Contour Dynamics Research Articles

Simulation of the instability of axisymmetric vortices using the contour dynamics method

The two-dimensional problem of the stability of the circular motion of an ideal incompressible fluid with given radial vorticity distribution is studied.

Contour dynamics—An interface method for studying the evolution of large density gradient ionospheric plasma clouds

A 2-D model is introduced of an ionospheric plasma cloud with a piecewise-constant density gradient and a diffusive-regularized boundary. Using contour dynamics (a boundary integral method) only the boundary between the regions of different densities need to be followed so that the 2-D model reduces to the 1-D motion of closed curves. Numerical results are shown including a comparison of the finite-difference and contour dynamics models.

Ionospheric plasma cloud dynamics via regularized contour dynamics. I. Stability and nonlinear evolution of one-contour models

The linear stability and nonlinear evolution of a regularized contour dynamical model of an ionospheric plasma cloud (or deformable dielectric) is examined. That is, the cloud is modeled by piecewise-constant ion density regions; and the regularization is accomplished with a tangential diffusion operator that models aspects of the diffusion operator in two dimensions. A complete linear stability analysis of a circular cloud shows that a single-mode excitation ‘‘cascades downward’’ in wavenumber as it grows in amplitude, a process that results from the symmetry-breaking electric field. Approximate formulas are derived for the amplitude growth and cascade-down phenomena and verified with precise numerical calculations. A simple rescaling shows that clouds with large λ (=cloud-ion density/ambient-ion density) evolve more slowly and appear more dissipative. The regularized contour-dynamical algorithm for computations in the nonlinear regime is validated against the linear analysis and truncation errors are assessed by using different spatial resolutions. Calculations in the nonlinear regime show the experimentally observed ‘‘backside’’ striations. Furthermore, at long times, a secondary structure arises on the sides of the primary striations. A comparison of simulations with different λ shows that nonlinear effects arise sooner in normalized time (but longer in real time) if λ is larger.

Stability and Nonlinear Evolution of Plasma Clouds Via Regularized Contour Dynamics.

A model is introduced of an ionospheric plasma cloud (deformable dielectric) with a piecewise-constant ion density and a diffusive-regularized boundary. The linear stability of a single-contour circular cloud is studied and a new evolution equation for modal amplitudes is obtained which has the property that the wave number of maximum amplitude decreases with time (downward cascade). Analytic expressions show that large clouds evolve more slowly and appear more dissipative.

Stability and Nonlinear Evolution of Plasma Clouds via Regularized Contour Dynamics

A model is introduced of an ionospheric plasma cloud (deformable dielectric) with a piecewise-constant ion density and a diffusive-regularized boundary. The linear stability of a single-contour circular cloud is studied and a new evolution equation for modal amplitudes is obtained which has the property that the wave number of maximum amplitude decreases with time (downward cascade). Analytic expressions show that large clouds evolve more slowly and appear more dissipative.

Contour dynamics for the Euler equations in two dimensions

We present a contour dynamics algorithm for the Euler equations of fluid dynamics in two dimensions. This is applied to regions of piecewise-constant vorticity within finite-area-vortex regions (FAVR's). Essentially, this reduces the dimensionality by one and we are treating the interaction of closed polygonal contours whose nodes are advected by the total fluid motion computed self-consistently. A leapfrog centered scheme is used for temporal advancement. Computer simulation results are given for two and four like-signed interacting FAVR's. In all cases wavelike surface deformations are observed. If the distance between FAVR's is comparable to their extent (“diameter”), these surface deformations are large. They play an essential role in the observed coalescence of FAVR's.

Contour dynamics for numerical fluid-flow calculations

The numerical solution of problems in fluid dynamics is discussed in terms of a Dynamics-of-Contours (DOC) methodology. Elements of the flow are represented by contour points (or lines or surfaces) that move relative to the fluid in a manner that rigorously conserves mass, momentum and energy, and also represents the transient dynamics. The principal advantage of this technique is that it automatically gives fine resolution of flow features in regions of detailed structure, and coarse resolution where not much is occurring. Several examples are presented to illustrate the results of test problems.