A fast contour dynamics method for many-vortex calculations in two-dimensional flows

A new computational method extending the contour dynamics/surgery (CS) algorithm is announced which gives typical speed-up factors of two orders of magnitude in calculations of flows involving many interacting vortices. The method makes use of an alternative expression for the velocity field in the exterior of a vortex that takes the form of a rapidly convergent series. Each term in this series can be expressed as a complex coefficient divided by the complex distance x + iy from the vortex center. The complex coefficient, or moment, is a real number pair that describes shape characteristics of the vortex (e.g., circulation, eccentricity, etc.). In numerical calculations, where accuracy is necessarily limited, it is frequently sufficient to retain only the leading-order terms in this series, particularly for a gas of well-separated vortices. The real computational gain is made, however, by reexpanding the series of all vortices that are sufficiently separated from a given vortex as a single, truncated series in positive powers of the complex distance from this vortex's center. The coefficients of this series involve only the moments of the other vortices and their centroid separation from the given vortex. The leading-order truncation, for instance, simply gives point vortex dynamics, except that self- or close-range interactions are computed using the full contour integral expression of contour dynamics (hence, all vortices retain nontrivial spatial structure, vital to a proper dynamical description of close-range interactions). In general, the optimal truncation depends on a dynamic balance between the cost of all moment computations and the cost of all contour integrations. This method, called "moment-accelerated contour surgery," which is briefly outlined above for the planar case, has a direct analog in spherical geometry. There are also extensions to generalized two-dimensional (2-D) flows having more general linear operator relationships between streamfunction and vorticity. Details are provided for quasigeostrophic flow.