Introduction to Vortex Dynamics

This book has been written for applied mathematicians, theoretical physicists and engineers who are interested in the modern developments of vortex dynamics, and their applications to turbulent as well as laminar flows in various natural and technological settings. Many recent developments in these fields have been reported in the free-wheeling style of much of theoretical physics, and the author has collected the diverse material and organized it into a coherent and readable account.The book is divided into four chapters: (i) introduction and overview; (ii) two-dimensional Hamiltonian models of vorticity, such as the Kirchhoff model for point vortices, and infinite-dimensional models for two-dimensional vortex patches; (iii) the dynamics and statistics of three-dimensional vortex models; and (iv) the dynamics and stability of swirling vortical flows. In the second chapter the Hamiltonian equations of motion for point vortices are discussed, and some of their dynamical properties, such as integrals of motion, symmetries, relative equilibria, and stability, are established. The third chapter considers the recent program of studying the statistical properties of three-dimensional vortex models with an eye on the infamous fluid turbulence problem. One of the main models is a finite three-dimensional Hamiltonian model for the Euler equations of ideal fluids. The last chapter concerns an important field that is largely known as vortex breakdown but is actually a complex of related vortical flows and their instabilities. Symmetrical vortex configurations such as axisymmetric and circular vortex rings are discussed and their properties calculated.