Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows

Abstract

The complex flow situations that regularly develop in a two-dimensional vortical flow h ave tradionally, indeed almost exclusively, been studied using Eulerian numerical methods, particularly spectral methods. These Eulerian methods have done remarkably well at modelling low to moderate Reynolds number flows.However, at the very high Reynolds numbers typical of geophysical flows, Eulerain methods run into difficulties, not the least of which is sufficient spatial resolutions. On the other hand, Lagrangian methods are and contour dynamics methods, are inherently inviscid. It would appear, therefore, that Lagrangian mehtods ideally suited for the modelling of flows at very high Reynolds numbers. Yet in practice, Lagrangian methods have themselves been limited by the frequent, extraordinary increase in the spatial complexity of inviscid flows. As a consequence, Lagrangian methods have been restricted to relatively simple flows which remain simple. Recently, an extension of contour dynamics, “contour surgery”, has enabled the modelling of complex inviscid flows in wholly Lagrangian terms, This extension overcomes the buildup of small-scale structure by truncating, in physical space, the modelled range of scales. The results of this truncation, or “surgery”, is to make feasible the computation of flows having a range of scales spanning four to five orders of magnitude, or one to two orders of magnitude greater than ever considered by Eulerian-Lagrangian methods. This paper discusses the history of contour dynamis which led to contour surgery, gives details of the contour surgery algorithm for planar, cylindrical, spherical, and quasi-geostrophic flow, presents new results obtained with high-resolutions calculations, including the first every comparison between contour surgery and a traditional pseudo- spectral method, and outlines some outstanding problems facing dynamics/ surgery.