-mode system. Solid line is stochastic model, and dashed line is barotropicmodel. Only diagonal correlations are used.


nonclimate modes. The stochastic mode reductionstrategy presented here is a generalized framework ofthat presented in MTV99, MTV01, MTV02, andMTV03, which is more suitable for implementation inother complex geophysical systems with minimal fittingof the unresolved modes.

The stochastic mode elimination procedure is justi-fied rigorously for 4 ! 1, where 4 measures the ratio ofcorrelation times of unresolved modes to the resolvedones. MTV02, MTV03 show that the strategy appliesfor values of 4 as large as 0.5 and even for 1 in manysituations. For the cases presented here 4 is relativelylarge; in particular for the 2-mode low-order model thisratio is 1.5, for the 4-mode model 1.1, and for the7-mode model 0.67. Despite these large ratios the low-order models perform reasonably well.

The low-order stochastic models reproduce the geo-graphical distribution of the climatological means, stan-dard deviation, and transient eddy forcing reasonablywell. On the other hand, they underestimate the ampli-tudes of the standard deviation and of the transienteddy forcing by a factor of about 2. Low-order modelswith four and seven modes capture the autocorrelationfunctions fairly well. These results provide evidence foreffective stochastic dynamics despite the minimal time-scale separation between resolved and unresolvedmodes.

Kwasniok (2004) points out that for deriving low-order models based on PIPs for a similar barotropicmodel climate, at least 15 PIP modes are needed toreproduce the means and variances reasonably well. Tocapture the temporal characteristics, the autocorrela-tion function and the PDFs, at least 40 PIPs are needed.A similar EOF model needs 100 EOFs to model thesystem as well as with 40 PIPs. In a baroclinic model,Achatz and Branstator (1999) report that their EOFmodel needs between 30 and 70 EOFs to simulate theclimate mean state and transient eddy fluxes. The pre-

sented systematic stochastic mode reduction strategyleads to much lower-order models with qualitativelycomparable results.

A budget analysis reveals that the minimal stochasticmodel dynamics consists of the linear part of the baretruncation and the augmented linearity. The aug-mented linearity accounts for the linear coupling be-tween the resolved and unresolved modes. It consistsof a linear correction term and additive noise. Thisminimal linear stochastic model reproduces the statis-tics of the full low-order stochastic model within minordiscrepancies. This linear stochastic model is differ-ent from previous proposed linear stochastic models(Newman et al. 1997; Whitaker and Sardeshmukh1998; Winkler et al. 2001). These studies approximatethe nonlinear part of the equations by a linear operatorand additive noise. This noise is typically white in timebut may be spatially correlated. In other words, theytruncate the dynamics on both the resolved and un-resolved modes, and add ad hoc damping, necessaryto stabilize the model (Whitaker and Sardeshmukh1998). The approach presented here truncates the dy-namics only on the unresolved modes and predictsall necessary corrections and noises; therefore, it alsopredicts the necessary damping. Furthermore, the sys-tematic approach does not assume anything a prioriabout the dominant dynamics, such as linearity. It israther a result of the stochastic mode reduction strategythat for the model climate considered here midlatitudelow-frequency variability can be modeled by a linearlow-order stochastic model. Further evidence for thedominance of linear dynamics in reduced barotropicmodels has been provided by Kwasniok (2004). In hisempirical low-order model based on PIPs, the linearinteraction terms dominated over the nonlinear inter-action terms.

Kwasniok shows that reduced PIP models are supe-rior to low-order models based on EOFs. On the other

FIG. 13. PDF for the 2-mode system. Solid line is stochastic model, and dashed line is barotropic model. The meansare not subtracted.


FIG. 14. Autocorrelation function for the7-mode system. Solid line is stochastic model,dashed line is barotropic model, and dashed–dotted line is bare truncation. Only diagonal cor-relations are used.


FIG. 15. PDF for the 7-mode system. Solidline is stochastic model, dashed line is barotro-pic model, and dashed–dotted line is bare trun-cation (note these figures are scaled to empha-size the stochastic and barotropic model re-sults and not the bare truncation). The meansare not subtracted.


hand, the first 10 or 15 PIPs in the barotropic modelsare virtually indistinguishable from the EOFs (Kwas-niok 1996). An interesting study by Farrell and Ioannou(2001) points to the potential inadequacy of EOFs as abasis for the dynamics. Their work refers to the study ofthe nonnormality of a stable linear operator and opti-mal representation of both the growing structures in thesystem and the structures into which these evolve. Thelinear operator at the climate mean state utilized here isunstable with a 14-dimensional unstable manifold un-less additional ad hoc dissipation is added at the outset;thus this basis is not directly applicable here. Recently,Kwasniok (2005) has introduced a simplified PIP strat-egy that only utilizes tendencies in the linear dynamicsbut allows for instantaneous linear instabilities. Obvi-ously, it is worthwhile to explore the basic stochasticmode reduction strategy developed here in other opti-mal bases besides EOFs. The authors plan to do this ina future publication.

Acknowledgments. We thank Rafail Abramov forproviding us with his barotropic model code and Grant

Branstator and Daan Crommelin for their comments.The manuscript was improved by the comments ofthree anonymous reviewers. We acknowledge theNOAA Climate Diagnostics Center for providing uswith the NCEP–NCAR reanalysis dataset. This re-search is funded in part through NSF-CMG GrantDMS-0222133 and NSF Grant DMS-0209959.

APPENDIX A

Determination of Forcing

The (time independent) forcing F is based on obser-vations. We are using the observed 300-hPa vorticityand wind fields, obtained from the National Centers forEnvironmental Prediction (NCEP)–NCAR reanalysisdata as our target climate. These data cover the years1958–97 for the months of December–January. The ini-tial forcing (first guess) is calculated from the NCEP–NCAR dataset according to

FIG. 16. Autocorrelation function for the 4-mode system with cross correlation. Solid line is stochastic model, anddashed line is barotropic model.


"F#1% ! " · 8vCL#"CL $ f $ h%9 $"CL

$F$ K76"CL

$ " · #v!"-%CL, #A1%

where the prime denotes deviations from a temporalmean X, and the subscript CL indicates observed cli-matological variables. With this forcing the model wasintegrated for 3600 days, and its variability has beenanalyzed. The standard deviation of low-pass filtered(periods :10 days) streamfunction is weaker and has itsmaximum in a different location when compared withthe observations. Therefore, an iterative procedure isused to determine a forcing field &F from &CL and" · (v!&,)CL that better matches the observations

"F#n $ 1% ! "F#n% $ 8"CL " "#n%9 $ 8. · #v!"-%CL

" . · #v!"-%#n%9, #A2%

where n indicates the iteration step. For this purposeruns over 3600 days have been carried out. From theseruns the actual forcing fields are calculated, and then

the forcing field is updated for the next run. The reduc-tion of the root-mean-square error equilibrates for themean after about 30 iterations.

APPENDIX B

Calculation of EOFs

EOFs are widely used in atmospheric science. Theleading EOFs extract the preferred circulation patternsof the atmosphere from a dataset. Since EOFs consti-tute a complete basis, the streamfunction 0(t) can bewritten as

##t% ! 1i

ai#t%ei, #B1%

where the principal components ai(t) are given by theprojection of 0(t) onto the EOFs ei:

ai#t% ! ###t%, ei%, #B2%

where (·, ·) denotes an inner product. To calculateEOFs that are orthogonal with respect to an inner

FIG. 17. Autocorrelation function for the 4-mode system consisting of only linear bare truncation and augmentedlinearity. Solid line is stochastic model, and dashed line is barotropic model. Only diagonal correlations are used.


product, the following eigenproblem needs to besolved:

CMei ! &iei, #B3%

where M is a diagonal matrix that depends on the innerproduct and C is the covariance matrix. The kineticenergy norm has the following inner product:

##, /% ! "12#%#/ dS !

12#.#./ dS

! "12##%/ dS. #B4%

Due to the representation of the spectral coefficientsand by using the kinetic energy norm, the resulting co-variance matrix for the eigenproblem is not necessarilysymmetric. By solving a slightly different eigenproblemthe covariance matrix can be made symmetric:

5MC5Mgi ! &igi. #B5%

By solving this eigenproblem the eigenvalues +i are thesame as for the original eigenproblem, only the eigen-vectors gi have to be transformed to give the eigenvec-tors ei of the original problem by

ei ! #5M%"1gi. #B6%

APPENDIX C

Interaction Coefficients

Here we derive the interaction coefficients. To sim-plify the derivation we transform the barotropic vortic-ity Eq. (1) into the streamfunction equation:

!#

!t! "%"1!J##, %# $ f $ h% "

%#

$$ D%4# $ F".

#C1%

By multiplying Eq. (C1) with the EOFs and using thekinetic energy norm (appendix B) we derive the fol-lowing interaction coefficients:

Hi ! #ei$F " J##, f $ h% " J##, %#% "%#

$

$ D%4#% dS, #C2%

Lij ! "#ei$J#ej, f $ h% $ J#ej, %#% $ J##, %ej% $%ej

$

" D%4ej% dS, #C3%

and

Bijk ! "#eiJ#ej, %ek% dS. #C4%

APPENDIX D

Effective Equations

We now derive effective equations for systems of thetype as given in (8) and (9). The mode elimination pro-cedure is general but uses the following properties ofthe coefficients Bxyz

ijk : it assumes that these coefficientsare symmetric in the last two indices, that is, Bxyz

ijk !Bxzy

ikj , and satisfy

Bijkxyz $ Bjki

xyz $ Bkijxyz ! 0 #energy conservation%, #D1%

1ijk

Bijkxyz !

!xiyjzk ! 0 #Liouville property%, #D2%

where x, y, and z each stand for 2 or 3.To derive effective equations for the slow modes 2,

we closely follow the mode elimination procedure usedand developed in MTV99, MTV01, MTV02, MTV03,but we give a more seamless version of the results inthese papers, which is more convenient for the compu-tation of the coefficients in the effective equations. Themode elimination procedure is based on the assumptionthat the fastest component of the dynamics of the fastmodes in Eq. (9); that is, the dynamical system

ci ! 1jk

Bijk***cjck #D3%

is ergodic and mixing with integrable decay of correla-tion. In other words, we assume that for almost all ini-tial conditions, and suitable functions f and g, we have

limT!0

1T #

0

T

f#c#t%% dt ! . f/, #D4%

where .·/ denotes expectation with respect to some ap-propriate invariant distribution, and

G#s% ! limT!0

1T #

0

T

g#c#t $ s%, c#t%% dt

" limT!0

1

T2 #0

T #0

T

g#c#t%, c#t-%% dt dt- #D5%

is an integrable function of s; that is, |;<0 G(s)ds| = <.

Under the above assumptions, it can be shown (Kurtz1973; Papanicolaou 1976) that in the limit as 4 ! 0 thedynamics of the slow modes 2i in Eq. (9) can be rep-resented by the following effective Itô stochastic differ-ential equation (SDE):

d)i#t% ! 1j

Lij)))j#t% dt $ 1

jkBijk

))))j#t%)k#t% dt

$ Gi dt $ 52 1j

,ij#)#t%% dWi. #D6%


The second line in this equation consists of the termsthat emerge to represent the effect of the fast modes on

the slow ones as 4 ! 0. Here W ! (W1, · · ·, Wm) is anm-dimensional Wiener process, and Gi is given by

Gi ! limT!0

1T #

0

T

dt#0

0

ds Kj*8c#t%9 1

j& !

!cj#t%Ki

)8c#t $ s%9' $ limT!0

1T #

0

T

dt#0

0

ds 1j

Kj)8c#t%9& !

!)jKi

)8c#t $ s%9',

#D7%

>ij is the Cholesky decomposition (i.e., Pij ! ?k>ik>jk)of

Pij ! limT!0

1T #

0

T

dt#0

0

ds Ki)#c#t%%Kj

)#c#t $ s%%, #D8%

and

Ki)#c% ! 1

jLij

)*cj $ 21jk

Bijk))*)jck $ 1

jkBijk

)**cjck, #D9%

Ki*#c% ! 1

jLij

*))j $ 1j

Lij**cj $ 21

jkBijk

*)*)jck

$ 1jk

Bijk*)))j)k. #D10%

The sequel of this appendix is to put the SDE (D6) ina form more suitable for computations. To do so, weshall make the following assumptions:

1) The statistics of c(t) solution of (D3) can be approxi-mated by the statistics of 3(t) solution of originalsystem in (9). This means all the c can be replaced by3 in (D7), (D8), and (D9).

2) The moments of 3(t) can be approximated as if 3(t)were a Gaussian process. This means in particularthat moments of order 3 vanish, whereas momentsof order 4 can be related to moment of order 2 as

limT!0

1T #

0

T

dt *i#t%*j#t%*k#t $ s%*l#t $ s%

! BiBk1ij1kl $ Bik#s%Bjl#s% $ Bil#s%Bjk#s%,

#D11%

where

Bij#s% ! Bji#"s% ! limT!0

1T #

0

T

dt *i#t%*j#t $ s% #D12%

and

Bij#0% ! Bi1ij. #D13%

Under these assumptions, straightforward manipula-tions show that the drift Gi and diffusion tensor Hijreduce to

Gi ! Hi $ 1j

Lij)j $ 1jk

Bijk)j)k $ 1jkl

Mijkl)j)k)l,

#D14%

Pij ! Qij $ 1k

Uijk)k $ 1kl

Vijkl)k)l. #D15%

The new forcing coefficients are given by

Hi ! H i#1% $ H i

#2%, #D16%

with

H i#1% ! 1

jkl#

"0

0

dt Ljl)*Bijk

))*Blk#t%, #D17%

H i#2% ! 1

klmn#

"0

0

dt Lkl**Bimn

)**Bkm#t%Bln#t% $ Bkn#t%Blm#t%

Bk.

#D18%

The new linear interaction coefficients are given by

Lji ! Lji#1% $ Lji

#2% $ Lji#3% #D19%

with

Lij#1% !

12 1

kl#

"0

0

dt Lkj*)Lil

)*Bkl#t%

Bk, #D20%

Lij#2% !

12 1

klmn#

"0

0

dt Bimn)**! Bljk

*)*

Bl$

Bkjl*)*

Bk"

8Bkm#t%Bln#t% $ Bkn#t%Blm#t%9, #D21%

Lij#3% ! 2 1

kln#

-0

0

dt Bkjl))*Bikn

))*Bln#t%. #D22%

The new quadratic nonlinear interaction coefficientsare given by

Bjik ! Bjik#1% $ Bjik

#2% #D23%

with

Bijk#1% !

12 1

mn#

"0

0

dt Bmjk*))Lin

)*Bmn#t%

Bm, #D24%

Bijk#2% !

12 1

mn#

"0

0

dt#Lmj*)Bikn

))* $ Lmk*) Bijn

))*%Bmn#t%

Bm.

#D25%

The new cubic nonlinear interaction coefficients aregiven by


Mijkl !13 1

mn#

"0

0

dt#Bnkl*))Bijm

))* $ Bnlj*))Bikm

))*

$ Bnjk*))Bilm

))*%Bnm#t%

Bn. #D26%

The diffusion coefficients are given by

Qij ! Qij#1% $ Qij

#2% #D27%

with

Qij#1% !

12 1

lk#

"0

0

dt Lil)*Ljk

)*Blk#t% #D28%

Qij#2% !

12 1

klmn#

"0

0

dt Bikl)**Bjmn

)**

8Bkm#t%Bln#t% $ Bkn#t%Blm#t%9 #D29%

and

Uijk ! 1ln#

"0

0

dt#Lil)*Bjkn

))* $ Ljl)*Bikn

))*%Bln#t% #D30%

Vjikl ! 2 1mn

#"0

0

dt Bikm))*Bjln

))*Bmn#t%. #D31%

The terms involving H(1)i and L(3)

ij are Itô drifts.Note that the effective equations in (D6) can be put

in the form

d)i#t% ! 1j

Lij)))j#t% dt $ 1

jkBijk

))))j#t%)k#t% dt $ Gi dt

$ 52 1j

,ij#1%8)#t%9 dWi

#1% $ 52 1j

,ij#2% dWi

#2%,

#D32%

where W(1) ! (W(1)1 , · · · , W(1)

m ) and W(2) ! (W(2)1 , · · · ,

W(2)m ) are two independent m-dimensional Wiener pro-

cesses and >(1)ji , >(2)

ji are defined so that

Qij#1% $ 1

kUijk)k#t%

$ 1k1

lVijkl)k#t%)l#t% ! 1

k,ik

#1%8)#t%9,jk#1%8)#t%9,

#D33%

Qij#2% ! 1

k,ik

#2%,jk#2%. #D34%

It is straightforward to verify that the left-hand sides inEq. (D34) are positive definite m - m tensors, so thetensors >(1)

ij [2(t)] and >(2)ij exist and can be calculated by

Cholesky decomposition.

REFERENCES

Achatz, U., and G. W. Branstator, 1999: A two-layer model withempirical linear corrections and reduced order for studies ofinternal climate variability. J. Atmos. Sci., 56, 3140–3160.

Barnston, A. G., and R. E. Livezey, 1987: Classification, season-

ality, and persistence of low-frequency atmospheric circula-tion patterns. Mon. Wea. Rev., 115, 1083–1126.

Berner, J., 2003: Detection and stochastic modeling of nonlinearsignatures in the geopotential height field of an atmosphericgeneral circulation model. Ph.D. thesis, Universität Bonn,157 pp.

Branstator, G. W., and S. E. Haupt, 1998: An empirical model ofbarotropic atmospheric dynamics and its response to tropicalforcing. J. Climate, 11, 2645–2667.

Charney, J. G., and J. G. DeVore, 1979: Multiple flow equilibriain the atmosphere and blocking. J. Atmos. Sci., 36, 1205–1216.

Crommelin, D. T., and A. J. Majda, 2004: Strategies for modelreduction: Comparing different optimal bases. J. Atmos. Sci.,61, 2206–2217.

Delsole, T., 2001: Optimally persistent patterns in time-varyingfields. J. Atmos. Sci., 58, 1341–1356.

——, 2004: Stochastic models of quasigeostrophic turbulence.Surv. Geophys., 25, 107–149.

Farrell, B. F., and P. J. Ioannou, 2001: Accurate low-dimensionalapproximation of the linear dynamics of fluid flow. J. Atmos.Sci., 58, 2771–2789.

Feldstein, S. B., 2000: The timescale, power spectra, and climatenoise properties of teleconnection patterns. J. Climate, 13,4430–4440.

Gardiner, C. W., 1985: Handbook of Stochastic Methods.Springer-Verlag, 442 pp.

Hasselmann, K., 1988: PIPs and POPs: The reduction of complexdynamical systems using principal interaction and oscillationpatterns. J. Geophys. Res., 93, 11 015–11 021.

Kurtz, T. G., 1973: A limit theorem for perturbed operator semi-groups with applications to random evolutions. J. Funct.Anal., 12, 55–67.

Kwasniok, F., 1996: The reduction of complex dynamical systemsusing principal interaction patterns. Physica D, 92, 28–60.

——, 2004: Empirical low-order models of barotropic flow. J. At-mos. Sci., 61, 235–245.

Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking andvariations in atmospheric predictability. J. Atmos. Sci., 42,433–471.

Majda, A. J., and I. Timofeyev, 2000: Remarkable statistical be-havior for truncated Burgers-Hopf dynamics. Proc. Natl.Acad. Sci. U.S.A., 97, 12 413–12 417.

——, and ——, 2004: Low dimensional chaotic dynamics versusintrinsic stochastic chaos: A paradigm model. Physica D, 199,339–368.

——, ——, and E. Vanden-Eijnden, 1999: Models for stochasticclimate prediction. Proc. Natl. Acad. Sci. U.S.A., 96, 14 687–14 691.

——, ——, and ——, 2001: A mathematical framework for sto-chastic climate models. Commun. Pure Appl. Math., 54, 891–974.

——, ——, and ——, 2002: A priori tests of a stochastic modereduction strategy. Physica D, 170, 206–252.

——, ——, and ——, 2003: Systematic strategies for stochasticmode reduction in climate. J. Atmos. Sci., 60, 1705–1722.

Marshall, J., and Coauthors, 2001: North Atlantic climate vari-ability: Phenomena, impacts and mechanisms. Int. J. Clima-tol., 21, 1863–1898.

Newman, M., P. D. Sardeshmukh, and C. Penland, 1997: Stochas-tic forcing of the wintertime extratropical flow. J. Atmos. Sci.,54, 435–455.

Papanicolaou, G. C., 1976: Some probalistic problems and meth-ods in singular perturbations. Rocky Mountain J. Math., 6,653–674.

Percival, D. B., J. E. Overland, and H. O. Mofjeld, 2001: Inter-pretation of North Pacific variability as a short- and long-memory process. J. Climate, 14, 4545–4559.

Reinhold, B. B., and R. T. Pierrehumbert, 1982: Dynamics of


weather regimes: Quasi-stationary waves and blocking. Mon.Wea. Rev., 110, 1105–1145.

Schubert, S. D., 1985: A statistical-dynamical study of empiricallydetermined modes of atmospheric variability. J. Atmos. Sci.,42, 3–17.

Selten, F. M., 1995: An efficient description of the dynamics ofbarotropic flow. J. Atmos. Sci., 52, 915–936.

Stephenson, D. B., V. Pavan, and R. Bojariu, 2000: Is the NorthAtlantic Oscillation a random walk? Int. J. Climatol., 20,1–18.

Thompson, D. W. J., and J. M. Wallace, 1998: The Arctic Oscil-lation signature in the wintertime geopotential height andtemperature fields. Geophys. Res. Lett., 25, 1297–1300.

——, and ——, 2001: Regional climate impacts of the NorthernHemisphere annular mode and associated climate trends. Sci-ence, 293, 85–89.

Wallace, J. M., and D. S. Gutzler, 1981: Teleconnections in the

geopotential height field during the Northern Hemispherewinter. Mon. Wea. Rev., 109, 784–804.

——, and D. W. J. Thompson, 2002: The Pacific center of actionof the Northern Hemisphere annular mode: Real or artifact?J. Climate, 15, 1987–1991.

Whitaker, J. S., and P. D. Sardeshmukh, 1998: A linear theory ofextratropical synoptic eddy statistics. J. Atmos. Sci., 55, 237–258.

Winkler, C. R., M. Newman, and P. D. Sardeshmukh, 2001: Alinear model of wintertime low-frequency variability. Part I:Formulation and forecast skill. J. Climate, 14, 4474–4494.

Wunsch, C., 1999: The interpretation of short climate records,with comments on the North Atlantic and Southern Oscilla-tions. Bull. Amer. Meteor. Soc., 80, 245–255.

Zhang, Y., and I. M. Held, 1999: A linear stochastic model of aGCM’s midlatitude storm tracks. J. Atmos. Sci., 56, 3416–3435.


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