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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

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Contents New class of superfluid turbulence. Experiment in superfluid He 3 B: Onset independent of the Reynolds number (Re s 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 Normal fluid Superfluid

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

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

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

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

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

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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

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Numerical results: High temperatures, high friction Parameters Rotation velocity rad/s Superfluid “Reynolds number” Temperature and the MF ratio

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

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Numerical simulations of vortex evolution in a rotating container

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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

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

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

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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

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

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Summary: Superfluid turbulence in other systems He-3 A: High vortex friction; q>>1 except for very low temperatures T<

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