| In this work a new model is proposed to describe the motion of N localized vortex structures in two dimensional incompressible viscous flow. We will use the velocity-vorticity representation, commonly known as the vorticity equation, of incompressible flow which has the particular advantage in two dimensions of reducing to a scalar equation.;An early model of two-dimensional inviscid flow in terms of moving point vortices was developed by Kirchhoff and Helmholtz. While the Helmholtz-Kirchhoff point vortex model captures many of the basic physical phenomena observed in 2D rotational flows, experiments with even simple vortex configurations exhibit complications far beyond the point vortex predictions. This new model replaces the Dirac point vortex with a Hermite expansion to represent each vortex. This is a natural choice of representation as the leading order term in the expansion is a Gaussian which is both a common regularization of a delta function and, even more importantly, is an exact solution of the 2D vorticity equation. Furthermore these Hermite functions have recently been shown by Gallay and Wayne to be related to invariant manifolds in the phase space of the vorticity equation which govern its long time asymptotics. With mild restrictions on the initial data we prove that the expansions are convergent for all time. The model reduces the 2D vorticity equation to a system of quadratic ordinary differential equations (ODEs) that track the centers of each vortex along with the evolution of each Hermite moment. As a first application of this approach a single Hermite expansion is used to model a co-rotating pair of vortices to greatly improve the accuracy of the far field acoustic pressure.;Additionally, leading-order corrections to the frequency of rotation due to viscosity and finite core size are derived in the case of two well separated like-signed vortices. Finally, combinatorial formulas for the coefficients of the resulting ODEs are derived to improve the numerical implementation of the model. With these combinatorial formulae classic numerical experiments such as vortex merger are computed with our new model and results are discussed. |
