This book presents fundamental concepts and seminal results to the study of vortex filaments in equilibrium. It also presents new discoveries in quasi-2D vortex structures with applications to geophysical fluid dynamics and magnetohydrodynamics in plasmas. It fills a gap in the vortex statistics literature by simplifying the mathematical introduction to this complex topic, covering numerical methods, and exploring a wide range of applications with numerous examples. The authors have produced an introduction that is clear and easy to read, leading the reader step-by-step into this topical area. Alongside the theoretical concepts and mathematical formulations, interesting applications are discussed. This combination makes the text useful for students and researchers in mathematics and physics.

We give a brief review of the asymptotic theory of slender vortex filaments with emphases on the choices of scalings characterizing the physical problems and the corresponding assumptions and/or restrictions introduced in the formation of the asymptotic theory of Callegari and Ting (1978) and its extension by Klein and Ting (1992). In particular, the slender filaments considered are assumed to be forming loops or tori. Because of this restriction, the theory is not applicable to the trailing vortex system of a rotorcraft. We describe the multiple length scales characterizing the vortex system, formulate the expansion scheme, derive the governing equations and then identify the assumptions or restrictions inherent in the multi-scale analysis and needed for the validity of the asymptotic theory of the trailing vortex system. Ting, Lu and Tung, Chee Ames Research Center

We give a brief review of the asymptotic theory of slender vortex filaments with emphases on the choices of scalings characterizing the physical problems and the corresponding assumptions and/or restrictions introduced in the formation of the asymptotic theory of Callegari and Ting (1978) and its extension by Klein and Ting (1992). In particular, the slender filaments considered are assumed to be forming loops or tori. Because of this restriction, the theory is not applicable to the trailing vortex system of a rotorcraft. We describe the multiple length scales characterizing the vortex system, formulate the expansion scheme, derive the governing equations and then identify the assumptions or restrictions inherent in the multi-scale analysis and needed for the validity of the asymptotic theory of the trailing vortex system. Ting, Lu and Tung, Chee Ames Research Center ...