Some results on the 1D linear wave equation with van der Pol type nonlinear boundary conditionsand the Korteweg-de Vries-Burgers equation

Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt − wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F (un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result

2010 program of study : swirling and swimming in turbulence

Swirling and Swimming in Turbulence was the theme at the 2010 GFD Program. Professors Glenn Flierl (M.I.T.), Antonello Provenzale (ISAC-CNR, Turin) and Jean-Luc Thiffeault (University of Wisconsin) were the principal lecturers. Together they navigated an elegant path through topics ranging from mixing protocols and efficiencies to ecological strategies, schooling and genetic development. The first ten chapters of this volume document these lectures, each prepared by pairs of this summer’s GFD fellows. Following on are the written reports of the fellows’ own research projects.Funding was provided by the Office of Naval Research under Contract No. N000-14-09-10844 and the National Science Foundation through Grant No. OCE 082463

Spatial eco-evolutionary dynamics along environmental gradients: multi-stability and cluster dynamics

International audienceHow the interplay of local adaptation and dispersal determines species appearance, distribution and range dynamics is still incompletely understood. Here we combine individual-based simulations and mathematical analysis of large-population approximation models to advance the analysis of spatial spread and phenotypic diversification of a single-species population along a one-dimensional resource gradient. Local competition shapes selection on heritable variation in the individual ecological trait (niche position) and the evolutionary response feeds back on the local ecological state of the population (abundance). Key parameters of spatial spread and phenotypic diversification are the individual dispersal rate, the size of the spatial competition neighborhood, and the phenotype mutational variance. From a focal location the population spreads by forming clusters in space and/or trait, or by spreading along a continuous cline in both space and trait. The conditions for clustering are broader than previously known. The spacing of clusters is determined by the spatial scale of competition. When the space-trait domain is bounded, multi-stability occurs, whereby small initial differences can lead to alternative spatial and trait distributions. The transient dynamics involve adaptational lags which cause a slow-down in cluster formation and population range expansion

POROUS MEDIUM CONVECTION AT LARGE RAYLEIGH NUMBER: STUDIES OF COHERENT STRUCTURE, TRANSPORT, AND REDUCED DYNAMICS

Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from carbon dioxide storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced-dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not ``true turbulence. The objective of this dissertation research is to quantitatively characterize the dynamics and heat transport in two-dimensional horizontal and inclined porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number $Ra$ by investigating the emergent, quasi-coherent flow. This investigation employs a complement of direct numerical simulations (DNS), secondary stability and dynamical systems theory, and variational analysis. The DNS confirm the remarkable tendency for the interior flow to self-organize into closely-spaced columnar plumes at sufficiently large $Ra$ (up to $Ra \simeq 10^5$), with more complex spatiotemporal features being confined to boundary layers near the heated and cooled walls. The relatively simple form of the interior flow motivates investigation of unstable steady and time-periodic convective states at large $Ra$ as a function of the domain aspect ratio $L$. To gain insight into the development of spatiotemporally chaotic convection, the (secondary) stability of these fully nonlinear states to small-amplitude disturbances is investigated using a spatial Floquet analysis. The results indicate that there exist two distinct modes of instability at large $Ra$: a bulk instability mode and a wall instability mode. The former usually is excited by long-wavelength disturbances and is generally much weaker than the latter. DNS, strategically initialized to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically-steady porous medium convection results from an interplay between the competing effects of these two types of instability. Upper bound analysis is then employed to investigate the dependence of the heat transport enhancement factor, i.e. the Nusselt number $Nu$, on $Ra$ and $L$. To solve the optimization problems arising from the ``background field upper-bound variational analysis, a novel two-step algorithm in which time is introduced into the formulation is developed. The new algorithm obviates the need for numerical continuation, thereby enabling the best available bounds to be computed up to $Ra\approx 2.65\times 10^4$. A mathematical proof is given to demonstrate that the only steady state to which this numerical algorithm can converge is the required global optimal of the variational problem. Using this algorithm, the dependence of the bounds on $L(Ra)$ is explored, and a ``minimal flow unit is identified. Finally, the upper bound variational methodology is also shown to yield quantitatively useful predictions of $Nu$ and to furnish a functional basis that is naturally adapted to the boundary layer dynamics at large $Ra$

Instability in Geophysical Flows

An Open Access overview of physical processes that generate instability in geophysical systems. It covers classical analytical approaches together with numerical methods for quick prediction of stability in a system. Including exercises and MATLAB® coding examples, it can be used for self-study or advanced courses in the environmental sciences

Tidal dynamics and dispersion around coastal headlands

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 1989The dynamics of shallow tidal currents and tide-induced dispersion are investigated around coastal headlands that have alongshore length scales that are comparable to or less than the tidal excursion. Depth-averaged shallow water equations forced by oscillatory flow are solved numerically for Gaussian headlands. The tidal flows around these headlands are shown to be characterized by flow separation and. transient eddy formation. Idealized models of flow separation and the transport and damping of vorticity away from the headland explain much of the observed behavior. The characteristics of the separated wake are compared with known results from the study of viscous flow around bluff bodies. The kinematics of particle dispersion in the numerical solutions is described and analyzed.This work was supported by NSF grant OCE-87-11031 and the National Center for Atmospheric Research. Additional funding was provided by the Woods Hole Oceanographic Institution's Ocean Venture Fund, Coastal Research Center, and Education Program

Stirring and mixing : 1999 Program of Summer Study in Geophysical Fluid Dynamics

The central theme of the 1999 GFD Program was the stirring, transport, reaction and mixing of passive and active tracers in turbulent, stratified, rotating fluids. The problem of mixing in fluids has applications in areas ranging from oceanography to engineering and astrophysics. In geophysical settings, mixing spans and unites a broad range of scales -- from micrometers to megameters. The mixing of passive tracers is of fundamental importance in environmental and industrial problems, such as pollution, and in determining the large-scale heat and salt balance of the worlds oceans. The transport of active tracers, on the other hand, such as vorticity, plays a key role in the turbulence that occurs in most geophysical and astrophysical fluids. William R. Young (Scripps Institution of Oceanography) gave a series of principal lectures, the notes of which as taken by the fellows, appear in this volume. Report of the projects of the student fellows makes up the second half of this volume.Funding was provided by the National Science Foundation under Grant No. OCE-9810647 and the Office of Naval Research under Grant No. NOO0l4-97-1-0934