Transcript Dynamics of superfluid He 3 and superconductors

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
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
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 W=0.21rad/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
lines
 Competition between vortex
multiplication and their extraction into
the bulk
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
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:
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)x , and low temperatures.
No turbulence.
 Superfluid He II: Low vortex friction: q<<1 except for
temperatures very close to Tl .
Unstable towards turbulence.