New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Mathematical test models for superparametrization in anisotropic turbulence

Contributed by Andrew J. Majda, February 6, 2009 (received for review January 8, 2009)
Abstract
The complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering turbulence and climate atmosphere ocean science requires novel computational strategies with the current and next generations of supercomputers. In these applications the smallerscale fluctuations do not statistically equilibrate as assumed in traditional closure modeling and intermittently send significant energy to the largescale fluctuations. Superparametrization is a novel class of seamless multiscale algorithms that reduce computational labor by imposing an artificial scale gap between the energetic smallerscale fluctuations and the largescale fluctuations. The main result here is the systematic development of simple test models that are mathematically tractable yet capture key features of anisotropic turbulence in applications involving statistically intermittent fluctuations without local statistical equilibration, with moderate scale separation and significant impact on the largescale dynamics. The properties of the simplest scalar test model are developed here and utilized to test the statistical performance of superparametrization algorithms with an imposed spectral gap in a system with an energetic −5/3 turbulent spectrum for the fluctuations.
The complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence and combustion (1–3) as well as climate atmosphere ocean science requires novel computational strategies even with the current and next generations of supercomputers. This is especially important since energy often flows intermittently from the smaller unresolved or marginally resolved scales to affect the largest observed scales in such anisotropic turbulent flows. For example, atmospheric processes of weather and climate cover ≈10 decades of spatial scales, from a fraction of a millimeter to planetary scales. Regarding atmospheric fluid dynamics, one is primarily concerned with spatial scales larger than tens of meters because the smaller scales fall whithin the inertial range of atmospheric turbulence. Spatial scales between 100 m and 100 km, referred to as small through mesoscale, show an abundance of processes associated with dry and moist convection, clouds, waves, boundary layer, topographic, and frontal circulations. A major stumbling block in the accurate prediction of weather and shortterm climate on the planetary and synoptic scales is the accurate parametrization of moist convection. These problems involve intermittency in space and time due to complex evolving, strongly chaotic, and quiescent regions without statistical equilibration and with only moderatescale separation so that traditional turbulence closure modeling fails (1–3). Cloudsystemresolving models realistically represent smallscale and mesoscale processes with fine computational grids. But because of the high computational cost, they cannot be applied to large ensemble size weather prediction or climate simulations within the near future.
A different modeling approach, the cloudresolving convection parametrization (CRCP) or superparametrization (SP) was recently developed (4–8). The idea is to use a 2D cloudsystemresolving model in each column of a largescale model to explicitly represent smallscale and mesoscale processes and interactions among them. It blends convectional parametrization on a coarse mesh with detailed cloudresolving modeling on a finer mesh. This approach has been shown to be ideal for parallel computations and it can easily be implemented on supercomputers. The method has yielded promising new results regarding tropical intraseasonal behavior (4–9) and has the potential for many other applications in climateatmosphereocean science (10, 11).
Systematic mathematical formulation and numerical analysis of such superparametrization algorithms has the potential to lead to algorithmic improvements as well as other applications in diverse scientific and engineering disciplines and this is the main topic of the present article. Recently (12) it has been shown in a precise fashion how superparametrization can be regarded as a multiscale numerical method. Such multiscale formulation (13) has been utilized recently to develop new highly efficient and 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 superparametrization algorithms and heterogenous multiscale methods (HMMs) developed recently in the applied mathematics literature (14). HMM algorithms are a mathematical synthesis of earlier work (see refs. 15 and 16, and references therein) as well as an abstract formulation that leads to new multiscale algorithms for complex systems with widely disparate time scales (14, 17–19). However, as noted recently (8), there are significant differences in the regimes of nonlinear dynamics being modeled by superparametrization algorithms as compared with HMM. Reduced HMM timesteppers have been analyzed and applied for various physical systems with wide scale separation, ɛ = 10^{−3},10^{−4}, with ɛ the scale separation ratio between large and small scales, and rapid local statistical equilibration in time (17, 19). However, the skill and success of superparametrization algorithms relies on intermittency in space and time due to complex evolving, strongly chaotic, and quiescent regions without statistical equilibration despite only modest values of scale separation,
The purpose of the present article is to introduce a class of mathematical test models for superparametrization that are simple enough to be analyzed with large confidence in a given physical context, yet reveal essential mechanisms and features of both superparametrization and HMM numerical algorithms for further improved development. This is done in detail in the next section. Subsequent sections of the article introduce the simplest scalar test model and analyze the skill of basic superparametrization algorithms for the simplest scalar test problem as a revealing demonstration. Here, the emphasis is on models with intermittent strongly unstable fluctuations and only moderate scale separation without statistical equilibration so that traditional numerical closure methods such as HMM cannot be applied. The article ends with a brief concluding discussion.
Test Models for Superparametrization
Algorithms for superparametrization are based on implicit separation of a physical system into its largescale, slowly varying mean, u, and smallerscale, more rapidly varying fluctuations, u′, akin to a Reynolds averaging formulation in turbulence (1–3, 5, 7, 12, 15). Here, this is made explicit by introducing two spatial scales, X and x ∈ ℝ^{N} with X = ɛx, and two timescales t,τ, with τ = t/ɛ with ɛ < 1 a scale separation parameter. See (refs. 12, 13, 20–23) for diverse physical examples of such a multiscale formulation. The physical field, U ∈ ℝ^{M}, then has the decomposition into its slowly varying mean u(X,t) and its fluctuations, u′(X, x, t, τ),
In Eq. 1 and in the models below, u′ is always a zero mean Gaussian 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 a function f(t, τ),
denotes the empirical time average over the fluctuations for a fixed value of ɛ. Next, we write down coupled equations for the largescale mean, u, and fluctuations u′, which are akin to those that occur in diverse complex physical applications (5, 7, 12, 13, 20, 22, 23). In the test models developed here, the largescale mean, u, is governed by the prototype LargeScale Dynamics
In Eq. 3, F, in general, is a nonlinear function of its arguments that represents fluctuations in turbulent scalars such as moisture through the argument, Cov(u′), and turbulent fluctuations from advection through the dependence on the gradient,
In the test models introduced here, the prototype dynamics for the fluctuations u′ are given by the SmallScale Dynamics
An essential feature in Eq. 4 is that we choose
The spatial Fourier transform for fixed largescale values, X,t, is the equivalent linear stochastic equation, with û′_{k} the spatial Fourier spectral representation of the Gaussian random field u′ (24), i.e., with dW(k) independent complex white noise. This spectral representation yields the formula From Eq. 5, C_{k}(X,t,τ) satisfies the linear equation with coefficients depending on u, where P′_{k} = P′(u,ik).
The equations in 3–8 summarize the essential features of the test models for superparametrization. Besides their analytic simplicity, these models have several attractive features. First, by choosing σk and γk to vary in a suitable fashion, any turbulent energy spectrum for the small scales can be modeled such as a
Superparametrization Algorithms in the Test Model
The main strategy in superparametrization algorithms is to retain the largescale equations in Eq. 3, but make various space–time discrete approximations for the smallscale model in Eq. 4 that involve highly reduced computational labor in solving Eq. 4; the goal is to retain suitable statistical accuracy in calculating the turbulent covariance that affects the largescale model in Eq. 3 in suitable turbulent regimes. The first approximation is to introduce an artificial scale gap through a length scale L and solve the smallscale problem in Eq. 4 in a periodic configuration (5, 7, 8). In the present context, the random stochastic integral in Eq. 6 is replaced by a discrete sum of random variables over the lattice with wavenumber
A Scalar Test Model for Superparametrization
Here, we develop the test model in detail in the simplest context for a realvalued scalar field, u, in a singlespace dimension. The largescale equations in 3 have the form,
where F_{ext} is a constant forcing. Here, the scalar differential operator

A. If λk = γk − f(u)A_{k} < 0, then no statistical equilibrium is achieved and

B. If λk = γk − f(u)A_{k} > 0, then the statistical equilibrium is the limiting behavior as τ → ∞.
The new statistical phenomena studied here occur because of the largescale impact in the region from Eq. 18, where traditional closure methods fail (see Eq. 22 below); the region from Eq. 19 is the standard situation where traditional equilibrium closure methods apply (see Eqs. 20 and 21). The design features in Eqs. 13, 14, 15 and 18 for a range of values for u are responsible for the small scale intermittency that impacts the large scales in a nontrivial fashion as shown below. This is the regime of interest for superparametrization. When the situation in Eq. 19 is satisfied for all values of u and the scale separation parameter, ɛ, is sufficiently small, the model is in a typical regime for application of HMM methods (14, 19). In fact, one can write down a formal closure in the limit ɛ → 0 for any value of u that satisifies Eq. 19 by utilizing the equilibrium statistical value depending on u in Eq. 19 to calculate < Cov(u′) >_{eq} (u) through Eqs. 7 and 17. The result is the Equilibrium Statistical Closure,
This closure is a useful benchmark for the behavior of multiscale numerical methods like HMM as well as superparametrization; of course, it is important for the phenomena discussed here that in the regime with Eq. 18, such an equilibriumlimiting closure, cannot even be defined for these values of u. For the specific choice of parameters in the test models listed above,
For the constant states u with u > −1.7, the equilibrium statistical closure in Eq. 20 is welldefined and one can do a straightforward linear stability analysis (27) of the closure equation in Eq. 19 to understand the impact of the small scales on the large scales in this regime. The result is that in this regime perturbations δu satisfy
and there is potential growth for a band of largescale unstable wave numbers only for
Intermittency and Superparametrization in the Test Model
Here, we simulate the scalar test model as well as superparametrization with an imposed spatial scale gap defined by L in Eq. 9 for a set of values of L = 2,1,0.5 in the regime of the test model with smallscale intermittency as described in detail in Eqs. 10–19 of the last section. We use the value ɛ = 0.1 with modest scale separation in the empirical time average from Eq. 2 needed in Eqs. 10 and 17. The goal here is to explore the statistical accuracy of superparametrization in the test model in this regime with smallscale intermittency and modest scale separation mimicking realistic physical systems. The numerical algorithm for 10 at every large time step, Δt, involves two discrete fast forward and backward Fourier 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 Δt. There are no stability constraints in this largescale time integrator and 129 discrete largescale mesh points are utilized in the simulations reported below. The formulas in 16 and 17 are utilized for the smallscale covariance at each wavenumber k with
In Fig. 1 Upper we display bubble diagrams of the emerging largescale solutions for the test model and the superparametrization approximation for the smallestscale gap with L = 2 or h = 0.5. Both largescale solutions are turbulent and closely resemble each other statistically by sight. This is confirmed by the excellent agreement of the long time mean value for the large scales,
Concluding Discussion
The main result in this article is the systematic development of simple test models that are mathematically tractable yet capture key features of anisotropic turbulence involving statistical smallscale intermittency with moderatescale separation and significant impact on the largescale dynamics. The properties of the simplest scalar test model have been developed extensively here and utilized to test the statistical performance of superparametrization algorithms with an imposed spectral gap in a system with an energetic −5/3 turbulent spectrum at small scales, without such a scale gap. Regimes of skill and failure of the superparametrization algorithm as the scale gap increases have been demonstrated in the test model. Such types of test models developed here are potentially useful in understanding and improving the statistical algorithmic performance of both superparametrization and HMM methods in problems with only moderate scale separation. In particular, understanding the statistical skill of smallscale dimensional reduction in superparametrization (5, 7, 15) is obviously an important future direction.
Acknowledgments
This work was supported in part by National Science Foundation Grant MDS0456713, Office of Naval Research Grant N00140511064, and Defense Advanced Research Projects Agency Grant N000140811080 (to A.J.M.). Housing for sabbatical visit to the Courant Institute was supported by Defense Advanced Research Projects Agency Grant N000140811080 (to M.J.G.).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: jonjon{at}cims.nyu.edu

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.
References
 ↵
 Hinze J
 ↵
 Monin AS,
 Yaglom AM
 ↵
 Townsend AA
 ↵
 ↵
 ↵
 ↵
 ↵
 Xing Y,
 Majda AJ,
 Grabowski WW
 ↵
 ↵
 Wyant MC,
 Khairoutdinov M,
 Bretherton CS
 ↵
 ↵
 ↵
 Majda AJ,
 Xing Y
 ↵
 Weinan E,
 Engquist B
 ↵
 ↵
 Majda AJ,
 Abramov R,
 Grote MJ
 ↵
 VandenEijnden E
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 Majda AJ,
 Biello JA
 ↵
 Yaglom AM
 ↵
 ↵
 Majda AJ,
 Grote MJ
 ↵
Citation Manager Formats
Sign up for Article Alerts
Jump to section
You May Also be Interested in
More Articles of This Classification
Physical Sciences
Related Content
 No related articles found.