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: T. Araki, M. Tsubota, Osaka University G. Volovik, HUT

Contents ØNew 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) ØTheoretical 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’

Force balance couples the velocities: 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 Cd. GM 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 velocity W=0. 21 rad/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 q Multiplication region: • high flow velocity, high density of entangled vortex loops, • vortex crossings and interconnections q Rest of the liquid (bulk): • low flow velocity, polarized vortex lines ð Competition between vortex multiplication and their extraction into the bulk

Multiplication region • Loop size l • 3 D vortex density n~l -3 • 2 D (vortex line) density L=nl=l -2 q Multiplication due to collisions and reconnections q 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: q. Low temperatures, Solution saturates at ØInstability towards turbulent vortex tangle q. Higher temperatures, ØNo multiplication of vortices

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