Vortex Nucleation in a Superfluid by Dominic Marchand B.Sc. i n Computer Engineering, Universite Laval, 2002 B.Sc. i n Physics, The University of B r i t i s h C o l u m b i a , 2004 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF Master of Science in T h e Faculty of Graduate Studies (Physics) T h e University O f B r i t i s h C o l u m b i a October 2006 © Dominic Marchand, 2006 Abstract Superfluids have very peculiar rotational properties as the Hess-Fairbank experiment spectacularly demonstrates. In this experiment, a rotating vessel filled w i t h helium is cooled down past the critical temperature. Remarkably, as the liquid becomes superfluid, it gradually stops its rotation. T h i s expulsion of vorticity, analogous to the Meissner effect, provides a fundamental experimental definition of superfluidity. A s a consequence, superfluids not only posses quasiparticles like phonons, but also quantized vortex excitations. T h i s thesis examines the creation mechanism of vortices, or nucleation, i n the low temperature limit. A t 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 i n 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 w i t h i n 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 w i t h standard hydrodynamics. T h e 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 w i t h one or more vortices. In this thesis, we carry out a semiclassical calculation of the nucleation rate exponent. We use the W K B method along the path of least action created by the presence of a bump or wedge. T h i s work is but a first approximation as fluctuations around this path can be added as well. T h e 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. T h i s 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 1 Introduction 1.1 1.2 1.3 1.4 1.5 2 viii 1 A Brief History of Superfluidity 1.1.1 T h e B i r t h of the Field of Quantum Fluids 1.1.2 Toward a Macroscopic Theory, the T w o - F l u i d M o d e l 1.1.3 T h e L a n d a u - T i s z a Controversy (1941-1947) 1.1.4 Towards a Microscopic Justification 1.1.5 Another Superfluid Theory of Superfluidity 1.2.1 Definition of Superfluidity 1.2.2 The Two-Fluid Model 1.2.3 Quantum Fluid 1.2.4 Similarities w i t h the Ideal Bose-Einstein Gas 1.2.5 Wave Function 1.2.6 Excitations and Quasiparticles 1.2.7 T h e Quantization of Circulation and the Vortex State Investigations of Vortices and Vortex Nucleation i n Superfluid H e 1.3.1 Rotational M o t i o n and Quantization 1.3.2 Observation of Vortices 1.3.3 Vortex Nucleation and the L i m i t a t i o n of Superflow 1.3.4 Nucleation of Quantized Vortex Rings 1.3.5 Q u a n t u m Tunnelling Theoretical Studies of Vortex Nucleation Our M o t i v a t i o n H y d r o d y n a m i c s and Superflow Configurations 2.1 Two-Dimensional Ideal F l u i d Flow Description 2.1.1 Convention for F l u x and Density i n T w o Dimensions 4 1 1 2 3 4 5 5 5 6 8 8 10 12 13 14 15 15 15 17 19 20 21 22 22 23 iii 2.2 2.3 2.1.2 Description of a Two-Dimensional Ideal F l u i d F l o w 2.1.3 T h e Velocity Potential $ 2.1.4 T h e Stream Function * 2.1.5 Boundary Conditions 2.1.6 Conformal M a p p i n g 2.1.7 M u l t i p l y - C o n n e c t e d Regions Description of Quantized Vortices 2.2.1 Properties of a Rectilinear Vortex 2.2.2 Velocity Potential and Stream Function 2.2.3 Example: Off-Center Vortex i n a Cylinder K i n e t i c Energy 2.3.1 Energy of an Off-Center Vortex i n a Cylinder 2.3.2 Line Integral on the Boundary 2.3.3 B r a n c h Cuts 23 24 25 25 26 27 29 29 30 31 35 35 40 41 3 Simple Yet Useful Configurations 3.1 Vortex i n a Cylinder W i t h a Wedge 3.1.1 Starting Configuration W i t h No Wedge 3.1.2 Conformal M a p p i n g 3.1.3 Corrections to the Position of the Vortex 3.1.4 K i n e t i c Energy 3.1.5 Rotation of the Cylinder 3.2 Circular B u m p on a Flat W a l l 3.2.1 Complex Velocity Potential 3.2.2 K i n e t i c Energy 43 43 44 44 50 51 55 56 57 60 4 Vortex Nucleation and Tunneling 4.1 T h e W K B A p p r o x i m a t i o n 4.1.1 Exponent 4.1.2 W K B i n Higher Dimensions 4.1.3 Vortex Nucleation 4.2 Tunneling Rate For a Perfect Cylinder 4.2.1 Properties of the Barrier 4.2.2 Approximations to the Potential 4.2.3 Numerical Integration 4.2.4 Curve F i t s and Approximations 4.3 Tunneling Rate For a Circular B u m p on a F l a t W a l l 4.3.1 Properties of the Barrier 4.3.2 Numerical Integration 4.3.3 Curve F i t s and Approximations 4.4 Effective Mass of a Vortex 4.4.1 Inertial Mass of a Vortex for a Non-Uniform Condensate 4.5 Future Work and Conclusions 66 66 66 67 67 67 67 69 72 72 74 74 75 76 85 86 88 iv Bibliography 90 v List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 Phase diagram and specific heat of H e Two-fluid model Energy spectrum of quasiparticles in superfluid helium Schematic of a superfluid gyroscope Possible metastates for the superfluid gyroscope Alternative models of vortex ring nucleation by ion 2 7 12 17 18 18 2.1 2.2 2.3 2.4 2.5 2.6 Simply-connected and multiply-connected regions M e t h o d of images for a vortex inside a cylinder Geometry for an off-center vortex w i t h an image vortex Equipotential lines and lines of force for a cylindrical capacitor Equipotential and stream lines for an off-center vortex B r a n c h cut for a vortex in a cylinder 27 32 33 38 39 42 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 M a p p i n g to a cylinder w i t h a wedge M a p p i n g from a circle to a circle with a wedge M a p p i n g from a circle to the upper half-plane M a p p i n g to the upper half-plane with a wedge M a p p i n g to a circle w i t h a wedge Coordinates offset due to the mapping Energy of a vortex i n a cylinder w i t h a wedge Energy of a vortex on the x axis i n a cylinder w i t h a wedge Modification to the energy of a vortex i n a rotating cylinder Stream functions of a flow past a cylinder and a moving cylinder Vortices configuration for a cylindrical bump on a flat wall Stream function of a vortex near a moving cylinder B r a n c h cut for the vortex configuration near a cylinder Energy of a vortex near a semi-circular bump Lines of constant energy for a vortex near a semi-circular bump 43 45 46 46 49 50 53 54 56 58 59 60 60 64 65 4.1 4.2 4.3 Potential for a vortex i n a perfect cylinder as a function of u Approximations of V(x) for a vortex i n a perfect cylinder j(u) for a perfect cylinder where the potential is defined i n 2 parts: linear and quadratic 68 70 4 71 vi 4.4 j(u) 4.5 and quadratic Curve fits of 7(7/) for a perfect cylinder using 7 = A | i z | ~ + 7 0 72 73 F i t parameters for a perfect cylinder w i t h 7 = A | ' u | + 70 F i t parameters for a perfect cylinder using 7 = A | i i | ~ + 70 and curve fit of these parameters Curve fits of 7(14) for a perfect cylinder using 7 = A | u | ~ + 70 and using the results of a fit for A and 70 Curve fits of j(u) for a circular bump using 7 = A | u | + 70 F i t parameters for a circular bump using 7 = A | u | ~ + 70 and curve fit of these parameters Curve fits of j(u) for a circular bump using 7 = A | u | + 7 0 and using the results of a fit for A, 70 and A F i t parameters for a circular bump using 7 = + 70 and curve fit of these parameters Curve fits of 7(14) for a circular bump using 7 = -f 70 and using the results of a fit for A and 70 77 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 for a perfect cylinder where the potential is defined i n 2 parts: quadratic A _ A 2 78 2 _ A 79 80 A 81 - A 82 83 84 vii Acknowledgements I would like to express my thanks to my supervisor, P h i l i p Stamp, for his invaluable help and advice, and to M o n a Berciu, for agreeing to be my second reader. Thanks to my friend and colleague Rodrigo Pereira w i t h w h o m 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 a m grateful to L a r a Thompson, Jason Penner, M y a Warren, M a t t h e w Hasselfield and A n d r e w Hines for their help. Special thanks to professor M a t t h e w C h o p t u i k and his group who lent me the computer on which I am typing this thesis. A n d last, but not least, thanks to my parents, Jacques and Helene, and to my partner, Genevieve, for their support and encouragement. viii 1. Introduction T h i s chapter is intended as a quick review of superfluidity i n H e starting w i t h a short historical 4 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 T h e following review is but a very summarized account and the interested reader will find a more exhaustive treatment i n 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 T h e B i r t h of the F i e l d of Q u a n t u m F l u i d s T h e study of matter at low temperatures started w i t h Heike Kamerlingh-Onnes (Nobel laureate of 1913) who both liquified helium 4 (1908) and discovered superconductivity. Decisive experimental evidence that helium 4 undergoes a drastic modification at 2.17 K was provided by W . H . Keesom and his collaborators w i t h measurements of such physical properties as the heat of vaporization, the dielectric constant and more importantly the heat capacity . T h e anomalies 1 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. K a p i t s a and, indepently, A l l e n and Misener discovered superfluidity i n 1937. P y o t r Leonidovich K a p i t s a 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 Figure 1.1b shows the shape of the specific heat which inspired Ehrenfest to name the transition temperature the A-point. 1 1 1 2 3 4 5 T(K) 1 (a) 2 3 4 T(K) (b) Figure 1.1: (a) Phase diagram of H e and (b) Specific Heat of H e 4 4 in 1978. He first demonstrated that liquid helium II can flow without resistance through narrow capillaries implying that the flow was inviscid. O n the other hand, other experiments involving the rotation of a disk immersed i n helium II showed that it could still behave as a viscous fluid under different experimental conditions. T h e peculiar properties of helium, the so-called phenomenon of superfluidity, founded the new field of quantum fluids. A s 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 T o w a r d a M a c r o s c o p i c Theory, the T w o - F l u i d M o d e l T h e development of methods to deal with collective phenomena was paramount to the explanation 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. T h i s 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 L o n d o n [29]. T h e discrepancy between experiments on the viscosity of helium II and the Bose-Einstein theory prompted T i s z a to introduce the two-fluid model i n 1938, a most impressive deviation 2 from classical hydrodynamics. T h e 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. T h i s 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. T h i s 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. T h i s is exactly what was offered by L a u d a u i n 1941 [24]. 1.1.3 T h e 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 i n which the binding forces are too weak to preserve the lattice structure. T h e London-Tisza model built on the theory of Bose-Einstein condensation of an ideal gas. Landau's approach was similar to that used i n solid state physics where one discusses the normal modes of motion of the solid as a whole. T h e rather v i v i d 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 ' t r u t h ' was likely a compromise between them. L a n d a u first postulated the existence of two independent motions i n 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 d i d 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 i n which excitation moves. T h e excitations are normal because they may be scattered and reflected thus showing viscosity. T h e fluid associated w i t h 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 i n 1946 [2] provided a confirmation of 3 Landau's two-fluid model. A year later, L a u d a u had to modify the roton spectrum and combined it to the phonon spectrum i n a single energy-momentum curve. T h i s modification had become necessary to better reproduce experimental results for the velocity of second sound measured by Peshkov and later by P e l l a m and Scott. T i s z a criticized Landau's modification of his theory but it was nevertheless i n 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. L a n d a u 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 w i t h an imperfect Bose-Einstein gas w i t h 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 i n 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 i n a gas. T h i s adjustment merely increases the effective mass of the moving atom. He also argued that phonons are the only low-lying states which allowed h i m to recover the correct low-temperature behavior of the specific heat. A n d 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. T h i s idea originated i n 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 i n a superfluid and provides a fundamental experimental definition of superfluidity. We will come back to this experiment i n the next section. 1.1.5 A n o t h e r Superfluid Ordinary helium has an extremely rare isotope found only i n one part i n ten million. Despite the identical electronic properties of the H e and H e atoms, they have completely different 4 3 properties at low temperatures due to the spin 1/2 fermion nature of H e and the bosonic 3 nature of H e . H e becomes superfluid only below 2 m K , three orders of magnitude lower i n 4 3 temperature than H e . Another surprising difference is that H e has two distinct superfluid 4 3 phase. T w o of the superfluid transitions were first observed by Osheroff, Richardson and Lee in 1972, work for which they received the 1996 Nobel prize. T h e phenomenon of superfluidity in H e is obviously quite different than i n H e and we will restrict the rest of this study to 3 4 helium II. 1.2 Theory of Superfluidity 1.2.1 Definition of Superfluidity One problem w i t h the word superfluidity is that it actually summarizes a number of different properties that are not necessarily intrinsically related. T h i s very word can be used in the literature alternatively to mean any of these properties or a l l of them without further explanation. A s mentioned before, helium II flows through small capillaries without friction and will flow over the r i m of the vessel containing it. It sustains persistent currents, has peculiar properties i n rotation and shows effects analogous to the Josephson effect i n superconductors. A very good account of these properties and to which extent they i m p l y each other was given by Leggett at the V l t h International Summer School i n Theoretical Physics i n Kiljava, F i n l a n d [28]. Here we w i l l mainly focus on the more fundamental rotational properties. Non-Classical Rotational Inertia under the name non-classical Leggett refers to the rotational properties of a superfluid rotational inertia or N C R I . To explain what N C R I is, consider a bucket filled w i t h a classical fluid which is then rotated. T h e fluid is stationary at first but eventually the liquid will be rotating w i t h 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 w i t h a microscopic container containing 5 only one atom the results would be radically different as the angular momentum of a quantum system is quantized i n units of h. T h e 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 i n its behaviour as a macroscopic quantum system should display an equivalent effect. In 1967, Hess and Fairbank [21] verified this experimentally. T h e y d i d more than start the container from rest and show that the helium d i d not rotate as the container was brought into rotation. T h i s could have been interpreted as a proof that helium II had low or no viscosity. Instead they started w i t h a rotating vessel filled w i t h helium i n its normal phase and then cooled it. A s the temperature was lowered, the helium gradually stopped rotating until as T —> 0 it was completely stationary. T h e y reported their findings as analogous to the Meissner effect found i n superconductors. O n l y 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 N C R I as it provides an experimental definition of this effect. Since N C R I 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 w i t h the following formulation for the problem of N C R I . 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 i n 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 i n d/R is I cl = J 2 p r dr « NmR . (1.1) 2 According to classical mechanics, if the walls rotate with an angular velocity LU, the equilibrium state has an angular momentum Ld = Iciu, (1.2) Ed = \hi^. (1-3) and an energy 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 J . 1 T h e moment of inertia of 6 the system can then be defined by • _d E du 2 I (1-4) 2 a)-»0 in which case the angular momentum is given by * = S7 = ' W < 1 5 ) T h e extent to which the system departs from its normal or classical behaviour can be expressed by n where n s is the superfluid fraction, s = ^ = l - f , P Id (1.6) p is the the superfluid density and p is the total density. s 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 p n and the normal fraction is n n Figure 1.2: Two-fluid model. n n = — = 1- n . P (1.7) s is the normal fraction and n s 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 t h i n metal disks immersed i n helium and suspended by a torsion fibre. T h e disks were sufficiently close so that above the superfluid phase transition the fluid was dragged completely w i t h the disks. B y 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 N C R I is characteristic of the equilibrium state and is quantum-mechanical i n 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 i n the usual way by is comparable to other typical length scales in the liquid. Most elements i n 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 \<IB « 0.4 n m while the interatomic distance is d ~ 0.27 n m such that quantum effects are extremely important. 1.2.4 Similarities w i t h the Ideal Bose-Einstein G a s 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 L o n d o n and Tisza's suggestion and compare the ideal Bose-Einstein gas to liquid helium II . 2 T h e Ideal Bose-Einstein Gas Let a macroscopic number N of H e atoms contained i n a 4 cubic box of volume V be at thermal equilibrium at temperature T . T h e average number of atoms i n each state i of energy ej is given by the Bose-Einstein distribution function exp [(ej - ii)/k T\ B - 1 where /u. is the chemical potential of the gas and /V = 2 5>( T)). ei) (1.10) This 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 i n (1.9) is small enough to allow the following approximation: ry ~ JCBZ_ ( L 1 1 ) eo - M It follows that ii is only slightly less than eoUsing periodic boundary conditions, the energy levels of individual atoms i n a box are 6 k l m 2m V^ = {k2 + l 2 + m 2 ) 4 w i t h k, I and m , being positive integers. ' { L 1 2 ) It will be convenient to define the zero of energy to be eo = e m . Furthermore, w i t h 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 i n (1.10), we can now integrate the product of the density of state X>(e) times the distribution function of (1.9). O n l y 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, Jo N (T) + N'{T). T))de (1.14) 0 P u t t i n g p, = 0 and evaluating the integral to obtain the upper b o u n d on the number of particles i n the excited states yields 7Y'(r) = 2.612V ( l ^ n V 2irh J 3 Setting N'(TB) / 2 . (1.15) = N gives the critical temperature 27rfi ( 2 N \ / 3 2 9 such that the fraction of particles i n the lowest energy level can be expressed as N (T) 0 T<T . (1.17) B N This macroscopic occupation of the lowest energy level is known as the condensate. Comparison to H e l i u m II Replacing N/V by the density of liquid H e i n (1.16) yields 4 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. T h e similarities suggest that the A-point marks the onset of B E C . T h e specific heat of both the ideal gas and helium show a cusp but the one i n 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 states . Even at absolute zero some particles will not be in the 3 lowest energy level but it remains nevertheless populated by a macroscopic number of particles suggesting the existence of a condensate. T h i s condensate and the excited levels are identified w i t h the superfluid fraction. In practice, less than ten percent of the particles i n the condensate will be i n 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 noninteracting quasiparticles as suggested by Bogoliubov and can be associated w i t h 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 i n t u r n 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 N C R I . 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 3 See [41] for more details. 10 these results to get the wave function. Figure 1.1b shows that at low temperature C v unlike T / 3 2 ~ T 3 for B E C . Close to the A transition it has a weak power law behaviour: 1 C{T) + A_\T -T \~ a c T <T , C where C(T) is a non-singular function of T and a is the critical exponent measured to be a « —0.009. A very good agreement can be found with predictions for the three-dimensional X Y model. Systems described by this model have an order parameter that can be represented by a two-dimensional unit vector where the angle of this vector is 9. Above the critical temperature the vectors have random angles but below T they develop a long range order and a common c direction. Solutions of the X Y - m o d e l have the general form ^ ( f ) = Voexp (i0(f)), (1.19) and, i n analogy to the wave function of a B E C , we can normalize it so that the density of particles i n the condensate is no = kA)(OI 2 (1-20) Above the critical temperature no is zero and iV'oCf)! = 0 such that the phase 6 cannot be defined. T h e macroscopic wave function looks actually similar to the order parameter of the system. T h e momentum of the condensate is obtained i n the usual way w i t h ptP = -iVip =ptp, (1.21) and p = hV6. (1.22) T h e superfluid velocity can then be defined as v = 8 m4 (1.23) W h e n the superfluid is at rest the phase has the same value throughout; for a constant superfluid velocity, the phase varies uniformly. T h e phase is thus a well behaved function that varies smoothly, or i n other words, it is coherent. T h i s 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. T h e 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 condensate. However, it also depends on the nature of the thermally excited states. T h e presence of interactions gives rise to collective motion of the atoms and we are interested i n the normal modes of this motion. Figure 1.3 shows the dispersion curve proposed by L a n d a u 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 i n superfluid helium Since a liquid can propagate longitudinal waves, the thermal excitations of helium II should definitely include phonons. A s for a crystal lattice, the low energy part of the phonon dispersion curve is approximated by a straight line w i t h e - cp, (1.24) where c is the first sound velocity. T h e 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 fi is the effective mass. L a n d a u shows that there r is no dissipation due to quasiparticles for a superfluid moving w i t h a velocity smaller than Cmi n (shown on Figure 1.3) explaining the necessity of this gap for superfluidity to occur. T h i s 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. T h e excitation spectrum proposed by L a u d a u can further be motivated by using an ideal gas of phonons and rotons to calculate the specific heat and other thermodynamic parameters. The agreement w i t h experimental values remains reasonable up to 1.2 K . Above this temperature, interactions between excitations need to be taken into account. 1.2.7 T h e Q u a n t i z a t i o n o f C i r c u l a t i o n a n d the V o r t e x S t a t e Circulation and Quantization T h e circulation is defined as the line integral around a closed path of the fluid velocity: T = i v - df. (1.26) T h e 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. T h e existence of a condensate and the form of its wave function lead to a quantization of the superflow circulation as predicted by Onsager. T h e curl of (1.23) prescribes V x u s = 0, (1.28) such that the flow is irrotational. Inserting (1.23) i n (1.26) yields r = — ivdm J d f = — A6, m (1.29) where A6 is the change i n the phase angle after going around a closed path. T h e single- valuedness of the wave function stipulates that ^ ( f ) =ij>(r)exp(iAO), (1.30) implying that A8 = 2irl w i t h I being an integer. T h e circulation is then K = 1-, m (1.31) where K is usually used for a quantized circulation or the circulation of a vortex and I is called the topological winding number. T h e quantum of circulation is h/m and has a value of 13 9.98 x 1 0 m s - . _ 8 2 1 T h e V o r t e x State It first seems like (1.28) would prohibit any rotational motion since the circulation of any closed loop i n the continuous fluid is zero (1 = 0). T h i s 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 i n two different situations where a hole in the superfluid is present and the circulation is evaluated for a path around that hole. T h e hole can be provided by a solid boundary like in the case of an annular container or it can appear spontaneously i n the superfluid in the form of an vortex core. T h e vortex is a state of rotational motion characterized by a quantized circulation. A circulating flow w i t h cylindrical symmetry will obey (1.28) if ~(rvt) = 0. (1.32) Therefore, the superfluid velocity of a vortex is v oc 1/r. T h e divergence is avoided by having s the density go to zero at the center such that the wave function is also zero and its phase is not defined. T h e 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 . O n l y 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. T h i s study focuses on the case of a rectilinear vortex. T h e vortex core can however be bent in general and is called a vortex line. W h e n the two ends of a vortex line are joined together, we have a vortex ring. O f course, more than one vortex can exist i n a system and they tend to organize i n a triangular array. T h i s 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 He 4 Little has been said yet on the subject that will be of utmost interest i n this work: vortices and vortex nucleation. There is a wealth of experiments on vortices i n superfluid helium, not to mention recent work done on rotating B E C ' s . T h i s 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 T i l l e y [41] provide a good 14 introduction to the subject. 1.3.1 Rotational M o t i o n and Quantization E a r l y experiments on rotation i n 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 i n 1950 and found that the meniscus had the same shape as that of a normal fluid. H a l l and V i n e n [19] would later propose that this can be explained by the presence of an array of vortices uniformly distributed. T h e first experiment designed to look for quantized circulation was carried out by V i n e n 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 T h e 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]. T h e Hess-Fairbank experiment d i d not demonstrate directly the appearance of separate vortices i n the vessel. T h i s was accomplished by Packard and Sanders in 1972 [33] using electrons trapped i n the vortex cores. T h i s technique was based on pioneering work on the trapping of ions by vortices by Careri, M c C o r m i c k and Scaramuzi i n 1962 [16]. A refinement of this technique using an optical detection system rotating w i t h the container would even allow the positions of the vortices to be photographed. Quantized vortex rings were first studied by Rayfield and Reif i n 1964 [36] and were nucleated by movings ions through superfluid H e . Despite the three-dimensional nature of this 4 effect, it is very similar to the nucleation of a vortex line i n the proximity of a circular bump that we shall consider later. 1.3.3 V o r t e x N u c l e a t i o n a n d the L i m i t a t i o n of Superflow T h e 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. T h i s limit on the superfluid velocity is called the critical velocity. It is understood that vortices actually play 15 a significant role i n this phenomenon, not rotons as was first believed. Indeed, the critical velocity at which nucleation occurs was calculated for a vortex by Feynman i n 1955 [14], and is much lower than that of a roton and in better agreement w i t h experiment. W h e n a vortex is nucleated, it does not only take energy from the superfluid when it is created, but as argued by V i n e n in 1963, it must first overcome a barrier. T h e calculation of a critical velocity w i t h the L a n d a u criterion does not take this into account. T h e 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 hypothesis 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 overcoming (or tunneling as we w i l l see shortly) a potential barrier. A review by Langer and Reppy [27] discusses this fluctuation theory using the results of experiments done w i t h a superfluid gyroscope. A schematic of this device and a short description of how it is operated is presented in Figure 1.4. T h e annulus packed w i t h porous material is filled by immersing it i n a bath of helium. T h e 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 L determined. We find that no matter how fast the annulus is initially rotating, the p angular momentum has a m a x i m u m that cannot be exceeded. T h i s critical value is a function of the temperature and the pore size. T h e critical angular momentum decreases w i t h increasing temperature, which can be a result of the reduction of the superfluid fraction, or the superfluid critical velocity, or both. T h e temperature dependence of p /p s can also be determined w i t h 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. T h i s is shown as a dashed line i n Figure 1.5. If the temperature is modified but does not cross the critical velocity line v u s cr 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 E a too big for transition to be likely at low temperature and small superfluid velocity. T h e rate at which the superfluid velocity 16 C O c Figure 1.4: Schematic of a superfluid gyroscope inspired of Langer and Reppy. A persitent current w i t h angular momentum L is set up in an annular container (A) filled w i t h a porous material (D). T h e annulus is mounted on a fiber (C). After rotation i n the horizontal plane at angular velocity UJ, the annulus is brought into vertical position and the detector (B) measures its deflection when it precesses. p 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 w i t h 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. F r o m 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 i n experiments but the qualitative agreement is poor. Also, i n experiments of superflow i n tortuous channels like i n jeweller's rouge, it is hard to justify how a local fluctuation could uniformly modify the superfluid velocity. 1.3.4 N u c l e a t i o n of Quantized V o r t e x Rings Experiments on nucleation are often concerned w i t h vortex rings creation by moving ions. T h e 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. T w o rival 17 \ \ \ \ Av \ •-, transition s t ' \ vsc(T) \ \ ;T i X / / / T t/ / Figure 1.5: Possible metastates for the superfluid gyroscope. T h e dashed line is the temperature dependence of the critical velocity v and the horizontal lines are quantized values of v . s s 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. T h i s 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 M u c h of the experimental data on vortex ring nucleation has come from a comprehensive series of investigation by M c C l i n t o c k and his co-workers (Bowley, Nancolas, Stamp, etc.) To 4 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. T h i s 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 i n the temperature range where the numer of rotons is finite. F i n a l l y 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 i n 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 V i n e n in 1963, M u i r h e a d et al [5, 6] treated both the pealing model and the quantum transition model as quantum tunelling processes. Bowley [37] dealt only w i t h the quantum transition model and treating the vortex as a quantum object i n a potential well. T h e 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 M c C l i n t o c k et al i n 1988 uses again quatum tunneling to interpret experimental nucleation rates and even calculates a barrier height i n very good agreement w i t h the theory of quantum tunnelling. T h e y also point out that most studies i n the past were addressing the slightly different question of the conditions under which preexisting vorticity expands to form dissipative tangles. of vorticity is extremely difficult. It turns out that preparing a sample of superfluid free 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 i n the critical velocity is not at 0.5 K anymore but at 0.2 K , thus ruling out the roton-driven vortex nucleation. 4 T h e rate that they obtain is See [38] and references therein. 19 accurately fitted w i t h v = v + Aexp(-j^j, (1.34) 0 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 w i t h quantum mechanical tunneling. Finally, they make an interesting observation about the possibility for the height of the barrier to vary w i t h pressure which could be a manifestation of the coupling to the phonon bath. Avenel and Varoquaux [42, 43] conducted a series of experiments i n a different configuration than the usual ion-ring. T h e y 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. T h e quantum tunneling rate is again found to be expressed by Arrhenius law. T h e vortices created i n these experiments are believed to be half-rings w i t h both ends travelling on the boundary. O n a slightly different topic, they also published an article i n 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 i n 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. A t 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. T h i s transition is more likely to happen where the barrier is weakest such that irregularities of the container are believed to play an important role. M a n y studies have been published presenting diverse configurations w i t h different boundary conditions. T h e y 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. T h e exponent of the nucleation rate is then calculated w i t h the W K B method. One of the first calculation of a nucleation rate based on the semiclassical approximation was carried out by Volovik i n 1971 [47]. He considered a vortex ring close to a circular obstacle. M a n y more investigations could be mentioned but none of them offer a realistic picture. T w o 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 i n 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 O h m i c dissipation, except at very high temperature. 1.5 Our Motivation T h i s study is mostly concerned w i t h the nucleation of rectilinear vortices i n a cylindrical container 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 w i t h and without a vortex i n the system. A semiclassical approximation is then used to get an approximate tunneling rate. N o t h i n g of this is new, of course, but few studies have taken into account the irregularities on the container walls, and even fewer w i t h an analytical treatment. Previous works on the subject are either numerical W K B calculations or phenomenological activation theories. T h e intent is to consider simple yet experimentally realistic configurations that will include a feature like a bump or a wedge. T h e 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 i n the treatment of the dissipative effects of couplings to the phonon (and possibly the roton) bath. T h i s would be done quite naturally i n our approach by using the Caldeira-Leggett dissipative tunneling effect [4]. Unfortunately, time constraints have not allowed us to include this i n the present document. 21 2. Hydrodynamics and Superflow Configurations In this chapter, we w i l l first introduce basic concepts of hydrodynamics related to the description of a two-dimensional ideal fluid flow. We then describe vortices i n this context, illustrating w i t h 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. A l l these concepts and techniques will be applied i n the next chapter w i t h more interesting configurations. 2.1 Two-Dimensional Ideal Fluid Flow Description W h e n 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. B u t 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. T h a t is, the fluid motion will take place in a series of planes parallel to the xy plane and that motion will be the same i n each plane. T h e velocity field is then a function of x and y only and has no component parallel to the z axis. A s 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 C o n v e n t i o n for F l u x a n d D e n s i t y i n T w o D i m e n s i o n s Since the motion is known everywhere if we have its description i n 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 w i l l actually refer to the kinetic energy per unit length. For the less obvious cases, we shall abide by the following conventions: • T h e 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. • T h e fluid density or number density, is defined as the usual three-dimensional fluid or number density. 2.1.2 D e s c r i p t i o n of a T w o - D i m e n s i o n a l Ideal F l u i d F l o w 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. T h e incompressibility of the fluid is ensured by requiring that the velocity potential is harmonic: A<& = V <& = 0. T h i s 2 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: ? = V x f . 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. T h e velocity field v = (vi, v ) according to the previous definitions is then given by 2 vx = — = —, ox ay and u = 2 — ay = - — . ox (2.5) These equations are simply the Cauchy-Riemann conditions which must be satisfied for the complex function, fi(z), to be differentiable. T h i s function is defined quite naturally as the complex analogue of <fr and is called the complex velocity potential: Sl{z) = ${x,y) + M(x,y), where z = x+iy. (2.6) T h i s function is differentiable and its derivative is called the complex velocity: Cl'{z) = vi(x, y) - iv {x, y). 2 (2.7) The analogue of the usual velocity vector is then given by taking the complex conjugate of this function: Cl'(z) = v\ + iv22.1.3 T h e V e l o c i t y P o t e n t i a l •$ 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 i n 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 $ = $+ f 0 v-dl. (2.8) J A W h e n P is moved to Q by an infinitesimal displacement, the flux along P($ is <5$ where <J(p — $. p~^. If P($ is parallel to the x axis, we find that v i = f f as stated i n (2.5). Similarly for P(^ taken along the y axis, we find that i>2 = T h e flux thus defined is the same as the velocity potential. Now let P move such that the value of <& remains constant. T h e curve that will be traced out will always be perpendicular to the fluid velocity and will define an equipotential line. 2.1.4 T h e Stream Function * Next, we consider the flux of a fluid crossing a path. B y convention we will consider a flux to be positive when, looking from point A i n the direction of point P, the fluid is flowing from left to right. A g a i n let A and P be any two points i n 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. W i t h h being the normal to the infinitesimal curve element dl, the flux across the path * = / JA v-ndl= vxdl. (2.9) 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. W h e n 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$ expressed i n terms of = Sx, the flux is —vi • by. So the velocity vector 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. T h i s is why we first defined to be a stream function. T h e stream lines and equipotential lines are two equivalent ways of describing the fluid flow configuration. We w i l l use one or the other depending which one is better suited for the problem. B y definition stream lines and equipotential lines are always perpendicular to each other. 2.1.5 Boundary Conditions Boundary conditions i n a two-dimensional ideal fluid flow can be specified as a vanishing condition on the derivative of $ at the boundary (no flow through the boundary). A n equivalent 25 and rather elegant condition imposes that ^> is constant on the boundary (the boundary is therefore a stream line). T h e first case is a Neumann problem whereas the second one is a Dirichlet problem. For problems w i t h 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. T h e 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. T h i s 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 w i l l preserve angles. T h i s 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. T h i s ensures that the new transformed boundary conditions are respected. T h e 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 w i t h an inverse method where one starts w i t h 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 CauchyRiemann 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 i n this study, we will start w i t h known problems on which we w i l l apply simple transformations whose effects are well understood already to 26 achieve the desired configurations. 2.1.7 Multiply-Connected Regions irreconcilable paths reconcilable paths reducible circuit irreducible circuit reconcilable paths Figure 2.1: Simply-connected and multiply-connected regions A Few Definitions T h e problems that will be considered shortly involve multiply-connected regions. Before reviewing the properties of a flow i n 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. A n y two paths lying i n the region that can be made to coincide by continuous variation, again without crossing any boundary, are said to be reconcilable. A n y 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, i n whole or i n 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 w i t h 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 M u l t i p l y - C o n n e c t e d Region Section 2.1.3 states that the flux along any two curves joining two points must have the same value. T h i s is always true for a simply-connected region. For the more general case this statement is restricted to reconcilable curves. T h i s also means that the circulation of a circuit i n 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 circuits. These circuits can be deformed to appear as the boundaries they are winding around. T h e circulation around each of these i n the counterclockwise direction is denoted « i , «£, • • • > n-iK If an arbitrary circuit winds around each boundary a certain number of times p\,p2,--- ,p -i n 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 M u l t i p l y - C o n n e c t e d Region Similar modifications to the state- ments of section (2.1.4) are also required for the stream function. T h e flux across two different paths joining any two points is necessarily the same only for reconcilable paths. T h e flux flowing outside of a circuit is always zero for an incompressible fluid i n a simplyconnected region. Equivalently, the flux across any reducible closed path will be zero. A g a i n , w i t h 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> • • • ) P n - i with the same convention as before, then one can write the flux through this circuit as n-l flux=5> . (2.11) iMi T h e n (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 w i l l make on its properties. To abide by our two-dimensional description, only rectilinear vortices parallel w i t h the z axis will be considered. A n off-center vortex inside a cylinder w i l l 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. A s a vortex is by definition a rotational motion we can therefore expect a singularity i n <&. However, our starting point w i l l be the striking property of quantization of the circulation. A s 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. T h e circulation V is therefore quantized i n units of h/m. From (1.26), we find the expression for the velocity field of a single vortex to be K, 6 (2.12) 2rrr' where the circulation of the vortex is K = l(h/m) w i t h 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. W h a t actually happens close to the core is a complex problem, but corrections to this approximation are only of order £ , where £ is considered small compared to 2 other lengthscales of the problem. T h e approximation used here corresponds to a vortex with 29 an empty core, which differs from calculations done in the book by L a m b , 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. T h i s vortex core is assumed to be a circle w i t h 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. U p to an integration constant we have $ = = — arctan 2TT \xj = — 6, 2vr -±]n(y/x*+y*) = -£\nr. y (2.13) J (2.14) $ is therefore multivalued but its gradient, the velocity field, is still single-valued. A s 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. T h e stream functions are circles and the stream constant diverges at the vortex center. T h e equipotential lines are straight lines meeting at the vortex center. T h e 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. T h e 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 i n section 2.1.6 that exchanging the stream functions and the velocity potential 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 w i t h flux /j,, where n(z) = /j,ln(z- 5 z ). 0 (2.16) See 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 i n a cylindrical container of radius R. T h e vortex is located at a distance b from the center of the cylinder and is located on the x axis without loss of generality. T h e vortex core radius is again £. T h e boundary consists of two cylinders: the vortex core and the container itself. M e t h o d of Images To solve this problem we can use the method of images and place vortices outside of the region we are interested i n to ensure that the boundary conditions are satisfied. Let's first look for a simple solution involving only one vortex. B y 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. F r o m (2.14) the stream function of two vortices is simply In r In r 27T (2.17) 27T If we choose K = —«i such that the image vortex has the same circulation but with opposite 2 s i g n , the condition for \5 to be constant on the boundary is that r/r' 6 should also be constant. We then consider the geometry summarized i n 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. T h i s 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\ = d, (2.18) \0'A'\ = d, 2 (2-19) \00'\ = 5. (2.20) x For the circles \OP\ = £ and | 0 ' P ' | = R to be stream lines, we want r/r' = constant. O n the small cylinder this ratio is found to be r_ r> 6 £-di = d -8-Z 2 = g + di d -6 2 + C (2.21) From now on, we shall refer to a vortex of negative circulation as an antivortex. 31 Figure 2.2: M e t h o d of images for a vortex inside a cylinder. T h e small circle w i t h radius £ is the vortex core and the big circle w i t h radius R is the container. and on the big cylinder r[ d -R = 2 r d + R = 2 R-6-di R + 6 + di' ' { 1 Equations (2.21) and (2.22) respectively reduce to - d (d -s)=e 1 2 d (d 2 5) = R 2 1 + - L , r = T> ( 2 - 2 3 ) t, ^ = % (2-24) Neglecting d\ i n (2.24) we find that the antivortex is at a distance of R /5. 2 In our problem, 6 + di is the position of the vortex which we defined as b such that we can approximate 5 by b. T h e antivortex is then at a distance of R /b 2 from the origin. We should also check that di is indeed small. Substituting 5 = b — di i n (2.23) and solving for di we have di = + 1 ^ - 6 ) 2 + ^ , (2.25) where we have kept only the positive solution for d i . If £ is small, we can substitute x = 4 £ 2 and do a Taylor expansion: 32 such that e d X With d = 2 R /b, 2 ~ (2.27) (d2-6)" {R b- (2.28) b )' 2 2 which is of order £ and should be neglected. 2 Figure 2.3: Geometry for an off-center vortex w i t h an image vortex Velocity F i e l d and C o m p l e x 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. T h e velocity field can then be obtained from (2.5). For illustration purposes, let's start w i t h 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 v + v' = — zn -y , y_\ ( , 2 + D )\D I2 2 D c + \ D' J 2 (2.29) From Figure 2.3 one obtains D = a + y, 2 2 2 (2.30) 33 D' =c +y, (2.31) a + b + c = R /b, (2.32) 2 2 2 2 and using the fact that a + b = x we get a =x-b, (2.33) c = R /b - x . (2.34) and 2 Inserting these results back into (2.29), we find that the velocity vector is ,n _ U l - JS_ -?/ i i 2,r \x*+v'+l'-2bz a (2.35) !g. 2 2_ j£ + +X 6 +Y 2 x —-x K n o w i n g that § f = t>i and | j | = v , one derives from (2.35) the following expression for 2 the velocity potential: K ( x — R lb <P = — arctan 2tx\ y 2 A n d similarly w i t h ^ = v and ^ 2 « = i 4TT arctan x — b\ y J „„. (2.36) . = —t>2, one can write the stream function as J 5 W ^ b + a; + y - 2bx 2 2 2 \ ( J M v 7 ) y T h i s is of course defined only up to a constant term. T h e constant term i n <E> does not matter at all here and the one i n \P simply defines the zero of the stream function. B y adding ^ - In we set the stream constant to zero on the cylinder which w i l l prove useful i n the next section. Together, those two results yield a complex velocity potential equivalent to the sum of the complex velocity potential (2.15) of the two vortices. A s expected, we have 34 2.3 Kinetic Energy T h e last topic of this chapter is devoted to the evaluation of the kinetic energy of the fluid. K n o w i n g the change i n kinetic energy of the fluid when adding a vortex to a specific configuration 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. T h e 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 v dS = p J J(v\ 2 + vl) dS. (2.39) T h i s 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. T h e n a more general approach will be presented. 2.3.1 Energy of an Off-Center Vortex in a Cylinder Evaluating (2.39) w i t h (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 i n R/£, the kinetic energy 7 is 2 2T n 2TT f In X) , + In (R -b 2 \ 2 R (2.40) 2 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 i n a straightforward manner. Analogy W i t h T w o Charged Cylinders Consider a capacitor made of two concentric cylinders as i n §3 of [22]. In this problem, the analog of the velocity field is the electric field E. Since there are no charges i n 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, 7 (2.41) This 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 = <t>-iA . z (2.45) From there, we could map this problem to other configurations. To illustrate, let's consider a charged straight line passing through the origin and perpendicular to the xy plane. T h e field is given by E r = 2 A / r and Eg = 0 w i t h A being the charge per unit length. Because of its symmetry, this problem proves much easier to solve i n polar coordinates. W e w i l l therefore use the Cauchy-Riemann conditions i n polar coordinates: du _ 1 dv dr dv r 88' 1 du dr r d9 (2.46) (2.47) We find ^ dr $ = (2.48) = Er = *i r r dr -2A / — = -2Alnr, A = (2.51) (2.49) and 2A j d9 = 2A6», 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 i n a capacitor is well known to be given by Energy = — = (2.53) where C is the capacitance and is defined as C = —= charge V potential difference e l e c t r i c In other words, the capacitance defines how big a charge can be stored per volt. T h e 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> E dl = - fjj^dl, n (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 E dl n BA = —dl al = A -A . 2 1 (2.58) We know that the flux through a closed contour is 47rQ. One can then express the capacitance and the field energy as ^ Energy Air fa - </>i ' = ^ - L ( A - Ai)(0 2 2 ^ ( '60) 2 07T Capacitance of T w o Parallel Cylinders We consider two infinite conducting cylinders w i t h the same geometry as i n Figure 2.2 and we want to calculate the capacitance and the energy of this configuration. T h e 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 = <t> = 2 -2Aln(%Y (2.61) 2Aln(^-J, (2.62) where </>i and fa are the potential on the small and big cylinder respectively. T h e ratios of r and r' on each is given by (2.23) and (2.24) such that & - & = 2 A l n ( ^ ) . Finally with A 2 — A\ = A 2V (2.63) — AQ = 4ir\, we write the capacitance is and Energy = A In ( ^ ) . 2 Energy of a V o r t e x Inside a Cylinder (2.65) T h e analogy between the simple electrostatic problem of two charged cylinders and the hydrodynamic description of a vortex inside a cylinder is straightforward. the container. T h e small cylinder corresponds to the vortex core and the big one to A s 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 1/2E we integrate l/2v . 2 B y analogy to the results i n the previous section, 2 we can write the kinetic energy of the fluid flow as Energy = | ( * - $ i ) ( M ' 2 - * i ) . (2-66) 2 T h e analogue of ( A — A\), which was the flux through a section of the potential line, is now 2 the potential difference ( $ 2 - $ 1 ) which is the circulation on a section of a stream function. T h e quantization of the circulation i n a superfluid implies that ($ TT ~ $ 0 ) — - We can check K 2 for the potential i n (2.36) of a vortex inside a cylinder. A s 6 —* 0, x —> R and y —> 0 + such that $n = ,. K ( lim — arctan 2ir\ V (bR-R \ —— — arctan . „ , 0+6 y V 0+ 2 l i m ^ - ^ arctan (—00) — arctan (00)^ K 7T I 2TTI _ 2 7T\ _ and as 9 —» 27r, x —> R and y —> 0 2) K = such that K / (bR-R \ I arctan I — - ^ — ^ — 2 $ 7T 2 = hm — (2.67) ~2' J— arctan (R-b 39 = l i m ^ - ( arctan (co) — arctan ( - c o ) = 2 ^ 2 + 2) = 2- ^ A s expected A<3> = K. T h e analogue of the potential difference (02 — <t>i) is the difference between the stream functions on each cylinder ^2 — ^ l - Since ^ has been chosen such that the stream constant is 0 on the cylinder, we can simply evaluate it on the vortex core. We need to substitute the following into (2.4). x = 6 + £cos0, (2.69) V = £sin0, (2.70) and we get , L T r/ = — " — 4TT l n (1 R + i b 2 ( b 2 + ? + % 2 c o s 9 ) ~ 2 R 2 b ( + £ b c o s °)] ( 27i) B?e \^ We then neglect terms i n the logarithmic function that are of higher order than 0 ( £ - 2 ) which yields (R* + b - 2R b \ 4 K 2 K 2 [R 2 + b\ 2 , 0 _ W i t h (2.66) we can now write the energy of a vortex i n 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 w i t h (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 v dxdy 2 = pj J{v\ + v ) dx dy 2 40 = pi f ( V • ( $ V $ ) - $ A $ ) dx dj/, (2.74) where the second term is zero since $ is homogeneous. T h e first term can be replaced by an integral over the boundary by the divergence theorem and we finally get 2T = pj>$V$>-hdl = p j ^ ^ d l , (2.75) where n is the unit vector normal to dt, as usual. Similarly we can also express the kinetic energy i n terms of the stream function. Since - 2 = ( f ) 2 + (^) weget 2 2T - p j ^ W -hdl = p j ^ ^ d l . (2.76) A s $ and \P are always perpendicular, one can generalize (2.5) and write 3$ 3* 3T=3r7 , a n d 3* dn- 3* = -dl- ( 2 7 7 ) Substituting i n (2.75) and (2.76) we get the forms we will be using: 2T = -pj$d$>, (2.78) 2T = pj-$d§. (2.79) and T h i s last expression is actually the one that has been used i n section 2.3.1 to calculate the energy of an off-center vortex. 2.3.3 Branch Cuts T h e two expressions for the kinetic energy derived i n the last section are equivalent but will require a different treatment of the branch cuts. W h e n the problem involves vortices and no source, we have a branch cut i n $ such that we will need to integrate around them if we use (2.78). T h i s is not the case if we use (2.79) as \P does not have any branch cut. T h e branch cut for the problem of an off-center vortex is illustrated i n Figure 2.6. Conversion from one expression to another involves integrating by parts. If there are branch 41 cuts i n one of these expression, boundary terms of the integration by parts will conveniently account for them. 42 3. Simple Yet Useful Configurations T h e objective of this chapter is to calculate the shape of the potential barrier for the nucleation of a vortex i n simple yet experimentally realistic geometries. T h e potential barrier is obtained by calculating the kinetic energy of the flow i n these configurations i n 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 w i t h one bump. We then t u r n 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 T h i s first configuration starts from the 'off-center vortex i n a cylinder' case that we presented as an example i n the previous chapter. A s shown i n 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. T h e vortex position is denoted as (xb, Ub) or (6cos/3, 6sin/3) i n cylindrical coordinates. Figure 3.1: M a p p i n g to a cylinder w i t h a wedge. T h e height of the wedge is h and the position of the vortex is (xb, Vb)- 43 3.1.1 Starting Configuration W i t h N o Wedge T h e vortex can now be located anywhere i n the cylinder such that (2.36) and (2.37) need to be modified accordingly. One obtains the velocity potential ,, . K ( x — (R lb) cos (3 x — bcos/3\ <P(x, y) = — arctan . — - — arctan -——— , ' 2ir\ y - (R /b)sm/3 y - 6sin/3 J' 2 y y j : 2 (3.1) v ; and stream function K f ^ ' > ~ 4^ V 3.1.2 n R+ b {x + 4 \R b 2 2 y ) - 2R b(x cos f3+ y sin/3) 2 2 + R (x 2 2 2 + y ) - 2R b{xcos(3 + y sin/3) 2 2 2 (3.2) Conformal Mapping Now that we have a basic configuration for which the complex velocity potential is known, the next step consists i n mapping it to the final geometry, which w i l l be done i n three steps. T h e first map transforms the interior of the cylinder to the upper half-plane. We then map to the upper half-plane w i t h a wedge and finally map back to a circle w i t h a wedge. Figure 3.2 summarizes these steps by showing what the stream lines are after each transformation. O n l y the stream functions are mapped here but potential lines can be obtained using the same changes of variables. M a p p i n g to the U p p e r Half-Plane T h e circle of radius R is first transformed into the upper half-plane w i t h z(x, y) —> w(u, v) as shown in Figure 3.3. T h i s mapping is given by « =<(£f). (3-3) which corresponds to the following change of variable: 2Ry ~TT> , To U {R + x) o a n d +y 2 R 2 V = - (x + y ) 2 — . (3.4) {R + x) 2 . 2 2 + y 2 Solving for x and y, one has R(l-(u 2 x = + v )) 7T~ and +u 2 — (1 + v) 2 2 2Ru y = — ^5 (l+v) +u 2 2 5-. (3.5) v 44 x •-> z plane w plane x o plane I plane Figure 3.2: M a p p i n g from a circle to a circle w i t h a wedge T h i s yields V(u,v) = •^ln\(u 2 + v )(R 2 + b + 2 2 + [R + b - 2Rbcos(3) + 2v{R 2 2 2 In (u + v )(R 2 47T + (R 2 2 2 + b - 2RbcosP) 2 (3.6) 2RbcosP) + b + 2 2Rbcosp) 2 - 2v(R M a p p i n g to the U p p e r Half-Plane W i t h a Wedge - b ) - 4«i?6sin/?] 2 - b ) - 4uRbsm/3 2 . T h e goal is to map the upper half- plane to the upper half-plane w i t h a wedge of height ho located at the origin, denoted by the map w(u, v) —> l(m, n). 45 v A u CL\ a 2 > 0,3 2 plane &4 05 to plane Figure 3.3: M a p p i n g from a circle to the upper half-plane. T h e points denoted by uppercase letters are mapped to their lowercase counterparts. A clever mapping shown i n 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\/w 2 — 1 (3.7) vA A 2 A 3 w plane A4 a 2 (14 I plane Figure 3.4: M a p p i n g to the upper half-plane with a wedge. T h e points denoted by uppercase letters are mapped to their lowercase counterparts. T h e factor ho, which is necessary to impose the height of the wedge, also acts as a scaling factor on the entire plane. T h i s 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 w i t h 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. T h e adequate change of variable can be derived as follows. T h e real and imaginary part of (3.8) give mn = uv 0 and 0 0 0 m—n=u—v— 0 h. n Solving for v yields 2 -1 V = — ( m - n + hi) ± \\]{m 2 2 - n + hi) 2 2 +4m n . 2 2 (3.9) 2 T h e second solution is ignored to ensure that v is real. T h e solution corresponding to the upper half-plane is -{m - n + h ) + ^{m 2 v = ^ 2 2 - n + h) 2 2 2 + 2 4m n 2 2 (3.10) and u = mn (3.11) \ -(m 2 - n + hi) + ^ ( m 2 2 - n + hi) 2 2 + 4m n ' 2 2 T h e reverse transformation is given by -(u - v - hi) + yj{u - v - hi) 2 2 2 2 + 4u v 2 2 2 (3.12) 2 and m = uv (3.13) \ -{u -v - 2 h ) + ^{u 2 2 2 0 -v 2 hi) 2 + 4u v ' 2 2 Defining the following variables to simplify these expressions: C = (m — n + hi) and D = C' = (u - v - hi) and D' = 2uv, 2 2 2 2 2mn, (3.14) (3.15) 47 we write J-C + ^/C + D 2 «= y 2 2 u = ^ + \J-2C = 2 ' , D' + u + x, = V C 2 ' (3.17) 2 + /J> , 2 ' (3.18) 2 and n = y m = , , • 7-2C + 2V C (3.19) (3.20) D r m +n 2 = VC' 2 2 / 2 + D' ' 2 + D' . (3.21) 2 T h e stream function becomes * ( m , n) = (3.22) In ( VC + D (R 2 47T 2 + b + 2Rbcos (3) + {R + b - 2Rb cos /3) 2 2 2 2 . \ + ^-2C + 2 ^ T & { R - — In [ VC An \ + D (R - J-2C 2 2 + 2 - b) - 2 2 , ^-2C 4 R *P + 2VC + b D s 2 + b + 2Rbcos (3) + (R 2 2 2VC T&(R 2 R b D s m / M a p p i n g Back to a Circle 2Rbcos0) 2 , ' ). / - 2 C + 2v C + D / 4 v 2 +b - 2 - b) - 2 ) D/ i P 2 2 J Finally, we map back to a circle w i t h a wedge, l(m, n) —> o(x, y), as shown i n Figure 3.5. For simplicity the variable x and y are used as i n the first configuration but it is important to note that they are not the same. T h e change of variable is essentially the same as i n (3.4) and (3.5). First, we have 2Ry M = TT; , v> o (R + x) +y 2 2 a n ( R 2 i n ~ ~7r, - (x 2 ^5 (R + x) 2 + y) 2 o~> +y 2 (3.23) v ' 48 n A x > m Ai A 2 A 3 A£ A 4 A / plane Figure 3.5: M a p p i n g from the upper half-plane w i t h a wedge to a circle w i t h a wedge. The points denoted by uppercase letters are mapped to their lowercase counterparts. w i t h the reverse mapping R{1- m + n 2 (l + n ) + m 2 2Rm 2 (l+n) 2 2 + m 2 v Substituting i n equations (3.14) yields C = ^ L + h, 2 0 [(R + x) 2 (3-25) + y) 2 and D = 4 ^ ^ - ^ ) . ((R + x) + y ) 2 2 2 (3.26) 2 T h e height of the wedge inside the circle is readily obtained from (3.23). Let ho be the height i n the upper half-plane and h be the height inside the circle, then "»- s r b <•*•"> 49 3.1.3 Corrections to the P o s i t i o n of the V o r t e x 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. T h i s is a very small effect for points far from the wedge, or when the wedge is very small. T h e vector plot i n 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 ) . W h e n asking what the energy of a vortex i n 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 i n 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 i n 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 i n 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). T h e answer is once again very complicated and the following functions are defined for convenience: fix, y) = ^C (x, 2 y) + D (x, 2 y) and g(x, y) = C(x, y), (3.29) where C(x, y) and D{x, y) are defined i n (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 f = / ( a cos a, a sin a ) , a (3.30) 50 and 9a = g{& cos a , a sin a). (3.31) We then have the following substitutions 2 _ P (/ 2 /i i £ + 2g + 1) 2 a ' a , A T T o ^ \ 9 >2 l^.OZJ (l + /a + V 2 / a - 2 f f ) ' a ^ ( 1 - /a) = , , * J: , , l + /a + V 2 / a - 2 f f ' { 1 (3.33) ] n a i V 2 / + 2g ) a y o i1 + • f/ a + • = VFTt 2/ a - l„2 ^ a ' ( ' ^ 3 34 a and (3.22) becomes *(*, y) = (3.35) ^ In ^(/ + / )(l + f + y/2f-2g(j a + s/2f - 2g ^j a a a ~9a + 0.5(1 + / a ) v 2 / a - 2 a ) / a 5 - v 27+2^( \ / / a - 5 + 0.5(1 + / „ ) y ^ / a + 2 / 3.1.4 2 - ^ h l ^(/ - y/2f-2g(f - y/2jT2gi^Jfl 2 5 a + / a ) ( l + /a + V 2 / a - 2 a ) / 5 a -g a + 0.5(1 + f )y/2f -2g^j a a -g% + 0.5(1 + / ) ^ / a + 2 a P a Kinetic Energy B u l d i n g on the example of section 2.3.1, we first find the kinetic energy of a vortex anywhere in the vessel. T h i s 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). T h e vortex core is assumed to remain a circle up to a quadratic correction i n £ such that $ 2 — $ 1 = « is still valid. Also, (3.35) is still zero at the container boundary. T h i s leaves \P to be calculated on 51 the vortex core w i t h (3.35). Using /? /(( = + 0 cos a , (a + £ ) s i n a ) , a (3.36) and <?£ = g((a + £) cos a , (a + £ ) s i n a ) , (3.37) the energy is 2T = (3.38) ~ In ( ( / ? + /a) ( l + / a + V 2 / a - + - - 2# ( / \ M + 2 - a ^ 5 a - 2 5 a ) + 0.5(1 + / ) v ^ / a - 2 a gl + 0.5(1 + f )y/2f a PK ^ In | t f + / ) ^ l + f 5 a + 2g a a 2 a - y/2fe-2gs(j -g a fa + 2 9i a (Jp a a -gl + y/2f a - 2 + 0.5(1 + 5 a f )y/2U^2g^ a + 0.5(1 + / ) y/2f a a + 2g ^j a T h i s result is accurate when the vortex distance to the wall is large i n 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 i n the limit where the vortex and the antivortex meet on the boundary. W h i l e 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. W i t h £ 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 m i n i m u m 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 w i t h the wedge is bounded by the energy of a vortex without a wedge. 52 Figure 3.7: K i n e t i c energy for a cylinder w i t h a wedge i n function of the position of the vortex Energy of a Vortex on the x Axis W h e n a = 0, (3.38) c a n b e s i g n i f i c a n t l y s i m p l i f i e d . Firstly, C ( a , 0 ) = -jf^ + ^ , (3.39) and D(a, w h e r e n o w a G [-R, R - h}. 0) = 0, (3.40) W i t h D = 0, (3.30) r e d u c e s t o f (3.28) i n (3.39) a n d s o l v i n g for t h e zeros gives a = -R = y/C' (a, 2 a 0). Substituting a n d a = R — h. T h u s C ( a , 0) is a l w a y s n e g a t i v e a n d we d e f i n e 7a = 5 a | Q = 0 = -/aL 0 = = C{tL, 0). (3.41) Similarly, C(a + C, 0) for a C{a - t, 0) for a e [-R + C, G [0, R - h - £] 0] 53 2T fa Figure 3.8: K i n e t i c energy of a vortex on the x axis in a cylinder w i t h and without a wedge in function of the position of the vortex. T h e height of the wedge is h and the radius of the container is R. where the domain for a is more restrictive. A s pointed out above, the results are not expected to hold when the vortex is i n the vicinity of the wall. Finally, (3.38) becomes 2n 2 T = ^ l n 47T fa - 4n In VJ? fa + f t + V fa - /| fafj (3.43) Terms of order £ should now be neglected. We substitute x for ± £ depending on the sign 2 of a and we do a Taylor expansion of 7^ and y^ToT^(R-a) 2 7£ = ^TaT? = 7a l a + -4 (R + a) - 2 - 2 R (R-a) \ + a7Tr-— (R + a) -\x + 2 3 / r i J 2 (R + a) 3 x + . ,. 0(x ^ (3.44) 2 (3.45) 0(x ). z W i t h £ much smaller than the other lengthscales, the part of the numerator of (3.43) proportional to £ can be neglected while the denominator is then proportional to £ . Thus the 2 following expression for the kinetic energy does not depend on the sign of a, and (3.43) now 'R\ , (R -a?\ pK In ( - ] + In ~2T7 R n 2T = 2 2 , / In (R-a) 2 (R-a) 2 -h (R 2 + a) 2 (3.46) 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 w i t h angular velocity Co, the velocity v of an element of fluid relative to the uniformly rotating frame is related to the velocity vo i n the inertial frame of reference by VQ = v + Co x r. (3.47) For a perfect cylinder, accounting for the rotation is easily done w i t h 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 i n the inertial frame and E is the energy i n the rotating frame. T h i s 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. T h e angular momentum of a vortex in a cylindrical container is 8 L = p J r dr J vgr d6, (3.49) where the angular part of the integral is actually the circulation §v • dl. T h e circulation depends on whether the vortex lies inside the integration path and T = K,Q(r — b). T h e angular momentum is then given by L = pK Jb/ r dr = — (R 2 2 - (3.50) b ), 2 and the energy (2.40) becomes IT pK , ~2T7 . R\ , (R?-b 2 - PKW(R 2 - b ). 2 (3.51) For a cylinder w i t h a wedge, (3.47) predicts there will be a fluid flow through the wedge, which is prohibited. One might t r y to take the rotation into consideration before applying the 8 This 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 i n a rotating cylinder. T h e angular velocity is u) = 0.5K/R and the radius of the container is R. 2 conformal maps. Unfortunately, the solid rotation of a fluid, where w a r , can't be described by a complex velocity potential, which precludes any use of conformal mapping. T h i s 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 w i t h a wedge. T h i s motivates our interest for the configuration presented in the next section, namely a flat wall w i t h 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 i n the neighborhood of the b u m p and the vortex. T h e 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 i n this configuration is the use of a semi-circular b u m p instead of a wedge. T h i s choice will prevent the need for a conformal map like (3.8) which introduces square roots i n 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 A l l 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. T h e 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 C y l i n d e r a n d M o v i n g Cylinder T h e 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 *( y) Xi = u(\ + -^j>)x, (3.53) while the stream function, shown i n Figure 3.10, is = « ( i - p ^ ) v . ( 3 - 5 4 ) T h e 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. T h i s amounts to a transformation to another inertial frame moving away from the first w i t h velocity u. T h e complex velocity potential is now n(z) =U-, z and (3.55) UR r 2 x + y z L while the stream function is ¥(*, y) = (- ) 3 x* + y A 57 Now the x axis is still a stream constant but the cylinder is not. A s mentioned i n 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. T h e fluid velocity normal to the surface is given by 3$ r\r=R V dr = -[/cos0, (3.58) 57 while the velocity of the boundary is —Ux. Thus the normal component to the surface is the same as i n (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 • — £ £ l n r , w i t h d/ds (3.59) being the space derivative i n the orientation of the doublet. recovered for fi = E q u a t i o n (3.56) is —2-KUR . Vortex Configuration 2 T h e vortex is located outside of the cylinder at (a cos a , a sin a) where a > R and a e [0, IT]. T h i s problem has already been solved i n section 2.2.3, except now the vortex is outside of the cylinder and the image vortex is inside. W i t h the method of images, the antivortex is located at ((R /a) 2 cos a, (R /a)sina). 2 T h e 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. there is also an antivortex at ( a c o s a , —asina) and a vortex at ((R /a) 2 cos a, Thus —(R /a)sma). 2 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. T h e radius of the bump is R. There is a vortex at ( a c o s a , a s i n a ) and ((R /a) cos a , — (R /a)sma) and an antovortex at ( a c o s a , —asina) and ((R /a) cos a , {R /a) sin a ) . 2 2 S u m of the T w o Configurations 2 2 Using the previous results (2.13) and (2.14) and adding the contribution from the moving cylinder i n (3.56) and (3.57), we have UR x K 2 ar + y 2ir z . y-yo arctan x — XQ ^ y + %-yo + arctan ^ R X - %-Xrj . y + yo arctan X — XQ ^ y arctan 2 X — (3.60) ^yo R 2 ^ X Q and UR y K 2 x +y 2 2n 2 ln^(x- x) 2 0 // 1 \n^(x- + (y- x) 2 0 R2 + {y + y ) 2 0 \ 2 / + lnJ(x2 R (3.61) y) 2 0 R \ ~^ Qj 2 x 2 ( + fV+ R -^Vo 2 \ 2' where XQ = a c o s a and yo = a s i n a . T h e stream functions of each component is shown i n 3.12. 59 u x X -> -> Figure 3.12: Stream function of a vortex near a moving cylinder. T h e cylinder is moving with velocity —Ux 3.2.2 Kinetic Energy T h e 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 w i t h one vortex. T h e n one can use either (2.78) or (2.79) on the boundary to calculate the energy of the vortex. T h i s boundary is the combination of the cylinder and the two vortex cores located outside of the bump. O f 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 — ad and w i t h $ > \ r = R = -URsin9, (3.62) r = R , the energy is pU R 2 2T = 0 /•2ir 2 / Jo sin 9 d$ = 2 (3.63) U TTR , 2 2 P which corresponds to the kinetic energy of a quantity of fluid moving at velocity U and having the same volume as the cylinder. T h i s quantity will need to be substracted from the total kinetic energy to isolate the vortex contribution. Integral on the C y l i n d e r O n the cylinder, it is convenient to use polar coordinates such R and that the boundary is r = the total derivative d<& — ^\r=Rd9. F r o m (3.60), and using the shorthand notation (XQ, yo) for (a cos a, a sin a ) , we have <9$ 39 UR . _ , K ( r -rx cos 9 - ry sin 9 \ _ — _ 2ryo 1 (3.64) r 9 + —[2TT \a — + r—— 2rxo cos6 sin9 ( r — cos 9 + ryo sin 9 2n V a 2TXQ cos 9 + 2ryo sin 9 ( r -r(R /'a )x cos9 + r(R /a )y< sin9 2lv\R /o? + r - 2r{R /a )x cos 9 + 2r(R /a )y sin9 K ( r — r(R /a )xQcos9 - r(R /a )yosin9 ~ 2TT\R /a + r - 2r(R /a )x cos9 - 2r(R /a )y sin9 2 2 0 sm 2 0 2 T o 3 a o 2 TXQ K 2 T + 2 — 2 2 2 2 K 2 0 + A 2 0 2 2 2 2 0 2 i 2 2 2 0 2 2 2 2 2 2 2 0 0 which, evaluated at r = R, gives a?-R 2 = -URsin9 + \R + a -2Rx cos9 + 2Ry sin9 a -R R + a - 2Rx cos 9 - 2Ry sin 9 ~d9, 2 =R 2 0 2 2 0 2 2 0 (3.65) 0 T h e contribution to the stream function on the cylinder from the vortices being zero, only (3.57) is left and the integral becomes 2T i = pU R j> sin 9 d9 2 2 2 (3.66) cy nUR(a - R ) r * -sing ^ 2TT % R + a - 2Rx cos 9 + 2Ry sin 9 2 + P 2 2 2 2 0 0 61 sin( KUR(Q?-R ) 2 + 2, P W + a — 2RXQ cos 9 — 2Ryo sin 9 % 2 d9. T h e first term is the same as the metastable state energy (3.63). T h e second term is equivalent to the t h i r d 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 w i t h one integral for which we need the following result taken from [18], 2.558: / A + B cos x + C sin x a + b cos x + c sin x Bc-Cb dx (3.67) . , . , 7T hi (a + o cos x + c sin x) 4- t —5 b + Bb-Cc ^r x -ro b + c l l d + (A °a\ I — V B + c J J a + b cos x + c sin x' B B + C 2 2 and / dx a + b cos x + c sin x (a-6) tan §+c arctan * / 2_ 2_c2 ^/ 2_ 2_ 2 a b c x 0 a i In ^ f - ^ ± ^ V P + W (a-6) tan f+c+v'fc +c -a t a n + c 2 = < 1 In (a + c t a n 2 > b + c 2 2 b 2 a <6 2 2 2 + c o? = b + c 2 ; Here A = 73 = 0, C = 1, a = R 2 (3.68) a — b : -2 I. c+(a—fc) tan 2 2 + a , b = - 2 . R x o and c = —2Rya. Since the integral is 2 evaluated on a closed path, any term having an even dependence on 9 goes to zero such that o m v W^c lX r e m a 2T m s . T h e total contribution to the kinetic energy from the cylinder is then c y l / 'i _ r>2\ - KU(—^—Wo = U nR 2 2 Integral on the V o r t e x C o r e = U nR 2 2 /a - KU( 2 — R\ J sin a . 2 (3.69) B y 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 -UR (r ; cos0 + x -) 2 o (r cos e + x ) 2 0 + {rsme + y ) 2 0 + ± 9 — arctan r sin e + 2ir 2j/o rcos# 62 rsin0 + ( ^ # \y rcos0 + ( )x rsin0 + ( 0 + arctan 12/0 (3.70) — arctan rcose+[ 0 iso and UR (r sin0 + y ) 2 * = — r cos 0 K o (3.71) (r sin 0 + yo) 2 + XQ) + 2 In r — In \Jr cos 0 + (r sin 0 + 2t/o) 2 2 2 2^ R\ a + R 2 2 + In r cos 9 + XQ) + XQ] + [r sin 9 + y a 2 -R \ r sin 0 + a -R \ 2 2 - In r cos 9 + Y a 2 2 2 a 2 ) yo Next we need 5$ K / W r + 2rsin9yo 2 (3.72) r + 4yQ + 4r sin 9yo 2 2TT\ r + rB cos 9XQ + rA sin 0yo 2 + r + B x\ 2 + A yl 2 2 + 2rB cos 0 x + 2r A sin 9y o 0 r + rB cos 9XQ + rB sin 0yo r + B a + 2rB cos 9x + 2rB sin 9y r sin 9 UR [r + a + 2r cos 8XQ + 2r sin9yo 2 2 2 2 Q - + where A = 0 , 2 0 2 2 2 ( r c o s 0 + XQ)(—2r sin0xo + 2r cos0yo) ( r + a + 2r cos 0xo + 2r sin 0yo) 2 2 2 a n d / J = - ^ r ~ - T h e previous result evaluated at r = £, while taking £ to be 2 much smaller than any other lengthscale, simplifies to 30 (3.73) 2~n' O n the other hand, (3.71) reduces to (3.74) K 2^ , 2a sin a 1, I n — - I n / ^cos , f a? + R \ a + ^ ^ j sin a 2 2 2 • sin a . 63 Finally, multiplying the the last two results and integrating yields the following contribution to the kinetic energy: pnUR 2 27}core pK" ~2n~ Kinetic Energy (3.75) sin a In 2a sin a -fiascos a + { 2 _ a R 2 ) s m a T h e 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 2T = KU P (a — 2 \ l + R\ . z a J sm a + OK 2 — Ait r 4a sin a hi 2 2 l n ^ c o s a+ [^—tf) s m Q T h e shape of the resulting potential barrier is shown i n Figure 3.14 and i n Figure 3.15. 2T^ (b) Figure 3.14: K i n e t i c 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 b u m p i n 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 i n a perfect cylinder and close to a circular bump to calculate the exponent of the tunnelling rate with the W K B method. T h e 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 W K B Approximation T h e W K B method is a semi-classical technique for obtaining approximate solutions to the time-independent Schrodinger equation i n one dimension. For a particle w i t h energy E moving through a region where the potential V(x) is constant, the wave function is of the form ij>(x) = A e ± i k x , with k= ^2m{E-V). (4.1) W i t h V constant, the particle moves w i t h a fixed wavelength A, and constant amplitude. If V{x) varies rather slowly i n 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 T h i s 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. T h i s probability is proportional to T«exp(-2 ), 7 (4.2) and the exponent is 1 = \ l j2m\V{x) n Ja - E\ dx. (4.3) T h i s 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 i n this chapter. 4.1.2 W K B in Higher Dimensions W K B 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 W K B 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. T h i s technique is shown to be consistent w i t h 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 w i t h 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 i n section 4.4. 4.2 Tunneling Rate For a Perfect Cylinder We start w i t h 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 B a r r i e r T h e potential barrier is obtained from the kinetic energy of the flow of a vortex in a perfect cylinder found i n (3.51). It can be written in the more convenient form (4.4) where £ = £/R is the core radius normalized w i t h 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). T h e 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 i n a perfect cylinder as a function of the velocity u We want to consider the tunneling through the barrier of a particle w i t h zero momentum. T h e W K B exponent (4.3) can then be expressed i n terms of V(x) such that (4.7) A g a i n for simplicity we define x) dx, Ja (4.8) 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 w i t h up to quadratic terms but even w i t h higher order terms the approximation is poor. A few more properties of this potential will be needed to find a better approximation. T h e potential has a m a x i m u m at — \; max x V U i (4-H) and its value is V =-l-\nul max (4.12) T h e 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 x — 1. T h e inflection point is located at b * _ 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 T h e calculation of the exponent (4.8) involves the difficult integral of \Jv{x). To carry out this integral, V{x) w i l l need to be approximated by a polynomial. Even though a Taylor expansion w i t h a small number of terms has already been ruled out, Figures 4.1 and 4.2 suggest that for small £ a n d small u, a quadratic expression can provide a good approximation between between x a and x m a x . For small £ and large u a linear approximation between the two same points is acceptable. Between x and x = 1 the potential falls very quickly and the area under b the curve is relatively small. A simple quadratic term w i t h 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 i n a perfect cylinder, (a) Taylor expansion at the origin w i t h terms up to second order, (b) A p p r o x i m a t i o n i n two parts: a line from (x , 0) to (x , V ) and a Taylor expansion around x w i t h terms up to quadratic order, (c) T h e potential V(x), shown as a dashed line. a max max m a x polynomial. Starting w i t h the linear part, the slope of the line joining (x , 0) and (x , a dV max -1-lnK) d£ \ =x x ^Ju=l a V ) is max (4.15) _ ^c-lng W i t h a change of variable so that x is at the origin, (4.8) yields a ^-fV^^(^-f^). To build the quadratic part, V max (4-16) is added to a quadratic term 1 fd V(x -x')\ 2 max 21 dx (4.17) 2 so y ( x ' ) « - 1 - In K ) - 2u(u - l ) x ' . 2 ' (4.18) To perform the integral we will need r J , VaTto?dx=l f %y/a 2 \ + bx + r % In (Vbx + Va + bx ) 6 > 0 ^ , rirr , 2 2 2 1 fv aT^ +^ / K arctan (j§^) (- ) 4 19 b < 0, 70 such that Iquad = - 1 - In « ) , , v arctan VMu -1) (i - ^ ) (4.20) / o A/8U(M — 1) - l - l n « ) - 2 « ( « - l ) 1- T h e approximation given by 7;j + 7 ad is shown in Figure 4.3. n gU a e o a x H CQ Figure 4.3: W K B 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 i n two parts. T h e 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- T h e numerical results are drawn i n solid lines for comparison. T h e second approximation w i t h two quadratic parts can be calculated similarly and is shown on Figure 4.4. A s we expected, the first approximation is more accurate at small u, whereas the second is more accurate at large u. 71 10 5 15 20 25 30 u Figure 4.4: W K B exponent j(u) for a vortex i n a perfect cylinder as a function of the angular velocity parameter u and calculated from a potential approximated i n two parts. T h e 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- T h e numerical results are drawn i n solid lines for comparison. 4.2.3 N u m e r i c a l Integration Because of the difficulty to evaluate the integral analytically even when using polynomial approximations, we now t u r n to numerical techniques. T h e procedure to evaluate 7 as a function of u and £ starts w i t h solving for the zeros of the function and then integrating between these two limits. T h e 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 T h e form of the curves j(u) for a constant value of £ suggests that a form like 7(u) = A | u | - A + 7 o , (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 w i t h the numerical data. 5 10 15 20 u 25 30 Figure 4.5: Curve fits of the W K B exponent -y(u) for a perfect cylinder and for different values of the core radius £ using 7 = A | i i | + 7o. A curve fit is done for each value of £. T h e numerical results are drawn i n solid lines for comparison. _ A T h e 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 w i t h 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 1 0 - 8 to 1 0 ~ . We therefore fix the exponent at an average 4 value for this range w i t h A = 2. T h e results for A and 70 are presented i n Figure 4.7. A can then be approximated by A = b{\n£f+ especially if the point £ = 1 0 ~ 10 b, 0 (4-22) is neglected. 70 is well approximated by a logarithmic function 73 of the form 7 0 = 5 In ( | ) + 5 0 , (4.23) except for an anomaly at £ = 1 0 . We then find that - 9 A » 0.28(ln<£) ' + 3, (4.24) 70 » 0.021 In | - 0.07, (4.25) 7(u, |) « (0.28(ln|) 2 6 + 3 ) | u | - + (0.021 l n £ - 0.07). 2 6 2 (4.26) T h e values of ( t t , £) are compared against the numerical results i n Figure 4.8. 7 T h e exponent 7 given by (4.24) is good for values of £ « 1 0 . T h i s gives us three different - 7 approximations w i t h varying applicability depending upon the parameter range of interest. 4.3 Tunneling Rate For a Circular Bump on a Flat Wall T h e 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 T h e barrier is obtained from the kinetic energy of the flow of a vortex close to a moving bump as derived i n (3.76). T h e barrier has a m i n i m u m along a = ir/2 and can be written as pn? where £ = /Ax (x -1) \ 2 2 2 fx 2 + l\ , A M . is the core radius normalized w i t h 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 i n (3.76). T h e dependence on bump velocity U is now included i n the unitless parameter u defined as (4-28) w i t h 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 T h e W K B exponent (4.3) at E = 0 can be expressed i n terms of V(x) and J \JV{X) (4.30) dx. Ja A g a i n for simplicity we define 7 [ y/V(x)dx, where K (4.31) 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 m a x i m u m is positive. Despite the fact that the shape of the barrier is quite similar to the one studied i n the previous section, its properties, i.e. zeros, m a x i m u m and inflexion points, are much harder to approximate. To find the exact m a x i m u m , one needs to solve a s i x t h order polynomial: u(a 6 (4.33) - a - a + 1) + a - 2 a - 8 a - a + 2a = 0. 4 2 6 5 3 2 A n approximate solution still has a complicated form. Solving for the zeros by doing a Taylor expansion of the polynomial i n the logarithmic function a n d keeping only a few terms also agrees poorly w i t h 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 i n expressing the potential as a polynomial, we are left with the only option of performing the integration numerically. T h e evaluation of 7(11, £) is similar to the case of a perfect cylinder. It starts by solving for the position of the m a x i m u m to be sure that there are zeros. We then look for a zero located between x = 1 and x = x located between x m a x and infinity. \jv(x) m a x and one is then integrated between these two limits. T h e results are drawn as solid lines i n 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 T h e results of the numerical integration look similar to those of a vortex i n a perfect cylinder. T h e 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 w i t h the numerical data. T h e results for A, 7 0 and A are presented i n Figure 4.10. A can then be approximated by a logarithmic function A = bln(i) + bo, (4-34) and 7 0 by the same form as well 7 0 = 9 In (0 + 90- (4.35) Even the exponent A finds an adequate approximation w i t h 1 A = d\£\ + d . (4.36) 5 0 Summarizing, we have A » - 1 2 . 0 In ( 0 - 1 8 , (4-37) 70 « 0.171 In ( | ) - 6.65, (4.38) A « 0.27|°- (4.39) 7(u, | ) « (-12.01n(O-18)H-°- ^ ' - - 21 + 1.041, 2 7 ) 2 1 1 0 4 1 + (0.1711n(|)-6.65). (4.40) T h e values of j(u, 0 are compared against the numerical results i n Figure 4.11. T h e agreement is best for | = 1 0 ~ . 8 T h e last fit was acceptable but a more accurate fit can be calculated using only two parameters. T h e results for A and 7 0 when A is fixed at A — 1 are presented i n Figure 4.12. A and 7 0 still have a logarithmic £ dependence and l{u, A « - 1 1 . 8 In (f) - 15, (4.41) 70 » 0.136 In ( 0 - 9 . 4 , (4.42) £) « (-11.81n (() - 1 5 ) 1 ^ + (0.1361n ( | ) - 9 . 4 ) . (4.43) T h e values of j(u, 0 are compared against the numerical results i n Figure 4.13. T h e agreement is better than (4.37) owing to the smaller number of parameters. T h e optimal results are again for ( = 1 0 ~ . 8 76 1 1 0.8 ' 1 Parameter 70 H U - H 100000 - 1 • I " ' Parameter'A ' 1 10000 0.6 • I 1000 I 0.4 * 0.2 0 i - * 100 • + • i le-10 3.5 I i 1 le-08 le-06 . 1 . 1 le-04 0.01 le-10 le-OB le-06 £ £ (a) (b) ' T i Parameter' A le-04 0.01 H-4—i - I 3 I 2.5 1 i 2 1.5 i le-10 le-06 le-OB le-04 0.01 £ (c) Figure 4.6: Curve fit parameters of the W K B exponent j(u) for a perfect cylinder as a function of £ using 7 = A | t t | + JQ. (a) Offset 7 0 . (b) A m p l i t u d e A. (c) Exponent A. _ A 77 ' 1 ' 1 Parameter 70 • 10000 1 Z T ' ' R Pararrieter' A H U - H : 0.4 1000 0.3 100 0.2 10 - 0.1 0 1 1 1 1 le-10 le-08 le-06 le-04 | 1 1 0.01 le-10 I le-08 le-06 i i (a) (b) . I . le-04 I 0.01 Figure 4.7: Curve fit parameters of the W K B exponent ( u ) for a perfect cylinder as a function of £ using 7 = A | u | + 70 and curve fit of these parameters, (a) Offset % approximated by 7o = g i n (£) + go- (b) A m p l i t u d e A approximated by A = 6 ( l n £ ) ^ + bo7 - 2 78 Figure 4.8: Curve fits of the W K B exponent j(u) for a perfect cylinder and for different values of the core radius £ using 7 = A | u | + 7 0 . Parameter A is approximated by A = 6 ( l n £ ) ^ + bo and 7 0 by 7 0 = 5 I n (£) + go- T h e numerical results are drawn i n solid lines for comparison. - 2 79 Figure 4.9: Curve fits of the W K B exponent j(u) for a circular b u m p on a flat wall and for different values of the core radius £ using 7 = A | u | + 7 0 . A curve fit is done for each value of £. T h e numerical results are drawn in solid lines for comparison. _ A 80 Parameter 70 ' ~I 1—1—' 1 1 - - - 1 le-10 1 le-08 1 . 1 le-06 . le-04 1 . 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 W K B exponent j(u) for a circular bump as a function of £ using 7 = A | u | ~ + 7o and curve fit of these parameters, (a) Offset 70 approximated by 70 = # l n ( £ ) + go. (b) A m p l i t u d e A approximated by A = 61n(£) + bo- (c) Exponent A approximated by A = d\£\ + do. A s 81 2 4 6 8 10 12 14 u Figure 4.11: Curve fits of the W K B exponent -7(11) for a circular b u m p and for different values of the core radius £ using 7 = A | u | + 7 . Parameters A is approximated by A = C l n (£) + Co, 7o by 70 = P i n (f) + P and A by A = D\^\ + D . T h e numerical results are drawn i n solid lines for comparison. _ A 0 s 0 0 82 Parameter 70 1 >—H—1 ' Parameter 'A H-+-H 1 + • iX\ - - : y" - - A'" - i ,T'' 1 le-10 le-08 . le-06 I . I le-04 . I 0.01 . r le-10 1 le-08 le-06 le-04 0.01 (b) Figure 4.12: Curve fit parameters of the W K B 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) A m p l i t u d e A approximated by A = 6In (£) + bo- 83 2 4 6 centering 8 10 12 14 u Figure 4.13: Curve fits of the W K B exponent ^(u) for a circular bump and for different values of the core radius £ using 7 = A | u | + 7 o . Parameters A is approximated by A = C In (£) + Co and 70 by 70 = -B In (£) + BQ. T h e numerical results are drawn i n solid lines for comparison. _ 1 84 4.4 Effective Mass of a Vortex T h e determination of the mass of a vortex i n superfluid H e is not a new problem albeit a very 4 intricate one. Too often its discussion i n 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 D u a n and Leggett [9, 10, 11, 12, 13]. A naive treatment w i l l only include the mass of the normal core given by m = 7rp£ , (4.44) 2 core where the mass is given per unit length of the vortex or container. T h e 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 « %, c (4.45) s for the inertial mass of the vortex, where eo is the static vortex energy and c is the sound s velocity. T h e same result was later derived by Popov i n 1973 [35] by mapping vortices and phonons into charged particles and photons i n relativistic electrodynamics. D u a n and Leggett also published a few papers on the subject. In [11], D u a n shows that due to gauge-symmetry breaking and the topology of a vortex, the condensate compressibility contribution to a vortex mass agrees w i t h (4.45). In the end, this result is only valid for a vortex i n 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/c s 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 eoT h i s lengthscale should be determined by R' ~ C T , S B (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 I n e r t i a l M a s s o f a V o r t e x for a N o n - U n i f o r m Condensate T h i s derivation follows the one by D u a n [11] which provides a more detailed treatment than the general dimensional arguments provided i n his publications w i t h Leggett [9, 13]. T h e following fills a few gaps left by D u a n but reproduces only the calculation of the inertial mass of a rectilinear vortex. T h e study of the uniform condensate mentioned in chapter 1 provides at most a qualitative picture of the microscopic justification of the superfluid flow associated w i t h the motion of the condensate. D u a n 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 T h e order parameter for a uniform condensate i n the zero temperature limit, w i t h a small deviation i n the superfluid density A(r, t), can be expressed as </>(f, t) = e ^ ' ^ l + A(f, t l l e - " " ' ^ 1 (4.48) where no and [io are the uniform condensate number density and the chemical potential. A t T = 0, one has no = N. T h e small deviation i n 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. T h e 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) « 2 m c A ( f , t), 2 (4.51) when neglecting the term proportional to A . 2 86 T h e dynamic equation for the superfluid, ^ + ( V ^ + M )=0, specifies that the flow must be potential at all time. (452) In terms of the phase of the order parameter, we have . dO oi + m.4 o ~i - M s = ( } T h e 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 i n bulk superfluid where the superfluid velocity is given by v {r) s = {h/m^e^ and we give the vortex a small velocity ve . x 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 v {f) s as t —> 0. From (4.53), the change i n the chemical potential is do - 5u = h— = -hvdt h VO = --v •*• • e^, (4.54) and from (4.51) the change i n the density is 8p(r) = (4.55) cjr which describes a dipolar density field. T h e energy change can then be calculated to second order i n the density fluctuation as 1 r dv 2 2j E= ^ 2 ^ ) d ' f ( 4 5 6 ) where u is the energy density which relates to p, and p w i t h d u _ J_dji 2 dp 2 UI4 dp _ c 2 Nrrii (4.57) This is the same as in the BCS theory of superconductors where the phase and number density are also conjugates. 9 87 Inserting this and (4.55) into (4.56) and integrating from the core radius £ to the radius of the system R we find v f Nh s i n 6 — dr, 2 J c% r 2 E = v 2 2 \irNh 2 c m 2 4 2 , R m £ „2 r R c m|) 2 £ (4.58) T h e coherence length due to the Heisenberg indeterminism is n (4.59) m^Cs where m$c is the characteristic momentum scale. In (4.58) this yields s E = — 2 2 R~\ In- r (4.60) and one can define the factor i n square brackets to be the inertial mass of the vortex i since E = H (4.61) (l/2)m v . 2 inert Finally, using K = h/1714 and our result (2.73) for the energy of a vortex i n a cylinder located at the center, we recover E = 4.5 pK? 4TT R e ln^ = ^ . 0 (4.62) £ 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 i n those configurations without resorting to numerical techniques. The tunneling rate could then be calculated in the semi-classical limit. W e had to limit ourselves to the calculation of the W K B exponent along the path of least action. F r o m there we can consider fluctuations around this path using a quadratic expansion about it. 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Vortex nucleation in a superfluid Marchand, Dominic 2006
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Title | Vortex nucleation in a superfluid |
Creator |
Marchand, Dominic |
Date Issued | 2006 |
Description | Superfluids have very peculiar rotational properties as the Hess-Fairbank experiment spectacularly 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 fundamental 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 temperature 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]. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085241 |
URI | http://hdl.handle.net/2429/18062 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2006-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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