FEDSM2003-45761 EMPIRICAL MODELING OF UNSTEADY FLOWS

ABSTRACT In this work we demonstrate the utility of stochastic empirical models. Linear stochastic models have been successful at reproducing responses to forcing of large-scale atmospheric flow. However, the linear forms of the models reach their limit on highly nonlinear problems. It is shown here that for simple low dimensional problems, a quadratically nonlinear empirical model is successful at reproducing a chaotic attractor. In addition, empirical models can be used to diagnose the important dynamics of a flow system and identify the essential terms to include in a forward dynamical model. Once the empirical model has been built, it can then be used to diagnose the underlying dynamics. INTRODUCTION Numerical modeling of unsteady flows has traditionally involved using some type of time stepping combined with known dynamics discretized from a partial differential equation. However, sometimes the details of the dynamics are not sufficiently known or we wish to develop a model that reproduces dynamic behavior without involving the details of the full equations of motion. In such cases, we can substitute an empirical model based on observed data. These stochastic empirical models are built from measured or simulated data and are based on a Markhov model. Given a sufficient amount of data to develop the model, the discretized dynamics of the traditional model can be replaced by a matrix of computed values that serve as a propagator matrix. Empirical models have gained popularity in recent years as an alternative to the more traditional dynamical models. They have proven powerful in many scientific and engineering areas to simulate and predict the behavior of dynamical systems. In contrast to the traditional dynamical models that are based on first principles, empirical models are based on data. Thus, they are stochastic in nature and hav

Double cnoidal waves of the Korteweg-de Vries equation: A boundary value approach

Double cnoidal waves of the Korteweg-de Vries equation are studied by direct solution of the nonlinear boundary value problems. These double cnoidal waves, which are the spatially periodic generalization of the well-known double soliton, are exact solutions with two independent phase speeds. The equation is written in terms of two phase variables and expanded in two-dimensional Fourier series. The small-amplitude solution is obtained via the Stokes' perturbation expansion. This solution is numerically extended to larger amplitude by employing a Newton-Kantorovich[+45 degree rule]continuation in amplitude[+45 degree rule] Galerkin algorithm. The crests of the finite amplitude solution closely match the sech2 solitary wave form and the three cases of solitary wave interaction described by Lax are identified for the double cnoidal waves. This simple approach reproduces specific features such as phase shift upon collision, distinction between instantaneous and average phase speeds, and a "paradox of wavenumbers".Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29346/1/0000414.pd

Modeling nonlinear resonance: A modification to the stokes' perturbation expansion

The Stokes' series is a small amplitude perturbation expansion for nonlinear, steadily translating waves of the form u(x - ct). We have developed a modification to the Stokes' perturbation expansion to cope with the type of resonance that occurs when two different wavenumbers have identical phase speeds. Although the nonlinear wave is smooth and bounded at the resonance, the traditional Stokes' expansion fails because of the often-encountered "small denominator" problem. The situation is rectified by adding the resonant harmonic into the expansion at lowest order. The coefficient of the resonant wave is determined at higher order. Near resonance is treated by expanding the dispersion parameter in terms of the amplitude. As an example, we have chosen the Korteweg de Vries equation with an additional fifth degree dispersion term. However, the method is applicable to the amplitude expansions of much more complicated problems, such as the double cnoidal waves of the Korteweg de Vries equation, the problem that motivated this study.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27469/1/0000510.pd

Towards implementing artificial intelligence post-processing in weather and climate: Proposed actions from the Oxford 2019 workshop

The most mature aspect of applying artificial intelligence (AI)/machine learning (ML) to problems in the atmospheric sciences is likely post-processing of model output. This article provides some history and current state of the science of post-processing with AI for weather and climate models. Deriving from the discussion at the 2019 Oxford workshop on Machine Learning for Weather and Climate, this paper also presents thoughts on medium-term goals to advance such use of AI, which include assuring that algorithms are trustworthy and interpretable, adherence to FAIR data practices to promote usability, and development of techniques that leverage our physical knowledge of the atmosphere. The coauthors propose several actionable items and have initiated one of those: a repository for datasets from various real weather and climate problems that can be addressed using AI. Five such datasets are presented and permanently archived, together with Jupyter notebooks to process them and assess the results in comparison with a baseline technique. The coauthors invite the readers to test their own algorithms in comparison with the baseline and to archive their results

Open weather and climate science in the digital era

The need for open science has been recognized by the communities of meteorology and climate science. While these domains are mature in terms of applying digital technologies, the implementation of open science methodologies is less advanced. In a session on “Weather and Climate Science in the Digital Era” at the 14th IEEE International eScience Conference domain specialists and data and computer scientists discussed the road towards open weather and climate science. Roughly 80 % of the studies presented in the conference session showed the added value of open data and software. These studies included open datasets from disparate sources in their analyses or developed tools and approaches that were made openly available to the research community. Furthermore, shared software is a prerequisite for the studies which presented systems like a model coupling framework or digital collaboration platform. Although these studies showed that sharing code and data is important, the consensus among the participants was that this is not sufficient to achieve open weather and climate science and that there are important issues to address. At the level of technology, the application of the findable, accessible, interoperable, and reusable (FAIR) principles to many datasets used in weathe

Solving nonlinear wave problems with spectral boundary value techniques.

Solitary waves are important in modeling geophysical flows. They have been the basis for successful models of coherent structures, such as the persistent high pressure systems in the atmosphere, Gulf Stream rings, and the Great Red Spot of Jupiter. Most previous numerical studies have solved the nonlinear evolution equations as initial value problems. Here, they are approached as spectral boundary value problems. One technique used is to analytically expand the equation in powers of an amplitude parameter via the Stokes' perturbation expansion, which is accurate for small amplitude. The solution may then be extended to larger amplitude using a Newton-Kantorovich iteration. We have applied these techniques to several problems. The first problem addressed is wave resonance; the phase speeds of two wave components are equal and the traditional Stokes' expansion fails. The situation is rectified by adding the resonant wave into the expansion at the appropriate lower order. The coefficient of the resonant wave is determined at higher order in the expansion, resulting in a modified expansion which agrees well with the numerical solution. The case of near resonance is treated by expanding the resonance parameter in terms of the amplitude as well. A second problem is computing double cnoidal waves (two independent waves on each period) of the integrable Korteweg de Vries equation and the nonintegrable Regularized Long Wave (RLW) equation. The nonlinear evolution equation is written in terms of two phase dimensions and the techniques described above are applied. The KdV solution compares well to the exact solution in the soliton limit. Although we cannot prove the double cnoidal waves of the RLW exist, the results agree well with initial value solutions for moderate amplitudes.Ph.D.Physics, Atmospheric SciencePure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128156/2/8812905.pd

Why Are There Fewer Women in Engineering?

This paper attempts to explain the paucity of women in engineering. While the percentage of women entering engineering and science careers has been increasing, the number at higher ranks has not increased as quickly, after considering the appropriate time lag. The differences in tenure rate due to gender alone were statistically insignificant. Instead, these were attributed to the fact that women who are married or have children are less successful than are men with matching characteristics. One solution proposed is to recognize that priorities might be different at differing stages of family life. It is also important to encourage more children of both genders to go into engineering by making it an appealing career

Practical genetic algoritms. Second edition

Hoboken, NJxvii, 253 p.: bibl., gloss., index; 24 c

1.1 ON GENETIC ALGORITHMS AND DISCRETE PERFORMANCE MEASURES Abstract

A relation exists between the manner in which a statistical model is developed and the measure employed for gauging its performance. Often the model is developed by optimizing some continuous measure of performance, while its final performance is assessed in terms of some discrete measure. The question then arises as to whether a model based on the direct optimization of the discrete measure may be superior to or significantly different from the model based on the optimization of continuous measure. Some Artificial Intelligence parameter estimation techniques allow the optimization of discrete measures. Genetic Algorithms constitute one such technique, and therefore, allow for an examination of this question. Here, one type of genetic algorithm is employed to optimize three discrete performance measures of a parametric model for the prediction of hail. A more conventional technique is then employed to optimize the same discrete measures. The former outperforms the latter. In other words, the direct optimization of three discrete measures via genetic algorithms yields better fits to the data than alternatives requiring the intermediate step of optimizing a continuous measure