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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 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
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
Superfluid turbulence. Results.[ Finne et al., Nature 424, 1022 (2002)]
Forces on vortices • Magnus force • Force from the normal component Mutual friction parameters d, d’
couples the velocities: Force balance Hall and Vinen mutual friction parameters Mutual friction force on the superfluid
Mutual friction parameters in He-3 B.Microscopic theory [Rep. Prog. Phys (2002)] Mutual friction parameters Distance between the CdGM states Effective relaxation time
Mutual friction parameters in He-3 B.Experiment [Hook, Hall, et al. (1997)]
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
Numerical results: High temperatures, high friction Parameters • Rotation velocityW=0.21rad/s • Superfluid “Reynolds number” • Temperature and the MF ratio
Numerical results:Low temperatures, low friction • Rotation velocity and the Reynolds number • Temperature and the MF ratio • Evolution time Enormous multiplication of vortices !
Numerical simulations of vortex evolution in a rotating container
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
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
Total vortex evolution MF parameters Superflow velocity The counterflow velocity The self-induced velocity Finally, Similarly to the Vinen equation (1957)
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
Vortex instability Evolution equation Two regimes of evolution: • Low temperatures, Solution saturates at • Instability towards turbulent vortex tangle • Higher temperatures, • No multiplication of vortices
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.