Paul K. Newton

The N-Vortex problem : analytical techniques

Introduction * N-Vortices in the Plane * Domains with Boundaries * Vortex Motion on a Sphere * Geometric Phases * Statistical Point Vortex Theories * Vortex Patch Models * Vortex Filament Models * References * Index

Motion of three point vortices on a sphere

Abstract We solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius R. The system is more general than the corresponding one in the plane [1,3,5,19,28,33,34,37,38] and reduces to it in the limit R → ∞, as long as the three vortices remain sufficiently close to each other during the course of their motion. Instead of using spherical coordinates, we use cartesian coordinates in which the vector χi points from the center of the sphere to the vortex with strength Γi. An important conserved quantity is the center of vorticity vector, c = (∑Γ i χ i ) (∑Γ i ) which, with no loss of generality, we align with the z-axis. Based on the size of this vector relative to the radius of the sphere, we classify the motion into one of five types: super-radial, radial, sub-radial, degenerate, or a limiting super-radial case. This categorization allows us to draw several conclusions about the qualitative motion of the vortices. We then fully characterize all fixed and relative equilibria on the sphere. For fixed equilibria, the vortices must lie on great circles (geodesics). If the strengths are equal, they form an equilateral triangle. Otherwise, the triangle shape is specified once the strength of the vortices is given. The relative, equilibria are classified as either degenerate (c = 0) or non-degenerate (c ≠ 0). For each type, the shape of the vortex triangle is described and the frequency of rotation is computed. As in the planar problem, it is possible to introduce trilinear coordinates and study the motion in a phase plane, which allows us to locate all the equilibria, as well as to characterize more complex relative dynamics. We then describe self-similar vortex collapse on the sphere, stating necessary and sufficient conditions for collapse to occur, computing the collapse times and vortex trajectories on the route towards collapse. Comparisons are made with the collapse formulas in the plane derived in [1,30].

Point vortex motion on a sphere with solid boundaries

We consider the motion of a point vortex on the surface of a sphere with solid boundaries. This problem is of interest in oceanography, where coherent vortex structures can persist for long times, and move over distances large enough so that the curvature of the Earth becomes important (see Gill [1982], Chaos [1994]). In this context, the boundary is a first step in modeling the presence of coastlines and shores using inviscid theory. Using the equations of motion for the vortex projected onto the stereographic plane, we construct the appropriate Green’s function using classical image method ideas, as long as the domain has certain symmetry properties. After the solution is obtained in the stereographic plane, it is projected back down to the sphere, yielding the sought after solution to the problem. We demonstrate the utility of the method by solving for the vortex trajectories and streamlines for several canonical examples, including a spherical cap, longitudinal wedge, half longitudinal wedge, channel,...

Dynamics of heavy particles in a Burgers vortex

This paper presents a linear stability analysis as well as some numerical results for the motion of heavy particles in the flow field of a Burgers vortex, under the combined effects of particle inertia, Stokes drag, and gravity. By rendering the particle motion equations dimensionless, the particle Stokes number, a Froude number, and a vortex Reynolds number are obtained as the governing three parameters. In the absence of gravity, the vortex center represents a stable equilibrium point for particles up to a critical value of the Stokes number, as the inward drag overcomes the destabilizing centrifugal force on the particle. Particles exceeding the critical Stokes number value asymptotically approach closed circular orbits. Under the influence of gravity, one or three equilibrium points appear away from the vortex center. Both their locations and their stability characteristics are derived analytically. These stability characteristics can furthermore be related to the nature of the critical points in a re...

The N-vortex problem on a rotating sphere. II. Heterogeneous Platonic solid equilibria

We describe a new method of constructing point vortex equilibria on a sphere made-up of N vortices with different strengths. Such equilibria, called heterogeneous equilibria, are obtained for the five Platonic solid configurations, hence for . The method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant. By symmetries inherent in the Platonic solid configurations, this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure. For the tetrahedron (N=4) and octahedron (N=6), the degree of heterogeneity is 4 and 6, respectively, hence we are free to choose each of the vortex strengths independently. For the cube (N=8), the degree of heterogeneity is 5, for the icosahedron (N=12) it is 7, while for the dodecahedron (N=20) it is 4. Thus, the entire set of equilibria based on the Platonic solid configurations is obtained, including substructures associated with each configuration constructed by taking different linear combinations of the basis elements.

A Stochastic Markov Chain Model to Describe Lung Cancer Growth and Metastasis

A stochastic Markov chain model for metastatic progression is developed for primary lung cancer based on a network construction of metastatic sites with dynamics modeled as an ensemble of random walkers on the network. We calculate a transition matrix, with entries (transition probabilities) interpreted as random variables, and use it to construct a circular bi-directional network of primary and metastatic locations based on postmortem tissue analysis of 3827 autopsies on untreated patients documenting all primary tumor locations and metastatic sites from this population. The resulting 50 potential metastatic sites are connected by directed edges with distributed weightings, where the site connections and weightings are obtained by calculating the entries of an ensemble of transition matrices so that the steady-state distribution obtained from the long-time limit of the Markov chain dynamical system corresponds to the ensemble metastatic distribution obtained from the autopsy data set. We condition our search for a transition matrix on an initial distribution of metastatic tumors obtained from the data set. Through an iterative numerical search procedure, we adjust the entries of a sequence of approximations until a transition matrix with the correct steady-state is found (up to a numerical threshold). Since this constrained linear optimization problem is underdetermined, we characterize the statistical variance of the ensemble of transition matrices calculated using the means and variances of their singular value distributions as a diagnostic tool. We interpret the ensemble averaged transition probabilities as (approximately) normally distributed random variables. The model allows us to simulate and quantify disease progression pathways and timescales of progression from the lung position to other sites and we highlight several key findings based on the model.

Periodic solutions of the Ginzburg-Landau equation

Abstract Spatially periodic solutions to the Ginzburg-Landau equation are considered. In particular we obtain: criteria for primary and secondary bifurcation; limit cycle solutions; nonlinear dispersion relations relating spatial and temporal frequencies. Only relatively simple tools appear in the treatment and as a result a wide range of parameter cases are considered. Finally we briefly treat the case of spatial bifurcations.

Vortex Lattice Theory: A Particle Interaction Perspective

Recent experiments on the formation of vortex lattices in Bose-Einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns, ones that are free of discrete symmetries and can form in a random environment. We give an overview of an

Streamline topologies for integrable vortex motion on a sphere

N

p21-Activated Kinase (PAK) Regulates Cytoskeletal Reorganization and Directional Migration in Human Neutrophils

-particle based Hamiltonian theory which, if formulated in terms of the