Joseph Nebus

Vortex dynamics, statistical mechanics, and planetary atmospheres

Planetary Atmospheres Statistical Mechanics Monte Carlo Methods The Coupled Barotropic Solid Sphere Model Phase Transitions and Relative Energy-Enstrophy Theory Extremal Free Energy in the Mean Field Theory Phase Transitions of Barotropic Flows Phase Transitions to Super-Rotation in a Coupled Field on the Rotating Sphere The Shallow-Water Model and Jupiters Great Red Spot First-Order Phase Transitions A Statistical Shallow-Water Model for the Jovian Atmosphere.

The spherical model of logarithmic potentials as examined by Monte Carlo methods

We examine Euler’s equations for inviscid fluid flow by a discretized version representing the fluid as a piecewise constant finite approximation based on Voronoi cells. The strengths of these cells are constrained to conserve circulation and enstrophy. With this model we examine by a Monte Carlo Metropolis–Hastings algorithm the dependence of the system on such parameters as the number of points, the statistical mechanics temperature, and the number of sweeps used in the simulation. Tools to examine the system include the mean nearest neighbor parity, energy, distance between extreme-valued sites, and statistical study of an individual site or of all mesh site values. In negative statistical mechanics temperatures a solid-body rotation state is found. The positive-temperature state is not as strongly organized. Numerical evidence supports our expectation of a single phase transition, between positive and negative temperatures.

A Monte Carlo algorithm for free and coaxial ring extremal states of the vortex N-body problem on a sphere

We show that the search for statistical equilibria at very low positive temperatures, using a Monte Carlo algorithm, can successfully locate dynamical equilibria of the N-vortex problem on a sphere. Numerical results are collected to show that for a wide range of particle numbers, this algorithm accurately and efficiently locates the ground state or lowest energy equilibrium. The extremal states found numerically are carefully compared with well-known exact configurations such as the regular polyhedra. Using an essential tool called the radial distribution function, we state a theorem that is useful for comparing N-vortex configurations which are related to one another by elements of the group O(3). It is found that by constraining the system to equally spaced latitudinal rings of vortices the computational cost may be reduced by an order of magnitude. Many of the results reported here apply directly to other N-body problems on a sphere, such as the distribution of N charges on a sphere.