Elementary models with probability distribution function intermittency for passive scalars with a mean gradient

Bourlioux, Anne; Majda, Andrew J.

doi:10.1063/1.1430736

Cited by 40 publications

(72 citation statements)

References 27 publications

(40 reference statements)

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“…Here we introduce a family of test models for climate change science which have direct qualitative relevance for actual observed features for tracers in the atmosphere (40,41) with the additional attractive feature of exactly solvable statistics for the mean and covariance (37,42,43) with many degrees of freedom despite the inherent statistical nonlinearity. Thus, they are physically relevant unambiguous test models for uncertainty in climate change science (37).…”

Section: Test Models For Climate Change Science Illustrating Featuresmentioning

confidence: 99%

Quantifying uncertainty in climate change science through empirical information theory

Majda

Gershgorin

2010

Proc. Natl. Acad. Sci. U.S.A.

111

146

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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 CO 2 . 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. model errors | practical algorithms | unbiased empirical estimates T he climate is an extremely complex coupled system involving significant physical processes for the atmosphere, ocean, and land over a wide range of spatial scales from millimeters to thousands of kilometers and time scales from minutes to decades or centuries (1, 2) Climate change science focuses on predicting the coarse-grained planetary scale long time changes in the climate system due to either changes in external forcing or internal variability such as the impact of increased carbon dioxide (3) For several decades the predictions of climate change science have been carried out with some skill through comprehensive computational atmospheric and oceanic simulation (AOS) models (1-4) which are designed to mimic the complex physical spatio-temporal patterns in nature. Such AOS models either through lack of resolution due to current computing power or through inadequate observation of nature necessarily parameterize the impact of many features of the climate system such as clouds, mesoscale and submesoscale ocean eddies, sea ice cover, etc. Thus, there are intrinsic model errors in the AOS models for the climate system and a central scientific issue is the effect of such model errors on predicting the coarse-grained large scale long time quantities of interest in climate change science.

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Section: Test Models For Climate Change Science Illustrating Featuresmentioning

confidence: 99%

Quantifying uncertainty in climate change science through empirical information theory

Majda

Gershgorin

2010

Proc. Natl. Acad. Sci. U.S.A.

111

146

View full text Add to dashboard Cite

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“…A natural question is to isolate the essential role played by stochasticity in these studies. To that end, Bourlioux and Majda 19 exhibited that a passive scalar advected by either a deterministic or a random shear layer while diffusing in the presence of a large scale scalar gradient could also develop a non-Gaussian PDF, with large probability of large deviations over a wide range of scalar values.…”

Section: Introductionmentioning

confidence: 99%

“…Through these simulations, we observe the longest lived, slowest decaying eigenmode, which retains features of both the initial random scalar field, and the prescribed fluid flow. The probability distribution functions (PDFs) will be defined in terms of the instantaneous spatial distribution of scalar values (generalizing to multiple dimensions those considered by Bourlioux and Majda 19 ) in contrast to more familiar ensemble or time averaged constructions. These numerical results should be taken not to discount the more complicated fluid motions and complicated analyses which identify the limiting PDF, but rather to present a documentation of the heavy-tailed distributions which may arise from simple fluid flows in order to give improved interpretation of empirically measured PDFs.…”

Section: Introductionmentioning

confidence: 99%

“…For such random flows, a host of theoretical results have helped to understand the origin of non-Gaussianity in scalar signals. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] Different types of stochastic flows have been shown to give rise to heavy tailed scalar distribution functions, some involving positive Lyapunov exponents, 9,16,23 and others with pure shear flows having no positive Lyapunov exponent. 5-7, 11, 14, 15, 20 The basic picture for such studies is that the PDF for large scalar fluctuations is composed of values associated with anomalously rare coherent fluid stretching events which outlive their incoherent counterparts under the action of diffusion.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

2007

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The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Péclet numbers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involving scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad tailed distribution is similar for shear flows and time-varying non-sheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also connect geometric features of the scalar field with the shape of the PDFs. We document that the PDF is constructed by a subtle balance between spatial regions of strong and weak shear in conjunction with the presence of

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“…There are extremely few mathematical analyses on which to base the validity of such simulations. For fluid flows which are random shear layers, several important mathematical results have been obtained, either for cases in which the scalar is fluctuating in the presence of a large scale, mean gradient [7] , or in the freely decaying scalar case [8,10,11,[20][21][22][23]33] . We focus upon the freely decaying case in this article.…”

Section: Introductionmentioning

confidence: 99%

Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

2007

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In 1993, Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-monotonic "breathing" phenomenon is discovered from these results and carefully defined, with a solid example for special initial data to predict such phenomenon. The signature of this phenomenon may persist at long time, characterized by the approach of the PDF core to its infinite time, invariant value. We find that this approach may be strongly dependent on the nondimensional Péclet number, of which the invariant measure itself is independent. Further, the "breathing" PDF is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit. Rigorous asymptotic analysis is also performed to identify the Gaussian core of the invariant measures, and the critical rate at which the heavy, stretched exponential regime propagates towards the tail as a function of time is calculated.

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Elementary models with probability distribution function intermittency for passive scalars with a mean gradient

Cited by 40 publications

References 27 publications

Quantifying uncertainty in climate change science through empirical information theory

Quantifying uncertainty in climate change science through empirical information theory

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

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