Presentation on theme: "Dynamics and Statistics of Quantum Turbulence at Low Temperatures"— Presentation transcript:
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 turbulenceQuantum 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 lawIn 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 lawIn 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 lawIn 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 rateP(k) : energy flux from large to small ke : 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 nucleatedEddies are broken up to smaller onesSmall 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 disksTemperature 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.
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 equationFourier 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 = 2t = 4t = 6t = 8t = 12t = 10Numerical simulation of GP turbulenceThe 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 excitationsWe 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 excitationsModified GP equation gives us ideal quantum turbulence only with quantized vortices!
24 Simulation of Quantum Turbulence We numerically studiedDecaying turbulence starting from the random phase with no energy injectionSteady turbulence with energy injection
25 Simulation of Quantum Turbulence : Numerical Parameters Space : Pseudo-spectral methodTime : 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 Turbulencevortexphasedensity0 < t < 6g0=0g0=1
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 rangeP(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 stageEnergy spectrum is consistent with the Kolmogorov law
46 Comparison with Energy Spectrum Middle stageConsistency breaks down from large k (vortex mean free path)
47 Comparison with Energy Spectrum Last stageThere 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 = 25Vortex 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 equationg12 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)1TriangularSquareDouble CoreSheetK.K, M.Tsubota, and M.Uedag12/g
54 Two-Component Quantum Turbulence g12 = gg12Kolmogorov lawKolmogorov law begins to break????Self-organization of turbulenceInter-component coupling may suppress the Richardson cascade and make another order of turbulence.
56 SummaryBy 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.