Quantized Vortex Dynamics and Superfluid Turbulence

✍ Scribed by C.F. Barenghi (editor), R.J. Donnelly (editor), W.F. Vinen (editor)

• Publisher: Springer
• Year: 2001
• Tongue: English
• Leaves: 433
• Category: Library

✦ Synopsis

This book springs from the programme Quantized Vortex Dynamics and Sup- ?uid Turbulence held at the Isaac Newton Institute for Mathematical Sciences (University of Cambridge) in August 2000. What motivated the programme was the recognition that two recent developments have moved the study of qu- tized vorticity, traditionally carried out within the low-temperature physics and condensed-matter physics communities, into a new era. The ?rst development is the increasing contact with classical ?uid dynamics and its ideas and methods. For example, some current experiments with - lium II now deal with very classical issues, such as the measurement of velocity spectra and turbulence decay rates. The evidence from these experiments and many others is that super?uid turbulence and classical turbulence share many features. The challenge is now to explain these similarities and explore the time scales and length scales over which they hold true. The observed classical aspects have also attracted attention to the role played by the ?ow of the normal ?uid, which was somewhat neglected in the past because of the lack of direct ?ow visualization. Increased computing power is also making it possible to study the coupled motion of super?uid vortices and normal ?uids. Another contact with classical physics arises through the interest in the study of super?uid vortex - connections. Reconnections have been studied for some time in the contexts of classical ?uid dynamics and magneto-hydrodynamics (MHD), and it is useful to learn from the experience acquired in other ?elds.

✦ Table of Contents

Chapter 1
1 The Two-Fluid Model
2 Quantized Vortex Lines
3 Modelling the Vortex Lines
3.1 Microscopic Model
3.2 Mesoscopic Model
3.3 Macroscopic Model
4 Turbulence
4.1 Turbulent Counterflows
4.2 Turbulent Coflows
5 Motion of Superfluid Vortices for a Given Normal Fluid
6 Motion of the Normal Fluid at Given Super .uid Vortices
7 Fully Coupled Motion of Super .uid Vortices and Normal Fluid
8 Discussion
References
Chapter 2
1 Introduction
2 Update on Pipe Flow
3 Update on Towed Grid Experiments
3.1 The Nature of Grid Turbulence in Helium II
3.2 Four Regimes of Decaying Grid Turbulence in Helium II [15 ]
4 Agenda for the Future
4.1 The University of Oregon 6 cm Wind Tunnel
4.2 Wind Tunnels for Model Testing
4.3 Tow Tanks
5 Challenges for the Future
5.1 The Challenge of Instrumentation
5.2 Challenges for Understanding Counterflow Turbulence
5.3 Challenges for Understanding Periodic Boundary Layer Experiments
5.4 Instrumentation to Detect Vortices Below 1 K
5.5 The Normal Fluid and the Vortex Tangle
5.6 Flow over Blunt Objects,TestingModels such as Submarines
Acknowledgements
References
Chapter 3
1 Single Vortex Nucleation
2 Multiple Slips and Collapses
References
Chapter 4
1 Introduction
2 Superconducting Systems That Use He II Cooling
2.1 Accelerator Magnet System for LHC
2.2 High Field Solenoid for the NHMFL 45-T Hybrid
2.3 RF Cavity Systems for the TESLA Electron Collider
3 Application Relevant He II Properties
3.1 Second Sound Pulse Transport
3.2 Transient and Steady Transport in the Mutual Friction Regime
3.3 The He II Energy Equation
3.4 Fluid Dynamics of Forced Flow He II
3.5 He II/Vapor Two Phase Flow
3.6 Fountain Effect (Fluid Management)
4 Conclusions
Acknowledgements
References
Chapter 5
1 Introduction
2 Experimental Apparatus and Protocol
3 Results and Discussion
4 Conclusion
Acknowledgments
References
Chapter 6
1 Background
2 Creation and Detection of Vortices
3 The Experiment
4 Preliminary Results
5 Discussion
6 Conclusions
Acknowledgements
References
Chapter 7
1 Introduction
2 Experimental Setup
Acknowledgements
References
Chapter 8
1 Experiment
2 Results
2.1 Stable Turbulent Flow
2.2 Intermittent Switching
2.3 Turbulent Phases
2.4 Laminar Phases
3 Conclusion
References
Chapter 9
1 Introduction
2 Vortex Filament Motion
2.1 The Biot–Savart Law and the Local Induction Approximation
2.2 Boundary Conditions
2.3 Meshing of the Filaments
3 Reconnections of Filaments
4 Analysis of the Super .uid Flow
5 Alternative Approaches
6 Conclusions: What Needs to Be Done
References
Chapter 10
1 HVBK Equations
2 Incompressible Renormalized HVBK Flows
3 Rotating Frame Renormalized HVBK Equations
Acknowledgments
Appendix: Lie-Poisson Hamiltonian Formulation
References
Chapter 11
1 Introduction
2 Gross–Pitaevskii Theory and Two-Fluid Hydrodynamics
3 Interaction of Phonons with a Vortex in Hydrodynamics
4 Momentum Balance in the Two-Fluid Hydrodynamics
References
Chapter 12
1 Introduction
2 Linear Theory
3 Nonlinear Solutions
3.1 In .nite Cylinder Assumption
3.2 Unit Aspect Ratio
4 Discussion
References
Chapter 13
1 Introduction
2 Counterflow Turbulence
3 Grid Turbulence in Superfluid Helium
3.1 Measurements of the Decay of Vortex Lines, and the Quasi-classical Model
3.2 Superfluid Turbulence on Length Scales Larger than the Vortex Line Spacing
3.3 The Turbulent Energy Spectra in Superfluid Grid Turbulence
3.4 Superfluid Turbulence at Very Low Temperatures
3.5 Dissipation at Higher Temperatures
4 Summary and Conclusions
Acknowledgements
References
Chapter 14
1 Introduction
2 The Self-Consistent Equation of Motion
3 Numerical Methods for 2-D Flows
3.1 The Normal-Fluid Flow in 2-D
3.2 Delta Function Forcing on Grid
3.3 Extrapolation of the Normal-Fluid Flow in the Neighbourhood of the Super .uid Vortex Line
3.4 Numerical Stability and Time Stepping
4 Results in 2-D Flows
5 Numerical Methods for 3-D Flows
5.1 The Free Normal-Fluid
5.2 The Superfluid
5.3 The Interaction Modelling
5.4 Preliminary Results in 3-D Flows
6 Discussion and Conclusion
Acknowledgements
References
Chapter 15
1 Introduction
2 Vortex Motion Following Reconnection
2.1 The Case v_{ns} = 0
2.2 The Case v_{ns} = const =/= 0
3 The Model
4 Results
References
Chapter 16
1 Introduction
1.1 The Two-Fluid Model
2 Boundary Layer Vortices
2.1 Properties of the Vortex Line Solutions
2.2 Discussion
3 Stability Analysis
3.1 Linear Stability
3.2 Stability Results
3.3 Discussion
References
Chapter 17
1 Introduction
2 The Spectral Decay Model
Acknowledgements
References
Chapter 18
1 Introduction
2 Vortex Wave Cascade Process
3 Cascade Process in the Vortex Tangle
3.1 Decay of the Vortex Tangle
3.2 Comparison with the Vinen ’s Equation
Acknowledgements
References
Chapter 19
1Introduction
2 Constructing the Trial Distribution Function
3 Hydrodynamic Impulse of the Vortex Tangle
4 Energy of the Vortex Tangle
5 Conclusion
References
Chapter 20
1Introduction and Scientific Background
2 Langevin Equation
3 Fokker–Planck Equation
4 Possible Violation of Thermal Equilibrium
References
Chapter 21
1Introduction
2 Analytical Investigation
3 Conservation Laws and Pair Correlators
4 Some Numerical Results
References
Chapter 22
1Introduction
2 Quenched Superfluid Transition
3 Superfluid Turbulence
4 Three Dimensions
References
Chapter 23
1Introduction
2 The Fluid Equations
3 Shortcomings of the GP Model
4 Vortices
5 Superfluid Turbulence; Vortex Line Reconnection
6 Intrinsic Vortex Nucleation
7 Capture of Impurities by Vortex Lines
8 Nonlocal Models
9 Conclusions
Acknowledgments
References
Chapter 24
1 Introduction and Formulation of the Problem
2 Critical Velocities
3 Flow Around a Disk via a Janzen–Rayleigh Expansion
4 Unstable Solutions
5 The Euler–Tricomi Equation near the Transonic Region
References
Chapter 25
1 Introduction
2 Applicability of the Generalized Gross–Pitaevskii Model
3 Nonlocal Nonlinear Schrödinger Equation
4 Vortex Nucleation and Roton Emission
5 Conclusions
References
Chapter 26
1 Introduction
2 Spin-Up and Nucleation of Vortices in Superfluid Helium
3 Stability of Multicharged Vortices
4 Nucleation of Vortices by Rapid Thermal Quench
References
Chapter 27
1 Motivation and Background
2 Weak Turbulence Theory for NLSE
3 Linear Dynamics of the GPE
3.1 Without a Condensate
3.2 With a Condensate
4 Applicability of WKB Descriptions
5 Weakly Nonlinear GPE Waves
References
Chapter 28
1 Basic Equations
2 Magnus Force
3 Three-Dimensional Effects
4 Failure of Mechanistic Reduction
References
Chapter 29
1 Introduction
2 Definition of the System
3 Numerical Methods
4 Bifurcation Diagram and Scaling in 2D
5 Subcriticality and Vortex-Stretching in 3D
References
Chapter 30
1 Introduction
2 Fluid Equations
3 Time-Independent Solutions in the Object Frame
4 The Critical Velocity
5 Vortex Shedding and Drag
6 The Critical Velocity in Inhomogeneous Condensates
7 Comparison to Ions in Helium
8 Conclusion
Acknowledgements
References
Chapter 31
1 Time-Dependent Gross–Pitaevskii Equation
1.1 Equivalent Hydrodynamics of Compressible Isentropic Fluid
1.2 Thomas–Fermi Limit for Large Condensates
2 Energy of a Vortex in a Large Rotating Trap
3 Small-Amplitude Excitation of a Vortex in a Rotating Trap
3.1 Stability of a Vortex
3.2 Splitting of Normal-Mode Frequencies Caused by a Vortex
4 Vortex Dynamics
4.1 Dynamics of Straight Vortex
4.2 Inclusion of Curvature
References
Chapter 32
1 Introduction
2 Kinetic Regime
3 Coherent Regime
4 External Potential
References
Chapter 33
1 Introduction
2 Effective Action
3 Finite Size Effects
References
Chapter 34
1 Introduction
2 Some Vortex Generalities
3 The Importance of Reconnection
4 Reconnection in Normal Fluids
5 Reconnection in Superfluids
6 Conclusions
References
Chapter 35
1 Introduction
2 Dissipation of Energy and Helicity
3 Interlocked Flux Rings
4 Orthogonal Vortex Tubes
References
Chapter 36
1 Vortex Structures and Tangles in Classical and Quantized Vortex Flows
2 Measures of Tropicity for Vortex Tangles: Tubeness, Sheetness and Bulkiness
3 Measures of Geometric Complexity: Directional Alignment and Writhing
4 Algebraic Measure of Structural Complexity: Average Crossing Number
5 Measures of Topological Entanglement: Kinetic Helicity and Directional Linking
6 Relationships Between Complexity Measures and Energy Levels
Acknowledgements
References
Chapter 37
1 Introduction
2 Magnetic Reconnection
3 Vortex Reconnection
4 Conclusions
Acknowledgment
References
Chapter 38
1 Introduction
2 Current-Sheet Formation at a Hyperbolic Magnetic Neutral Line in a Stagnation-Point Plasma Flow
3 Effect of a Uniform Shear–Strain in the Plasma Flow
4 Discussion
Acknowledgements
References
Chapter 39
1 Introduction and Simple Examples
2 Different Aspects of Nonlocality
2.1 Direct Coupling Between Large and Small Scales
3 Concluding Remarks
References
Chapter 40
1 Superfluid {^3}He
2 Order-Parameter Texture and Superflow in {^3}He-A
3 Double-Quantum Vortex Line
4 Vortex Sheet
5 Dynamic Response
6 Summary and Future Work
Acknowledgements
References
Chapter 41
1 Wetting Properties of {^4}He on Weak-Binding Alkali Metals
1.1 Wetting Transitions of Liquid Helium
1.2 Interface Model Description of Wetting Transitions
2 Lifetime of an Undercooled Film
3 Application to {^4}He/Cs
4 Conclusions
Acknowledgments
References
Chapter 42
1 Introduction
2 Model and Comparison with Experiment
3 Analogies with Classical and Quantum Fluids
4 The Breakdown Steps and Their Relation to Other Types of QHE Breakdown
5 Summary
Acknowledgement
References
Chapter 43
1 Introduction
2 Order Parameter and Lagrangian
3 Hydrodynamical Equation
4 Vortex State
4.1 The Profile of a Single Vortex
4.2 Vortex Dynamics
5 Summary
References
Chapter 44
1 Introduction
2 Canonical Quantization of Planar Vortices
2.1 The Spectrum of Unbounded Vortex Pairs
3 Pair Quantum Dynamics in a Circular Box
3.1 Spectral Structure of Low Energy States
References