scholarly journals Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification

Entropy ◽  
2018 ◽  
Vol 20 (7) ◽  
pp. 509 ◽  
Author(s):  
Nan Chen ◽  
Andrew Majda
Keyword(s):  
Large Scale ◽  
Stochastic Systems ◽  
Planck Equation ◽  
Reaction Diffusion ◽  
Small Scale ◽  
Model Errors ◽  
Nonlinear Stochastic Systems ◽  
Hybrid Strategy ◽  
Highly Nonlinear ◽  
Non Gaussian

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

scholarly journals Isotropic non-Gaussian gNL-like toy models that reproduce cosmic microwave background anomalies

2019 ◽  
Vol 626 ◽  
pp. A13 ◽  
Author(s):  
F. K. Hansen ◽  
T. Trombetti ◽  
N. Bartolo ◽  
U. Natale ◽  
M. Liguori ◽  
...  
Keyword(s):  
Cosmic Microwave Background ◽  
Large Scale ◽  
Statistical Significance ◽  
Common Mechanism ◽  
Small Scale ◽  
Cold Spot ◽  
Microwave Background ◽  
Significance Level ◽  
Non Gaussian ◽  
Cmb Fluctuations

Context. Based on recent observations of the cosmic microwave background (CMB), claims of statistical anomalies in the properties of the CMB fluctuations have been made. Although the statistical significance of the anomalies remains only at the ∼2−3σ significance level, the fact that there are many different anomalies, several of which support a possible deviation from statistical isotropy, has motivated a search for models that provide a common mechanism to generate them. Aims. The goal of this paper is to investigate whether these anomalies could originate from non-Gaussian cosmological models, and to determine what properties these models should have. Methods. We present a simple isotropic, non-Gaussian class of toy models that can reproduce six of the most extensively studied anomalies. We compare the presence of anomalies found in simulated maps generated from the toy models and from a standard model with Gaussian fluctuations. Results. We show that the following anomalies, as found in the Planck data, commonly occur in the toy model maps: (1) large-scale hemispherical asymmetry (large-scale dipolar modulation), (2) small-scale hemispherical asymmetry (alignment of the spatial distribution of CMB power over all scales ℓ = [2, 1500]), (3) a strongly non-Gaussian hot or cold spot, (4) a low power spectrum amplitude for ℓ <  30, including specifically (5) a low quadrupole and an unusual alignment between the quadrupole and the octopole, and (6) parity asymmetry of the lowest multipoles. We note that this class of toy model resembles models of primordial non-Gaussianity characterised by strongly scale-dependent gNL-like trispectra.

Kolmogorov’s contributions to the physical and geometrical understanding of small-scale turbulence and recent developments

1991 ◽  
Vol 434 (1890) ◽  
pp. 183-210 ◽  
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Inertial Range ◽  
Small Scale ◽  
Small Scales ◽  
Recent Developments ◽  
Self Similar ◽  
Scale Turbulence ◽  
High Reynolds ◽  
Non Gaussian

This paper reviews how Kolmogorov postulated for the first time the existence of a steady statistical state for small-scale turbulence, and its defining parameters of dissipation rate and kinematic viscosity. Thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes. We present theoretical arguments and experimental evidence to indicate when the small-scale motions might tend to a universal form (paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic), and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the - 5/3 inertial range spectrum. These may be particular forms of eddy or ‘eigenstructure’ such as spiral vortices, which may not be unique to turbulent flows. Also, they tend to lead to the notable spiral contours of scalars in turbulence, whose self-similar structure enables the ‘box-counting’ technique to be used to measure the ‘capacity’ D K of the contours themselves or of their intersections with lines, D' K . Although the capacity, a term invented by Kolmogorov (and studied thoroughly by Kolmogorov & Tikhomirov), is like the exponent 2 p of a spectrum in being a measure of the distribution of length scales ( D' K being related to 2 p in the limit of very high Reynolds numbers), the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the Reynolds number. Thus Kolmogorov & Tikhomirov provide the basis for a more widely applicable measure of the self-similar structure of turbulence. Finally, we also review how Kolmogorov’s concept of the universal spatial structure of the small scales, together with appropriate additional physical hypotheses, enables other aspects of turbulence to be understood at these scales; in particular the general forms of the temporal statistics such as the high-frequency (inertial range) spectra in eulerian and lagrangian frames of reference, and the perturbations to the small scales caused by non-isotropic, non-gaussian and inhomogeneous large-scale motions.

scholarly journals pth Mean Practical Stability for Large-Scale Itô Stochastic Systems with Markovian Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Yan Yun ◽  
Huisheng Shu ◽  
Yan Che
Keyword(s):  
Comparison Principle ◽  
Large Scale ◽  
Stochastic Systems ◽  
Markovian Switching ◽  
Practical Stability ◽  
Nonlinear Stochastic Systems ◽  
Deterministic Systems ◽  
The One

Motivated by the study of a class of large-scale stochastic systems with Markovian switching, this correspondence paper is concerned with the practical stability in thepth mean. By investigating Lyapunov-like functions and the basic comparison principle, some criteria are derived for various types of practical stability in thepth mean of nonlinear stochastic systems. The main contribution of these results is to convert the problem of practical stability in thepth mean of stochastic systems into the one of practical stability of the comparative deterministic systems.

scholarly journals Sinking microplastics in the water column: simulations in the Mediterranean Sea

Ocean Science ◽  
2021 ◽  
Vol 17 (2) ◽  
pp. 431-453
Author(s):  
Rebeca de la Fuente ◽  
Gábor Drótos ◽  
Emilio Hernández-García ◽  
Cristóbal López ◽  
Erik van Sebille
Keyword(s):  
Vertical Distribution ◽  
Mediterranean Sea ◽  
Water Column ◽  
Large Scale ◽  
Circulation Model ◽  
Small Scale ◽  
The Mediterranean ◽  
Scale Turbulence ◽  
Non Gaussian ◽  
The Vertical Distribution

Abstract. We study the vertical dispersion and distribution of negatively buoyant rigid microplastics within a realistic circulation model of the Mediterranean sea. We first propose an equation describing their idealized dynamics. In that framework, we evaluate the importance of some relevant physical effects (inertia, Coriolis force, small-scale turbulence and variable seawater density), and we bound the relative error of simplifying the dynamics to a constant sinking velocity added to a large-scale velocity field. We then calculate the amount and vertical distribution of microplastic particles on the water column of the open ocean if their release from the sea surface is continuous at rates compatible with observations in the Mediterranean. The vertical distribution is found to be almost uniform with depth for the majority of our parameter range. Transient distributions from flash releases reveal a non-Gaussian character of the dispersion and various diffusion laws, both normal and anomalous. The origin of these behaviors is explored in terms of horizontal and vertical flow organization.

scholarly journals Instantaneous variance scaling of AIRS thermodynamic profiles using a circular area Monte Carlo approach

2018 ◽  
Vol 11 (5) ◽  
pp. 2717-2733
Author(s):  
Jesse Dorrestijn ◽  
Brian H. Kahn ◽  
João Teixeira ◽  
Fredrick W. Irion
Keyword(s):  
Monte Carlo ◽  
Large Scale ◽  
Specific Humidity ◽  
Time Variability ◽  
Small Scale ◽  
Circular Area ◽  
Scaling Exponents ◽  
Cloud Parameterization ◽  
Non Gaussian ◽  
The Tropics

Abstract. Satellite observations are used to obtain vertical profiles of variance scaling of temperature (T) and specific humidity (q) in the atmosphere. A higher spatial resolution nadir retrieval at 13.5 km complements previous Atmospheric Infrared Sounder (AIRS) investigations with 45 km resolution retrievals and enables the derivation of power law scaling exponents to length scales as small as 55 km. We introduce a variable-sized circular-area Monte Carlo methodology to compute exponents instantaneously within the swath of AIRS that yields additional insight into scaling behavior. While this method is approximate and some biases are likely to exist within non-Gaussian portions of the satellite observational swaths of T and q, this method enables the estimation of scale-dependent behavior within instantaneous swaths for individual tropical and extratropical systems of interest. Scaling exponents are shown to fluctuate between β=-1 and −3 at scales ≥500 km, while at scales ≤500 km they are typically near β≈-2, with q slightly lower than T at the smallest scales observed. In the extratropics, the large-scale β is near −3. Within the tropics, however, the large-scale β for T is closer to −1 as small-scale moist convective processes dominate. In the tropics, q exhibits large-scale β between −2 and −3. The values of β are generally consistent with previous works of either time-averaged spatial variance estimates, or aircraft observations that require averaging over numerous flight observational segments. The instantaneous variance scaling methodology is relevant for cloud parameterization development and the assessment of time variability of scaling exponents.

Vorticity and mixing in Rayleigh–Taylor Boussinesq turbulence

2016 ◽  
Vol 802 ◽  
pp. 395-436 ◽  
Author(s):  
Nicolas Schneider ◽  
Serge Gauthier
Keyword(s):  
Large Scale ◽  
Mixing Layer ◽  
Boussinesq Approximation ◽  
The Self ◽  
Small Scale ◽  
Nonlinear Growth ◽  
Small Scales ◽  
Self Similar ◽  
Order Quantities ◽  
Non Gaussian

The Rayleigh–Taylor instability induced turbulence is studied under the Boussinesq approximation focusing on vorticity and mixing. A direct numerical simulation has been carried out with an auto-adaptive multidomain Chebyshev–Fourier–Fourier numerical method. The spatial resolution is increased up to $(24\times 40)\times 940^{2}=848\,M$ collocation points. The Taylor Reynolds number is $\mathit{Re}_{\unicode[STIX]{x1D706}_{zz}}\approx 142$ and a short inertial range is observed. The nonlinear growth rate of the turbulent mixing layer is found to be close to $\unicode[STIX]{x1D6FC}_{b}=0.021$. Our conclusions may be summarized as follows.(i) The simulation data are in agreement with the scalings for the pressure ($k^{-7/3}$) and the vertical mass flux ($k^{-7/3}$).(ii) Mean quantities have a self-similar behaviour, but some inhomogeneity is still present. For higher-order quantities the self-similar regime is not fully achieved.(iii) In the self-similar regime, the mean dissipation rate and the enstrophy behave as $\langle \overline{\unicode[STIX]{x1D700}}\rangle \propto t$ and $\langle \overline{\unicode[STIX]{x1D714}_{i}\,\unicode[STIX]{x1D714}_{i}}^{1/2}\rangle \propto t^{1/2}$, respectively.(iv) The large-scale velocity fluctuation probability density function (PDF) is Gaussian, while vorticity and dissipation PDFs show large departures from Gaussianity.(v) The pressure PDF exhibits strong departures from Gaussianity and is skewed. This is related to vortex coherent structures.(vi) The intermediate scales of the mixing are isotropic, while small scales remain anisotropic. This leaves open the possibility of a small-scale buoyancy. Velocity intermediate scales are also isotropic, while small scales remain anisotropic. Mixing and dynamics are therefore consistent.(vii) Properties and behaviours of vorticity and enstrophy are detailed. In particular, equations for these quantities are written down under the Boussinesq approximation.(viii) The concentration PDF is quasi-Gaussian. The vertical concentration gradient is both non-Gaussian and strongly skewed.

How essential of the balance between large and small scale features to reproduce precipitation during a sudden sharp turn from drought to flood

2020 ◽  
Author(s):  
Yuanyuan Ma
Keyword(s):  
Extreme Events ◽  
Large Scale ◽  
Climate Model ◽  
Regional Climate ◽  
Small Scale ◽  
Model Errors ◽  
Continuous Simulation ◽  
Spectral Nudging ◽  
Sharp Turn ◽  
The Difference

&lt;p&gt;Sudden turn from drought to flood (STDF) is a unique representation of intra-seasonal extreme events and occurs frequently. However, it is notoriously difficult to represent in climate simulations due to the accumulation of model errors. This study uses a regional climate model (RCM) with different initialization and nudging schemes to explore effective approaches for capturing a STDF event. Results show that the conventional continuous integration with single initialization cannot reproduce the STDF event, while nudging or re-initialization can. Furthermore, spectral nudging and re-initialization outperform the conventional continuous simulation in reproducing precipitation features, but grid nudging induces the largest biases for precipitation though it has the smallest biases for other meteorological elements. Scale separation analysis shows that the large-scale features of the conventional continuous simulation drift far from the actual fields and force erroneous small-scale features, whereas the nudging and re-initialization successfully prevent the model from drifting away from the forcing fields at large-scales. The different performance for simulating precipitation among spectral nudging, re-initialization and grid nudging can be attributed to that the former two methods generate their own small-scale information via the RCM, while grid nudging over-suppresses the small-scale information while retaining the large-scale features. The difference in small-scale features affects the simulation of different moisture fluxes and convergences, as well as clouds, and then results in diverse precipitation. These results illustrate that both the consistency with large-scale features and the local variability from small-scale features are both robust factors for reproducing precipitation features during extreme events using RCMs.&lt;/p&gt;

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