Dynamics and Statistics of Quantum Turbulence at Low Temperatures

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Dynamics and Statistics of Quantum Turbulence at
Low Temperatures

• Michikazu Kobayashi

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Contents

1. Motivation and Introduction.
2. Model Gross-Pitaevskii Equation.
3. Simulation of Quantum Turbulence.
4. 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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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
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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 gt 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 gt 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)
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

1. Decaying turbulence starting from the random
   phase with no energy injection
2. 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.
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Decaying Turbulence
vortex
phase
density
0 lt t lt 6
g00
g01
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Decaying Turbulence
vortex
phase
density
t 5
g00
g01
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Decaying Turbulence
density
t 5
g00
Small structures in g0 0 are dissipated in g0
1 Dissipation term dissipates only
short-wavelength excitations.
g01
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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 lt t lt 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 lt k lt
2p/x Non-dissipating range
g0 1 h -5/3 at 4 lt t lt 10 g0 0 h
-5/3 at 4 lt t lt 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.

1. P(k) is almost constant in the inertial range
2. 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,
150406 (2003)
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Triangular
Square
Double Core
Sheet
K.K, M.Tsubota, and M.Ueda
0
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.