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Magnetoelastic interactions in the earth's core 1973
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Title | Magnetoelastic interactions in the earth's core |
Creator |
Crossley, David John |
Publisher | University of British Columbia |
Date Created | 2011-03-01 |
Date Issued | 2011-03-01 |
Date | 1973 |
Description | Previous calculations of the interaction of plane elastic waves in a uniform magnetic field in the Earth's liquid core showed negligible damping of such waves. Subsequent extensions of the theory have treated separately the damping of radial oscillations in a uniform field, and the effect of a field gradient on plane waves. It has been speculated that enhanced attenuation would take place for standing waves in a field gradient. An additional effect might also be expected from a proper treatment of the field geometry, as within the Earth both magnetic and free-oscillation fields can be expanded in spherical harmonics. In the present thesis a rigorous evaluation of magnetoelastic interactions in a spherical conductor is given, with a view to clarifying these predictions. The results show that within the Earth's core and at seismic frequencies the interaction is indeed weak. Typical values of the Q of the damping due to magnetic effects are at least 10¹³. Consideration of a wide range of harmonics in the interaction fails to find a significant effect due to field geometry. The role of viscous damping is evaluated using a recent value for the core viscosity and typical viscous Q's were about 10¹⁶. The possibility of gaining useful information from magnetic or viscous damping of the free oscillations is thus remote, but the importance of the results lies in their extension to core oscillations of longer periods. Such oscillations will also be underdamped and their velocity fields may be suitable for the new turbulent dynamo theories of the Earth's main magnetic field. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-03-01 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0052614 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geophysics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/31928 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0052614/source |
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\510°I MAGNETOELASTIC INTERACTIONS IN THE EARTH'S CORE by DAVID JOHN CROSSLEY M.Sc. University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Geophysics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1973 I n 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 t h e r e q u i r e m e n t s f o r a n a d v a n c e d 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 m a k e 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 a n d 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 b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y 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 n o t b e 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 . D e p a r t m e n 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 V a n c o u v e r 8, C a n a d a CD ABSTRACT Previous calculations of the interaction of plane elastic waves in a uniform magnetic field in the Earth's liquid core showed negligible damping of such waves. Subsequent extensions of the theory have treated separately the damping of radial oscillations in a uniform field, and the effect of a field gradient on plane waves. It has been speculated that enhanced attenuation would take place for standing waves in a field gradient. An additional effect might also be expected from a proper treatment of the field geometry, as within the Earth both magnetic and free-oscillation fields can be expanded in spherical harmonics. In the present thesis a rigorous evaluation of magnetoelastic interactions in a spherical conductor is given, with a view to clarifying these predictions. The results show that within the Earth's core and at seismic frequencies the interaction is indeed weak. Typical values of the Q of the 13 damping due to magnetic effects are at least 10 . Consideration of a wide range of harmonics in the interaction fails to find a significant effect due to field geometry. The role of viscous damping is evaluated using a recent value for the core viscosity and typical viscous Q's were about 10"^. The possibility of gaining useful information from magnetic or viscous damping of the free oscillations is thus remote, but the importance of the results lies in their extension to core oscillations of longer periods. Such oscillations will also be underdamped and their velocity fields may be suitable for the new turbulent dynamo theories of the Earth's main magnetic field. (ii) TABLE OF CONTENTS ABSTRACT (i) TABLE OF CONTENTS (ii) LIST OF TABLES (iv) LIST OF FIGURES (v) ACKNOWLEDGMENTS (vi) SECTION 1 - INTRODUCTION 1.1 Magnetoelastic Interactions 1 1.2 Review of the Free Oscillations 4 1.3 Review of the Geomagnetic field 7 1.4 Viscosity 10 SECTION 2 - BASIC FORMALISM 2.1 Physical Assumptions 2.2 Mathematical Notation 2.3 Elasic Equations 2.4 Magnetic Equations 2.5 Boundary Conditions SECTION 3 - THE INDUCTION EQUATION 3.1 Linearisation 25 3.2 Field Expansions 29 3.3 Selection Rules 34 3.4 Solution in the Outer Core 37 13 16 17 20 22 ( i i i ) SECTION 4 - ENERGY CONSIDERATIONS 4.1 Energy Equations 45 4.2 Ohmic Dissipation 50 4.3 Viscous Dissipation 51 4.4 Q 53 SECTION 5 - PARTICULAR INTERACTIONS 5.1 Radial Oscillations 56 5.2 Results for Radial Oscillations 64 5.3 Non-Radial Oscillations 68 5.4 Viscous Damping 69 SECTION 6 - SUMMARY AND CONCLUSIONS 6.1 Comparison with Previous Results 74 6.2 Geophysical Implications 75 REFERENCES 80 APPENDIX A - THE GAUNT AND ELSASSER INTEGRALS 85 APPENDIX B - COMPUTING THE FREE OSCILLATIONS B.l Starting Conditions 89 B.2 Normalisation 9 7 B.3 The Numerical Earth Model 98 Civ) LIST OF TABLES Table 1. Core Parameters 12 Table 2. Integrals for Viscous Dissipation and Elastic 52 Energy. Table 3. Main Magnetic Field Parameters 62 Table 4. Radial Oscillations in a Uniform Field 65 Table 5. Radial Oscillations in a Field with a Uniform Gradient 66. Table 6. Radial Oscillations in a Sinusoidal Field 67 Table 7. Non-Radial Oscillations in a Sinusoidal Field (n = 1) 71 Table 8. Viscous Damping 73 Table 9. Gaunt and Elsasser Integrals 88 Table 10. Parameters for Earth Model JAB1 9.9_ Cv) LIST OF FIGURES Fig. 1. Fundamental Free Modes of a Sphere 5 Fig. 2. Earth Model 14 Fig. 3. Amplitudes of Spheroidal Displacements in the Earth 28 Fig. 4. Interactions with the Toroidal Quadrupole Magnetic Field 36 Fig. 5. Electromagnetic Skin Depths for the Core 39 Fig. 6. Induced Field Regions in the Outer Core 44 Fig. 7. Amplitudes of the Radial Oscillations in the Earth 57 Fig. 8. Radial Functions for the Toroidal Quadrupole Field 63 Fig. 9. Amplitudes of Spheroidal Overtones in the Earth 70 ACKNOWLEDGMENTS I wish to thank my supervisor, Dr. D.E.Smylie for suggesting the problem and providing well-appreciated guidance when required. Grateful thanks also go to the many friends in the Department of Geophysics and Astronomy at the University of British Columbia for providing such an enjoyable research and social environment. The excellent f a c i l i t i e s at the Computing Centre of this University also deserve my gratitude. While 1 was at York University, Ontario, Pat Pooley and Olive Lambert helped in the typing of the thesis and I thank them both. The Aurora Institute of Advanced Studies is also to be thanked for providing amenable f a c i l i t i e s during the writing of the manuscript. This work was supported through operating grants provided by the National Research Council of Canada. SECTION 1 INTRODUCTION This thesis is principally concerned with the interaction between magnetic fields and elastic waves in a spherical electrically conducting body. Magnetoelasticity is a term often used with reference to the exchange of elastic and magnetic energy in ferromagnetic crystals (Landau and Lifshitz, 1960, p.155). In the present context the term is used, in a macroscopic sense, to describe the effect of a magnetic field on elastic deformations of a continuous medium. The treatment is directed towards evaluating such effects in the core of the Earth. 1.1 Magnetoelastic Interations The Earth is known to have a surface magnetic field which is predominantly dipole and of internal origin (Ride and Roberts, 1961). This field cannot be a relic of the past and is considered to be generated continuously by some form of induction process in the Earth's core (Roberts, 1967, Ch.3). To maintain the associated currents the core must be electrically conducting. A large earthquake produces two types of elastic waves within the Earth. One type consists of two travelling body waves P,S, the other is a harmonic series of standing waves. These standing waves are called equivalently normal modes, free oscillations or eigenvibrations, and in recent years their study has considerably refined seismic models of the Earth's interior (Wiggins, 1972). Because a moving electrical conductor in a magnetic field experiences a Lorentz force, i t is natural to ask two questions. Are elastic waves, particularly free oscillations, attenuated by the magnetic field in the Earth's core? What does a measure of that attenuation indicate physically about either the magnetic field or the elastic properties within the core? The first important attempt to assess the interaction was made by Knopoff (1955). He calculated the effect of a static, uniform magnetic field on the propagation of plane waves in a semi-infinite medium. For values of magnetic field strength and conductivity probable in the Earth's core (Table 1), i t was found that negligible attenuation takes place. However propagation of (a) plane waves in (b) a semi-infinite medium in the presence of (c) a uniform magnetic field is not realistic for a l l interactions within the core. Subsequent developments, including the present investigation, have been aimed at removing these limitations. The next step was taken by Kraut (1965) who discussed the attenuation of the radial oscillations of a homogeneous conducting fluid sphere, retaining condition (c). Again the interaction was very weak, the longest period of oscillation, taken to be the fundamental radial mode observed for the real Earth, is decreased by less than two parts in 10 . This represents an effective Q of the oscillation of order 10 1 7. Nevertheless a further calculation was made by Lilley (1967) who conjectured that there is enhanced attenuation in the presence of a non-uniform field. This is caused by magnetic induction due to translation of an elastic element through a field gradient in addition to the induction due to volume dilatation of the element present in a uniform field. For travelling waves the effective damping was increased by an 3 order of magnitude within half a wavelength of the origin of the coordinate system (Lilley and Smylie, 1968). Although condition Cc) had now been relaxed, conditions Ca) and Cb) were reinstated. Lilley transferred the result to estimate the damping of the long-period eigenvibrations and deduced a Q of about 10^ might be reached. This is s t i l l a factor of 10 above indicated Q's for the normal modes and at least 10^ above current detection levels (Dratler et al., 1971). The geometry of the Earth's magnetic field in the core can be treated as a combination of spherical harmonic field components CBullard and Gellman, 1954) in an analogous way to the representation of free oscillations of a spherical elastic body CAlterman et al., 1959). This similarity in geometry between the magnetic and elastic fields suggests the possibility of a resonance interaction between field components which might enhance the damping effect. Further, the magnetic field in some parts of the core must be quite non-uniform, notably near the core- mantle boundary because of the conductivity contrast between the outer core and lower mantle (Rochester and Smylie, 1965). The present thesis utilises the geometry of the core with spherical harmonic expansions of the magnetic and elastic fields and relaxes a l l three conditions (a), (b), (c) noted above. It is then possible to discuss magnetoelastic interactions in the Earth's core using realistic seismic models and taking advantage of current developments in geomagnetic dynamo mechanisms. The generality of the mathematical approach allows any harmonic displacement field to be substituted in place of the free oscillations e.g. Chandler wobble deformations and earth tide deformations, both of which are forced vibrations of degree two. 4 A recent study on the dynamical stability of the fluid core (Higgins and Kennedy, 1971) has raised serious doubts about the existence of large scale convective flow as required by conventional dynamo theory. As a result there is considerable interest in the possibility of turbulent dynamo action, possibly sustained by an oscillatory mechanism. The low viscosity predicted by Gans (1972) therefore gives further stimulus to a detailed study of magnetoelastic interactions in the outer core, as a mechanism of energy- dissipation. 1.2 Review of the Free Oscillations Following a large earthquake, the Earth continues to vibrate in a set of normal modes, three of which are indicated schematically in Fig.l. The early history of the theory of these free oscillations is reviewed by Stoneley (1961). It is convenient to begin here with the work of Lamb (1882) on the free modes of a uniform, incompressible, homogeneous elastic sphere. Lamb's analysis showed that there are two basic modes of vibration which are now referred to as torsional and spheroidal vibrations. The former have no radial displacement and the simplest motion is a twisting about a polar axis. The lowest degree spheroidal mode is a radial contraction and expansion, and the degree two mode oscillates between a prolate and oblate spheriod (Fig.l). If the material of the Earth is allowed to be compressible and self gravitation is taken into account, the analysis becomes more complicated (Love, 1911, Ch.VII). Gravity not only acts as a restoring force to shorten the eigenperiods but also causes a large i n i t i a l state of stress throughout the Earth. The method of including this i n i t i a l stress in the equations of motion was presented originally by Rayleigh (1906). The Earth was by then known to be non-homogeneous 6 (Oldham, 1906) and Hoskins (1920). extended Love's theory to a radially heterogenous Earth, taking the elastic parameters to be simple algebraic functions of radius. Takeuchi (1950) used hand calculating machines on a closely related problem, that of determining the response of the Earth to tidal deformation. At the same time Bullen (1950) produced his second whole Earth model, model B. Benioff 0-954) then announced that a 57 min. period had been detected on seismic records following the Kamchatka earthquake of 1952. Although this was subsequently questioned as being an eigenvibration (e.g. Bullen, 1963, p.264), several workers began numerical calculations of the eigenperiods using computers (Pekeris and Jarosch, 1958; Jobert, 1957). The most widely quoted results of that time were published by Alterman et al. (1959) using a step-by-step integration of the equations of motion, modified from Love's analysis. The eigenperiods obtained by Alterman et al. were confirmed when seismic records of the Chilean earthquake of 1960 were examined (e.g. Benioff et al., 1961). In the last decade there has been considerable effort to improve the agreement between theoretical and observed eigenperiods; only a few developments need be cited in this review. The records of Benioff et al. showed very close multiplets of lines where a single frequency was expected theoretically, but there was l i t t l e hesitation in attributing this effect to the rotation of the Earth (e.g. Pekeris et al . , 1961). In effect, the standing nodal patterns on the Earth's surface drift relative to a seismic station. This mode then appears split into 2n+l components, where n is the degree of the mode. The effect of Earth oblateness on the eigenperiods has been studied by Dahlen (1968), 7 and recently Madariaga (1972) has given an analysis of the effect of large scale lateral heterogeneities on the torsional eigenperiods. These second order departures from spherical symmetry destroy the complete separation of the torsional and spheroidal free oscillations, and result in a certain amount of coupling between modes. Generally eigenperiods have been obtained by straightforward power spectral analysis, usually performed at each seismic laboratory following an earthquake of sufficient magnitude 6.5 ). Results to 1968 were summarized by Derr (1969). However, by combining a large set of existing records from a single earthquake, Dziewonski and Gilbert (1972) have improved the identification of the eigenperiods. Following the Colombian earthquake of 1970 the records of Dratler et al. (1971) show very clearly the persistence of many overtones of the free oscillations. Such improvements in the quantity and quality of the observed eigenperiods have enabled Earth models to be developed consistent with the mode identifications (e.g. Haddon and Bullen, 1969). The recently developed Earth models are also better constrained, within the quality of the data, by improved fitting techniques (Backus and Gilbert, 1970) . Damping of the free oscillations is generally attributed to anelasticity in the upper mantle (Jackson and Anderson, 1970). Although estimates of Q are somewhat uncertain due to the splitting and coupling of modes mentioned earlier, a typical Q is 300 but for the radial oscillation and most overtones a Q of 10^ seems indicated (Dratler et al. 1971) . 1.3 Review of the Geomagnetic Field In contrast to the generally excellent confirmation of theory by observations on the free oscillations, there is s t i l l considerable uncertainty as to how the Earth's magnetic field is sustained. Hide and Roberts (1961) have given a thorough description of the observations and basic theory, and a review by Weiss (19 71) summarises the current position. To explain the origin of the magnetic field associated with sunspots, Larmor (1919) first suggested a self-generating dynamo mechanism. This was criticised by Cowling (1934) who proved what is now known as Cowling's Theorem, which states that an axisymmetric field cannot be maintained. This ruled out the model proposed by Larmor. Elsasser (1946), in the first of a series of papers on magnetic induction in the Earth's core, discussed the physical conditions required for dynamo action to take place. Bullard (1949) initiated his own extensive contribution to dynamo theory by proposing a particular model suggested by dynamical motions likely within the core. Initially a dipole field is assumed to exist. The combination of the Earth's rotation and radial convection, driven thermally from the deep core, cause a non-uniform fluid rotation which turns the dipole field around the axis. The field lines then l i e in circles of latitude, in opposite sense on either side of the equatorial plane. The field generated is toroidal, with quadrupole symmetry, and has field lines similar to the displacement field T° of torsional oscillations (Fig.l. and Section 3.2). Because the conversion is an efficient process, driven continuously by rotation and convection, a strong toroidal field can be produced from a weak dipole field. To sustain the dipole field, the convective flow is imagined to 9 rise in columns and twist the toroidal field into loops through the Coriolis force. The loops are then supposed to coalesce and largely cancel leaving a dipole field as described in detail by Parker (1955). If the whole cycle is efficient then the dipole field can be maintained against ohmic dissipation. A detailed mathematical formulation of the dynamo process was given by Bullard and Gellman Q.954). A general method of expanding both the magnetic and flow velocity fields in spherical harmonics was initiated and the dynamo of Bullard was given exhaustive numerical treatment. Unfortunately a stable solution was not achieved, and there was some indication that energy would be passed to the higher harmonics of the magnetic field instead of returning to the dipole field. Improved numerical techniques (Gibson and Roberts, 1969) indicated the problem was not tr i v i a l , which seemed to confirm a demonstration by Braginskii (1964) that the velocity fields being used were too symmetrical. Noting Braginskii's result, Lilley (1970) introduced a third velocity component into the flow and this seemed to provide a more stable dynamo action. Subsequent calculations by Gubbins (1972) unfortunately showed this dynamo was also unstable. Meanwhile dynamo mechanisms had been investigated which were proved rigorously to work (e.g. Backus, 1958) but at the expense of rather a r t i f i c i a l velocity fields and a new approach was begun by Steenbeck et al. (1966) on the possibility of dynamo action in a turbulent fluid medium. Their work has been reviewed and extended by Moffatt(1970a) who's latest contribution indicates a random driving force can generate dynamo action under certain conditions (Moffatt, 1972). In a fluid of infinite extent a steady state can be reached in which the magnetic energy density is maintained above the level of kinetic energy density on the assumption of no mean flow. The presence of core boundaries tends to induce a mean flow, so further work is required before a successfu mechanism can be claimed. The large scale flow necessary for the original Bullard dynamo now is challenged on thermodynamic grounds by Higgins and Kennedy (1971). On the basis of new data for the effect of pressure on the melting point of metals and other solids, they argue that the melting point curve lies well above the adiabatic curve. The excess is estimated at 500°C at the core-mantle boundary if the two curves are coincident at the inner-core boundary. The outer core is considered fluid and so its temperature cannot be anywhere less than that given by the melting point curve. Simple dynamical arguments then lead to the conclusion that the outer core is everywhere near the melting point and is quite stable against radial convection. The effect of this result on dynamo theory has been mentioned briefly by Bullard and Gubbins (1971) and i t appears that oscillatory dynamos and those excited by inertial waves (Moffat, 1970b) hold some promise for the future. 1.4 Viscosity If the outer core of the Earth behaves as a perfect fluid, then anelasticity cannot be a mechanism for attenuating seismic waves. The role of viscosity in the outer core then becomes the only alternative to magnetic dissipation of seismic energy. Although this thesis is mainly concerned with magnetic damping, the effect of viscosity is straight- forward to calculate and will perform a minor role in the ensuing discussion. From the passage of P waves through the outer core, Jeffreys (1970, p.323) has estimated 5 x 10^ poise as an upper limit for the dynamic viscosity. As Jeffreys points out, this also includes the effect of bulk viscosity. The torsional vibrations of the Earth are confined to the mantle because the fluid core cannot support pure shear', this implies the core-mantle boundary is a free surface for the mantle vibrations. MacDonald and Ness (1961) have given a detailed account of the modification of the eigenperiod of 1° due to viscous and magnetic stresses across this boundary. It was found that there was negligible attenuation of the oscillation due to stiffening of the boundary by these stresses. Similar results were obtained by Sato and Espinoza (1967). Another approach to the viscosity of the outer core is to compare the relative attenuation of seismic waves which are reflected by the core, S.cS and transmitted through the core, SKS. Pairs of rays are chosen to have identical mantle paths. The results again give an upper limit on core viscosity in the region of 10"^ poise (Suzuki and Sato, 1970). These seismic estimates of viscosity are markedly higher than those from other sources (Malkus, 1968) probably because they represent upper limits. In an attempt to settle the question Gans (1972) makes use of Andrade's formula to determine the viscosity of iron at the melting point and arrives at the surprisingly low result of about 10 ^ poise for the dynamic viscosity. The kinematic viscosity is therefore about 12 the same as that of water at 20°C. This result indicates strongly that if the outer core is mainly molten iron, then i t behaves as a true liquid. Some of the quantities discussed in this section are presented in Table 1. Table 1 Core Parameters Inner core radius 1215 km Appendix B b Outer core radius 3485 km II Frequency of typical oscillation 'v-lO-2 -1 sec it Velocity of typical oscillation 5̂ x 10~4 -1 cm sec ii ft* Maximum toroidal field strength 480 gauss Bullard and Gellman(1954) cr Electric conductivity 3 x 105 ohm m II 1 Dynamic shear viscosity 0.08 poise Gans (1972) Permeability 4TTX 10~7 henry m ̂ - 13 SECTION 2 BASIC FORMALISM In this section the equations governing the motion of the medium and the behaviour of the magnetic field are reviewed. It is usual either to combine viscous and elastic forces and form the equation of viscoelasticity (e.g. Bland, 1960), or to add a magnetic field to hydrodynamics and call i t magnetohydrodynamics (e.g. Roberts, 1967). Because viscous and magnetic effects are expected to perturb the free oscillations only slightly, for reasons given in Section 1.1, the basic equations are those of elasticity (love, 1911, Ch.VII). The equations might well be referred to as those of magnetoviscoelasticity. 2.1 Physical Assumptions The Earth is to be treated as a spherical, radially heterogeneous, self-gravitating, compressible elastic body. With a quarter of i t removed, i t appears schematically as in Fig.2. The outer core, discovered first by Oldham (1906), extends just over half way to the surface: the inner core, proposed by Lehmann (1936), occupies about a third of the outer core radius. The outer core is traditionally assumed to be a liquid with zero rigidity (Jeffreys, 1970, p.285). From the P wave attenuation data already mentioned, Suzuki and Sato (1970) concluded that the outer core behaves more like a viscous liquid than a low-rigidity solid. Evidence on the stiffness of the core mantle boundary from the eigenperiods of torsional 10 —2 oscillations indicates an upper limit of about 10 dynes cm (Sato and Espinosa, 1967). This is in agreement with the limit obtained by Takeuchi (1950) from earth tide calculations. By contrast, the effect the 14 Fig. 2. Earth Model inertia of the core has on the amplitude of the 19 yr lunar nutation Q is shown by Jeffreys (1970,p.295) to place an upper limit of 10 dynes -2 cm on the core rigidity. In comparison, the -mantle rigidity averages 12 - 9 about 10 dynes cm L. The inner core has been generally recognised to be solid (Bullen, p.242) and evidence from the eigenperiods of low degree oscillations definitely favours a normal solid rigidity (Dziewonski and Gilbert, 1972, p.409). There is s t i l l some doubt as to the constancy of seismic properties within the inner core. The model used by Dziewonski and Gilbert (1972, Table 6) has both P and S velocities constant, whereas the model used in the present problem (Appendix B) has a small gradient in S velocity, and therefore also in rigidity. Because there is l i t t l e reason to do otherwise at this stage, the composition of the inner core is taken to be the same as the outer core (Jeffreys, p.203). The composition of the outer core is of interest because i t influences the electrical conductivity which in turn influences ohmic dissipation (Section 4.2). Recent work on the conductivity of liquid iron and metallic alloys by Gardiner and Stacey (19 71) and Jain and Evans (19 72), extrapolated to core pressures and temperatures, have confirmed Bullard's 1949 estimate (Table 1). The question of time scales is important. After an earthquake most oscillations die away within a few days of their excitation because of the finite Q (Dratler et al, 1971) whereas dynamo processes associated with variations of the geomagnetic field are by comparison stationary (Hide and Roberts, 1961). This enables the relative rotation of the mantle and core to be ignored and in Fig.2 both parts of the Earth are taken as fixed by the geographic coordinate system (Munk and MacDonald, 1960, p. 11). The rotation of the Earth ensures that the main magnetic field is orientated with the geographic polar axis by dynamo action. The polar axis of the coordinate system for the free oscillations passes through the epicentre of the earthquake and will be inclined to the geographic polar axis in most cases. However there is no loss of generality in assuming the two axes coincident, because for a given earthquake location a simple transformation of colatitude 9 allows for the subtended angle between the two axes. For example, for an earthquake at the equator no transformation is necessary for the 0S 2 oscillation, and the radial mode has no preferred axis of symmetry (Fig.l). 2.2 Mathematical Notation Elasticity theory is usually expressed in Cartesian tensor notation, and electrodynamics is nearly always formulated in vector notation. It has been decided to retain the appropriate usage to aid physical understanding, although in some equations both forms will be found. This may look inelegant but is s t i l l rigorous for an isotropic medium. As usual, repeated indices imply summation and i . . s I for i = j , otherwise S.. = 0. A displacement field is represented throughout by u_ and a velocity field by v, the three coordinates at a reference point are denoted by the components x.. The notation for a harmonic component of the magnetic field is T̂ or and for the spheroidal oscillations is 0 S m where s,n s s v n are the degree and p,m are the order of the associated Legendre functions and tf is the overtone. 17 2.3 Elastic Equations The most general form of the equations of motion for a volume element of density j> is ot a*. J where the substantive derivative refers to a fixed material element (Roberts, 1967, p. 16). The tensor TF„ represents the total stress field and F is the body force per unit mass. For the Earth, the body forces may be gravitational or rotational; the gravitational forces arise from either self gravitation or lunar and solar tidal effects. In the present treatment F^ is taken as entirely due to self gravitation and is derived from a potential V. The total stress can be expressed as 7 T*j. S X ; j •+ X;j + «Y1;j where X. . is the elastic sress.T. . is the viscous stress and m. . is i j i j i j the magnetic stress. In a linear isotropic medium nu , the magnetic part of the Maxwell stress tensor (Stratton, 1941,p.98), is related to the components of magnetic induction by m Ho where is the permeability. Because B is a vector field, the magnetic stress acting over the surface of a volume is equivalent to a body force within that volume; in the present case the body force per unit volume can be written c ( I * H > (Roberts, p. 11) where J_ is the current density. The electric part of 'the Maxwell stress can be neglected by the argument of no free charge (e.g. Elsasser, 1956, p.137). The viscous stress can be written in several forms, one which separates the different viscosities is (Landau and Lifshitz, p.214) where ^ are the coefficients of dynamic shear and bulk viscosities. The bulk viscosity is often neglected because the rate of shear deformation usually greatly exceeds the rate of compression; i t represents resistance to pure expansion (Jeffreys, p.3). As discussed by Love (1911, p.89) and in a more recent treatment by Smylie and Mansinha (1971), every point within the Earth is under a hydrostatic stress >?•• which balances the self gravitation Here £ > $0 an^ P0 a r e respectively the density, gravity and hydrostatic pressure in the i n i t i a l state, assumed to vary only with the radius r. Because a volume element carries its i n i t i a l stress with i t during a deformation (Rayleigh, 1906) , the element at a reference point had an i n i t i a l stress P;. - a. = - ( f t , -j> o3 0u f) * , where u r is the radial displacement. This viewpoint is not immediately obvious and originally caused some confusion in the treatment of self gravitation. In addition to the hydrostatic stress there is an additional elastic stress T.̂ . related to the deformation u^ by a general form of Hooke's Law (Love, 1944, p.102) where A, »vi are Lame's elastic parameters and A is the cubical dilatation, & s du^/dx . Physically, the meaningful parameters are the rigidity modulus and k ( = A } the bulk modulus. The total elastic stress tensor is To first order in displacements, the density change of the element is given by which leads to an additional gravitational potential satisfying Poisson's equation where G is the gravitational constant. Because F^ is derived from the total gravitational potential, f _ , V = V0 + V, and q = M . In taking the equations of motion (1) to represent the vibrations of an elastic medium the displacements can be considered infinitesimal, and to first order J ot J dt* The elastic equations are then obtained by substituting for TTj! and F from the above relations. P & - I -r £T;j - r ^ J l j + , (2) at1- c dx. ax. dat: ' where $'• represents the equivalent gravitational force per unit volume correct to fi r s t order in quantities small with the displacements (Hoskins, 1920, pp.7-8). Equations (2) have been given in symbolic form by Smylie and Mansinha (1971, p.332). It is convenient here to first write them where and then in the vector form L(u) = £ (3) where L is a form of vector operator. The terms in f_ are the two viscous volume forces, the second of which is zero if the medium is incompressible, and the magnetic volume force. 2.4 Magnetic Equations In the core of the Earth the displacement current can be neglected so the field equations are, strictly considered, in pre- Maxwell form, Vx E = - i i , X7.8 - 0 , VxH - 3 , B = M 0 H (4) (Elsasser, 1956). The vectors are E_, the electric field strength, B_, the magnetic induction, H, the magnetic field strength and jJ the conduction current density. Ohm's law is taken in a form suitable for the relation between current and field in a medium moving with velocity v, T = <r (E + vfx 6) } < 5 > where er is the electrical conductivity, (Landau and Lifshitz, p.205). By a suitable combination of (4) and (5) the current and electric field can be excluded. The result is known as the induction equation where is the magnetic diffusivity (ju00") -I The induction equation is of central concern in the next section so i t is useful to consider the role of the three terms. Remarks similar to the following can be found in most textbooks on the subject of magnetohydrodynamics (e.g. Roberts, 1967, Ch.2). The terms on the right hand side of (6) can be compared by dimensional analysis |V , ( *x_6)| = ^ IVx(V*8)l where, i f ITand d£ are a velocity and a length typical of the system, defines the magnetic Reynolds number. In a medium of high conductivity, R̂ is large and there is negligible diffusion of the field; the field is said to be frozen into the medium. Conversely, for small conductivity, R is small and the field will diffuse faster than i t can be maintained m by the flow. If the medium is stationary, so R̂ is zero, (6) is a diffusion equation and in a bounded medium the solution is an infinite series of decay modes CElsasser, 1946). In a spherical conductor of radius R the mode of longest decay has the form &(r,t) - 6 ( r ) e " t / c , I , RVcr/V l . (7) If, however, the field is stationary diffusion and induction are balanced. Since Elsasser (1946) and Bullard (1949) speculate!on the possibility of a geodynamo, there has been considerable effort to find a solution B_ which, is stationary and which has a steady dipole component. Even if a solution for B_ was found, v would have to obey the equations of motion to complete the dynamo mechanism. The combination of (2) and C6) presents an impressively difficult problem; (6) alone is not simple, as is well known from dynamo theory . Equations (2) have been linearised by Alterman et al (1959) without the viscous and magnetic terms. It is, in principle, possible to treat the additional forces in a manner similar to the Coriolis force (Pekeris et al, 1961) and compute the change in eigenperiod. A simpler approach is to solve the linearised induction equation and determine the energy lost by ohmic dissipation and thus obtain a measure of Q, 2.5 Boundary Conditions It is appropriate to complete this section with a review of the boundary conditions for the magnetic field and the free oscillations. Because both fields span bounded media, they are modal in character and the boundary conditions are influential. For the free oscillations Pekeris and Jarosch (1958) ;give the following conditions; (a) Regularity of the displacements and stresses at r= 0 . (b) Zero surface stress on the deformed surface of the Earth. (c) The interior and exterior gravitational potentials and their gradients are continuous at the deformed surface of the Earth. In addition; (d) Within a liquid or at a liquid-solid interface, the transverse stress is zero. And, at any discontinuity within the Earth; 23 Ce) Displacements, gravitational potentials and their gradients^ and normal stresses are continuous. These are dynamic boundary conditions and apply to a l l free oscillations of periods an hour or less. If the frequency decreases to zero (the static limit), certain of the equations of motion become degenerate. To satisfy the surface conditions (b), (c), condition (e) has to be relaxed to allow one extra free constant within the Earth CSmylie and Mansinha, 1971, pp. 342-344). There is s t i l l some doubt as to which of the conditions in (e) have to be modified and not a l l authors agree (e.g.' Pekeris and Accad, 1972, p.241). Fortunately the difficulty does not arise for the normal modes. Condition (a) will be discussed further in Appendix B because i t has often been satisfied only approximately (e»g> Alsop,1963,p. 486). The boundary conditions for the magnetic field at a discontinuity in f o o r °~ a r e g i v e n by Stratton (pp.34-37) and are derived from (4). They are : (f) Normal component of B_ continuous, unconditionally. (g) Transverse component of B_ continuous if there is no surface current. (h) Transverse component of E_ continuous, unconditionally. The first condition is derived fromV.B*0. The second condition is derived from ^X H - T t from which 7.7=. 0 and is valid only if the conductivity is finite.on both sides of the boundary. Otherwise for an infinite conductivity on one side a surface current density exists. In the present treatment of the Earth, the mantle can be considered insulating as the time scale of core-mantle coupling, using a finite lower-mantle conductivity, greatly exceeds the time scale of the free oscillations. At the core-mantle boundary the normal components of B_ are continuous, and because there are no sources outside the core, by - 3 dynamo hypothesis, the field _B falls off as r in the mantle. No electric currents can flow in the mantle and thus the normal component of must vanish at the boundary. SECTION 3 THE INDUCTION EQUATION A solution of the induction equation is now obtained by a linearisation procedure. Sections 3.2 and 3.3 follow the treatment of the dynamo problem by Bullard and Gellman (1954, Sections 4 and 6). 3.1 Linearisation The induction equation (6) is quite hard to solve unless i t is first linearised by a perturbation method. The elastic equations (3) can be included in the scheme, although mainly for completeness because they are not solved directly. Quantities are expanded in two ways, •& ' +fe > I J e i (8a) and i f = a */$t -To 4 i f , +ifa The following interpretations are implied; } (8b) B̂ Main core magnetic field, sustained by currents J_ 0 b_ Perturbation of JB 0 by energy from v^ , sustained by j_ t^.v^ Displacement and velocity associated with fluid flow in the core. u_̂ ,v̂ , Displacement and velocity associated with the elastic free oscillations. —2'—2 Displacement and velocity fields caused by the interaction being perturbations of u.̂ , v^ f Body force for fluid flow. —o f_̂ Body force for free oscillations, zero by definition. f_2 Body force due to the interaction Because of time scales B , J and f are considered static. - o ' - o - o Perturbation quantities b_, j_, u_2> y_2» a n <^ —2 have the harmonic dependence >'uJt of the free oscillations, i.e. v/J=w,t£. 4,fcc. To perform the linearisation i t is necessary to assume two conditions |b| « |S 0I (9a) Ib|/IB0I « l»f.l /»!f.l , ( 9 b ) and define the zero-order induction equation to be -\7xVx| 0=: /vrcyio -^<r Vx(v;x80) . d o ) The unperturbed elastic equations are given by equation (3) for the free oscillations Equations (8a) and (8b) are now substituted into (3) and (6), conditions (9a) and (9b) are applied and equation (10) subtracted. To first order in small quantities the resulting equations are at and where assuming | V / ^ | ^ < 1 VT| I » unconditionally. In a conducting medium a perturbation of the steady field B_Q is equivalent to disturbing the field lines and producing Alfven waves (Roberts, Ch.5). In the present situation the elastic forces are controlling the behaviour of the medium and the field b_ is assumed to have the time dependence of v^. The hydromagnetic waves are thus in step with the free oscillations and are themselves standing waves in the core. The effect of finite viscosity and electrical conductivity is of course to attenuate the waves. If required, the equations following (11) can be used to compute the perturbation in the free oscillation velocity. Condition (9a) can be verified in the sequel; condition (9b) requires that the magnitudes of _v̂ , v^ must be compared. Bullard and Gellman (1954, p.273) assumed that the transverse component of v^ is a measure of the westward drift of the non-dipole surface field. As such i t is taken to represent the motion of the outer core past the mantle. An extrapolation of the observed drift of 0.18° per year leads to maximum radial and transverse velocities of 0.014 cm sec and 0.04 cm sec ^ respectively. The amplitude of v^ in the core requires the variation of amplitude of displacement with depth to be computed for a realistic Earth-model, as in Fig.3. In this calculation, the radial surface displacement is usually normalised to be unity, here 1 cm. In a recent study of the source mechanism of the 1964 Alaskan earthquake, Ben-Menahem et al.(1972) have given observed surface amplitudes. When suitably corrected for instrument response and displacement of the epicentre from the station, radial displacements at the surface are of the order of 1 mm for the low-degree spheroidal oscillations. This agrees with Nowroozi's (1965) estimate for QS 2 for the same earthquake. For those oscillations with appreciable kinetic energy in the core (Figs. 3,7 and 9) periods range from 3233 sec for QS 2 to 244 sec for 4S Q . Free oscillation velocities in the radial direction are thus in the range 10-^ to 10"^ cm sec-"*" with transverse velocities an order of magnitude smaller. 28 Fig. 3. Amplitudes of Spheroidal Displacements in the Earth. Computed from model JAM (Table 10, Appendix B) . Ful l lines are radial displacements, dashed lines are transverse displacements. Thus - . | 0 ~ * < I if, I / l \ f o l < I and conditions C9a) and (9b) are satisfied simultaneously i f lb I << l 8 o l * [ 0 ~ l ( 1 2 ) 3.2 Field Expansions At first' sight equation (11) looks similar to the induction equation (10) but there are differences to be noted. In (11) the free oscillation velocity v^ replaces the flow velocity v^ of (10) and contains an extra part to the field which is lamellar, v^ being solenoidal. Second, i f (11) is written in the form 7 XV X b -f p 0 cr H - <u 0 o- Vx (V, x§J (13) the left hand side contains the field b_ induced by the interaction term (^"t ~f §>o) o n the right. Because of the perturbation approximation (9a) i t is unlikely that dynamo action will take place, particularly as ]B0 is taken to be stationary with respect to b_. Nevertheless because of the spherical symmetry of the problem, field expansions can be used in a similar approach to that of Bullard and Gellman (1954, p.220). As discussed by Smylie (1965), the magnetic field is solenoidal and can be written as the sum of a poloidal vector S_ and a toroidal sector T_, by a theorem of Backus (1958). Thus where S and T are the defining scalars. The degree and order of a particular harmonic are s and p for the. main field B̂ 0 , and and k for ' the induced field b_. The defining scalars are expanded in spherical 3 0 harmonics QO S (15) for ji0and similarly for b_, the summation notations in (15) being equivalent. Following Smylie (1965) the associated Legendre functions (16) (17) are defined with the normalisation of Hobson (1955, p.93)7 P/u) »(-i) po -H.V W l I P,V> , f * » P / V . P.V> , • For reference the components of the vector elements are given for 8 0 (Smylie, 1965). Again (16) and (17) are similar for b_. The components of the free oscillation displacement field u_ are well known (e.g. Alterman et al., 1959, pp.84, 86), but it is useful to indicate how they are derived. The Helmholtz separation theorem (Morse and Feshbach, 1953, p.53) allows the displacement field to be written where L is a scalar and A a zero-divergence vector field. If the degree and order of a harmonic of the displacement field are denoted by n and m, the scalar L can be expanded as and the components of L : \ ? L , a lamellar vector( are 3r (tr). = Because V x A is solenoidal, the defining scalars can be expanded as in (15) and the components written as in (17). The lamellar and poloidal fields have the same angular functions and they can be combined into a spheroidal vector field with components ( s ; ) r - u r t r , f c ) p;v w (s:i = u i - f . t J d S T e * " * ' / <18> ' StlA W s;*t> The toroidal part T m of V*A has the same radial components as the toroidal —n magnetic field, (17) although the radial function tjj| is usually written wm . The total displacement field is then the sum of two vector fields, n r and for a harmonic n, m where S n = r u. r t r „ e ^ -+ r i r n V ( K n e w is a spheroidal vector displacement and ir= w.-rj . v l P.V""*)' is a torsional vector displacement. The expansion of the displacement field in this way has been used by many authors, but a rigorous justification rests on the result of Backus' theorem. Because the rigidity of the outer core is negligible a purely transverse elastic motion has no restoring force. Therefore only spheroidal oscillations sample the core and the vector components of _u,̂ are given by the components of alone. expansions (14) - (18) into (13) leads to a rather lengthy manipulation of the angular functions. The procedure closely follows that described by Bullard and Gellman (1954, p.224) and Smylie (1965, p.172). Essentially the angular functions are grouped together in such a way that the poloidal and toroidal parts of (13) are separated. This is accomplished by multiplying (13) f i r s t by a poloidal vector S'̂ then by a toroidal k _ i vector T*£ whose radial functions are chosen to be r and 1 respectively. On integration over a spherical surface the orthogonality relations for the associated Legendre functions ensure the separation of the vector equation into poloidal and toroidal parts. toroidal parts each of which contains triple angular integrals K, L. A discussion of these integrals can be found in an appendix by Scott in Gibson and Roberts (1969). They are generally referred to as Gaunt and Elasser integrals and they depend on the six indices of the three fields* Substitution of the fields B_0 b_ and v. in the form of the The term also becomes separated into poloidal and by definition To ensure that k, m, p are a l l positive integers i t is necessary to have one superscript negative. Definitions of K and L and some of their relevant properties are given in Appendix A. The equation for the poloidal field is found to be - X C i . ( rS / ) C n h 4 - i ) - ^ ( £ 4 l ) - S ( S + . ) J K - J - t r ^ s ( 5 4 l ) S / [> -s(5-r<) + i ( i f ) ) ] K [ ( 1 9 ) and for the toroidal field (20) where v^has been written ifc)u_^. Although these two equations appear complicated, they have a simple physical interpretation which depends on the properties of K, L. Imagine a particular component of the velocity field P P interacting with a component of the main magnetic field, or Tg. Selection rules, Section 3.3, then determine the values of I, k which give non-zero K,L. The right hand sides of (19) and (20) are then summed over a l l n, m, s, p to provide the source term for each permissible t, k of the , perturbation field. The equation (13) can then be solved for the radial k k functions s^ and t^ to determine the harmonic of b_, the induced field. 3.3 Selection Rules The selection rules are quoted from Scott in Gibson and Roberts (1969, p.588) for k, m, p and I, n, s positive definite integers. 1. For K, L non zero; - k-+m+p = 0 2. For K non zero; (i) £, + n + s is even. (ii) ji, n, s can form the sides of a triangle. 3. For L non zero; (i) X + n + s is odd. (ii) JI, n, s can form the sides of a triangle, ( i i i ) No two superfixes zero, e.g. m = p = 0. (iv) No two superfix-suffix pairs equal, e.g. m=p, n=s. Rules 2(i), 3(i) indicate that either K or L vanishes for any particular selection of I , n, s. Rules 2(ii), 3(ii) ensure that the sums on the right hand sides of (19) and (20) are finite, and this simplifies con- siderably the evaluation of the perturbation fields. Beyond these observations further illustration of the selection rules is best left to a particular example. Before this is done however a remark should be made concerning the expansions (15) in comparison with the expansions of Bullard and Gellman. In (15) the spherical harmonies are complex to simplify the algebra. However, the radial functions are not necessarily real, but are only required to satisfy the relation (Smylie, 1965), where a * signifies complex'conjugation. This ensures that the field B is real providing a similar condition also holds for 35 the toroidal components. The dynamo expansions of Bullard and Gellman are in terms of sines and cosines and the two forms are of course m c m s equivalent. For example, for two real poloidal harmonic's S ' , S ' n n of the Bullard-Gellman dynamo, the related complex radial functions are where the superscripts c,s refer to cosine and sine functions. For the dynamo theory the radial functions of the velocity field are real to represent actual flows in the outer core. Because the free oscillations are usually formulated from complex harmonics the magnetic fields for non-zero order are also complex. It would be impractical to investigate a l l possible interactions in order to solve (19) and (20) completely. Instead the main magnetic field B Q will be assumed to consist of a single component, the axial quadrupole toroidal field T°.. Arguments have been forwarded in the Introduction as to why this component is expected to have appreciable strength in the outer core. Restriction to one component considerably simplifies the right hand sides of (19) and (20) as only the sums over n and m have to be considered. An interaction diagram can be constructed similar to that of Gibson and Roberts (1969, p.584) but with a fundamental difference. Instead of specifying a velocity field and examining magnetic field interactions, the main field is specified and the magnetoelastic interactions are evaluated. The diagram for T2 is shown in Fig.4. The presence of an s or t in. the diagram indicates a spheroidal oscillation of degree n and order m producing an induced magnetic field of poloidal 36 SPHEROIDAL OSCILLATION Sln n 0 0 -I 0 1 I I I -2-10 12 2 2 2 2 2 -3 -2-1 0/23 3 3 3 3 3 3 3 -4-3-2-1 0/234 4 4 4 4 4 4 4 4 4 I-1 1° I1 t t t S S t t t 2T' 2° 2l 2 2 t S s t t t t t S S S S 3-3 3-2 3-' 3° 31 3 2 33 t t t S S S S t t t t t S s s s s s 4-4 4-3 4-2 4-' 4° 4' 42 43 44 t t t t t s s s s s s t t t t t t t t t Fig. 4 . Interactions with the Toroidal Quadrupole Magnetic Field. The induced f i e l d is shown s for poloidal, t for toroidal. or toroidal type with an k harmonic. If a square is blank it means either K or L is zero by one or more of the selection rules. Harmonics up to 4,4 only are shown although the diagram continues indefinitely. Writing the radial function of T ° by the capital letter T to distinguish i t from the toroidal perturbation field, equations (19) and (20) then reduce to £ ( r S k ) -2(iil)(rs>) - lopr r s / J(0 dt* 1 <x \ (21) i ! ( r t / ) - 4 1 ^ ( * t / ) - - . * u ) ^ f l r r t / o » f where, for s = 2, p = 0, J 4ffAU-»l) C 2 dr 4 l T m + 0 [ » < ^ 0 -M**) - 6J j K £# rt| a In (21) the time dependence of the perturbation fields has been iu>b recognised as e and in (22) the Gaunt and Elasser integrals are given their indicial dependence to illustrate the formalism. 3.4 Solution in the Outer Core Equations (21) now have to be solved in the outer core. At the inner-core boundary r is of the order of 10̂ m and for low degree spheroida modes CO is typically 10 ̂ rad. sec"^ The degree t of the induced field is likely to be a low integer, say typically $,= 5, then 0 << |i'tOjA0er| ? to about IO - 8. (22) The equations are thus well approximated by dr l \ (23) i l O t / ) - 3 Lv> where ot = loOp̂ o*. i t is convenient at this point to introduce the electromagnetic skin depth S (Landau and Lifshitz, p.195) and its relation to C K , A graph of S against period T of the field b_, or y_, is shown in Fig.5 for the periods involved in the present discussion. Independent solutions to the homogeneous form of (23) are e , e* for both fS^ and f ^ . Particular integrals can be found from these solutions and their Wronskian by a well known method (e.g. Morse and Feshbach, p.528). The solutions of (23), valid for < 5> CL are obtained in the usual way? a -e 'jVW^j ^ (24, where *<co (3 •= focCr -O , iS'= i'* (<v~a). The constants in (24) can be found by matching the solutions at the boundaries r = a, r = b to solutions determined for the inner core and the mantle." At the inner-core boundary a choice must be made on the induction probable in the inner core. Using equation (7) the longest r s / = C e 4 De - { e J e. 39 In T (SECS) Electromagnetic Skin Depths for the Core. decay time for a static field in the outer core is of the order of 15000 yrs and for the inner core is of the order of 2000 yrs- However the main dipole field is at present decreasing, has polarity reversals which occur in less than 10^ yrs, and can be characterised by variations of the order of 10^ yrs (Kaula, 1968, p.133). Any harmonic component of these variations with a period of about 10^ yrs will penetrate through the entire inner core (Fig.5). This assumes the inner core has the same conductivity as the outer core. There is thus some evidence for a leakage of the field into the inner core, but the nature of the field is uncertain because the temporal variations in the outer core are not well analysed at the present time. Subsequently the inner core field will be ignored and the equations (21) are solved as a homogeneous system with f(r) = g(r) = 0. Letting OC r o t / * , y= r"Ŝ the first equation of the pair (21) is a Bessel equation with a solution u s A x j (x), regular at the origin. A prime denotes radial differentiation and i' (x) is a spherical Bessel function. The solutions are to be evaluated at r = a, so X ~>7 ) . and for x large, (Abramowitz and Stegun, 1965, p.364). Near r = a, & and thus the solutions are (25) The boundary conditions for continuity in normal and transverse fi e l d s (Section 2.5) imply, using (17), Continuity in , L ( r ^ k ) a n d tji ? (*"-<K) ( 2 6a) Assuming an insulating mantle, the radial component of j : - (Vxb) — / < • * © must vanish, thus t / = 0 ? ( <r =b) (26b) The poloidal f i e l d within the mantle is then obtained from (21) , and i t satisfies with solution p. i , where (d) is the f i e l d value at the Earth's surface. The poloidal f i e l d then satisfies d (raj 4 ) 4 1st - o , C***l>) . (26c) fir ; There are now five boundary conditions (26a) - (26c) for the six constants in (24) and (25). The sixth condition is supplied by the requirement that the tangential electric f i e l d i s continuous at r = a. Using (4) and (5) and for the perturbation part of E_ this implies continuity in the tangential part of _ ^ f V )< b ) - * § 0 ) . If \r, is continuous , then Continuity in i ( f t f c ) ? ( r =. A) (26d) supplies the last required condition. A discontinuity in v_̂ at the inner core boundary leads to a boundary toroidal f i e l d generated by shear (Smylie, 1965, p.175). This will not be treated in the present work. The boundary conditions (26a) - (26d) serve to determine the six constants A - F in (24) and (25). Omitting the algebra, the resulting expressions for the perturbation fields are - j ) - , * U . e j W d r - j a e 3 W * J In each of the integrals in these equations, the exponential becomes large at one end of the integration. The asymptotic evaluations can be easily obtained, almost by inspection (Jeffreys and Jeffreys, 1956, p.503), J<J, J l ' « Writing e - €- e for r"-ft. > ft-h where n is a small integer, With these approximations the perturbation fields are 7 (27) applicable everyshere in the outer core except for the boundary layer near r = a (Region III, Fig.6). The solutions f a l l naturally into two parts, the first is for Region I (Fig.6) and the exponential is for Region II. Equations (27) indicate that rs^ continues on into the mantle across r = b, while rt^ drops to zero within the boundary layer. Fig. 6 . Induced Field Regions in the Outer Core. 4> SECTION 4 ENERGY CONSIDERATIONS The perturbation magnetic fields derived in the last section are generated by the kinetic energy of the free oscillations. To formulate exactly how the transfer of energy takes place, the energy balance is considered. The role of viscosity is treated as an integral part of the discussion. 4.1 Energy Equations If the Earth were perfectly elastic there would be no damping of the free oscillations. The total energy of each harmonic is then divided equally between kinetic and potential energies, the potential energy being the sum of elastic strain energy and gravitational energy (Kovach and Anderson, 1967, p. 2162). It is useful to remember that over a cycle of an oscillation, velocity dependent forces result in dissipation of energy, whereas amplitude dependent forces do not. It is clear that one simple model cannot describe the behaviour of the Earth under a l l conditions within its interior (Jeffreys, pp. 6-13). The mantle behaves as a nearly perfectly elastic medium and the outer core has the properties of a fluid, notably low (or zero) rigidity. To describe the departures from elastic behaviour, the Kelvin-Voigt model (Bland, p. 2) is taken for the outer core because i t represents closely the frequency dependence of the attenuation of elastic waves in liquids (Knopoff, 1964, p. 635). The total stress field is taken as simply the sum of the elastic stress, which depends linearly on the strain, and the viscous stress which depends linearly on the strain rate. In this model, when the medium is stressed, there is a delay in the attainment of the strain that would occur in a purely elastic medium. The subsequent loss in elastic energy can be accounted for by the usual expression for dissipation in a viscous medium (Lamb,1932, p. 579). Consider now, within the core, a small element of material of volume JV, bounded by a surface &S and containing an internal heat energy U per unit mass. With reference to Section 3.2 for the interpretation of quantities, the rate at which body forces do work on the element is and the rate at which surface stresses do work is these forms are obtained by following Love (1944, pp. 93-94). Adding these two expressions, the total rate at which work is done on the element per unit time is fV. p -v T . djr, + T,*j to (28) J o a t J $Xj J d o - using the equations of motion (2). The equation of continuity, or mass conservation, requires that so neglecting the products of quantities small with the velocity, This is the rate of change of kinetic energy density. To interpret the second term of (28), the strain energy function W (Love, 1944, p. 94) is introduced and defined by (30) where is the strain tensor and C ,'• the strain deviator (Bullen, p. 32). The terms on the right hand side are the compressional and shear elastic energy densities. Denoting a time averaged quantity, by an overbar, W The tr.ird term in (28) is the rate of viscous dissipation per unit volume (Landau and Lifshitz, 1960, p.214). Combining the above relations (2), (28), (29) and (30) - - ^ ^ ^ - ^ ' l / h ^ ^ - ^ - r ^ ̂ {3i) The rate of change of magnetic energy density can be written in a similar form (Roberts, p. 18), i / J - aM = -JL v. (£ * 8 ) ->r. f j x g j - I T 1 C32) where the terms on the right are respectively, the outflow of energy from <W given by the Poynting vector, the rate at which work is done by the magnetic volume force against the deformation and the energy lost by ohmic dissipation. Elastic deformations are assumed adiabatic (Bullen, p. 74), so that heat conduction between the element and its surroundings can be ignored. The energy balance is maintained by the rate of increase of internal energy expressing the First Law of Thermodynamics. Equations (31) - (33) are now added and integrated throughout a volume V to obtain the change in total energy per unit time. +J(T j + x ;j ) IT, n, A5 - j X (| x8). n dS C 3 4 ) In (34) Gauss' theorem has been used to get the surface integrals, n are components of the normal vector n. and S is the surface bounding V. The terms on the right hand side of (34) are interpreted as follows. The first term is the gravitational energy passing into V, the second term is the work done by the elastic and viscous stresses over the boundary S and the last term is the electromagnetic energy flowing out of V. If V is now taken to be the core, S is the core-mantle boundary. It can be shown that, because the mantle is considered an insulator, the Poynting vector vanishes just within the mantle. The mantle also has a high rigidity by comparison with the outer core and this indicates there is negligible transfer of energy across S by the viscous stresses. The energy within the core is thus maintained by the flow of gravitational and elastic energy. For plane xvave motion, the elastic energy flow is called the intensity of the wave (Morse and Feshbach, p. 151). Within the core the elastic and gravitational energy is stored as the potential energy of the displacement field. The internal heat energy of the core contains contributions from viscous and ohmic dissipation which are positive definite and therefore result in a net loss of energy per cycle. Finally there is a conservation of the energy transferred between the displacement field and the magnetic field through the action of the Lorentz force. Writing E, and E for the kinetic energy and the magnetic energy K. tn. within the volume V, and t r , - J 2)4. it J <r ( ' where is the rate of change of energy due to viscous dissipation. 4 9 In (35) v Is taken as the free oscillation velocity and the two equations of (36) can be linearised by using (8a) in order to assess the contributions of the main and pertubation magnetic fields. Consider the magnetic energy arising from this linearisation, The first term on the right hand side gives the magnetic energy stored in the main field and the second term is linear in b_ and so averages to zero over a cycle. The last term is the magnetic energy stored in the perturbation field, and is denoted subsequently by e . The ohmic r m dissipation rate reduces in the same way, where the first term is the dissipation in the main field and the second term averages to zero over a cycle. For the perturbation field 4fUo and <ru> J ~~ m „ • - - OS) where a dot signifies time differentiation. Also averaged over a cycle, the kinetic energy and viscous dissipation are respectively, r , » w x j j * , . **cJV C 3 9 ) 4 J J ~ - 1 and I - i f & dV t 40) Equations (38) - (40) are the basis for assessing the attenuation of the free oscillations. 4.2 Ohmic Dissipation Consider first the kinetic energy. For spheroidal oscillations the displacement u. is given by the vector components (18) and these are substituted into (39) . The angular part of the volume integral can be evaluated using the spheroidal vector orthogonality properties (Smylie and Mansinha, 1971, p. 338). For a particular harmonic the time averaged kinetic energy in the outer core is C41) The quantities U and V denote the radial and transverse displacements for a mode where they are taken as In terms of the magnetic field, equation (38) for the ohmic dissipation per cycle is /u 0 ViO J Because the curl of a poloidal vector is a toroidal vector and vice- versa, the angular part of this integral follows from the orthogonality properties of these vectors on a sphere (Smylie, 1965, p. 172). The energy dissipated per cycle by the perturbation field is Y a X ^ ^ Tr' 1 V 1 1 J C42) where Because only self interactions are involved, i t is clear that a l l pert- urbation fields dissipate energy. In a similar way, the magnetic energy averaged over a cycle becomes ^ M 2 & 0 '-<- k '! (43) Expressions (42) and (43) are to be evaluated in the next section for particular interactions. 4.3 Viscous Dissipation The stress-strain rate relations for viscous deformation can be written where It is easily shown that which can be compared directly v/ith the strain energy function W, 52 equation (30). Using (40) the viscous dissipation can ba expressed as • . where the time dependence of and £ j j i s that of CP£ . For comparison the elastic energy stored per cycle i s showing the close equivalence between Ey» and If the elastic and viscous parameters are constant within a given volume V, the ratio of the rate of viscous dissipation to elastic strain energy can be expressed in terms of these constants and the angular frequency only. To i l l u s t r a t e this property the following table i s presented, quantities being averaged over a cycle. Table 2 Integrals for Viscous Dissipation and Elastic Energy Deformation Viscous Dissipation (joules) Elastic Strain Energy (joules) Ratio Compressional Shear 2 ^ ; ^ K 4fTtJ \ ~7T Explicit expressions for the elastic strain energy are given by Kovach and Anderson (1967 , p. 2157). The f i n a l form for the viscous dissipation can be written down by the analogy given above (44) where a prime denotes radial differentiation. Higgins and Kopal (1968) have derived the dissipation rate for shear viscosity but their final expression contains a typographical error. 4.4 Q To measure the energy dissipation per cycle of an oscillation, a specific dissipation function Q is introduced where E is the peak energy and 5 _ the dissipation rate (Knopoff, 1964, dt p. 626). The particular energy to which E refers depends on the damping mechanism. For instance, in the mantle the damping is due to anelastic effects and so E refers to the strain energy stored per cycle (e.g. Anderson and Archambeau, 1964). Ohmic dissipation can be considered as an imperfection in inertia and thus E must strictly be taken as the kinetic energy. In practice, because energy dissipation is here only expected to be a small effect, i t makes l i t t l e difference which inter- pretation is placed on E. One advantage of using Q as opposed to Q is that for a layered 54 system with energy per layer, Q ^ for a l l the layers is a linear combination of Q_^for each layer, -I i (Jackson and Anderson, 1970, p. 4). For a two-layered Earth, a core (c) and mantle (m), - 1 (45) where E is the total energy. To observe a for the core from an observed Q for a mode, suppose a change of 10% can be detected seismically. i.e. A Q ^ 1 0 " * ' . Then from the previous equation, this implies If the mantle has an infinite Q or there is no kinetic energy in the m mantle, the observed Q will be the Qc for the core. In this case d^c (46) db so that the kinetic energy integral given by (41) can be extended over the whole Earth, for a Q due to dissipation only in the core. Following Knopoff (1964, p. 626), the Q defined by (46) can be related to a damping factor, for an oscillation with displacement u™ and angular frequency ui by n (47) where is applicable for the attenuation of a standing wave. In Table 2, the ratio of viscous dissipation to elastic strain energy is given for medium uniform in the viscous and elastic constants. Regarding viscous dissipation as an imperfection in elasticity, the ratio of energy lost to energy stored defines a Q. For shear the Q is and this agrees with the expression given by Knopoff (1964, p. 635) for acoustic loss in Kelvin-Voigt solid. SECTION 5 PARTICULAR INTERACTIONS The discussion in the previous sections can now be applied to calculate the Q of an oscillation due to magnetic and viscous damping. The toroidal quadrupole field component has been chosen as probably the strongest field in the core, and therefore of greatest geophysical interest. Table 9 in Appendix A indicates there are no resonance inter- actions to be considered and so the choice of a velocity field can be made on dynamical grounds. 5.1. Radial Oscillations It is well known (Ness et a l . , 1961) that the fundamental radial oscillation has a high Q, which is explained by the small amount of shear energy, relative to compressional energy, in the mantle (Kovach and Anderson, 1964, p.2162). The radial overtones also have high Q's (Dratler et a l . , 1971), and i t can be seen from Fig.7 that a l l these oscillations have appreciable energy within the core. It is therefore natural to treat first the radial oscillations; they also have no transverse displacement and this simplifies the mathematical treatment. From the interaction diagram (Fig 4), the only field induced by the oscillations S 0 is the. toroidal quadrupole t° . Defining T(r) as the radial function of the harmonic T° and U(r) as the radial displacement of the eth overtone of a radial oscillation, o.0r) - ; u > f * r ( r U r y > ( 4 8 ) 5 7 Fig. 7 . Amplitudes of the Radial Oscillations in the Earth. In this and subsequent equations a prime denotes a radial differentiation. The radial function for the induced field becomes rt° t « - i ( r U T ) ' 4 i (rVl) * ( 4 9 ) using (27) and (48). The subscript b indicates the term (rUT) is to be evaluated at r = b. Similarly and so .CosLc=±) 4 (rVT) h e j (5Q) The relative strengths of the induced and main magnetic fields can be estimated using (49). For a uniform field, T(r) is constant, Ifcal , U + U' V 1 C51) The perturbation expansion is thus invalid near the origin r = 0, and wherever the gradient of the displacement field is of order unity. Neither of these conditions hold in the outer core, the first does not because simply r a. The gradient of the radial displacement is generally of order 10 (Fig.7) and because the radial displacement is required to be continuous across r = a,b, large gradients are unlikely to occur at the boundaries. This limitation on the linearisation is similar to that described by Lilley and Smylie (1968, p.6529). The energy in the perturbation magnetic field, averaged over a cycle, is obtained from (43) b ^ 5ju0 J a fca C* * ' <52> and can be computed directly using (50). Contributions from the boundary layer, (Region II, Fig.6) can be obtained analytically. The exponential terms in (50) decrease rapidly to zero within a few skin depths of r = b. Providing the term (rUT) does not grow exponentially within the boundary layer, and this is arranged by the choice of T, the term can be considered constant in the boundary layer. The contributions to (52) from the boundary layer are then where 3 - (tZ& and a - lb" -b) On integration this contribution becomes and the magnetic energy is now (53) For the main field the magnetic energy density is @* /«2jW which gives for T° alone, The total energy dissipated per cycle is obtained from (42)f where (50) is appropriate for the first term in the integrand. The second term can be readily reduced by substituting with (49) and the rate of ohmic dissipation then follows, + 2(rVT)' f^T^f l + Vl)'ccS\ + {«VT)"(u>si k j L r2 ic 7 The approximations are found to be valid near r = b and the rate of ohmic dissipation per cycle reduces to (55) The kinetic energy per cycle is simply (56) Following the discussion in Section 4.4, the limits to the last integral can be taken as 0 and d if the dissipation within the core is considered as the only energy loss in the Earth. The effective Q associated with the magnetic damping is thus, from (46), ^ • « (4TI E H ) " im , (57) where the peak kinetic energy is used in accordance with the definition of Knopoff (1964, p.626). Equations (53) to (57) are now used to compute the various quantities for the interaction. It remains to specify the radial functions U, T within the regions of the integrations. Consider first the radial displacement U. In Fig.7 the amplitude of the radial displacement is plotted versus radius for the purely radial mode and the first four overtones. A recent publication on the modes of the Alaskan earthquake (Dziewonski and Gilbert, 1972) does not l i s t observed eigenperiods above 4S Q for the radial oscillations (their Table 7). The amplitudes of U in Fig.7 were established using a recent Earth model supplied by Jordan and Anderson (1972). Some details of the computations for the displacement field are to be found in Appendix B. As mentioned in the Introduction, the agreement between theoretical and observed eigenperiods gives considerable confidence in the broad properties of the Earth models currently in use. For the present purpose the choice of model is not critical because amplitudes are not very sentitive to minor changes in Earth structure. Choice of a radial function for the toroidal magnetic field is open to some speculation. Only approximate indications of the strength of T ° have been obtained by dynamo theory. Following Bullard and Gellman (1954, p. 275) the maximum field strength B is taken as 480 gauss. To cover the simple types of f i e l d , three elementary fuctions are used in the computations. The f i r s t f i e l d , called Type A, assumes a constant value throughout the outer cere. The two functions called Types B and C have uniform gradients, and the fourth f i e l d , Type D, is a sinusoidal function with a variable number of oscillations in the radial direction (given by the index n). These functions are shown in Fig.8 together with their radial forms and f i r s t and second derivatives. In Table 3 the magnetic energy within the core is shoxra for the various f i e l d s , computed from equation (54), together with the f i e l d and i t s f i r s t two derivatives at the core-mantle boundary. Table 3 Main Magnetic Field Parameters Field Type Energy E J m Field at Core-Mantle (joules) Boundary (m.k.s . units) T T' T" x 10" 4 x 10" 8 x 10 A 1.17 x 10 1 8 480 0.0 0.0 B 5.72 x 10 1 7 480 2.1 0.0 C 17 2.19 x 10 0.0 -2.1 0.0 D (n = 1) 5.61 x 10 1 7 0.0 -6.6 -.1.4 Cn = 5) 5.86 x 10 1 7 0.0 -33.2 15.5 (n =10) 5.86 x 10 1 7 0.0 66.4 1629.3 63 FIELD AMPLITUDE IN TYPE OUTER CORE RADIAL FUNCTION T(r)° Bm T' = O T" - O T(r)=Bm(r-a)(b-a) T' -Bm /(b-a) T" * O T(r) - Bm (b-r)/(b~a) T' =-Bm/(h-a) r = O T(r)s BmSlnft T's n?r Bm cos ft T''=-}j^)2BmSin,f ft ~ rnr (r-a)/(b~a) F i g . 8. Radial Functions for the Toroidal Quadrupole Field. 5.2 Results for the P^adial Oscillations. The results of the computations of Q values are presented in Tables 4, 5 and 6. It is immediately clear that the Q's for the interactions are so high that the oscillations are virtually unattenuated. Observations of these Q's can probably be safely dismissed under a l l conditions. .Further evaluation and discussion of the results is interesting from a physical, rather than a practical, point of view. A comparison of Tables 4 and 5 for the fundamental radial oscillation confirms the gradient effect (Lilley and Smylie, 1968). From Table 3, field Type A has about twice the energy of Type B and when this difference is allowed for, the energy dissipated in the Type B field is an order of magnitude larger than that dissipated in the Type A field. For the overtones however the gradient effect is not evident, e.g. 3S Q has a relatively large amount of dissipation for both field types. In Fig.7 it can be seen that 3S Q has the steepest gradient at r = b in Region II. The damping is thus seen to be dependent on the combinations of gradients in both T and U, as expected from the terms appearing in equation (55). It is also clear that most of the ohmic dissipation takes place in Region II where there is a large gradient in the induced field due to the requirement of no induction in the mantle. In Table 5, Region I dissipation has been excluded because i t is negligible. In Table 6, where the field Type D is sinusoidal, again most of 2 the dissipation occurs in Region II, and i t is proportional to n Table 4 Radial Oscillations in a Uniform Field Oscillation Period Kinetic Skin Induced Energy Effective (sees) Energy Depth Field Dissipated per Q per cycle (m) Energy cycle (joules) (joules) per cycle (joules) Region I Region II oSo 1227.65 2.9 x 10 1 5 32.2 4.4 x 105 0.0085 52.0 7xl0 1 4 l S o 614.15 1.6 x 10 1 6 22.8 1.9 x IO6 0.035 430.0 14 5x10 2 o 398.56 2.3 x 10 1 6 18.3 4.7 x IO6 0.030 200.0 lx l O 1 5 3 o 305.62 3.6 x 10 1 6 16.1 6.8 x 106 0.088 1000.0 14 4x10 / s 243.64 5.5 x IO 1 6 14.3 1.2 x IO7 0.11 64.0 lx l O 1 6 4 o ON On Table 5 Radial Oscillations in a Field with a Uniform Gradient Oscillation Induced Field Energy per cycle (joules) Energy Dissipated per cycle (joules) Region II Type B Type C Effective Q Type B Type C S 1.0 x 105 200.0 49.0 14 1.8 x 10 7.6 x io 14 o o ,S 8.9 x 105 210.0 39.0 14 9.6 x 10 5.2 x io 15 1 0 oS 9.4 x 105 420.0 39.0 6.8 x 10 1 4 7.4 x io15 2 o 2.1 x 106 930.0 3.0 14 4.8 x 10 1.4 x io 17 3 o /S 3.4 x 106 180.0 30.0 15 3.8 x 10 2.2 x io 16 4 o Table 6 Radial Oscillations in a Sinusoidal Field Oscillation n Energy per cycle inlnduced Field(joules) Energy Dissipated per cycle(joules) Region I Region II Effective Q 1 9.3xl05 0.03 4.8xl02 13 2.6x10 •o s o 5 2.2xlO ? 7.3 1.2xl04 12 3.0x10 10 . ... . 8.7xlO? 110.0 4.8xl04 2.6X1011 1 6.6xl06 0.11 3.9xl02 14 5.2x10 i s o 5 1.3x108 21.0 9.7x10 3 13 2.0x10 10 5.1xl08 320.0 3.9xl04 12 5.2x10 1 3.9xl06 0.09 3.8xl02 14 7.4x10 2S0 5 4.6xl0 7 6.1 9.6xl03 13 3.0x10 10 1.8xl08 79.0 3.8xl04 12 7.4x10 1 5.5xl06 0.11 32.0 1.4xlOib 3S0 5 5.0xl07 5.4 8.0xl0 2 14 5.6x10 10 1.9xl08 66.0 3.2x103 14 1.4x10 1 6.5xl06 0.15 3.0xl02 2.4xl015 4S0 5 4.0xl07 4.4 7.5xl03 9.2xl0 13 10 1.4xl08 43.0 3.0x104 2.4xl0 1 3 68 This can easily be verified by substituting the radial function into (55). Thus Q is proportional to n , a clear indication of the gradient effect. It should be noticed however that for a sinusoidal field, the energy dissipated in Region I increases faster than n and for n greater than about 104, Region I becomes the dominant region for dissipation. This would be a rough field indeed, with a peak-to- peak amplitude of nearly 10 gauss, and a wavelength about 200m. If a comparison is to be made between the fields here and the field functions resulting from dynamo theory, the choice of a sinusoidal field, with n = 1 is probably the closest (Bullard and Gellman, 1954, p.246). A typical Q for the radial modes can therefore be taken as 10"^ for this type of field. 5.3 Non-Radial Oscillations The discussion of Section 5.1 can be easily extended to the case of non-radial oscillations. With the notation used in Appendix A, equations (22) can be written. where, for the non-radial oscillations, the transverse displacement is added to the term on the right hand side of (48). The previous theory can then be extended by substituting in place of in the expressions (53) and (55). According to Table 9 (Appendix A), a, is zero for a l l the fundamental oscillations and so only toroidal fields will be induced. Several oscillations are chosen as representative motions in the outer core. Fig.9 shows a plot of the amplitudes with depth for the radial displacements of several overtones. The fundamental oscillations S , S , and S have been shown in Fig.3. The overtones of the O 2 ' O 5 ' 0 10 6 spheroidal oscillations S2 are seen to have appreciable energy within the inner core, while the amplitudes of the overtones 2S 2 and S are large at the core-mantle boundary. Such oscillations, with most of their energy at the core-mantle interface, are referred to as Stoneley waves (e.g. Alsop, 1963, pp.498-499). In Table 7 results are presented for the ohmic dissipation of these modes with the energy dissipated given mainly for Region II as before. The energy dissipated is shown for each of the induced fields produced by the interactions in Fig.4 and Table 9 (Appendix A). To obtain the Q for the mode these energies are then summed to find the total dissipation. 5.4 Viscous Damping Expression (44) gives the energy dissipated as viscous heating for an oscillation in the outer core. It is straightforward to program *7 A RADIUS (! 0$ KM) F i g . 9. A m p l i t u d e s o f S p h e r o i d a l O v e r t o n e s i n t h e E a r t h . Table 7 Non-Radial Oscillations in a Sinusoidal Field (n=l) Oscillation Period Kinetic Energy Dissipated per cycle Effective (sees) Energy per cycle(joules) for each Induced Field(joules) Q t° 14 1.9x10 C4 13 1.5x10 0S2 3232.35 36.9 119.0 1S2 1469.34 15 5.7x10 33.7 109.0 14 5.0x10 2S2 904.32 2.7xl015 1.56 5.06 5.1xl0 1 5 o t- r° 3 • t5 0S5 1190.64 14 5.2x10 28.7 0.81 14 2.2x10 1S5 729.40 14 6.6x10 5.53 0.16 15 1.5x10 2S5 660.06 6.1xl016 584 16.5 14 1.2x10 o ho o fc12 0S10 580.20 14 3.8x10 0.28 2.5xl0~ 3 0.24 9.6xl0 1 5 1S10 466.09 2.4xl015 21.2 0.18 17.7 7.7xl0 1 4 2S10 416.03 14 9.2x10 0.45 3.9xlO~ 4 0.37 1.4xl0 1 6 this expression and insert the estimates for the viscous coefficients assumed for the core. Gans' (1972) estimate for - ) is used (Table 1) but there is no available estimate for % s o It arbitrarily taken equal to ^ . To calculate the Q for the dissipation only shear energy is used. The evaluations are given separately for the contributions from the bulk and shear viscosities and are shown in Table 8, for those oscillations considered for magnetic dissipation. As expected, those oscillations with large amplitudes at the core-mantle boundary have more shear dissipation than the other modes. This does not necessarily lead to a lower Q for these modes because Q is amplitude independent (the energy of the mode also depends on amplitude). It can be seen that the shear dissipation for the fundamental radial mode Q S 0 is small compared to the other oscillations. This accords with the case of elastic strain energy in the mantle (Kovach and Anderson, 1967), as i t should since the basic integral is the same (Table 2). Aside from, the radial oscillation a typical Q for shear viscous dissipation can be taken as 10''""', of the same order of magnitude as the magnetic Q. Table 8 Viscous Damping Oscillation s V n Degree Overtone Energy Dissipated per cycle (joules) Shear Bulk Effective Q for Shear Dissipation Q. 0 1 2 3 4 4.0 x 10 9.6 22.9 48.7 82.3 0.95 7.2 16.1 25.5 55.7 9.2 x 1 0 ^ 2.1 x 1 0 ^ ° 1.2 x 1 0 ^ 9.2 x 10^1 8.4 x 10 4 2 0 1 2 0.41 1.78 2.64 2.8 x IO-2, 4.9 x 10 ~ 0.85 5.8 x 1 0 ^ 4.0 x 10 b 5 0 1 2 1.53 0.76 100.64 2.5 x IO - 3 2.0 x 10 4.8 4.3 x IO1.5 1.1 x 107)? 7.6 x 10 ^ 10. 0 1 ' 2 0.24 134.55 0.70 2.8 x 10" 3 3 - ° - 2 1.4 x 10 2.0 x 1 0 ^ 2.3 x l o r * 1.7 x 10 1 6 SECTION 6 SUMMARY AND CONCLUSIONS The outcome of the discussion in the previous Section is that the free oscillations of the Earth are negligibly attenuated by either the magnetic f i e l d or the f l u i d viscosity within the core. It remains now to give a brief summary of the results in comparison with former investigations, to indicate the direct geophysical consequences, and to offer some speculation on possible implications. 6.1 Comparison with Previous Results As reviewed in the Introduction, Knopoff (1955) calculated that magnetoelastic interactions were unimportant within the Earth's f l u i d core. The work of Kraut (1965) and the refinements of L i l l e y and Smylie (1968) did not significantly alter this conclusion. The suggestion of Lilley- (19.67) that the gradient effect might lead to enhanced damping of standing elastic waves has now been followed up with the refined analysis presented in this thesis. The Q's obtained by Kraut and L i l l e y and Smylie were of the order of IO"*" 7 . The Q's determined from the present study are variable, depending, as:expected, on the oscillation chosen and the radial function of the toroidal magnetic f i e l d . For a f i e l d suggested by the Bullard-Gellman dynamo model, that is a half-sinusoid within the outer 13 core, the lowest Q obtained was 10 for the fundamental spheroidal oscillation 0 S 2 . At i t s strongest the interaction is thus a few orders of magnitude larger than previously obtained. The Q can be arbitrarily lowered by making the radial function rougher, but the j u s t i f i c a t i o n is minimal, and there does not appear to be a further mechanism for lowering the Q values obtained here. The reason for the difference between the Q's here and those obtained before l i e s in the rigorous treatment of the geometry of the fields and their gradients. However there can be no hope of measuring such high Q's seismically because of the domination of the mantle 2 3 dissipation, evident by the observed Q's of 10 to 10 . The speculation of L i l l e y i s seen to be unsubstantiated, indicating, as he himself recognised, the caution necessary in interpreting order-of-magnitude estimates. Due to the lowering of the viscosity of the outer core to the value suggested by Gans (19 72), the role of viscous dissipation is also insignificant. For the 0 S 2 o s c i l l a t i o n the Q due to shear viscosity is 16 now of the order of 10 , even higher than the magnetic Q. This does not allow for the effect of the second coefficient, or bulk, viscosity. 6.2 Geophysical Implications Several comments can be made concerning the direct implications of the theory. It w i l l be assumed in the following that a 'normal' 3 free oscillation refers to the mode Q S 2 which has a period of 3 x 10 13 sees, and a Q of 10 , obtained from ohmic dissipation in the boundary layer (Region II) of a magnetic f i e l d of Type D with n = 1 (Fig. 8). Using equation (47), the damping time of such an oscillation in the core can be written U) (58 8 and i s equal to about 10 years. The dependence of the kinetic energy and ohmic dissipation on the pertinent parameters in the core, can be obtained by considering equations (55) to (57). Denoting dependency by the symbol^, from (55), and for Region I (M 0 a cruJ<c ( 5 9 ) where U and B are typical values of oscillation amplitude and magnetic f i e l d strength, and r ^ is a representative core radius. Similarly from thus using (57), (59) and (60) W - l ^ — 7—~ 6 For a Q in the boundary layer ^0 CT to & (60) (61) Q-tr ~ . (62) These expressions for Q, (61) and (62) , show a strong dependency on angular frequency such that, as the frequency is lowered, the dissipation in the. bulk of the core becomes more important. The skin depth also increases. Considering the boundary layer dissipation, the relation of Q with period is Period Q z ~ Period , (63) so that for a displacement f i e l d Q S 2 at the period of the Chandler 3 Wobble fo-r example, the Q is about 10 . This is not directly applicable to the actual Chandler Wobble because at such a long period the displacement f i e l d is not expected to have the character of a free o s c i l l a t i o n . The results so far have been discussed for their negative aspect, that i s the unlikely use of the damping in determining the structure of the magnetic f i e l d within the core. However interesting implications can be suggested in connection with the behaviour of the interaction at periods longer than the free modes and their overtones . It has long been known that free oscillations can exist at longer periods than the fundamental for each degree of the harmonic expansion of the displacement f i e l d (Alterman et a l . , 1959). The presence of these oscillations depends on the density distribution within the core and Alterman et a l . found the oscillations for only one of their t r i a l Earth models, Bullen B. These oscillations have most of their energy in the core, and so by the previous reasoning might be expected to be undamped by magnetoelastic or viscous interactions. With damping times for two oscillations given by the ratio obtained from (58) and (63), the damping time of a core oscillation might be about 10 7 years i f there is no energy dissipation in the mantle. Is is difficult', as pointed out by Alterman et a l . , to understand how these core modes can be excited by a source in the mantle because of the very fact which makes them interesting here, namely their being confined to the core. 78 At periods corresponding to the bodily tides of the Earth, that is at periods at multiples of 12 hours, the Earth responds in a second degree forced oscillation. The theory of the oscillations of a real Earth model at such frequencies is not as straightforward as for the normal free modes because the Earth's diumal rotation might be expected to be important. Also, in the static limit (when the frequency is allowed to be zero), the behaviour of the liquid core becomes quite different from the dynamical case. As Smylie and Mansinha (19 71) have shown, the system of equations (equation (Bl), Appendix B) degenerates to a second order system with the motion determined entirely by gravitational forces. That i s , the core responds passively to the gravitational perturbations and the elastic stresses are no longer important. Pekeris and Accad (1972) have discussed the behaviour of the liquid core at tidal frequencies, but there is some doubt as to the correctness of their asymptotic theory in the manner of letting the angular frequency go to zero (Smylie, personal communication). The nature of the oscillations in the core at these frequencies is at the present stage unclear. The damping of these oscillations can however be inferred in a cautious manner from (63) and (64) without knowing precisely the displacements in the core, at least for an order of magnitude estimate. At periods of 24 hours the magnetic Q is found to be of the order of 9 5 10 and the damping time is about 10 years. These oscillations will then also be considerably underdamped and may be expected to persist as long as the longest temporal variations of the geomagnetic field. Such a coincidence of time scales is in a l l probability fortuitous, and could be dismissed entirely but for the following speculation. It has already been mentioned that the arguments of Higgins and Kennedy (19 71) have raised serious doubts as to the existence of the large scale convection of the core required for the conventional dynamo theory. The recent emphasis on turbulent induction processes (Moffatt, 19 72) and the development of models with a cellular flow in the core (Gubbins, 19 72) has indicated that alternative mechanisms are possible. A suitable energy source has yet to be established. As Moffatt indicated, the requirements on the velocity f i e l d are that i t should have no mean flow and that the motion lack reflexional symmetry. The time scale of the fluctuations also has to be long compared to the diurnal rotation, at periods of a month or more (Moffatt, 19 72, p. 398). If the core does respond to oscillations of such long period, and this w i l l depend on the adiabaticity, or degree of s t a b i l i t y (Pekeris and Accad, 1972), the associated velocity fields may well satisfy the condition noted above. The results of the present work then ensure that the damping of such oscillations is only appreciable over geologic times, which may be an important contribution to future studies of the behaviour of the core of the Earth. 80 REFERENCES ABRAMOWITZ, M. and STEGUN, I. 1965 Handbook of Mathematical Functions. Dover. ALSOP, L.E. 1963 Free Spheroidal Vibrations of the Earth at Very Long Periods, Part I-Calculation of Periods for Several Earth Models. B u l l . Seim. Soc. Am. 53 (3) 483-501. ALTERMAN, Z., JAROSCH, H. and PEKERIS, C.L. 1959 Oscillations of the Earth. Proc. Roy. Soc. Lond. A 252 80-95. ANDERSON, D. and ARCHAMBEAU, CB. 1964 The Anelasticity of the Earth. J . Geophys. Res. 69_ (10) 2071-2084. BACKUS, G.E. 1958 A Class of Self-Sustaining Dissipative Spherical Dynamos. Ann. Phys. N.Y. _4 372-447. BACKUS, G.E. and GILBERT, F. 1970 Uniqueness in the Inversion of Inaccurate Gross Earth Data. P h i l . Trans. Roy. Soc. Lond. 266 123-192. BENIOFF, H. 1954 Progress Report, Seimological Laboratory. Trans. Am. Geophys. Un. 35 985-987. BENIOFF, H., PRESS, F. and SMITH, S. 1961 Excitation of the Free Oscillations of the Earth by Earthquakes. J . Geoph-s. Res. 66_ 605-619. BEN-MENAHEM, A., ROSENMAN, M. and ISRAEL, M. 1972 Source Mechanism of the Alaskan Earthquake qf 1964 from Amplitudes of Free Oscillations and Surface Waves. Phys.'Earth and Planet. Int. 5_ (1) 1-29. BLAND, D.R. 1960 The Theory of Linear Viscoelasticity. Pergamon Press. BOLT, B.A. and DORMAN, J . 1961 Phase and Group Velocities of Rayleigh Waves in a Spherical Gravitating Earth. J . Geophys. Res. 66_ (9) 2965-2979. BRAGINSKII, S.I. 1964 (Trans. 1965: Self-Excitation of a Magnetic Field During the Motion of a Highly Conducting Fluid. Soviet Phys. J.E.T.P. 20 (3) 726-759). BULLARD, E.C. 1949 The Magnetic Field Within the Earth. Proc. Roy. Soc. Lond. A 197 433-453. BULLARD, E.C. and GELLMAN, H. 1954 Homogeneous Dynamos and Terrestrial Magnetism. P h i l . Trans. Roy. Soc. Lond. 247 213-278. BULLARD, E. and GUBBINS, D. 1971 Geomagnetic Dynamos in a Stable Core. Nature 232 (5312) 548-549. BULLEN, K.E. 1950 An Earth Model Based on a Compressibility-Pressure Hypothesis. Mon. Not. Roy. Astron. Soc. Geophys. Supp. 6_ 50-59. BULLEN, K.E. 1963 An Introduction to the Theory of Seismology. Third Edition. Cambridge University Press. 8 1 COWLING, T.G. 1934 The Magnetic Field of Sunspots. Mon. Not. Roy. Astron. Soc. 94 39. CHRISTIANSEN, J. 1970 Numerical Solution of Ordinary Simultaneous Differential Equations of the 1st Order Using a Method for Automatic Step Change. Numer. Math. 14 317-324. DAHLEN, F.A. 1968 The Normal Modes of a Rotating E l l i p t i c a l Earth. Geophys. J . Roy. Astron. Soc. 16 329-367. DERR, J.S. 1969 Free Oscillation Observations Through 1968. B u l l . Seism. Soc. Am. 5_9 (5) 2079-2099. DRATLER, J . , FARRELL, W.E., BLOCK, B. and GILBERT, F. 1971 High-Q Overtone Modes of the Earth. Geophys. J . Roy. Astron. Soc. 2_3 399-410. DZIEWONSKI, A.M. and GILBERT, F. 1972 Observations of Normal Modes from 84 Recordings of the Alaskan Earthquake of 1964 March 28. Geophys. J . Roy. Astron. Soc. 21_ 293-446. ELSASSER, W.M. 1946 Induction Effects in Terrestrial Magnetism. Phys. Rev. 69 (3) 106-116. ELSASSER, W.M. 1950 The Earth's Interior and Geomagnetism. Rev. Mod. Phys. 22 1-35. ELSASSER, W.M. 1956 Hydromagnetic Dynamo Theory. Rev. Mod. Phys. ̂ 8 (2) 135-163. GANS, R.F. 1972 Viscosity of the Earth's Core. J . Geophys. Res. _77 (2) 360-366. GARDINER, R.B. and STACEY, F.D. 1971 E l e c t r i c a l Resistivity of the Core. Phys. Earth and Planet. Int. 4_ 406-410. GIBSON, R.D. and ROBERTS, P.H. 1969 The Bullard-Gellman Dynamo: in The Application of Modern Physics to the Earth and Planetary Interiors. Wiley and Son. 577-602. GUBBINS, D. 1972 Kinematic Dynamos and Geomagnetism. Nat. Phys. Sci. 238 119-122. HADDON, R.A.W. and BULLEN, K. 1969 An Earth Model Incorporating Free Oscillation Data. Phys. Earth and Planet. Int. 2_ 35-49. HIDE, R. and ROBERTS, P.H. 1961 The Origin of the Main Geomagnetic Field: in Physics and Chemistry of the Earth. _4 27-98. HIGGINS, G. and KENNEDY, G.C. 19 71 The Adiabatic Gradient and the Melting Point Gradient in the Core of the Earth. J . Geophys. Res. 76_ (3) 1870 -1878. HIGGINS, T.P. and KOPAL, Z. 1968 Volume Integrals of the Products of Spherical Harmonics and their Application to Viscous Dissipation Phenomena in Fluids. Math. Note 563, Math. Res. Lab., Boeing Sci. Res. Lab. HOBSON, E.W. 1955 The Theory of Spherical and E l l i p s o i d a l Harmonics. Chelsea. HOSKINS, L.M. 1920 The Strain of a Gravitating Sphere of Variable Density and E l a s t i c i t y . Trans. Am. Math. Soc. _21 1-43. INFELD, L. and HULL, T.E. 1951 The Factorisation Method. Rev. Mod. Phys. 23 (1) 21-68. JACKSON, D.D. and ANDERSON, D.L. 1970 Physical Mechanisms of Seismic-Wave Attenuation. Rev. Geophys. and Space Phys. _8 (1) 1-63. JAIN, A. and EVANS, R. 19 72 Calculation of the E l e c t r i c a l Resistivity of Liquid Iron in the Earth's Core. Nat. Phys. Sci. 235 165-166. JEFFREYS, H. 19 70 The Earth. Fifth Edition. Cambridge University Press. JEFFREYS, H. and JEFFREYS, B.S. 1950 Methods of Mathematical Physics. Second Edition. Cambridge University Press. JOBERT, N. 195 7 Sur l a Period des Oscillations Spheroidales de l a Terre. C. R. Acad. Sci. Paris 244 921-922. JORDAN, T. and ANDERSON, D. 1972 (Earth Model given in Ph.D. Thesis of Jordan, formerly at Seismological Laboratory, Division of Geological Sciences, California Institute of Technology). KAULA, W.A. 1968 An Introduction to Planetary Physics. Wiley and Son. KNOP0FF, L. 1955 The Interaction Between Elastic Wave Motions and a Magnetic Field in E l e c t r i c a l Conductors. J . Geophys. Res. 60_ (4) 441-456. KN0P0FF, L. 1964 Q. Rev. Geophys. 2 (4) 625-660. K0VACH, R.L. and ANDERSON, D.L. 1967 Study of the Energy of the Free Oscillations of the Earth. J . Geophys. Res. 72_ (8) 2155-2168. KRAUT, E.A. 1965 Free Radial Modes in a Compressible Conducting Fluid Sphere Containing a Uniform Internal Magnetic Field. J . Geophys. Res. 70 (16) 3927-3933. . LANDAU, L.D. and LIFSHITZ, E.M. 1960 Electrodynamics of Continuous Media. Addison Wesley. LEHMANN, I. 1936 P'. Bur. Centr. Seism. Internat. A 14 3-31. LAMB, H. 1882 On the Vibrations of an Elastic Sphere. Proc. Lond. Math. Soc. 13 189-212. LAMB, H. 1932 Hydrodynamics. Sixth Edition. Cambridge University Press. LARMOR, J . 1919 How Could a Rotating Body Such As the Sun Become a Magnet? Rep. B r i t . Assn. 159-160. LILLEY, F.E.M. 1967 Magnetoelastic Effects in a Non-Uniform Field. Ph.D. Thesis, University of Western Ontario. 83 LILLEY, F.E.M. and SMYLIE, D.E. 1968 Elastic Wave Motion and a Non-Uniform Magnetic Field in E l e c t r i c a l Conductors. J . Geophys. Res. ̂73 (20) 6527-6533. LILLEY, F.E.M. 1970 On Kinematic Dynamos. Proc. Roy. Soc. Lond. A 316 153-167. LOVE, A.E.H. 1911 Some Problems of Geodynamlcs.Dover. LOVE, A.E.H. 1944 A Treatise on the Mathematical Theory of E l a s t i c i t y . Fourth Edition. Dover. MacDONALD, G.J.F. and NESS, N.F. 1961 A Study of the Free Oscillations of the Earth. J . Geophys. Res. 66 (6) 1865-1911. MADARIAGA, R.I. 1972 Toroidal Free Oscillations of a Laterally Heterogeneous Earth. Geophys. J . Roy. Astron. Soc. 27_ 81-100. MALKUS, W.V.R. 1968 Precession of the Earth as the Cause of Geomagnetism. Science 160 259-264. MOFFATT, H.K. 1970a Turbulent Dynamo Action at Low Magnetic Reynolds Number. J . F l . Mech. 41 (2) 435-452. MOFFATT, H.K. 19 70b Dynamo Action Associated with Random Inertial Waves in a Rotating Conducting Fluid. J . F l . Mech. 44 (.4) 705-719. MOFFATT, H.K. 1972 An Approach to a Dynamic Theory of Dynamo Action in a Rotating Conducting Fluid. J . F l . Mech. 53 (2) 385-399. MORSE, P.M. and FESCBACH, H. 1953 Methods of Theoretical Physics. Part I. McGraw-Hill. MUNK, W.H. and MacDONALD, G.J.F. 1960 The Rotation of the Earth. Cambridge University Press. NESS, N.F., HARRISON, J.C.and SLLCHTER, L.B. 1961 Observations of the Free Oscillations of the Earth. J . Geophys. Res. 66̂ (2) 621-629. NOWROOZI, A.A. 1-65 Eigenvibrations of the Earth after the Alaskan Earthquake. J . Geophys. Res. 70_ (20) 5145-5156. OLDHAM, R.D. 1906 Constitution of the Interior of the Earth as Revealed by Earthquakes. Quart. J . Geol. Soc. 62_ 456-475. PARKER, E.N. 1955 Hydromagnetic Dynamo Models. Astr. J . 122 293-314. PEKERIS, C.L. and JAROSCH, H.'1958 The Free Oscillations of the Earth : in Contributions in Geophysics. Pergamon. PEKERIS, C , ALTERMAN, Z. and JAROSCH, H. 1961 Rotational Multiplets in the Spectrum of the Earth. Phys. Rev. 122 1692-1700. PEKERIS, C.L. and ACCAD, Y. 1972 Dynamics of the Liquid Core of the Earth. Proc. Roy. Soc. Lond. A 273 237-260. RAYLEIGH, Lord 1906 On the Dilatational Stability of an Elastic Sphere. Proc. Lond. Math. Soc. 13 189- ROBERTS, P.H. 196 7 An Introduction to Magnetohydrodynamlcs. American Elsevier. ROCHESTER, M.G. and SMYLIE, D.E. 1965 Geomagnetic Core-Mantle Coupling and the Chandler Wobble. Geophys. J. Roy. Astron. Soc. 10 289-315. SATO, R. and ESPINOSA, A.F. 1967 Dissipation Factor of the Torsional Mode QT2 for a Homogeneous-Mantle Earth with a Soft-Solid Core or a Viscous- Liquid Core. J . Geophys. Res. 72_ (6) 1761-1767. SMYLIE, D.E. 1965 Magnetic Diffusion in a Spherically Symmetric Conducting Mantle. Geophys. J . Roy. Astron. Soc. 9_ 169-184. SMYLIE, D.E. and MANSINHA, L. The E l a s t i c i t y Theory of Dislocations in Real Earth Models and Changes in the Rotation of the Earth. Geophys. J . Roy. Astron. Soc. 2_3 329-354. STEENBECK, M., KRAUSE, F. and RADLER, K.H. 1966 (Trans: Computation of the Average Lorentz Field Strength v x B_ for an E l e c t r i c a l l y Conducting Turbulent Medium Through the Action of the Coriolis Force). Zeit. Natur. 21a 369-376. STONELY, R. 1961 The Oscillations of the Earth: in Physics and Chemistry of the Earth. _4 239-250. STRATTON, J.A. 1941 Electromagnetic Theory. McGraw-Hill. SUZUKI, Y. and SAITO, R. 1970 Viscosity Determination in the Earth's Outer Core from ScS and SKS Phases. J . Phys. Earth 18 (2) 157-170. TAKEUCHI, H. 1950 On the Earth Tide in the Compressible Earth of Varying Density and E l a s i c i t y . Trans. Am. Geophys. Un. 3_1 651-689. TAKEUCHI, H. and SAITO, R. 19 71 Seismic Surface Waves: in Methods of Computational Physics. 11 Academic Press. WEISS, N.D. 1971 The Dynamo Problem. Quart. J . Roy. Astron. Soc. 12 432-446. WIGGINS, R.A. 1968 Terrestrial Variational Tables for Periods and Attenuation of the Free Oscillations. Phys. Earth and Planet. Int. 1 201-266. WIGGINS, R.A. 1972 The General Linear Inverse Problem: Implication for Surface Waves and Free Oscillations for Earth Structure. Rev. Geophys. and Space Phys. 10 (1) 251-285. 85 APPENDIX A THE GAUNT AND ELSASSER INTEGRALS In section 3.2 tv/o integrals K L are introduced which are the product of three angular functions. They are defined by K . c * J M i K *s * s**9dBdf 3 -'D following Bullard and Gellman (1954, p.224). A l l the indices in these expressions are positive definite and the associated Legendre functions have been defined according to Hobson's normalisation. The selection rules, Section 3.3, follow from the behaviour of these integrals for various combinations of the indices. From the azimuthal parts of the integrals, 2tT C O unless -k+^-r p a. 0 t when this integral has the value 2tT : this i s the f i r s t selection rule. The other rules follow from a more detailed examination of the properties of the associated Legendre functions (e.g., Infeld and Hull, 1951, pp.52-54). As noted in Section 3.3, the properties of K L are used to establish the fact that only a f i n i t e number of interactions contribute to a particular induced f i e l d . Because this is an important constraint on the allowed interactions producing a self-inducing magnetic f i e l d , i t is not surprising to find K L discussed at some length in dynamo theory (e.g., Bullard and Gellman, 1954). An appendix by Scott in Gibson and Roberts (1969) contains a discussion of the properties of K L and a l i s t i n g of the values of two closely related integrals G + ,E' for various indices. These are defined by fr* U,-v, hs.l) - J ' P/ ?;P/ d r where the associated Legendre functions are used in the Ferrer form as + + used by Scott. The relations between K # L (Hobson's form) and G , E (Ferrer form) are found to be K L" -k, m,)s> i , rt, S f lu til for - |< ; m 4 f> ?i 0 , and K I , n , s I, n, 5 J for - k s r v w p & 0 • Equations (22), defining the source functions for the induced f i e l d s , can be written where U,T are the.radial functions for a displacement of harmonic n,m 0 and for a main magnetic f i e l d T 2 respectively. The constants a^, b^ + + and b̂ , are related to G , E by a, - c b, c 4 [>-*(*•»») -aiiW] G- + where . . > . . , , i In order to evaluate the magnetoelastic interactions, the values - + + of a^, b^ and b2 are determined for the various indices of G , E This + + involves knowing G and E and these integrals were programmed using the formulae given by Scott (Gibson and Roberts, 1969). Table 9̂ shows the values obtained for G , E t a b^ and b^ for indices up to (4,4) for (n,m); only combinations which satisfy the selection rules are included. The constants a t̂ b^ and b^ for negative"m are found from Cr*(f>tS, rw-p, in,I) ~ pj $ t V It can be seen from the table that, although G"*~,E + can become large, the constants are generally a l l of order unity. This indicates there are no resonance interactions which might be expected to produce a large perturbation magnetic f i e l d . A search for a l l indices up to (10,10) for (n,m) and (s,p) also did not reveal a resonance value. It must be concluded that the geometry of the fields does not produce any particularly interesting effects. Table 9 Gaunt and Elsasser Integrals ro n P s 1 G+ E + a l b l b2 0 0 0 2 2 0.40 2.00 0.00 0 1 0 2 1 0.27 1.20 2.40 0 1 0 2 3 0.17 0.80 -2.40 0 2 0 2 2 0.11 0.29 1.71 0 2 0 2 4 0.11 0.51 -4.11 0 3 0 2 1 0.17 -0.51 4.11 0 3 0 2 3 0.08 0.13 1.60 0 3 0 2 5 0.09 0.38 -5.71 0 4 0 2 2 0.11 -0.38 5.71 0 4 0 2 4 0.06 0.08 1.56 0 4 0 2 6 0.07 0.30 -7.27 1 1 0 2 1 -0.27 -0.60 -1.20 1 1 0 2 2 -2.40 -0.33 1 1 0 2 3 0.69 0.27 -0.80 1 2 0 2 1 -2.40 -1.80 1 2 0 2 2 0.34 0.14 0.86 1 2 0 2 3 -4.11 -0.20 1 2 0 2 4 1.14 0.26 -2.06 1 3 0 2 1 0.69 -1.03 8.23 1 3 0 2 2 -4.11 -0.57 1 3 0 2 3 0.69 0.10 1.20 1 3 0 2 4 -5.71 -0.13 1 3 0 2 5 1.56 0.23 -3.43 1 4 0 2 2 1.14 -0.63 9.52 1 4 0 2 3 -5.71 -0.28 1 4 0 2 4 0.98 0.07 1.32 1 4 0 2 5 -7.27 -0.09 • 1 4. •o 2 6 1.96 0.20 -4.85 2 2 0 2 2 -2.74 -0.29 -1.71 2 2 0 2 3 -41.4 -0.20 2 2 0 2 4 6,86 0.09 -0.69 2 3 0 2 2 -41.14 -1.43 2 3 0 2 4 -137.14 -0.17 2 3 0 2 5 21.82 0.11 -1.71 2 4 0 2 2 6.86 -0.95 14.29 . 2 4 0 2 3 -137.14 0.67 2 4 0 2 4 8.31 0.03 0.62 2 4 0 2 5 -305.45 -0.13 2 4 0 2 6 46.99 0.12 -2.91 3 3 0 2 3 -68.57 -0.17 -2.00 3 3 0 2 4 -1440.00 -0.13 3 3 0 2 5 174.55 0.04 0.57 3 4 0 2 3 -1440.00 -1.17 3 4 0 2 4 -101.82 -0.03 -0.55 3 4 0 2 5 -7330.91 -0.13 3 4 0 2 6 845.87 0.06 -1.45 4 4 0 2 4 -3258.18 -0.11 -2.18 4 4 0 2 5 -87970. 0.09 4 4 0 2 6 8458.74 0.02 -0.48 89 APPENDIX B COMPUTING FREE OSCILLATIONS Numerical integration of the equations of motion of the free oscillations of a model Earth generally follows the treatment of Alterman et a l . (1959). The basic equations, formulated by Love (1911, Ch.VII), are f i r s t written in spherical polar coordinates and then transformed into linear first-order d i f f e r e n t i a l equations. With routine computing f a c i l i t i e s these equations can be integrated simultaneously using a step-by-step procedure, such as a Runge-Kutta algorithm. The method is well-known and details of the computations have been presented several times (e.g., Bolt and Dorman, 1961; Alsop, 1963). The purpose of this Appendix is to present details of the Earth model used in the computations of Section 5, and to examine the condition of regularity at the origin for the integration of the equations. B.l Starting Conditions Referring to Section 2.3 the equations of e l a s t i c i t y (2) can be written in the form of six coupled linear equations (Alterman et a l . . 1959) 90 4- «iS±L) ^ - j>J. X<5 ? ( B 1 ) where w is the angular frequency, and The six variables Yi-'-y^ have the following interpretations; (j( - \J radial displacement y a \X •+ 2^*U' change in normal stress _ \J transverse displacement u M T Y V ' - V ^ U J transverse shear stress p decrease in gravitational potential - P - 4-fi"G-J'pU change in gravitational flux density. The displacements U, V are identical with the radial functions Un ' Vn °^ e c l u a t i 0 1 1 (18), as the equations (Bl) refer to a spheroidal displacement of a particular degree and order. The cubical dilatation A and the perturbation in the gravitational potential are each written in spherical harmonic form thus defining the radial functions X,P. Two properties possessed by (Bl) C 3 n readily be identified: the first is that no derivatives of the elastic moduli occur, the second is that many of the coefficients on the right hand side are singular at the origin, r=0. The elastic moduli are not known accurately within the Earth and i t is to avoid errors involved in taking their derivatives that the form (Bl) is preferred. The singularity at the origin is avoided by several devices. The simplest technique is to begin the integration away from the origin and to establish starting values of the variables y_̂ by repeated integration. Three of the variables y^, y^, and y^ are zero at the origin and for a solid Earth i t would require only three integrations to obtain the values of y^, y^ and y^ at the starting depth. In practice an error is introduced by this method for those oscillations with displacements near the origin (Fig.3). A depth r 0 i s sought, generally by t r i a l and error (Bolt and Dorman, 1961, p. 2963), below which y^, y^ and y<- can be considered zero to the order of accuracy of the integration method. Such a starting set can be represented by f\ o o \ •JjK.) - ( 0 I O j , for i = 2,4,6 , and y^(r 0) = 0 for i= 1,3,5 . Clearly rQ'cannot be chosen as zero, for the derivatives of the y's are then singular. An alternative approach is to choose an r 0 below which the Earth may be considered to be homogeneous (e.g. Wiggins, 1968). Within the sphere r = r Q , the equations (Bl) are solved for c. homogeneous medium and the solutions , obtained analytically, are Bessel functions (Love, 1911, Ch.VII). The integration i s then begun at r = r Q with the values for y_̂ determined by the solution for r < r 0 . These solutions are discussed by Takeuchi and Saito (1971) and are here reproduced in the same notation as in (Bl). One independent solution is given here by 3.* y 3 r $0 a A -U>\]v J noting that Takeuchi and' Saito define y^ differently from above. Two other independent solutions are given by €.11 | i n h * > ) + f # U l U ) ( } (B2) (B3) 93 where and as given by Takeuchi and Saito. In the present work the singularity is treated by a power-series expansion about the origin for each of the six variables. The series are given by CO M. « ** L A (y f v (B4) for i = 1, 6 where of,V are integer variables and A,'yare constant coefficients to be determined. Gravity has the expansion where is the value of gravity within an arbitrarily small uniform sphere of density § 0 L°) near r = 0. Because of the complexity of the coefficients in (Bl), the derivations of the equations resulting from a substitution of (B4) are quite tedious. When lowest powers of r are compared for each equation, three independent values for W are allowed; -A s ̂ , fit v. ft -1 and For each (A there are six solutions to the simultaneous equations and a l l eighteen equations can be written by the form The index k determines the r-dependence of the f i r s t term in the series \c~o -for t > 2 , 4 (<* = n. or *-2) , L - L (S> a - i ) , 1 C - 5 C - 1,3 It i s found that for the case eisfHonly one constant i s required for the solution; ford--n,ft-2, two constants each are required giving a total of five independent constants. However the general solution near the origin must be taken as the linear combination and then the constants are reduced to three. The solution with these constants is 9l A / ^ n - H + 8 ' f n lJS> % c F ̂ n " 1 where two constants are given by The third constant i s contained in the system c > c, (B5) ( B 6 ) ( 3 7 ) 95 where , . The set (B7) can be expressed in terms of only one constant by allowing the relation F s * ( 3-n ;Y ) A ( B 8 ) and with A = n, the constants in (B6) are identical with those in (B2). To show the equivalence between (B3) and (B5) with (B7) , the set (B3) is approximated by letting r-$0, (x=»0) whence % - -(nj> 4 t ) _L_ t** x % £2/4h^n + 0 ] f jL. 4 . - % v 2' 4 3 - - u f r A ^ U i ) «L - f % ... r 2 * 4 * = - [ U ^ M 4 ^ 4 0 ^ 4 ^ ^ 2 1 2 * 4 3 ; (B9) It is then found that (B5) with (B6) to ( B 8 ) is identical to (B2) and (B9), as they should be in the limit r-fcO. t The system (B5) has y and y_̂ zero for n^3 and is thus unsuitable for a Runge-Kutta integration procedure at the origin. Following the method of Smylie and Mansinha (1971, p. 344) a change of variable is now 96 maae (BIO) giving the expansions near the origin as £ i ~ Ar 4 AV-3 • 0 , / £ . A ^ 2 2 - g 4 B ' * l - v a , 2a' •= 0 * i C r 4 C V * 4 . 0 , *; 2 , = 0 4 J ) V i 4 D , - o 0 , 0 h -- p r ^ fr'-r* 4 £fe = 0 , F > J valid at r = O. The system to be integrated is then i ; \ (Bll) where C„ is the matrix of coefficients for the right hand side of (Bl). Two integrations throughout the Earth are required to obtain the constants A, F in (Bll); the most convenient choice is (1,0) and (0,1) respectively. The system (B12) can be integrated up to some radius r when the reverse transformation of (BIO) restores the system to (Bl). To isolate the third constant, a transformation of the form is required. Then, with A-F zero, £, •=. A' r a 4 . . . Z2< = 6 ' f + •*• z 4 valid at r = 0. I, •=• o , 2,' - o 97 The equations of motion are in this case and one integration i s sufficient to find the constant. This method is then a rigorous way of starting the Runge-Kutta integration for the system (Bl). In practise i t has been found appropriate for low-degree spheroidal os c i l l a t i o n s , but i s inefficient for other oscillations because of error accumulation in the integration. The radial oscillations are simple to integrate because only three equations are involved y,' = - r l P > •* P 3» if/ - [-u/v4M* - 2 J J 3, - * r J ax The starting solutions are 3a * 6 -* 6'"*+-- with B r (SX-r2^)A . For the variables y^, y^ a n d y,- the starting set is (0,1,0) at r = 0, and at the eigenfrequency, where y^ changes sign, this set becomes i f d denotes surface values. B.2 Normalisation To stabilise the system (Bl) and prevent overflow in the computations for large n, the coefficients and the variables can be arranged to be of order unity by a simple scaling operation (Wiggins, 98 personal Communication), The appropriate changes are for the variables, and A x ID~ ,a , ̂ y ID"'25 <r * 10" \ % * ' 0"3 , G * U>\ u> x lO3 , for the Earth parameters, the original values being in c.g.s. units. B.3 The Numerical Earth Model An Earth model was supplied by Jordan and Anderson (1972). A free o s c i l l a t i o n routine was written to obtain the amplitudes of the oscillations i n the core rather than rely on published amplitudes or a packaged program. The Jordan-Anderson model (denoted here by JAB1) was used for three reasons. It (a) has a solid inner core, (b) was supplied with a complete l i s t of properties and eigenperiods useful for checking the integrations, and (c) has a good least-squares f i t to observed travel-time data and observed eigenperiods. The properties of the model are shown in Table 10. Linear interpolation was used to interpolate the model as the results were found similar to those obtained by cubic spline interpolation. A fourth-order Runge-Kutta routine was used, supplied by the Computing Centre at the University of Br i t i s h Columbia and incorporating an automatic error control on the step size, after Christiansen (1970). Table 10 Parameters for Earth Model JAB1 Radius km Depth km Density -3 gm cm Gravity -2 cm sec X Bulk Modulus 10^ 2 dyne cm Rigidity xlO 1 2 dyne cm 0 6371 12.58 0.0 12.694 1.540 100 6271 12.57 52.0 12.689 1.541 200 6171 12.56 78.0 12.674 1.539 300 6071 12.53 110.0 12.648 1.535 400 59 71 12.52 144.0 12.643 1.532 500 5871 12.51 178.0 12.642 1.531 600 5771 12.51 .212.0 12.639 1.528 700 5671 12.50 247.0 12.633 1.523 800 5571 12.50 281.0 12.632 1.517 900 5471 12.49 316.0 12.623 1.510 1000 5371 12.46 350.0 12.610 1.499 1100 5271 12.39 385.0 12.581 1.485 1215 5156 12.28 423.0 12.513 1.467 1215 5156 12.11 423.0 12.460 0.0 1300 5071 12.08 450.0 12.444 0.0 1400 49 71 12.04 482.0 12.411 0.0 1500 4871 11.99 514.0 12.334 0.0 1600 4771 11.93 546.0 12.219 0.0 1700 4671 1.1.87 578.0 12.042 0.0 1800 45 71 11. 80 609.0 11.805 0.0 1900 4 471 11. 72 640.0 11.561 0.0 2000 4371 11.64 671.0 11.307 0.0 2100 4271 11.56 701.0 11.048 0.0 2200 4171 11.47 731.0 IQ.785 0.0 2300 4071 11.39 760.0 10.542 0.0 2400 39 71 11.30 790.0 10.309 0.0 2500 3871 11.21 818.0 10.032 0.0 2600 3771 11.11 846.0 9. 718 0.0 2700 36 71 11.00 874.0 9.388 0.0 2800 3571 10.88 901.0 9.023 0.0 2900 3471 10. 76 928.0 8.628 , 0.0 3000 3371 10.62 954.0 8. 209 0.0 3100 32 71 10.48 9 79.0 7. 797 0.0 3200 3171 10. 33 1003.0 7.400 0.0 3300 30 71 10.19 1026.0 7.026 0.0 3400 29 71 .10.04 1049.0 6.676 0.0 3485 2886 2.90 1068.0 6.373 0.0 3485 2886 5.58 1068.0 4.523 2.948 3510 2861 5.56 1064.0 4.510 2.934 3550 2821 5.54 1059.0 4. 49 8 2.916 3625 2746 * 5.50 1049.0 4.471 2. 871 3700 2671 5.46 1041.0 4.412 2. 822 3775 ' 2596 5.42 1034.0 4.309 2. 775 3850 2521 5. 38 1027.0 4.194 2. 730 3925 2446 5.34 1021.0 4.079 2.686 4000 2371 5.30 1016.0 3.966 2.642 Table 10 (continued) 4075 2296 5.26 1011.0 3.863 2.601 4150 2071 5.22 1Q08.0 3. 765 2.555 4225 2146 5.19 1001.0 3.583 2.465 4375 1996 5.11 999.0 3.487 2.422 4450 1921 5.07 997.0 3.391 2.380 4525 1846 5.04 996.0 3.309 2.336 4600 1771 5.00 994.0 3.226 2.293 46 75 1696 4.96 994.0 3.136 2.253 4750 1621 4.92 993.0 3.058 2.214 4825 1546 4.89 993.0 2.977 2.174 4900 1471 4.85 993.0 2.888 2.136 49 75 1396 4.81 993.0 2.819 2.089 5050 1321 4.77 993.0 2.740 2.044 5125 1246 4.72 994.0 2.651 1.998 5200 1171 4.68 994.0 2.549 1.956 52 75 1096 4.64 995.0 2.435 1.920 5350 1021 4.59 996.0 2.346 1.871 5425 946 4.55 997.0 2.253 1. 824 5500 871 4.50 998.0 2.154 1.778 5550 821 4.47 999.0 2.087 1.746 5600 771 4.44 1000.0 2.020 1. 712 5650 721 4.41 1000.0 1.957 1.677 5700 671 4.38 1001.0 1.901 1.642 5 700 6 71 4.05 1001.0 1.911 1.09 8 5725 646 4.02 1001.0 1.846 1.092 5750 621 4.00 1000.0 1.777 1.088 5775 596 3.97 1000.0 1. 706 1.086 5800 571 3.95 1000.0 1.632 1.084 5825 546 3.92 999.0 1.558 1.084 5850 521 3.90 999.0 1.482 1.084 5875 496 3.87 999.0 1. 408 1.084 5900 471 3.85 998.0 1.338 1.083 5925 446 3.82 998.0 1.2 72 1.079 5951 420 3.80 997.0 1.210 1.074 ' 5951 420 3.58 997.0 1.186 0. 780 59 75 396 3.57 997.0 1.151 0. 775 6000 371 3.54 996.0 1.118 0. 768 6050 321 3.49 994.0 1.076 0. 739 6100 271 3.44 992.0 1.067 0.695 6150 221 3.39 990.0 1.066 0.648 6175 196 3.37 989.0 1.051 0.633 6200 171 3.35 988.0 1.009 0.633 6225 146 3.34 987.0 0.962 6.637 6250 121 3.33 986.0 0.889 0.656 6271 100 3.32 986.0 0.810 0.681 62 71 100 3.32 986.0 0.810 0.6 81 6290 81 3.32 9 85.0 0. 730 0. 708 6310 61 • 3.32 984.0 0.643 0. 738 6330 41 3.31 984.0 0.568 0. 762 6350 21 3.30 983.0 0.524 0. 769 6350 21 2.79 983.0 0.427 0.322 6360 11 2. 79 982.0 0.427 0.322 6371 0 2. 79 981.0 0.427 0. 322
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