3. Vortex dynamics, knots and links - MatAppricca/teaching/Torino3.pdf · 3. Vortex dynamics, knots and links 3.1 Elements of vortex dynamics: Navier-Stokes and Euler equations, localized

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3. Vortex dynamics, knots and links

3.1 Elements of vortex dynamics: Navier-Stokes and Euler equations, localized vorticity,Poisson equation, Biot-Savart law, dynamics of thin filaments, vortex ring solutionand localized induction approximation (LIA).

3.2 Vortex knots and links: vortex links, Thomson’s stability results, torus knot solutionsunder LIA, Kida class and stability criterium, direct numerical simulations.

3.3 Conservation laws and topology: Helmholtz conservation laws, scalar and vectorinvariants (Lagrangian quantities, frozen-in fields, Frobenius invariants), kinetichelicity and linking numbers.

3.4 Articles iincluded:Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech.

35,117-129.Arnold, V.I. 1974 The asymptotic Hopf invariant and its application. Proc. Summer

School in Differential Equations, Akademiya ArmSSR., Erevan [translated in:1986 Sel. Math. Sov. (5) 4, 327-345].

Further reading: a good introduction to vortex dynamics is:Saffman, P.G. 1992 Vortex Dynamics. Cambridge University Press.The role of invariants in ideal fluid mechanics is discussed in:Tur, A. & Yanovsky, V. 1993 Invariants in dissipationless hydrodynamic media. J.

Fluid Mech. 248, 67-106.A very first discussion of the invariance of kinetic helicity is given by:Moreau, J.J. 1977 Sur les integrales premieres de la dynamique d’un fluide parfait

barotrope et le theorème de Helmholtz-Kelvin. Seminaire d’Analyse Convexe(Montpellier) 7, 1-25.

A review on the role of kinetic helicity in real fluids is given by:Moffatt, H.K. & Tsinober, A. 1992 Helicity in laminar and turbulent flow. Annu.

Rev. Fluid Mech. 24, 281-312.

Chapter 3 – Vortex dynamics, knots and links

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3.1 Elements of vortex dynamics


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3. Vortex dynamics, knots and links - MatAppricca/teaching/Torino3.pdf · 3. Vortex dynamics, knots and links 3.1 Elements of vortex dynamics: Navier-Stokes and Euler equations, localized