\u2022 '-->\u2022'\u2022*> A simple example of a stable B is a uniform field B = B z, where r o o p V B Q - O \" . The B ^ required to balance the Poincare force is displayed in Figure 5. For this calculation, B . ^ = 1, ti is that of the Crab (see Table 15 \u20143 ' II) and the total central density is 10 gm cm (the model of Figure' 4)\u2022 \u2022 ' This result for B ' can be easily scaled for other values of B . . , ti and

<0>. For the remainder of this thesis, only those B * 's which are stable in o the sense just explained are considered. It is then convenient to absorb the B ' required to balance the quasi-steady Poincare\" force into the definition of B q and to drop the quasi-steady Poincare force from the equations. The resulting equations are valid only over time scales short compared to the time over which ti varies. For the Crab this time is roughly ti\/ti - a thousand years (see Table II). Any Poincarg force due to precession with a period \u00ab ti\/ti should be retained in the equations as a term of the form -p ti x r if i t is not small compared to other retained terms, co p However, precession will not be considered in this thesis. 44 . The Viscous term The charged fluid experiences viscous effects due to electron-electron scattering. The charged fluid shear viscosity n in the absence of a magnetic field has been calculated by Heinzman and Nitsch (1972). Flowers and Itoh (1975) calculated the electron fluid shear viscosity n e which is related to n by n lp = n lp . The two results for n are in good agreement. The charged fluid kinematic shear viscosity V \u00a3 is defined as follows: 1 9 1D -9 9 _1 [11.61] v - n lp \u00ab.\u00b1r c = 3 x 10 Tn cm sec . c c c 3 e 7 -2 Note that the shear viscosity is a T . On the other hand, the Landau \"theory 2 of Fermi liquids predicts that the bulk viscosity is \u00ab T (Abrikosov and Khalatnikov, 1958), which vanishes in the extreme degeneracy limit. Accord-ingly, the bulk viscosity is neglected. Since V \u00a3 depends on T, which fluctuates slightly due to the fluid flow, v c experiences slight fluctuations. The effect of these viscous fluctu-ations on the flow is negligible i f the temperature fluctuations are small. The size of the temperature fluctuations is discussed in Chapter III. The estimate [11.61] for is based on the assumption that the magnetic field vanishes. This is a good assumption if the electron cyclotron fre-quency io is much smaller than the electron-electron collision frequency T\" 1. From [11.26] and [11.49] i t follows that [11.62] oo T = 106 B-.T\"2 \u00bb 1 , ce e 11 7 so i t is a poor approximation to neglect the effect of the magnetic field on the viscosity. This was noted by Heinzman et al.(1973). 45 In the limit <\u00ab> T \u00b0\u00b0> which applies to the charged fluid in view of.[11.62], the isotropic viscosity tensor with non-zero components - 'n for the B = 0* case is replaced by an anisotropic tensor. This tensor depends on the direction of B but not on its magnitude. Its non-zero components are s t i l l = n c (Chapman and Cowling, 1939). Thus, the order of magnitude 2 of the viscosity term is equal to n V \/X in both the u' x ~ > 0 limit and the OJ x \u2014> \u00b0\u00b0 limit. The ratio of the size of the viscosity term to the ce e J size of the Coriolis term is to within a factor of two the Ekman number E: [11.63] E = v \/Q X 2 3 x io-4 n~ 1T\" 2x~ 2 c o 2 \/ 0 where the dimensionless length scale X, is used (see Table III). Since o E is very small for X, - 1 and T., ^ .1, the viscosity should have only a o 7 small effect on large scale charged fluid flow within pulsars which are not extremely did. In such cases where the viscosity has l i t t l e effect the anisotropy in the viscosity will not have an Important effect on the motion. The superfluid drag Electrons in the charged fluid can scatter off the normal neutrons in the superfluid vortex cores, as can the small fraction of protons which are not superconducting. As a result, there is a drag force which tends to reduce the relative velocity v - v between the neutron superfluid and the charged fluid. For small relative, velocities, this drag force may be represented by a term P c ( v g ~ V c ) \/ T n t^ i e charged fluid momentum equation. Feibelman (1971) estimated the relaxation time required to reduce the relative velocity between the electrons and the neutrons, and showed that 46 the electron drag is larger than the proton drag. Baym et al.(1975) discuss the relationship of this relaxation time with the pqst-glitch relaxation time of pulsars. For reasonable values of the adjustable parameters, two of which are the temperature and the neutron energy gap size A, the theore-tically predicted post-glitch relaxation times can be the order of days (as in the Crab) or a year (as in the Vela). The fluid momentum equation has been discussed term by term, and a number of approximations have been made and justified. Given below are the resulting linearized equations: [11.64] |-(p v ) = -2ft xp v - Vp' - p'vV 3t co c o co c \"c c o; .-p v*' - (1\/4TT)B x(V, x B ' ) - (1\/4TT)B' x(V x B ) CO o o -v\" \u2022\"'\"a' + p (v - v ) \/ - r n . co s c D The Gravitational Source Equation The gravitational potential perturbation satisfies [11.65] V2*' = 4TTGP' . c This equation is coupled to the momentum equation through the term -pcQV$', which as previously noted is only about 1% of the size of the buoyancy term for long wavelength motions. In that case\", i t is a fairly good approximation to neglect the term -p^V*' and not to solve [11.65]. This is done in a l l the free oscillation calculations in this thesis. 47 \u2022 - ^ v The Magnetic Induction Equations The magnetic field evolves according to [11.66] I? = -cv\" x E \u00b0 t subject to the constraint [11.67] V \u2022 B = 0 . For processes happening so slowly that electromagnetic radiation can be neglected, the current J is given by the pre-Maxwell equation -h -> A-jr n T\u00bb ' 1 [11.68] V x B = - J c To close these equations, a relationship must be found between the electric field E and the other variables in the problem, such as and p' , i.e., a generalized Ohm's law appropriate to the charged fluid must be found. A standard procedure (Krall and Trivelpiece, 1973) used in forming a generalized Ohm's law begins with the momentum equations for the electron and proton fluids. Not a l l terms in these equations will be written out; those that are omitted are analogous to terms in the one-fluid equations which are small. A term that appears in the two-fluid equations which has no analogue in the one-fluid equations is the electron-proton collision term. The resulting linearized two-fluid equations are 3v [11.69] n m - ~ = -2ft x n m v - Vp' - n'm V$ oa a 9t o o a a a a a a o v ^ 6v + n q ( E ' + - ^ x B ) + n m - ^ coll. pa a c o oa a fi.t In [11.69], a = {e,p} labels electrons and protons. Subscripts 'o' refer to unperturbed quantities and primes refer to perturbations, as in the 4 8 o n e - f l u i d e q u a t i o n s . The symbols n , v , q and m r e f e r t o t he number d e n s i t y , v e l o c i t y , cha r ge and e f f e c t i v e mass, r e s p e c t i v e l y , o f p a r t i c l e s o f t ype a . Because t h e e l e c t r o n s a r e r e l a t i v i s t i c and the p r o t o n s n o n -* * * r e l a t i v i s t i c , m = ym and m - m . I n terms o f t h e s e v a r i a b l e s t h e p e r t u r -e e p p * b a t i o n s i n J , i n t h e cha r ge d e n s i t y and i n the c e n t r e o f mass v e l o c i t y a r e d e f i n e d by [11.70] J * =\\q n v \/ ^ot oct a a [11.71] p ' = ^ n ' q q \/_ a a [11.72] v = ^ \\ n m c p \/ ~\"oct~q V a c a * ft M u l t i p l y i n g [11 .69] by

; i z + \u2022 w n i s h e 8 i n t h i s limit. The total crust angular acceleration is the sum of a term ftQ due to quasi-steady external torques and any precessional motion, and a term ft' due to the shear stresses and the perturbations in the magnetic stresses communicated by the fluid interior: [ 1 1 . 9 6 ] ft = $ + ft' . o If the crust is treated as a spherical rigid body with a scalar moment of inertia I, then [ 1 1 . 9 7 ] ft' = (N^ + N^\/I . The power spectrum of the perturbations ft'(t) in the angular acceleration of the crust due to charged fluid flow will be used in Chapters V and VI to determine the periodicities in some computer simulated flows. 57 Figure 2. Charged f l u i d mass f r a c t i o n as a function of t o t a l density i n the f l u i d region. The d ensity i s i n grams. 58 59 Figure 3. Logs of squares of sound speeds as a fu n c t i o n of t o t a l density i n the f l u i d region. The change i n slope of the s 2 graph above I O 1 5 gm r e l a t i v i s t i c . 15 -3 \u00b0 10 cm i s due to the neutrons becoming more 60 CO o L d LU Q_ CO =) O CO Li_ O CO LU CC < ZD o CO LL. O 20.4, 20.0 19.6 19.2 18.8 CO O 18.4 - j 13.0 1 1 1 13.5 14.0 14.5 LOG DENSITY J 15.0 15.5 6 1 Figure 4. p*'\"'' and pK\u00a31 as functions of radius i n a r o t a t i n g neutron o o star model. P Q \u00b0 ^ ^ s t n e s p h e r i c a l l y symmetric part of the t o t a l mass density, p^ i s the r a d i a l part of the density perturbation due to rota t ion. The radius i s measured i n km. 62 63 Figure 5. Toroi d a l magnetic f i e l d perturbation required to balance the quasi-steady Poincare\" f o r c e . The unperturbed magnetic f i e l d i s assumed to be a uniform one p a r a l l e l to the r o t a t i o n axis 11 and to be 10 gauss. The r o t a t i o n axis i s v e r t i c a l ; the equator i s ho r i z o n t a l . The t o r o i d a l f i e l d i s given i n a contour p l o t . Contour plots with s i m i l a r l a b e l i n g conventions are used throughout the thesis f o r the azimuthal parts of B' and v . The maximum of the absolute value of the quantity ( i n t h i s case, max|B^|, which i s 4.93 x 10 5 gauss) i s given on each p i c t u r e . A value of +10 i s assigned to the point where t h i s maximum occurs i f the quantity i s p o s i t i v e there. Otherwise, -10 i s assigned to i t . MAX B A = 4 .93 x IO5 GAUSS 1 65 CHAPTER I I I : TYPES OF CHARGED FLUID WAVES In the l a s t chapter, the equations of the charged f l u i d dynamics were derived. The present chapter uses these equations to f i n d the general c h a r a c t e r i s t i c s of charged f l u i d waves. Dispersion Relations When the length scale A of a free o s c i l l a t i o n i n the charged f l u i d i s small compared to the distance R over which the equilibrium s o l u t i o n v a r i e s , d i s p e r s i o n r e l a t i o n s may be derived r e l a t i n g the frequency w and the l o c a l ->-wavenumber k of the o s c i l l a t i o n . Even when A - R, the dispersion r e l a t i o n s derived upon neglecting Inhomogeneities i n the equilibrium s o l u t i o n can serve as us e f u l guides i n determining the r e l a t i o n between the length and time scales of the o s c i l l a t i o n s . In the present section, such inhomoge-n e i t i e s are neglected and the charged f l u i d i s treated as i n f i n i t e i n extent. The motivation for t h i s somewhat crude treatment of the waves i n the f l u i d i s that i t enables one to determine e a s i l y the possible types of waves and th e i r prime c h a r a c t e r i s t i c s . I t i s assumed throughout t h i s thesis that the length scales A and time scales T of charged f l u i d disturbances are such that the conditions derived i n the previous chapter for the charged p a r t i c l e s to be treated as a f l u i d are obeyed. These conditions are s a t i s f i e d f o r an extremely wide range of A and T. In p a r t i c u l a r , they are e a s i l y s a t i s f i e d i n the case A = R. To a large extent, the work i n th i s section i s an adaptation to neutron stars of the general theory of waves i n r o t a t i n g e l e c t r i c a l l y conducting f l u i d s i n the presence of a magnetic f i e l d , which has recently been reviewed by Acheson and Hide (1973). When i t i s l i n e a r i z e d and s p a t i a l g r a d i e n t s i n p c Q a r e n e g l e c t e d , the mass c o n s e r v a t i o n e q u a t i o n becomes [ I I I . l ] s \" 2 | | \u2014 + V . v = 0 . co 9 t c The n o t a t i o n P ' = p ' \/ p has been u sed In [ I I I . l ] f o r t he r e d u c e d p r e s s u r e C CO p e r t u r b a t i o n . I s h a l l n e g l e c t s e v e r a l terms i n t he l i n e a r i z e d momentum e q u a t i o n . One i s t h e Po incare \" t e r m , w h i c h i s o m i t t e d b e c a u s e o n l y f r e e o s c i l l a t i o n s a r e o f i n t e r e s t h e r e ; . I s h a l l a l s o n e g l e c t -p 7*' and -p'v\"i|i . The g r a v i t a t i o n a l \u00b0 CO C O term - p c o $ * V was shown i n t he p r e v i o u s c h a p t e r t o be f a i r l y s m a l l . A l s o , s i n c e * ' s a t i s f i e s an e l l i p t i c e q u a t i o n , t h i s g r a v i t a t i o n a l te rm canno t be r e s p o n s i b l e f o r any new t y p e s o f waves . In t h e buoyancy te rm \" P ^ v ^ * ^ Q v a r i e s r o u g h l y l i n e a r l y w i t h d i s t a n c e to t he c e n t r e o f t he s t a r ( see C h a p t e r I V ) . T h u s , buoyancy can be p r o p e r l y t r e a t e d i n t h e d i s p e r s i o n r e l a t i o n a p p r o a c h o n l y i f A << R. However, i n t h i s s h o r t w a v e l e n g t h l i m i t , ~ P ^ ^ 0 i s r o u g h l y A\/R t imes t h e s i z e o f t he p r e s s u r e g r a d i e n t term - V p ' and i s thus n e g l i g i b l e . T h e q u e s t i o n o f the c e f f e c t s o f buoyancy on modes w i t h l e n g t h s c a l e s A = R i s a d d r e s s e d i n C h a p t e r s IV t h r o u g h V I f o r each c l a s s o f o s c i l l a t i o n i n d i v i d u a l l y . The damping terms ( v i s c o s i t y and s u p e r f l u i d d rag ) a r e t e m p o r a r i l y n e -g l e c t e d ; they a r e r e - i n t r o d u c e d i n a l a t e r s e c t i o n where t h e y a r e shown t o damp most o s c i l l a t i o n s o n l y l i g h t l y . The r e s u l t i n g h i g h l y s i m p l i f i e d momentum e q u a t i o n s a r e 3v [III.2] -r-S- = -2&n x \u00a3 - VP ' - T - T \u2014 B x (V x f \u2022\u2022)\u2022\u2022 . 3t o c Atrp o co The l i n e a r i z e d i n d u c t i o n e q u a t i o n i s [ I I I .3] |f- = V x ( v , x t ) . at c o 67 - > \u2022 - > ' \u2022 If a l l v a r i a b l e s vary i n time and space as r , then [III.4] -iw s ~ 2 P' + i l k \u2022 v = 0 C O c [III.5] -iojv = -2ft x v - ikP' - -.\u2014-\u2014- B x ( i k x B * ) c o c 4irp o co [ I I I . 6 ] -ioJB' = i k \u2022 B v - i t \u2022 v B O C C O Sound Waves An equation f o r P' may be derived by taking the sc a l a r product of k with [III.5] and using [ I I I . 4 ] : [III.7] ( k 2 - u) 2s\" 2)P' = -it \u2022 [-2ft x v - . 1 B x (ik x B')] . V co o c 4irp o \u2022 co \u2022\u2022; \u2022 Note than i n the l i m i t ft , B \u2014> 0, [III.7] reduces to the Fourier analyzed o o form of a wave equation f o r P'; i n the absence of r o t a t i o n and a magnetic f i e l d , the only kind of wave which can propagate i s a sound wave.with d i s -persion r e l a t i o n [III.8] a) - s k (ft = B = 0) . co o o If the magnetic body force and the C o r i o l i s force are s u f f i c i e n t l y small, sound waves with the dispersion r e l a t i o n [III.8] can propagate. The t o t a l a cceleration, C o r i o l i s a c c e l e r a t i o n and magnetic acceleration have 2ft 2 2 \u2014 2 sizes i n the r a t i o 1 : : V \u2022 k co , res p e c t i v e l y , where V. i s the Alfv\u00a3n OJ A r A speed: 68 If to = s kj then the three accelerations are in the ratio 1 : 2ft Is k : co o co 2 2 -2 -12 2 V ^ \/ S C Q - 1 : 10 ^2*6 : ^11' S\u00b0 t* i e c o n ^ i t i o n s necessary for the propagation of sound waves which are not much affected by rotation or the magnetic field are' [III.10] ft >, \u00ab 102 ;, B.r \u00ab 106 . These conditions are well satisfied unless the magnetic field or. the angular velocity are so large that they have a substantial effect on the equilibrium structure of the star. Such cases are not considered in this thesis. Chapter IV is.devoted to a fuller examination of sound waves in the charged fluid. Inertial Waves. And Hydromagnetic-inertial Waves 2 If ID \u00ab s k, then the left-hand side of [III.7] becomes k P' to a co 2 2 2 good approximation. Dropping to Is In comparison to k is equivalent to co replacing the wave equation for P' with an elliptic equation in which pressure effects are instantaneously propagated. In such an approximation, equation [III.4] becomes the incompressibility condition [III.11] ik \u2022 v - 0 c and [III.6] simplifies to [III.12] -iwf' = ik \u2022 B v . . o c Multiplying [III.5] by ik and using [III.11-12] produces [III.13] [(-ioo)2 - (V. \u2022 i t ) 2 ] it x v = (2ft \u2022 it)(-ito)v . A c o c Since the l i n e a r operation ( k x ) has the eigenvalues 0, l i k , of which the f i r s t i s excluded by the i n c o m p r e s s i b i l i t y condition, the dispersion r e l a t i o n which follows from [III.13] i s : (2ft \u2022 k) [III.14] O J Z + ~ OJ - (V \u2022 kr = 0 , which i s equation (4.11a) of Acheson and Hide (1973). A c h a r a c t e r i s t i c dimensionless parameter associated with [III.14] i s (2ft \u2022 k) , , [III.15] 5 = \u2014 \u00b0 \u2014 - 10 ft9Aft B..7 . 1 VT t 2 o i i kV. \u2022 k A Note that the estimate of the magnitude of t% i n [III.15] i s not accurate i f k and ft*Q are almost perpendicular. However, the estimate Is accurate f o r almost a l l d i r e c t i o n s of lc. In order to save space, statements such as these about the dependence of the s i z e of various quantities and e f f e c t s on the d i r e c t i o n of k w i l l be omitted f o r the remainder of t h i s chapter. If \u00a3 >> 1, then the roots of [III.14] s p l i t into two branches, widely separated i n frequency. The high frequency branch i s formed by i n e r t i a l waves: 2ft\" \u2022 \u00a3 [III.16] co = \u00b1 \u2014 \u00b0 k ^ 2 0 0 Q 2 s e c \" The lower frequency branch i s formed by hydromagnetic-inertial waves: (V. * t ) 2 k , 0 _ 9 _\u2022\u2022 _-, [III.17] OJ = t \u2014 = 10. B,, \\, ft0 sec ^ . 2ft \u2022 t 1 1 6 2 o Hide (1971) made an important contribution to the understanding of the charged f l u i d dynamics when he noted that \u00a3 i s generally >> 1 for o s c i l l a t i o n s with length scales A - R i f the magnetic f i e l d i s not very 12 much larger than 10 gauss. As a r e s u l t , two modes of the charged f l u i d are the i n e r t i a l modes and the hydromagnetic-inertial modes. 70 The parameter E, can be \u00ab 1 i f the magnetic f i e l d i s extremely strbng 4 -4 ^ B l l > : > 1 0 } o r i f t h e w a v e l e n 8 t n i s extremely small ' ( A << 10 ). Under such conditions r o t a t i o n has only a minor e f f e c t on the waves, which are e s s e n t i a l l y pure hydromagnetic (Alfveh) waves. In this thesis, I am pr i m a r i l y concerned with the less extreme conditions i n which \u00a3 \u2022>>'.!. I n e r t i a l Waves The periods of i n e r t i a l waves are t y p i c a l l y h a l f the r o t a t i o n period P, or s l i g h t l y longer: [III.18] \" T j . X \\ p \u2022 _4 Since the periods of sound waves are \u00a3 R \/ S C Q ~ 10 s e C > there i s no overlap between the frequency range of sound waves and that of i n e r t i a l waves i n neutron s t a r s . It follows from [III.12] that over the time scales T T the r a t i o of the magnetic energy of these waves to t h e i r k i n e t i c energy i s ; extremely small: [IH.19] - f ^ f - T 2 ~- IO\" 8 ft!2 A \" 2 B 2 - P v 2 6 11 2 co c The r a t i o of the magnetic body force to the C o r i o l i s force over time scales _2 i s also equal to $ . Under these circumstances the e f f e c t of the mag-netic f i e l d can be completely neglected. Hydromagnetic-Inertial Waves The periods of these highly d i s p e r s i v e waves can be extremely long: [III.20] = B\" 2 A 2 ft2 months. o If \u00a3 >> 1, there i s no overlap between the periods of these waves and the periods of i n e r t i a l waves. Unlike pure Alfv\u00a3n waves, the group v e l o c i t y 3to\/3k of hydromagnetic-inertial waves i s not n e c e s s a r i l y p a r a l l e l to The magnetic energy i n hydromagnetic-inertial waves i s very much larger than the k i n e t i c energy: I m . 2 1 ] I B ^ \/ | , , ? 2 \u201e 1 0 8 fi-2 n 2 x 2 _ 2 V c i n contrast to the i n e r t i a l waves [III.19] where t h i s r a t i o i s reversed. Over time scales T , the magnetic body forces are roughly the same size as the C o r i o l i s f o r c e : | [B x (V x B ' ) ] \/ 4 T T | [III.22] \u00b0 \u2014 ; \u00ab 1 , 12ft x p v I . ' O C O c 1 whereas the t o t a l force on a f l u i d element i s smaller than either by the 2 factor \u00a3 . Damping Due To V i s c o s i t y And Superfluid Drag For order of magnitude estimates of viscous damping times, the use of an i s o t r o p i c v i s c o s i t y tensor i s s u f f i c i e n t . For incompressible flow, iso-t r o p i c v i s c o s i t y contributes an extra term v \u00a3V to [ I I I i 2 ] . The e f f e c t of t h i s term on the dispersion r e l a t i o n i s to change equation [III.14] to 2ft\" \u2022 t + \u00b1\\ c [III.23] to2 + + . i v k 2 to - (V A \u2022 k ) 2 = 0 . For i n e r t i a l waves, i t follows from [III.23] that \u201e->- ->-2ft \u2022 k . [III.24] to = + \u2014 \\ i v k 2 . .. k c The viscous damping time f o r i n e r t i a l waves i s thus [III.25] T,(TI) = 1\/v k 2 = 10 A 2 T 2 sec. V c 6 7 It i s straightforward to show that the viscous damping time f or sound waves 2 i s also about l \/ v c k . For hydromagnetic-inertial waves, the dispersion r e l a t i o n which follows from [III.23] i s ( v . - k ) 2 k(V. \u2022 k ) 2 _ . [III.26] co = \u2014 ~ 2 \u00ab' + \u201e A V - U \" v k Z \u2022 +(2fi \u2022 k)\/k + i v k 2ft \u2022 k c o c o (HI) 2 (I) The viscous damping time f o r these waves i s thus about \u00a3 times T y : [III.27] T T ( 7 H I ) = 5 2\/v k 2 - 100 ft2 \\t B 7 2 T 2 years. The very long viscous damping time f or hydromagnetic-inertial waves when 2 -2 ^ i s >> 1 r e s u l t s from the f a c t that only a small f r a c t i o n \u00a3 of the energy i n these waves resides i n f l u i d motion. Equation [III.23] can also be used to determine the viscous damping times f o r waves with wavelengths so small that \u00a3 < K 1, i . e . when k >> ft^\/V^. -4-Under such conditions the fa c t o r l(2ft \u2022 k)\/k i n [III.23] may be dropped. o 2 The viscous damping time f o r such Alfven waves i s - 2\/v ck , which for 2 2 \u2014 6 2 2 \u2014 2 k >> ft \/VA i s << 2V.\/v ft = 10 B , . T., ft. sec. Such waves are thus damped o A A c o 11 7 2 i n a very short time. To analyze the e f f e c t s of su p e r f l u i d drag on the three basic types of waves, I assume that v = 0 i n the ro t a t i n g frame. These e f f e c t s may then 2 be determined by formally replacing v^k i n [III.23-27] by 1 \/ T Q - For example, the i n e r t i a l wave damping time due to su p e r f l u i d drag i s j u s t [III.28] T D I } = T D \u2022 For hydromagnetic-inertial waves the superfluid drag damping time is [III.29] T < H I ) = ? 2x D . Since x D can be days to years, T ^ H ' 1 \" ^ can be up to 108 years if \u00a3 = 104. 2 When T q >> l\/v ck , which is true for a wide range of conditions, superfluid drag is a less important damping mechanism than is viscosity. Sound waves can be damped by collisionless processes such as Landau damping, but for wavelengths longer than the electron mean free path, collisionless damping is insignificant (Pippard, 1955). Wave Damping By Heat Conduction Since the charged fluid is not completely degeneratej the mass density p- depends on the temperature. If the temperature is perturbed by T', there will be a density perturbation (9p \/9T) T', which is acted upon by gravity to produce a thermal buoyancy force -(9p \/9T) T'V$ .Because temperature C Pc \u00b0 perturbations which are created by charged fluid motion can be damped by heat conduction, the thermal buoyancy force can damp waves in the charged fluid. The appropriate form of the heat flux equation (Acheson and Hide, 1973) when linearized is [III.30] ct | r ' + v \u2022 VT I =

d i r e c t i o n with an extremely large v e l o c i t y shear near the crust, p a r t i c u l a r l y near the equator. Af t e r only two rotation s , v i s c o s i t y has not yet had time to reduce the shear. Apart from the shear which a r i s e s from the presence of no - s l i p boundaries, the computed s o l u t i o n i s p r a c t i c a l l y i d e n t i c a l to the s o l u t i o n [V.42-44] of the simple model equations [V.39-41], which do not take boundary conditions into account. This agreement i s consistent with the soundness of the numerical method. Because the numerical c a l c u l a t i o n s included buoyancy whereas the simple model [V.39-41] does not, the agreement between the two i s also consistent with the conclusion reached e a r l i e r i n thi s chapter that the buoyancy force has l i t t l e q u a l i t a t i v e e f f e c t on axisymmetric i n e r t i a l flow. The undulations which are apparent i n the contour i i n e s near the crust i n Figure 14 a r i s e because the numerical g r i d i s coarsest at these large r a d i i (see Figure 13) and because c o i n c i d e n t a l l y the shear i n v^ i s extremely large i n the same region. These are the most extreme conditions possible. Thus the s i z e of these a r t i f i c i a l wiggles gives some i n d i c a t i o n of the maximum numerical errors which can possibly a r i s e i n the numerical simulations. The c a l c u l a t i o n was not ca r r i e d further than two rotations because i t was apparent that nothing of i n t e r e s t was going to happen for a very long time. The question of what happens to the charged f l u i d i f the Poincare force i s \"turned on\" suddenly i s returned to i n Chapter VI, where the flow i 8 i s followed f o r a time - 10 times longer than i t was here. 110 Second Flow Simulation A second simulation was performed to see what type of charged f l u i d flow develops from a simple i n i t i a l v e l o c i t y condition. In th i s simulation, i t was assumed that the quasi-steady Poincare force was balanced by magnetic body forces, as explained i n Chapter I I : [V.45] Q = fi . The i n i t i a l v e l o c i t y f i e l d was chosen to be a simple azimuthal flow which vanishes on the crust and core surfaces: [V.46] V x(r,e,t=0) \u00ab (r - R ) ( r - R ) sine ,

> R \/v^, v i s c o s i t y would have slowed t h i s steady flow. The second part of the flow i s o s c i l l a t o r y i n character, with the amplitude of the meridional and azimuthal o s c i l l a t i o n s equal to about a half of the magnitude of the steady azimuthal flow. At .15 of a r o t a t i o n period a f t e r t = 0, a large scale meridional flow has developed, consisting of a large s w i r l centered near r = .75Rc> 6 = 3ir\/8, plus a small counterswirl near the core. The azimuthal flow i s developing a large shear near the crust. At .31 ro t a t i o n s , the counterswirl has grown. By .47 rotatio n s , the o r i g i n a l s w i r l i n the meridional flow has disappeared. I l l What remains i s the counterswirl, which i s now very large. The shear i n the azimuthal flow near the crust has begun to decrease, and by .78 r o t a t i o n s , the flow i s much l i k e i t was sh o r t l y a f t e r t = 0. The o s c i l l a t o r y part of the flow thus has a period of roughly .7 of a r o t a t i o n period. From the large s p a t i a l scale of the O s c i l l a t i o n , i t can be reasonably concluded that the o s c i l l a t i o n i s e i t h e r the fundamental mode or another one very close to i t i n frequency. The flow patterns at about 11 and 16 r o t a t i o n periods give no i n d i c a t i o n of the f l u i d switching to another mode. To show more c l e a r l y the gross features of the flow, the angular accelerations of the crust and the core due to f l u i d shear stress are plotted i n Figure 16. The accelerations of the s o l i d parts of the star have two components, one a steady p o s i t i v e a c c e l e r a t i o n due to the steady part of the flow i n the + -> applying the operator \"(V + ViJ> ) x\" to them. The result is o [VI.17] (v\" + V> ) x (-z x u) = -(v\" + Vifi ) x Q . o o Note that although p^ does not appear in [VI.17], the effects of buoyancy are s t i l l taken into account through the terms involving v\\ji . . The

Q vanishes. This eliminates the arbitrariness associated with the possible presence of geostrophic flow (Greenspan, 1968). This arbitrariness arises from dropping the 9(p c V c)\/3t term in the momentum equation, and is analogous to the arbitrariness in the pressure solution in incompressible hydrodynamic flow. In incompressible flow, an arbitrary constant may be added to the pressure solution. There is a simple yet instructive example of a solution to [VI.15-18] for the case in which 5 q is a uniform magnetic field in the z direction, and in which the magnetic body force is linearized. For this case, the charged fluid velocity is given by B \\ [VI.19] p v CO C \\87Tft , O\/ V x B which clearly satisfies the anelastic condition. The flow field [VI.19] has a number of unusual features. The fluid velocity is everywhere parallel to the current J ' . The linearized Coriolis force 2ft x p v and the o co c linearized magnetic body force - ^ Q x (V x exactly cancel so that in the linear approximation the pressure perturbation p^ vanishes everywhere. 145 These features are possible because B q and fiQ are parallel, and will probably not be present in the non-aligned case. -v ->-Note that for an arbitrary B', the charged fluid velocity v c given in [VI.19] does riot necessarily satisfy the no-slip boundary conditions [VI.5]. Thus there can be a boundary layer, whose existence seems to be in contra-diction with statements made in Chapter III that there are no oscillatory hydromagnetic-inertial boundary layers where B \u2022 n[^ ^ 0. However, there \/is no contradiction because any such boundary layers are transient. An - . -y extremely large shear in v c near E will cause B' to grow in such a way that the shear in v^ is rapidly reduced. The Charged Fluid Response Time Problem One of the essential simplifying approximations which have been made in two-component models of post-glitch timing behaviour of pulsars (Shaham et al,, 1973; Baym et al., 1975) is that the charged fluid co-rotates with the crust following a glitch. In support of this approximation, i t has been argued (Baym et al., 1969c) that within the time i t takes ah Alfve*n wave to cross the star,, the charged fluid can adjust to the new angular velocity of the crust. Since this crossing time is small compared to the post-glitch relaxation time, i t is argued, one may safely neglect the internal motions in the charged fluid and simply assume i t co-rotates with the crust. However, I showed in Chapter III that unless the magnetic field is -1 -A so large that &2 is \u00ab 10 (i.e. that E, \u00ab 1 for A - R), the waves which couple strongly to the magnetic field are hydromagnetic-inertial waves, which behave like Alfven waves only i f A \u00ab R. Such very short wavelength Alfven waves are generally damped by viscosity in very much less than a second, and so cannot propagate across the star. Thus, the assumption which has been made in two-component models that the charged fluid as a whole responds to a disturbance in the Alfven wave crossing time is not well founded, unless B q is ^ 10 gauss. What then is the charged fluid response time? By definition, i t is the time for the fluid as a whole to adjust after the magnetic field lines have been sheared. Thus, i t is probably the time scale of the longest wave-length oscillations that are strongly coupled to the magnetic field, i.e., roughly the period T\u201e of hydromagnetic-inertial modes with X R. This Hi period can be the order of a month, for example, if B^ is roughly - 1 within the star. The observed post-glitch relaxation time for the Crab is several days (Boynton et al., 1972). Thus, the charged fluid response time need not be small compared to the post-glitch relaxation time. The magnetostrophic equations can be used to determine the behaviour of the charged fluid in response to changes in the angular velocity of the crust. One possible computer simulation involves a sudden change in the angular velocity of the crust (a glitch). A second possible simulation involves a sudden change in the angular acceleration of the crust. In this chapter, the results of a computer simulation of the second kind are reported. The choice of this simulation over a glitch simulation was based on numerical convenience, and not because of a lack of interest in the details of charged fluid post-glitch flow. However, the charged fluid response time, which is of great interest, should be almost the same in the chosen simulation as in a glitch simulation. 147 The Magnetic Field Structure The charged fluid flow which develops following a disturbance depends on the general structure and strength of the magnetic field in the Interior. The choice of magnetic field structure and strength in any magnetostrophic calculations must be guided by a number of considerations including pulsar observations, theories of neutron star formation, magnetohydrodynamic stability and simplicity. One can learn very little about the interior magnetic field from pulsar observations. Its mean value < B > is probably << 10^ gauss: if < B > were o \u00b0 o ^ 10 gauss, the star would be so distorted that the main energy loss mechanism would be the emission of gravitational radiation, in disagreement with timing observations (Ruderman, 1972). As regards the field structure, > 10 I deduced from pulsar slowing down rates that if B q . ^ 10 gauss at the inner crustal surface, the magnetic field there is almost purely poloidal or almost purely toroidal (see [11.25]). Theories of neutron star formation are of little help in determining the interior field structure and strength. The value of

R\u00a3 excluded.1 The simulation ran for 125 days of simulated time. There are two possible descriptions of the i n i t i a l disturbance which are equivalent in the sense that the flow which follows is the same in both cases. In the first description, the magnetic field is uniform and the star is not decelerating prior to the in i t i a l time t =0. At t = 0 a steady external torque is suddenly applied to the crust, i.e., the Poincare force is \"turned on\". In the second description, the star is decelerating and the magnetic field B q is nearly uniform prior to t = 0. The non-uniformity in B q arises from the relatively small toroidal field required to balance the quasi-steady PoincarS force. This toroidal field was calculated in Chapter II and is displayed in Figure 5. At t = 0, the small balancing toroidal field is \"turned off\", that is, the i n i t i a l perturbation B ' ( t = 0) in the magnetic field is the negative of the field in Figure 5. This i n i t i a l perturbation B ' ( t = 0) clearly satisfies the Maxwell equation [VI.2] and the magnetic field boundary condition [VI.6]. non-linearity in the subsequent flow is of order B ' \/ B , which is o ee [11.59]). If terms of this size had been neglected compared to 1 Appendix C for further details of the grid and the numerical methods terms of order 1, which they were not i n the actual c a l c u l a t i o n , the r e l a t i o n -ship between vV_ and B' would be that given i n [VI. 19]. Thus to a f i r s t approximation, the properties of the flow f i e l d v c i n [VI.19] hold for the simulated flow. In p a r t i c u l a r , the pressure perturbation p^ ar i s e s only from the small n o n - l i n e a r i t i e s , and thus buoyancy has very l i t t l e e f f e c t on the f l u i d flow. Basic Time Scales Of The Flow In order to show more c l e a r l y some of the basic time scales of the flow, the perturbation i n the c r u s t a l angular a c c e l e r a t i o n due to viscous stress i s displayed i n Figure 18. An i s o t r o p i c viscous tensor was chosen, as i n 11 2 -1 Figure 16, and V \u00a3 was taken to be 2 x 10 cm sec , which corresponds to a temperature T_ - .4. Any other value of v could have been chosen, or the 7 c anisotropy i n the v i s c o s i t y taken into account, or the magnetic torque f l u c t u a t i o n s included, but none of these would have changed the time scales. I t w i l l be shown l a t e r that the most prominent mode i n the flow simula-t i o n has a wavelength X^ = .3. Using ti^ = 1.9 and T^ = .4, equation [VI.7] can be used to estimate that N'\/N' i s - 20 for t h i s flow. The t o t a l pertur-B V bation ft' i n the.crust angular a c c e l e r a t i o n i s therefore - 20 times the -13 -2 acceleration shown i n Figure 18, i . e . , ft' i s - 4 x 10 sec , which i s -4 . about 10 times the s i z e of ft .. Thus, the f l u c t u a t i n g Poincard force -*\u2022. -y -P cft x r i s r e l a t i v e l y n e g l i g i b l e , and i t was a good approximation not to include i t i n the f l u i d momentum equation [VI.4]. The most s t r i k i n g feature i n Figure 18 i s the o s c i l l a t i o n s i n ft'(t). There appears to be one predominent period of about a day. On closer inspection, i t can be seen that there are two quite d i f f e r e n t phases to ft'(t). The f i r s t phase, which I c a l l the charged f l u i d response phase, 154 lasts a few days after t = 0. It is distinguished by a positive time average of ft'. During this phase the charged fluid responds to the crustal slowing down in such a way as to oppose i t . The charged fluid response time in this simulation is thus a few days, which is comparable to the observed post-glitch relaxation time for the Crab pulsar (Boynton et al., 1972). The important point is not that the two times are almost equal, but that the charged fluid response time is not very much smaller than the post-glitch relaxation time. ' \u2022 \u2022 \u2022 * . The charged fluid response phase is followed by a phase in which fi' fluctuates about a mean of zero. In the latter phase the fluid oscillates about a state of co-rotation with the decelerating crust. The oscillations are undamped because viscosity and superfluid drag have been neglected in the fluid equations. A power spectrum analysis of the crustal acceleration graph in Figure 18 was performed in order to examine more quantitatively this periodicities in the flow.1 The resulting power spectrum is given in Figure 19, which shows a large number of well defined peaks. These peaks are not the result of fluctuations at a single point in the discrete Fourier analysis, because each is composed of at least several points (there; are 2,000 points in the discrete frequency domain). There can be no doubt that the peaks are not the result of numerical \"leakage\" from the main peak at a period of 1.2 days. If there were a significant amount of such leakage, one would expect i t to produce peaks in the frequency region > 25 rad\/day, where there are no peaks. In fact, there should be no peaks in this region because waves of that high a frequency cannot propagate on the computational grid, whereas waves whose frequencies are lower than this can. 1 The program EOURT from the U.B.C. Computing Centre was used in the Fourier analysis. The original 50,000 data points were averaged in blocks of 25 prior to Fourier analysis. The power spectrum was smoothed with a running average of 3 points. I thank G. Fahlman for recommending this procedure to me. It is thus quite certain that these peaks are not art i facts of the Fourier analysis. The peaks are due to the osc i l la t ion of the charged f lu id in hydromagnetic-inertial modes with frequencies equal to the frequencies at the peak maxima. Some of the prominent modes have periods of about .30, .60, .96, 1.2, 2.4, 3.8, 4.9, 6.3, 10 and 13 days. These normal mode periods were computed for, values of = ^\u00bb ^2 = ^\"* 9 , a n < * ^c = ^ a n\" The periods for other magnetic f i e ld strengths, crustal angular velocit ies and inner crustal rad i i may be obtained by scaling, because i t follows from [III.20] -2 2 that a l l normal mode periods are \u00ab \u00a31^ t R c C'^111)\/8.9 ] . Details Of The Flow Pictures of the charged f lu id velocity V \u00a3 and the perturbation B' in the magnetic f i e l d at a number of different times are given in Figure 20, which extends over many pages. Each picture is labeled with the time in days at which the situation is depicted. The meridional and azimuthal parts of v \u00a3 are separately displayed, as are the poloidal and toroidal parts of B' It was previously noted that B'(t = 0) is the negative of the toroidal f i e ld depicted in Figure 5, so i t is unnecessary to provide a separate picture of i t in Figure 20. The i n i t i a l azimuthal flow v^Ct = 0) vanishes, so only the meridional component of v (t = 0) i s displayed. Since the magnetic body force Q at t = 0 is equal to ~ P C ^ 0 x r by the the choice of i n i t i a l conditions, i t follows from [VI.15] that [VI.22] v x ( t = 0) . o The z component of v ' ( t = 0) can be determined from [VI.16]. Because is 156 smaller at larger r a d i i , v (t = 0) i s larger near the crust than near the z centre. These features of v \u00a3 ( t = 0) are displayed i n the f i r s t picture of Figure 20. Because v c ( t = 0) does not satisfy the no-slip boundary condition, a thin transient boundary layer is formed at the crust * After .06 days, an azimuthal flow v^ has begun to develop through the action of the Coriolis force on the meridional flow v . This azimuthal flow is in such a direction as to oppose the crustal slowing down. At this time v, i s s t i l l small in comparison to the meridional flow. Because of the large shear near the crust in the i n i t i a l meridional flow, the developing azimuthal flow has a similar shear. The magnetic f i e l d perturbation has a non-zero poloidal part B*^ at .06 days. It can be seen that the component of B^ normal to the crust vanishes as required by the magnetic f i e l d boundary condition [ V I .6]. On the other hand, i t can be seen from the pictures of B' in Figure 20 that the tangential component of B at any point on the crust i s not fixed, because the boundary conditions in the limit a -*-'\u00ab> do not place any restrictions on n x B' ] ^. BV l i e s nearly parallel to the unperturbed magnetic f i e l d because 9B' 9v 9v . [VI.23] r \u2014 - B -r\u2014^- - B -\u2014\u2014 z , 9t o 9z o 9z ' where the dependence of v^(t = 0) only on x (see [VI.22]) has been used in the last step of [VI.23]. As a result, |9B'\/\u00b0t| i s << |9B'\/9t| for some X z time after t =0. Since B z = B x = 0 i n i t i a l l y , |Bx| is << |B^| during this time. After a quarter of a day, and B^ have grown larger. The largest growth i n v^ i s near the crust at the equator since the unperturbed magnetic f i e l d is tangent to the crust there and as a result large shears in the x direction cannot be effectively resisted there. 157 The r e g i o n i n w h i c h v , and B l have l a r g e s h e a r s i s l a r g e r than i t was a t .06 d a y s . T h e r e i s a s m a l l r e g i o n n e a r the r o t a t i o n a x i s i n wh i ch v , i s \u2022

d i r e c t i o n , and ft' i s > 0. The p o l o i d a l and t o r o i d a l components o f each o f v \u00a3 and B ' a r e by now comparab le i n m a g n i t u d e . A f t e r 1.5 d a y s , t h e cha r ged f l u i d i s n e a r i n g the end o f i t s r e s p o n s e phase bec au se the l o n g e r w a v e l e n g t h waves have begun t o r e a c h the c e n t r e . The s i t u a t i o n a t t he l a s t day o f the s i m u l a t i o n , day 125, d i s p l a y s c h a r a c t e r i s t i c s w h i c h were p r e s e n t i n t h e f i r s t two days o f t h e f l o w , i n c l u d i n g the s m a l l n e s s o f |B ' | compared to |B ' | and the a l i g n m e n t o f the c o n t o u r s o f e q u a l v ^ and B^ a l o n g t h e d i r e c t i o n o f B * q . The l e n g t h s c a l e 158 for changes in v c and B' is about Rc\/3, so that the period of the oscillations 2 '\" should be about (1\/3) times the period for longest wavelength waves. This is the order of days, which is compatible with the calculated predominant period of 1.2 days. At 125 days, there are indications of waves with, wavelengths which are very small compared to R^ . Such small scale structures would not be present after 125 days if viscosity had been taken into account and if T^ were ^ 1, because the damping times of such waves can be the order of a few days or (HI) less (see [111.27]). Since is an extremely sensitive function of A, the large scale structures present at 125 days would probably persist for several years. The results of the computer simulation can be summarized and interpreted as follows. The sudden onset of the crust deceleration shears the magnetic field lines near the crust, particularly near the equator; This generates a variety of hydromagnetic-inertial waves. Because the shorter wavelength waves travel more quickly, they reach the stellar interior after only a fraction of a day, at which time the longer wavelength waves are s t i l l near the crust, widening the shear region. After several days, the longer wave-length waves reach the interior. At this time, the charged fluid is oscil-lating about a state of rigid rotation with the decelerating crust. The oscillations persist for some time thereafter. 159 Figure 18. Perturbation i n crust angular a c c e l e r a t i o n due to viscous torque. The perturbations i n the crust angular a c c e l e r a t i o n due to v a r i a t i o n s i n the magnetic torque induced by the f l u i d flow are larger than the v i s c o s i t y induced perturbations. However, i t i s not the amplitude of the perturbations but only t h e i r character as a function of time that i s of i n t e r e s t here. There are two d i s t i n c t phases apparent i n s i ' ( t ) . One i s a charged f l u i d response phase l a s t i n g a few days i n which the time average of fi'(t) i s p o s i t i v e . This i s followed by a second phase, i n which the time average of fi'(t) i s zero. 161 Power spectrum analysis of crust angular a c c e l e r a t i o n data i n Figure 18. The zero of the v e r t i c a l scale i s a r b i t r a r y . Peaks i n the spectrum correspond to frequencies to of hydro-ma g n e t i c - i n e r t i a l normal modes of the charged f l u i d . The 2 -1 -2 frequencies to are \u00b0= tt^ [R (km)\/8.9] , and these were computed f o r B.. - 1, ft = 1.9, and R (km)\/8.9 = 1. The 11 z c lack of structure at frequencies above 25 rad\/day i s a numerical a r t i f a c t due to the f i n i t e g r i d s i z e . 163 \\ Figure 20. Magnetostrophic flow of charged f l u i d induced by c r u s t a l d eceleration. In these p i c t u r e s , the charged f l u i d v e l o c i t y v^ and the perturbation B' i n the magnetic f i e l d are displayed at various times. Both v\"c and B*' are s p l i t i nto p o l o i d a l parts v , B*' and t o r o i d a l parts v., B', which are separately p p 9

> 3 T^ cm (the electron mean free path) and i f the -10 -2 time scale T i s >> 10 T^ sec (the electron mean-free time). Under such conditions, q u a s i - n e u t r a l i t y i s an excellent approximation. The magnetic f i e l d evolves i n time according to the i n f i n i t e conductivity induction equation. The creation or destruction of charged f l u i d by the acti o n of the weak i n t e r a c t i o n on density perturbations i s completely n e g l i g i b l e , and the usual mass conservation equation holds. Off-diagonal terms i n the material pressure tensor due to f i n i t e Larmor radius e f f e c t s are r e l a t i v e l y small. Other r e l a t i v e l y small terms i n the f l u i d momentum equation are the viscous term and the s e l f - g r a v i t a t i o n a l term. To a very good approximation, the heat f l u x equation i s decoupled from the other equations. To my knowledge, there has been no previous attempt to answer t h i s f i r s t set of questions. What types of small amplitude waves can e x i s t i n the charged f l u i d ? What can damp them? How much are they damped? How do the answers to these questions depend on the gross parameters of the problem, such as Q, B and T? ' ' 182 The answers to these questions are given i n Table X. The most s i g n i f i c a n t r e s u l t s are the long periods of hydrbmagnetic-iriertial waves and t h e i r extremely long damping times. Two a d d i t i o n a l damping mechanisms not l i s t e d i n the table are heat conduction and e l e c t r i c a l r e s i s t i v i t y , which are s i g n i f i c a n t only i f the star i s so young that T^ i s >> 1. The possible existence of i n e r t i a l waves and hydromagnetic-inertial waves ins i d e neutron stars was noted by Hide (1971). However, he did not consider the question of damping or investigate further the magnetohydro-dynamics of the charged f l u i d . 3. What kinds of mode-mode coupling can occur? Can t h i s weakly non-linear e f f e c t lead to heavy damping of a wave which i s only l i g h t l y damped i n the l i n e a r approximation? The second of these questions concerns hydromagnetic-inertial waves. If such a wave can resonantly couple to two other waves, one of which i s a sound wave or an i n e r t i a l wave, the p o s s i b i l i t y should be considered that energy can be transferred to the sound or i n e r t i a l wave and dissipated i n a time short compared to the period of the hydromagnetic-inertial wave. I found that the only such resonant coupling involves a hydromagnetic-i n e r t i a l wave and e i t h e r a p a i r of i n e r t i a l waves or a p a i r of sound waves. However, I showed that resonant energy transfer was not possible i n t h i s s i t u a t i o n . 183 Table X Charged F l u i d Waves And Their C h a r a c t e r i s t i c s TYPE PERIOD VISCOUS DAMPING TIME SUPERFLUID DRAG DAMPING TIME SOUND 10~ A X\u00a3 sec o 2 2 10 X, T^ o 7 sec days INERTIAL1\" 1 r o t a t i o n 2 period 2 2 1 0 X 6 T 7 sec days HYDROMAGNETIC-INERTIAL1\" B U X 6 \" n 2 months 100 B~ 2 ft2 years 4 2 X6 T7 106 ft2 x2 v years NON-DIMENSIONALIZED VARIABLES NOTATION WAVELENGTH X, = X(cm)\/10 6 CRUST ANGULAR SPEED Q2 = ft(rad\/sec)\/102 MAGNETIC FIELD B l l \"-^(sauss)\/^ 1 INTERIOR TEMPERATURE T ? = T(\u00b0K)\/10 7 SUPERFLUID DRAG TIME* x 1 = T^day)\/!. It i s assumed above that.B \u2022 ft. X, >> 10 11 2 6 TD d e f i n e d by;drag f o r c e \/ u n i t volume on charged f l u i d c s c D 184 4. What are some normal modes of oscillation of the charged fluid? I calculated some of the sound modes and their eigenfrequencies in the inviscid approximation. Neither the variations in the charged fluid sound speed nor the buoyancy force had any qualitative effect on the modes. I calculated some inertial eigenmodes and eigenfrequencies in a crude representation of the interior consisting of two co-axial rotating cylinders of finite length with a gravitational field perpendicular to the rotation axis. Viscosity was neglected. A l l axisymmetric modes were stable and relatively unaffected by the gravitational buoyancy force. I also calcu-lated a number of non-axisymmetric modes which are independent of the distance along the rotation axis. The frequencies of these buoyancy-inertial modes, as I call them, become zero if the gravitational field is turned off 5. How do charged fluid oscillations affect the crust? What is the relationship, i f any, between the charged fluid and the quasi-sinusoidal residuals? The motion of the charged fluid can change the angular velocity of the crust either through the viscous torque or through variations in the magnetic torque. If the fluid is oscillating at frequencies of inertial waves or higher, the viscous torque is the more effective of the two mechanisms. At hydromagnetic-inertial frequencies, the fluctuations in the magnetic torque are generally more effective. It appears unlikely that charged fluid oscillations can account for the residuals. A mean interior magnetic field of ^ 2 x lO 1^ gauss would be required. Furthermore, there seems to be no natural way that hydro--4 magnetic-inertial modes can produce an to timing residual power spectrum. 185 6. How does the charged f l u i d respond under a v a r i e t y of i n i t i a l conditions and external forces? In p a r t i c u l a r , how does i t respond and how quickly does i t respond to changes i n the crust angular v e l o c i t y ? ; I simulated the charged f l u i d flow over two quite d i f f e r e n t time scales on a computer to see how the f l u i d responds. Two simulations were performed i n which the f l u i d motion was followed over the time scales of i n e r t i a l waves. In one of these simulations, the charged f l u i d i n i t i a l l y has a small azimuthal flow as seen i n a reference frame co-rotating with.the cru s t . A meridional flow develops through the action of the C o r i o l i s f orce, and the f l u i d begins to o s c i l l a t e p r i m a r i l y i n an i n e r t i a l mode with a period - .75 times the r o t a t i o n period. One i n t e r e s t i n g feature of the flow i s a type of Taylor-Proudman column p a r a l l e l to the r o t a t i o n axis above a s o l i d core which o s c i l l a t e s r e l a t i v e to the crust. I also simulated the f l u i d motion over the long time scales of hydro-magn e t i c - i n e r t i a l waves. I t i s only over these time scales that the magnetic f i e l d and the charged f l u i d are strongly coupled. P r i o r to the i n i t i a l time t = 0, the magnetic f i e l d is. a uniform 11 10 gauss f i e l d p a r a l l e l to the r o t a t i o n axis. The star i s not decelerating, At t = 0, a steady external torque i s applied to the crust so i t decelerates s t e a d i l y . The angular v e l o c i t y and deceleration chosen were those of the Crab pulsar-Although the simulation performed i s not a g l i t c h simulation, the response time of the charged f l u i d should be roughly the same i n both cases. It takes several days for the longer wavelength (A = R \/3) hydromagnetic-i n e r t i a l waves formed at the crust near the equator to reach the centre of the s t a r . At the end of t h i s time, the charged f l u i d i s o s c i l l a t i n g about a state of co-rotation with the decelerating crust. The observed p o s t - g l i t c h 186 r e l a x a t i o n time f o r the Crab pulsar i s sdveral days. The major conclusion of my thesis i s that the charged f l u i d response time i s not necessa r i l y very much shorter than p o s t - g l i t c h r e l a x a t i o n times. In view of t h i s conclusion, simple two-component models of p o s t - g l i t c h behaviour which assume that the charged f l u i d response time i s much shorter than post-g l i t c h r e l a x a t i o n times need to be re-examined. 187 BIBLIOGRAPHY Abrikosov, A.A. and Khalatnikov, I.M. Sov. Phys. Usp. 1, 68 (1958) Acheson, D.J. and Hide, R. Rep. Prog. Phys. _36, 159 (1973) Bahcall, J.N. and Wolf, R.A. Phys. Rev. 140, B1452 (1965) Batchelor, G.K. Quart. J . Roy. Meteorol. Soc. 29, 224 (1953) Baym, G., Lamb, D.Q., and Lamb, F.K. \"Influence of External Torques on the the Wobble Motion of Two-Component Neutron Stars\", preprint (1975) Baym, G., Pethick, C , and Pines, D. Nature 224, 673 (1969a) . Baym, G., Pethick, C , and Pines, D. Nature 224, 674 (1969b) Baym, G., Pethick, C., and Pines, D. Nature 224, 872 (1969c) * Baym, G., Pethick, C , and Sutherland, P. Ap. J . 170, 299 (1971) Baym, G. and Pines, D. Ann. Phys. 66_, 816 (1971) Boynton, P.E., Groth, E.J., Hutchison, D.P., Nanos, J r . , G.P., Partridge, R.B., and Wilkinson, D.T. Ap. J. 175, 217 (1972) Canuto, V. and Chitre, S.M. Phys. Rev. D 9, 1587 (1974) Chanmugan, G. M.N.R.A.S. 169, 353 (1974) . ' Chapman, S. and Cowling, T.G. The Mathematical Theory Of Non-Uniform Gases, Cambridge U n i v e r s i t y Press (1939) Cohen, J.M. pp. 87-95, R e l a t i v i t y And G r a v i t a t i o n , Gordon and Breach (1971) Drake, F. Talk at 61\"*1 Texas Symposium, New York (1972) Feibelmah, P.J. Phys. Rev. D 4_, 1589 (1971) Flowers, E. and Itoh, N. \"Transport Properties of Dense Matter\", preprint (1975) Fujimoto, M. and Murai, T. Publ. Astron. Soc. Japan 24, 269 (1972) Ginzberg, V.L. Sov. Phys. Uspekhi 14, 83 (1971) Greenspan, H.P. The Theory Of Rotating F l u i d s , Cambridge U n i v e r s i t y Press (1968) Greenstein, G. and McClintock, J.E. Science 185, 487 (1974) Groth, E.J. \"Timing of the Crab Pulsar I I . Method of Analysis\", preprint (1975a) 188 Groth, E.J. \"Timing of the Crab Pulsar I I I . The Slowing Down and the Nature of the Random Process\", pr e p r i n t (1975b) Harlow, F.H. and Welch, J.E. Phys. FI. 8_, 2182 (1965) Harrison, B.K., Thorne, K.S., Wakano, M., and Wheeler, J.A. G r a v i t a t i o n a l Theory And G r a v i t a t i o n a l Collapse, Univ. of Chicago Press (1965) Heinzman, H., Kundt, W., and Schrufer, E. Astron. and Astrophys. 27, 45 (1973) Heinzman, H. and Nitsch, J . Astron. and Astrophys. 21, 291 (1972) Hide, R. Nature Phys. S c i . 229, 114 (1971) K r a l l , N.A. and T r i v e l p i e c e , A.W. P r i n c i p l e s Of Plasma Physics, McGraw-Hill (1973) Landau, L.D. and L i f s h i t z , E.M. Electrodynamics Of Continuous Media, Pergamon Press (1966) Landau, L.D. and L i f s h i t z , E.M. S t a t i s t i c a l Physics, Pergamon Press (1969) Le Blanc, J.M. and Wilson, J.P. Ap. J . 161, 541 (1970) Malkus, W.V.R. Mathematical Problems In The Geophysical Sciences, p. 207, Amer. Math. Soc. (1970) Manchester, R.N. and Taylor, J.H. Ap. J . 191, L63 (1974) Manchester, R.N., Taylor, J.H., and Van, Y.Y. Ap. J . 189, L119 (1974) Markey, P. and Tayler, R.J. M.N.R.A.S. 163, 77 (1973) Nelson, J . , H i l l s , R., Cudaback, D., and Wampler, J . Ap. J . 161, L235 (1970) Ostriker, J.P. and Gunn, J.E. Ap. J . 157, 1395 (1969) Pandharipande, V.R. and Smith, R.A. Nuc. Phys. A237, 507 (1975) Pippard, A.B. P h i l . Mag. 46, 1104 (1955) Reichley, P.E. Talk at 6 t h Texas Symposium, New York (1972) Roache, J.P. Computational F l u i d Dynamics, Hermosa Pu b l i c a t i o n s , Albuquerque New Mexico (1972) 189 Roberts, D.H. and Sturrock, P.A. Ap. J . 173, L33 (1972) Ruderman, M. Nature 223, 597 (19o9) Ruderman, M. Nature 225, 619 (1970) Ruderman, M. Ann. Rev. Astron. Astrophys. 10, 427 (1972) Ruderman, M. and Sutherland, P.G. Nature Phys. S c i . 246, 93 (1973) Ruderman, M. and Sutherland, P.G. Ap. J . 190, 137 (1974) Sagdeev, R.Z. and Galeev, A.A. Nonlinear Plasma Theory, W.A. Benjamin (1969) Shaham, J . , Pines, D., and Ruderman, M. Ann. of the N.Y. Acad. S c i . 224, 190 (1973) Tayler, R.J. M.N.R.A.S. 161, 365 (1973a) Tayler, R.J. M.N.R.A.S. 162, 17 (1973b) ter Haar, D. Physics Reports 3_, 57 (1972) Tsuruta, S. and Cameron, A.G.W. Nature 207, 364 (1965) Vandakurov, Yu.V. Ap. L e t t . 5., 267 (1970) Vandakurov, Yu.V. Sov. Astr. 16, 265 (1972). 1 9 0 APPENDIX A;, NUMERICAL METHODS FOR SOUND WAVES In b r i e f , the d i f f e r e n t i a l equation f o r was solved by approximating i t by a f i n i t e matrix eigenvalue problem. The i n t e r v a l [a,l] was divided into n + 1 equal i n t e r v a l s of length Ar: [A.l] ( i ) r = a + iAr so that r ( 1>,..., r ( n> are the i n t e r i o r points, and r ( o ) and r ( n + 1 > a r e the boundary points. The function f, and i t s f i r s t and second d e r i v a t i v e s were approximated as follows: [A.2] [A.3] [A.4] , f f ( r ( 1 ) ) - > f, ( 1> f ; ( r - o -> (f (i+l) _ f ( i - l ) ')\/2Ar f ' ' ( r ( i > ) - > ( f J i + 1 > . 2 f i 1 > + f C i - D ) \/ ( A r ) 2 The d i f f e r e n t i a l equation f o r f^ then becomes a set of n l i n e a r equations i n n + 2 unknowns, the i t h equation r e l a t i n g \" f ^ 1 \" 1 ^ , f ^ and f . The boundary conditions were handled by solving t h e i r d i f f e r e n c e analogues f o r fj\u00b0^ and f j - n + 1 ) j a n ( j then s u b s t i t u t i n g these expressions t i l into the f i r s t and n equations. The r e s u l t i s a matrix eigenvalue problem: [A.5 ] A \u2022 F = -O\/F , where A i s a non-symmetric, t r i - d i a g o n a l , n x n r e a l matrix, and _F i s the column vector (f . . \u2022, f ^ ) \u2022 Equation [A.5] was solved with the use of routines FIGI2 and IMTQL2 of the subroutine package EISPACK.1 The c a l c u l a t i o n s were performed i n 1 Obtained from the U.B.C. Computing Centre. 191 double p r e c i s i o n on an IBM 370 machine. To test convergence of the answers, the equations were solved with n = 25, 50 and 100. The dependence of the lowest f i v e eigenf requencies of the 1, a = 0 case on n i s displayed i n Table X. I t can be seen from Table XI that the numerical method i s converging w e l l . The f i n a l r e s u l t s were a l l calculated with n = 100. Table XI Convergence Of Sound Wave Eigenfrequencies n = 25 n = 50 n = 100 1.750 1.742 1.738 5.018 4.966 4.937 7.888 7.823 7.781 10.65 10.59 10.55 13.33 13.32 13.28 APPENDIX B: NUMERICAL METHODS FOR INERTIAL FLOW 192 The equations were solved by replacing d e r i v a t i v e s by di f f e r e n c e s , and solving the r e s u l t i n g algebraic equations. S p a t i a l Mesh Systems A set of four staggered grids i n the upper right-hand quadrant of the z = 0 plane was used, s i m i l a r i n many respects to the MAC mesh system of Harlow and Welch (1965). These four grids can be generated by the product of a p a i r of r grids and a pa i r of 6 g r i d s : [B.l] r . = R g + ( i - |)Ar [B.2] r i + l \/ 2 = R s + ( i \" [B.3] 6 j ' - (j - |)AG [B .4] 3 j + l \/ 2 = ( j \" 1 ) A 9 where i - {1,...,^} , j = {1,...^} , R g i s the radius of the s o l i d c e n t r a l core, and [B.5] Ar = (R - R ) \/ ( N - 2) c s r [B.6] A6 = ( T T\/2)\/(N - 2) . 6 The four meshes may be labeled by ( i , j ) , ( i , j + -|), ( i + - | , j ) , and ( i + -T, j +4). In the ca l c u l a t i o n s reported i n t h i s thesis i both N and N 2 2 r x i were taken to be 15. A representation of the four grids i s given i n Figure 13. 193 The three components ^ v r* V0\u00bb v (j )) \u00b0f t n e v e l o c i t y were placed on the ( i ( i , j +-|\"), and ( i , j ) meshes, re s p e c t i v e l y . The magnetic f i e l d components were s i m i l a r l y placed. The va r i a b l e s p' and $' were placed on c the ( i , j ) mesh. Of the stress tensor components, a , a . . , , and a,, were r r ; \u2022 00