Recent Progress in Topological Fluid Dynamics

Slides from the Ravello School on Mathematical Physics (2017)

  1. Topological interpretation of helicity
    • Coherent structures and topological fluid mechanics
    • Diffeomorphisms and topological equivalence
    • Kinetic and magnetic helicity of flux tubes
    • Gauss linking number
    • Călugăreanu invariant and geometric decomposition
  2. Vortex knots dynamics and momenta of a tangle
    • Localized Induction Approximation (LIA) and Non-Linear Schroedinger (NLS) equation
    • Integrable vortex dynamics and LIA hierarchy
    • Torus knot solutions to LIA
    • Linear and angular momentum in terms of signed area information
  3. Magnetic knots and groundstate energy spectrum
    • Magnetic relaxation
    • Topology bounds the energy
    • Inflexional instability of magnetic knots
    • Constrained minimization of magnetic energy
    • Groundstate energy spectra of magnetic knots and links
    • Bending energy spectra: Magnetic vs. elastic systems
  4. Topological transition of soap films
    • Seifert surfaces and soap films
    • The Plateau problem: the catenoid
    • Topological transition from 1-sided to 2-sided surface
    • Local analysis of the twisted saddle catastrophe
    • Geometry and energy considerations during transition
  5. Helicity change under reconnection: the GPE case
    • Reconnection and change of helicity
    • Application to solar cornal loops
    • Writhe conservation under reconnection
    • Cascade of quantum vortex links under Gross-Pitaevskii equation
    • Iso-phase surfaces as Seifert surfaces
    • Evolution of iso-phase surfaces of minimal area
  6. Topological decay measured by knot polynomials
    • Knot polynomials as new tools in turbulence research
    • Polynomial skein relations
    • Adapted polynomials as new invariants of fluid mechanics
    • Vortex knot cascade detected by monotonically decreasing sequence of polynomial numerical values
    • HOMFLYPT as best quantifier of topological complexity