Mathematical test models for superparametrization in anisotropic turbulence

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 smaller-scale fluctuations do not statistically equilibrate as assumed in traditional closure modeling and intermittently send significant energy to the large-scale fluctuations. Superparametrization is a novel class of seamless multi-scale algorithms that reduce computational labor by imposing an artificial scale gap between the energetic smaller-scale fluctuations and the large-scale 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 large-scale 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 short-term 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 moderate-scale separation so that traditional turbulence closure modeling fails (1–3). Cloud-system-resolving models realistically represent small-scale 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 cloud-resolving convection parametrization (CRCP) or superparametrization (SP) was recently developed (4–8). The idea is to use a 2D cloud-system-resolving model in each column of a large-scale model to explicitly represent small-scale and mesoscale processes and interactions among them. It blends convectional parametrization on a coarse mesh with detailed cloud-resolving 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 climate-atmosphere-ocean 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 time-steppers 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, $\varepsilon =\frac{1}{6}$ to $\frac{1}{10}$ (8, 12, 13, 20).

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 large-scale, slowly varying mean, u, and smaller-scale, 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 large-scale 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 large-scale mean, u, is governed by the prototype Large-Scale 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, $\frac{\partial }{\partial X}\text{Cov}(u\text{'})$. The quantity F_ext is prescribed external forcing. Here, in the test models, we choose $P\left(\frac{\partial }{\partial X}\right)$ to be a fixed constant coefficient PDE of 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 turbulent dynamics at the large scales only through the interaction with the statistics of the fluctuations and this feature provides a clean test model for this effect. The goal of superparametrization is to develop cheap, seamless, “on-the-fly” numerical algorithms that capture the essential statistical impact of the fluctuations on the large scales (5, 7, 8, 12).

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

An essential feature in Eq. 4 is that we choose $P\text{'}\left(u,\frac{\partial }{\partial x}\right)$ to be a constant coefficient differential operator in the small-scale variables with coefficients that depend explicitly on the large-scale variables with varying stability and instability features without statistical small-scale equilibration. This happens in practice (5, 7, 8, 12) and is an essential ingredient to allow for spatiotemporal intermittency in the test models. In general, the equations for the fluctuations involve strong nonlinear interactions. Here, in order to achieve analytic tractability in the test models, these nonlinear effects are replaced by additional turbulent damping, Γu′, and spatially correlated white noise forcing, $\sigma (x)\stackrel{.}{W}$, in accordance with the simplest quasi-linear turbulence models (25, 26). Such closure approximations have been utilized in the statistically stable regime with some skill in modeling large-scale atmospheric turbulence (25). This approximation has the advantage that the small-scale equations for the fluctuating spatial covariance can be solved explicitly as a function 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,

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 $-\frac{5}{3}$ spectrum (26); in particular, such energy spectra can include significant energy for |k| << 1 so there is very little practical separation of spatial scales. Secondly, the value of ɛ in Eq. 3 is explicit through the empirical time average in Eq. 2 so that algorithms with moderate timescale separation, ɛ ≃ $\frac{1}{6},\frac{1}{10}$, or extreme timescale separation, ɛ = 10⁻³,10⁻⁴ << 1 can be analyzed as well as the limiting behavior ɛ↓0. Finally the nonlinear dependence of the nonlinear covariance equation in 8 on the large-scale variables can be designed so that $C_k(\tau ,\bar{u})$ achieves a statistically stable equilibrium value as τ → ∞ for some values of the large-scale variable, u, while $C_k(\tau ,\bar{u})$ grows in amplitude without statistical stability for other values of u. These are the crucial ingredients to create intermittency in the present test models without local statistical equilibration and moderate scale separation as occurs in the applications involving anisotropic geophysical turbulence. All three of these facets of the models will be illustrated in the simplest scalar test model in a subsequent section of this article.

Superparametrization Algorithms in the Test Model

The main strategy in superparametrization algorithms is to retain the large-scale equations in Eq. 3, but make various space–time discrete approximations for the small-scale 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 large-scale 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 small-scale 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 $k_j=\frac{\overrightarrow{j}}{L}$ where $\overrightarrow{j}$ is a vector of integers so that an approximate covariance Cov_L to Eq. 7 is constructed from the sum

with C_kj satisfying the same equations in 8 for this discrete set of spatial wavenumbers. In pragmatic terms for the test model, Eq. 9 is the trapezoidal approximation to Eq. 7 with h = L⁻¹. A second more stringent approximation in superparametrization is to somehow exploit the intermittency and solve suitable reduced 1-dimensional versions of the small-scale test problem in Eq. 4, which are L -periodic in space rather than the full N -dimensional small-scale periodic problem (5, 7, 15). Finally, a third type of statistical approximation that attempts to exploit the intermittency and chaotic instability in the small scales is to systematically reduce the 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 a test model for superparametrization because of the lack of space available here.

A Scalar Test Model for Superparametrization

Here, we develop the test model in detail in the simplest context for a real-valued scalar field, u, in a single-space dimension. The large-scale equations in 3 have the form,

where F_ext is a constant forcing. Here, the scalar differential operator $P\left(\frac{\partial }{\partial X}\right)$ is chosen to have advection, dispersion, and dissipation as in typical anisotropic systems in applications with

The dispersion and advection coefficients are given the value A = 1, c = 1 while the viscosity and damping have the values ν = 10⁻⁸, d = 10⁻² in the simulations reported below. For the small-scale equations from 4 in their Fourier representation in 5, we choose uniform damping, γk ≡γ = 1 and variance σ_k² to yield a $-\frac{5}{3}$ turbulent energy spectrum; in other words, σ_k² satisfies

with the value E₀ = 0.1 for the preconstant. Thus, if the interaction with the large-scale field, u, is ignored in the equations for the small scales, the statistical equilibrium state is an energetic turbulent field without scale separation. A crucial design feature for the model is to build in intermittency by changing the large time behavior of the small-scale dynamic covariance equation in 8 as the value of u varies so that there are regions for u without statistical equilibration of the small-scale fluctuations. Here, we specify the symmetric part of P′_k with the form

with

and

In Eq. 15 the value δ = 0.1 is utilized and $\bar{A}$ is a fixed constant chosen to guarantee max_kA_k = 1. Note from Eq. 8 that the effect of the large scales on the small scales is to amplify the small-scale variance for f(u) > 0, i.e., for $u\&lt;-\sqrt{2}$ or $0\&lt;u\&lt;\sqrt{2}$, and to diminish the small-scale variance otherwise when f(u) < 0 while A_k in Eq. 15 governs this rate for various wavenumbers. This is one ingredient for intermittency because the small scales impact the large scales through this covariance in Eq. 10 and its magnitude is influenced strongly by the value of u. In the present context, the covariance equation in 8 has the explicit solution for fixed X, t given by

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

with T = ɛ⁻¹. There are two different large time behaviors possible for the covariance function, C_k(τ), as τ → ∞; namely,

• 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 large-scale 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 equilibrium-limiting 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 well-defined 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 large-scale unstable wave numbers only for $\left|u\right|\&lt;\sqrt{\frac{3}{2}}$ as predicted by the statistical closure solely due to interaction with the small scales. This is another important typical feature in the model dynamics. Even in the regime from Eq. 19 with a limiting statistical closure, there can be nontrivial pattern formation on the large scales (27) created solely by interaction with the small scales.

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 small-scale 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 small-scale 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 large-scale time integrator and 129 discrete large-scale mesh points are utilized in the simulations reported below. The formulas in 16 and 17 are utilized for the small-scale covariance at each wavenumber k with $C_k(0)=\frac{{\sigma }_k^2}{2{\gamma }_k}$, the equilibrium spectrum without large-scale interaction. Adaptive quadrature is utilized for the exact model to compute the covariance integral in Eq. 7, whereas, for the superparametrization algorithms as in Eq. 9, the trapezoidal method is utilized with spacing 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 the intermittent regime from Eq. 19 studied here, the covariance is sensitive to small changes in u that can lead to numerical instability in the quadrature; to avoid this problem, the covariance is always truncated below a maximal value set to 10 here. Statistics of the chaotic solution at large scales are always computed for the last 4,000 large-scale time units starting from random initial data on the large scales with a total integration time of 5,000 (see ref. 16 for similar details). The results shown below utilize a value of F_ext = −0.4 for the external forcing in Eq. 10.

In Fig. 1 Upper we display bubble diagrams of the emerging large-scale solutions for the test model and the superparametrization approximation for the smallest-scale gap with L = 2 or h = 0.5. Both large-scale 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, $\bar{u}$ = −0.973 in the exact model and $\bar{u}$_L = −0.953 for superparametrization as well as the statistical standard deviation at large scales, $\bar{\sigma }$ = 0.647 for the true model and $\bar{\sigma }$_L = 0.628 for superparametrization with L = 2. Snapshots of the exact and superparametrization solutions are given at the representative times t = 3,156 and t = 4,997 in Fig. 1 Lower as well as the graphs of < Cov(u′) >, < CovL(u′) > at those times for both the exact and superparametrization. Note the large values of < Cov(u′) > that are produced only in the regions with u < −1.7 in both Upper and Lower in agreement with the effects of small-scale intermittency from Eqs. 18 and 22. In such a turbulent large-scale solution one can expect only statistical agreement in an approximation rather than agreement at each time due to instabilities and Fig. 1 Lower demonstrates this. Increasing the scale gap to L = 1 yields noticable statistical errors in superparametrization with the superparametrization statistical mean, $\bar{u}$_L = −0.935 compared with −0.973 and large-scale standard deviation $\bar{\sigma }$_L = 0.568 for superparametrization, which is 10% smaller than $\bar{\sigma }$ = 0.647 for the exact model; thus, there is very significant skill for superparametrization with L = 1. Increasing the scale gap in superparametrization with L = 0.5 or h = 2 yields an extremely different large-scale statistical mean state $\bar{u}$_L = −1.47 for the superparametrization algorithm compared with $\bar{u}$ = −0.973 in the exact model and furthermore there is negligible variance in the superparametrization solution. According to Eq. 22, the value of $\bar{u}$_L = −1.47 sits in the regime where the closure in Eq. 20 can be utilized to provide guidelines; at the value $\bar{u}$_L = −1.47 with F_ext = −0.4 the right-hand side of Eq. 20 vanishes identically and according to the linear stability analysis in Eq. 23, $\bar{u}$_L is a stable fixed point of Eq. 20. The superparametrization algorithm with the largest scale gap, L = 0.5, apparently produces small-scale fluctuations that are too weak to access the intermittent regime from Eq. 22 in the small-scale dynamics. Increasing the magnitude of the external forcing to F_ext = −0.5 creates larger values for the turbulent fluctuations at large scales, while decreasing the magnitude to F_ext = −0.3 creates a chaotic large-scale pattern without intense small-scale intermittency; these results are not developed in detail here but superparametrization with a moderate scale gap can capture all these features.

Fig. 1.

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

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 small-scale intermittency with moderate-scale separation and significant impact on the large-scale 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 small-scale dimensional reduction in superparametrization (5, 7, 15) is obviously an important future direction.