Stability analysis of a class of two-dimensional multipolar vortex equilibria

Published versio

Polygonal N-vortex arrays: A Stuart model

Published versio

The construction of exact multipolar equilibria of the two-dimensional Euler equations

Published versio

Analytical solutions for distributed multipolar vortex equilibria on a sphere

Published versio

Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure

Diffusion-Limited Aggregation on Curved Surfaces

We develop a general theory of transport-limited aggregation phenomena occurring on curved surfaces, based on stochastic iterated conformal maps and conformal projections to the complex plane. To illustrate the theory, we use stereographic projections to simulate diffusion-limited-aggregation (DLA) on surfaces of constant Gaussian curvature, including the sphere ($K>0$) and pseudo-sphere ($K<0$), which approximate "bumps" and "saddles" in smooth surfaces, respectively. Although curvature affects the global morphology of the aggregates, the fractal dimension (in the curved metric) is remarkably insensitive to curvature, as long as the particle size is much smaller than the radius of curvature. We conjecture that all aggregates grown by conformally invariant transport on curved surfaces have the same fractal dimension as DLA in the plane. Our simulations suggest, however, that the multifractal dimensions increase from hyperbolic ($K0$) geometry, which we attribute to curvature-dependent screening of tip branching.Comment: 4 pages, 3 fig

Scaling Limits for Internal Aggregation Models with Multiple Sources

We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.Comment: 74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in v2: added "least action principle" (Lemma 3.2); small corrections in section 4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version); expanded section 6.

Oxygen uptake and denitrification in soil aggregates

A mathematical model of oxygen uptake by bacteria in agricultural soils is presented with the goal of predicting anaerobic regions in which denitrification occurs. In an environment with a plentiful supply of oxygen, microorganisms consume oxygen through normal respiration. When the local oxygen concentration falls below a threshold level, denitrification may take place leading to the release of nitrous oxide, a potent agent for global warming. A two-dimensional model is presented in which one or more circular soil aggregates are located at a distance below the ground-level at which the prevailing oxygen concentration is prescribed. The level of denitrification is estimated by computing the area of any anaerobic cores which may develop in the interior of the aggregates. The oxygen distribution throughout the model soil is calculated first for an aggregated soil for which the ratio of the oxygen diffusivities between an aggregate and its surround is small via an asymptotic analysis. Second, the case of a non-aggregated soil featuring one or more microbial hotspots, for which the diffusion ratio is arbitrary, is examined numerically using the boundary-element method. Calculations with multiple aggregates demonstrate a sheltering effect whereby some aggregates receive less oxygen than their neighbours. In the case of an infinite regular triangular network representing an aggregated soil, it is shown that there is an optimal inter-aggregate spacing which minimises the total anaerobic core area