Mathematical test models for superparametrizationin anisotropic turbulenceAndrew J. Majdaa,1 and Marcus J. Groteb

aDepartment of Mathematics and Climate, Atmosphere, Ocean Science, Courant Institute of Mathematical Sciences, New York, NY 10012; and bDepartmentof Mathematics, University of Basel, CH-4052 Basel, Switzerlandb;

Contributed by Andrew J. Majda, February 6, 2009 (sent for review January 8, 2009)

The complexity of anisotropic turbulent processes over a widerange of spatiotemporal scales in engineering turbulence andclimate atmosphere ocean science requires novel computationalstrategies with the current and next generations of supercomput-ers. In these applications the smaller-scale fluctuations do not sta-tistically equilibrate as assumed in traditional closure modeling andintermittently send significant energy to the large-scale fluctua-tions. Superparametrization is a novel class of seamless multi-scalealgorithms that reduce computational labor by imposing an artifi-cial scale gap between the energetic smaller-scale fluctuations andthe large-scale fluctuations. The main result here is the system-atic development of simple test models that are mathematicallytractable yet capture key features of anisotropic turbulence inapplications involving statistically intermittent fluctuations with-out local statistical equilibration, with moderate scale separationand significant impact on the large-scale dynamics. The propertiesof the simplest scalar test model are developed here and utilizedto test the statistical performance of superparametrization algo-rithms with an imposed spectral gap in a system with an energetic–5/3 turbulent spectrum for the fluctuations.

intermittency | moderate-scale separation | multiscale algorithms

T he complexity of anisotropic turbulent processes over a widerange of spatiotemporal scales in engineering shear turbu-

lence and combustion (1–3) as well as climate atmosphere oceanscience requires novel computational strategies even with the cur-rent and next generations of supercomputers. This is especiallyimportant since energy often flows intermittently from the smallerunresolved or marginally resolved scales to affect the largestobserved scales in such anisotropic turbulent flows. For example,atmospheric processes of weather and climate cover ≈10 decadesof spatial scales, from a fraction of a millimeter to planetary scales.Regarding atmospheric fluid dynamics, one is primarily concernedwith spatial scales larger than tens of meters because the smallerscales fall whithin the inertial range of atmospheric turbulence.Spatial scales between 100 m and 100 km, referred to as smallthrough mesoscale, show an abundance of processes associatedwith dry and moist convection, clouds, waves, boundary layer,topographic, and frontal circulations. A major stumbling block inthe accurate prediction of weather and short-term climate on theplanetary and synoptic scales is the accurate parametrization ofmoist convection. These problems involve intermittency in spaceand time due to complex evolving, strongly chaotic, and quiescentregions without statistical equilibration and with only moderate-scale separation so that traditional turbulence closure modelingfails (1–3). Cloud-system-resolving models realistically representsmall-scale and mesoscale processes with fine computational grids.But because of the high computational cost, they cannot be appliedto large ensemble size weather prediction or climate simulationswithin the near future.

A different modeling approach, the cloud-resolving convec-tion parametrization (CRCP) or superparametrization (SP) wasrecently developed (4–8). The idea is to use a 2D cloud-system-resolving model in each column of a large-scale model to explicitlyrepresent small-scale and mesoscale processes and interactions

among them. It blends convectional parametrization on a coarsemesh with detailed cloud-resolving modeling on a finer mesh. Thisapproach has been shown to be ideal for parallel computationsand it can easily be implemented on supercomputers. The methodhas yielded promising new results regarding tropical intraseasonalbehavior (4–9) and has the potential for many other applicationsin climate-atmosphere-ocean science (10, 11).

Systematic mathematical formulation and numerical analysisof such superparametrization algorithms has the potential tolead to algorithmic improvements as well as other applicationsin diverse scientific and engineering disciplines and this is themain topic of the present article. Recently (12) it has been shownin a precise fashion how superparametrization can be regardedas a multiscale numerical method. Such multiscale formulation(13) has been utilized recently to develop new highly efficientand skillful versions of superparametrization on mesoscales (8)as seamless multiscale numerical algorithms. In this recent work(8, 12) a theoretical link has been established between superpa-rametrization algorithms and heterogenous multiscale methods(HMMs) developed recently in the applied mathematics litera-ture (14). HMM algorithms are a mathematical synthesis of earlierwork (see refs. 15 and 16, and references therein) as well as anabstract formulation that leads to new multiscale algorithms forcomplex systems with widely disparate time scales (14, 17–19).However, as noted recently (8), there are significant differencesin the regimes of nonlinear dynamics being modeled by super-parametrization algorithms as compared with HMM. ReducedHMM time-steppers have been analyzed and applied for vari-ous physical systems with wide scale separation, ε = 10−3, 10−4,with ε the scale separation ratio between large and small scales,and rapid local statistical equilibration in time (17, 19). How-ever, the skill and success of superparametrization algorithmsrelies on intermittency in space and time due to complex evolving,strongly chaotic, and quiescent regions without statistical equili-bration despite only modest values of scale separation, ε = 1

6 to 110

(8, 12, 13, 20).The purpose of the present article is to introduce a class of math-

ematical test models for superparametrization that are simpleenough to be analyzed with large confidence in a given physi-cal context, yet reveal essential mechanisms and features of bothsuperparametrization and HMM numerical algorithms for fur-ther improved development. This is done in detail in the nextsection. Subsequent sections of the article introduce the simplestscalar test model and analyze the skill of basic superparametriza-tion algorithms for the simplest scalar test problem as a revealingdemonstration. Here, the emphasis is on models with intermittentstrongly unstable fluctuations and only moderate scale separationwithout statistical equilibration so that traditional numerical clo-sure methods such as HMM cannot be applied. The article endswith a brief concluding discussion.

Author contributions: A.J.M. designed research; A.J.M. and M.J.G. performed research;and A.J.M. wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: jonjon@cims.nyu.edu.

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Test Models for SuperparametrizationAlgorithms for superparametrization are based on implicit sepa-ration of a physical system into its large-scale, slowly varying mean,u, and smaller-scale, more rapidly varying fluctuations, u′, akin toa Reynolds averaging formulation in turbulence (1–3, 5, 7, 12, 15).Here, this is made explicit by introducing two spatial scales, X andx ∈ R

N with X = εx, and two timescales t, τ , with τ = t/ε withε < 1 a scale separation parameter. See (refs. 12, 13, 20–23) fordiverse physical examples of such a multiscale formulation. Thephysical field, U ∈ R

M , then has the decomposition into its slowlyvarying mean u(X , t) and its fluctuations, u′(X , x, t, τ ),

U = u(X , t) + u′(X , x, t, τ ). [1]

In Eq. 1 and in the models below, u′ is always a zero meanGaussian random field that is stationary in x for fixed (X , t, τ )with Cov(u′)(X , t, τ ) its M × M covariance matrix (24). For afunction f (t, τ ),

< f > (t) = ε

∫ ε−1

0f (t, τ )dτ [2]

denotes the empirical time average over the fluctuations for a fixedvalue of ε. Next, we write down coupled equations for the large-scale mean, u, and fluctuations u′, which are akin to those thatoccur in diverse complex physical applications (5, 7, 12, 13, 20, 22,23). In the test models developed here, the large-scale mean, u, isgoverned by the prototype Large-Scale Dynamics

ut + P(

∂

∂X

)u = F

(< Cov(u′) >,

∂

∂X< Cov(u′) >

)

+ Fext(X , t). [3]

In Eq. 3, F, in general, is a nonlinear function of its argumentsthat represents fluctuations in turbulent scalars such as moisturethrough the argument, Cov(u′), and turbulent fluctuations fromadvection through the dependence on the gradient, ∂

∂X Cov(u′).The quantity Fext is prescribed external forcing. Here, in the testmodels, we choose P( ∂

∂X ) to be a fixed constant coefficient PDEof interest in the given physical context such as, for example,the hydrostatic primitive equations at a background rest state(12, 20, 23); clearly such test models develop nontrivial turbu-lent dynamics at the large scales only through the interaction withthe statistics of the fluctuations and this feature provides a cleantest model for this effect. The goal of superparametrization is todevelop cheap, seamless, “on-the-fly” numerical algorithms thatcapture the essential statistical impact of the fluctuations on thelarge scales (5, 7, 8, 12).

In the test models introduced here, the prototype dynamics forthe fluctuations u′ are given by the Small-Scale Dynamics

u′τ + P′

(u,

∂

∂x

)u′ = −σ (x)W (x, τ ) + �

(∂

∂x

)u′. [4]

An essential feature in Eq. 4 is that we choose P′(u, ∂∂x ) to be a con-

stant coefficient differential operator in the small-scale variableswith coefficients that depend explicitly on the large-scale variableswith varying stability and instability features without statisticalsmall-scale equilibration. This happens in practice (5, 7, 8, 12) andis an essential ingredient to allow for spatiotemporal intermittencyin the test models. In general, the equations for the fluctuationsinvolve strong nonlinear interactions. Here, in order to achieveanalytic tractability in the test models, these nonlinear effects arereplaced by additional turbulent damping, �u′, and spatially corre-lated white noise forcing, σ (x)W , in accordance with the simplestquasi-linear turbulence models (25, 26). Such closure approxima-tions have been utilized in the statistically stable regime with someskill in modeling large-scale atmospheric turbulence (25). Thisapproximation has the advantage that the small-scale equations

for the fluctuating spatial covariance can be solved explicitly as afunction of the large-scale fluctuations as demonstrated next.

The spatial Fourier transform for fixed large-scale values, X , t,is the equivalent linear stochastic equation,

∂

∂τu′

k + P′(u, ik)u′k = −γku′

k + σkWk [5]

with u′k the spatial Fourier spectral representation of the Gaussian

random field u′ (24), i.e.,

u′(X , x, t, τ ) =∫

RNeix·k u′

k(X , t, τ )dW (k) [6]

with dW (k) independent complex white noise. This spectralrepresentation yields the formula

Cov(u′) =∫

RNCkdk, Ck = Cov(u′

k). [7]

From Eq. 5, Ck(X , t, τ ) satisfies the linear equation with coeffi-cients depending on u,

∂

∂τCk +

(P′

k + (P′

k

)∗)2

Ck = −(γk + γ ∗

k

)2

Ck + σkσ�k , [8]

where P′k = P′(u, ik).

The equations in 3–8 summarize the essential features of the testmodels for superparametrization. Besides their analytic simplicity,these models have several attractive features. First, by choosing σkand γk to vary in a suitable fashion, any turbulent energy spectrumfor the small scales can be modeled such as a − 5

3 spectrum (26);in particular, such energy spectra can include significant energyfor |k| << 1 so there is very little practical separation of spatialscales. Secondly, the value of ε in Eq. 3 is explicit through theempirical time average in Eq. 2 so that algorithms with moderatetimescale separation, ε � 1

6 , 110 , or extreme timescale separation,

ε = 10−3, 10−4 << 1 can be analyzed as well as the limitingbehavior ε ↓ 0. Finally the nonlinear dependence of the non-linear covariance equation in 8 on the large-scale variables canbe designed so that Ck(τ , u) achieves a statistically stable equilib-rium value as τ → ∞ for some values of the large-scale variable,u, while Ck(τ , u) grows in amplitude without statistical stabilityfor other values of u. These are the crucial ingredients to createintermittency in the present test models without local statisticalequilibration and moderate scale separation as occurs in the appli-cations involving anisotropic geophysical turbulence. All three ofthese facets of the models will be illustrated in the simplest scalartest model in a subsequent section of this article.

Superparametrization Algorithms in the Test ModelThe main strategy in superparametrization algorithms is to retainthe large-scale equations in Eq. 3, but make various space–timediscrete approximations for the small-scale model in Eq. 4 thatinvolve highly reduced computational labor in solving Eq. 4; thegoal is to retain suitable statistical accuracy in calculating the tur-bulent covariance that affects the large-scale model in Eq. 3 insuitable turbulent regimes. The first approximation is to intro-duce an artificial scale gap through a length scale L and solve thesmall-scale problem in Eq. 4 in a periodic configuration (5, 7, 8).In the present context, the random stochastic integral in Eq. 6 isreplaced by a discrete sum of random variables over the latticewith wavenumber kj = j

L where j is a vector of integers so thatan approximate covariance CovL to Eq. 7 is constructed from thesum

CovL(u′) = L−N∑

j

Ckj , kj =jL

[9]

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with Ckj satisfying the same equations in 8 for this discrete setof spatial wavenumbers. In pragmatic terms for the test model,Eq. 9 is the trapezoidal approximation to Eq. 7 with h = L−1. Asecond more stringent approximation in superparametrization isto somehow exploit the intermittency and solve suitable reduced1-dimensional versions of the small-scale test problem in Eq. 4,which are L-periodic in space rather than the full N-dimensionalsmall-scale periodic problem (5, 7, 15). Finally, a third type of sta-tistical approximation that attempts to exploit the intermittencyand chaotic instability in the small scales is to systematically reducethe required time interval of integration of the small-scale model(8) compared with the time step in the large-scale model. Below,we only examine the effects of spatial periodic discretization in atest model for superparametrization because of the lack of spaceavailable here.

A Scalar Test Model for SuperparametrizationHere, we develop the test model in detail in the simplest contextfor a real-valued scalar field, u, in a single-space dimension. Thelarge-scale equations in 3 have the form,

∂u∂t

+ P(

∂

∂X

)u = < Cov(u′) > (X , t) + Fext [10]

where Fext is a constant forcing. Here, the scalar differential oper-ator P( ∂

∂X ) is chosen to have advection, dispersion, and dissipationas in typical anisotropic systems in applications with

P(

∂

∂X

)= A

∂3

∂X 3 − ν∂2

∂X 2 + c∂

∂X+ d. [11]

The dispersion and advection coefficients are given the valueA = 1, c = 1 while the viscosity and damping have the valuesν = 10−8, d = 10−2 in the simulations reported below. For thesmall-scale equations from 4 in their Fourier representation in 5,we choose uniform damping, γk ≡ γ = 1 and variance σ 2

k to yielda − 5

3 turbulent energy spectrum; in other words, σ 2k satisfies

σ 2k

2γk= E0

(|k| + 1)β, β = 5

3[12]

with the value E0 = 0.1 for the preconstant. Thus, if the interac-tion with the large-scale field, u, is ignored in the equations forthe small scales, the statistical equilibrium state is an energeticturbulent field without scale separation. A crucial design featurefor the model is to build in intermittency by changing the largetime behavior of the small-scale dynamic covariance equation in8 as the value of u varies so that there are regions for u withoutstatistical equilibration of the small-scale fluctuations. Here, wespecify the symmetric part of P′

k with the form

P′k + (

P′k

)∗

2= −f (u)Ak [13]

with

f (u) = u(

1 − u2

2

)[14]

and

Ak = Ae−δ|k||k|2. [15]

In Eq. 15 the value δ = 0.1 is utilized and A is a fixed constantchosen to guarantee maxk Ak = 1. Note from Eq. 8 that the effectof the large scales on the small scales is to amplify the small-scalevariance for f (u) > 0, i.e., for u < −√

2 or 0 < u <√

2, and todiminish the small-scale variance otherwise when f (u) < 0 whileAk in Eq. 15 governs this rate for various wavenumbers. This is oneingredient for intermittency because the small scales impact thelarge scales through this covariance in Eq. 10 and its magnitude is

influenced strongly by the value of u. In the present context, thecovariance equation in 8 has the explicit solution for fixed X , tgiven by

Ck(τ ) = σ 2k

2λk(1 − e−2λkτ ) + e−2λkτ Ck(0),

λk = γk − f (u)Ak. [16]

The empirical time average of this covariance from Eq. 2, whichis needed for Eq. 10 via Eq. 7 for the exact solution and via Eq. 9for superparametrization is given exactly by

1T

∫ T

0Ck(τ )dτ = 1

2λkT(1 − e−2λkT )

[Ck(0) − σ 2

k

2λk

]+ σ 2

k

2λk[17]

with T = ε−1. There are two different large time behaviorspossible for the covariance function, Ck(τ ), as τ → ∞; namely,

A. If λk = γk − f (u)Ak < 0, then no statistical equilibrium isachieved and

Ck(τ ) grows without bound as τ → ∞. [18]

B. If λk = γk − f (u)Ak > 0, then the statistical equilibrium

Ceqk (u) = σ 2

k

2(γk − f (u)Ak)[19]

is the limiting behavior as τ → ∞.

The new statistical phenomena studied here occur because of thelarge-scale impact in the region from Eq. 18, where traditional clo-sure methods fail (see Eq. 22 below); the region from Eq. 19 is thestandard situation where traditional equilibrium closure methodsapply (see Eqs. 20 and 21). The design features in Eqs. 13, 14, 15and 18 for a range of values for u are responsible for the smallscale intermittency that impacts the large scales in a nontrivialfashion as shown below. This is the regime of interest for super-parametrization. When the situation in Eq. 19 is satisfied for allvalues of u and the scale separation parameter, ε, is sufficientlysmall, the model is in a typical regime for application of HMMmethods (14, 19). In fact, one can write down a formal closure inthe limit ε → 0 for any value of u that satisifies Eq. 19 by utilizingthe equilibrium statistical value depending on u in Eq. 19 to cal-culate < Cov(u′) >eq (u) through Eqs. 7 and 17. The result is theEquilibrium Statistical Closure,

ut + P(

∂

∂X

)u = < Cov(u′) >eq (u) + Fext [20]

with < Cov(u′) >eq (u) =∫

σ 2k

2(γk − f (u)Ak)dk

=∫

Ceqk (u)dk. [21]

This closure is a useful benchmark for the behavior of multiscalenumerical methods like HMM as well as superparametrization; ofcourse, it is important for the phenomena discussed here that inthe regime with Eq. 18, such an equilibrium-limiting closure, can-not even be defined for these values of u. For the specific choiceof parameters in the test models listed above,

γk − f (u)Ak < 0 and 18 is satisfied for [22]some k if and only if u < −1.7.

For the constant states u with u > −1.7, the equilibrium statisticalclosure in Eq. 20 is well-defined and one can do a straightforwardlinear stability analysis (27) of the closure equation in Eq. 19 to

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understand the impact of the small scales on the large scales in thisregime. The result is that in this regime perturbations δu satisfy

∂

∂tδu + P

(∂

∂X

)δu = f ′(u)

[∫σ 2

k dk2(γk − f (u)Ak)2

]δu [23]

and there is potential growth for a band of large-scale unstable

wave numbers only for |u| <

√32 as predicted by the statisti-

cal closure solely due to interaction with the small scales. Thisis another important typical feature in the model dynamics. Evenin the regime from Eq. 19 with a limiting statistical closure, therecan be nontrivial pattern formation on the large scales (27) createdsolely by interaction with the small scales.

Intermittency and Superparametrization in the Test ModelHere, we simulate the scalar test model as well as superparame-trization with an imposed spatial scale gap defined by L in Eq. 9 fora set of values of L = 2, 1, 0.5 in the regime of the test model withsmall-scale intermittency as described in detail in Eqs. 10–19 of thelast section. We use the value ε = 0.1 with modest scale separationin the empirical time average from Eq. 2 needed in Eqs. 10 and17. The goal here is to explore the statistical accuracy of superpa-rametrization in the test model in this regime with small-scaleintermittency and modest scale separation mimicking realisticphysical systems. The numerical algorithm for 10 at every largetime step, �t, involves two discrete fast forward and backwardFourier transforms for u and < Cov(u′) > (X , t), respectively.In Fourier space the linear ordinary differential equation for the

kth Fourier mode of u is solved exactly under the assumption that< Cov(u′) > (X , t) is constant over the small time interval �t.There are no stability constraints in this large-scale time integratorand 129 discrete large-scale mesh points are utilized in the simula-tions reported below. The formulas in 16 and 17 are utilized for the

small-scale covariance at each wavenumber k with Ck(0) = σ2k

2γk,

the equilibrium spectrum without large-scale interaction. Adap-tive quadrature is utilized for the exact model to compute thecovariance integral in Eq. 7, whereas, for the superparametriza-tion algorithms as in Eq. 9, the trapezoidal method is utilized withspacing h = 0.5, 1, 2 corresponding to L = 2, 1, 0.5, respectively;the spectral integral is truncated in both cases to |k| ≤ 103. In theintermittent regime from Eq. 18 studied here, the covariance issensitive to small changes in u that can lead to numerical insta-bility in the quadrature; to avoid this problem, the covariance isalways truncated below a maximal value set to 10 here. Statisticsof the chaotic solution at large scales are always computed for thelast 4,000 large-scale time units starting from random initial dataon the large scales with a total integration time of 5,000 (see ref.16 for similar details). The results shown below utilize a value ofFext = −0.4 for the external forcing in Eq. 10.

In Fig. 1 Upper we display bubble diagrams of the emerginglarge-scale solutions for the test model and the superparame-trization approximation for the smallest-scale gap with L = 2or h = 0.5. Both large-scale solutions are turbulent and closelyresemble each other statistically by sight. This is confirmed by theexcellent agreement of the long time mean value for the largescales, u = −0.973 in the exact model and uL = −0.953 for

Fig. 1. Comparison of large-scale solutions. (Upper) Snapshots of the numerical solution for 1,000 < t < 3000. (Lower) The exact solution (plain) andthe superparmetrization (dashed) are shown at t = 3,156, 5,997 with Fext = −0.4 and L = 2, together with the covariances < Cov(u′) > (dash-dot) and< CovL(u′) > (dotted).

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superparametrization as well as the statistical standard deviationat large scales, σ = 0.647 for the true model and σL = 0.628for superparametrization with L = 2. Snapshots of the exactand superparametrization solutions are given at the representativetimes t = 3,156 and t = 4,997 in Fig. 1 Lower as well as the graphsof < Cov(u′) >, < CovL(u′) > at those times for both the exactand superparametrization. Note the large values of < Cov(u′) >that are produced only in the regions with u < −1.7 in both Upperand Lower in agreement with the effects of small-scale intermit-tency from Eqs. 18 and 22. In such a turbulent large-scale solutionone can expect only statistical agreement in an approximationrather than agreement at each time due to instabilities and Fig. 1Lower demonstrates this. Increasing the scale gap to L = 1 yieldsnoticable statistical errors in superparametrization with the super-parametrization statistical mean, uL = −0.935 compared with−0.973 and large-scale standard deviation σL = 0.568 for superpa-rametrization, which is 10% smaller than σ = 0.647 for the exactmodel; thus, there is very significant skill for superparametrizationwith L = 1. Increasing the scale gap in superparametrization withL = 0.5 or h = 2 yields an extremely different large-scale statisti-cal mean state uL = −1.47 for the superparametrization algorithmcompared with u = −0.973 in the exact model and furthermorethere is negligible variance in the superparametrization solution.According to Eq. 22, the value of uL = −1.47 sits in the regimewhere the closure in Eq. 20 can be utilized to provide guidelines;at the value uL = −1.47 with Fext = −0.4 the right-hand side ofEq. 20 vanishes identically and according to the linear stabilityanalysis in Eq. 23, uL is a stable fixed point of Eq. 20. The super-parametrization algorithm with the largest scale gap, L = 0.5,apparently produces small-scale fluctuations that are too weakto access the intermittent regime from Eq. 22 in the small-scaledynamics. Increasing the magnitude of the external forcing to

Fext = −0.5 creates larger values for the turbulent fluctuationsat large scales, while decreasing the magnitude to Fext = −0.3creates a chaotic large-scale pattern without intense small-scaleintermittency; these results are not developed in detail here butsuperparametrization with a moderate scale gap can capture allthese features.

Concluding DiscussionThe main result in this article is the systematic development ofsimple test models that are mathematically tractable yet capturekey features of anisotropic turbulence involving statistical small-scale intermittency with moderate-scale separation and significantimpact on the large-scale dynamics. The properties of the sim-plest scalar test model have been developed extensively here andutilized to test the statistical performance of superparametriza-tion algorithms with an imposed spectral gap in a system with anenergetic −5/3 turbulent spectrum at small scales, without sucha scale gap. Regimes of skill and failure of the superparametriza-tion algorithm as the scale gap increases have been demonstratedin the test model. Such types of test models developed here arepotentially useful in understanding and improving the statisticalalgorithmic performance of both superparametrization and HMMmethods in problems with only moderate scale separation. Inparticular, understanding the statistical skill of small-scale dimen-sional reduction in superparametrization (5, 7, 15) is obviously animportant future direction.

ACKNOWLEDGMENTS. This work was supported in part by National Sci-ence Foundation Grant MDS-0456713, Office of Naval Research GrantN0014-05-1-1064, and Defense Advanced Research Projects Agency GrantN00014-08-1-1080 (to A.J.M.). Housing for sabbatical visit to the CourantInstitute was supported by Defense Advanced Research Projects Agency GrantN00014-08-1-1080 (to M.J.G.).

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5474 www.pnas.org / cgi / doi / 10.1073 / pnas.0901383106 Majda and Grote

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