Vortex instability and the onset of superfluid turbulenceN.B. Kopnin
Low Temperature Lab., HUT, Finland,L.D. Landau Institute for Theoretical Physics, Moscow.
Collaboration
Experiment: M. Krusius, V. Eltsov, A. Finne, HUTL. 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
(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
Normal fluid Superfluid
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 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 velocity 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 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
linesCompetition between vortex
multiplication and their extraction into the bulk
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
Total vortex evolution
MF parameters
Superflow velocity
The counterflow velocityThe self-induced velocity
Finally,
Similarly to the Vinen equation (1957)
Another approach: Vorticity equationNavier—Stokes equation
Vorticity equation
In superfluids: mutual friction force instead of viscosity
Superfluid vorticity equation
Averaged over random vortex loops assuming
Vortex instabilityEvolution 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) , and low temperatures.
No turbulence. Superfluid He II: Low vortex friction: q<<1 except for
temperatures very close to T .
Unstable towards turbulence.
