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The stability of body centered polygonal vortex line configurations Mertz, Gordon J. 1977

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THE STABILITY OF BODY CENTERED POLYGONAL VORTEX LINE CONFIGURATIONS by < GORDON J. MERTZ B.Sc, University of B r i t i s h Columbia,1975 A THESIS SU3MITTSD IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1977 Copyright Gordon J, Mertz 1977 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date -6 i i ABSTRACT This thesis is concerned with the s t a b i l i t y of rectilinear vortex l i n e configurations. A method for consideration.of stab-i l i t y problems, using the equations of motion for small pertur-bations about equilibrium, i s developed; and the free energy criterion for s t a b i l i t y stated. The former method i s used i n analyzing the s t a b i l i t y of the body centered polygonal configur-ation, which i s a Thomson polygon centered by a vortex of arbit-rary relative circulation; the latter criterion i s applied to the Thomson heptagon s t a b i l i t y question. The main results are that a Thomson polygon of any order can be stabilized by a large enough central circulation and that the Thomson heptagon i s stable. 111 CONTENTS ABSTRACT .............. i i CONTENTS ................... i i i ACKNOWLEDGMENTS ............................. iv 0. INTRODUCTION . ....... 1 1. GENERAL CONSIDERATIONS REGARDING CLASSICAL VORTEX MOTION 3 2.. RIGIDLY ROTATING CONFIGURATIONS 5 3., THE METHOD OF SMALL PERTURBATIONS ............. 7 4. STABILITY ANALYSIS OF THE BODY CENTERED POLYGONAL (B.C.P.) CONFIGURATION 9 5., STABILITY OF THE THOMSON HEPTAGON ......... ........ . 19 r 6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 NOTES .............................. ............... 26 LITERATURE CITED ..................... 27 APPENDIX A: CALCULATION OF THE TRIGONOMETRIC SUMS S M 28 APPENDIX B: CALCULATION OF E AND L 30 FIGURES: (4.1) Stability of the B. CP. Configuration 17 (B.I) The Contour for Integrals (B.3),(B.,4) ........ 31 ACKNOWLEDGMENTS I thank Dr. R A., Kaempffer for suggesting this research topic, and for his considerable guidance throughout the produc-tion of this thesis. I am also indebted to D.. Chapman and D. Hally for their many helpful comments. I am grateful to the National Research Council and to the University of British Columbia for their financial assistance 1 0. INTRODUCTION In his Adams Prize Essay of 1882 J. J. Thomson considered the stability problem posed by K rectilinear vortex lines of equal strength forming a regular polygon,, as a special case of the three-dimensional motion of vortex rings in an ideal fluid.. Thomson was motivated by the then popular search for a hydro-dynamical account of the contemporary atomic theory. The evol-ution of atomic theory since 1882 seems to preclude any such classical accounts, and'this'thesis makes no pretense to contri-, bute to atom models; rather i t reflects a renewal of interest in the theory of classical vortex .motion brought about by the } "observation" of Onsager vortices* in superfluid helium which, like an ideal f l u i d , i s essentially inviscid and incompressible. The helium atoms of the said vortices move coherently in macro-scopic orbits of the same angular momentum nn (n an integer) defining the rectilinear vortex velocity f i e l d . In polar (r,6) coordinates a single vortex located at r=0 is described by ( 0 -D ve = i l = 2*r - V = 0 ' where m i s the mass of a helium atom and t i s the circulation since t 4 v»ds_ (line integral enclosing the vortex) = nh m and is the strength parameter of the vortex. The aforementioned revival of interest is readily eviden-ced by recent works appearing on. this subject including 2 Tkachenko (1966!) and Chapman (1977); their work was particularly relevant to the writing of this thesis. This thesis addresses i t s e l f to questions of stability and generalizes Thomson's work. Specifically, the modification of --stability due to the addition of a central vortex to the Thomson polygon i s examined. Also the stability question of the Thomson heptagon (thought to be unstable by Thomson, but shown by o Havelock (1931) and Morton (1935) to be indeterminate in the f i r s t order perturbation theory employed) is settled using the free energy criterion. The work presented here contains as new contributions to knowledge proofs that (1.) the modified Thomson polygon shows non-trivial stability dependence on the ratio of the circulation of the central vortex to that of the polygonal vortices, C2.) the Thomson heptagon is in fact stable. 3 1. GENERAL CONSIDERATIONS REGARDING CLASSICAL VORTEX MOTION Equations (O.l) may be expressed in terms of a complex velocity f i e l d w(z) = v x - i v y as r w(z) =-rj ; z = x+iy ; the residue of w(z) representing the circulation When more than one vortex line is present, linear superposition of the individual velocity fields is assumed; i.e., given a config-uration of N vortex ;lines located at zk = ak +*kk ' k = l,2....N , each of circulation t^r then the velocity f i e l d of the f l u i d is (1*1) w(z) ±1-i k z-z^Ct) The equations of motion relating the positions z n(t) of a sys-tem of vortex lines may be derived at once by assuming the Helmholtz-Kirchhoff condition;' that is,, the velocity of a vortex line (dz n/dt) is just that of the flu i d at z n without the said vortex line present. Thus (1.2) ^ ;. ^ denotes IZ # a t i k z n(t)-z k(t.) * k * n Equations (1.2) admit of four integrals-,; generally known as 4 The angular moment of circulation ( 1 . 5 ) y (z^dz^/d t-z^dz^/dt) = I = const. ' ^ n n n n n The vortex stream function (1.6) l^IltnVk l n[( z n~ 2k ) ^ "V] = W = c o n s t-These constants of the motion w i l l play an important role in the fourth section where they w i l l be used in deciding the relevance to stability considerations of certain eigenmodes in the spect-rum of a given configuration. The center of circulation (1*3) / Y z =|~Z = c o n s t . n n n The moment of circulation const. f i II II n 5 2. RIGIDLY ROTATING CONFIGURATIONS A configuration of vortices is considered to be rigid (rigidly rotating) i f the vortex positions z n(t) satisfy (2.1) zh(t.) = z°°e i f t t ; z°°,0 are constants. This thesis deals with a certain class of rigi d configurations only. The reason for restricting stability considerations to these configurations is twofold. Fi r s t l y , in an historical vein, this work is to be an addendum to the stability analysis of certain rigid systems started by Thomson and continued by Morton and Havelock. Secondly, it.: is known that any vortex system present in. liquid helium at thermal equilibrium must be in rigid rotation. This may be understood as follows: liquid helium may be regarded as a superposition of a normal fluid and a nondissipative superfluid; the superfluid vortex must have a normal f l u i d core since the singular nature of vortex flow may not be physically realized. As is well known,, a normal f l u i d in thermal equilibrium must rigidly rotate; hence the vortex cores are carried in-rigid rotation. It is thus hoped that the con-siderations to follow maintain a certain degree of historical and modern interest. Specifically, the general body centered polygon (b.c.p.) is to be examined. -This is just a Thomson polygon (N vortices of circulation 2trY at the vertices of of a regular polygon of radius R so that (2.2) z°° = R e i ( 2 T T / N ) n ; n=l,....N ) and a central vortex of circulation 2lfY =2rf*& (^arbitrary) located so that (2.3) z°° = 0 That this configuration is rigid may be verified as follows: Substitute (2.1) into (1.2), then (2.4) - n n —• —00 ^-rr 00 00 Z k Z — Z v n n k. must hold for a l l n i f the system is ri g i d . For.the b.c.p., using natural units (R is the length unit and R /Y is the time unit), ; one finds (upon setting n=N) N-I 1 M I-ttos[(27T/N)k] + isin[(277/N)k] implying ^ N-l (2.5) A=^T+2Z (1/2) = * + (N-D/2 k=l since the imaginary part of the subject sum vanishes (because cos [(2*VN)k] =cos [(27TAT) (N-k)] and sin[(2*T/N)k] =-sin[(2Tr/N) (N-k)J .) Thus b.c.p. rig i d i t y is established and only the question of stability remains. 7 3. THE METHOD OF SMALL PERTURBATIONS The criterion for stability adopted here is that used by Thomson et a l , namely i f a configuration is to be considered stable then infinitesimal deviations about equilibrium must not grow, and hence must oscillate since there is no dissipation in an ideal f l u i d . Only perturbed configurations capable of evolv-ing "naturally" (that i s , without the agency of external forces) into the equilibrium configuration are considered; thus they must have the equilibrium constants of the motion. In section 5. the alternative but equivalent stability criterion of Uhlenbeck and Putterman (1969) w i l l be employed. If the vortex position i s s p l i t into an equilibrium (z°) and perturbational (6CN) component, then (3.1) z n(t) = z°(t) +<*n(t) Substituting (3.1) into (1.2) and expanding in (^n~^)/(zfl-z£) to f i r s t order yields (3.2) —i?;= i 2 _ - * n * dt ~TT (z9-z£)2 For rigid systems i t i s desirable.v to transform to the co-rotating reference frame, so that in the stationary coordinates specified by (3.33k z°(t) = z ^ V ^ , oc n( t) = ^ n ( t ) e i t t t (.3.2) becomes the constant coefficient equation 8 (3.4) at = i ( z 0 0 - z 0 0 ) 2 k Kzn zk ; The solutions to (3.4) are constrained by the conservation laws (1.3),(1.4),(1.5),(1.6), since as stated, the perturbed and equilibrium configurations share the same constants of the motion. In the f i r s t order deviations these laws read (3.5) (3.6) (3.7) (3.8) y y G?°/8 +2°°/) = o A=r n v n "n n "n y n n k n * OO _ 0 0 n z - "»k -oo -oo z n ~ zk = 0 Clearly, (3.8) i s identical to (3.6) since by the definition (2.4) for A the double sum in equation (3.8) becomes 2 P ^ y n(z°°^ n+2°°^ n). Thus the possible motions of the perturbed n general rigid system are given by (3.4) and are constrained by the three independent conservation laws (3.5),(3.6),(3.7). 9 4. STABILITY ANALYSIS OF THE BODY CENTERED POLYGONAL CONFIGURATION The equations developed in the previous section are now applied to the b.c.p. j.corrf^ and C2.3). The equations of motion become (in natural units) C4.1a) -£S=H dt (4.ib) [ k [ei(2n'/N)n_eir27I/Jr)k]2 [ ei(2"/tr)nl2 dt = l .i(2Tr/N)nJ2 and are constrained by (4 .2) (4.3) C4.4) n p ^ e - i ( 2 f r / ! T ) n + /^ ei(2fT/N)n = 0 % -i(2*r/N)n ai(2ff/N)n dt dt = 0 If equations (4 . 1 ) are differentiated, and substitutions are performed so as to eliminate the complex conjugate terms, then they may be cast in the form of a second order eigenvalue prob-lem 4- easily solved by standard methods. However, great simplif-ication is achieved by directly exploiting the symmetry of the system. This N-fold symmetry dictates that the displacements o f the vortices forming the polygon must be a superposition of the modes specified by 10 C4.5) /?n(M,ft) = b M ( t ) e i C 2 f r M / W ) n (1*1,...10 , which contains a phase convention making the vortex n=TT the reference vortex, located on the positive real axis, C4.6J /3R(Mrt.) « bM(t) Denoting the displacement of the central vortex by (4.7) /?0(M,t) = c M ( t ) when the polygonal vortices are in the mode M, the equations of motion (4.1a) reduce to an equation for the reference vortex (4.8) dt N--1 1_ei(2TrM/N)k and in that mode the equation (4.1b) reads (4.9) dc M dt „ h.i(2^M/N)n n=l [ei(27r/W)nj2 + H e M The solutions to the above, for the mode M, are constrained by the conservation laws (4.10) i(27T/JT)Mn _ = 0 ( 4 - n ) \ Y l e i ( 2 ^ m i U - 1 ) n + h M Y]e-i(2^/N).(M-l)n = Q n n ' (4.12) _ J 8 i . y ~ e l ( ^ H > ( M - l ) n _ dt *"rr db, dt "n Y"e-i'(277/N;)(M-l)n = 0 11 The machinery of the stability analysis i s now set up; in applying i t special cases in which the conservation laws play an important role are f i r s t considered: M=l: Since 2Texpi|2n/N)nJ=0, the constraint (4.10) becomes (4.10-L) ^ = 0 Similarly (4.11) now reads flKb^+b-^^O, implying (4.11^ Heft^) •= 0 The constraint (4.12) yields F»lm(db.j/dt)=0, thus (4„12 1) (dbx/dt) = 0 According to (4.5) , ^ n ( l ) = i b e 1 ( 2 7 r / N ) n (b is a real constant), describes a rotation of the rigid configuration through a fixed angle. Such a static mode places no restrictions on the stabil-i t y of the system. M=N; The constraint (4.10) becomes , (4„10N) ^ c N + NbN = 0. The constraints (4.11) and (4.12) place no restrictions on b^ since £expij(2W/N-) (N-l)n] =0. In the case ^=0 the constraint (4.10^) requires (4.13) b N = 0 (*=0) , clearly forbidding this mode. To investigate the. case*T^ 0, 12 substitute (4.10^) into equation (4.8), yielding (4.8N) — i ( ^ + Njb^ + i Q b N . dt k similar substitution using equations (4.10N) and (4.9) yields, with the notation (4.14) S = X r [ e i ( 2 ^ ) n j - 2 = . n=l ' 2 for N=2 . 0 for N>2 db.T (4.9^) — a = i(^+N)(S/W)bTT + ±P.bN There are two subcases to be considered: •*£=-N: The constraint (4.10^)' requires of this subcase that cN=bN; this and the definition (4.5) show that the subject mode consists of a displacement of the entire configuration, because equations (4.8^) and(4..9^) both imply /4^(N)=const.exp(-i^ t ) . Then the definitions (3.3) give 0^(N)=const. Hence in the non-rotating frame this mode is static and thus not relevant to stability considerations. The displacement of an entire config-uration is only possible when (as in this subcase) the total circulation is zero, so that the center of circulation is at inf i n i t y . •*i*-N: For N>2 the value of S (equation (4.14) ) is zero so that equations (4.8^) and (4.9^) are inconsistent unless b^=0; thus this mode is forbidden. These equations are identical i n the case N-2, they both read 13 db0 (4.82) — * = i(^c + 2)b 2 + iQb g d t which used in conjunction with i t s complex conjugate db 0 (4.82) —& - -i(^T+2)b 2 - i P b 2 dt yields the second order equation (4.15) = [(^+2) 2-P 2]b 2 = (*+2-£n(*+2+n )b 2 . If the solutions of equation (4.15) w^t (4.16) b 2(t) = b 2(0)e a are substituted into (4.15), then the stability criterion (that the mode must oscillate) becomes (4.17) w2 = (^+2-p)(^+2+P)<0 i.e. w2=*i<o2, o>2 real , or explicitly (using the value of fl given in equation (2.5) ) (4.18) w| = (3/4)(4«+5)< 0 or ^<-(5/4) Thus for -^f<-(5/4) the mode is stable, oscillating according to (4.10^ = 2) c 2 = -(2/5^)b2>(8/5)b2 , which describes the aligned, cophasal oscillation of the central vortex (of circulation X0^~(b/'A))[) and the two satellite vortices (of circulation X)• 14 The preceding analysis shows that forJT>2 the cases M=l  and M=N cannot give rise to unstable modes. Thus the cases rele-vant to stability" considerations (for N"->:2) are those of M=2,.,. .....N~—l: Under-this specification, T^exp i(277/TT)Mn =0, so that the constraint (4.10) reads (4.19) c M = 0 and the constraints (4.11) and C4f12) place no restrictions on bj^ » With the notation N-l 2 - e 1 ( 2 7 r / N ) M c (4.20) ^ = 21; k=l [ 1 - ei(2T7/K)tJ2 one can write the equation of motion (4.8) 04.21) - J = + if i l f c and i t s complex conjugate _ db„ _ _ (4.21) = -i(S M+-^)b M - i« from which follow the second order equation a 2 -<4-22) [<s M^) ( s M ^ - f t 2 ] v ^ ( V V H + 1 ) + W ( ^ ) ] h . where again the value of Q as given i n (2.5) has been used. Substitution of the solutions (4.23) b ^ t ) = bM(0)e M 15 yields (4.24) 4 = *<sM+SirJr+i> + - ( ^ i ) 2 . If the mode i s stable (oscillatory) then w^  must be imaginary, i . e . W j ^ =±iCx)M ( ^ j y j real). This requirement and (4.24) yield a general Stability Criterion for the Body Centered Polygonal  Conf igurationnamely (4.25) *( SM+SM-M) + VM - <0 • The trigonometric sums SM, calculated in appendix A, are given by (4 26) q - (M-2HM-N) On account; of the reality of Sj^,(4.25) may 'be .-written as t and since S M i s nonpositive ( M=2,....N-l) this criterion becomes (4.27) ^ > l [ | S „ l - ¥ ] Clearly, i f (4.27) is satisfied for the mode M; in which IS^I is greatest, i t w i l l be satisfied for a l l other modes. This stabil-i t y determining mode (denoted M*) is specified by 1 6 ( N+2 (4.28) M* = for N" even I N±1,N±3 for W odd 2 2 and the corresponding values of IS^ I given by (4.29) N2- •4N+4 8 for N even N2- •4N+3 for N odd 8 Then from (4.27) and>::(4.29) follows the stability criterion for the stability determining mode M* (and thus for any b.c.p. configuration of N>2) (4.30) '£=$4 for N even ^=fP for N odd Thus, a large enough value of -K w i l l stabilize the configuration. The results of this section (criteria (4.18) and (4.30) ) are illustrated by figure (4.1). The boundary points of the "region of stability" represent the case where the stability determining mode has a zero eigenvalue. Thus the system represented by this point must be regarded as being of indeterminate st a b i l i t y in the f i r s t order theory presented here. In the limit ^=0 one should recover Thomson's results; this is in fact the case i f Havelock's and Morton's corrections are taken into account. These corrections show that in the l i n -17 • CO \« Stabili F L t y of t igure (4.1) tie B.C.P. Confif ^juration »<-»ri\« \* • ft ft A k ft ft 1 I t 1 1 >\ \ \> \ \ \ > Ml V j ft. \ \ \ \ U U i 00 . V ft \ »\N %\ V o > > t 1 i « t « d / M •* w i W W 0 ^ * * . . ft ft ft ft Al \ \ \ \ \\ \\ > % . to \ \ 1 1 a » x\ » T 1 1 1 . . . . ! * •« I i i " " ; 1 1" B ^ (*| f f i ..kftvVftll ft PI I ft I i Hi II i W %^ \\W\^ 1 <? II 18 ear approximation the Thomson heptagon is not unstable (as Thomson thought) but i s of indeterminate stability (as substitu-tion into condition (4.30) w i l l readily verify). Because this case is historically interesting, the next section w i l l resolve the indeterminateness of stability associated with. i t . In closing, one might note that the preceding considera-tions seemingly illustrate the d i f f i c u l t y associated with guessing (applying "intuition" to) stability dependence on sys-tem parameters. For example-, the author never guessed ahead of time that a polygon of arbitrarly large IT could always be stabilized by a strong enough central circulation, as is in fact the case. \ 19 5. STABILITY OF THE THOMSON HEPTAGON The stability determining modes of the Thomson heptagon are, by (4.28), M*=4,5. An arbitrary superposition of these modes i s , by the definition (4*5), (5J) Bn(4,5) = a e i < W 7 ) n + b ei(l< W 7)n . equation (4.24) reveals that the characteristic frequency of such displacements i s zero (w^=Wg=0) implying ihdeterminateness of stability (in this linear approximation). This indeterminate-ness can be resolved, in principle, by solving the equations of motion expanded to beyond f i r s t order in the perturbations. This appears, however, to be a formidable task. Stability problems can usually be solved by location of the minima of a generalized energy function, and fortunately 3 u c h a function, appropriate to the considerations here, has been given in the literature. Putterman and Uhlenbeck (1969) showed that a rigidly rotating configuration of vortex lines in thermodynamic equilibrium is stable only-if the velocity f i e l d v satisfies the condition (5.2) ^ v 2 - £}• (r x v)jdr = Min. The f i r s t term of the integrand is the kinetic energy per unit mass, and r x v is the angular momentum per unit mass of the fl u i d . With the notations (5.3) E = -|v2 dr 20 (5.4) L = J ( r x v) dr the stability criterion becomes E - A»L = Min. or, since L is parallel to ft for two (5.5) — ~ — dimensional conf igurations, E - fi. L = Min. In Appendix B i t is shown that (5.6) E = -§"W = -IT/XI ' V n y k l n [ ( Z n - z k ) ( i . ^ ) ] n 1c and (5.7) L = -7TA= - T r T ^ l z ^ z . /— n n fc n when divergent terms not dependent on stability parameters are ignored. This quantity E- QL is referred to as the "free energy" (F). The stability question of the Thomson heptagon w i l l now be resolved using the criterion ( 5 . 5 ) . If the free energy i s expan-ded in terms of small displacements about equilibrium, then the criterion (5.5) requires that the f i r s t order terms vanish and that the dominant higher order terms be positive definite. The expansion may be carried out separately for each characteristic mode (displacements of unique characteristic frequency w^), since they are physically independent. Thus a small displacement in the indeterminate mode (M=4,5) is considered, with the usual notation and z )° being the vortex position in the co-rotating 21 reference firame (note that F i s invariant under the transform-ation V * z n > 0 e i 0 t > , (5.8) 2 n)°=^ 0 +B n(4,5) = •^2tr/7)n + a | f ci(87^)n + b ei(ICWr7)n . Then ^ ) ° - ^ ) < W ( 2 ^ n - e i ( W 7 ) k > r ei(8w/7)n_ ei(8nr/7)ki 1+a i (2*7/7 )n i(2w/7)k 6 i (107r/7)n_oi (1077/7 )k ei(27r/7)n_ei(2rr/7)k = (ei(27r/7)n^ei(2rr/7)k ) ( 1 + a p + b q ) i f p = e i (6^/7)n + ei (477/7)nei (27T/7)k+ei (27r/7)nei"(47r/7)k+ei (677/7 )k and q=ei(8^/7)n+ei(677/7 Jn^i(27r/7)k+ei(40/7)nei(47T/7)k + e i (2rr/7)nei (6rr/7)k + ei (87T/7 )k Thus (upon expansion) .9) E = 4 7 r X / ( l n ( e i ( 2 ^ 7 ) n - e i ( 2 ^ 7 ^ ) + ( a p + b q ) 4 ( a p + b q ) 2 ^ kn < ^ 1 3 \ +g(ap+bq) +... +complex conjugate of same termsj L may be expanded in closed form, L = - 7 r X { l + a e i C ^ n _ • /Ow/OA_ — 1 f n-rr/n N _ — thus yielding 22 (5.10) L - -N7r(l+aa+bb) (W~7) Thus L contributes only second order terms (other than unimport-ant constant terms).. Explicit calculation of E requires evalii-ation of double sums of the form ei(2Tx/7)n ei(2>ry/7)k nk . Consider f i r s t the case y ^ r7 (r an integer), then the above sum becomes ' 0 i f (x+y) X 7(integer) ei(27T/7)(X+y)n = [ - j r (=-7) i f (x+y) = 7(integer) If y = ,r7 then the sums under consideration become 0 i f x / 7(integer) « . N(N-l) i f x - 7(integer) X ( N - l ) e i ( 2 T n c / 7 ) n = n* Thus these double sums, are non-zero only i f (x+y) = 7(integer). Therefore, with the definitions of p and q, i t is readily seen that only terms of the form (pq) w i l l yield non-zero contribu-tions to E. A l l odd! orders in the kinetic energy expansion must then vanish; hence the f i r s t order terms of E vanish (as they do in L) so that in f i r s t order F=0, which is required for an extremum. The second order term of E is I ^ X ' (ab)(PQ) • I ^ X ' (5b) kn c kn 23 ^abyiV2™ •+ 2 e i ( l ^ / 7 ) n e i ( 2 7 r / 7 ) k + 3 e i (10^/7 ) n e i (4^/7 )k 2 IE1 + 4 ei(8^/7)n ei (6v7/7)k + 4 ei(6^/7)n ei(87r/7)k" + 3ei(47r/7)nei(107r/7)k + 2 ei(2^/7)n ei(12rr/7)k + e i 2 * k ] + c ^ =|abTr[2W(N--l)-18NJ + CC. =^|aWT(N-l) - |ablW-l) Thus F i s given in second order (using equation (2.5) forQ) by (5..12) F ( 2 ) - E ( 2 ) - L ( 2 ) = ilT(lT-l){aa-ab-ab+bbj= |WCN-I) |a-b| 2 (o) F is then positive definite everywhere except in the plane a=b, where the free energy surface is f l a t . It is this flatness which leads to the indeterminateness of stability in the linear approximation to the equations of motion. The question of stability must be decided by examination of (at least) the fourth order terms. Since l/4^=0, F ^ = E ^ . Therefore, on account of E ( 4 ) = § ( a b ) 2 7 r X o m ) 2 + C* C* and 8 nk 21 '(pq) 2 = 34N" kn C5.12) F ( 4 ) = ^ N ( F-l)[(ab) 2+Ub) 2] . In the plane a=b fF^ 4^~(const.)|a| 4, which is positive definite. 24 In conclusion then* Outside the plane a=b the positive definite second order terms dominate;- in the plane a=b the second order terms vanish but the fourth order terms are posit-ive definite, so that the overall free energy surface i s posit-ive definite. Thus the Thomson heptagon must be stable. / 25 . CONCLUSION* Thomson's results concerning the stability of regular vortex polygons have been generalized to include the case of the polygonal center being occupied by a vortex of arbitrary relat-ive circulation. Of most signifigance is the fact that a large enough central circulation can stabilize a Thomson polygon of arbitrarly large N. This may be a helpful hint i n the search for stable configurations of large numbers of vortices. The efficacy of the free energy criterion has been demonstrated, particularly in cases where " f i r s t order" stability is indeterminate..The Thomson heptagon is stable. 26 NOTES 1. Onsager (1949) postulated the quantization of circulation for a superflow, which leads directly to the vortex concept. "Observationw refers to a large body of indirect experimen-tal evidence, a thoroughly referenced account of which may be found in Putterman (1974). 2. The author is indebted to D. Chapman for these references. 3. These integrals are given by Lamb (1932), although in a different but equivalent form. The f i r s t three constants are analogous respectively to center of mass, moment of inertia, and angular momentum. The last constant is named by way of analogy to the stream function of classical f l u i d theory. 4. In fact the general equations of motion (3.4) for a pertur-bed configuration may be cast in such form, see Tkachenko (1966). / 5. The configuration described by ^ C=0, N=2 is stable in the sense that i t is always in equilibrium , as solution of the equations of motion (1.2) w i l l readily verify. 27 LITERATURE CITED Chapman, D. 1977. M..Sc. thesis (University of British Columbia) Hansen, E. R. 1975. A Table of Series and Products Havelock, T. H. 1931. Phil. Magr. 11, 617 Lamb, K. 1932. Hydrodynamics, pp 219-224 Morton,, W. B. 1935. Procedings of the Royal Irish Academy XLII A, 21 Onsager, L. 1949. Nuovo Cimento Suppl. no.. 2 to Vol. 6, 249 Putterman, S. J. 1974. Superfluid Hydrodynamics (North-Holland Amsterdam) Putterman, S„ J. and Uhlenbeck„ G. E. 1969. Phys. Fluids 12, 2229 Thomson, J . J. 1882*. On the Motion of Vortex Rings, Adams Prize Essay,, pp 95-107 Tkachenko,, V» K. JETP 22_, 1282 and 23,, 1049 28 APPENDIX A: CALCULATION OF THE TRIGONOMETRIC SUMS S M The sums S M defined by equation (4.20) are easily reduced to — 1 -i(277/N)k,, i(277M/N)fe x g — \ -" vx—e ) -T (A.l) k = 1 2 [l-COS ( 277/10 k] N~l ' k=l -cos(277/N)k+cos(277tM-l)/N)k +i 2 [l-cos (277/TDk] rsin(2r7/N)k+sin(27T(M-I)/N)k t 2[l-cos(277/N)kJ J The imaginary part of (A.l) must vanish since cos(2vr/N)k = cos [(2?VN) (N-k)] and sin(27rp/N)k = -sin[(277P/N) (N-k)} (P an integer). The f i r s t part of the real sum in (A.l) i s given in Thomson (1882) (page 97), (A.2) cos(27r/N)k (N-1HN-5) T-fer 1-c k=l ^ c o s ( 2 7 r / N ) k 6 The latter part of the real sum in (A.l) is evaluated from a formula given in E.R. Hansen (1975) (equation (41.2.18) ), cos(27rp/N)k . r N > ; = Ncschx.csch^*coshU5-P+N k-zj coshx-cos(2n/N)k • I c )x R N-1,P=0,1,2,.... and £• In. ^' '"roundout function")=0 i f P<CN. 29 When the divergent k=N (for coshx=l) term is subtracted off then (setting P=M-1) .{A>a) ^ ,0swn-um* m r , 2 _ ^  l-cos(27r/N)k 12 6 Combining the results of (A.3) and (A.2) = | l . M ] 2 - ( . | - ! M H M l ) | (A.4) _ (M-2KM-N) 30 APPENDIX B: CALCULATION OF E AND L It is now useful to introduce the velocity potential for a single vortex (located at (a &,b k) ) CB..1) = Y k arctg x-a k so that (as is easily verified) ^ k *^k (B.2) v (due to vortex k) = — , v (due to vortex k) = — - . It is helpful to write E and L ("definitions (5.4) and (5.5) ) as contour integrals by use of Green's Theorem 1 L dx Ay J Fydy+Fxdx= I | - 7 * - party In (S is an area in the x,y plane and C is a curve enclosing i t ) , With straightforward calculation (B.3) L=5lL k , L ^ - ^ t^y^rdv (rdr=xdx+ydy) , (B the contour C is as shown on the next page. The small circles of radius £ surround each of the vortices present in the configur-ation. They exclude integration over the singular center of each vortex; £ i s considered to be very small, physically represent-32 ing the breakdown of ideal vortex flow at atomic dimensions. Any terms of f i r s t or higher order in € are considered neglig-. ible. The integrals around the small circles represent the ener-gy or angular momentum excluded from-consideration by the c i r c -les. Clearly the excluded energy or angular momentum due to vortices outside the £-circle must vanish when £ does (and is thus of f i r s t or higher order in 6 ) ; the only important contrib-ution is from the vortex inside. However, only configuration dependent quantities are relevant to the stability considera-tions for which these calculations are to be used. For L, the integral on the large circle obviously vanishes since dr=0 (on r=R). For E, said integral vanishes as R*-oo, Thus only the integrals on the straight lines yield important contributions. On x=av, dx=0, so that since b (x,y) = v. (x,y) , which go to - zero as R-ND. thus yielding for L (B.5) L*£>k ¥[2R2-2ak- (vo2- (v^2] If terms in € and the non-configuration dependent R term are dropped 33 (B. 6) L=-TTZ V4+bk) The corresponding calculation for E is also elementary, 4lo«l(vf-bj)2+(aj-ak)2|] -f&i°s (v«-bj»2+<arak)2|-llo«|-[<R2-ak'*-bjl +<arak): The f i r s t and last terms combine to yield zero in the limit R-*CD. When the sum E = ^ E - V is taken the terms j=k yield the jk J * "self energy" 21 log£.. The remaining terms are the j=k * relevant ones E = X' 8'kyj(-i)7rlog[(arak)2+(bJ-bk)2] (B.7) 


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