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Kinematic and thermodynamic stability of vortices in superfluids Chapman, David M. F. 1977

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THE KINEMATIC AND THERMODYNAMIC STABILITY OF VORTICES IN SUPERFLUIDS BY DAVID M.F. CHAPMAN B. S c , U n i v e r s i t y of Ottawa, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (DEPARTMENT OF PHYSICS) We accept t h i s t h e s i s as conforming to the required standard. A p r i l , 1977 Copyright, David M.F. Chapman, 1977 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l ica t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of P h y s l c s The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e A p r i l 26, 1977 ( i ) ABSTRACT The equations of motion of v o r t i c e s i n an i d e a l two-dimensional f l u i d are derived, and the laws of conservation associated with the motion are discussed. Two regions of flow are considered, an i n f i n i t e region and a region bounded on the outside by a c i r c l e . The k i n e t i c energy and angular momentum of a vortex f l u i d i n these two regions are c a l c u l a t e d . A Lagrangian formalism i s introduced i n order to discuss the symmetry transformations of vortex systems and t h e i r associated conservation laws. The kinematic s t a -b i l i t y of r i g i d l y r o t a t i n g polygonal configurations of quantized v o r t i c e s i s determined, and the states of thermodynamic equilibrium of a r o t a t i n g s u p e r f l u i d are found for low angular v e l o c i t i e s , r e s u l t i n g i n the c a l c u l a t i o n of the spectrum of c r i t i c a l angular v e l o c i t i e s f or the creation of one, two, and three v o r t i c e s . ( i i ) TABLE OF CONTENTS Page 1. INTRODUCTION 1.1 A B r i e f Review of the Relevant L i t e r a t u r e 1 1.2 The Two-Fluid Model of L i q u i d Helium 2 1.3 Some Mathematical P r e l i m i n a r i e s 4 1.4 The Object of t h i s Work 6 2. MOTION OF VORTICES IN A TWO-DIMENSIONAL FLUID 8 2.1 Equations of Motion and Conservation Laws i n an I n f i n i t e F l u i d 8 2.2 The V e l o c i t y P o t e n t i a l of a Vortex F l u i d Inside the Unit C i r c l e 11 2.3 Equations of Motion and Conservation Laws Inside the Unit C i r c l e 12 3. DYNAMICAL QUANTITIES OF A VORTEX FLUID 14 3.1 K i n e t i c Energy 14 3.2 . Angular Momentum 16 3.3 Renormalizing the Energy 17 3.4 In the Limit of an I n f i n i t e F l u i d 18 4. THE LAGRANGIAN MECHANICS OF VORTEX SYSTEMS 20 4.1 The Lagrangian Formalism 20 4.2 The Symmetry Transformations and Conservation Laws of V.ortex Motion 22 5. THE STABILITY OF RIGIDLY ROTATING CONFIGURATIONS 25 5.1 The Equilibrium State of a Rotating Vortex F l u i d 25 5.2 The Linearized Equations of Motion of the Perturbations 27 5.3 The S t a b i l i t y of the Polygonal Configurations 30 ( i i i ) TABLE OF CONTENTS (continued) Page 6. THERMODYNAMIC EQUILIBRIUM OF A ROTATING SUPERFLUID 33 6.1 One Vortex in a Rotating Cylinder 34 6.2 The Spectrum of C r i t i c a l Angular Velocities in He II 35 7. CONCLUSION 39 BIBLIOGRAPHY 41 APPENDIX A: STREAM FUNCTION APPROXIMATION FOR ENERGY CALCULATION 43 APPENDIX B: SOME RELEVANT TRIGONOMETRIC SUMS 44 APPENDIX C: THE FREE ENERGY OF REGULAR POLYGONAL CONFIGURATIONS 45 APPENDIX D: SOME MISPRINTS IN HAVELOCK'S PAPER ' 46 LIST OF TABLES TABLE I Some Relevant Trigonometric Sums 32 TABLE II The Values of r and fi . for the Stable Polygons 32 c mm LIST OF FIGURES Figure 1: The equilibrium free energies of one, two, and three vortices 38 (iv) ACKNOWLEDGMENTS I thank F.A. Kaempffer for h i s suggestion of my research top i c and for h i s supervision during my period of study at UBC. I am also grate-f u l f o r the knowledge and understanding of physics I have gained from the informal discussions we have had, and from h i s l e c t u r e s . I am indebted to my co-students G. Mertz and D. Hally for the valuable suggestions and ideas they forwarded during our many discussions on the subject of vortex motion. I thank the National Research Council of Canada for t h e i r f i n a n c i a l support i n the form of a Postgraduate Scholarship. y 1 1. INTRODUCTION 1.1 A B r i e f Review of the Relevant L i t e r a t u r e Interest i n the motion of v o r t i c e s i n a two-dimensional i d e a l f l u i d was i n i t i a t e d i n the l a t e 19*"*1 century by mathematicians and p h y s i c i s t s such as Helmholtz, K i r c h o f f , Lord K e l v i n , Stokes, and Routh. Much of t h i s e arly work i s covered i n the c l a s s i c texts on hydrodynamics by Lamb(1932) and Milne-Thomson(1968). While considerable e f f o r t was invested i n f i n d i n g solutions of the equations of motion of v o r t i c e s i n various configurations with various boundaries, i t became apparent that c e r t a i n configurations of v o r t i c e s moved as a r i g i d body, preserving t h e i r r e l a t i v e displacements, generally by r o t a t i n g with a constant angular v e l o c i t y . The s t a b i l i t y of these r i g i d l y r o t a t i n g configurations against small deformations became of i n t e r e s t . Thomson(1883) studied the s t a b i l i t y of v o r t i c e s of equal strength placed at the v e r t i c e s of regular polygons. He found that such configurations are stable up to and in c l u d i n g N=6 (N being the number of v o r t i c e s i n the configuration) independent of the s i z e of the polygon, whereas a l l configurations with N>7 are unstable, N=7 being of indeterminant s t a b i l i t y to f i r s t order i n pertur-bation theory. Further work on the s t a b i l i t y of various r i g i d l y r o t a t i n g con-f i g u r a t i o n s has been done by Morton(1934,1935) and Havelock(1931). A t r e a t i s e on the motion of v o r t i c e s i n two dimensions with a r b i t r a r y boundaries has been produced by Lin(1943) who also reviews some of the early l i t e r a t u r e . Interest i n the motion of v o r t i c e s was rekindled by the conjecture by Onsager(1949) and Feynman(1955) that l i q u i d helium i n the s u p e r f l u i d state allowed p e r s i s t e n t currents of c i r c u l a t o r y flow whose c i r c u l a t i o n would be quantized i n u n i t s of h/m, m being the mass of the helium atom. This c u r l - f r e e 2 flow with net c i r c u l a t i o n could only be supported by singular v o r t i c e s thread-ing the f l u i d , such that the flow would be i r r o t a t i o n a l everywhere except at the vortex l i n e i t s e l f . This conjecture was v e r i f i e d experimentally by Vinen(1961) who demonstrated that the c i r c u l a t i o n i n He I I could indeed only be i n units of h/m. The presence of vortex l i n e s cleared up many mysteries about H e ' l l i n r o t a t i o n , e s p e c i a l l y the " r o t a t i o n paradox", discussed by Wilks(1970) i n chapters 7 and 8 of h i s introductory book on l i q u i d helium. The discovery p r e c i p i t a t e d much research on r o t a t i n g superfluids and on the motion of v o r t i c e s i n c y l i n d r i c a l containers. The t h e o r e t i c a l work of Hess (1967) and the experimental work of Reppy, Depatie, and Lane(1961) and Reppy and Lane(1965) i s relevant to t h i s work. Tkachenko(1966) determined the s t a -b i l i t y of i n f i n i t e , doubly p e r i o d i c vortex l a t t i c e s . Putterman and Uhlenbeck (1969) produced the d e f i n i t i v e paper on the thermodynamics of r o t a t i n g super-f l u i d s upon which t h i s work i s based. The detection of the presence of d i s -crete quantized v o r t i c e s i n He II was accomplished by Packard and Sanders(1969) and the s p a t i a l p o s i t i o n s of v o r t i c e s i n r o t a t i n g He II were photographed by Williams and Packard(1974), showing no discernable pattern. The work of Packard and Sanders(1969) also demonstrated that there e x i s t s a c r i t i c a l angular v e l o c i t y of r o t a t i o n , below which no v o r t i c e s may be present i n the f l u i d at equilibrium. 1.2 The Two-Fluid Model of L i q u i d Helium It i s not intended to cover the theory of l i q u i d helium i n t h i s work, but some of the r e s u l t s and basic concepts of the two-fluid model due to Landau(1941) w i l l be stated. The reader i s again refered to Wilks(1970) f o r a more complete introduction. In the two-fluid model of He II i t i s postulated that the actual s u p e r f l u i d may be described by two no n - i n t e r a c t i n g , i n t e r p e n e t r a t i n g f l u i d s , so that at each point there i s a "normal" f l u i d with density p and v e l o c i t y V n -n and also a " s u p e r f l u i d " with density and v e l o c i t y V^-. The t o t a l density of the f l u i d i s p=p +p , and the t o t a l momentum density i s i=p V +p V . The n s J - n-n s-s equilibrium r a t i o - P g / P i s a function of temperature alone, being unity at T=0, and dropping to zero as T->-T , the lambda t r a n s i t i o n temperature (above which helium i s not a s u p e r f l u i d ) . The..superfluid component may not experience v i s c o s i t y , i t s entropy i s postulated to be zero, whereas the normal component does experience v i s c o s i t y and the entropy of the f l u i d resides i n the normal component alone. The s u p e r f l u i d flow i s also postulated to have no turbulence, t h i s condition being s a t i s f i e d by the statement curlV"s=0 (that i s , the flow i s " i r r o t a t i o n a l " ) . The r e s u l t s which are needed i n t h i s work concern the conditions on the flow of both normal and s u p e r f l u i d components for a r o t a t i n g v e s s e l of He II to be i n thermal equilibrium. Putterman and Uhlenbeck use a v a r i a t i o n a l p r i n c i p l e to determine the equilibrium conditions, maximizing the entropy of the f l u i d while maintaining t o t a l mass, t o t a l energy, and t o t a l angular momen-tum constant. The conditions on the v e l o c i t y f i e l d s are: (a) The normal f l u i d rotates as a r i g i d body with the angular v e l o -c i t y of the container, i . e . V = fixr (1-1) -n - -(b) The s u p e r f l u i d v e l o c i t y f i e l d i s stationary i n the frame of reference i n which the normal f l u i d i s at r e s t , i . e . 3V — g , -z— = 0 (in the r o t a t i n g frame) (1-2) o t (c) The s u p e r f l u i d v e l o c i t y f i e l d i s i r r o t a t i o n a l , i . e . VxV = 0 (1-3) -s (d) With the approximation that .p and are everywhere constant, the condition f o r thermodynamic equilibrium of the f l u i d i s that the "free energy" of the s u p e r f l u i d component be at an absolute minimum, i . e . F = E-fi.L = minimum (1-4) In which E i s the k i n e t i c energy of the s u p e r f l u i d and L i s the angular momentum of the s u p e r f l u i d . This statement i s equivalent to the statement that, i n the r o t a t i n g frame, the energy of the s u p e r f l u i d i s at an absolute minimum. Conditions (b) and (c) are s a t i s f i e d by singular v o r t i c e s of the c l a s s i c a l type being present i n the s u p e r f l u i d and ro t a t i n g i n r i g i d configurations with angular v e l o c i t y Q. If the vortex strengths are not quantized, but take any value i n a continuous range, condition (d) would be s a t i s f i e d by an i n f i n i t e number of v o r t i c e s of i n f i n i t e s i m a l strength, approaching a state of s o l i d body r o t a t i o n . The problem of s a t i s f y i n g condi-t i o n (d) becomes i n t e r e s t i n g when i t i s recognized that the vortex strengths are i n r e a l i t y quantized, and that there i s a minimum .non-zero strength. 1.3 Some Mathematical Pr e l i m i n a r i e s The experiments on r o t a t i n g He II generally contain the l i q u i d i n c y l i n d r i c a l vessels whose diameter i s much smaller than t h e i r length, so that the e f f e c t s of the vapour-liquid i n t e r f a c e at the top and the s o l i d boundary at the bottom are ignorable. Furthermore, a t h e o r e t i c a l assumption i s made that the vortex l i n e s are r e c t i l i n e a r and p a r a l l e l i n the f l u i d , not s p a g h e t t i - l i k e tangle, so that the f l u i d has t r a n s l a t i o n a l symmetry along the axis of r o t a t i o n of the container. A cross-section of the f l u i d perpen-d i c u l a r to the axis may be regarded as t y p i c a l of the f l u i d as a whole, and the hydrodynamics of the f l u i d may be reduced to a two-dimensional problem, i n whichvortex l i n e s are singular points i n the two-dimensional v e l o c i t y f i e l d . The c o n d i t i o n that the s u p e r f l u i d v e l o c i t y f i e l d i s i r r o t a t i o n a l allows the i n t r o d u c t i o n of a v e l o c i t y p o t e n t i a l 4>,the d e r i v a t i v e s of which give the components of : V! = | i 1 v 2 = | i d-5) 3xj 5x2 In a d d i t i o n , the approximation P s=constant i s tantamount to V.V = 0 , (1-6) which i s s a t i s f i e d i n two dimensions by the i n t r o d u c t i o n of a stream function ij» and w r i t i n g dip 3i!i . . v l = ^ v2 = " (1-7) A 3x2 9xj Equating (1-6) and (1-7) gives equations which may be viewed as the Cauchy-Riemann equations governing the conjugate functions <f> and which are the r e a l and imaginary parts of a complex v e l o c i t y p o t e n t i a l 0 whose complex d e r i v a t i v e i s the complex v e l o c i t y f i e l d w: d<5> — = V ! - i v 2 = w(z) (1-8) Consequently, the d i f f e r e n t i a l equation governing the p o t e n t i a l $ i s Laplace's equation V 2$ = 0 (1-9) with the boundary condition that any s o l i d boundary must be a streamline of the flow, i . e . ^ c o n s t a n t on the boundary. A f l u i d whose v e l o c i t y f i e l d may be expressed i n such a manner i s c a l l e d an i d e a l f l u i d . The hydrodynamics of a two-dimensional, i d e a l f l u i d may now be approached using the techniques of complex v a r i a b l e theory and a n a l y t i c functions. This i s the s t a r t i n g point of s e c t i o n 2 on vortex motion. 1.4 The Object of t h i s Work The questions which must be answered are "what configurations of vortex l i n e s i n s i d e a r o t a t i n g c y l i n d e r rotate r i g i d l y with the c y l i n d e r and are stable against small perturbations, and which of these represent the state of thermodynamic equilibrium at a given angular v e l o c i t y of r o t a t i o n ? " The f i r s t question i s answered f o r configurations of equal strength v o r t i c e s forming regular polygons: configurations with N>7 are unstable, and configura-tions with N<7 are stable i f the radius of the c i r c l e formed by the v e r t i c e s of the polygon i s smaller than a Certain f r a c t i o n of the cyl i n d e r radius, that f r a c t i o n depending on N. This t r a n s l a t e s into a minimum angular v e l o c i t y above which the configuration i s s t a b l e . The state of thermodynamic e q u i l i -brium as a function of ft i s determined f o r low values of ft, and the c a l c u l a t i o n of the c r i t i c a l angular v e l o c i t i e s f o r the cre a t i o n of one, two, and three v o r t i c e s i s performed. In section 2 the equations of motion of v o r t i c e s i n an i n f i n i t e f l u i d and i n a f l u i d contained within a c i r c u l a r boundary are derived and discussed. The k i n e t i c energy and angular momentum of a vortex f l u i d are c a l -culated i n section 3, showing t h e i r r e l a t i o n to the constants of the motion of vortex systems. In section 4 a Lagrangian formalism i s introduced, allowing the constants of the motion to be derived from symmetries of the Lagrangian. The equations of motion of v o r t i c e s perturbed s l i g h t l y from r i g i d l y r o t a t i n g configurations are derived i n section 5 and solved for the regular polygonal configurations. In section 6 the state of thermodynamic equilibrium of He II at low values of ft i s determined. Section 7 contains the conclusion of t h i s work. 7 Various portions of the material contained i n t h i s work can be found i n the l i t e r a t u r e c i t e d . The o r i g i n a l contributions of the author include the method of c a l c u l a t i n g the k i n e t i c energy and the angular momentum i n sections 3.1 and 3.2, and the Lagrangian technique i n se c t i o n 4.2. The method of c a l c u l a t i o n of the spectrum of c r i t i c a l angular v e l o c i t i e s i n s e c t i o n 6.2 i s due to Putterman(1974), but the act u a l c a l c u l a t i o n s are not a v a i l a b l e i n the l i t e r a t u r e . 8 2. MOTION OF VORTICES IN A TWO-DIMENSIONAL IDEAL FLUID 2.1 Equations of Motion and Conservation Laws i n an I n f i n i t e F l u i d The v e l o c i t y f i e l d of a two-dimensional i d e a l f l u i d ( i n v i s c i d and incompressible) can be expressed as the d e r i v a t i v e of a complex, s c a l a r p o t e n t i a l function *(z) = <|>(z)+iiKz). (2-1) tj) and IJJ being real-valued functions. The v e l o c i t y f i e l d i s obtained from w(z) = v 1 ( z ) - i v 2 ( z ) =^ |^ 1 . (2-2) dz Of i n t e r e s t i s a v e l o c i t y p o t e n t i a l of the form $(z) = l n ( z ) (2-3) which gives r i s e to the v e l o c i t y f i e l d w(z) = (2-4) which, i n polar coordinates, i s v r= 0, v Q= J . (2-5) This i s the v e l o c i t y f i e l d of a vortex situated at z=0 i n an i n f i n i t e f l u i d . The flow i s i r r o t a t i o n a l , since c u r l v = 0 everywhere except at the pole at the o r i g i n . The c i r c u l a t i o n of the f l u i d about any closed contour enclosing the pole i s T = o y.d£ = 2TTY (2-6) hence y i s c h a r a c t e r i s t i c of the vortex and i s c a l l e d the strength of the vortex. 9 Of more i n t e r e s t i s a configuration of N v o r t i c e s s i t u a t e d at time t at p o s i t i o n s z ^ { t ) (k=l,2,...N) with strengths y. •, whose v e l o c i t y p o t e n t i a l appears as the' sum ' ( z ) = i E Y k l n ( z - z k ) (2-7) and whose corresponding instantaneous v e l o c i t y f i e l d i s k " k The v o r t i c e s themselves move i n t h e . f l u i d , due.the presence of the other v o r t i c e s . The usual p r e s c r i p t i o n f o r obtaining the equations of motion of the v o r t i c e s , a v a i l a b l e i n any of the c l a s s i c texts on hydrodynamics, i s to assign to the vortex at p o s i t i o n the v e l o c i t y obtained by a superposition of the v e l o c i t y contributions of a l l the other v o r t i c e s , evaluated at z=z . n That i s , • d ' n ( t ) y' Y k (2-9) 1 dt f z (t)-z, (t) k n k where Z' i s the sum over a l l k=l,2,...N excluding k=n. Equation (2-9) repre-sents N ordinary d i f f e r e n t i a l equations, of f i r s t order i n t, whose sol u t i o n s are the vortex t r a j e c t o r i e s . N i n i t i a l conditions are required, the i n i t i a l p o s i t i o n s of the v o r t i c e s . Unlike the mechanics of point p a r t i c l e s , the i n i -t i a l v e l o c i t i e s are not needed, since the equations are of f i r s t order i n t. The i n i t i a l v e l o c i t i e s of the v o r t i c e s are uniquely determined by t h e i r i n i -t i a l p o s i t i o n s by the equations of motion (2-9) . The equations of motion can be put into canonical form through the intro d u c t i o n of the vortex stream function V - ^ l V k ^ W ( 2 - 1 0 ) .n k along with the canonical coordinates and momenta V V p k = V k ( 2 _ 1 1 ) 10 It i s e a s i l y v e r i f i e d that equations (2-9) and t h e i r complex conjugates may be wr i t t e n An (2-12a) . dq. 1 n k = o dt 9Pi . dp, 84< I k = - o dt (2-12b) suggesting that be i d e n t i f i e d as the Hamiltonian of the vortex system. This claim w i l l be substantiated i n sec t i o n 4, which deals with symmetries and conservation laws of the vortex system from a Lagrangian viewpoint. By a simple a p p l i c a t i o n of the chain r u l e of d i f f e r e n t i a t i o n and the canonical equations of motion, the time ra t e of change of any function A(q^,p^,t) may be written 1 dt 1 A'V + X 9 t 3A i n which 9A o 3A o L 3«k 3 p k 3^k 9 V (2-13) (2-14) i s the Poisson Bracket (P.B.) of A and ¥ . Functions A which have no e x p l i -c i t time dependence are conserved q u a n t i t i e s of the motion i f {A,¥ o} = 0 The following conserved quantities of vortex motion can be found: 1 r Centre of C i r c u l a t i o n Z = — )y z o y i k k o k Moment of C i r c u l a t i o n 0 = TY, o ~ k k Vortex Stream Function f = -Y Y'v Y , In k n k z -z n k (2-15) (2-16a) (2-16b) (2-16c) Angular Moment of C i r c u l a t i o n 1 v w = "T7 ZYI ( Z I Z . - Z . Z ) o 2 i ,L k k k k k k As an independent check, these can be shown to be constant by d i r e c t s u b s t i t u t i o n of the equations of motion. (2-16d) 11 2.2 The V e l o c i t y P o t e n t i a l of a Vortex F l u i d Inside the Unit C i r c l e The previous section treated v o r t i c e s i n an i n f i n i t e f l u i d . A more r e a l i s t i c s i t u a t i o n i s to have the v o r t i c e s contained i n a bounded f l u i d , and of p a r t i c u l a r i n t e r e s t i n t h i s work i s the case of a c i r c u l a r boundary, since many experiments on l i q u i d He use c y l i n d r i c a l v e s s e l s . For an i d e a l f l u i d , the v e l o c i t y p o t e n t i a l must be the s o l u t i o n of the d i f f e r e n t i a l equation V 2$(z) = 0 (2-17) with the condition at the boundary that there be no normal component of the v e l o c i t y f i e l d w(z). This i s equivalent to r e q u i r i n g that the boundary be a streamline of the flow, i . e . i j j(z) = 0 along the boundary. Since the a d d i t i o n of a constant to the p o t e n t i a l does not a f f e c t the v e l o c i t y f i e l d , the cons-tant which parametrizes the boundary streamline may be chosen to be zero. It has also been assumed that the boundary i s continuous and closed. The boun-dary condition may then be expressed as $(z) =$(z) along the boundary. (2-18) The problem of f i n d i n g the appropriate v e l o c i t y p o t e n t i a l f o r a vortex f l u i d with a r b i t r a r y boundaries has been approached by Lin(1943), but for simple boundaries such as the c i r c u l a r boundary the technique of conformal mapping i s more convenient. The boundary value problem (2-17) with (2-18) can be mapped into a region f o r which the s o l u t i o n i s already known or i s easier to solve; the inverse mapping gives the correct s o l u t i o n i n the o r i g i n a l region. In t h i s case, the region in s i d e the unit c i r c l e can be mapped onto the upper h a l f plane using the transformation ' l -g(z) = i (2-19) 1 + zj The same transformation maps poles i n the z-plane into poles i n the g-plane, and the c i r c u l a r boundary i s mapped into the r e a l a x i s . The s o l u t i o n to (2-17) i n the upper h a l f plane i s well-known, c f . Milne-Thomson(1968), and i s *(g) = I l Y k l n < 8 - g k ) - J l Y k l n ( g - I k ) '(2-20) k k Except at the poles, t h i s i s a s o l u t i o n to the p o t e n t i a l equation (2-17), and i t i s straightforward to show that, on the r e a l axis (g=g), the s o l u t i o n i s t o t a l l y r e a l , s a t i s f y i n g the condition (2-18). Using the transformation (2-19) gives the required s o l u t i o n i n s i d e the u n i t c i r c l e : *(z) = j £Y l n ( z - z k ) - ± l Y k l n ( l - z \) (2-21) k k where a r e a l constant has been dropped. This s o l u t i o n also s a t i s f i e s the boundary condition, which can be checked using the f a c t that z=l/z on the boundary. It i s i n t e r e s t i n g to note that the conformal mapping method i s not u s e f u l i n determining the p o t e n t i a l outside the u n i t c i r c l e , that i s , i n deriving the c i r c l e theorem of Milne-Thomson (page 157 of reference). The transformation (2-19) maps the outside of the u n i t c i r c l e into the lower hal f g-plane, but the d e r i v a t i v e (dg/dz) vanishes at z-*° (g=-i), so the jacobian of the transformation vanishes there and the inverse mapping i s not defined at that point. 2.3 Equations of Motion and Conservation Laws Inside the Unit C i r c l e Once the v e l o c i t y p o t e n t i a l for a vortex f l u i d has been deter-mined for any region, the equation of motion of the vortex at z , say, i s n obtained by subtracting from the p o t e n t i a l the c o n t r i b u t i o n of the vor-tex i n an i n f i n i t e f l u i d , and assigning the v e l o c i t y of the r e s u l t i n g f i e l d at z=z to the vortex at z . This i s expressed mathematically as n n J dt n dz '(z) - -ry l n ( z - z ) i n n z=z (2-22) 13 Using the v e l o c i t y p o t e n t i a l from (2-21), the equations of motion of v o r t i c e s i n s i d e the unit c i r c l e are This may be interpreted as each vortex having induced an image vortex of op-posite sign at the r e c i p r o c a l point. Note that the second sum contains the term k=n, representing the i n t e r a c t i o n of the n*"^ 1 vortex with i t s image. As i n s e c t i o n 2.1, the equations of motion can be put i n t o canoni-c a l form by the i n t r o d u c t i o n of the vortex stream function T = "I I'Y Y,ln z -z. + 7 h Yi l n o ^ ^ ' n ' k n k ^ f ' n ' k n k n k 1-z z. (2-24) n k The equations of motion (2-23) and t h e i r complex conjugates follow from the canonical equations (2-12a) and (2-12b) . Using r e l a t i o n , (2-15), the conserv-ed q u a n t i t i e s of the motion can be determined. A l l of the q u a n t i t i e s defined i n (2-16) remain constants of the motion excluding the centre of c i r c u l a t i o n . (This i s because the equations of motion are no longer i n v a r i a n t under trans-l a t i o n i n space; see section 4.) 14 3. DYNAMICAL QUANTITIES OF A VORTEX FLUID The k i n e t i c energy.and angular momentum of a two-dimensional vortex f l u i d w i l l be calculated f o r the case of the f l u i d contained i n s i d e the u n i t c i r c l e . The density i s assumed constant throughout the f l u i d , which i s a reason-able assumption f o r the s u p e r f l u i d component of l i q u i d He, t h i s density dropping r a p i d l y to zero only within microscopic distances from vortex cores and boundaries.(see Putterman,1974). I t w i l l be shown that the k i n e t i c ener-gy i s r e l a t e d to the vortex stream f u n c t i o n , ¥ , and that the angular momen-tum i s r e l a t e d to the moment of c i r c u l a t i o n , 0 q, and so these dynamical quantities of the f l u i d are functions only of the c o n f i g u r a t i o n of v o r t i c e s i n the f l u i d . 3.1 K i n e t i c Energy The k i n e t i c energy of an i d e a l f l u i d of constant density p i s *3P HP (Vijj)2dxdy R V. (ipVijOdxdy R = hp Hm.fC)c\i (3-D !c i n which ij; i s the stream function of the flow (Im$(z)), R i s the region of flow, and C i s the contour enclosing R, with outward normal fi. The vector i d e n t i t y V-(AyB) = yA.yB + Av2B along with the f a c t that V 2 IJJ=0 gives the second l i n e , while a p p l i c a t i o n of Green's Theorem gives the f i n a l l i n e . The contour C i s chosen to be the unit c i r c l e , and small c i r c l e s of radius e<<l surrounding the poles at z=z ; the s t r a i g h t segments j o i n i n g the outer K. and inner contours are ignorable, since t h e i r c o n t r i b u t i o n to the i n t e g r a l cancels. Small disks of radius £ have been excluded from the region of flow . i n c a l c u l a t i n g the i n t e g r a l , but t h i s i s reasonable on p h y s i c a l grounds since r e a l v o r t i c e s have f i n i t e cores containing no l i q u i d . Even without t h i s f a c t , one would be u n j u s t i f i e d i n shrinking e to below an inter-atomic distance i n the f l u i d , the distance at which the use of c l a s s i c a l hydrodynamics becomes questionable. The approximation used here i s acceptable as long as the vor-t i c e s do not approach each other or the boundary at distances of the order e, since then the energy of deformation of the vortex cores must be taken into account. The parameter e becomes important i n the c a l c u l a t i o n of the c r i t i -c a l angular v e l o c i t y f or the appearance of a s i n g l e quantized vortex l i n e i n l i q u i d He I I , i n section 6. The j u d i c i o u s choice of $(z) i n (2-21), i n which ^(z)=0 on the unit c i r c l e , reduces the i n t e g r a l (3-1) to m E = -%pl <n|;(V^.n)dS, (3-2) m which consists of the sum of a l l the contour i n t e g r a l s around the vortex p o s i t i o n s ( i . e . C_ surrounds z j ) . In order to evaluate one of these i n t e g r a l s m m about the vortex at z , say, i t i s convenient to define the q u a n t i t i e s z-z mk m z -z m k r k = Z k mk z -1/z, m k (3-3) so that ijj(z) may be expanded i n powers of z about z . The d e t a i l s of t h i s m c a l c u l a t i o n are found i n Appendix A. In a n t i c i p a t i o n of allowing e to become very small, only lne and zero order terms i n e are retained i n t h i s expan-sion, obtaining * = - Y m l n £ - ] ' Y k l n r m k + j ^ l n r ^ + j ^ l n ^ + 0(e) (3-4) S i m i l a r l y , i n the v i c i n i t y of z , m ViJj.ndJl = 1^ - ed6 = -(y + 0(e))d8 - d e m (3-5) Combining these i n (3-1), and again r e t a i n i n g only zero order and lne terms, *3P ZY m m 2TT ijjd6 (3-6) = P T T { - J y y Y-, Inr , + J Jy y, l n r ' + T jy y. lnr, - yy2ln£:l 1 ^ ^  1m k mk L f ' m' k mk 1 f'm'k k L m ' m k m k m k m which acquires a f a m i l i a r form when complex notation i s adopted: E = pu{- I I'Y mY kln m k = PTT(¥ - Ey 2lne) z -z m k + T Ty y. In m k 1-z z, m k - l Y2 l n e } L m J (3-7) Thus the energy of a vortex f l u i d appears as the sum of what has been i d e n t i f i e d as the Hamiltonian of the vortex system and a term which, f o r a given set of v o r t i c e s (y's constant), does not depend on the co n f i g u r a t i o n . The f i r s t sum i n (3-7) represents the energy of i n t e r a c t i o n between the v o r t i c e s the second sum i s the energy of i n t e r a c t i o n between the v o r t i c e s and a l l the images, while the t h i r d sum may be considered as the self-energy of the v o r t i c e s . This term diverges l o g a r i t h m i c a l l y as e-»-0, but an argument has a l -ready been given that t h i s parameter must remain f i n i t e , although small. 3.2 Angular Momentum The angular momentum of a two-dimensional i d e a l f l u i d i s L = p (rxV)r.drde = p Re{izw}r.drd9 (3-8) R . R Using the v e l o c i t y f i e l d for v o r t i c e s i n s i d e the unit c i r c l e , the integrand ReUzw} = ly (Re{z/(z-z )} + Re{ zz / (1-zz )}) (3-9) 17 The f i r s t set of terms contain s i n g u l a r i t i e s i n the region of i n t e g r a t i o n at the points z = z k > s o they must be expanded i n d i f f e r e n t convergent s e r i e s i n the two regions |z|'<|z^| and |z|>|z^|: Re{z/(z-z k)} = - I n=! CO = I n=0 cos n(e-e ) K. cos n(e-e,) k z < z. z > z k 1 (3-10) The remaining terms do not contain poles i n the region of i n t e g r a t i o n |z|<l, so one se r i e s s u f f i c e s , namely Re{zz /(1-zz )} = I ( r r ) n c o s n(0-6 ) |z|<l n=l (3-11) The only terms i n these s e r i e s that contribute to the angular momentum are those f o r which n=0, due to the f a c t that 2TT cos n(0-6 )d6 = { ~. rt k 0 2TT n=0 n^O (3-12) so the angular momentum i n t e g r a l becomes rl L = p£y k 2TT r dr = p u ^ U - r ^ ) k ''r k k = PTT(Y -0 ) (3-13) o o The angular momentum i s the sum of a conf i g u r a t i o n - f r e e term .(total c i r c u -l a t i o n ) and the negative of the moment of c i r c u l a t i o n . Note that the poles of the v e l o c i t y f i e l d do not present any divergences i n the f i n a l r e s u l t . 3.3 Renormalizing the Energy Expression (3-7) for the k i n e t i c energy d i f f e r s from the vortex stream function (2-24) only by what has been i d e n t i f i e d as the self-energy contributions -lysine from the v o r t i c e s . In cases for which the vortex stream function serves as the hamiltonian of the system, the y 's are K. 18 regarded as being f i x e d , hence the energy may be "renormalized" by sub-t r a c t i n g the s e l f - e n e r g i e s , the energy and vortex stream function becoming i d e n t i c a l . Since Ey 2 i s a p o s i t i v e d e f i n i t e quantity, i t may not vanish by a m s u i t a b l e choice of y's. For use i n a p p l i c a t i o n s to l i q u i d helium physics, the c o n t r i b u t i o n of the s e l f - e n e r g i e s may not be ignored, since Putterman's c r i t e r i o n f o r the thermodynamic equilibrium of s u p e r f l u i d s (Putterman and Uhlenbeck(1969)) requires the comparison of free energies of vortex l i n e configurations with unequal t o t a l c i r c u l a t i o n s . Even f o r quantized systems with each y, an i n t e g r a l m u l t i p l e of some unit strength, the £y2 may change due to a change i n t o t a l c i r c u l a t i o n , or a r e - d i s t r i b u t i o n of vortex strengths giving the same t o t a l c i r c u l a t i o n . Thus the renormalized energy i s u s e f u l only i n determining the r e l a t i v e s t a b i l i t y of configurations having the same t o t a l c i r c u l a t i o n and d i s t r i b u t i o n of vortex strengths, that i s , for those which d i f f e r only i n t h e i r geometrical configuration. In t h i s work (section 6) only quantized v o r t i c e s of u n i t strength w i l l be considered, so that y <*N and o Zy 2 o :N, where N i s the t o t a l number of v o r t i c e s , ra 3.4 In the Limit of an I n f i n i t e F l u i d The k i n e t i c energy and angular momentum of an i n f i n i t e vortex f l u i d may be obtained by repeating the previous c a l c u l a t i o n s , r e p l a c i n g the bound-ary at |z|=l by a boundary at |z|=a, and taking the l i m i t as a->-°°. I t i s not necessary to repeat the c a l c u l a t i o n , since the appropriate expressions r e -s u l t from replacing z^ by z^/a, a n d e by e/a i n expressions (3-7) and (3-13) revealing the behaviour at large values of a to be E/p-rr = -T Y'Y Y I l n l z ~z, I ~ IVlne + y 2 l n a + 0(a ) (3-14a) m k m L/pu = -Ty |z I 2 + Y a 2 (3-14b) u m1 m1 o m 19 Unless the t o t a l c i r c u l a t i o n i s zero, the k i n e t i c energy and angular mom-entum of an i n f i n i t e vortex f l u i d contain i n f i n i t e terms which behave as lna 2 ' and a , r e s p e c t i v e l y . These diverging terms appear i n the form of the energy and angular momentum of a si n g l e vortex of strength y , due to the net c i r -o c u l a t i o n of the flow at large distances . If the condition Y o = 0 i s imposed, the i n f i n i t e terms vanish, corresponding to the c a n c e l l a t i o n of the v e l o c i t y f i e l d s of the v o r t i c e s at large distances. The presence of these terms i n the case Y Q ^0 i s tantamount to the statement that no net c i r c u l a t i o n may be i n t r o -duced into an i n f i n i t e vortex f l u i d due to the i n f i n i t e energy and angular momentum that would be required to do so. Apart from these i n f i n i t e terms, the only d i f f e r e n c e remaining be-tween the energy and the vortex stream f u n c t i o n (2-10) for an i n f i n i t e f l u i d i s the contr i b u t i o n from the se l f - e n e r g i e s of the v o r t i c e s . Again, the energy may be "renormalized" , although the procedure seems uncertain i n t h i s case, since the question of renormalizing the energy to exclude the i n f i n i t e terms i s r a i s e d . This question, and whether i t makes sense to discuss such i n f i n i t e q u a n t i t i e s s e r i o u s l y , w i l l be l e f t open. 20 4. THE LAGRANGIAN MECHANICS OF VORTEX SYSTEMS 4.1 The Lagrangian Formalism It i s po s s i b l e to view the equations of motion of point objects as a r i s i n g from a p r i n c i p l e of le a s t a c t i o n , i n cases f o r which a Lagrangian function may be defined f o r the system. Once a Lagrangian has been defined, the symmetry prop e r t i e s of the system and the conservation laws of the motion may be extracted with a minimum of e f f o r t . For a review of the r e l a t i o n between Hamilton's p r i n c i p l e of le a s t a c t i o n and the conservation theorems of mathematical physics, see H i l l ( 1 9 5 1 ) . The basic concepts of the formalism are introduced below. of the coordinates q(t) and t h e i r time d e r i v a t i v e s q ( t ) , and po s s i b l y of t i t s e l f . (In the. f o l l o w i n g , a l l the coordinates have been denoted g e n e r i c a l l y by q, and the sums over coordinate l a b e l are omitted, f o r c l a r i t y . ) The action i s the f u n c t i o n a l computed between the two points (q^ ( t^),q^(t^)) and (q^(t^),q^(t^)) of c o n f i -guration space along a l l curves j o i n i n g the po i n t s . The t r a j e c t o r y f o r which the value of S i s a minimum, compared with a l l other t r a j e c t o r i e s , i s the ac1-t u a l motion of the system between the points. The techniques of v a r i a t i o n a l c a l c u l u s , along with t h i s p r i n c i p l e , show that the equations of motion may be obtained from the Lagrangian by the Euler-Lagrange equations The Lagrangian f o r a system of point objects i s a function L(q,q,t) (4-1) 3L d 3Li = 0 (4-2) 3q dt[3qj 21 Under a transformation t->-t' , q->-q', the f u n c t i o n a l form of the Lagrangian must a l t e r to preserve i t s numerical invariance, however, some transformations leave the Lagrangian f.orm-invariant, or at most add the t o t a l d e r i v a t i v e of some function of the coordinates (which leaves <SS invar i a n t ) . That i s , Such transformations are symmetry transformations, since they r e s u l t i n the transformation of a s o l u t i o n of the equations of motion i n t o another p o s s i b l e s o l u t i o n of the equations of motion. As shown by H i l l , the te s t f o r a symmet-ry transformation i s that where 61, 6q, and 6q are i n f i n i t e s i m a l q u a n t i t i e s . The i n t e r p r e t a t i o n of (4-4) i s that the RHS must be expressible as the t o t a l d e r i v a t i v e of some i n f i n i -tesimal function 6A(q,t). A given i n f i n i t e s i m a l symmetry transformation of L, with the equations of motion derived from an ac t i o n p r i n c i p l e , gives r i s e to an associated conservation law, w r i t t e n In t h i s way, each symmetry transformation of L leads to a conservation law, although the reverse i s not n e c e s s a r i l y true. There e x i s t conservation laws which do not correspond to symmetries of the Lagrangian, such as the Runge-Lenz vector of the Kepler problem, which i s discussed by Greenberg(1966). L'(q') = L(qJ)+ ~TT"A (q ' ) (4-3) (4-4) (4-5) 1 22 4.2 The Symmetry Transformations and Conservation Laws of Vortex Motion That the equations of motion (2-8) of v o r t i c e s i n an i n f i n i t e f l u i d r e s u l t from an a c t i o n p r i n c i p l e using the Lagrangian L ( v v v V • i i zkV -1 J * V k l n l v z k l ( 4 _ 6 ) k m k v i a the Euler-Lagrange equations (regarding z and z as independent v a r i a b l e s ) i s straightforward, i n f a c t , t h i s Lagrangian has been constructed so that t h i s would be so. The Lagrangian(4.-6) displays some unusual features: The v e l o c i t i e s do not appear i n quadrature, but i n b i l i n e a r combination with the coordinates, ensuring that the equations of motion are f i r s t order i n time . The Lagrangian . i s the sum of two constants of the motion (previously shown), W and ¥ , so the •o o Lagrangian i t s e l f i s a constant of the motion. In Newtonian point mechanics, the Lagrangian i s the d i f f e r e n c e between the k i n e t i c and p o t e n t i a l energies, which i s not conserved; t h i s Lagrangian allows no such i n t e r p r e t a t i o n , the t o t a l energy of the system having been shown to be ¥ q alone. With these comments i n mind, the Lagrangian may be investigated f or i t s symmetries, which are already known from section 2, with the intent of a s s o c i a t i n g a conservation law with each symmetry transformation. The t e s t f o r a symmetry transformation of t h i s Lagrangian i s written, from (4-4), 4 t - ( 6 t ) + k K ( V i k A f i z k V v z A ) k (6z -6z, 6z -6z, -\ m k m k n k z -z z -z . m k m k ' = - -^(SA) (4-7) and the associated conservation law i s o T ^ o 6 t + k K (V + 6A> =0 ( A - 8 ) k These expressions w i l l now be used to t e s t proposed i n f i n i t e s i m a l symmetry transformations and to f i n d t h e i r associated conservation laws. 23 (a) Space T r a n s l a t i o n Symmetry and Conservation of Centre of C i r c u l a t i o n The transformation St = 0, Sz = a, 6z = 0 (a i n f i n i t e s i m a l ) (4-9) i s a symmetry transformation on L, but does not leave L i n v a r i a n t i n form, r e q u i r i n g the intr o d u c t i o n of the i n f i n i t e s i m a l function 6A - k ( a E k V k - 5 z k V k ) ( 4 - 1 0 ) The associated conservation law follows from allowing a to be a r b i t r a r y . L e t t i n g a be completely r e a l or completely imaginary r e s u l t s i n the conser-vati o n of the imaginary or r e a l parts, r e s p e c t i v e l y , of the centre of c i r c u l a t i o n , Z . o (b) Rotational Symmetry and the Conservation of Moment of C i r c u l a t i o n The transformation 6t = 0, ^ z ^ ^ - a z i c ' ^ z k = ^ a z k ^ a i n i t e s i m a l ) (4-11) leaves L form-invariant, r e s u l t i n g i n the conservation of the moment of c i r c u l a t i o n 0 . o (c) Time t r a n s l a t i o n Symmetry and Conservation of Vortex Stream Function The transformation 6t = x, 6z k= 0, 6z k= 0 ( T i n f i n i t e s i m a l ) (4-12) leaves L form i n v a r i a n t , r e s u l t i n g i n the conservation of vortex stream function, Y . o These three symmetries of L are the ones which, i n Newtonian mechanics, lead to the conservation of l i n e a r momentum, angular momentum, and energy, r e s p e c t i v e l y . It i s easy to see that i n a vortex f l u i d , the l i n e a r momentum i s zero, due to the symmetry of the v e l o c i t y f i e l d contributed by each vortex. 24 It has been shown that the moment of c i r c u l a t i o n , 0 , i s a measure of the o angular momentum i n an i n f i n i t e f l u i d , and that the vortex stream funct i o n , ¥ , o i s the energy of the f l u i d (minus the s e l f energy of the v o r t i c e s ) . Also, i n Newtonian mechanics, i t i s the Hamiltonian which generates t r a n s l a t i o n s i n time, so t h i s j u s t i f i e s the i d e n t i f i c a t i o n of ¥ with the Hamiltonian of the J o vortex system. (d) The Absence of a Symmetry Transformation Leading to Conservation of WQ At the time of w r i t i n g , i t has not been po s s i b l e to f i n d an extra symmetry transformation associated with the conservation of W . There i s one remaining symmetry of the equations of motion, namely, the transformation : t' = e 2 3 t , z k = e 6 z k , z k = e " e £ k (g r e a l ) (4-13) which i s a sc a l e transformation, and does not lead to a conservation law. Since Wq i s a constant of the motion which apparently does not r e f l e c t a sym-metry of L, i t may belong to that cl a s s of conservation laws exemplified by the Runge-Lenz vector, mentioned e a r l i e r . This question remains open. As a f i n a l note, the Lagrangian for the vortex system bounded by the unit c i r c l e i s obtained by adding to L the terms necessary to a l t e r 4* , i . e -I Iv Y , l n | l - z z. | (4-14) k ^ m k ' m k 1 . m k The correct equations of motion are obtained from the Euler-Lagrange equations. The a d d i t i o n of these terms reduces the symmetry of the system, since the Lagrangian i s no longer inv a r i a n t under t r a n s l a t i o n s i n space, as i s e a s i l y v e r i f i e d . This agrees with the r e s u l t stated i n section 2.3, that the centre of c i r c u l a t i o n , Z , i s not conserved i n a f l u i d bounded by the un i t c i r c l e , o This r e s u l t may be generalized to a l l bounded f l u i d s . 25 5. THE STABILITY OF RIGIDLY ROTATING VORTEX CONFIGURATIONS In t h i s section, i t w i l l be shown that the state of kinematic e q u i l i b r i u m of the vortex f l u i d i s one of r i g i d r o t a t i o n of the vortex c o n f i g u r a t i o n . These configurations must then be examined f o r s t a b i l i t y against small perturbations from kinematic equilibrium, which i s e f f e c t e d by expanding the equations of motion i n small q u a n t i t i e s about a r i g i d l y r o t a t i n g configuration, obtaining l i n e a r equations of motion of the perturbations (as a f i r s t approximation). This method i s used i n the case of regular polygonal configurations of quantiz-ed v o r t i c e s l y i n g equally spaced on a c i r c l e of radius r<l i n s i d e the unit c i r -c l e . The symmetry of these configurations makes the normal modes of v i b r a t i o n . apparent, and the c r i t e r i o n of s t a b i l i t y i s reduced to f i n d i n g the roots of 2 two polynomials i n r of degree 2N, where N i s the number of v o r t i c e s . 5.1 The Equilibrium State of a Rotating Vortex F l u i d The thermodynamic equ i l i b r i u m of r o t a t i n g superfluids based on the Landau(1941,1947) macroscopic, two-fluid model has been treated by Putterman and Uhlenbeck(1969). Their r e s u l t s show that, for a s u p e r f l u i d contained i n a vessel r o t a t i n g with angular v e l o c i t y ft, the equilibrium state of the normal f l u i d component i s one of r i g i d r o t a t i o n (Yn= £ x^)> while the s u p e r f l u i d must be i r r o t a t i o n a l (curlV = 0) and s t a t i o n a r y i n the frame of reference r o t a t i n g s at ft, and that the "free energy" of the s u p e r f l u i d , F=E-ftL, must be at a mini-mum. If the condition of i r r o t a t i o n a l flow i s s a t i s f i e d by the presence of vortex l i n e s i n the f l u i d , the c o n d i t i o n of stationary V i n the r o t a t i n g frame requires that the configuration of vortex l i n e s rotate r i g i d l y with angular v e l o c i t y ft. This can be shown e a s i l y f o r a two-dimensional vortex f l u i d , f o r 26 which the equations of motion and the constants of the motion are already known. The conserved q u a n t i t i e s f o r the vortex f l u i d i n s i d e the u n i t c i r c l e are V , the energy, 0 , and W . The state of e q u i l i b r i u m of the system must minimize o o o ¥ , keeping 0 and W constant as a u x i l i a r y conditions. It i s s u f f i c i e n t to o o o rel a x these conditions , r e t a i n i n g only 0 q constant i n the v a r i a t i o n a l problem as the s o l u t i o n obtained thereby automatically guarantees the conservation of Wq. The state of s t a b l e equilibrium i s given by F = y + ft0 = minimum (5-1) o o i n which ft i s , f o r the moment, an undetermined m u l t i p l i e r . The f i r s t d e r i v a-t i v e s of F with respect to the independent v a r i a b l e s z^ and z^ must vanish, leading immediately to the d i f f e r e n t i a l equations z k= i f t z k (5-2) and t h e i r complex conjugates, whose solutions are > = z° Jk Z k i n which the z£ are constants, and ft may now be i d e n t i f i e d as the angular v e l o c i t y of a r i g i d l y r o t a t i n g c o n f i guration of v o r t i c e s . The value of ft i s c a l c u l a t e d by s u b s t i t u t i n g (5-3) into the equations of motion, obtaining a constraint on the z° i n the bargain, namely z°ft = E.'Y; (ZO-Z. 0)" 1 + Z - V ^ U - z 0 ^ , 0 ) " 1 . (5-4) n k'k n k k'k k n k * This i s a kind of eigenvalue problem, the general s o l u t i o n to which would enumerate a l l p o s s i b l e configurations which rotate r i g i d l y and t h e i r angu-l a r v e l o c i t i e s . If the problem i s r e s t r i c t e d to the i n f i n i t e f l u i d case with unit strength v o r t i c e s , i t can be shown that the r i g i d l y r o t a t i n g configura tions of N v o r t i c e s can be put into one-to-one correspondence with the N*N symmetric matrices with zero determinant. z , = z i , e (5-3) 27 Returning again to W^ , i t i s evident that f o r r i g i d l y r o t a t i n g systems, W =fi0 , and i s therefore automatically conserved, as expected, o o r 5.2 The Linea r i z e d Equations of Motion of the Perturbations The stable r i g i d l y r o t a t i n g configurations are now to be found by looking for. o s c i l l a t o r y motions of the v o r t i c e s about t h e i r e q u i l i b r i u m p o s i t i o n s . This i s eff e c t e d by expanding the equations of motion i n terms of small perturbations about the r i g i d l y r o t a t i n g configuration, and terminating the expansion at the f i r s t order terms, so that l i n e a r equations of motion of the perturbations r e s u l t . The s u b s t i t u t i o n z k = ( z k + V t ) ) e ± n t ( 5 _ 5 ) i s made i n (2-23), and the equations of motion are k k k Due to the s u b s t i t u t i o n (5-5), the v o r t i c e s are now being viewed from the frame of reference r o t a t i n g at i l , so the equilibrium p o s i t i o n s z£ are stationary. It i s necessary that the perturbed system leave the constants of the motion ¥ , 9 , and W unchanged to f i r s t order i n the b's, so the solutions to o o o k (5-6) need some co n s t r a i n t s . The constants of the motion may be expanded about a r i g i d l y r o t a t i n g c o n f i g u r a t i o n i n terms of the b^s, and the f i r s t order terms must vanish f o r the conservation laws to remain v a l i d . The r e s u l t i n g equations of constraint are V wo : Vk(zkV2kV = 0 °o: ^\(ikVzkV = 0 28 which combine to give the simpler, s i n g l e constraint J k V k b k - ° ' ( 5 " 8 ) The general s o l u t i o n to (5-6) i s not r e a d i l y apparent, e s p e c i a l l y i f the y's are of a r b i t r a r y value, I t has already been pointed out, however, that i f a l l the v o r t i c e s are of some u n i t strength, a c e r t a i n symmetry i s imposed on the system. For example, one c l a s s of r i g i d l y r o t a t i n g configura-tions i s that of N v o r t i c e s of equal strength l y i n g equally spaced on a c i r c l e of radius r<l (since the boundary i s at r = l ) , forming an N-sided polygon.The high degree of symmetry of these configurations suggests that the normal modes of v i b r a t i o n w i l l possess a s i m i l a r symmetry. The solutions to (5-6) for the polygonal configurations w i l l be i n v e s t i g a t e d . The p o s i t i o n s of the unperturbed v o r t i c e s forming an N-gon are z 0 = r e i ( 2 T r / N ) k ( 5 _ 9 ) The N coupled equations (5-6) may be uncoupled by proposing the solutions , . . i(2Tr/N)kr iwt i(2Tr/N)Lk , - i u t -i(2 Tr/N)Lk 1 D k(.L,t; = e l a L e e ° L e e J (L=1,2,..,N) (5-9) i n which oo i s r e a l , L l a b e l s the mode of v i b r a t i o n , and a and c are r e a l Li L constants to be determined by the i n i t i a l conditions. Since the functions e l a J t and e l t J t are l i n e a r l y independent, s u b s t i t u t i o n of (5-9) into (5-6) y i e l d s the two homogeneous equations ( P T 2 - p _ l s L + l ) a L " ( TN-L+1 " n " U ) C L = ° (L=l,2,..,N-1) (5-10) - ( T L + 1 -ft - . ) a L + ( P T 2 - P _ 1 S N _ L + 1 ) c L = 0 2 i n which p=r and the trigonometric sums T , S , and ft are defined and given i n closed form i n Table I, the c a l c u l a t i o n s f o r which appear i n Appendix B. For n o n - t r i v i a l s o l u t i o n s to e x i s t f o r a and c , OJ must be the s o l u t i o n Lt Li of the quadratic equation 29 " 2 - » W W + ( P T 2 - P _ 1 S L + 1 ) 2 " T N - L + r T L + l + fi(TN-L+l+ W = ° ( 5 " U ) The roots of t h i s equation are o 03 = C ± /AS (5-12) where 2pA = N 2p N- L-^±2J- 9 + N L * - ^ 4N - E — - 2(N-1) + L(N-L) ( 1 - p V l-p W l-p N 0 „ T L.2 L N-L 2 PB = N 2 p N - L - ^ _ L + NLP - P n - L(N-L) (5-13) (1-p ) 1-P 2 p C . . H ^ i ^ l 1-P (1-P ) Applying the constraint (5-8) to the so l u t i o n s (5-9) gives , ioit . - i o j t . p i(2iT/N)Lk _ /c. ... (a^e + c e ))e = 0 (5-14) L L k The constraint i s i d e n t i c a l l y s a t i s f i e d f o r L=1,2,...N-1 and places no conditions on a and c f o r these modes. However, for L=N, i t i s necessary that a =c =0 for ,the constraint to be s a t i s f i e d . In other words, the L=N mode must be removed from consideration i f the perturbations are not to a l t e r the constants of the motion. Havelock(1931) has used a s i m i l a r approach to the problem and h i s expressions giving the frequencies of the normal modes are the same, apart from some misprints i n the p u b l i c a t i o n . These misprints are l i s t e d i n Appendix D. % 30 5.3 The S t a b i l i t y of the Polygonal Configurations For the polygonal configurations to have stable o s c i l l a t o r y motion about equilibrium, i t i s necessary that a> be r e a l f o r a l l allowed modes, so the product AB must be p o s i t i v e f o r these modes. To demonstrate i n s t a b i l i t y , i t s u f f i c e s to f i n d one mode which i s unstable. From the quadratic equation (5-11) : governing u>, A and B may be wri t t e n i n the form 0 - ^ ( T N - L + 1 + W " f i ± ( P T 2 -P"lsW (5"15) This expression i s more convenient than (5-13) i n demonstrating the f a c t that A and B are symmetric under the interchange of L and N-L (recognizing that S has t h i s symmetry i t s e l f ) . The consequence of t h i s i s that the l e a s t stable mode or modes are. the " c e n t r a l " ones,that i s , L=N/2 f o r N even and L=(N±l)/2 f o r N odd. The s t a b i l i t y of the polygonal configurations i s then determined by the s t a b i l i t y of these modes, which are to be examined separ-at e l y f o r N even and N odd. (a) N even Let N=2n, L=n, N-L=n (n=l,2,...) (5-16) Sub s t i t u t i n g these i n t o (5-13) gives, f o r the le a s t s t a b l e mode a = 2p(l-p N) 2A= (n2-4n+2) + 4 n 2 p n + 2 ( 3 n 2 - 2 ) p 2 n + 4 n 2 p 3 n + (n 2+4n+2)p 4 n (5-17) „ , N.2 2 n.4 b = 2p(l-p ) B = -n (1-p ) Since b<0 always, the c r i t e r i o n f o r i n s t a b i l i t y i s a>0, which i s s a t i s f i e d f or a l l p f o r n>4. Since a(p=0)<0 for n=l,2,3 and ^ ™ a(p)>0, there e x i s t s a root P c ( n ) of a. For p<p c(n), the configurations n=l,2,3 are st a b l e ; for p>p c(n), they are unstable. (b) N odd Let N=2n+1, L=n, N-L=n+1 (n=l,2,...) (5-18) 31 Then, for the l e a s t stable modes, a = n(n-3) + n ( 2 n + l ) P n + (n+1)(2n+l)p n + 1 + 2 ( 3 n 2 + 3 n - l ) p 2 n + 1 + ( n + l ) ( 2 n + l ) p 3 n + 1 + n ( 2 n + l ) p 3 n + 2 - <n+l)(n+4)p 4 n + 2 (5-18) b = -n(n+l) + n(2n+l)p n + (n+1)(2n+l)p n + 1 -2(3n 2+3n+l)p 2 n + 1 + ( n + l ) ( 2 n + l ) p 3 n + 1 + n ( 2 n + l ) p 3 n + 2 - n ( n + l ) p 4 n + 2 Again, b<0 always i n the region 0<p<l, reducing the c r i t e r i o n f o r i n s t a b i l i t y to a>0, which i s s a t i s f i e d for a l l p f o r n>3. Again, a sign change i n a occurs for the s p e c i a l cases n=l,2, so there e x i s t s a root P c ( n ) such that these configurations are stable i f p<p c(n), and are unstable i f p>p c(n). In summary, the regular polygonal configurations i n s i d e the u n i t c i r c l e are unstable f o r a l l values of r, except f o r N=l,2,3,4 ,5 , and 6. For these values of N there e x i s t s a c r i t i c a l value of r, c a l l e d r , such that c i n the region x<*c> the configuration i s s t a b l e , and f o r * >* c> the c o n f i -guration i s unstable. The value of r c f o r the d i f f e r e n t polygons i s a v a i l a b l e from a numerical c a l c u l a t i o n of the roots of the polynomial a i n each case. The r e s u l t s of such a computation appear i n Table I I . In a d d i t i o n , r ^ gives a minimum value of ft f o r which the polygon i s s t a b l e , v i a the r e l a t i o n i n Table I. These values of ft . for each N also appear i n Table I I . mm N-l 1 _ e i ( 2 T r / N ) L k J l ( 1 _ l i ( 2 W N ) k ) 2 N i(2-rr/N)Lk J l ( 1 _ p e i ( 2 , / N ) k ) 2 p 1 S 1 + T1 - P T 2 -Js(L-2)(N-L) N-L 2-±—T, - N(L-l)-£ N-L ri N^2 d-p ) 1-p N hp 1 ( N - l ) + N N- l TABLE I SOME RELEVANT TRIGONOMETRIC SUMS (defined and i n closed form) N r c n . mm 2 .462 2.79 3 .567 3.43 4 .574 4.70 5 .588 5.86 6 .547 8.37 TABLE II THE VALUES OF r AND . FOR THE STABLE POLYGONS c mm 33 6. THERMODYNAMIC EQUILIBRIUM OF A ROTATING SUPERFLUID The states of stable kinematic e q u i l i b r i u m of quantized v o r t i c e s forming regular polygons were found' i n the l a s t s e c t i o n , however, t h i s i s i n s u f f i c i e n t to determine the state of thermodynamic e q u i l i b r i u m of the r o t a t i n g vortex f l u i d . The c r i t e r i o n f o r thermodynamic e q u i l i b r i u m according to Putterman and Uhlenbeck(1969) i s that the vortex f l u i d must have an absolute minimum of the free energy F=E-ftL. Experimentally, vessels of H e l l are rotated at some constant angular v e l o c i t y ft, so the minima of F f o r the various c o n f i -gurations must be found for given ft, and these minima compared i n order to f i n d that configuration which represents the absolute minimum of F f o r that ft. As ft i s increased from zero, there i s a range of ft for which the absence of v o r t i c e s represents the state of thermodynamic equilibrium. Above the c r i t i c a l angular v e l o c i t y ftj, the free energy of one vortex at the centre i s l e s s than the free energy of zero v o r t i c e s , so t h i s becomes the state of thermodynamic equilibrium. As ft i s increased f u r t h e r , there appears a spectrum of c r i t i c a l angular v e l o c i t i e s as i t becomes e n e r g e t i c a l l y favourable to introduce new v o r t i c e s into the f l u i d . That such i s the case i n r e a l i t y has been shown by the work of Packard and Sanders(1969). The purpose of t h i s section i s to show that such a spectrum can be reproduced from theory, at l e a s t f o r small numbers of v o r t i c e s . Since the energy of the vortex f l u i d involves products of the vortex strengths y^, i t i s reasonable to assume that the equilibrium state of the f l u i d would require that a l l the v o r t i c e s have the minimum non-zero quantum strength ( i . e . ti/m for h'elium) . For small numbers of v o r t i c e s , a t t e n t i o n i s r e s t r i c t e d to the regular polygons, whose s t a b i l i t y has been determined, or regular polygons with an extra vortex at the centre. The s t a b i l i t y of the l a t t e r has not been investigated. The c a l c u l a t i o n s of the minimum of 34 F as a function of Q, i s a task i n numerical a n a l y s i s f o r every configuration except i n the case of one vortex, which may be treated a n a l y t i c a l l y . This important c a l c u l a t i o n r e l a t e s the c r i t i c a l angular v e l o c i t y f o r the c r e a t i o n of one vortex, &i, to the vortex core radius parameter, e. The unit of length has already been chosen to be a, the c y l i n d e r r adius. The u n i t of time w i l l be ma2/Y> where y i s the u n i t quantum strength. A dimensionless F i s introduced by choosing the u n i t of energy to be p 7 r y 2 £ , I being the depth of the l i q u i d , and p being the density of the l i q u i d ( i n t h i s case the density used i s the density of the s u p e r f l u i d component of H e l l ) . The r a t i o of the vortex core radius to the c y l i n d e r radius i s e,,so the vortex core radius i s R=ea. The r a t i o of the polygon radius to the c y l i n -der radius w i l l continue to be r . 6.1 One Vortex i n a Rotating Cylinder The free energy of a s i n g l e vortex i n the c y l i n d e r i s , from(3-7) and (3-13), F = E-ftL = l n ( l - r 2 ) - lne - fi(l-r2) (6-1) For a given ft, e q u i l i b r i u m i s given by f = ° -> f £ i - i , B • «-» The second extremum does not e x i s t f o r ft<l. The f i r s t extremum i s a minimum for ft>l, whereas the second extremum i s always a maximum. The stable p o s i t i o n of a s i n g l e vortex i s at the centre, for ft>l. The equilibrium free "energy of a s i n g l e vortex as a function of ft becomes F1 = -ft -lne (6-3) The free energy of zero v o r t i c e s i n the f l u i d i s zero, so the condition that 35 one vortex at the centre i s thermodynamically stable over zero v o r t i c e s present i s Tl = -ft -lne < 0 (6-4) leading to the d e f i n i t i o n of the c r i t i c a l angular v e l o c i t y f o r the c r e a t i o n of one vortex &l = -lne (6-5) (In conventional u n i t s , t h i s i s 1^ = ( y / a 2 ) l n ( a / R ) . ) In the region l<ft<fti, one vortex at the centre i s said to be metastable, since that c o n f i g u r a t i o n i s stable against small perturbation's from e q u i l i b r i u m , but i t does not represent the state of thermodynamic equi l i b r i u m , which must minimize F absolutely. In t h i s region of angular v e l o c i t y , zero v o r t i c e s i n the f l u i d represents the state of thermodynamic equilibrium: the s u p e r f l u i d component i s at r e s t . The quantity ft^ has been measured by Packard and Sanders(1969) to be 1.6 sec ^(25.2 i n dimensionless u n i t s ) with a=0.05 cm, g i v i n g R=5*10 5 £ . A more r e a l i s t i c value of the vortex core radius i s R=0.3 X, from Wilks(1970), giving fti=1.0 sec 1(16.6 i n dimensionless units) with the same c y l i n d e r radius. The measurement of ft^ cannot be considered a good determination of R since R i s very s e n s i t i v e to small v a r i a t i o n s i n ft}. Also, there i s evidence i n Packard and Sanders' work that t h e i r experimental system was subject to strong e f f e c t s of m e t a s t a b i l i t y and the measurement of ftj i s i n some doubt. The measurement of subsequent ft2,ft3, etc -was c e r t a i n l y not reproducible. 6.2 The Spectrum of C r i t i c a l Angular V e l o c i t i e s i n He II The free energy of N v o r t i c e s i n a regular polygonal c o n f i g u r a t i o n i n a vessel r o t a t i n g at ft may be c a l c u l a t e d from (3-3) and (3-13) using (5-9) and (6 -4) . The d e t a i l s appear i n AppendixC, the r e s u l t being F (ft.p) = N l n ( l - p N ) -!sN(N-l)lnp -Nft(l-p) -NlnN +Nftx (6-6) 36 For N=l,2,3,4,5,6 the polygons are s t a b l e , so t h i s expression represents a 2 minimum of F with respect to deformations of the polygon. The v a r i a b l e s p=r and 0, are independent i n t h i s expression, but i t was shown that i n equilibrium the polygon must rotate with the same angular v e l o c i t y as the v e s s e l , g i v i n g the r e l a t i o n 2p p i - P N For a given N and ft, (6-6) i s the free energy of the c o n f i g u r a t i o n , p being the s o l u t i o n to (6-7), taking care that ft i s greater than ft . given i n mxn Table I I , section 5. Operationally, (6-7) may be solved for a s p e c i f i e d value of ft by an i t e r a t i v e technique; the value of p obtained i s substituted into (6-6) with ft to obtain F. Thus a curve of the e q u i l i b r i u m value of F vs ft,may be computed f o r each c o n f i g u r a t i o n . The a d d i t i o n of a vortex at the centre a l t e r s the form of the free energy and angular v e l o c i t y . The free energy of an. N-gon with a c e n t r a l vortex i s F N ( f t , p) = N l n ( l - p N ) -JsN(N+l)lnp -NlnN + (N+l)ft! -ft(l+N(l-p>) (6-8) with the new angular v e l o c i t y , due to the c e n t r a l vortex 2P P 1- P N The curves of F vs ft may be obtained i n the same way as above. Once these curves have been p l o t t e d , the states of thermal e q u i l i -brium can be determined by v i s u a l i n s p e c t i o n , and the c r i t i c a l angular v e l o c i t i e s obtained by the i n t e r s e c t i o n of the appropriate curves. The free energies of one vortex (F^), two v o r t i c e s (F^) , three v o r t i c e s i n an equi-l a t e r a l t r i a n g l e (F^)> and three c o l i n e a r v o r t i c e s O^) & r e plotted as a function of ft i n Figure 1, using R=0.3 X. The i n t e r s e c t i o n of these curves give the t h e o r e t i c a l values of fii,^, a n < l % t ° be 16.6, 20.1, and 22.1 37 r e s p e c t i v e l y (1.05 sec ^, 1.27 sec and 1.40 sec * i n conventional u n i t s ) . Figure 1 also shows that, i n t h i s region of angular v e l o c i t y , the t r i a n g u l a r configuration of three v o r t i c e s i s thermodynamically stable over three co-l i n e a r v o r t i c e s , although both configurations are stable with respect to small perturbations from equ i l i b r i u m . 38 Figure 1: The equilibrium free energies of one, two, and three v o r t i c e s . F and ft are p l o t t e d i n dimensionless u n i t s (see t e x t ) . The curves F^, "F^, F^, and F^ are the eq u i l i b r i u m f r e e energies of one vortex, two v o r t i c e s , three c o l i n e a r v o r t i c e s , and three v o r t i c e s i n an e q u i l a t e r a l t r i a n g l e , r e s p e c t i v e l y . The q u a n t i t i e s ft}, 0,2, and 03 are the c r i t i c a l v e l o c i t i e s f o r the creation of one, two, and three v o r t i c e s , r e s p e c t i v e l y . 39 7. CONCLUSION The r e s u l t s of s e c t i o n 5 concerning the s t a b i l i t y of the regular polygonal configurations i n an i d e a l f l u i d bounded by a c i r c l e are a departure point for the s o l u t i o n of the general problem of the s t a b i l i t y of a l l r i g i d l y r o t a t i n g configurations. Before t h i s i s attempted, the sub-problem of c l a s -s i f y i n g a l l possible r i g i d l y r o t a t i n g configurations must be solved. The acceptance of quantized vortex strengths s i m p l i f i e s t h i s task somewhat, since i t i s expected that the r e s u l t i n g configurations w i l l e x h i b i t some degree of symmetry; these symmetries would i n d i c a t e the normal modes of v i b r a t i o n and the s o l u t i o n of the equations of motion of the perturbations would proceed i n a straightforward manner. One question which has been nagging the author throughout t h i s research i s the nature of the law of conservation of angular moment of c i r c u l a t i o n . It has been demonstrated that the other conservation laws a r i s e n a t u r a l l y from the symmetries of the system of v o r t i c e s , but that no symmetry associated to W has been found. It may be the case that t h i s i s not an inde-o pendent conservation law, but that i t i s a consequence of the others. If t h i s i s a f a c t , i t . h a s not emerged i n the 100 years of l i t e r a t u r e on the subject. It would be s a t i s f y i n g to have t h i s cleared up once and f o r a l l . The approach used here may prove cumbersome when large numbers of v o r t i c e s are being considered. Recent work by Su(1973) and P o i n t i n and Lundgren(1976) i n v o l v i n g methods of s t a t i s t i c a l mechanics may bear more f r u i t for workers i n l i q u i d helium physics. The relevance of t h i s work to the a c t u a l behaviour of vortex l i n e s i n He II depends on the v a l i d i t y of the assumptions made concerning the properties of He I I : that the two-fluid model i s adequate i n describing the macroscopic properties of He I I , that the problem i s e s s e n t i a l l y two- dimen-40 s i o n a l , that the normal f l u i d and s u p e r f l u i d may be treated as incompres-s i b l e , and that the d e t a i l s of what occurs at the vortex core i s ignorable. The a b i l i t y to reproduce q u a n t i t a t i v e l y the spectrum of c r i t i c a l angular v e l o c i t i e s f o r vortex creation at low angular v e l o c i t y seems to support the assumptions made - more accurate experimental r e s u l t s are needed before a stronger statement can be made. 41 BIBLIOGRAPHY R.P. Feynman, Progress i n Low Temperature Physics, 'C.J. Gorter,ed., (North-Holland,Amsterdam,1955),Vol.1,p.36 D. F. Greenberg, "Accidental Degeneracy",Am. J . Phys. I l l ,1101,(1966) T.H. Havelock, "The S t a b i l i t y of R e c t i l i n e a r V o r t i c e s i n Ring Formation", P h i l . Mag. ,21,617,(1931) G. B. Hess, "Angular Momentum of Supe r f l u i d Helium i n a Rotating Container", Phys. Rev. 161 ,189 , (1967) E. L. H i l l , "Hamilton's P r i n c i p l e and the Conservation Theorems of Mathematical Physics", Rev. Mod. Phys.. 23 ,253 , (1951) H. Lamb, Hydrodynamics,p.219-236,(1932) L.D. Landau, J . Phys. USSR ,5_,71, (1941) C.C. L i n , "On the Motion of Vo r t i c e s i n Two Dimensions",University of Toronto Studies, Applied Math Series. No.5, (U. of Toronto Press,1943) L.M. Milne-Thomson, T h e o r e t i c a l Hydrodynamics, (1968) W.B. Morton, "On the S t a b i l i t y and O s c i l l a t i o n s of Certain Permanent Arrangements of P a r a l l e l V o r t i c e s " , Proc. Royal- I r i s h Acad. 42A,3,(1934); "Vortex Polygons", , 42A,6,(1935) L. Onsager, Nuovo Cimento Suppl. 6_,249, (1949) R.E. Packard and T.M. Sanders,Jr., "Detection of Single Quantized Vortex Lines i n Rotating He I I " , Phys. Rev. Let. 22,823,(1969) Y.B. P o i n t i n and T.S. Lundgren, " S t a t i s t i c a l Mechanics of Two-Dimensional Vo r t i c e s i n a Bounded Container", Phys. F l u i d s 19,1459, (1976) S.J. Putterman, Superfluid Hydrodynamics, (North-Holland,Amsterdam,1974) S.J. Putterman and G.E. Uhlenbeck, "Thermodynamic Equilibrium of Rotating Superfluids", Phys. F l u i d s 1_2,2229 , (1969) J.R. Reppy and C T . Lane, "Angular-Momentum Experiments with Li q u i d Helium", Phys. Rev. 140,106,(1965) J.D. Reppy, D. Depatie, and C T . Lane, "Helium II i n Rotation", Phys, Rev. Let. 5^,541,(1960) C H . Su, "System of Vo r t i c e s i n Rotating L i q u i d Helium", Phys. Fluidsl6,182, (1973) J . J . Thomson,"Motion of Vortex Rings", Adams P r i z e Essay (1883) BIBLIOGRAPHY (continued) V. K. Tkachenko, " S t a b i l i t y of Vortex L a t t i c e s " , Sov. Phys. JETP 23,1049,(1966) W.F. Vinen, Proc. Roy. Soc. A260,218,(1961) J. Wilks, An Introduction to L i q u i d Helium,(Clarendon Press, Oxford,1970) G.A. Williams and R.E. Packard, "Photographs of Quantized Vortex Lines i n Rotating He I I " , Phys. Rev. Let. 33_, 280, (1974) A3 APPENDIX A: STREAM FUNCTION APPROXIMATION FOR ENERGY CALCULATION The stream f u n c t i o n for a vortex f l u i d i n s i d e the u n i t c i c l e i s 4)(z) = - Z y In | z-z. | + Z , y , l n | l - z i ' The following notation i s introduced: i g ' i6. z-z = ee m z, = r, e k m k k i6 . . , i e ' z -z, = r , e mk z -1/z. = r ,e mk m k mk m k mk In the v i c i n i t y of z , the two terms i n the stream f u n c t i o n may be wri t t e n m l n l z - z , I = l n l (z-z )+(z -z, ) I = l n l e e ^ m + r . e1^mk| 1 k 1 1 m m k ' 1 mk 1 = hln(e 2 + r 2 + 2 e r , cos(g -6 .)) mk mk m mk r lne (k=m) i l n r . + 0(e/r . ) (k#n) mk mk l n | l - z i k | = l n | z k | + l n | z - l / i k | = ln|z" k| + l n | ( z - z ^ + C z ^ l / z " ^ | = lnr, + l n l e e ^ m + r',e^mk| k 1 mk 1 = lnr, + ^ l n ( e 2 + r ' 2 + 2 e r \ cos(B -6', )) k mk mk m mk = lnr, + ln r ' , + 0(e/r \ ) k mk mk If e<<r , and e<<r', then the stream f u n c t i o n i n the v i c i n i t y of z may be mk mk ' m written * = "Vn£ - 'k Vnrmk + EkYklnrmk+ Ek^klnrk + °(£) 44 APPENDIX B: SOME RELEVANT TRIGONOMETRIC SUMS It i s necessary to make use of the r e l a t i o n f e i ( 2 W N ) m k 1=12 and the serie s oo oo ( 1 - x ) " 2 = £(n+l)x n x ( l - x ) " 1 = I x n n=0 n=l oo x ( l - x ) = £ nx 1 1 n=l Evaluation of T : N i(2Tr/N)Lk T L = J l ( l - p ^ 2 ^ ) 2 P < 1 L = 1 ' 2 > - - ' N f i ( n + l ) p n e 1 ( 2 7 T / N ) ( L + n ) k k=l n=0 N l < n + l ) p n 6 * = 1,2,... n=0 ' = N 2 p " L I *p N* - N ( L - l ) p - L I p N £ A=l £=1 0 N-L N-L (1-P ) 1-P Evaluation of S^: N- l i(2Tr/N)Lk y l - e _ l i m r , L k=l ( l - e i ( 2 7 7 / N ) k ) 2 P^ 1 N L = lim r N + N(N-l)p N - N 2p N L + N(L-l) p H ( l - p " ) , 1 P+l l' n N.2 I (1-P ) = 2^ JiT ( N L ( N - L ) P ~ N ~ L - 2N(n-L)(L-l)p" L} (using l ' H o p i t a l ' s ^ ( N - L ) ( L - 2 ) r U l 6 ) APPENDIX C: THE FREE ENERGY OF REGULAR POLYGONAL CONFIGURATIONS It i s necessary to use the same r e l a t i o n used i n Appendix B and the s e r i e s oo n x l n ( l - x ) = -I -1 1 1 n=l to f i n d the following sums N-l j l n . ] l - e 1 ( 2 T r / N > m | . = ^ {Yl4l-Pe 1 ( 2 i r / N ) m|j m=l P m=l , lim 2 F * l °° N-l n { I I J e 1 ( 2 T r / N ) m n + c.c.} n= 1 m=1 and oo n { 7 £ (N6 - 1)1 £ = 1 2 lim P+l 1 L,n v" un,£N n=l ' ,. £N £ lim r V P_ _ P_ 1 p+1 1 z t l £ £ lim P+l N N I I m=l k=l In 1-p = InN (using l ' H o p i t a l ' s rule) I I l n | l - p e i ( 2 7 T / N ) ( m - k ) N N °° n = -Jj £ £ J £ ei(2ir/N)(m-k)n + c. c, m=l k=l n=l , n n,£N n=l ' £ = 1,2,. = Nln(l-p ) Using these i t i s straightforward to show that (with z = e i ( 2 7 T / N ) k )  E " - Zm Z k l n I V Z k l + W ^ - V J - V n £ %N(N-l)lnp -NlnN + Nln(l-p ) - Nine and that L = 1 (1-z z ) = N(l-p) m mm v ' 46 APPENDIX D: SOME MISPRINTS IN HAVELOCK'S PAPER The published v e r s i o n of the work by Havelock(1931) on the s t a b i l i t y of various configurations of v o r t i c e s contains some misleading m i s p r i n t s . The important ones are l i s t e d here(the equation numbers are Havelock's). (i ) In (23), the second term i n the l a s t l i n e should read ... , 2 n-1 p -f-p d-p ) ( i i ) In (25), the second term i n Q should read " -2(n-l) ", and there i s a missing term: _ 4 n i n 1-P ( i i i ) In (28), the second term should read " -2(n-l) ", but the term missing 2 from (25) has been replaced. The f i r s t term should read" (%n) ". 

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