Edward A. Overman

Evolution and merger of isolated vortex structures

Numerical simulations of the instability, merger, and breaking of two piecewise‐constant finite‐area vortex regions (FAVR’s) are presented. An improved contour dynamical algorithm with node insertion‐and‐removal to maintain the a priori accuracy is used. Corotating ’’V states’’ (symmetric steady‐state FAVR’s) were found to be unstable when properly perturbed if their centroid‐effective radius ratio, x/R, is <1.6, thereby verifying an estimate of Saffman and Szeto. This causes the FAVR’s to approach at an exponential rate, merge, and reform into a stable perturbed elliptical structure with filamentary arms (to conserve angular momentum). For larger x/R ratios, regular perimeter oscillations were observed and estimates of an eigenfrequency of the perturbed stable V states were obtained. When regions of different vorticity density merge, the larger‐density region is eventually entrained within the smaller‐density region. These simulations elucidate the self‐consistent close interactions of isolated vortex ...

Steady-state solutions of the euler equations in two dimensions: Rotating and translating V-states with limiting cases. I. Numerical algorithms and results

New second- and third-order algorithms are presented for calculating translating and rotating steady-state solutions of the 2D incompressible Euler equations (which we call V-states). These are piecewise constant regions of vorticity and the contours bounding them are obtained by solving iteratively a nonlinear integro-differential equation. New limiting contours with corners are obtained and compared with local analytical solutions. The precise results correct mistakes for limiting contours that were previously given.

Coherence and chaos in the driven damped sine-Gordon equation: measurement of the soliton spectrum

Abstract A numerical procedure is developed which measures the sine-Gordon soliton and radiation content of any field (φ, φt) which is periodic in space. The procedure is applied to the field generated by a damped, driven sine-Gordon equation. This field can be either temporally periodic (locked to the driver) or chaotic. In either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized (soliton wave-train) modes. The numerical procedure quantitatively identifies the presence, number and properties of these soliton wave-trains. For example, an increase of spatial symmetry is accompanied by the injection of additional solitons into the field.

Regularization of contour dynamical algorithms. I. tangential regularization

Abstract Contour dynamical methods are being applied to a variety of inviscid incompressible flows in two dimensions. These generalizations of the “waterbag” method provide simplified models for following the evolution of contours x (J) that separate regions of constant density which are the sources of the flow. The inviscid evolution of contour j , x t (j) is usually an area-preserving map. For physically unstable problems, a piecewise-constant initial condition may result in an ill-posed problem. That is, contours may rapidly grow in perimeter and/or develop singularities and numerically induced small-scale structures in a finite time. To avoid such problems and model realistic weakly dissipative or weakly dispersive flows, contour regularization procedures are required. Dissipative and dispersive tangential regularization procedures for one contour are introduced. A special case of the former, namely ẋ t = μx ss , corresponds in lowest order to a linear diffusion operator in two dimensions. The contour is parameterized with arc length using cubic splines and an adaptive curvature controlled node adjustment algorithm is used. A modified Crank-Nicolson method is used to solve the discrete representation of the full system, x t = x t +μx ss . Numerical results are given for the evolution of initially elliptical shapes according to prescribed area-preserving maps. The numerical results for area evolution agree with analytical results.

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 as...

Contour dynamics for the Euler equations: curvature controlled initial node placement and accuracy

Abstract We have performed a systematic study of several contour dynamical algorithms for the Euler equations for short times. We have used the Kirchhoff elliptical vortex alone and subject to weak perturbations. We have found that if the initial placement of nodes is such that the internodal distance is proportional to (curvature) −P where p ≈ 1 3 , then errors in short time calculations are minimized. This follows because the node density is invariant in time.

Computational vortex dynamics in two and three dimensions

Abstract We present a discussion of some solutions and outstanding problems of dissipationless and nearly dissipationless flows in two and three dimensions. In particular, we review two-dimensional steady-state solutions, linear stability and nonlinear evolutions including axisymmetrization of isolated vortex regions and merger or condensation of like-signed vorticity regions. These results are obtained from pseudo-spectral codes, contour dynamical (CD) codes and a new Hamiltonian moment-model of the Euler equations.

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

Abstract 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.