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Vortex instability and the onset of superfluid turbulence

Vortex instability and the onset of superfluid turbulence. N.B. Kopnin Low Temperature Lab., HUT, Finland, L.D. Landau Institute for Theoretical Physics, Moscow. Collaboration Experiment: M. Krusius, V. Eltsov, A. Finne, HUT L. Skrbek, Charles University, Prague Theory:

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Vortex instability and the onset of superfluid turbulence

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  1. Vortex instability and the onset of superfluid turbulence N.B. Kopnin Low Temperature Lab., HUT, Finland, L.D. Landau Institute for Theoretical Physics, Moscow. Collaboration Experiment: M. Krusius, V. Eltsov, A. Finne, HUT L. Skrbek, Charles University, Prague Theory: T. Araki, M. Tsubota, Osaka University G. Volovik, HUT

  2. Contents • New class of superfluid turbulence. Experiment in superfluid He 3 B: Onset independent of the Reynolds number Superfluid Normal fluid (Res is the ratio of superflow and the Feynman critical velocity) • Theoretical model for vortex instability • Results of numerical simulations • Mutual-friction controlled onset of turbulence

  3. Superfluid turbulence. Results.[ Finne et al., Nature 424, 1022 (2002)]

  4. Forces on vortices • Magnus force • Force from the normal component Mutual friction parameters d, d’

  5. couples the velocities: Force balance Hall and Vinen mutual friction parameters Mutual friction force on the superfluid

  6. Mutual friction parameters in He-3 B.Microscopic theory [Rep. Prog. Phys (2002)] Mutual friction parameters Distance between the CdGM states Effective relaxation time

  7. Mutual friction parameters in He-3 B.Experiment [Hook, Hall, et al. (1997)]

  8. Numerical simulations • Vortex evolution is integrated from [Schwartz 1988] • The local superflow: all the Biot—Savart contributions • Boundary conditions: Image vortices • Vortex interconnections for crossing vortices

  9. Numerical results: High temperatures, high friction Parameters • Rotation velocityW=0.21rad/s • Superfluid “Reynolds number” • Temperature and the MF ratio

  10. Numerical results:Low temperatures, low friction • Rotation velocity and the Reynolds number • Temperature and the MF ratio • Evolution time Enormous multiplication of vortices !

  11. Numerical simulations of vortex evolution in a rotating container

  12. Model for the onset of turbulence Distinguish two regions • Multiplication region: • high flow velocity, high density of entangled vortex loops, • vortex crossings and interconnections • Rest of the liquid (bulk): • low flow velocity, polarized vortex lines • Competition between vortex multiplication and their extraction into the bulk

  13. Multiplication region • Loop size l • 3D vortex density n~l -3 • 2D (vortex line) density L=nl=l -2 • Multiplication due to collisions and reconnections • Extraction due to inflation

  14. Total vortex evolution MF parameters Superflow velocity The counterflow velocity The self-induced velocity Finally, Similarly to the Vinen equation (1957)

  15. Another approach: Vorticity equation Navier—Stokes equation Vorticity equation In superfluids: mutual friction force instead of viscosity Superfluid vorticity equation Averaged over random vortex loops assuming

  16. Vortex instability Evolution equation Two regimes of evolution: • Low temperatures, Solution saturates at • Instability towards turbulent vortex tangle • Higher temperatures, • No multiplication of vortices

  17. Summary:Superfluid turbulence in other systems • He-3 A: High vortex friction; q>>1 except for very low temperatures T<<Tc . No turbulence. • Superconductors: High vortex friction; q>>1 except for very clean materials, l>>(EF /Tc)x , and low temperatures. No turbulence. • Superfluid He II: Low vortex friction: q<<1 except for temperatures very close to Tl . Unstable towards turbulence.

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