Biot-Savart induction effects of torus knots and unknots.

by Dr Chiara Oberti

Tuesday 30 Oct 2018, 12:00 → 13:00 Asia/Shanghai

3304 (北京工业大学数理学院)

In this talk we discuss in detail the geometric and topological aspects associated with the Biot-Savart induction effects due to a steady field in the shape of torus knots and unknots. The physical knot is identified with a filament of negligible cross-section in an ideal fluid, and torus knots are parametrized by standard equations. Since in ideal flows the Biot-Savart law is applied when the source field is either vorticity, electric current or magnetic field, this work is of interest for applications in fluid dynamics as well as magnetohydrodynamics. I will show that the writing number, a geometric quantity that can be estimated by algebraic computation of apparent crossings, is a good proxy to the helicity of magnetic torus knots/unknots with dominant toroidal coils, that are a good model for solar coronal loops. Since in the limit when the induction point approaches the source field, the Biot-Savart integral becomes singular, I will apply the analytical prescription of Moore and Saffman (1972) to regularize it, and determine the influence of knot geometry and topology on the dynamics of vortex torus knots/unknots.