1. Two scale modeling of superfluid turbulence Tomasz Lipniacki
He II:
-- normal component,
-- superfluid component,
-- superfluid vortices.
Relative proportion of normal fluid and superfluid as a function of temperature.
Solutions of single vortex motion in LIA
Fenomenological models of anisotropic
turbulence
vortex line core diameter ~ 1Å

2. Localized Induction Approximation
c - curvature
- torsion

3. Quasi static solutions:
Ideal vortex
Totally integrable, equivalent to non-linear Schrödinger equation.
Countable (infinite) family of invariants:
Quasi static solutions:

4. Hasimoto soliton on ideal vortex (1972): F i g u r e 1 2

5. Kida 1981
General solutions of
in terms of elliptic integrals including shapes such as
circular vortex ring
helicoidal and toroidal filament
plane sinusoidal filament
Euler’s elastica
Hasimoto soliton

6. Quasi static solutions for quantum vortices ?
Quantum vortex shrinks:

7. Quasi-static solutions
Lipniacki JFM2003
Frenet Seret equations

8. In the case of pure translation we get analytic solution
For
where
Solution has no self-crossings

9. Quasi-static solution: pure rotation
Vortex asymptotically
wraps over cones
with opening angle
for in cylindrical coordinates
where

10. Quasi-static solution: general case
For vortex wraps over paraboloids

11. Results in general case
I. 4-parametric class of solutions, determined by
initial condition
II. Each solution corresponds to a specific isometric transformation G(t) related analytically to initial condition.
III. The asymptotic for
is related analytically via transformation G(t) to initial condition.

12. Shape-preserving solutions
Lipniacki, Phys. Fluids. 2003
Equation
is invariant under the transformation
If the initial condition is scale-free, than the solution is self-similar
In general scale-free curve is a sum of two logarithmic spirals on coaxial cones; there is four parametric class of such curves.

13. Shape-preserving solutions
solutions with
decreasing scale

14. Analytic result: shape-preserving solution
In the case when transformation is a pure homothety we get analytic solution in implicit form:

15. Shape preserving solution: general case
Logarithmic spirals on cones

16. Results in general case
I. 4-parametric class of solutions, determined by initial condition
II. Each solution corresponds to a specific similarity transformation
III. The asymptotics for
is given the initial condition of the original problem (with time).

17. Self-similar solution for vortex in Navier-Stokes?
N-S is invariant to the same transformation
The friction coefficient corresponds to

18. Wing tip vortices

19. Macroscopic description
-- Euler equation for superfluid component
-- Navier-Stokes equation for normal component
Coupled by mutual friction force
where
and L is superfluid vortex line density.

20. Microscopic description of tangle superfluid vortices
Aim
Assuming that quantum tangle is “close” to statistical equilibrium
derive equations for quantum line length density L and anisotropy
parameters of the tangle.
Statistical equilibrium?
Two cases
I – rapidly changing counterflow, but uniform in space
II – slowly changing counterflow, not uniform in space

21. Model of quantum tangle
“Particles” = segments of vortex line of length equal to characteristic
radius of curvature.
Particles are characterized by their tangent t, normal n, and binormal b vectors.
Velocity of each particle is proportional to its binormal (+collective motion).
Interactions of particles = reconnections.

22. Reconnections 1. Lines lost their identity:
two line segments are replaced by two new line segments
2. Introduce new curvature to the system
3. Remove line-length

23. Motion of vortex line in the presence of counterflow
Evolution of its line-length is

24. Evolution of line length density
where
average curvature
average curvature squared I
Average binormal vector (normalized)

25. Rapidly changing counterflow
Generation term: polarization of a tangle by counterflow
Degradation term: relaxation due to reconnections

26. Generation term: polarization of a tangle by counterflow
evolution of vortex ring
of radius R and orientation
in uniform counterflow
unit binormal
Statistical equilibrium assumption: distribution of f(b) is the most
probable distribution giving right I i.e.

27. Degradation term: relaxation due to reconnections
Reconnections produce curvature but not net binormal
Total curvature of the tangle
Reconnection frequency
Relaxation time:

28. Finally
Lipniacki PRB 2001

29. Model prediction: there exists a critical frequency
of counterflow above which tangle may not be sustained

30. Quantum turbulence with high net macroscopic vorticity
Vinen-Schwarz equation
We assume now
Let
denote anisotropy of the tangent
Vinen type equation for turbulence with net macroscopic vorticity

31. Drift velocity
where

32. Mutual friction force
drag force
where
finally

33. Helium II dynamical equations
Lipniacki EJM 2006

34. Specific cases: stationary rotating turbulence
Pure heat driven turbulence
Pure rotation
„Sum”
Fast rotation
Slow rotation

35. Plane Couette flow V- normal velocity U-superfluid velocity
q- anisotropy
l-line length density

36. “Hypothetical” vortex configuration

37. Summary and Conclusions
Self-similar solutions of quantum vortex motion in LIA
Dynamic of a quantum tangle in rapidly varying counterflow
Dynamic of He II in quasi laminar case.
In small scale quantum turbulence, the anisotropy of
the tangle in addition to its density L is key to analyze
dynamic of both components.

39. Non-linear Schrodinger equation (Gross, Pitaevskii)
For
Madelung Transformation
superfluid velocity
superfluid density

40. Modified Euler equation
With
pressure
Quantum stress

41. Mixed turbulence
Kolgomorov spectrum for each fluid
- energy flux per unit mass

44. Two scale modeling of superfluid turbulence Tomasz Lipniacki
He II:
-- normal component,
-- superfluid component,
-- superfluid vortices.
Relative proportion of normal fluid and
superfluid as a function of temperature.
vortex line