Vortex Nucleation in a Superfluid
by
Dominic Marchand
B.Sc. in Computer Engineering, Universite Laval, 2002 B.Sc. in Physics, The University of British Columbia, 2004
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
Master of Science
in
The Faculty of Graduate Studies
(Physics)
The University Of British Columbia
October 2006
© Dominic Marchand, 2006 Abstract
Superfluids have very peculiar rotational properties as the Hess-Fairbank experiment spectac• ularly demonstrates. In this experiment, a rotating vessel filled with helium is cooled down past the critical temperature. Remarkably, as the liquid becomes superfluid, it gradually stops its rotation. This expulsion of vorticity, analogous to the Meissner effect, provides a funda• mental experimental definition of superfluidity. As a consequence, superfluids not only posses quasiparticles like phonons, but also quantized vortex excitations. This thesis examines the creation mechanism of vortices, or nucleation, in the low temper• ature limit. At these temperatures, thermal activation of vortices is ruled out and nucleation must be a tunneling effect. Unfortunately, there is no theory to describe this nucleation process. Vortex nucleation is believed to more likely occur in the vicinity of irregularities of the vessel. We therefore consider a few simple, yet experimentally realistic, two-dimensional configurations to calculate nucleation rates. Close to zero temperature and within a certain approximation, the superfluid is inviscid and incompressible such that it can naturally be treated as an ideal two-dimensional fluid flow. Calculating the energy of static vortex configurations can then be done with standard hydrodynamics. The kinetic energy of the flow as a function of the position of the vortex then describes a potential barrier for vortex nucleation. Under rotation, the vortex-free state becomes metastable and can decay to a state with one or more vortices. In this thesis, we carry out a semiclassical calculation of the nucleation rate exponent. We use the WKB method along the path of least action created by the presence of a bump or wedge. This work is but a first approximation as fluctuations around this path can be added as well. The main purpose has been to lay down the groundwork required to include the dissipative effect of the coupling to phonons, which is paramount to an accurate description of the phenomenon. This effect could then be included using the Caldeira-Leggett dissipative tunneling effect [4].
ii Table of Contents
Abstract ii
Table of Contents iii
List of Figures vi
Acknowledgements viii
1 Introduction 1 1.1 A Brief History of Superfluidity 1 1.1.1 The Birth of the Field of Quantum Fluids 1 1.1.2 Toward a Macroscopic Theory, the Two-Fluid Model 2 1.1.3 The Landau-Tisza Controversy (1941-1947) 3 1.1.4 Towards a Microscopic Justification 4 1.1.5 Another Superfluid 5 1.2 Theory of Superfluidity 5 1.2.1 Definition of Superfluidity 5 1.2.2 The Two-Fluid Model 6 1.2.3 Quantum Fluid 8 1.2.4 Similarities with the Ideal Bose-Einstein Gas 8 1.2.5 Wave Function 10 1.2.6 Excitations and Quasiparticles 12 1.2.7 The Quantization of Circulation and the Vortex State 13 1.3 Investigations of Vortices and Vortex Nucleation in Superfluid 4He 14 1.3.1 Rotational Motion and Quantization 15 1.3.2 Observation of Vortices 15 1.3.3 Vortex Nucleation and the Limitation of Superflow 15 1.3.4 Nucleation of Quantized Vortex Rings 17 1.3.5 Quantum Tunnelling 19 1.4 Theoretical Studies of Vortex Nucleation 20 1.5 Our Motivation 21
2 Hydrodynamics and Superflow Configurations 22 2.1 Two-Dimensional Ideal Fluid Flow Description 22 2.1.1 Convention for Flux and Density in Two Dimensions 23
iii 2.1.2 Description of a Two-Dimensional Ideal Fluid Flow 23 2.1.3 The Velocity Potential $ 24 2.1.4 The Stream Function * 25 2.1.5 Boundary Conditions 25 2.1.6 Conformal Mapping 26 2.1.7 Multiply-Connected Regions 27 2.2 Description of Quantized Vortices 29 2.2.1 Properties of a Rectilinear Vortex 29 2.2.2 Velocity Potential and Stream Function 30 2.2.3 Example: Off-Center Vortex in a Cylinder 31 2.3 Kinetic Energy 35 2.3.1 Energy of an Off-Center Vortex in a Cylinder 35 2.3.2 Line Integral on the Boundary 40 2.3.3 Branch Cuts 41
3 Simple Yet Useful Configurations 43 3.1 Vortex in a Cylinder With a Wedge 43 3.1.1 Starting Configuration With No Wedge 44 3.1.2 Conformal Mapping 44 3.1.3 Corrections to the Position of the Vortex 50 3.1.4 Kinetic Energy 51 3.1.5 Rotation of the Cylinder 55 3.2 Circular Bump on a Flat Wall 56 3.2.1 Complex Velocity Potential 57 3.2.2 Kinetic Energy 60
4 Vortex Nucleation and Tunneling 66 4.1 The WKB Approximation 66 4.1.1 Exponent 66 4.1.2 WKB in Higher Dimensions 67 4.1.3 Vortex Nucleation 67 4.2 Tunneling Rate For a Perfect Cylinder 67 4.2.1 Properties of the Barrier 67 4.2.2 Approximations to the Potential 69 4.2.3 Numerical Integration 72 4.2.4 Curve Fits and Approximations 72 4.3 Tunneling Rate For a Circular Bump on a Flat Wall 74 4.3.1 Properties of the Barrier 74 4.3.2 Numerical Integration 75 4.3.3 Curve Fits and Approximations 76 4.4 Effective Mass of a Vortex 85 4.4.1 Inertial Mass of a Vortex for a Non-Uniform Condensate 86 4.5 Future Work and Conclusions 88
iv Bibliography 90
v List of Figures
1.1 Phase diagram and specific heat of 4He 2 1.2 Two-fluid model 7 1.3 Energy spectrum of quasiparticles in superfluid helium 12 1.4 Schematic of a superfluid gyroscope 17 1.5 Possible metastates for the superfluid gyroscope 18 1.6 Alternative models of vortex ring nucleation by ion 18
2.1 Simply-connected and multiply-connected regions 27 2.2 Method of images for a vortex inside a cylinder 32 2.3 Geometry for an off-center vortex with an image vortex 33 2.4 Equipotential lines and lines of force for a cylindrical capacitor 38 2.5 Equipotential and stream lines for an off-center vortex 39 2.6 Branch cut for a vortex in a cylinder 42
3.1 Mapping to a cylinder with a wedge 43 3.2 Mapping from a circle to a circle with a wedge 45 3.3 Mapping from a circle to the upper half-plane 46 3.4 Mapping to the upper half-plane with a wedge 46 3.5 Mapping to a circle with a wedge 49 3.6 Coordinates offset due to the mapping 50 3.7 Energy of a vortex in a cylinder with a wedge 53 3.8 Energy of a vortex on the x axis in a cylinder with a wedge 54 3.9 Modification to the energy of a vortex in a rotating cylinder 56 3.10 Stream functions of a flow past a cylinder and a moving cylinder 58 3.11 Vortices configuration for a cylindrical bump on a flat wall 59 3.12 Stream function of a vortex near a moving cylinder 60 3.13 Branch cut for the vortex configuration near a cylinder 60 3.14 Energy of a vortex near a semi-circular bump 64 3.15 Lines of constant energy for a vortex near a semi-circular bump 65
4.1 Potential for a vortex in a perfect cylinder as a function of u 68 4.2 Approximations of V(x) for a vortex in a perfect cylinder 70 4.3 j(u) for a perfect cylinder where the potential is defined in 2 parts: linear and quadratic 71
vi 4.4 j(u) for a perfect cylinder where the potential is defined in 2 parts: quadratic and quadratic 72 4.5 Curve fits of 7(7/) for a perfect cylinder using 7 = A|iz|~A +70 73 4.6 Fit parameters for a perfect cylinder with 7 = A|'u|_A + 70 77 4.7 Fit parameters for a perfect cylinder using 7 = A|ii|~2 + 70 and curve fit of these parameters 78 4.8 Curve fits of 7(14) for a perfect cylinder using 7 = A|u|~2 + 70 and using the results of a fit for A and 70 79 4.9 Curve fits of j(u) for a circular bump using 7 = A|u|_A + 70 80 4.10 Fit parameters for a circular bump using 7 = A|u|~A + 70 and curve fit of these parameters 81 4.11 Curve fits of j(u) for a circular bump using 7 = A|u|-A+70 and using the results of a fit for A, 70 and A 82 4.12 Fit parameters for a circular bump using 7 = + 70 and curve fit of these parameters 83 4.13 Curve fits of 7(14) for a circular bump using 7 = -f 70 and using the results of a fit for A and 70 84
vii Acknowledgements
I would like to express my thanks to my supervisor, Philip Stamp, for his invaluable help and advice, and to Mona Berciu, for agreeing to be my second reader. Thanks to my friend and colleague Rodrigo Pereira with whom I had fruitful discussions on my research topic. In the proofreading of this thesis, I have had the help and advice of many friends and colleagues. I am grateful to Lara Thompson, Jason Penner, Mya Warren, Matthew Hasselfield and Andrew Hines for their help. Special thanks to professor Matthew Choptuik and his group who lent me the computer on which I am typing this thesis. And last, but not least, thanks to my parents, Jacques and Helene, and to my partner, Genevieve, for their support and encouragement.
viii 1. Introduction
This chapter is intended as a quick review of superfluidity in 4He starting with a short historical account of relevent experiments and discoveries. A definition of superfluidity based on the non-classical rotational inertia and the Hess-Fairbank experiment is then given before briefly reviewing the theory of superfluidity, quantized vortices and vortex nucleation. Finally, we summarize the reasons that motivate yet one more study of phenomena that have been studied for almost a century.
1.1 A Brief History of Superfluidity
The following review is but a very summarized account and the interested reader will find a more exhaustive treatment in the excellent work of Gavroglu and Goudaroulis [17]. Unfortunately, for clarity of exposition, explanation of many of the concepts presented here must be delayed to the next section.
1.1.1 The Birth of the Field of Quantum Fluids
The study of matter at low temperatures started with Heike Kamerlingh-Onnes (Nobel laureate of 1913) who both liquified helium 4 (1908) and discovered superconductivity. Decisive experi• mental evidence that helium 4 undergoes a drastic modification at 2.17 K was provided by W. H. Keesom and his collaborators with measurements of such physical properties as the heat of vaporization, the dielectric constant and more importantly the heat capacity1. The anomalies at 2.17 K led Keesom to postulate the existence of two different phases and he called helium I the phase found at temperature between 2.17 K and the boiling point 4.2 K. Helium II exists at temperatures lower than 2.17 K and would soon be found to have exceptional properties. Kapitsa and, indepently, Allen and Misener discovered superfluidity in 1937. Pyotr Leonidovich Kapitsa went on to firmly establish the existence of superfluidity and published a large amount of experimental observations on liquid helium II, work for which he received a late Nobel prize
1 Figure 1.1b shows the shape of the specific heat which inspired Ehrenfest to name the transition temperature the A-point.
1 1 2 3 4 5 T(K) 1 2 3 4 T(K) (a) (b)
Figure 1.1: (a) Phase diagram of 4He and (b) Specific Heat of 4He
in 1978. He first demonstrated that liquid helium II can flow without resistance through narrow capillaries implying that the flow was inviscid. On the other hand, other experiments involving the rotation of a disk immersed in helium II showed that it could still behave as a viscous fluid under different experimental conditions. The peculiar properties of helium, the so-called phenomenon of superfluidity, founded the new field of quantum fluids. As put by London [30] in a review of the subject:
... these phenomena represent more than just another subject of physics. There seems to be a good reason to suspect that they are manifestations of quantum mechanics on a macroscopic scale... a case where quantum mechanics would directly reach into the macroscopic world.
1.1.2 Toward a Macroscopic Theory, the Two-Fluid Model
The development of methods to deal with collective phenomena was paramount to the explana• tion of superfluidity as a macroscopic phenomenon as most of the concepts used to understand the system on an atomic scale could not be used to explain the macroscopic system. A heuristic principle that would prove important in the following developments is that the entire liquid should be treated as a gigantic molecule. This concept is referred to today as the macroscopic wave function or off diagonal long range order and its connection to the postulated phenomenon of Bose-Einstein condensation was suggested as early as 1938 by London [29]. The discrepancy between experiments on the viscosity of helium II and the Bose-Einstein theory prompted Tisza to introduce the two-fluid model in 1938, a most impressive deviation
2 from classical hydrodynamics. The two-fluid concept postulated the existence of both a normal or classical fluid and a superfluid or quantum fluid and the specific mechanism to go from one to the other. This phenomenological model was extremely successful and accounted for most of the known properties of helium and even anticipated thermal waves or second sound. These waves consist of an oscillation of the partial densities of the normal fluid and superfluid while the total density remains constant. This oscillation also implies a variation of the entropy and the temperature of the fluid. Because of the success of the two-fluid model, there was often a false tendency to think of the separation of the atoms between two components as a physical reality. Such a separation is impossible, of course, and Tisza's model still had to be modified to recover the same qualitative explanations without this division of atoms into different components. This is exactly what was offered by Laudau in 1941 [24].
1.1.3 The Landau-Tisza Controversy (1941-1947)
Although there were well established theories for the solid and gaseaous states, the theory of the liquid state was less than satisfactory. A liquid could then be treated as an imperfect gas in which the interactions become important or a broken solid in which the binding forces are too weak to preserve the lattice structure. The London-Tisza model built on the theory of Bose-Einstein condensation of an ideal gas. Landau's approach was similar to that used in solid state physics where one discusses the normal modes of motion of the solid as a whole. The rather vivid controversy between partisans of the two theories that followed focused on the differences of the two theories neglecting their considerable overlap and the fact that the 'truth' was likely a compromise between them. Landau first postulated the existence of two independent motions in helium II, each with its own effective density so that the sum of the two equals the total density of the fluid. Landau's theory was therefore also called a two-fluid model but did not suggest any real division of the fluid into two parts. He started by considering the fluid at zero temperature which is the ground state and was assumed to be free of vorticity. He then constructed the spectrum of liquid helium from the two types of excitations which describe the collective motion of the particles, the first one being the usual phonon and the second one being a roton which he defined as the elementary rotational excitation. Helium is then pictured as a background fluid in which excitation moves. The excitations are normal because they may be scattered and reflected thus showing viscosity. The fluid associated with the ground state is superfluid because it cannot absorb a phonon or a roton unless the fluid is flowing faster than some critical velocity. Andronikashvili's mesurements of the superfluid fraction in 1946 [2] provided a confirmation of
3 Landau's two-fluid model. A year later, Laudau had to modify the roton spectrum and combined it to the phonon spectrum in a single energy-momentum curve. This modification had become necessary to better reproduce experimental results for the velocity of second sound measured by Peshkov and later by Pellam and Scott. Tisza criticized Landau's modification of his theory but it was nevertheless in better agreement than the ideal Bose-Einstein gas. Furthermore, Bogoliubov found that quasi-particles could be used to express the energy spectrum and that no division into two types was possible, thus justifying the recent modification. Landau provided seminal contributions to our understanding of superfluidity, work for which he later receives the 1962 Nobel prize, but a microscopic justification of this theory was still eluding low-temperature physicists.
1.1.4 Towards a Microscopic Justification
A first attempt to establish a connection between Landau's theory and that of Bose-Einstein condensation was carried out by Bogoliubov. He started with an imperfect Bose-Einstein gas with a weak inter-particle interaction and showed that second quantization followed by an approximation procedure allowed the expression of the low-lying excited states as an ideal Bose-Einstein gas of quasi-particles. Unfortunately, the interaction between helium atoms is not weak enough to justify a nearly perfect Bose-Einstein gas and Landau's spectrum could not be justified yet. It was Feynman [14] who showed that the two-fluid model could be justified from first principles using physical intuition to make plausible guesses at the nature of the solutions. He also remedied to a large extent the model's major failures. Feynman first explained that the interatomic potential does not alter the existence of Bose-Einstein condensation in helium II because the loose liquid structure does not oppose the motion of an atom as the other atoms simply move out of the way to make room for it as they would in a gas. This adjustment merely increases the effective mass of the moving atom. He also argued that phonons are the only low-lying states which allowed him to recover the correct low-temperature behavior of the specific heat. And finally, he obtained the right shape for the energy spectrum, albeit with poor numerical agreement. To get a better energy spectrum, Feynman and Cohen proposed another wave function [15] which sugested that the roton was the quantum mechanical analogue of a microscopic vortex ring. This idea originated in Onsager's hypothesis [31] about the existence of vortex sheets, vortex lines and the quantization of the superfluid circulation. Developed by Feynman, it would also address the failure of Landau's two-fluid model in its treatment of rotational motion.
4 The experiment of Hess and Fairbank [21] would later provide a spectacular demonstration of the properties of the rotational motion in a superfluid and provides a fundamental experi• mental definition of superfluidity. We will come back to this experiment in the next section.
1.1.5 Another Superfluid
Ordinary helium has an extremely rare isotope found only in one part in ten million. Despite the identical electronic properties of the 4He and 3He atoms, they have completely different properties at low temperatures due to the spin 1/2 fermion nature of 3He and the bosonic nature of 4He. 3He becomes superfluid only below 2 mK, three orders of magnitude lower in temperature than 4He. Another surprising difference is that 3He has two distinct superfluid phase. Two of the superfluid transitions were first observed by Osheroff, Richardson and Lee in 1972, work for which they received the 1996 Nobel prize. The phenomenon of superfluidity in 3He is obviously quite different than in 4He and we will restrict the rest of this study to helium II.
1.2 Theory of Superfluidity
1.2.1 Definition of Superfluidity
One problem with the word superfluidity is that it actually summarizes a number of different properties that are not necessarily intrinsically related. This very word can be used in the literature alternatively to mean any of these properties or all of them without further expla• nation. As mentioned before, helium II flows through small capillaries without friction and will flow over the rim of the vessel containing it. It sustains persistent currents, has peculiar properties in rotation and shows effects analogous to the Josephson effect in superconductors. A very good account of these properties and to which extent they imply each other was given by Leggett at the Vlth International Summer School in Theoretical Physics in Kiljava, Finland [28]. Here we will mainly focus on the more fundamental rotational properties.
Non-Classical Rotational Inertia Leggett refers to the rotational properties of a superfluid under the name non-classical rotational inertia or NCRI. To explain what NCRI is, consider a bucket filled with a classical fluid which is then rotated. The fluid is stationary at first but eventually the liquid will be rotating with the container. Some energy is dissipated as heat in the process but after a while equilibrium is reached where the liquid and the container rotate together. If the same experiment was to be carried with a microscopic container containing
5 only one atom the results would be radically different as the angular momentum of a quantum system is quantized in units of h. The atom will rotate only at angular velocities corresponding to units of these quanta and under some critical velocity will remain stationary. It was pointed out by London that liquid helium in its behaviour as a macroscopic quantum system should display an equivalent effect. In 1967, Hess and Fairbank [21] verified this experimentally. They did more than start the container from rest and show that the helium did not rotate as the container was brought into rotation. This could have been interpreted as a proof that helium II had low or no viscosity. Instead they started with a rotating vessel filled with helium in its normal phase and then cooled it. As the temperature was lowered, the helium gradually stopped rotating until as T —> 0 it was completely stationary. They reported their findings as analogous to the Meissner effect found in superconductors. Only here it is the vorticity of the system which is expelled instead of the magnetic field. The Hess-Fairbank experiment is more than a demonstration of NCRI as it provides an experimental definition of this effect. Since NCRI is the most fundamental effect understood under the name of superfluidity, it also defines superfluidity as well.
1.2.2 The Two-Fluid Model
To introduce the two-fluid model one can start with the following formulation for the problem of NCRI. We take a large number N of identical atoms of mass m enclosed between two coaxial cylindrical walls of mean radius R and spacing d. d is assumed to be much greater than the atomic spacing in the system while d -C R such that all terms of order d/R can be neglected. The classical moment of inertia of the system to lowest order in d/R is
2 2 Icl = J pr dr « NmR . (1.1)
According to classical mechanics, if the walls rotate with an angular velocity LU, the equilibrium state has an angular momentum Ld = Iciu, (1.2) and an energy Ed = \hi^. (1-3)
For a quantum-mechanical system this will not be true of course but symmetry considerations suggest that for small u> the energy will still be proportional to J1. The moment of inertia of
6 the system can then be defined by 2 • I_d E
2 (1-4) du a)-»0 in which case the angular momentum is given by
* = S7 = 'W- <15)
The extent to which the system departs from its normal or classical behaviour can be expressed by
ns = ^ = l-f, (1.6) P Id
where ns is the superfluid fraction, ps is the the superfluid density and p is the total density. The superfluid fraction goes to zero when the system behaves classically and rotates with the walls, while it goes to 1 when L = 0 and the system stays at rest. Similarly the normal density
is denoted pn and the normal fraction is
nn = — = 1 - ns. (1.7) P
Figure 1.2: Two-fluid model. nn is the normal fraction and ns is the superfluid fraction.
This approach evokes Andronikashvili's mesurements of the superfluid fraction [2]. Instead of two cylindrical walls he used a pile of equally spaced thin metal disks immersed in helium and suspended by a torsion fibre. The disks were sufficiently close so that above the superfluid phase transition the fluid was dragged completely with the disks. By measuring the effect of temperature on the period of oscillations, he could measure the change of the moment of inertia and therefore deduce the superfluid fraction (see Figure 1.2). One important difference however
7 is that NCRI is characteristic of the equilibrium state and is quantum-mechanical in origin, whereas this is a persistent current experiment and is a metastable effect. Nevertheless, they both predict the same partial densities.
1.2.3 Quantum Fluid
A quantum fluid is essentially a substance which remains fluid at such low temperatures that the effects of quantum mechanics become significant. Essentially, quantum effects are significant when the thermal de Broglie wavelength, defined in the usual way by
is comparable to other typical length scales in the liquid. Most elements in the periodic table actually have much greater interatomic distances than their de Broglie wavelength. Helium clearly stands apart due to its much weaker interatomic potential which causes it to liquify only at 4.2 K. Due to its small mass, the de Broglie wavelength is as large as \ 1.2.4 Similarities with the Ideal Bose-Einstein Gas Interestingly, the same criteria concerning the de Broglie wavelength and the cubic root of the density of a boson gas are used to predict the onset of Bose-Einstein condensation. We now follow London and Tisza's suggestion and compare the ideal Bose-Einstein gas to liquid helium II2. The Ideal Bose-Einstein Gas Let a macroscopic number N of 4He atoms contained in a cubic box of volume V be at thermal equilibrium at temperature T. The average number of atoms in each state i of energy ej is given by the Bose-Einstein distribution function exp [(ej - ii)/kBT\ - 1 where /u. is the chemical potential of the gas and /V = 5>(ei)T)). (1.10) 2This treatment follows closely [41]. 8 For very low T, all N particles are in the ground state such that N ~ (n(eo, T)) and the exponent in (1.9) is small enough to allow the following approximation: ry ~ JCBZ_ (L11) eo - M It follows that ii is only slightly less than eo- Using periodic boundary conditions, the energy levels of individual atoms in a box are 6klm= {k2 + l2+m2) {L12) 2m4V^ ' with k, I and m, being positive integers. It will be convenient to define the zero of energy to be eo = em. Furthermore, with V being macroscopic the spacing between energy levels is very small and e can be treated as a continuous variable. One can then express the number single-particle states between e and e + de as X>(e)de where V(e) is the density of state. From (1.12), we have Instead of summing over states like in (1.10), we can now integrate the product of the density of state X>(e) times the distribution function of (1.9). Only the sum includes particles in the ground state whereas the integral misses them because 2?(0) = 0. Taking this correction into account, (1.10) becomes N = (n(0, T)) + / V(e)(n(e, T))de Jo = N0(T) + N'{T). (1.14) Putting p, = 0 and evaluating the integral to obtain the upper bound on the number of particles in the excited states yields 7Y'(r) = 2.612V (l^n3/2. (1.15) V 2irh J Setting N'(TB) = N gives the critical temperature 27rfi2 ( N \3/2 9 such that the fraction of particles in the lowest energy level can be expressed as N0(T) T This macroscopic occupation of the lowest energy level is known as the condensate. Comparison to Helium II Replacing N/V by the density of liquid 4He in (1.16) yields an approximate value for the temperature of the A transition of Tg = 3.1 K compared to the experimental value of 2.17 K. There is also an interesting correspondance between N'(T)/N and the temperature dependance of the superfluid fraction. The similarities suggest that the A-point marks the onset of BEC. The specific heat of both the ideal gas and helium show a cusp but the one in helium is more dramatic. We are thus reminded that the attractive forces are not to be neglected. The states of the ideal gas are not true eigenstates of helium II but can shed some light on the subject. In this paragraph the lowest energy level and the excited states refer to the states of an ideal gas. In this picture, the interactions will effectively deplete the condensate and populate other excited states3. Even at absolute zero some particles will not be in the lowest energy level but it remains nevertheless populated by a macroscopic number of particles suggesting the existence of a condensate. This condensate and the excited levels are identified with the superfluid fraction. In practice, less than ten percent of the particles in the condensate will be in the lowest 'ideal-gas' level. Interactions also alter the nature of the excited levels. The thermal excitations of helium II are now collective excitations that can be treated as non- interacting quasiparticles as suggested by Bogoliubov and can be associated with the normal fraction. 1.2.5 Wave Function The last section showed that superfluid helium possesses a condensate giving credence to the postulated existence of a macroscopic quantum state, which is in turn supported by a wealth of experimental evidence. It is the wave function of this state that will now be considered. Feynman [14, 15] started from symmetry considerations on the microscopic level to build his wave function. Leggett [28] uses a similar approach formulating properties of the wave function implied by NCRI. Here a simpler approach will be used [3], starting from the experimental results on the specific heat to find which universality class of thermodynamic phase transition reproduce 3See [41] for more details. 10 3 these results to get the wave function. Figure 1.1b shows that at low temperature Cv ~ T unlike T3/2 for BEC. Close to the A transition it has a weak power law behaviour: 1 a C{T) + A_\T -Tc\~ T the vectors have random angles but below Tc they develop a long range order and a common direction. Solutions of the XY-model have the general form ^(f) = Voexp (i0(f)), (1.19) and, in analogy to the wave function of a BEC, we can normalize it so that the density of particles in the condensate is no = kA)(OI2- (1-20) Above the critical temperature no is zero and iV'oCf)! = 0 such that the phase 6 cannot be defined. The macroscopic wave function looks actually similar to the order parameter of the system. The momentum of the condensate is obtained in the usual way with ptP = -iVip =ptp, (1.21) and p = hV6. (1.22) The superfluid velocity can then be defined as v8 = (1.23) m4 When the superfluid is at rest the phase has the same value throughout; for a constant superfluid velocity, the phase varies uniformly. The phase is thus a well behaved function that varies smoothly, or in other words, it is coherent. This ensures that the particles keep a uniform motion that can be maintained for a long period as a change of the velocity would require the simultaneous change of a macroscopically large number of particles. The wave function is said 11 to wave some 'rigidity'. 1.2.6 Excitations and Quasiparticles We have seen that the existence of superfluidity strongly depends on the presence of a conden• sate. However, it also depends on the nature of the thermally excited states. The presence of interactions gives rise to collective motion of the atoms and we are interested in the normal modes of this motion. Figure 1.3 shows the dispersion curve proposed by Landau and later verified by neutron scattering. <(!>) 1- 'main 0.5- 0.5 1 1.5 2 2.5 p'/h Figure 1.3: Energy spectrum of quasiparticles in superfluid helium Since a liquid can propagate longitudinal waves, the thermal excitations of helium II should definitely include phonons. As for a crystal lattice, the low energy part of the phonon dispersion curve is approximated by a straight line with e - cp, (1.24) where c is the first sound velocity. The second part of the spectrum, called the roton part, can be approximated by + (1.25) where A is the energy gap for the roton and fir is the effective mass. Landau shows that there is no dissipation due to quasiparticles for a superfluid moving with a velocity smaller than Cmin (shown on Figure 1.3) explaining the necessity of this gap for superfluidity to occur. This is because a low energy quasiparticle cannot be scattered into any state for a small velocity. In 12 practice, the critical velocity of superfluid flow is much lower than Cmin due to other excitations like vortices. The excitation spectrum proposed by Laudau can further be motivated by using an ideal gas of phonons and rotons to calculate the specific heat and other thermodynamic parameters. The agreement with experimental values remains reasonable up to 1.2 K. Above this temperature, interactions between excitations need to be taken into account. 1.2.7 The Quantization of Circulation and the Vortex State Circulation and Quantization The circulation is defined as the line integral around a closed path of the fluid velocity: T = iv - df. (1.26) The circulation is closely related to vorticity since Stokes' theorem allows (1.26) to be expressed as T=U(Vxv)dS, (1.27) where V x v is the vorticity of the flow. The existence of a condensate and the form of its wave function lead to a quantization of the superflow circulation as predicted by Onsager. The curl of (1.23) prescribes Vxus = 0, (1.28) such that the flow is irrotational. Inserting (1.23) in (1.26) yields r= — ivd- df= — A6, (1.29) m J m where A6 is the change in the phase angle after going around a closed path. The single- valuedness of the wave function stipulates that ^(f) =ij>(r)exp(iAO), (1.30) implying that A8 = 2irl with I being an integer. The circulation is then K = 1-, (1.31) m where K is usually used for a quantized circulation or the circulation of a vortex and I is called the topological winding number. The quantum of circulation is h/m and has a value of 13 9.98 x 10_8m2s-1. The Vortex State It first seems like (1.28) would prohibit any rotational motion since the circulation of any closed loop in the continuous fluid is zero (1 = 0). This state of helium II, from which superfluid rotation is completely absent, is known as the Laudau state. In reality, quantization of the rotational motion can occur in two different situations where a hole in the superfluid is present and the circulation is evaluated for a path around that hole. The hole can be provided by a solid boundary like in the case of an annular container or it can appear spontaneously in the superfluid in the form of an vortex core. The vortex is a state of rotational motion characterized by a quantized circulation. A circulating flow with cylindrical symmetry will obey (1.28) if ~(rvt) = 0. (1.32) Therefore, the superfluid velocity of a vortex is vs oc 1/r. The divergence is avoided by having the density go to zero at the center such that the wave function is also zero and its phase is not defined. The shape of the vortex core is approximated by an empty cylinder as the density typically falls to zero inside a characteristic distance £ called the healing length or coherence length. In superfluid helium the vortex core has a typical size of the order of 1 A. Only vortices of circulation ±/i/m have been observed as a vortex of greater circulation is unstable to the decay to a number of singly quantized vortices. This study focuses on the case of a rectilinear vortex. The vortex core can however be bent in general and is called a vortex line. When the two ends of a vortex line are joined together, we have a vortex ring. Of course, more than one vortex can exist in a system and they tend to organize in a triangular array. This happens for a very carefully prepared system; otherwise, they might form a vortex tangle that produces superfluid turbulence. 1.3 Investigations of Vortices and Vortex Nucleation in Superfluid 4He Little has been said yet on the subject that will be of utmost interest in this work: vortices and vortex nucleation. There is a wealth of experiments on vortices in superfluid helium, not to mention recent work done on rotating BEC's. This review is by no mean exhaustive and we shall only mention a number of ideas and experiments that have been influential. For more information, the book of Donnelly [8] and the book of Tilley and Tilley [41] provide a good 14 introduction to the subject. 1.3.1 Rotational Motion and Quantization Early experiments on rotation in superfluid helium include Andronikashvili's experiment where the normal fraction of the fluid would rotate but the superfluid fraction stayed at rest. In contrast, Osborne [32] rotated a cylindrical bucket of helium II in 1950 and found that the meniscus had the same shape as that of a normal fluid. Hall and Vinen [19] would later propose that this can be explained by the presence of an array of vortices uniformly distributed. The first experiment designed to look for quantized circulation was carried out by Vinen in 1961 [46]. He studied the conditions for the presence of quantized circulation and free vortices in a rotating annulus. He found that at low velocity the circulation inside the container increases in a series of equal, quantum steps K. 1.3.2 Observation of Vortices The contribution of Hess and Fairbank in 1967 [21] has already been mentioned. It has also provided an experimental verification that there is an upper limit on the angular velocity for the rotation without vortices. Later the same year, Hess would carry out theoretical calculations of the rotational speeds at which various equilibrium configurations of vortices would appear in the rotating container [20]. The Hess-Fairbank experiment did not demonstrate directly the appearance of separate vortices in the vessel. This was accomplished by Packard and Sanders in 1972 [33] using electrons trapped in the vortex cores. This technique was based on pioneering work on the trapping of ions by vortices by Careri, McCormick and Scaramuzi in 1962 [16]. A refinement of this technique using an optical detection system rotating with the container would even allow the positions of the vortices to be photographed. Quantized vortex rings were first studied by Rayfield and Reif in 1964 [36] and were nu• cleated by movings ions through superfluid 4He. Despite the three-dimensional nature of this effect, it is very similar to the nucleation of a vortex line in the proximity of a circular bump that we shall consider later. 1.3.3 Vortex Nucleation and the Limitation of Superflow The factors which limit the velocity of superflow have generated a lot of interest but as for the nucleation of vortices, no theory can account for the complete picture. This limit on the superfluid velocity is called the critical velocity. It is understood that vortices actually play 15 a significant role in this phenomenon, not rotons as was first believed. Indeed, the critical velocity at which nucleation occurs was calculated for a vortex by Feynman in 1955 [14], and is much lower than that of a roton and in better agreement with experiment. When a vortex is nucleated, it does not only take energy from the superfluid when it is created, but as argued by Vinen in 1963, it must first overcome a barrier. The calculation of a critical velocity with the Landau criterion does not take this into account. The critical velocity is defined as the velocity for which the barrier disappears and the vortex nucleation becomes thermally activated. Langer and Fisher proposed such a theory [26] by starting from the rather different hy• pothesis that superflows are metastables states and the transition from this state to another is blocked by a potential barrier which can be overcome by thermal fluctuations. Thus the creation of a vortex is simply the transition from a metastable state to another by overcom• ing (or tunneling as we will see shortly) a potential barrier. A review by Langer and Reppy [27] discusses this fluctuation theory using the results of experiments done with a superfluid gyroscope. A schematic of this device and a short description of how it is operated is presented in Figure 1.4. The annulus packed with porous material is filled by immersing it in a bath of helium. The material allows the superfluid to flow but clamps the normal fluid. A superfluid current is set up by rotating the annulus before cooling through the A-point to some temperature. It is then tipped into a vertical position where its precession can be measured and its angular momentum Lp determined. We find that no matter how fast the annulus is initially rotating, the angular momentum has a maximum that cannot be exceeded. This critical value is a function of the temperature and the pore size. The critical angular momentum decreases with increasing temperature, which can be a result of the reduction of the superfluid fraction, or the superfluid critical velocity, or both. The temperature dependence of ps/p can also be determined with the gyroscope for different pore sizes such that this contribution to the reduction of angular momentum can be substracted and the temperature dependence of the critical velocity itself can be calculated. This is shown as a dashed line in Figure 1.5. If the temperature is modified but does not cross the critical velocity line vs cru the flow is a metastable state and the superfluid velocity will remain unchanged such that moving along one of the horizontal lines is a reversible change of temperature. Langer and Fisher point out that due to the metastable nature of these states, transitions can be induced by thermal fluctuations. A transition then has an activation energy Ea too big for transition to be likely at low temperature and small superfluid velocity. The rate at which the superfluid velocity 16 CO c Figure 1.4: Schematic of a superfluid gyroscope inspired of Langer and Reppy. A persitent current with angular momentum Lp is set up in an annular container (A) filled with a porous material (D). The annulus is mounted on a fiber (C). After rotation in the horizontal plane at angular velocity UJ, the annulus is brought into vertical position and the detector (B) measures its deflection when it precesses. decays is then (1.33) The fluctuation theory predicts that if the system is held at point very close to the critical velocity curve, the decay time will be logarithmic with time which is indeed observed. Finally, Langer and Fisher suggest that the thermal fluctuation required to decrease the superfluid velocity might be the formation of a ring. From this, the energy and velocity of a vortex ring allow the calculation of the activation energy and the coresponding critical velocity. In general, qualitative features of the model are observed in experiments but the qualitative agreement is poor. Also, in experiments of superflow in tortuous channels like in jeweller's rouge, it is hard to justify how a local fluctuation could uniformly modify the superfluid velocity. 1.3.4 Nucleation of Quantized Vortex Rings Experiments on nucleation are often concerned with vortex rings creation by moving ions. The ions are more controllable than irregularities on a vessel's wall and they provide a mean of detection for a single vortex. There is evidence that it is a stochastic process governed by a nucleation rate v. Two rival 17 \ \ \ \ \ •-, transition Avs t ' \ vsc(T) \ \ ;TX T i t / / / / / Figure 1.5: Possible metastates for the superfluid gyroscope. The dashed line is the temperature dependence of the critical velocity vs and the horizontal lines are quantized values of vs. models for the ion-vortex transition have been proposed as shown on Figure 1.6: the peeling model and the quantum transition model. The peeling model was first suggested by Rayfield and is essentially a process where a ring continuously grows from a 'proto-ring' attached to the ion. It predicts no true critical velocity and a strongly temperature-dependent nucleation rate. The quantum transition model of Schwarz and Jang (1973) propose an analogous calculation to Landau's critical velocity for roton emission and predicts a sharp critical velocity as well as no temperature dependence. This later model seems to account at least qualitatively for most results but a temperature dependence suggest that both mechanisms are probably only very approximate descriptions of what really happens. peeling model quantum transition model Figure 1.6: Alternative models of vortex ring nucleation by ion 18 Much of the experimental data on vortex ring nucleation has come from a comprehensive series of investigation by McClintock and his co-workers4 (Bowley, Nancolas, Stamp, etc.) To mention only a few contributions, they were able to calculate a nucleation rate and the critical velocity from the average nucleation rate measured in experiment. This was done under the assumption that the nucleation rate is of the form of the nucleation rate is constant for a velocity greater than the the critical velocity. Moreover, the variation of the nucleation rate as a function of temperature suggested that there was actually two contributions: an intrinsic rate independent of temperature and a thermal rate which operates only in the temperature range where the numer of rotons is finite. Finally they managed to predict a critical velocity closer to its experimental value by considering a 'roton-assisted' process where a roton is absorbed during the vortex creation, thus hinting at the very important role of thermal excitations in nucleation. 1.3.5 Quantum Tunnelling If as mentioned, vortices are favourable states when the superfluid velocity is greater than some critical value, and a potential barrier separates these states, it is natural to consider the possibility of tunnelling through that barrier. Results calculated from tunelling theories are usually of the same order of magnitude but comparisons to measured quantities are hampered by the ignorance of the exact size, structure and energy of the vortex core. Following a suggestion made by Vinen in 1963, Muirhead et al [5, 6] treated both the pealing model and the quantum transition model as quantum tunelling processes. Bowley [37] dealt only with the quantum transition model and treating the vortex as a quantum object in a potential well. The critical velocity is calculated by setting the bare ion energy to be equal to discrete level of the vortex state which reproduces rather well experimental measures. A more recent paper by McClintock et al in 1988 uses again quatum tunneling to interpret experimental nucleation rates and even calculates a barrier height in very good agreement with the theory of quantum tunnelling. They also point out that most studies in the past were addressing the slightly different question of the conditions under which preexisting vorticity expands to form dissipative tangles. It turns out that preparing a sample of superfluid free of vorticity is extremely difficult. Their technique involves using smaller ions that are small enough to be negligibly influenced by remanent vorticity. Also, experiments conducted at lower pressure revealed that the important rise in the critical velocity is not at 0.5 K anymore but at 0.2 K, thus ruling out the roton-driven vortex nucleation. The rate that they obtain is 4See [38] and references therein. 19 accurately fitted with v = v0 + Aexp(-j^j, (1.34) where VQ is the temperature independent part, and A and e are constant parameters. The sesond part, also called the Arrhenius law, is of course associated with quantum mechanical tunneling. Finally, they make an interesting observation about the possibility for the height of the barrier to vary with pressure which could be a manifestation of the coupling to the phonon bath. Avenel and Varoquaux [42, 43] conducted a series of experiments in a different configuration than the usual ion-ring. They studied a superflow through a micro-orifice and observed a transition from a quantum tunneling to a thermally activated nucleation regime exactly like in the vortex-ring experiment. The quantum tunneling rate is again found to be expressed by Arrhenius law. The vortices created in these experiments are believed to be half-rings with both ends travelling on the boundary. On a slightly different topic, they also published an article in 1998 [44] where they argue that pinning and unpinning of vortices also occurs by quantum tunnelling at very low temperatures. 1.4 Theoretical Studies of Vortex Nucleation We have seen that the conditions for the presence of vortices in the system has been studied, but there is no satisfactory theory for their creation mechanism yet. Experimental studies have suggested two mechanisms for the appearance of vortices, namely, thermal activation and quantum tunneling. At very low temperatures the only possibility is for nucleation by quantum tunneling. Basically, the non rotating state becomes metastable and can decay into a state with one vortex. This transition is more likely to happen where the barrier is weakest such that irregularities of the container are believed to play an important role. Many studies have been published presenting diverse configurations with different boundary conditions. They all use more or less the same technique where a potential barrier is calculated by evaluating the kinetic energy of the superflow in function of the position of the vortex. The exponent of the nucleation rate is then calculated with the WKB method. One of the first calculation of a nucleation rate based on the semiclassical approximation was carried out by Volovik in 1971 [47]. He considered a vortex ring close to a circular obstacle. Many more investigations could be mentioned but none of them offer a realistic picture. Two things that are generally poorly treated are the vortex mass and the dissipative effect of the coupling to the bath. A recent article by Avenel and Varoquaux in 2003 [45] maps the vortex nucleation 20 problem to the escape problem of a Brownian particle from a metastable cubic potentiel. This is an interesting attempt but the model of a Brownian particle seems implausible since a bath of phonons and rotons will not give Ohmic dissipation, except at very high temperature. 1.5 Our Motivation This study is mostly concerned with the nucleation of rectilinear vortices in a cylindrical con• tainer at very low temperatures. Vortices are assumed to be nucleated by quantum tunneling through a potential barrier calculated by evaluating the kinetic energy of the flow with and without a vortex in the system. A semiclassical approximation is then used to get an approxi• mate tunneling rate. Nothing of this is new, of course, but few studies have taken into account the irregularities on the container walls, and even fewer with an analytical treatment. Previous works on the subject are either numerical WKB calculations or phenomenological activation theories. The intent is to consider simple yet experimentally realistic configurations that will include a feature like a bump or a wedge. The inclusion of the irregularities of the vessel should bring the predicted nucleation rate closer to its experimental value. Where the real novelty of this study lies, however, is in the treatment of the dissipative effects of couplings to the phonon (and possibly the roton) bath. This would be done quite naturally in our approach by using the Caldeira-Leggett dissipative tunneling effect [4]. Unfortunately, time constraints have not allowed us to include this in the present document. 21 2. Hydrodynamics and Superflow Configurations In this chapter, we will first introduce basic concepts of hydrodynamics related to the descrip• tion of a two-dimensional ideal fluid flow. We then describe vortices in this context, illustrating with the example of an off-center vortex inside a cylindrical container. Finally, different ways of calculating the kinetic energy of a vortex configuration are presented using the same example. All these concepts and techniques will be applied in the next chapter with more interesting configurations. 2.1 Two-Dimensional Ideal Fluid Flow Description When a superfluid's temperature reaches absolute zero, it becomes completely superfluid. Its flow is then perfectly inviscid, and to a certain approximation, incompressible and irrotational. Of course, it is not perfectly imcompressible as low-energy excitations like phonons can be created under compression. Neither is the superfluid irrotational as vortices can be nucleated in it. But a proper treatment will allow us to restrict the vorticity to regions located outside of the boundary of the fluid. Phonons do have an important impact on the nucleation of vortices but they are ignored altogether at this stage of the calculation. In other words the superfluid will be quite naturally described by the simple case of an ideal fluid flow. Rather than resorting to numerical techniques, we shall restrict our investigation to problems that can be described in two dimensions to further simplify the analysis. That is, the fluid motion will take place in a series of planes parallel to the xy plane and that motion will be the same in each plane. The velocity field is then a function of x and y only and has no component parallel to the z axis. As will be shown shortly, the two-dimensional ideal fluid flow has a few analytical peculiarities that allow for relatively simple exact solutions to many problems of great interest. 22 2.1.1 Convention for Flux and Density in Two Dimensions Since the motion is known everywhere if we have its description in one plane, we shall consider only the plane z = 0. For extensive quantities which are usually defined by a three-dimensional integral, we will define them per unit length along the z axis. For example, the kinetic energy of a fluid will actually refer to the kinetic energy per unit length. For the less obvious cases, we shall abide by the following conventions: • The flux through a curve, is defined as the flux through the surface comprised between the plane z = 0 and z = 1 of the cylindrical surface having this curve as base. • The fluid density or number density, is defined as the usual three-dimensional fluid or number density. 2.1.2 Description of a Two-Dimensional Ideal Fluid Flow Let us restate the two usual properties that a two-dimensional ideal fluid possesses: 1. incompressibility (divergence of velocity vanishes) 2. irrotationality (curl of velocity vanishes) where v = (^1,^2^3) and V3 was taken to be 0. These conditions can be expressed more simply by defining a velocity potential The velocity is then given at any point by the gradient of this potential, u = V$. (2.3) It follows immediately that V x v = 0 so that the flow is irrotational. The incompressibility of the fluid is ensured by requiring that the velocity potential is harmonic: A<& = V2<& = 0. This condition is also referred to as the continuity equation; it prescribes that the system should not be gaining or losing matter. Such a velocity potential can be defined even for a three-dimensional ideal fluid. The peculiarity of the two-dimensional case arises from the fact that we can also define a vector 23 potential ty, such that the velocity is equivalently defined by the curl of this vector potential: ?=Vxf. Since V3 = 0, only the z component of the vector potential is non-zero and we have "-(<*•-*)• <2-4) where, for the sake of simplicity, the z component of \t has been denoted by \& and will be called the stream function for reasons that will be explained shortly. Here the incompressibility follows directly since the divergence of a curl is zero. The irrotationality of the fluid will then be ensured by requiring that the stream function is also harmonic: A\l/ = 0. The velocity field v = (vi, v2) according to the previous definitions is then given by vx = — = —, and u = — = - —. (2.5) ox ay 2 ay ox These equations are simply the Cauchy-Riemann conditions which must be satisfied for the complex function, fi(z), to be differentiable. This function is defined quite naturally as the complex analogue of Sl{z) = ${x,y) + M(x,y), (2.6) where z = x+iy. This function is differentiable and its derivative is called the complex velocity: Cl'{z) = vi(x, y) - iv2{x, y). (2.7) The analogue of the usual velocity vector is then given by taking the complex conjugate of this function: Cl'(z) = v\ + iv2- 2.1.3 The Velocity Potential •$ We want to define the flux of a fluid along a path as the line integral of the velocity along that path. For the special case of a closed path, this integral has already been defined as the circulation of the fluid for this closed path. Let A and P be any two points in the plane. The flux along any curve joining these two points has to be the same for the continuity equation to hold. Let A be fixed and P vary, then the flux along the path is a function of P such that $ = J% v • dl. If the lower limit was to be changed for another point 73, a constant term corresponding to Jg v • dl would be added. Thus, $ can be seen as being defined up to a 24 constant term and $ = $0+ f v-dl. (2.8) J A When P is moved to Q by an infinitesimal displacement, the flux along P($ is <5$ where 2.1.4 The Stream Function * Next, we consider the flux of a fluid crossing a path. By convention we will consider a flux to be positive when, looking from point A in the direction of point P, the fluid is flowing from left to right. Again let A and P be any two points in the plane and let A be fixed and P vary. The flux across any curve joining these two points has to be the same for the continuity equation to hold. With h being the normal to the infinitesimal curve element dl, the flux across the path * = / v-ndl= vxdl. (2.9) J A J A \I/ is defined up to a constant since changing the lower limit adds a constant term to the integral corresponding to f£ v • h dl. When P is moved to Q by an infinitesimal displacement parallel to the y axis, the flux across P~C$, for P$ = Sy, is v\ • 5x. IfP$ = Sx, the flux is —vi • by. So the velocity vector expressed in terms of has the same definition as (2.5). If P is again moved such that the flux remains constant, the path will be parallel to the velocity of the fluid everywhere and the result is a stream line. This is why we first defined to be a stream function. The stream lines and equipotential lines are two equivalent ways of describing the fluid flow configuration. We will use one or the other depending which one is better suited for the problem. By definition stream lines and equipotential lines are always perpendicular to each other. 2.1.5 Boundary Conditions Boundary conditions in a two-dimensional ideal fluid flow can be specified as a vanishing con• dition on the derivative of $ at the boundary (no flow through the boundary). An equivalent 25 and rather elegant condition imposes that ^> is constant on the boundary (the boundary is therefore a stream line). The first case is a Neumann problem whereas the second one is a Dirichlet problem. For problems with an infinite domain, some boundary condition must also be specified at infinity. Usually the velocity is required to be uniform or to vanish. The more complicated case of a moving boundary will call for some modifications to these conditions. Basically, the velocity of the fluid normal to the boundary should be equal to the normal component of the velocity of the boundary. 2.1.6 Conformal Mapping Conformal mapping is a powerful tool to solve problems that would otherwise prove much more difficult. The problem reduces to finding a complex quantity describing the solution in one simple known configuration that can easily be mapped to the configuration of the actual problem. This mapping then prescribes how to transform this complex function or solution. For the two-dimensional ideal fluid flow, this quantity is the previously defined complex velocity potential (2.6). More precisely, a conformal map is a transformation that will preserve angles. This means that the stream lines and the equipotential lines will remain perpendicular after applying a conformal map. Thus the Cauchy-Riemann conditions are still satisfied and the velocity field in the new configuration is still given by (2.5) albeit using the transformed <£• and 9. Also, the transformed boundaries are still stream lines. This ensures that the new transformed boundary conditions are respected. The determination of the exact motion of a fluid subject to given boundary conditions can be extremely difficult which makes conformal mapping only useful for certain categories of problem. In general, very simple problems involving circular boundaries can be solved directly. Also, when the boundaries consist of fixed straight walls, a method of transformation developed by Schwarz and Christoffel can be used, see [1], section 5.6, pp345-365. However, most problems are actually solved with an inverse method where one starts with some known complex velocity potential and inquires what boundary conditions it can be made to satisfy. We can also look at $ = const, as being the stream functions and $ = const, as the equipotential lines. Conditions (2.1) and (2.2) are still satisfied. To make sure that the Cauchy- Riemann conditions (2.5) are respected, we actually replace <3? by — VP and VP by Thus a conformal map can actually give us the solution to two different irrotational motions. For simple problems like the ones considered in this study, we will start with known problems on which we will apply simple transformations whose effects are well understood already to 26 achieve the desired configurations. 2.1.7 Multiply-Connected Regions reconcilable paths irreconcilable paths irreducible circuit reducible circuit reconcilable paths Figure 2.1: Simply-connected and multiply-connected regions A Few Definitions The problems that will be considered shortly involve multiply-connected regions. Before reviewing the properties of a flow in such regions a few definitions will be needed. In a connected region, it is possible to pass from any point to any other point of the region by an infinity of paths without crossing the boundaries. Any two paths lying in the region that can be made to coincide by continuous variation, again without crossing any boundary, are said to be reconcilable. Any closed path, or circuit, that can be contracted to a point without crossing a boundary is said to be reducible. Obviously a circuit formed by two reconcilable paths is necessarily reducible. One can also distinguish between simple and multiple irreducible closed paths. A multiple circuit is one that can be made to appear, in whole or in part, as a repetition of another circuit a certain number of times. Consider, for example, a boundary composed of a certain number of circular boundaries. A path that winds once around only one of these boundaries is a simple circuit. A path that winds around one boundary many times, or that winds around more than one boundary, is said to be a multiple irreducible circuit. Now for a simply-connected region all paths joining any two points are reconcilable, or equivalently, all closed paths are reducible. For a doubly-connected region, one can draw two irreconcilable paths joining any two points and all other paths are going to be reconcilable with 27 one of these. A doubly-connected region also has one simple closed path that is not reducible and all other simple irreducible closed paths are reconcilable with this one. In general, a region is n-ply-connected if any two points can be joined by n irreconcilable paths, or equivalently, n—l simple irreducible and irreconcilable closed paths can be drawn. Figure 2.1 summarizes some of these concepts. The connectivity of the region we are considering has important implications on both the velocity potential and the stream function such that a few of the previous statements will need to be modified. Velocity Potential in a Multiply-Connected Region Section 2.1.3 states that the flux along any two curves joining two points must have the same value. This is always true for a simply-connected region. For the more general case this statement is restricted to reconcilable curves. This also means that the circulation of a circuit in a simply-connected region is zero for an irrotational flow. More generally, the circulation of any reducible circuit is zero. Furthermore, two reconcilable closed paths have the same circulation. If the order of connection of a region is n, there are n — l independent simple irreducible cir• cuits. These circuits can be deformed to appear as the boundaries they are winding around. The circulation around each of these in the counterclockwise direction is denoted «i, «£, • • • >K n-i- If an arbitrary circuit winds around each boundary a certain number of times p\,p2,--- ,pn-i where pi is the number of windings around the i^ boundary such that a counterclockwise loop counts as +1 and a clockwise loop counts as —1, then one can write the circulation for this circuit as n-l r = (2.10) i=i Also, if the path is not specified, (2.8) is undetermined up to a constant of the form (2.10). Stream Function in a Multiply-Connected Region Similar modifications to the state• ments of section (2.1.4) are also required for the stream function. The flux across two different paths joining any two points is necessarily the same only for reconcilable paths. The flux flowing outside of a circuit is always zero for an incompressible fluid in a simply- connected region. Equivalently, the flux across any reducible closed path will be zero. Again, with n — l independent simple circuits corresponding to n — 1 boundaries, the flux through each of these, taken to be along the counterclockwise direction, is denoted ji2, • • •, Mn- 28 If an arbitrary circuit winds around each boundary a certain number of times pi,P2> • • • )Pn-i with the same convention as before, then one can write the flux through this circuit as n-l flux=5>iMi. (2.11) Then (2.9) for a path that is not specified is undetermined up to a constant of the form (2.11). 2.2 Description of Quantized Vortices We now present how a vortex is described in the context of an ideal fluid flow as well as the assumptions we will make on its properties. To abide by our two-dimensional description, only rectilinear vortices parallel with the z axis will be considered. An off-center vortex inside a cylinder will serve as a demonstrative example throughout the rest of this chapter. 2.2.1 Properties of a Rectilinear Vortex The assumption that the flow is a potential flow, as in (2.3), has very restrictive consequences on the possible motions of the fluid. Foremost is the irrotationality of the velocity field unless the velocity potential has a singularity. As a vortex is by definition a rotational motion we can therefore expect a singularity in <&. However, our starting point will be the striking property of quantization of the circulation. As mentioned in the introduction, the single-valuedness of the wave function requires that the change in its phase along a closed contour to be an integer multiple of 2n. The circulation V is therefore quantized in units of h/m. From (1.26), we find the expression for the velocity field of a single vortex to be K, (2.12) 6 2rrr' where the circulation of the vortex is K = l(h/m) with I being an integer. We see that the velocity diverges for r —> 0 which is evidence enough that the structure of the core is very different from the surrounding liquid. To avoid the divergence, we assume that p —> 0 as r —> 0, where p is the density of the fluid. We further assume that the density falls from its bulk liquid value to zero along the typical lengthscale £ which we call the core radius or the healing length. What actually happens close to the core is a complex problem, but corrections to this approximation are only of order £2, where £ is considered small compared to other lengthscales of the problem. The approximation used here corresponds to a vortex with 29 an empty core, which differs from calculations done in the book by Lamb, for example. In the problems we will consider, the vortex core is part of the boundary which also means that the region occupied by the superfluid is a multiconnected region. This vortex core is assumed to be a circle with the singularity at its center, again up to corrections of order £2. 2.2.2 Velocity Potential and Stream Function Using (2.12) and integrating (2.5) yields an expression for and * of a vortex at the origin. Up to an integration constant we have $ = — arctan = — 6, (2.13) 2TT \xj 2vr y J = -±]n(y/x*+y*) = -£\nr. (2.14) $ is therefore multivalued but its gradient, the velocity field, is still single-valued. As far as the velocity field is concerned, it does not matter if we simply use the principal value of the arctan for For considerations related to the evaluation of the energy, however, we will take the arctan to be defined from — n/2 to 7r/2 and we will add a correction of ir when the argument is between n/2 and 37r/2. The stream functions are circles and the stream constant diverges at the vortex center. The equipotential lines are straight lines meeting at the vortex center. The more general case of a vortex located at (xo, yo) is easily obtained by the change of variable x —> x — XQ and y —> y — yo. The complex velocity potential is readily obtained from (2.13) and (2.14) and is Sl(z) = -i£-]n{z-zo), (2.15) where z$ = XQ + iyo. We mentioned in section 2.1.6 that exchanging the stream functions and the velocity po• tential give the solution to a different problem. Here the stream lines are straight lines meeting at the center while the equipotential lines are circles centered around ZQ. Substituting fj, for K/2TT we get the complex velocity potential of a source with flux /j,, where n(z) = /j,ln(z- z0). (2.16) 5See reference [23]. Many examples involving vortices are solved therein. 30 2.2.3 Example: Off-Center Vortex in a Cylinder To illustrate, we now study a simple problem that will prove useful to our investigation of more complicated configurations. Consider an off-center vortex in a cylindrical container of radius R. The vortex is located at a distance b from the center of the cylinder and is located on the x axis without loss of generality. The vortex core radius is again £. The boundary consists of two cylinders: the vortex core and the container itself. Method of Images To solve this problem we can use the method of images and place vortices outside of the region we are interested in to ensure that the boundary conditions are satisfied. Let's first look for a simple solution involving only one vortex. By symmetry, this vortex should be on the same axis as the first one. Furthermore, remember that for the boundary conditions to be respected we want the stream function to be constant on the boundary. From (2.14) the stream function of two vortices is simply In r In r (2.17) 27T 27T If we choose K2 = —«i such that the image vortex has the same circulation but with opposite sign6, the condition for \5 to be constant on the boundary is that r/r' should also be constant. We then consider the geometry summarized in Figure 2.2 to determine where the antivortex should be placed. Note that for a solution to be found we have to allow for an offset between the center of the vortex core and the actual vortex position where the singularity is located. This distance should however be negligible. Let point A and point A' be the positions of the vortices, O the center of the vortex core and O' the center of the container. First, we define \OA\ = dx, (2.18) \0'A'\ = d2, (2-19) \00'\ = 5. (2.20) For the circles \OP\ = £ and |0'P'| = R to be stream lines, we want r/r' = constant. On the small cylinder this ratio is found to be r_ £-di g + di = = (2.21) r> d2-8-Z d2-6 + C 6From now on, we shall refer to a vortex of negative circulation as an antivortex. 31 Figure 2.2: Method of images for a vortex inside a cylinder. The small circle with radius £ is the vortex core and the big circle with radius R is the container. and on the big cylinder r[ = d2-R = d2 + R r R-6-di R + 6 + di' { ' 1 Equations (2.21) and (2.22) respectively reduce to d (d -s)=e L, = T> (2-23) 1 2 - r t, 2 d2(d1 + 5) = R - ^ = % (2-24) Neglecting d\ in (2.24) we find that the antivortex is at a distance of R2/5. In our problem, 6 + di is the position of the vortex which we defined as b such that we can approximate 5 by b. The antivortex is then at a distance of R2/b from the origin. We should also check that di is indeed small. Substituting 5 = b — di in (2.23) and solving for di we have di = + 1^-6)2 + ^, (2.25) where we have kept only the positive solution for di. If £ is small, we can substitute x = 4£2 and do a Taylor expansion: 32 such that e d (2.27) X 2 ~ (d2-6)" With d2 = R /b, (2.28) {R2 b- b2)' which is of order £2 and should be neglected. Figure 2.3: Geometry for an off-center vortex with an image vortex Velocity Field and Complex Velocity Potential Figure 2.3 summarizes the variables and the geometry used. Since the stream function and the velocity potential for a vortex are already known, see (2.13) and (2.14), we can directly write the solution of this problem by adding the contribution from each vortex. The velocity field can then be obtained from (2.5). For illustration purposes, let's start with the velocity field and its property of additivity. Using v as the contribution due to the actual vortex and v' as the contribution from the image vortex, the total velocity field is simply the sum of the two. Since each is given by (2.12), we have -y , y_\ (, c \ v + v' = — I2 2 + 2 (2.29) zn D2 + D )\D D' J From Figure 2.3 one obtains D2 = a2 + y2, (2.30) 33 D'2 = c2 + y2, (2.31) a + b + c = R2/b, (2.32) and using the fact that a + b = x we get a = x-b, (2.33) and c = R2/b - x. (2.34) Inserting these results back into (2.29), we find that the velocity vector is ,n _ JS_ i -?/ i a U + (2.35) l - 2,r \x*+v'+l'-2bz !g.+X2+Y2_2j£x 6 —-x Knowing that §f = t>i and |j| = v2, one derives from (2.35) the following expression for the velocity potential: K ( x — R2lb x — b\ „„. And similarly with ^ = v2 and ^ = —t>2, one can write the stream function as « = iJ5W^\ (M7) 4TT b2 + a;2 + y2 - 2bx J v y This is of course defined only up to a constant term. The constant term in 34 2.3 Kinetic Energy The last topic of this chapter is devoted to the evaluation of the kinetic energy of the fluid. Knowing the change in kinetic energy of the fluid when adding a vortex to a specific configura• tion will yield the actual energy of the vortex. If we express it as a function of the position of the vortex, we effectively find the potential barrier that must be overcome for vortex nucleation to occur. The kinetic energy, T, can be evaluated by integrating the square of the velocity field over the region occupied by the fluid: 2T = p J J v2dS = p J J(v\ + vl) dS. (2.39) This integral can be complicated however, and it is often advantageous to reduce it to a contour integral. We will first calculate the energy of a vortex inside a cylinder by comparing it to a similar problem. Then a more general approach will be presented. 2.3.1 Energy of an Off-Center Vortex in a Cylinder Evaluating (2.39) with (2.35) is still easily done for this problem. We first carry out the angular part of the integral. For the radial part, we consider a circle of radius b — £ and then an annulus of small radius b + £ and big radius R. Neglecting higher order terms in R/£, the kinetic energy7 is 2f , (R2-b2 2T n In - + In (2.40) 2TT X) \ R2 Before using a simpler method to solve for the kinetic energy of a vortex, it is helpful to consider a well known problem that can be mapped to the vortex case in a straightforward manner. Analogy With Two Charged Cylinders Consider a capacitor made of two concentric cylinders as in §3 of [22]. In this problem, the analog of the velocity field is the electric field E. Since there are no charges in the region of interest, one can equivalently define a potential or a vector potential for the electric field. In the two-dimensional case, the vector E lies in the xy plane while A is chosen to be perpendicular (along the z axis). E = -V0, (2.41) 7This yields the same result as in section 9.2.3 of [34]. 35 £ = Vxi. (2.42) From these, one can calculate the two components of E: These conditions are again the Cauchy-Riemann conditions. One can also define a complex potential, similarly to the ideal two-dimensional flow: w = From there, we could map this problem to other configurations. To illustrate, let's consider a charged straight line passing through the origin and perpen• dicular to the xy plane. The field is given by Er = 2A/r and Eg = 0 with A being the charge per unit length. Because of its symmetry, this problem proves much easier to solve in polar coordinates. We will therefore use the Cauchy-Riemann conditions in polar coordinates: du _ 1 dv (2.46) dr r 88' dv 1 du (2.47) dr r d9 We find ^ = Er = *i (2.48) dr r r dr $ = -2A / — = -2Alnr, (2.49) and A = 2A j d9 = 2A6», (2.51) such that the complex potential is w = -2Alnr-2iA6' = -2Aln2. (2.52) 36 Assuming that we know what the complex potential is, how can the energy be calculated? The energy stored in a capacitor is well known to be given by Energy = — = (2.53) where C is the capacitance and is defined as C = — = electric charge V potential difference In other words, the capacitance defines how big a charge can be stored per volt. The electric flux is defined as j>E-dS. (2.55) The flux of the electric field through a section of an equipotential line is j> Endl = - fjj^dl, (2.56) where dl is an element of length along the equipotential line and n is the normal to it. According to (2.43) and (2.44), dn- = -dl- (2-5?) Therefore, (2.56) simplifies to BA E dl = —dl = A -A . (2.58) n al 2 1 We know that the flux through a closed contour is 47rQ. One can then express the capacitance and the field energy as ^ Air fa - i ' ^ ^ 2 Energy = -L(A2 - Ai)(02 - ( '60) 07T Capacitance of Two Parallel Cylinders We consider two infinite conducting cylinders with the same geometry as in Figure 2.2 and we want to calculate the capacitance and the energy of this configuration. The field produced between the cylinders is the same as the one produced by two charged wires passing through point A and A'. For the cylinders to be 37 Figure 2.4: Equipotential lines and lines of force for a cylindrical capacitor equipotential lines we want r/r' to remain constant on them. Using (2.49), the potential on the cylinders is found to be 01 = -2Aln(%Y (2.61) &-&=2Aln(^). (2.63) Finally with A2 — A\ = A2V — AQ = 4ir\, we write the capacitance is and Energy = A2 In (^). (2.65) Energy of a Vortex Inside a Cylinder The analogy between the simple electrostatic problem of two charged cylinders and the hydrodynamic description of a vortex inside a cylinder is straightforward. The small cylinder corresponds to the vortex core and the big one to the container. As we have seen, the boundary conditions are similar since the field velocity 38 perpendicular to the cylinders has to vanish (instead of the parallel component of the electric field). Figure 2.5 shows the complex velocity potential components of an off-center vortex. For the energy, instead of 1/2E2 we integrate l/2v2. By analogy to the results in the previous section, we can write the kinetic energy of the fluid flow as Energy =|(*2-$i)(M'2-*i). (2-66) The analogue of (A2 — A\), which was the flux through a section of the potential line, is now the potential difference ($2 - $1) which is the circulation on a section of a stream function. K The quantization of the circulation in a superfluid implies that ($2TT ~ $0) — - We can check for the potential in (2.36) of a vortex inside a cylinder. As 6 —* 0, x —> R and y —> 0+ such that ,. K ( (bR-R2\ $n = lim — arctan —— — arctan . „ , 2ir\ V 0+6 y V 0+ lim ^- ^ arctan (—00) — arctan (00)^ K I 7T 7T\ K (2.67) _ = 2TTI _ 2 2) ~2' and as 9 —» 27r, x —> R and y —> 0 such that K / (bR-R2\ (R-b $27T = hm — I arctan I —-^—^— J — arctan 39 = lim ^- ( arctan (co) — arctan (-co) = 2^2 + 2) = 2- ^ As expected A<3> = K. The analogue of the potential difference (02 — x = 6 + £cos0, (2.69) V = £sin0, (2.70) and we get Ri b2 b2 2 cos 9 2R2b b cos ,r = " ln ( + ( + ? + % ) ~ ( + £ °)] (27i) LT/ — — 1 4TT \^ B?e We then neglect terms in the logarithmic function that are of higher order than 0(£-2) which yields 4 2 2 2 2 K (R* + b - 2R b \ K [R + b \ ,0_ With (2.66) we can now write the energy of a vortex in a cylinder: which is the expected result (2.40). 2.3.2 Line Integral on the Boundary If we want to study more complicated configurations, for example by applying a few conformal maps to the previous problem, the kinetic energy as evaluated with (2.39) yields a surface integral that can prove difficult to work out. To simplify, one can express the kinetic energy using (2.5) and 2T = p J J v2dxdy = pj J{v\ + v2) dx dy 40 = pi f (V • ($V$) - $A$) dx dj/, (2.74) where the second term is zero since $ is homogeneous. The first term can be replaced by an integral over the boundary by the divergence theorem and we finally get 2T = pj>$V$>-hdl = pj^^dl, (2.75) where n is the unit vector normal to dt, as usual. Similarly we can also express the kinetic energy in terms of the stream function. Since -2=(f)2 + (^)2weget 2T - pj^W -hdl = pj^^dl. (2.76) As $ and \P are always perpendicular, one can generalize (2.5) and write 3$ 3* , 3* 3* 3T=3r7 and dn- = -dl- (277) Substituting in (2.75) and (2.76) we get the forms we will be using: 2T = -pj$d$>, (2.78) and 2T = pj-$d§. (2.79) This last expression is actually the one that has been used in section 2.3.1 to calculate the energy of an off-center vortex. 2.3.3 Branch Cuts The two expressions for the kinetic energy derived in the last section are equivalent but will require a different treatment of the branch cuts. When the problem involves vortices and no source, we have a branch cut in $ such that we will need to integrate around them if we use (2.78). This is not the case if we use (2.79) as \P does not have any branch cut. The branch cut for the problem of an off-center vortex is illustrated in Figure 2.6. Conversion from one expression to another involves integrating by parts. If there are branch 41 cuts in one of these expression, boundary terms of the integration by parts will conveniently account for them. 42 3. Simple Yet Useful Configurations The objective of this chapter is to calculate the shape of the potential barrier for the nucleation of a vortex in simple yet experimentally realistic geometries. The potential barrier is obtained by calculating the kinetic energy of the flow in these configurations in function of the position of the vortex. Since vortex nucleation is suspected to occur near irregularities of a rotating vessel, we first consider a perfect container with one bump. We then turn to the problem of a flow past a bump on a flat wall, which is equivalent to an infinitely large cylinder. 3.1 Vortex in a Cylinder With a Wedge This first configuration starts from the 'off-center vortex in a cylinder' case that we presented as an example in the previous chapter. As shown in Figure 3.1, the additional bump is infinitely narrow and we will refer to it as a wedge. It has height h and is located on the x axis. The vortex core still has radius £ and the cylinder has radius R. The vortex position is denoted as (xb, Ub) or (6cos/3, 6sin/3) in cylindrical coordinates. Figure 3.1: Mapping to a cylinder with a wedge. The height of the wedge is h and the position of the vortex is (xb, Vb)- 43 3.1.1 Starting Configuration With No Wedge The vortex can now be located anywhere in the cylinder such that (2.36) and (2.37) need to be modified accordingly. One obtains the velocity potential ,, . K ( x — (R2 lb) cos (3 x — bcos/3\ and stream function K f R4+ b2{x2+ y2) - 2R2b(x cos f3+ y sin/3) (3.2) ^' V> ~ 4^ n \R2b2 + R2(x2 + y2) - 2R2b{xcos(3 + y sin/3) 3.1.2 Conformal Mapping Now that we have a basic configuration for which the complex velocity potential is known, the next step consists in mapping it to the final geometry, which will be done in three steps. The first map transforms the interior of the cylinder to the upper half-plane. We then map to the upper half-plane with a wedge and finally map back to a circle with a wedge. Figure 3.2 summarizes these steps by showing what the stream lines are after each transformation. Only the stream functions are mapped here but potential lines can be obtained using the same changes of variables. Mapping to the Upper Half-Plane The circle of radius R is first transformed into the upper half-plane with z(x, y) —> w(u, v) as shown in Figure 3.3. This mapping is given by « = <(£f). (3-3) which corresponds to the following change of variable: 2Ry , R2 - (x2 + y2) . . U~TT> To o and V = — (3.4) {R + x)2 + y2 {R + x)2 + y2 Solving for x and y, one has R(l-(u2 + v2)) 2Ru x = — 7T~ and y = — ^5 5-. (3.5) (1 + v)2 +u2 (l+v)2+u2 v 44 x •-> z plane w plane x I plane o plane Figure 3.2: Mapping from a circle to a circle with a wedge This yields V(u,v) = •^ln\(u2 + v2)(R2 + b2 + 2RbcosP) (3.6) + [R2 + b2- 2Rbcos(3) + 2v{R2 - b2) - 4«i?6sin/?] In (u2 + v2)(R2 + b2 + 2Rbcosp) 47T + (R2 + b2 - 2RbcosP) - 2v(R2 - b2) - 4uRbsm/3 . Mapping to the Upper Half-Plane With a Wedge The goal is to map the upper half- plane to the upper half-plane with a wedge of height ho located at the origin, denoted by the map w(u, v) —> l(m, n). 45 v A u > CL\ a2 0,3 &4 05 2 plane to plane Figure 3.3: Mapping from a circle to the upper half-plane. The points denoted by uppercase letters are mapped to their lowercase counterparts. A clever mapping shown in Figure 3.4, see [1], pp. 355-356, that can be used to map a linear flow to a flow blocked by a wedge of height ho is a very good starting point: / = ho\/w2 — 1 (3.7) vA A2 A3 A4 a2 (14 w plane I plane Figure 3.4: Mapping to the upper half-plane with a wedge. The points denoted by uppercase letters are mapped to their lowercase counterparts. The factor ho, which is necessary to impose the height of the wedge, also acts as a scaling factor on the entire plane. This has no consequence for the problem of linear flow proposed by Ablowitz and Fokas [1]. In the problem studied here, the wedge has to be a small perturbation 46 with little effect on the stream lines away from the wall. A more sound mapping is then given by (3.8) such that I —> w for w 3> 1. The adequate change of variable can be derived as follows. The real and imaginary part of (3.8) give 0 0 0 0 0 mn = uv and m—n=u—v— hn. Solving for v2 yields -1 V = — (m2 - n2 + hi) ± \\]{m2 - n2 + hi)2 +4m2n2. (3.9) The second solution is ignored to ensure that v is real. The solution corresponding to the upper half-plane is -{m2 - n2 + h2) + ^{m2 - n2 + h2)2 + 4m2n2 v = ^ (3.10) and u = mn (3.11) \ -(m2 - n2 + hi) + ^(m2 - n2 + hi)2 + 4m2n2' The reverse transformation is given by -(u2 - v2 - hi) + yj{u2 - v2 - hi)2 + 4u2v2 (3.12) 2 and m = uv (3.13) 2 2 2 2 2 2 2 2 \ -{u -v - h 0) + ^{u -v - hi) + 4u v ' Defining the following variables to simplify these expressions: C = (m2 — n2 + hi) and D = 2mn, (3.14) C' = (u2 - v2 - hi) and D' = 2uv, (3.15) 47 we write J-C + ^/C2 + D2 «= y 2 ' ' ' u = ^ =, (3.17) \J-2C + + D2' u2 + x,2 = VC2 + /J>2, (3.18) and n = y , (3.19) m = , • D (3.20) 7-2C + 2VrC/2 + D'2 ' m2 + n2 = VC'2 + D'2. (3.21) The stream function becomes *(m, n) = (3.22) In ( VC2 + D2(R2 + b2 + 2Rbcos (3) + {R2 + b2 - 2Rb cos /3) 47T . \ + ^-2C + 2^T&{R2 - b2) - , 4RbDsi*P ) ^-2C + 2VC2 + D2/ - — In [ VC2 + D2(R2 + b2 + 2Rbcos (3) + (R2 + b2 - 2Rbcos0) An \ 2 2 2 4RbDs mP - J-2C + 2VC T&(R - b ) - , ' ). / 2 2 v/-2C + 2v C + JD / Mapping Back to a Circle Finally, we map back to a circle with a wedge, l(m, n) —> o(x, y), as shown in Figure 3.5. For simplicity the variable x and y are used as in the first configuration but it is important to note that they are not the same. The change of variable is essentially the same as in (3.4) and (3.5). First, we have 2Ry , R2 - (x2 + y2) an( n M = TT; v> o i ~ ~7r, ^5 o~> (3.23) (R + x)2+y2 (R + x)2 + y2 v ' 48 n A x > m Ai A2 A3 A£ A4 A / plane Figure 3.5: Mapping from the upper half-plane with a wedge to a circle with a wedge. The points denoted by uppercase letters are mapped to their lowercase counterparts. with the reverse mapping R{1- m2 + n2 2Rm (l + n)2+m2 (l+n)2 + m2 v Substituting in equations (3.14) yields L 2 C=^ + h 0, (3-25) [(R + x)2 + y2) and D = 4^2^2-^). (3.26) ((R + x)2 + y2) The height of the wedge inside the circle is readily obtained from (3.23). Let ho be the height in the upper half-plane and h be the height inside the circle, then "»- srb <•*•"> 49 3.1.3 Corrections to the Position of the Vortex Even if one starts from a circle and maps to a circle of the same radius, the addition of the wedge modifies the coordinates of every point inside the circle. This is a very small effect for points far from the wedge, or when the wedge is very small. The vector plot in Figure 3.6 shows how positions are shifted. Figure 3.6: Coordinates offset due to the mapping Clearly, the position of the vortex is also shifted by a small offset: a vortex at (b cos /?, 6 sin (3) is shifted to (a cos a, a sin a). When asking what the energy of a vortex in a cylinder with a wedge is when the vortex is at a specific position, care must be taken to determine the right position of the vortex in the starting configuration to take this offset into account. In fact, with the velocity varying roughly as 1/r close to the vortex core, a slight shift in position is enough to have a big impact on the shape of the kinetic energy. To determine the kinetic energy, we want to express b and (3 in terms of a and a. Starting with a vortex at position (acos(a),asin(o!)), we map this position back to a cylinder without a wedge using (3.5), (3.17), (3.16), (3.14) and (3.23). The answer is once again very complicated and the following functions are defined for convenience: fix, y) = ^C2(x, y) + D2(x, y) and g(x, y) = C(x, y), (3.29) where C(x, y) and D{x, y) are defined in (3.25) and (3.26). In a more compact notation, only / and g will be used and the subscript a will indicate that they are evaluated at the vortex position such that fa = /(a cos a, a sin a), (3.30) 50 and 9a = g{& cos a, a sin a). (3.31) We then have the following substitutions 2 2 2 _ P (/a + 2ga + 1) ' /i i £ , ATT o^\9> 2 l^.OZJ (l + /a + V2/a-2ffa) ' ^(1 - /a) {1 ] = , , * J: n, , (3.33) l + /a + V2/a-2ffa' iV2/a + 2ga) yo = i • f • FTt l„ ' (3'34^ 1 + /a + V2/a - 2^a and (3.22) becomes *(*, y) = (3.35) ^ In ^(/ + /a)(l + fa + s/2fa - 2ga^j / + y/2f-2g(ja ~9a + 0.5(1 + /a)v 2/a-25a) / 2 2 - v 27+2^( \//a - 5 + 0.5(1 + /„) y^/a + 25a / - ^hl ^(/ + /a)(l + /a + V 2/a-25a) - y/2f-2g(fa -ga + 0.5(1 + fa)y/2fa-2g^j - y/2jT2gi^Jfl -g% + 0.5(1 + /a)^/a + 2Pa 3.1.4 Kinetic Energy Bulding on the example of section 2.3.1, we first find the kinetic energy of a vortex anywhere in the vessel. This solution lends itself to further simplifications for the case of a vortex on the same axis as the wedge. General Expression For this problem, it is advantageous to reuse the technique presented in section 2.3.1 for an off-center vortex where the energy is given by (2.66). The vortex core is assumed to remain a circle up to a quadratic correction in £ such that $2 — $1 = « is still valid. Also, (3.35) is still zero at the container boundary. This leaves \P to be calculated on 51 the vortex core with (3.35). Using /? = /((a + 0 cos a, (a + £)sina), (3.36) and 2T = (3.38) ~ In ((/? + /a) (l + /a + V2/a - 25a) + - 2# (/a - 5a + 0.5(1 + /a) v^/a - 25a - \M + 2^- gl + 0.5(1 + fa)y/2fa + 2ga PK2 ^ In |tf + /a)^l + fa + y/2fa - 25a - y/2fe-2gs(ja -ga + 0.5(1 + fa)y/2U^2g^ - fa + 29i (Jpa -gl + 0.5(1 + /a) y/2fa + 2ga^j This result is accurate when the vortex distance to the wall is large in comparison to the vortex core parameter. Otherwise, the assumption that the vortex remains perfectly circular and that it is a stream function breaks down. In reality, the energy should go to zero in the limit where the vortex and the antivortex meet on the boundary. While the behaviour as the vortex approaches this limiting case might not be rendered accurately, (3.38) still drops to zero when the vortex is inside a distance of £ from the container. With £ considered small compared to other lengthscales, this proves to be an acceptable approximation. Figure 3.7 shows the energy of the vortex. One clearly sees that T(a) is minimum at a = 0 when the vortex is on the same axis as the wedge. Vortex nucleation will therefore be more likely to occur on the x axis because the potential barrier is lower. Figure 3.8 shows the shape of the barrier for a = 0 and emphasizes that the energy with the wedge is bounded by the energy of a vortex without a wedge. 52 Figure 3.7: Kinetic energy for a cylinder with a wedge in function of the position of the vortex Energy of a Vortex on the x Axis When a = 0, (3.38) can be significantly simplified. Firstly, C(a,0) = -jf^ + ^, (3.39) and D(a, 0) = 0, (3.40) 2 where now a G [-R, R - h}. With D = 0, (3.30) reduces to fa = y/C' (a, 0). Substituting (3.28) in (3.39) and solving for the zeros gives a = -R and a = R — h. Thus C(a, 0) is always negative and we define 7a = 5a|Q=0 = -/aL=0 = C{tL, 0). (3.41) Similarly, C(a + C, 0) for a G [0, R - h - £] C{a - t, 0) for a e [-R + C, 0] 53 fa 2T Figure 3.8: Kinetic energy of a vortex on the x axis in a cylinder with and without a wedge in function of the position of the vortex. The height of the wedge is h and the radius of the container is R. where the domain for a is more restrictive. As pointed out above, the results are not expected to hold when the vortex is in the vicinity of the wall. Finally, (3.38) becomes 2n 2T=^ln In fa + ft+ V fafj (3.43) 47T fa - VJ? 4n fa - /| Terms of order £2 should now be neglected. We substitute x for ±£ depending on the sign of a and we do a Taylor expansion of 7^ and y^ToT^- 2 2 (R-a) (R-a) \ /ri. ,. 7£ = 7a + -4 J 2 (3.44) (R + a)3 + a7Tr-—(R + a)2 -\x + 0(x ^ ^TaT? = -2-2R x + 0(xz). (3.45) la (R + a)3 With £ much smaller than the other lengthscales, the part of the numerator of (3.43) pro• portional to £ can be neglected while the denominator is then proportional to £2. Thus the following expression for the kinetic energy does not depend on the sign of a, and (3.43) now 2 2 pKn 'R\ , (R -a?\ , / (R-a) 2T = In ( - ] + In In (3.46) ~2T7 R2 (R-a)2 -h2(R + a)2 54 3.1.5 Rotation of the Cylinder Figures 3.7 and 3.8 do not have a local minimum and a metastable state does not seem to be allowed. In fact, the container needs to be rotating as in the Hess-Fairbank experiment for the vortex state to be favourable. If the container is rotating with angular velocity Co, the velocity v of an element of fluid relative to the uniformly rotating frame is related to the velocity vo in the inertial frame of reference by VQ = v + Co x r. (3.47) For a perfect cylinder, accounting for the rotation is easily done with the transformation rule, see §39 of [25], for the energy between a uniformly rotating frame and an inertial frame: E = EQ - L • Co, (3.48) where EQ and L are the energy and the angular momentum in the inertial frame and E is the energy in the rotating frame. This rule is derived from the fact that the rotation of the frame adds a centrifugal potential energy term \(Co x v)2 where v is given by (3.47). One can then start from a non-rotating configuration, calculate the energy, and then add a correction term accounting for the rotation of the configuration, provided the angular momentum can readily be calculated.8 The angular momentum of a vortex in a cylindrical container is L = p J r dr J vgr d6, (3.49) where the angular part of the integral is actually the circulation §v • dl. The circulation depends on whether the vortex lies inside the integration path and T = K,Q(r — b). The angular momentum is then given by L = pK / r dr = — (R2 - b2), (3.50) Jb 2 and the energy (2.40) becomes pK , . R\ , (R?-b2 IT - PKW(R2 - b2). (3.51) ~2T7 For a cylinder with a wedge, (3.47) predicts there will be a fluid flow through the wedge, which is prohibited. One might try to take the rotation into consideration before applying the 8This is used in [34] to calculate the critical angular velocity for a vortex state to be favourable. 55 Figure 3.9: Modification to the kinetic energy of a vortex in a rotating cylinder. The angular velocity is u) = 0.5K/R2 and the radius of the container is R. conformal maps. Unfortunately, the solid rotation of a fluid, where war, can't be described by a complex velocity potential, which precludes any use of conformal mapping. This makes it difficult to determine what the effect of the wedge would be on the velocity field. One solution is to consider a container which is big enough compared to the vortex and the wedge such that the problem reduces to a flat wall with a wedge. This motivates our interest for the configuration presented in the next section, namely a flat wall with a bump. However, the present calculation remains worthwhile as Figure 3.9 and 3.8 already hint at what the exact potential must be for this configuration. 3.2 Circular Bump on a Flat Wall Since the rotation of the container is difficult to account for, we now consider a container large enough to be approximated by a flat wall in the neighborhood of the bump and the vortex. The motion due to the rotation is now replaced by a linear flow that has been mapped to satisfy the new boundary. Similarly to the rotating case, when this flow is strong enough the vortex state becomes a metastable state. Another difference in this configuration is the use of a semi-circular bump instead of a wedge. This choice will prevent the need for a conformal map like (3.8) which introduces square roots in the equations and accounts for most of the complexity of our previous solutions. A circular bump is a valid approximation for most cases where the irregularities of the container are small. 56 3.2.1 Complex Velocity Potential All the elements we need to express the complex velocity potential are well known results and we will only need to add up their contributions. We first consider a full circle and solve for the energy of the entire xy plane. The x axis has to be a stream function as well for this to be equivalent to a semi-circular bump on a wall when the energy is divided by two. Flow Past a Cylinder and Moving Cylinder The complex velocity potential for a flow past a cylinder is Q.{z) = u(z +— \, (3.52) where R is the radius of the cylinder and the velocity field tend to ux away from it. From (3.52), the velocity potential is *(Xiy) = u(\ + -^j>)x, (3.53) while the stream function, shown in Figure 3.10, is = «(i-p^)v. (3-54) The stream constant is zero on the cylinder and the x axis and (2.5) gives v\ —> U as r S> 1. Equivalently, one can consider a cylinder moving with constant velocity —Ux. This amounts to a transformation to another inertial frame moving away from the first with velocity u. The complex velocity potential is now n(z) = U-, (3.55) z and UR2r xz + yL while the stream function is 3 57 ¥(*, y) = A ( - ) x* + y Now the x axis is still a stream constant but the cylinder is not. As mentioned in section 2.1.5, the component of the fluid velocity normal to the surface must be equal to the normal component of the velocity of the boundary. The fluid velocity normal to the surface is given by 3$ V = -[/cos0, (3.58) r\r=R dr 57 while the velocity of the boundary is —Ux. Thus the normal component to the surface is the same as in (3.58) and the boundary conditions on the cylinder are satisfied. flow past a cylinder moving cylinder Figure 3.10: Stream functions of a flow past a cylinder and a moving cylinder. In the first case the velocity of the flow at infinity is Ux. In the other, the velocity of the cylinder is —Ux. Figure 3.10 shows that the stream function is the analogue of the potential for an electric dipole. In the context of a fluid flow this object is called a doublet source with a velocity potential given by •—££lnr, (3.59) with d/ds being the space derivative in the orientation of the doublet. Equation (3.56) is recovered for fi = —2-KUR2. Vortex Configuration The vortex is located outside of the cylinder at (a cos a, a sin a) where a > R and a e [0, IT]. This problem has already been solved in section 2.2.3, except now the vortex is outside of the cylinder and the image vortex is inside. With the method of images, the antivortex is located at ((R2/a) cos a, (R2/a)sina). The boundary conditions on the cylinder are satisfied but there is a flow across the x axis. Using the method of images, we know that the lower half-plane has to be an image of the configuration of the upper half-plane. Thus there is also an antivortex at (acosa, —asina) and a vortex at ((R2/a) cos a, —(R2/a)sma). Figure 3.11 summarizes this distribution. Interestingly, this configuration is the same as a two-dimensional description of a vortex ring located close to a sphere and it should allow us to give some results for the nucleation rate of vortex rings produced by moving ions. 58 Figure 3.11: Vortices configuration for a cylindrical bump on a flat wall. The radius of the bump is R. There is a vortex at (acosa, asina) and ((R2/a) cos a, — (R2/a)sma) and an antovortex at (acosa, —asina) and ((R2/a) cos a, {R2/a) sin a). Sum of the Two Configurations Using the previous results (2.13) and (2.14) and adding the contribution from the moving cylinder in (3.56) and (3.57), we have UR2x K . y-yo . y + yo (3.60) z arctan arctan ar + y 2ir x — XQ X — XQ ^ y + %-yo ^ y - ^yo + arctan R^ 2 arctan R2 X - %-Xrj X — ^XQ and UR2y K 2 2 ln^(x- x0) + (y- y0) (3.61) x2 + y2 2n 1 / / R2 \2 ( R2 2 2 x \n^(x- x0) + {y + y0) + lnJ(x- ~^ Qj + f V + -^Vo R2 \ 2 / R2 \ 2' where XQ = acosa and yo = asina. The stream functions of each component is shown in 3.12. 59 u x X -> -> Figure 3.12: Stream function of a vortex near a moving cylinder. The cylinder is moving with velocity —Ux 3.2.2 Kinetic Energy The configuration when there are no vortices and only the moving cylinder is the starting configuration, and its energy must be calculated first. It is a metastable state that can decay into a state with one vortex. Then one can use either (2.78) or (2.79) on the boundary to calculate the energy of the vortex. This boundary is the combination of the cylinder and the two vortex cores located outside of the bump. Of course both equations yield the same but in (2.78) the branch cuts of (see Figure 3.13), need to be taken care of. Here (2.79) will be used to keep the calculation more succinct. Recall that (2.79) states that 2T = p§^ d$. Figure 3.13: Branch cut for the vortex configuration near a cylinder 60 Metastable State We need d$ on the boundary r = R. Using (3.57), one get — = -URsin9, (3.62) ad r=R and with $>\r=R, the energy is /•2ir 2 2 2 2 2 2T0 = pU R / sin 9 d$ = PU TTR , (3.63) Jo which corresponds to the kinetic energy of a quantity of fluid moving at velocity U and having the same volume as the cylinder. This quantity will need to be substracted from the total kinetic energy to isolate the vortex contribution. Integral on the Cylinder On the cylinder, it is convenient to use polar coordinates such that the boundary is r = R and the total derivative d<& — ^\r=Rd9. From (3.60), and using the shorthand notation (XQ, yo) for (a cos a, a sin a), we have 2 2 <9$ UR . _ , K ( r -rx0 cos 9 - ry0 sin 9 \ sm 2 2 39 r 9 + —[-2TT \aT— + r3— —o 2rxoa coso_6 —_ 2ryo1 sin9 (3.64) 2 K ( r — TXQ cos 9 + ryo sin 9 2 2n V a + T2 — 2TXQ cos 9 + 2ryo sin 9 2 2 2 2 2 K ( r -r(R /'a )x0cos9 + r(R /a )y<0 sin9 + A 2 2 2 2 2 2lv\R /o? + r - 2r{R /a )x0 cos 9 + 2r(R /a )y0 sin9 K ( r2 — r(R2/a2)xQcos9 - r(R2/a2)yosin9 i 2 2 2 2 2 2 ~ 2TT\R/a + r - 2r(R /a )x0 cos9 - 2r(R /a )y0 sin9 which, evaluated at r = R, gives a?-R2 = -URsin9 + 2 2 ~d9, =R \R + a -2Rx0cos9 + 2Ry0sin9 2 2 a -R (3.65) 2 2 R + a - 2Rx0 cos 9 - 2Ry0 sin 9 The contribution to the stream function on the cylinder from the vortices being zero, only (3.57) is left and the integral becomes 2 2 2 2Tcyi = pU R j> sin 9 d9 (3.66) nUR(a2 - R2) r2* -sing ^ + P 2 2 2TT % R + a - 2Rx0 cos 9 + 2Ry0 sin 9 61 KUR(Q?-R2) sin( d9. + P 2, % W + a2 — 2RXQ cos 9 — 2Ryo sin 9 The first term is the same as the metastable state energy (3.63). The second term is equivalent to the third one if we substitute 9^—9 such that the denominator is identical and the numerator pick a (—1) from sinf3, ( — 1) from d9 and (—1) from the inversion of the limits. We are left with one integral for which we need the following result taken from [18], 2.558: A + B cos x + C sin x dx (3.67) / a + b cos x + c sin x Bc-Cbt . , . , Bb-Cc —5 7T hi (a + o cos x + c sin x) 4- -ro ^rx bl + bl + cd + (A - BB + C°a\ I — V B2 + c2 J J a + b cos x + c sin x' and dx / a + b cos x + c sin x (a-6) tan §+c arctan a2 > b2 + c2 ^/a2_b2_c2 * x/02_b2_c2 tan +c 2 2 2 i In ^ f -^±2 2^ 2 a <6 + c VP+W (a-6) tan f+c+v'fc +c -a (3.68) < = : 1 In (a + ctan a — b -2 o? = b2 + c2 I. c+(a—fc) tan ; Here A = 73 = 0, C = 1, a = R2 + a2, b = -2.Rxo and c = —2Rya. Since the integral is evaluated on a closed path, any term having an even dependence on 9 goes to zero such that omv W^clX remams. The total contribution to the kinetic energy from the cylinder is then / 'i _ r>2\ /a2 — R2\ 2 2 2 2 2Tcyl = U nR - KU(—^—Wo = U nR - KU( J sin a. (3.69) Integral on the Vortex Core By symmetry, the contribution from the vortex and the antivortex are the same and only one has to be evaluated. Also, it is advantageous to move the origin to (XQ, yo) such that (3.60) and (3.61) are UR2(r cos0 + x ) r sin e + 2j/o - ; -o + ± 9 — arctan 2 2 rcos# (r cos e + x0) + {rsme + y0) 2ir 62 rsin0 + ( ^# \y0 rsin0 + ( 12/0 + arctan — arctan (3.70) rcos0 + ( )x0 rcose+[ iso and 2 UR (r sin0 + yo) * = — (3.71) 2 r cos 0 + XQ)2 + (r sin 0 + yo) K In r — In \Jr2 cos2 0 + (r sin 0 + 2t/o)2 2^ R2\ a2 + R2 r cos 9 + 2 XQ) + r sin 0 + + In a y -R2\ a2-R2\ - In r cos 9 + XQ] + [r sin 9 + yo a2 Y a2 ) Next we need 5$ K / r2 + 2rsin9yo (3.72) W 2TT\ r2 + 4yQ + 4r sin 9yo 2 r + rB cos 9XQ + rA sin 0yo + 2 2 2 r + B x\ + A yl + 2rB cos 0xo + 2r A sin 9y0 2 r + rB cos 9XQ + rB sin 0yo 2 2 2 r + B a + 2rB cos 9xQ + 2rB sin 9y0 r sin 9 - UR2 [r2 + a2 + 2r cos 8XQ + 2r sin9yo + (rcos0 + XQ)(—2r sin0xo + 2r cos0yo) (r2 + a2 + 2r cos 0xo + 2r sin 0yo)2 where A = 0,2and /J = 2-^r~- The previous result evaluated at r = £, while taking £ to be much smaller than any other lengthscale, simplifies to (3.73) 30 2~n' On the other hand, (3.71) reduces to (3.74) 2 2 K , 2a sin a 1, / , f a? + R \ In—-In ^cos a+^^j sin2 a • sin a. 2^ 63 Finally, multiplying the the last two results and integrating yields the following contribution to the kinetic energy: pnUR2 27}cor e sin a (3.75) pK" 2a sin a In -fiascos a+{ 2_ ) sm a ~2n~ a R2 Kinetic Energy The total kinetic energy should be twice the contribution from one vortex core, from (3.75), and the contribution from the cylinder, from (3.69). One must also substract the metastable state energy (3.63) and divide the result by two since the original configuration only includes the upper half-plane. So the energy for a vortex close to a moving semi-circular bump on a flat wall is l z 2 r 2 2 PKU (a + R \ . OK 4a sin a 2T = — sm a + — hi ln^cos a+ [^—tf) sm Q 2 \ a J Ait The shape of the resulting potential barrier is shown in Figure 3.14 and in Figure 3.15. 2T^ (b) Figure 3.14: Kinetic energy of a vortex near a semi-circular bump in function of the position of the vortex for U = K/R where —Ux is the velocity of the bump, K the circulation of the vortex and R the radius of the bump, (a) Shape of the potential barrier (b) Shape of the barrier along the y axis (a = TT/2) 64 Figure 3.15: Lines of constant energy for a vortex near a semi-circular bump in function of the position of the vortex 65 4. Vortex Nucleation and Tunneling In this chapter, we use our results for the kinetic energy of a vortex in a perfect cylinder and close to a circular bump to calculate the exponent of the tunnelling rate with the WKB method. The exponent is calculated using different polynomial approximations as well as numerical integration techniques. We then provide a brief discussion of the effective mass of a vortex, which remains an interesting open question. 4.1 The WKB Approximation The WKB method is a semi-classical technique for obtaining approximate solutions to the time-independent Schrodinger equation in one dimension. For a particle with energy E moving through a region where the potential V(x) is constant, the wave function is of the form ij>(x) = Ae±ikx, with k= ^2m{E-V). (4.1) With V constant, the particle moves with a fixed wavelength A, and constant amplitude. If V{x) varies rather slowly in comparison to the wavelength then the potential is almost constant and V remains 'practically' sinusoidal except that the amplitude and the wavelength will now depend on x. 4.1.1 Exponent This is a well know technique from quantum mechanics and we shall only state the result for the tunneling probability of a particle through a barrier from x = a to x = b. This probability is proportional to T«exp(-27), (4.2) and the exponent is 1=\l j2m\V{x) - E\ dx. (4.3) n Ja This only gives the exponent of the probability of decay, or tunneling rate, but since T 66 depends exponentially on 7, the variation of the prefactor represents only a small correction. Our focus will be on the exponent in this chapter. 4.1.2 WKB in Higher Dimensions WKB is a technique which works essentially for one-dimensional potentials or problems that are spherically symmetric and can thus be reduced to one dimension. In higher dimensions, the decay is more likely to occur along the path of least action, that is, where the potential is weakest. One can therefore use the standard WKB method along this one-dimensional path. However, fluctuations around this path will modify the exponent and the amplitude of the tunneling rate. Schmid [39] presents a good review of how the quasiclassical technique is used in higher dimensions. This technique is shown to be consistent with the instanton technique used by Caldeira and Leggett [4]. See also the book by Coleman [7]. 4.1.3 Vortex Nucleation For the vortex nucleation by quantum tunneling, one considers a particle with an effective mass m and an energy E, going through a barrier given by the kinetic energy of the vortex as a function of its position. For a particle with zero momentum the energy is simply E — 0. Calculating the effective mass of the vortex is a complicated problem which deserves its own discussion, contained in section 4.4. 4.2 Tunneling Rate For a Perfect Cylinder We start with the simpler of the two problems, considering possible polynomial approximations. These prove to be valid only for a small range of the parameters. To extend the result beyound this range we will have to resort to numerical integration techniques. 4.2.1 Properties of the Barrier The potential barrier is obtained from the kinetic energy of the flow of a vortex in a perfect cylinder found in (3.51). It can be written in the more convenient form (4.4) where £ = £/R is the core radius normalized with respect to the vessel radius R to obtain a unitless parameter, x is the position of the vortex which is also normalized and corresponds to 67 b = Rx in (3.51). The dependence on the angular velocity of the container is now included in the unitless parameter u defined as with cu the angular velocity of the container and K the circulation of the vortex. For simplicity we also define a dimensionless potential V(x) such that Figure 4.1: Potential for a vortex in a perfect cylinder as a function of the velocity u We want to consider the tunneling through the barrier of a particle with zero momentum. The WKB exponent (4.3) can then be expressed in terms of V(x) such that (4.7) Again for simplicity we define x) dx, (4.8) J a where K imp _ (4.9) 68 For the tunneling at zero momentum, the barrier must drop to zero at some distance and its value at the origin is therefore V(0) < 0 or u>-ln£. (4.10) A Taylor expansion is not well suited to approximate this potential especially close to the wall of the vessel. Figure 4.2 shows a Taylor expansion with up to quadratic terms but even with higher order terms the approximation is poor. A few more properties of this potential will be needed to find a better approximation. The potential has a maximum at xmax — \; i (4-H) V U and its value is Vmax = -l-\nul (4.12) The first zero can be approximated Taylor expansion about the origin and solving for the root, leading to and the second zero is assumed to be at xb — 1. The inflection point is located at * _ y" + 2°-/^, (4.14) where V(x) goes from a positive curvature for x < x^ to a negative curvature for x > X{. 4.2.2 Approximations to the Potential The calculation of the exponent (4.8) involves the difficult integral of \Jv{x). To carry out this integral, V{x) will need to be approximated by a polynomial. Even though a Taylor expansion with a small number of terms has already been ruled out, Figures 4.1 and 4.2 suggest that for small £ and small u, a quadratic expression can provide a good approximation between between xa and xmax. For small £ and large u a linear approximation between the two same points is acceptable. Between xb and x = 1 the potential falls very quickly and the area under the curve is relatively small. A simple quadratic term with the same curvature as V(x)\Xmax would be sufficient. Thus, a piecewise polynomial approximation could do better than a single 69 Figure 4.2: Approximations of the potential for a vortex in a perfect cylinder, (a) Taylor expansion at the origin with terms up to second order, (b) Approximation in two parts: a line from (xa, 0) to (xmax, Vmax) and a Taylor expansion around xmax with terms up to quadratic order, (c) The potential V(x), shown as a dashed line. polynomial. Starting with the linear part, the slope of the line joining (xa, 0) and (xmax, Vmax) is dV -1-lnK) (4.15) d£ \x=xa ^Ju=l _ ^c-lng With a change of variable so that xa is at the origin, (4.8) yields ^-fV^^(^-f^). (4-16) To build the quadratic part, Vmax is added to a quadratic term 1 fd2V(x -x')\ max (4.17) 21 dx2 so y(x') « -1 - In K) - 2u(u - l)x'2. ' (4.18) To perform the integral we will need r , f %y/a + bx2 + r% In (Vbx + Va + bx2) 6>0 VaTto?dx=l 2 \ 2^ K , rirr , (4-19) J 1 fv/aT^ + ^ arctan (j§^) b < 0, 70 such that -1 - In «) VMu -1) (i - ^) Iquad = /o , ,v arctan (4.20) A/8U(M — 1) -l-ln«)-2«(«-l) 1- The approximation given by 7;jn + 7gUad is shown in Figure 4.3. a e o a x H CQ Figure 4.3: WKB exponent 7(u) for a vortex in a perfect cylinder as a function of the angular velocity parameter u and calculated from a potential approximated in two parts. The first part is a linear approximation and the second part is a quadratic approximation. Results are shown for different values of the core healing radius £ = i/R- The numerical results are drawn in solid lines for comparison. The second approximation with two quadratic parts can be calculated similarly and is shown on Figure 4.4. As we expected, the first approximation is more accurate at small u, whereas the second is more accurate at large u. 71 5 10 15 20 25 30 u Figure 4.4: WKB exponent j(u) for a vortex in a perfect cylinder as a function of the angular velocity parameter u and calculated from a potential approximated in two parts. The first part is a quadratic approximation and the second part is a different quadratic approximation. Results are shown for different values of the core healing radius £ = £/R- The numerical results are drawn in solid lines for comparison. 4.2.3 Numerical Integration Because of the difficulty to evaluate the integral analytically even when using polynomial ap• proximations, we now turn to numerical techniques. The procedure to evaluate 7 as a function of u and £ starts with solving for the zeros of the function and then integrating between these two limits. The results are drawn as solid lines in Figures 4.3, 4.4, 4.5 and 4.8 as a comparison for the other approximations. 4.2.4 Curve Fits and Approximations The form of the curves j(u) for a constant value of £ suggests that a form like A 7(u) = A|u|- + 7o, (4.21) 72 would provide an adequate fit. Figure 4.5 presents the results of this fit for each value of £ and indicates a very good agreement with the numerical data. 5 10 15 20 25 30 u Figure 4.5: Curve fits of the WKB exponent -y(u) for a perfect cylinder and for different values of the core radius £ using 7 = A|ii|_A + 7o. A curve fit is done for each value of £. The numerical results are drawn in solid lines for comparison. The parameters A, 70 and A as a function of £ for each fit is shown on Figure 4.6. A good approximation for the parameters that would reproduce the numerical results proves difficult so that a different fit is tried instead. We still use (4.21) but with the exponent fixed. For a core radius of the order of one A and for a size of the vessel varying between a centimeter and a micrometer, £ varies between 10-8 to 10~4. We therefore fix the exponent at an average value for this range with A = 2. The results for A and 70 are presented in Figure 4.7. A can then be approximated by A = b{\n£f+ b0, (4-22) especially if the point £ = 10~10 is neglected. 70 is well approximated by a logarithmic function 73 of the form 70 =5 In (|)+50, (4.23) except for an anomaly at £ = 10-9. We then find that A » 0.28(ln<£)2'6 + 3, (4.24) 70 » 0.021 In |- 0.07, (4.25) 7(u, |) « (0.28(ln|)26 + 3)|u|-2 + (0.021 ln£- 0.07). (4.26) The values of 7(tt, £) are compared against the numerical results in Figure 4.8. The exponent 7 given by (4.24) is good for values of £ « 10-7. This gives us three different approximations with varying applicability depending upon the parameter range of interest. 4.3 Tunneling Rate For a Circular Bump on a Flat Wall The exponent of the tunneling rate for a semicircular bump on a flat wall is now studied. Note that these results apply also to the nucleation of vortex rings by a moving ion, up to a multiplicative constant, if one ignores the recoil of the ion when the vortex nucleates (ie. in the limit of a very heavy ion). Once again we will resort to numerical integration techniques. 4.3.1 Properties of the Barrier The barrier is obtained from the kinetic energy of the flow of a vortex close to a moving bump as derived in (3.76). The barrier has a minimum along a = ir/2 and can be written as 2 2 2 2 pn? /Ax (x -1) \ fx + l\ ,AM. where £ = is the core radius normalized with respect to the radius of the bump R to define a unitless parameter, x is the position of the vortex which is also normalized and corresponds to a = Rx in (3.76). The dependence on bump velocity U is now included in the unitless parameter u defined as (4-28) with K the circulation of the vortex. For simplicity we also define a unitless potential V(x) such that V(x) = ^V(x). (4.29) 74 The WKB exponent (4.3) at E = 0 can be expressed in terms of V(x) and J \JV{X) dx. (4.30) J a Again for simplicity we define 7 [ y/V(x)dx, (4.31) where K Imp (4.32) For the tunneling at zero momentum, the barrier must be positive over some region and we will need to check that its maximum is positive. Despite the fact that the shape of the barrier is quite similar to the one studied in the previous section, its properties, i.e. zeros, maximum and inflexion points, are much harder to approximate. To find the exact maximum, one needs to solve a sixth order polynomial: u(a6 - a4 - a2 + 1) + a6 - 2a5 - 8a3 - a2 + 2a = 0. (4.33) An approximate solution still has a complicated form. Solving for the zeros by doing a Taylor expansion of the polynomial in the logarithmic function and keeping only a few terms also agrees poorly with the exact values. A Taylor expansion is not well suited to approximate this potential especially close to the bump because of the logarithmic divergence at x = 1. A Taylor expansion at another point could provide a decent prediction for some range of x, if only a point to expand around could be choosen away from the divergence without resorting to the inspection of the barrier at specific values of £ and u. 4.3.2 Numerical Integration After the difficulties encountered in expressing the potential as a polynomial, we are left with the only option of performing the integration numerically. The evaluation of 7(11, £) is similar to the case of a perfect cylinder. It starts by solving for the position of the maximum to be sure that there are zeros. We then look for a zero located between x = 1 and x = xmax and one located between xmax and infinity. \jv(x) is then integrated between these two limits. The results are drawn as solid lines in Figures 4.9, 4.11 and 4.13 and are used for comparison with the various fits. 75 4.3.3 Curve Fits and Approximations The results of the numerical integration look similar to those of a vortex in a perfect cylinder. The curves j(u) for constant values of £ can likely be approximated by (4.21). Figure 4.9 indicates that there is indeed a very good agreement with the numerical data. The results for A, 70 and A are presented in Figure 4.10. A can then be approximated by a logarithmic function A = bln(i) + bo, (4-34) and 70 by the same form as well 70 = 9 In (0 + 90- (4.35) Even the exponent A finds an adequate approximation with 1 5 A = d\£\ + d0. (4.36) Summarizing, we have A » -12.0 In (0-18, (4-37) 70 « 0.171 In (|)- 6.65, (4.38) A « 0.27|°-21 + 1.041, (4.39) 7(u, |) « (-12.01n(O-18)H-°-27^)'21-1-041 + (0.1711n(|)-6.65). (4.40) The values of j(u, 0 are compared against the numerical results in Figure 4.11. The agreement is best for |= 10~8. The last fit was acceptable but a more accurate fit can be calculated using only two param• eters. The results for A and 70 when A is fixed at A — 1 are presented in Figure 4.12. A and 70 still have a logarithmic £ dependence and A « -11.8 In (f) - 15, (4.41) 70 » 0.136 In (0-9.4, (4.42) l{u, £) « (-11.81n (() - 15)1^+ (0.1361n (|)-9.4). (4.43) The values of j(u, 0 are compared against the numerical results in Figure 4.13. The agreement is better than (4.37) owing to the smaller number of parameters. The optimal results are again for ( = 10~8. 76 1 1 ' Parameter 70 HU-H ' Parameter'A 1 100000 - 1 " ' 0.8 • I 1 10000 0.6 • I I i 0.4 I 1000 - 0.2 * 100 • * 0 • + i i 1.1.1 le-10 le-08 le-06 le-04 0.01 le-10 le-OB le-06 le-04 0.01 £ £ (a) (b) ' Parameter' A H-4—i T i 3.5 - I 3 I 2.5 1 i 2 i 1.5 le-10 le-OB le-06 le-04 0.01 £ (c) Figure 4.6: Curve fit parameters of the WKB exponent j(u) for a perfect cylinder as a function of £ using 7 = A|tt|_A + JQ. (a) Offset 70. (b) Amplitude A. (c) Exponent A. 77 1 10000 1 ' ' ' Parameter 70 • ZT R ' Pararrieter' A HU-H: 1 0.4 1000 0.3 100 0.2 10 - 0.1 1 1 1 1 | 1 I.I.I 0 1 le-10 le-08 le-06 le-04 0.01 le-10 le-08 le-06 le-04 0.01 i i (a) (b) Figure 4.7: Curve fit parameters of the WKB exponent 7(u) for a perfect cylinder as a function of £ using 7 = A|u|-2 + 70 and curve fit of these parameters, (a) Offset % approximated by 7o = gin (£) + go- (b) Amplitude A approximated by A = 6(ln£)^ + bo- 78 Figure 4.8: Curve fits of the WKB exponent j(u) for a perfect cylinder and for different values of the core radius £ using 7 = A|u|-2 + 70. Parameter A is approximated by A = 6(ln£)^ + bo and 70 by 70 = 5In (£) + go- The numerical results are drawn in solid lines for comparison. 79 Figure 4.9: Curve fits of the WKB exponent j(u) for a circular bump on a flat wall and for different values of the core radius £ using 7 = A|u|_A + 70. A curve fit is done for each value of £. The numerical results are drawn in solid lines for comparison. 80 1 - Parameter 70 ' 1—1—' ~I 1 - - 1 1 1.1.1. le-10 le-08 le-06 le-04 0.01 le-10 le-' le-06 le-04 0.01 1 (b) le-10 le le-06 le-04 0.01 1 (c) Figure 4.10: Curve fit parameters of the WKB exponent j(u) for a circular bump as a function of £ using 7 = A|u|~A + 7o and curve fit of these parameters, (a) Offset 70 approximated by 70 = #ln(£) + go. (b) Amplitude A approximated by A = 61n(£) + bo- (c) Exponent A approximated by A = d\£\s + do. 81 2 4 6 8 10 12 14 u Figure 4.11: Curve fits of the WKB exponent -7(11) for a circular bump and for different values _A of the core radius £ using 7 = A|u| + 70. Parameters A is approximated by A = Cln (£) + Co, s 7o by 70 = Pin (f) + P0 and A by A = D\^\ + D0. The numerical results are drawn in solid lines for comparison. 82 ' 1 Parameter 70 >—H—1 1 Parameter 'A H-+-H + • : - iX\ - y" - - A'" ,T'' - i 1 .I.I.I. r 1 le-10 le-08 le-06 le-04 0.01 le-10 le-08 le-06 le-04 0.01 (b) Figure 4.12: Curve fit parameters of the WKB exponent 7(14) for a circular bump as a function of £ using 7 = + 70 and curve fit of these parameters, (a) Offset 70 approximated by 70 = g In (£) + go. (b) Amplitude A approximated by A = 6In (£) + bo- 83 2 4 6 8 10 12 14 centering u Figure 4.13: Curve fits of the WKB exponent ^(u) for a circular bump and for different values of the core radius £ using 7 = A|u|_1+7o. Parameters A is approximated by A = C In (£) + Co and 70 by 70 = -B In (£) + BQ. The numerical results are drawn in solid lines for comparison. 84 4.4 Effective Mass of a Vortex The determination of the mass of a vortex in superfluid 4He is not a new problem albeit a very intricate one. Too often its discussion in the literature is poor or incomplete arguably because of the lack of a good answer. It was also subject to some controversy between Thouless et al. and Duan and Leggett [9, 10, 11, 12, 13]. A naive treatment will only include the mass of the normal core given by 2 mcore = 7rp£ , (4.44) where the mass is given per unit length of the vortex or container. The core of a vortex at finite temperature is occupied by the normal fraction of the fluid. However, the vortex core is extremely small and the core mass is much lower than the actual effective mass. Nevertheless, previous work from Suhl [40], which does give a much larger inertial mass for the vortex, is seldom used. Suhl gets m,ine.rt « %, (4.45) cs for the inertial mass of the vortex, where eo is the static vortex energy and cs is the sound velocity. The same result was later derived by Popov in 1973 [35] by mapping vortices and phonons into charged particles and photons in relativistic electrodynamics. Duan and Leggett also published a few papers on the subject. In [11], Duan shows that due to gauge-symmetry breaking and the topology of a vortex, the condensate compressibility contribution to a vortex mass agrees with (4.45). In the end, this result is only valid for a vortex in an infinite system, moving either very slowly or at a constant velocity. If the tunneling time scale is much shorter than the time scale of the superfluid R/cs where R is the system size, then the whole system does not have time to notice the presence of a vortex which would make for a smaller mass then the one given by (4.45). One might have to use a different value for the size of the system when calculating eo- This lengthscale should be determined by R' ~ CSTB, (4.46) where TB is the bounce time, or the time that the particle takes to cross the potential barrier [7, 39]. 85 4.4.1 Inertial Mass of a Vortex for a Non-Uniform Condensate This derivation follows the one by Duan [11] which provides a more detailed treatment than the general dimensional arguments provided in his publications with Leggett [9, 13]. The following fills a few gaps left by Duan but reproduces only the calculation of the inertial mass of a rectilinear vortex. The study of the uniform condensate mentioned in chapter 1 provides at most a qualitative picture of the microscopic justification of the superfluid flow associated with the motion of the condensate. Duan mentions that the condensate compressibility plays an important role such that an extension to the uniform theory is needed which allows for both spatial and time variation of the condensate. We have seen that a superfluid possesses an order parameter whose phase 9(f, t) describes the superfluid velocity field by (1.23), which we rewrite for convenience (4.47) 7714 The order parameter for a uniform condensate in the zero temperature limit, with a small deviation in the superfluid density A(r, t), can be expressed as (f, t) = e^'^l + A(f, tlle-""1'^ (4.48) where no and [io are the uniform condensate number density and the chemical potential. At T = 0, one has no = N. The small deviation in the density will create a difference in the chemical potential Ndp, = VdP, (4.49) where P is the pressure, N = VN the number of particle and V the volume. Note that there are no entropy term as the superfluid does not carry any. The sound velocity can be expressed as (4.50) where p is the density of the non-uniform condensate. For a small variation of the density, (4.49) becomes 5/i = ^-6p(r, t) « 2mc2A(f, t), (4.51) when neglecting the term proportional to A2. 86 The dynamic equation for the superfluid, ^+V(^ + M)=0, (452) specifies that the flow must be potential at all time. In terms of the phase of the order parameter, we have . dO m.4 o oi+ ~i s = M- ( } The first term in (4.53) is absent for classical fluid. From (4.51) this term implies that a time variation of 0 will also generate a superflow and correspond to a superfluid density change. This is due to the fact that the 0 and p are a pair of conjugate variables, a result of gauge-symmetry breaking.9 In simple terms, this is a manifestation of the rigidity of the condensate which 'impedes' modifications of the phase and thus increases the effective mass of a vortex. We now consider the example of a rectilinear vortex in bulk superfluid where the superfluid velocity is given by vs{r) = {h/m^e^ and we give the vortex a small velocity vex. Using the adiabatic phase assumption where the vortex moves slowly enough for the phase to adjust and 0(f, t) — 9(f — vt) such that superfluid velocity is simply vs{f) as t —> 0. From (4.53), the change in the chemical potential is do h - 5u = h— = -hv- VO = --v • e^, (4.54) dt •*• and from (4.51) the change in the density is 8p(r) = (4.55) cjr which describes a dipolar density field. The energy change can then be calculated to second order in the density fluctuation as 1 r d2v E= 2j ^2^)df' (456) where u is the energy density which relates to p, and p with d2u _ J_dji _ c2 (4.57) dp2 UI4 dp Nrrii 9This is the same as in the BCS theory of superconductors where the phase and number density are also conjugates. 87 Inserting this and (4.55) into (4.56) and integrating from the core radius £ to the radius of the system R we find v2 f Nh2 sin2 6 E = — dr, 2 J c% r v2 \irNh2 , R m 2 c m4 £ „2 r R (4.58) c2m|) £ The coherence length due to the Heisenberg indeterminism is n (4.59) m^Cs where m$cs is the characteristic momentum scale. In (4.58) this yields -2 r R~\ E = — In- (4.60) 2 and one can define the factor in square brackets to be the inertial mass of the vortex i H (4.61) 2 since E = (l/2)minertv . Finally, using K = h/1714 and our result (2.73) for the energy of a vortex in a cylinder located at the center, we recover pK? R e0 E = ln^ = ^. (4.62) 4TT £ 4.5 Future Work and Conclusions In this study we considered simple but realistic configurations to evaluate the nucleation rate of vortices by quantum tunnelling. We were able to derive a good approximation for the static energy of a vortex in those configurations without resorting to numerical techniques. The tunneling rate could then be calculated in the semi-classical limit. We had to limit ourselves to the calculation of the WKB exponent along the path of least action. From there we can consider fluctuations around this path using a quadratic expansion about it. The kinetic energy of a vortex close to a bump (3.76) depends on the angle a through sin2 a 88 such that a quadratic approximation is readily obtained to calculate the correction due to these fluctuations [7, 39]. The next very important modification will be to include the coupling to the bath of phonons using the Caldeira-Leggett [4] dissipative tunneling formalism. 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