A geometrical viewpoint of inviscid incompressible fluid dynamics highlights vorticity as the key field which generates the velocity field and is in turn transported, stretched and rotated, that is Lie-dragged, in the fluid flow. In this setting it is most natural to consider the velocity as a vector field, the momentum as a one-form (or co-vector) field, and the vorticity as a two-form field, making use of the metric and corresponding volume form. Such a view point is not only helpful in the abstract, but also gives practical ways of writing down the equations for vortex motion in a Lagrangian framework, where the coordinate system follows the evolution of a slender vortex. This talk will describe how one can write down the equations for vortex motion using such a coordinate system, which is general is both non-orthogonal and time-dependent. We will apply the framework to recover classic results on the motion of slender vortex rings.
By Geoffrey Vallis
By Luca Melzi
By Beatrice Pelloni
