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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 Normal fluid Superfluid (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

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**Superfluid turbulence. Results. [ Finne et al**

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

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**Forces on vortices Mutual friction parameters d, d’ Magnus force**

Force from the normal component Mutual friction parameters d, d’

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**Hall and Vinen mutual friction parameters**

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**

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**

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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 W=0.21rad/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 -3 2D (vortex line) density L=nl=l -2 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<<Tc . No turbulence. Superconductors: High vortex friction; q>>1 except for very clean materials, l>>(EF /Tc)x , and low temperatures. Superfluid He II: Low vortex friction: q<<1 except for temperatures very close to Tl . Unstable towards turbulence.

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Carlo F. Barenghi School of Mathematics University of Newcastle, UK Exotic turbulence opportunities in superfluid helium.

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