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**Dynamics and Statistics of Quantum Turbulence at Low Temperatures**

Michikazu Kobayashi

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**Contents Motivation and Introduction.**

Model : Gross-Pitaevskii Equation. Simulation of Quantum Turbulence. Quantum Turbulence of Two Component Fluid.

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**1, Motivation and Introduction**

Question : Does quantum turbulence have a similarity with that of conventional fluid? ?

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**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!

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**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

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**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

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**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

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**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

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**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

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**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.

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**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).

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**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.

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**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.

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**Kolmogorov Law of Superfluid Turbulence**

They found the Kolmogorov law not only in 4He-I but also in 4He-II.

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**Kolmogorov Law of Superfluid Turbulence**

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

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**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.

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**Model : Gross-Pitaevskii Equation**

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**Model : Gross-Pitaevskii Equation**

We numerically investigate GP turbulence. Quantized vortex

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**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

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**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

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**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

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**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

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**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!

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**Simulation of Quantum Turbulence**

We numerically studied Decaying turbulence starting from the random phase with no energy injection Steady turbulence with energy injection

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**Simulation of Quantum Turbulence : Numerical Parameters**

Space : Pseudo-spectral method Time : Runge-Kutta-Verner method

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**Simulation of Quantum Turbulence : 1, Decaying Turbulence**

There is no energy injection and the initial state has random phase. 3D

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

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

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**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

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

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

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**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)

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**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

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**Comparison with Classical Turbulence : Energy Spectrum**

By removing compressible excitations, quantum turbulence show the similarity with classical turbulence

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**Simulation of Quantum Turbulence : 2, Steady Turbulence**

Steady turbulence with the energy injection enables us to study detailed statistics of quantum turbulence.

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**Energy Injection as Moving Random Potential**

X0 : characteristic scale of the moving random potential ®Vortices of radius X0 are nucleated

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**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

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**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

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**Flow of Kinetic Energy in Steady Turbulence**

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**Energy Dissipation Rate and Energy Flux**

Energy dissipation rate e is obtained by switching off the moving random potential

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**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

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**Flow of Kinetic Energy in Steady Turbulence**

Our picture of energy flow is correct!

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**Energy Spectrum of Steady Turbulence**

Energy spectrum shows the Kolmogorov law again ®Quantum turbulence has the similarity with classical turbulence/

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**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

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**Comparison with Energy Spectrum**

Initial stage Energy spectrum is consistent with the Kolmogorov law

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**Comparison with Energy Spectrum**

Middle stage Consistency breaks down from large k (vortex mean free path)

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**Comparison with Energy Spectrum**

Last stage There is no consistency with the Kolmogorov law

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**Dependence on Energy Injection**

V0 = 50 (Strong Injection) Kolmogorov law is clear

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**Dependence on Energy Injection**

V0 = 25 Vortex line length is fluctuating

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**Dependence on Energy Injection**

V0 = 10 (Weak Injection) There is no Kolmogorov law

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**Phase Diagram of Quantum Turbulence**

There is a little dependence on the energy dissipation g0

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**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 !

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**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

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**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.

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**Two-Component Quantum Turbulence**

g12 = 0.2 g

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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.

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