1. 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

2. 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

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

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

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

6. 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

7. Mutual friction parameters in He-3 B. Experiment [Hook, Hall, et al

8. 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

9. Numerical results: High temperatures, high friction
Parameters
Rotation velocity W=0.21rad/s
Superfluid “Reynolds number”
Temperature and the MF ratio

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

11. Numerical simulations of vortex evolution in a rotating container

12. 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

13. 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

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

15. 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

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

17. 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.