1. Dynamics and Statistics of Quantum Turbulence at Low Temperatures
Michikazu Kobayashi

2. Contents Motivation and Introduction.
Model : Gross-Pitaevskii Equation.
Simulation of Quantum Turbulence.
Quantum Turbulence of Two Component Fluid.

3. 1, Motivation and Introduction
Question : Does quantum turbulence have a similarity with that of conventional fluid? ?

4. Quantum turbulence has the similarity with classical turbulence
Why is This Important?
Quantum turbulence has the similarity with classical turbulence
Quantum turbulence with quantized vortices can be an ideal prototype of turbulence!

5. Turbulence and Vortices : Kolmogorov Law of Classical Turbulence
Classical turbulence has the Kolmogorov law
In the energy-containing range, energy is injected to system at scale l0

6. Turbulence and Vortices : Kolmogorov Law of Classical Turbulence
Classical turbulence has the Kolmogorov law
In the inertial range, the scale of energy becomes small without being dissipated and having Kolmogorov energy spectrum E(k).
C : Kolmogorov constant

7. Turbulence and Vortices : Kolmogorov Law of Classical Turbulence
Classical turbulence has the Kolmogorov law
In the energy-dissipative range, energy is dissipated by the viscosity at the Kolmogorov length lK

8. Turbulence and Vortices : Kolmogorov Law of Classical Turbulence
Classical turbulence has the Kolmogorov law
: energy injection rate
: energy transportation rate
P(k) : energy flux from large to small k
e : energy dissipation rate

9. Turbulence and Vortices : Richardson Cascade
Kolmogorov law is believed to be sustained by the self-similar Richardson cascade of eddies.
Large eddies are nucleated
Eddies are broken up to smaller ones
Small eddies are dissipated

10. Difficulty of Studying Classical Turbulence
Classical eddies are indefinite!
Vorticity w = rot v takes continuous value.
Circulation k = ∳ v ・ds takes arbitrary value for arbitrary path.
Eddies are nucleated and annihilated by the viscosity.
® There is no global way to identify eddies.

11. Quantized Vortex Is Definite Topological Defect
Circulation k = ∳ v ・ds = h/m around vortex core is quantized.
Quantized vortex is stable topological defect.
Quantized vortex can exist only as loop.
Vortex core is very thin (the order of the healing length).

12. Quantum Turbulence Is an Ideal Prototype of Turbulence
Quantum turbulence can give the real Richardson cascade of definite topological defects and clarify the statistics of turbulence.
We study the statistics of quantum turbulence theoretically.

13. Experimental Study of Turbulent State of Superfluid 4He
J. Maurer and P. Tabeling, Europhys. Lett. 43 (1), 29 (1998)
Two counter rotating disks
Temperature in experiment : T > 1.4 K
® It is high to study a pure quantum turbulence.

14. Kolmogorov Law of Superfluid Turbulence
They found the Kolmogorov law not only in 4He-I but also in 4He-II.

15. Kolmogorov Law of Superfluid Turbulence
T > 1.4 K : 4He-II have much viscous normal fluid.
®Dynamics of normal fluid turbulence may be dominant.

16. Motivation of This Work
We study quantum turbulence at the zero temperature by numerically solving the Gross-Pitaevskii equation.
By introducing a small-scale dissipation, we create pure quantum turbulence not affected by compressible short-wavelength excitations.

17. Model : Gross-Pitaevskii Equation

18. Model : Gross-Pitaevskii Equation
We numerically investigate GP turbulence.
Quantized vortex

19. Introducing a dissipation term
To remove the compressible short-wavelength excitations, we introduce a small-scale dissipation term into GP equation
Fourier transformed GP equation

20. Compressible Short-Wavelength Excitations
Compressible excitations of wavelength smaller than the healing length are created through vortex reconnections and through the disappearance of small vortex loops.
®Those excitations hinder the cascade process of quantized vortices!
Vortex reconnection

21. Without dissipating compressible excitations⋯
C. Nore, M. Abid, and M. E. Brachet, Phys. Rev. Lett. 78, 3896 (1997)
t = 2
t = 4
t = 6
t = 8
t = 12
t = 10
Numerical simulation of GP turbulence
The incompressible kinetic energy changes to compressible kinetic energy while conserving the total energy

22. Without dissipating compressible excitations⋯
The energy spectrum is consistent with the Kolmogorov law in a short period
®This consistency is broken in late stage with many compressible excitations
We need to dissipate compressible excitations

23. Small-Scale Dissipation Term
Quantized vortices have no structures smaller than their core sizes (healing length x)
®Dissipation term g(k) dissipate not vortices but compressible short-wavelength excitations
Modified GP equation gives us ideal quantum turbulence only with quantized vortices!

24. Simulation of Quantum Turbulence
We numerically studied
Decaying turbulence starting from the random phase with no energy injection
Steady turbulence with energy injection

25. Simulation of Quantum Turbulence : Numerical Parameters
Space : Pseudo-spectral method
Time : Runge-Kutta-Verner method

26. Simulation of Quantum Turbulence : 1, Decaying Turbulence
There is no energy injection and the initial state has random phase. 3D

27. Decaying Turbulence
vortex
phase
density
0 < t < 6
g0=0
g0=1

28. Decaying Turbulence
vortex
phase
density
t = 5
g0=0
g0=1

29. Decaying Turbulence density t = 5 g0=0
Small structures in g0 = 0 are dissipated in g0 = 1
®Dissipation term dissipates only short-wavelength excitations.
g0=1

30. Decaying Turbulence
We calculate kinetic energy of vortices and compressible excitations, and compare them

31. Decaying Turbulence
g0 = 0 : Energy of compressible excitations Ekinc is dominant
g0 = 1 : Energy of vortices Ekini is dominant

32. Comparison with Classical Turbulence : Energy Dissipation Rate
g0 = 1 : e is almost constant at 4 < t < 10 (quasi steady state)
g0 = 0 : e is unsteady (Interaction with compressible excitations)

33. Comparison with Classical Turbulence : Energy Spectrum
Straight line fitting at Dk < k < 2p/x
: Non-dissipating range
g0 = 1 : h = -5/3 at 4 < t < 10
g0 = 0 : h = -5/3 at 4 < t < 7

34. Comparison with Classical Turbulence : Energy Spectrum
By removing compressible excitations, quantum turbulence show the similarity with classical turbulence

35. Simulation of Quantum Turbulence : 2, Steady Turbulence
Steady turbulence with the energy injection enables us to study detailed statistics of quantum turbulence.

36. Energy Injection as Moving Random Potential
X0 : characteristic scale of the moving random potential
®Vortices of radius X0 are nucleated

37. Realization of Steady Turbulence
Steady turbulence is realized by the competition between energy injection and energy dissipation
vortex
density
potential
Energy of vortices Ekini is always dominant

38. Realization of Steady Turbulence
Steady turbulence is realized by the competition between energy injection and energy dissipation
vortex
density
potential
Energy of vortices Ekini is always dominant

39. Flow of Kinetic Energy in Steady Turbulence

40. Energy Dissipation Rate and Energy Flux
Energy dissipation rate e is obtained by switching off the moving random potential

41. Energy Dissipation Rate and Energy Flux
Energy flux P(k) is obtained by the energy budget equation from the GP equation.
P(k) is almost constant in the inertial range
P(k) in the inertial range is consistent with the energy dissipation rate e

42. Flow of Kinetic Energy in Steady Turbulence
Our picture of energy flow is correct!

43. Energy Spectrum of Steady Turbulence
Energy spectrum shows the Kolmogorov law again
®Quantum turbulence has the similarity with classical turbulence/

44. Time Development of Vortex Line Length
W. F. Vinen and J. J. Niemela, JLTP 128, 167 (2002)
GP turbulence also show the t -3/2 behavior of L

45. Comparison with Energy Spectrum
Initial stage
Energy spectrum is consistent with the Kolmogorov law

46. Comparison with Energy Spectrum
Middle stage
Consistency breaks down from large k (vortex mean free path)

47. Comparison with Energy Spectrum
Last stage
There is no consistency with the Kolmogorov law

48. Dependence on Energy Injection
V0 = 50 (Strong Injection)
Kolmogorov law is clear

49. Dependence on Energy Injection
V0 = 25
Vortex line length is fluctuating

50. Dependence on Energy Injection
V0 = 10 (Weak Injection)
There is no Kolmogorov law

51. Phase Diagram of Quantum Turbulence
There is a little dependence on the energy dissipation g0

52. Application of Quantum Turbulence : Two-Component Quantum Turbulence
Two-component GP equation
g12 may be the important parameter for the consistency with the Kolmogorov law !

53. Phase Diagram of Vortex Lattice
K. Kasamatsu, M. Tsubota and M. Ueda, PRL 91, (2003) 1
Triangular
Square
Double Core
Sheet
K.K, M.Tsubota, and M.Ueda
g12/g

54. Two-Component Quantum Turbulence
g12 = g
g12
Kolmogorov law
Kolmogorov law begins to break?
???
Self-organization of turbulence
Inter-component coupling may suppress the Richardson cascade and make another order of turbulence.

55. Two-Component Quantum Turbulence
g12 = 0.2 g

56. Summary
By using modified GP equation with the small scale dissipation term, we find the Kolmogorov law in quantum turbulence which means the similarity between quantum and classical turbulence.
Future works : Two-component quantum turbulence.