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This book focuses on universal nonlinear dynamics model of mesoscale eddies. The results of this book are not only the direct-type applications of pure mathematical limit cycle theory and fractal theory in practice but also the classic combination of nonlinear dynamic systems in mathematics and the physical oceanography. The universal model and experimental verification not only verify the relevant results that are obtained by Euler's form but also, more importantly, are consistent with observational numerical statistics. Due to the universality of the model, the consequences of the system are richer and more complete. The comprehensive and systematic mathematical modeling of mesoscale eddies is one of the major features of the book, which is particularly suited for readers who are interested to learn fractal analysis and prediction in physical oceanography. The book benefits researchers, engineers, and graduate students in the fields of mesoscale eddies, fractal, chaos, and other applications, etc.
The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describe nonlinear phenomena better than traditional mathematical models. In many cases they have been used for solving inverse problems arising in models described by systems of differential equations and dynamical systems. "Fractal-Based Methods in Analysis" draws together, for the first time in book form, methods and results from almost twenty years of research in this topic, including new viewpoints and results in many of the chapters. For each topic the theoretical framework is carefully explained using examples and applications. The second chapter on basic iterated function systems theory is designed to be used as the basis for a course and includes many exercises. This chapter, along with the three background appendices on topological and metric spaces, measure theory, and basic results from set-valued analysis, make the book suitable for self-study or as a source book for a graduate course. The other chapters illustrate many extensions and applications of fractal-based methods to different areas. This book is intended for graduate students and researchers in applied mathematics, engineering and social sciences. Herb Kunze is a professor of mathematics at the University of Guelph in Ontario. Davide La Torre is an associate professor of mathematics in the Department of Economics, Management and Quantitative Methods of the University of Milan. Franklin Mendivil is a professor of mathematics at Acadia University in Nova Scotia. Edward Vrscay is a professor in the department of Applied Mathematics at the University of Waterloo in Ontario. The major focus of their research is on fractals and the applications of fractals.
The long-term evolution of Gaussian eddies is studied in an equivalent barotropic model using both linear and nonlinear quasi-geostrophic theory in an attempt to understand westward propagating satellite altimetry tracked mesoscale eddies. By examining both individual eddies and a large basin seeded with eddies, it is shown that long term eddy coherence and the zonal wavenumber-frequency power spectral density are best matched by the nonlinear model. Individual characteristics of the eddies including amplitude decay, length decay, zonal and meridional propagation speed of a previously unrecognized quasi-stable state are examined to provide baseline properties for comparison with extended models. An analytical technique is then used for evaluating scales of motion of typical mesoscale eddies in order evaluate the success of existing models and find other more appropriate theories. Starting from the spherical shallow water equations and assuming geostrophic dominance, a potential vorticity conservation law is derived in terms of all four non-dimensional parameters inherent in the equations while retaining the spherical geometry. By retaining freedom in the parameters, the scales can be determined at which various theories remain valid. It is argued that the FP equation equation and a new extension to the FP equation are required to describe the mid-latitude mesoscale eddies. Analytical solutions to the FP equation are sought using the classical and exterior differential systems methods of group foliation. Both methods of group foliation are used to find the cnoidal solution of the Korteweg-de Vries equation, a one-dimensional form of the FP equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest, and a reduced quasi-linear hyperbolic system is derived for the two-dimensional FP equation. The forces driving the slow westward propagation of mesoscale eddies also underly a particle constrained to the surface of the earth, but are quantitatively misunderstood. Starting with a free particle and successively adding constraints, it is shown that the particle's motion is inertial, despite literature to the contrary, and that an accelerometer trapped in inertial motion would not measure an acceleration.
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Published by the American Geophysical Union as part of the Geophysical Monograph Series, Volume 177. This monograph is the first to survey progress in realistic simulation in a strongly eddying regime made possible by recent increases in computational capability. Its contributors comprise the leading researchers in this important and constantly evolving field. Divided into three parts Oceanographic Processes and Regimes: Fundamental Questions Ocean Dynamics and State: From Regional to Global Scale, and Modeling at the Mesoscale: State of the Art and Future Directions The volume details important advances in physical oceanography based on eddy resolving ocean modeling. It captures the state of the art and discusses issues that ocean modelers must consider in order to effectively contribute to advancing current knowledge, from subtleties of the underlying fluid dynamical equations to meaningful comparison with oceanographic observations and leading-edge model development. It summarizes many of the important results which have emerged from ocean modeling in an eddying regime, for those interested broadly in the physical science. More technical topics are intended to address the concerns of those actively working in the field.
A new method of modeling the atmosphere, synthesizing data analysis techniques and multifractal statistics, for atmospheric researchers and graduate students.