2-D Vortices and Contour Dynamics

P.V.C. Vosbeek

The contour dynamics method is a numerical method for simulating the evolution of vortices (regions of rotating fluid) in two-dimensional flows. In the oceans and atmosphere vortices are abundant. Examples are the high and low-pressure areas on weather maps and the polar vortex. The motion on larger scales of these vortices can be considered two-dimensional. Under typical atmospheric and oceanic conditions this two-dimensionality is governed by planetary rotation or by density stratification (which e.g. in the oceans is caused by salinity and temperature gradients and in the atmosphere by a temperature gradient). Experiments and numerical simulations play an important role in understanding these two-dimensional flows.

Example: formation of a tripole

Contour dynamics is based on the observation that, for an incompressible, inviscid fluid, the evolution of a patch of uniform vorticity is fully described by the evolution of its bounding contour. The method is not limited to a patch of uniform vorticity: several contours can be nested, in order to approximate a given continuous vorticity distribution by a step-function. The velocity in the fluid, and in particular at the contours of vorticity-discontinuity, can be computed using Green's function of the associated Laplace operator. This amounts to calculating a sum of contour integrals. From the velocity field, the contours at a next time point can be determined. In general, the shape of the contours becomes increasingly complex with time. During the calculations points have to be added and removed in a proper way to approximate the contours nicely.

The project is a co-operation between the Scientific Computing Group of the Department of Mathematics and Computing Science, and the Vortex Dynamics Group of the Department of Physics, within the framework of the Stevin Centre.

For related subjects see the site of the Vortex Dynamics Group.