Motion of a non-axisymmetric particle in viscous shear flow

We examine the motion in a shear flow at zero Reynolds number of particles with two planes of symmetry. We show that in most cases the rotational motion is qualitatively similar to that of a non-axisymmetric ellipsoid, and characterised by a combination of chaotic and quasiperiodic orbits. We use Kolmogorov–Arnold–Moser (KAM) theory and related ideas in dynamical systems to elucidate the underlying mathematical structure of the motion and thence to explain why such a large class of particles all rotate in essentially the same manner. Numerical simulations are presented for curved spheroids of varying centreline curvature, which are found to drift persistently across the streamlines of the flow for certain initial orientations. We explain the origin of this migration as the result of a lack of symmetries of the particle’s orientation orbit.</jats:p

Contributions of plasma physics to chaos and nonlinear dynamics

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

Chaotic advection, mixing, and property exchange in three-dimensional ocean eddies and gyres

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 June 2018This work investigates how a Lagrangian perspective applies to models of two oceanographic flows: an overturning submesoscale eddy and the Western Alboran Gyre. In the first case, I focus on the importance of diffusion as compared to chaotic advection for tracers in this system. Three methods are used to quantify the relative contributions: scaling arguments including a Lagrangian Batchelor scale, statistical analysis of ensembles of trajectories, and Nakamura effective diffusivity from numerical simulations of dye release. Through these complementary methods, I find that chaotic advection dominates over turbulent diffusion in the widest chaotic regions, which always occur near the center and outer rim of the cylinder and sometimes occur in interior regions for Ekman numbers near 0.01. In thin chaotic regions, diffusion is at least as important as chaotic advection. From this analysis, it is clear that identified Lagrangian coherent structures will be barriers to transport for long times if they are much larger than the Batchelor scale. The second case is a model of the Western Alboran Gyre with realistic forcing and bathymetry. I examine its transport properties from both an Eulerian and Lagrangian perspective. I find that advection is most often the dominant term in Eulerian budgets for volume, salt, and heat in the gyre, with diffusion and surface fluxes playing a smaller role. In the vorticity budget, advection is as large as the effects of wind and viscous diffusion, but not dominant. For the Lagrangian analysis, I construct a moving gyre boundary from segments of the stable and unstable manifolds emanating from two persistent hyperbolic trajectories on the coast at the eastern and western extent of the gyre. These manifolds are computed on several isopycnals and stacked vertically to construct a three-dimensional Lagrangian gyre boundary. The regions these manifolds cover is the stirring region, where there is a path for water to reach the gyre. On timescales of days to weeks, water from the Atlantic Jet and the northern coast can enter the outer parts of the gyre, but there is a core region in the interior that is separate. Using a gate, I calculate the continuous advective transport across the Lagrangian boundary in three dimensions for the first time. A Lagrangian volume budget is calculated, and challenges in its closure are described. Lagrangian and Eulerian advective transports are found to be of similar magnitudes.Funding for this work was provided by Woods Hole Oceanographic Institution and the Office of Naval Research. The ONR support was through the MURI on Dynamical Systems Theory and Lagrangian Data Assimilation in 4D and the CALYPSO project