Interfacial Vorticity Research Articles

Finite-Amplitude Long Waves on Coastal Currents

Abstract Two models of coastal currents are described that allow fully nonlinear wavelike solutions for the limit of long waves. The first model is an adaptation of a model used by Yi and Warn for finite-amplitude βplane Rossby waves in a channel. It utilizes a particular choice of continental shelf topography to obtain a nonlinear evolution equation for long waves of finite amplitude. The second model describes the waves that form at the vorticity interface between two regions of constant potential vorticity. Again a nonlinear evolution equation is obtained for long waves of finite amplitude. For both model equations, numerical results are presented and compared with the corresponding results for the BDA equation, which is the weakly nonlinear limit for both models.

Instability and filamentation of finite-amplitude waves on vortex layers of finite thickness

We study the instability of finite-amplitude waves on uniform vortex layers of finite thickness bounded by a plane rigid surface. A weakly nonlinear analysis of vorticity interface perturbations, and spectral stability calculations using the full equations of motion, together show that steady progressive waves are unstable to general subharmonic pertubations in the range 0.094 < d/λ < 1.7, where d is the mean layer thickness and λ is the primary wavelength. The relevance of this instability to ultimate interface filamentation is tested by performing several numerical contourdynamical simulations of the nonlinear interface evolution for initial disturbances consisting of the finite amplitude wave plus eigenfunctions obtained from the spectral calculations. The results indicate that within the band of unstable wavelengths, small perturbations to the steady non-uniform flow given by the finite amplitude wave motion (vortex equilibrium) are able to grow in magnitude, until at a time tf, the wave extremum encounters a hyperbolic critical point of the velocity field after which filamentation occurs. Arguments are put forward based on the unsteady simulations with the purpose of identifying the preferred frame of reference for viewing the kinematical events controlling the filamentation process. An estimate for tf is then made, and the mechanism of filamentation found is discussed in relation to the recently proposed nonlinear-cascade mechanism of Dritschel (1988a).

The repeated filamentation of two-dimensional vorticity interfaces

Undular disturbances of arbitrarily small steepness to the boundary of a circular patch of uniform vorticity are found to give way to a qualitatively different type of behaviour after a time inversely proportional to the initial wave steepness squared. Repeatedly, thin filaments are drawn out at a frequency of nearly half the vorticity jump across the interface (the intrinsic frequency of linear waves on the interface), and the interaction between the filaments and the undulating boundary generate new disturbances from which still greater numbers of filaments of increasingly complicated shape are drawn out. The vortex boundary thereby experiences an extraordinary, continual and apparently irreversible growth in complexity.Essentially the same phenomenon occurs on all vortex-patch equilibria, e.g. the Kirchhoff elliptical vortex, with even greater complexity. The steepening of a disturbance on a non-circular vortex proceeds faster than that on a circular vortex, because of the varying mean strain and shear seen by the disturbance as it travels around the vortex boundary.Generalizations to more than one vorticity interface, to flows on the surface of a sphere, and to sharp but not infinitely sharp vorticity gradients are also discussed. The results support the view that almost any two-dimensional, inviscid, incompressible flow with large vorticity gradients will exhibit repeated filamentation.