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A major challenge in contemporary data science is the development of statistically accurate particle filters to capture non-Gaussian features in large-dimensional chaotic dynamical systems. Blended particle filters that capture non-Gaussian features in an adaptively evolving low-dimensional subspace through particles interacting with evolving Gaussian statistics on the remaining portion of phase space are introduced here. These blended particle filters are constructed in this paper through a mathematical formalism involving conditional Gaussian mixtures combined with statistically nonlinear forecast models compatible with this structure developed recently with high skill for uncertainty quantification. Stringent test cases for filtering involving the 40-dimensional Lorenz 96 model with a 5-dimensional adaptive subspace for nonlinear blended filtering in various turbulent regimes with at least nine positive Lyapunov exponents are used here. These cases demonstrate the high skill of the blended particle filter algorithms in capturing both highly non-Gaussian dynamical features as well as crucial nonlinear statistics for accurate filtering in extreme filtering regimes with sparse infrequent high-quality observations. The formalism developed here is also useful for multiscale filtering of turbulent systems and a simple application is sketched below.

A framework for low-order predictive statistical modeling and uncertainty quantification in turbulent dynamical systems is developed here. These reduced-order, modified quasilinear Gaussian (ROMQG) algorithms apply to turbulent dynamical systems in which there is significant linear instability or linear nonnormal dynamics in the unperturbed system and energy-conserving nonlinear interactions that transfer energy from the unstable modes to the stable modes where dissipation occurs, resulting in a statistical steady state; such turbulent dynamical systems are ubiquitous in geophysical and engineering turbulence. The ROMQG method involves constructing a low-order, nonlinear, dynamical system for the mean and covariance statistics in the reduced subspace that has the unperturbed statistics as a stable fixed point and optimally incorporates the indirect effect of non-Gaussian third-order statistics for the unperturbed system in a systematic calibration stage. This calibration procedure is achieved through information involving only the mean and covariance statistics for the unperturbed equilibrium. The performance of the ROMQG algorithm is assessed on two stringent test cases: the 40-mode Lorenz 96 model mimicking midlatitude atmospheric turbulence and two-layer baroclinic models for high-latitude ocean turbulence with over 125,000 degrees of freedom. In the Lorenz 96 model, the ROMQG algorithm with just a single mode captures the transient response to random or deterministic forcing. For the baroclinic ocean turbulence models, the inexpensive ROMQG algorithm with 252 modes, less than 0.2% of the total, captures the nonlinear response of the energy, the heat flux, and even the one-dimensional energy and heat flux spectra., Competing Interests: The authors declare no conflict of interest.

Understanding and improving the predictive skill of imperfect models for complex systems in their response to external forcing is a crucial issue in diverse applications such as for example climate change science. Equilibrium statistical fidelity of the imperfect model on suitable coarse-grained variables is a necessary but not sufficient condition for this predictive skill, and elementary examples are given here demonstrating this. Here, with equilibrium statistical fidelity of the imperfect model, a direct link is developed between the predictive fidelity of specific test problems in the training phase where the perfect natural system is observed and the predictive skill for the forced response of the imperfect model by combining appropriate concepts from information theory with other concepts based on the fluctuation dissipation theorem. Here a suite of mathematically tractable models with nontrivial eddy diffusivity, variance, and intermittent non-Gaussian statistics mimicking crucial features of atmospheric tracers together with stochastically forced standard eddy diffusivity approximation with model error are utilized to illustrate this link.

Abstract: In this article we expand and develop the authors' recent proposed methodology for efficient stochastic superparameterization algorithms for geophysical turbulence. Geophysical turbulence is characterized by significant intermittent cascades of energy from the unresolved to the resolved scales resulting in complex patterns of waves, jets, and vortices. Conventional superparameterization simulates large scale dynamics on a coarse grid in a physical domain, and couples these dynamics to high-resolution simulations on periodic domains embedded in the coarse grid. Stochastic superparameterization replaces the nonlinear, deterministic eddy equations on periodic embedded domains by quasilinear stochastic approximations on formally infinite embedded domains. The result is a seamless algorithm which never uses a small scale grid and is far cheaper than conventional SP, but with significant success in difficult test problems. Various design choices in the algorithm are investigated in detail here, including decoupling the timescale of evolution on the embedded domains from the length of the time step used on the coarse grid, and sensitivity to certain assumed properties of the eddies (e.g. the shape of the assumed eddy energy spectrum). We present four closures based on stochastic superparameterization which elucidate the properties of the underlying framework: a ‘null hypothesis’ stochastic closure that uncouples the eddies from the mean, a stochastic closure with nonlinearly coupled eddies and mean, a nonlinear deterministic closure, and a stochastic closure based on energy conservation. The different algorithms are compared and contrasted on a stringent test suite for quasigeostrophic turbulence involving two-layer dynamics on a β-plane forced by an imposed background shear. The success of the algorithms developed here suggests that they may be fruitfully applied to more realistic situations. They are expected to be particularly useful in providing accurate and efficient stochastic parameterizations for use in ensemble-based state estimation and prediction. [Copyright &y& Elsevier]

Abstract: This is a research expository paper regarding superparameterization, a class of multi-scale numerical methods designed to cope with the intermittent multi-scale effects of inhomogeneous geophysical turbulence where energy often inverse-cascades from the unresolved scales to the large scales through the effects of waves, jets, vortices, and latent heat release from moist processes. Original as well as sparse space–time superparameterization algorithms are discussed for the important case of moist atmospheric convection including the role of multi-scale asymptotic methods in providing self-consistent constraints on superparameterization algorithms and related deterministic and stochastic multi-cloud parameterizations. Test models for the statistical numerical analysis of superparameterization algorithms are discussed both to elucidate the performance of the basic algorithms and to test their potential role in efficient multi-scale data assimilation. The very recent development of grid-free seamless stochastic superparameterization methods for geophysical turbulence appropriate for “eddy-permitting” mesoscale ocean turbulence is presented here including a general formulation and illustrative applications to two-layer quasigeostrophic turbulence, and another difficult test case involving one-dimensional models of dispersive wave turbulence. This last test case has randomly generated solitons as coherent structures which collapse and radiate wave energy back to the larger scales, resulting in strong direct and inverse turbulent energy cascades. [Copyright &y& Elsevier]

Understanding the complexity of anisotropic turbulent processes in engineering and environmental fluid flows is a formidable challenge with practical significance because energy often flows intermittently from the smaller scales to impact the largest scales in these flows. Conceptual dynamical models for anisotropic turbulence are introduced and developed here which, despite their simplicity, capture key features of vastly more complicated turbulent systems. These conceptual models involve a large-scale mean flow and turbulent fluctuations on a variety of spatial scales with energy-conserving wave-mean-flow interactions as well as stochastic forcing of the fluctuations. Numerical experiments with a six-dimensional conceptual dynamical model confirm that these models capture key statistical features of vastly more complex anisotropic turbulent systems in a qualitative fashion. These features include chaotic statistical behavior of the mean flow with a sub-Gaussian probability distribution function (pdf) for its fluctuations whereas the turbulent fluctuations have decreasing energy and correlation times at smaller scales, with nearly Gaussian pdfs for the large-scale fluctuations and fat-tailed non-Gaussian pdfs for the smaller-scale fluctuations. This last feature is a manifestation of intermittency of the small-scale fluctuations where turbulent modes with small variance have relatively frequent extreme events which directly impact the mean flow. The dynamical models introduced here potentially provide a useful test bed for algorithms for prediction, uncertainty quantification, and data assimilation for anisotropic turbulent systems. [ABSTRACT FROM AUTHOR]

Abstract: Order-reduction schemes have been used successfully for the analysis and simplification of high-dimensional systems exhibiting low-dimensional dynamics. In this work, we first focus on presenting generic limitations of order-reduction techniques in systems with stable mean state that exhibit irreducible high-dimensional features such as non-normal dynamics, wide energy spectra, or strong energy cascades between modes. The reduced-order framework that we consider to illustrate these limitations is the dynamically orthogonal (DO) field equations. This framework is applied to a series of examples with stable mean state, including a linear non-normal system, and a nonlinear triad system in various dynamical configurations. After illustrating the weaknesses and generic limitations of order reduction, we develop a novel, two-way coupled, blended approach based on the quasilinear Gaussian (QG) closure and the DO field equations. The new method (QG–DO) overcomes the limitations of its two ingredients and achieves exceptional performance in the examples described previously as well as in other configurations with strongly transient character without using any tuned or adjustable parameters. [Copyright &y& Elsevier]

Efficient computation of geophysical turbulence, such as occurs in the atmosphere and ocean, is a formidable challenge for the following reasons: the complex combination of waves, jets, and vortices; significant energetic backscatter from unresolved small scales to resolved large scales; a lack of dynamical scale separation between large and small scales; and small-scale instabilities, conditional on the large scales, which do not saturate. Nevertheless, efficient methods are needed to allow large ensemble simulations of sufficient size to provide meaningful quantifications of uncertainty in future predictions and past reanalyses through data assimilation and filtering. Here, a class of efficient stochastic superparameterization algorithms is introduced. In contrast to conventional superparameterization, the method here (i) does not require the simulation of nonlinear eddy dynamics on periodic embedded domains, (ii) includes a better representation of unresolved small-scale instabilities, and (iii) allows efficient representation of a much wider range of unresolved scales. The simplest algorithm implemented here radically improves efficiency by representing small-scale eddies at and below the limit of computational resolution by a suitable one-dimensional stochastic model of random-direction plane waves. In contrast to heterogeneous multiscale methods, the methods developed here do not require strong scale separation or conditional equilibration of local statistics. The simplest algorithm introduced here shows excellent performance on a difficult test suite of prototype problems for geophysical turbulence with waves, jets, and vortices, with a speedup of several orders of magnitude compared with direct simulation. [ABSTRACT FROM AUTHOR]

Superparameterization is a fast numerical algorithm to mitigate implicit scale separation of dynamical systems with large-scale, slowly varying "mean" and smaller-scale, rapidly fluctuating "eddy" term. The main idea of superparameterization is to embed parallel highly resolved simulations of small-scale eddies on each grid cell of coarsely resolved large-scale dynamics. In this paper, we study the effect of model errors in using superparameterization for filtering multiscale turbulent dynamical systems. In particular, we use a simple test model, designed to mimic typical multiscale turbulent dynamics with small-scale intermittencies without local statistical equilibriation conditional to the large-scale mean dynamics, and simultaneously force the large-scale dynamics through eddy flux terms. In this paper, we consider the Fourier domain Kalman filter for filtering regularly spaced sparse observations of the large-scale mean variables. We find high filtering and statistical prediction skill with superparameterization (identical to the skill with perfect model), beyond conventional approaches such as the "bare truncation model" that ignores completely the eddy fluxes and the "equilibrium closure" model that crudely approximates the eddy fluxes with classical averaging theory. We show that this high filtering skill is robust even for very sparse observation networks and turbulent signals with a very steep, -6, spectrum. This is a counterexample to naive thinking that the small-scale processes are not so important in multiscale turbulent dynamics with steep energy spectrum. We find that the high filtering skill with superparameterization is robust for small enough scale gap, provided that the filter prior model satisfies the classical linear controllability condition. We will demonstrate a spectacular failure of filtering deterministically forced true signals with the exactly perfect model that does not satisfy controllability condition and a dramatic improvement when the controllability condition is restored with additional stochastic forcings. This result reconfirms and justifies the counterintuitive viewpoint that judicious model errors (or noises) can help filtering turbulent signals. [ABSTRACT FROM AUTHOR]

An information-theoretic framework is developed to assess the long-range coarse-grained predictive skill in a perfect-model environment. Central to the scheme is the notion that long-range forecasting involves regimes; specifically, that the appropriate initial data for ensemble prediction is the affiliation of the system to a coarse-grained partition of phase space representing regimes. The corresponding ensemble prediction probabilities, which are computable using ergodic signals from the model, are then used to quantify through relative entropy the information beyond climatology in the partition. As an application, the authors study the predictability of circulation regimes in an equivalent barotropic double-gyre ocean model using a partition algorithm based on K-means clustering and running-average coarse graining. Besides the established rolled up and extensional phases of the eastward jet, optimal partitions for triennial-scale forecasts feature a jet configuration dominated by the second empirical orthogonal function (EOF) of the streamfunction, as well as phases in which the jet interacts with eddies in higher EOFs. Due to mixing dynamics, the skill beyond three-state models is lost for forecast lead times longer than three years, but significant skill remains in the energy and the leading principal component of the streamfunction for septennial forecasts. [ABSTRACT FROM AUTHOR]

Quantifying the uncertainty for the present climate and the predictions of climate change in the suite of imperfect Atmosphere Ocean Science (AOS) computer models is a central issue in climate change science. Here, a systematic approach to these issues with firm mathematical underpinning is developed through empirical information theory. An information metric to quantify AOS model errors in the climate is proposed here which incorporates both coarsegrained mean model errors as well as covariance ratios in a transformation invariant fashion. The subtle behavior of model errors with this information metric is quantified in an instructive statistically exactly solvable test model with direct relevance to climate change science including the prototype behavior of tracer gases such as CO2. Formulas for identifying the most sensitive climate change directions using statistics of the present climate or an AOS model approximation are developed here; these formulas just involve finding the eigenvector associated with the largest eigenvalue of a quadratic form computed through suitable unperturbed climate statistics. These climate change concepts are illustrated on a statistically exactly solvable one-dimensional stochastic model with relevance for low frequency variability of the atmosphere. Viable algorithms for implementation of these concepts are discussed throughout the paper. [ABSTRACT FROM AUTHOR]

Filtering sparsely turbulent signals from nature is a central problem of contemporary data assimilation. Here, sparsely observed turbulent signals from nature are generated by solutions of two-layer quasigeostrophic models with turbulent cascades from baroclinic instability in two separate regimes with varying Rossby radius mimicking the atmosphere and the ocean. In the “atmospheric” case, large-scale turbulent fluctuations are dominated by barotropic zonal jets with non-Gaussian statistics while the “oceanic” case has large-scale blocking regime transitions with barotropic zonal jets and large-scale Rossby waves. Recently introduced, cheap radical linear stochastic filtering algorithms utilizing mean stochastic models (MSM1, MSM2) that have judicious model errors are developed here as well as a very recent cheap stochastic parameterization extended Kalman filter (SPEKF), which includes stochastic parameterization of additive and multiplicative bias corrections “on the fly.” These cheap stochastic reduced filters as well as a local least squares ensemble adjustment Kalman filter (LLS-EAKF) are compared on the test bed with 36 sparse regularly spaced observations for their skill in recovering turbulent spectra, spatial pattern correlations, and RMS errors. Of these four algorithms, the cheap SPEKF algorithm has the superior overall skill on the stringent test bed, comparable to LLS-EAKF in the atmospheric regime with and without model error and far superior to LLS-EAKF in the ocean regime. LLS-EAKF has special difficulty and high computational cost in the ocean regime with small Rossby radius, which creates stiffness in the perfect dynamics. The even cheaper mean stochastic model, MSM1, has high skill, which is comparable to SPEKF for the oceanic case while MSM2 has significantly worse filtering performance than MSM1 with the same inexpensive computational cost. This is interesting because MSM1 is based on a simple new regression strategy while MSM2 relies on the conventional regression strategy used in stochastic models for shear turbulence. [ABSTRACT FROM AUTHOR]

Climate change science focuses on predicting the coarse-grained, planetary-scale, longtime changes in the climate system due to either changes in external forcing or internal variability, such as the impact of increased carbon dioxide. The predictions of climate change science are carried out through comprehensive, computational atmospheric, and oceanic simulation models, which necessarily parameterize physical features such as clouds, sea ice cover, etc. Recently, it has been suggested that there is irreducible imprecision in such climate models that manifests itself as structural instability in climate statistics and which can significantly hamper the skill of computer models for climate change. A systematic approach to deal with this irreducible imprecision is advocated through algorithms based on the Fluctuation Dissipation Theorem (FDT). There are important practical and computational advantages for climate change science when a skillful FDT algorithm is established. The FDT response operator can be utilized directly for multiple climate change scenarios, multiple changes in forcing, and other para- meters, such as damping and inverse modelling directly without the need of running the complex climate model in each individual case. The high skill of FDT in predicting climate change, despite structural instability, is developed in an unambiguous fashion using mathematical theory as guidelines in three different test models: a generic class of analytical models mimicking the dynamical core of the computer climate models, reduced stochastic models for low-frequency variability, and models with a significant new type of irreducible imprecision involving many fast unstable modes. [ABSTRACT FROM AUTHOR]

Superparameterization (SP) is a large-scale modeling system with explicit representation of small-scale and mesoscale processes provided by a cloud-resolving model (CRM) embedded in each column of a large-scale model. New efficient sparse space–time algorithms based on the original idea of SP are presented. The large-scale dynamics are unchanged, but the small-scale model is solved in a reduced spatially periodic domain to save the computation cost following a similar idea applied by one of the authors for aquaplanet simulations. In addition, the time interval of integration of the small-scale model is reduced systematically for the same purpose, which results in a different coupling mechanism between the small- and large-scale models. The new algorithms have been applied to a stringent two-dimensional test suite involving moist convection interacting with shear with regimes ranging from strong free and forced squall lines to dying scattered convection as the shear strength varies. The numerical results are compared with the CRM and original SP. It is shown here that for all of the regimes of propagation and dying scattered convection, the large-scale variables such as horizontal velocity and specific humidity are captured in a statistically accurate way (pattern correlations above 0.75) based on space–time reduction of the small-scale models by a factor of 1/3; thus, the new efficient algorithms for SP result in a gain of roughly a factor of 10 in efficiency while retaining a statistical accuracy on the large-scale variables. Even the models with 1/6 reduction in space–time with a gain of 36 in efficiency are able to distinguish between propagating squall lines and dying scattered convection with a pattern correlation above 0.6 for horizontal velocity and specific humidity. These encouraging results suggest the possibility of using these efficient new algorithms for limited-area mesoscale ensemble forecasting. [ABSTRACT FROM AUTHOR]

The low-frequency response to changes in external forcing for the climate system is a fundamental issue. In two recent papers the authors developed a new blended response algorithm for predicting the response of a nonlinear chaotic forced-dissipative system to small changes in external forcing. This new algorithm is based on the fluctuation–dissipation theorem and combines the geometrically exact general response formula using integration of a linear tangent model at short response times and the classical quasi-Gaussian response algorithm at longer response times. This algorithm overcomes the inherent numerical instability in the geometric formula arising because of positive Lyapunov exponents at longer times while removing potentially large systematic errors from the classical quasi-Gaussian approximation at moderate times. Here the new blended method is tested on the low-frequency response for a T21 barotropic truncation on the sphere with realistic topography in two dynamical regimes corresponding to the mean climate behavior at 300- and 500-hPa geopotential height. The 300-hPa regime has robust strongly mixing behavior with a nearly Gaussian equilibrium state distribution, whereas the 500-hPa regime is weakly chaotic with slowly decaying time autocorrelation functions and strongly non-Gaussian climatology. It is found that the blended response algorithm has significant skill beyond the classical quasi-Gaussian algorithm for the response of the climate mean state and its variance. Additionally, the predicted response of the T21 barotropic model in the lowfrequency regime for these functionals does not seem to be affected by the model's structural instability. Thus, the results here suggest the use of the fluctuation–dissipation theorem–based blended response algorithm for more complex nonlinear geophysical models. [ABSTRACT FROM AUTHOR]

Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and physical instabilities on both large and small scales. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Here, explicit off-line test criteria for stable accurate discrete filtering are developed for use in the above context and mimic the classical stability analysis for finite difference schemes. First, constant coefficient partial differential equations, which are randomly forced and damped to mimic mesh scale energy spectra in the above problems are developed as off-line filtering test problems. Then mathematical analysis is used to show that under natural suitable hypothesis the time filtering algorithms for general finite difference discrete approximations to an s × s partial differential equation system with suitable observations decompose into much simpler independent s-dimensional filtering problems for each spatial wave number separately; in other test problems, such block diagonal models rigorously provide upper and lower bounds on the filtering algorithm. In this fashion, elementary off-line filtering criteria can be developed for complex spatially extended systems. The theory is illustrated for time filters by using both unstable and implicit difference scheme approximations to the stochastically forced heat equation where the combined effects of filter stability and model error are analyzed through the simpler off-line criteria. [ABSTRACT FROM AUTHOR]

Develops a numerical algorithm designed for efficient coupling of the equatorially trapped planetary waves with two-dimensional cloud-resolving models.Reports on news and developments related to in Chicago, Illinois as of January 22, 2001. Impact of convection in the tropics on short-term climate; Parabolic cylinder functions, Hermite polynomials and Gauss-Hermite quadrature.

Explains the development of crude closure algorithms based on equilibrium large scale statistical theories involving only a few constraints. Testing of crude closure algorithms through numerical experiments with the Navier-Stokes equations in two dimensions on a rectangular domain with stress-free boundary conditions.

A nonlinear optimal smoother and an associated optimal strategy of sampling hidden model trajectories are developed for a rich class of complex nonlinear turbulent dynamical systems with partial and noisy observations. Despite the strong nonlinearity and the significant non-Gaussian characteristics in the underlying systems, both the optimal smoother estimates and the sampled trajectories can be solved via closed analytic formulae. Thus, they are computationally efficient and the methods are applicable to high-dimensional systems. The nonlinear optimal smoother is able to estimate the hidden model states associated with various non-Gaussian phenomena and is particularly skillful in capturing the onset, demise and amplitude of the observed and hidden extreme events. On the other hand, the sampled hidden trajectories succeed in recovering both the dynamical and statistical features of the underlying nonlinear systems, including the fat-tailed non-Gaussian probability density function and the temporal autocorrelation function. In the situations with only a short period of partially observed training time series, the optimal sampling strategy can be used to efficiently create a sufficient number of samples in an unbiased fashion that facilitates an accurate prediction of important non-Gaussian features in both the observed and hidden variables. In addition, the information provided by the sampled trajectories based on imperfect models allows an effective way of quantifying the model error. It also offers a systematic approach to improve approximate models and stochastic parameterizations in highly non-Gaussian systems and thus advances the real-time forecasts. • A nonlinear optimal smoother is derived for a large class of nonlinear systems. • An optimal sampling strategy is design for recovering the hidden trajectories. • Accurately predict the hidden extreme events. • Provide useful information for improving stochastic parametrizations. [ABSTRACT FROM AUTHOR]

• A new method to recover the hidden trajectories in an optimal and deterministic way. • Judicious block decomposition for parallel computation in high-dimensional systems. • Applying statistical symmetry for reducing the computational cost. Parameter estimation for high-dimensional complex nonlinear turbulent dynamical systems with only partial observations is an important and practical issue. However, most of the existing parameter estimation algorithms are computationally expensive in the presence of a large number of state variables or parameters. In this article, a parameter estimation algorithm is developed for high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. This algorithm exploits the closed analytical form of the conditional statistics to recover the unobserved trajectories in an optimal and deterministic way, which facilitates the calculation of the likelihood function and circumvents the computationally expensive data augmentation approach in sampling the unobserved trajectories as widely used in the literature. Such an efficient method of recovering the unobserved trajectories is then incorporated into a standard Markov chain Monte Carlo (MCMC) algorithm to estimate parameters in complex dynamical system using only a short period of training data. Next, in light of the dynamical features, two effective strategies are developed and incorporated into the algorithm that facilitates the parameter estimation of many high-dimensional systems. The first strategy involves a judicious block decomposition of the state variables such that the original problem is divided into several subproblems coupled in a specific way that allows an extremely cheap parallel computation for the parameter estimation. The second strategy exploits statistical symmetry for a further reduction of the computational cost when the system is statistically homogeneous. The new parameter estimation algorithm is applied to a two-layer Lorenz 96 model with 80 state variables and 162 parameters and the model mimics the realistic features of atmosphere wave propagations and excitable media. The efficient algorithm results in an accurate estimation of the parameters, which further allows a skillful prediction by the model with estimated parameters. Other simple nonlinear models are also used to illustrate the features of the new algorithm. [ABSTRACT FROM AUTHOR]