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Preferred reversal paths caused by a heterogeneous conducting layer at the base of the mantle Costin, Simona E.O. 2003

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P R E F E R R E D R E V E R S A L PATHS CAUSED B Y A H E T E R O G E N E O U S CONDUCTING LAYER AT T H E BASE OF T H E M A N T L E By Simona E . O . Costin B.A.Sc.(Geological and Geophysical Engineering), University of Bucharest, R o m a n i a  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE  in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF E A R T H A N D O C E A N SCIENCES (GEOPHYSICS)  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA  August 2003 © Simona E . O . Costin, 2003  In presenting this thesis i n partial fulfillment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of m y department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of E a r t h and Ocean Sciences (Geophysics) T h e University of B r i t i s h Columbia 129-2219 M a i n M a l l Vancouver, Canada V 6 T 1Z4  Date:  Abstract  Paleomagnetic data from sedimentary and volcanic rocks suggest that the positions of the V i r t u a l Geomagnetic Pole ( V G P ) during polarity reversals over the past few million years are confined to longitudes through the Americas and A s i a . In this study I examine the possibility that lateral conductivity variations i n the lowermost region of the mantle contribute to the geographical distribution of the reversal paths. T h e conductivity model consists of a thin layer of material w i t h variable electrical properties which accumulates i n the core-mantle boundary region. The pattern of conductivity relies on the geodynamic predictions of the boundary topography and is i n good agreement w i t h the seismological observations of heterogeneities at the core-mantle boundary. Temporal variations i n the dipole field during a reversal generate electric currents i n the conductive layer, which in t u r n produce a secondary magnetic field.  Superposition of the secondary field on  a transition field affects the declination and inclination of the magnetic field at the surface and thereby changes the position of the V G P during a reversal.  M y results  predict preferred reversal paths over N o r t h A m e r i c a for uniform sampling of testing sites. This corresponds to one preferred path previously observed i n the geological records. In addition, preferred paths over the Americas and A s i a are predicted when using the same testing sites as the paleomagnetic database, compatible w i t h the distribution of paths given by the observations.  u  Table of Contents  Abstract  ii  List of Tables  v  List of Figures  vi  Acknowledgements  ix  1 INTRODUCTION  1  2 THEORETICAL BACKGROUND  11  2.1  Definition and Calculation of V G P  11  2.2  Electromagnetic Background: Maxwell's Equations  13  2.3  Potential Fields, Spherical Harmonics and Legendre Functions  14  2.4  2.5  Orthogonality of Spherical Harmonics, Schmidt Normalization and Gauss Coefficients  16  Vector Spherical Harmonics  17  3 THE PROBLEM 3.1  20  A Conceptual M o d e l  20  3.2  Formulation of the Problem  25  3.3  Boundary Conditions  26  3.4  Calculation of the Electric Current i n the Layer  27  3.5  Solution for the Magnetic Perturbation Using a T h i n Sheet A p p r o x i m a t i o n 32  iii  4 CONDUCTIVITY MODELS AND RESULTS .  37  4.1  V G P Paths Based on C M B Topography  4.2  Observational Constraints on the Conductivity M o d e l  40  4.3  V G P Paths Obtained w i t h a Constrained M o d e l  51  5 ADDITIONAL RESULTS  37  54  5.1  Tests on the Paleomagnetic Database  54  5.2  Brief Description of the Transition Field M o d e l  58  5.3  V G P Paths Obtained W h e n a Background Non-Dipole F i e l d Is Included  61  6 DISCUSSION A N D CONCLUSIONS  67  References  72  iv  List of Tables Table 4.1 Coefficients for the conductivity models  52  Table 4.1 Gauss coefficients of the perturbation field  52  v  List of Figures 1.1. Reversal paths as suggested by observations  4  1.2 Possible structures at C M B  7  2.1 The main elements of the magnetic field  12  3.1 Schematic illustration of the conductive layer  21  3.2 The conductivity model ( T C ) based on the geodynamic topography  of the  boundary  23  3.3 Complex structures at C M B  35  4.1 Geographical distribution of the sites used to test the V G P paths  37  4.2 V G P paths obtained using the T C model  38  4.3 Secular variation observations  40  4.4 Observations and model predictions for the g\ Gauss coefficients  41  4.5 Observations and model predictions for the h\ Gauss coefficients  42  4.6 Observations and model predictions for the equatorial dipole  44  4.7 Historical field model  45  4.8 Comparison between the historical field and model predictions for the Y% spherical harmonic coefficients  46  4.9 Comparison between the historical field and model predictions for the  spher-  ical harmonic coefficients  47  4.10 The conductivity model ( C C ) obtained using new constraints  49  4.11 V G P paths obtained using the C C model  50  5.1 V G P paths obtained for volcanic database  54  vi  5.2 V G P paths obtained for O D P database  55  5.3 V G P paths obtained from a compiled database  56  5.4 Geomagnetic spectrum for the non-dipole field  60  5.5 V G P paths obtained with a random background field  61  5.6 V G P paths obtained w i t h a random background field  62  5.7 V G P paths obtained for different reversal periods  64  vii  'Keep true to the dreams of thy youth" Friederich von Schiller  vin  Acknowledgements  Here are the acknowledgement  nominees:  Category: Best Director:  Bruce Allen Buffett, Allen Buffett Bruce, Buffett Bruce Allen For being the wisest, nicest, most helpful, incredible supervisor. Category: Best Supportive Professors:  Michael Bostock, Doug Oldenburg, Stuart Sutherland, Tad Ulrych For helping and trusting me, for your natural kindness and sense of humor, for all the interesting things I have learned from you. Special people like you make the E O S department wonderful. Category: Best Supportive Peers:  Charly Georg Bank, Christophe Hyde, Daniel Trad, Olga Zatsepina For  Friendship has many names.  Category: Best Office Mates:  Matt Keith Davie, Nicolas Lhomme, Joanna Kim Welford For all the inspired conversations we had, for your pleasant companionship i n the  office and ailleurs. Category: Best Support Across-the-Miles:  Barry Curtis Zelt For the mystic and real you. ix  • Category: The-Last-But-Not-the-Least:  Jon Edwin Mound For being such a nice bridge partner, for Samwise of the Shire, the solitude and the .tex files. • Category: Most Special Thanks  T.A. & T.M., and D.P., and 0. & V. Costin For the hot dinners you've been waiting me with. For your unabridged understanding, loving, support and help.  x  Chapter 1  INTRODUCTION  Planetary magnetic fields are widely believed to be generated and maintained by convective flows i n the internal regions of the planets, resulting i n the well known dynamo action. In the case of the E a r t h , the dynamo effect is believed to be generated i n the fluid outer core, where churning and twisting flows of highly conductive material continually regenerate the magnetic field. In the absence of regeneration the magnetic field would decay i n about 20,000 years. T h e convection i n the outer core is driven by thermal and compositional bouyancy sources as the core slowly cools and solidifies. These buoyancy forces drive fluid motions, which are acted on by the Coriolis force to produce helical flows. It is thought that these helical motions re-generate the magnetic field. F l u i d flows i n the core are highly non-linear and chaotic, causing a complex temporal variability of the geomagnetic field [Glatzmaier and Roberts, 1995]. A l t h o u g h the time variation of the magnetic field is precisely observed i n our days, we have limited insight into this variation from direct observations due to the slow rate of change i n the field. Most of our knowledge about the behavior of the magnetic field on a geological, time scale comes from the paleomagnetic records. W h e n a rock containing magnetic minerals cools through the Curie temperature, it retains the characteristics of the ambient magnetic field. This primary magnetization provides information about the direction and intensity of the field at the time of rock formation. T h e rock may subsequently acquire a secondary magnetization, however, the characteristics of the primary magnetization are important to paleo-magnetists.  1  The  Chapter 1.  INTRODUCTION  2  primary magnetization is usually thermo-remanent magnetization i n igneous rocks and depositional, post-depostional, or occasionally chemical remanent magnetization, i n sedimentary rocks  [Merrill and McElhinny, 1983]. Generally speaking, records from volcanic  rocks are easier to interpret, since the radioactive dating of the sample provides a reliable age. Lava records usually give information about one particular geomagnetic event. In contrast, observations from sediments provide a continuous record, which sometimes captures several paleomagnetic events, but determination of the magnetization time is more difficult and ocasionally unreliable  [Merrill and McElhinny, 1983].  One' of the most puzzling features of the magnetic field was determined by looking at the magnetization i n rocks and archeological clay objects. Measurements of the magnetization revealed that the Earth's magnetic field varies i n strength and direction, occasionally changing its polarity. Reversals of the magnetic field polarity were observed for the first time at the beginning of the last century. The initial observations were made by D a v i d and later Brunhes i n rocks baked by lava flows. These authors were the first to report the discovery of remanent magnetization that was opposite to the orientation of the present field. Over the folowing decades, subsequent studies on volcanic rocks have revealed that the Earth's magnetic field has reversed itself i n the past, e.g. M e r c a t o n [1926], M a t u y a m a [1929], Roche [1951] (see review).  Merrill and McElhinny, 1983 for a complete  Systematic studies have defined a polarity time-scale through joint measure-  ments of magnetic polarity and radioactive age on lava flows. T h e first time scale, put together by Cox et al. [1964], showed magnetic polarity reversals w i t h a rough periodicity of about one million years.  Opdyke et al. [1966] later found evidence for magnetic  reversals i n deep-sea sediment cores, with timing and magnetic reversals pattern similar to the one inferred from volcanic rocks on land. However, as new data was made available, it became apparent that there was no simple periodicity of the magnetic reversals.  Chapter 1.  INTRODUCTION  3  Records have revealed that the time intervals for the same magnetic polarity are sometimes long, up to one million years, or quite short, to less than a hundred thousand years [Merrill  and McElhinny,  1983]. Within a long period of predominant polarity there are  much shorter intervals of reversed polarity. According to their length, the intervals have been named chrons, subchrons or superchrons. The paleomagnetic record has also shown events known as excursions during which the reversal was either aborted or of extremely short duration. The currently accepted geomagnetic polarity time-scale was produced by Cande and Kent  [1995] based on magnetic anomalies determined in sea-floor sediments.  During a polarity reversal, the direction of the geomagnetic field changes in time by about 180°. The Virtual Geomagnetic Pole (VGP), conventionally defined as the pole of a dipolar field which gives the same observables at a site as the natural field, follows a particular geographic path from one hemisphere to another. In the last decades, several compilations have drawn attention to a more subtle morphology of the transition fields [e.g. Laj et al., 1991, 1992; Clement, 1991; Hoffman, 1992]. These works have revealed that the VGP paths during a reversal are not random, but are confined to two particular zones of longitude, covering Asian and American regions. Initially, the compilations showing preferred VGP paths came from measurements on deep-sea cores (Figure 1.1). The existence of preferred paths was disputed by a number of authors who questioned the reliability of the sedimentary records [e.g., Langereis et al,  1992] and the adequacy of the geographic distribution of sites [Valet et al, 1992].  The suggestion of preferred paths was subsequently challenged by Prevot and Camps [1993], using data from volcanic rocks. However, more recent compilations made by Love on lava data have found that volcanic records give transitional VGPs which tend to fall along American and Asian longitudes [Love, 1998; Gubbins and Love, 1998; Love, 2000], roughly consistent with the sedimentary data. The veracity of the confined VGP paths has not been resolved yet. But if the preferred  Chapter 1.  INTRODUCTION  4  (Fran "Geomagnetic Reversal Pattts" 1991, Nature, v. 351.)  Figure 1.1: Reversal paths as suggested by Laj et al. [1991].  paths really exist, then their persistence over millions of years suggests that structures in the mantle, with large characteristic time scales control this phenomenon [e.g., Gubbins, 1994]. Since the idea of confined reversal paths was first reported, numerous explanations have been proposed. Laj et al. [1991] noted a correlation between the VGP paths and P-wave anomalies in the lower mantle, while Constable [1992] observed that one of the preferred paths could be explained by decreasing the axial dipole of the present-day field; a change in sign in the non-axial-dipole terms yields the other preferred path. The flux patches in the present-day field that contribute to the preferred paths in the study of Constable appear to be persistent in the paleomagnetic field [Gubbins and Kelly, 1993; Johnson and Constable, 1998]. This could account for the consistency of the reversal paths in the past. Gubbins and Coe [1993] also attributed longitude-confined VGP paths  Chapter 1.  INTRODUCTION  5  to flux patches that migrate during a reversal. They concluded that the American and Asian paths are equally probable. Other studies appeal directly to core-mantle coupling to explain the tendency of the V G P paths to follow the same meridians. Gubbins and Sarson [1994] showed that heterogeneous heat flow at the core-mantle boundary ( C M B ) can influence the pattern of flow in the core, which would affect the transitional field morphology and possibly yield preferred V G P paths along the longitudes where downwelhngs occur. A n alternate model, proposed by Runcorn [1992], attributes the geometrical confinement of the reversal paths to the presence of a conducting shell in the D" region of the mantle, beneath the Pacific Ocean. Runcorn argued that the existence of the conducting layer would explain the observed low level of secular variation in the Pacific by electromagnetic screening. Moreover, the electrical currents induced in the shell by a reversing dipole can generate an electromagnetic torque that rotates the core until the reversing dipole path coincides with one of the meridians that border the conducting shell [Runcorn, 1992; Herrero-Bervera and Runcorn, 1997]. Aurnou et al. [1996] quantitatively assessed Runcorn's idea using a distribution of electrical conductivity based on shear-wave tomography. They found that preferred reversal paths could be obtained through this mechanism, although this result was later challenged by Brito et al. [1999], who concluded that the electromagnetic torque due to the heterogeneity of D" is too small to rotate the core and thus no preferred paths would emerge. Holme [2000] also argued against Runcorn's mechanism for V G P paths, on the grounds that the required conductance is unrealistically high. A l l of the aforementioned studies were motivated by observations of complex, heterogeneous structures in the lowermost part of the mantle (see Figure 1.2). Seismological observations show increased levels of heterogeneity in the lowermost 200 km, indicating lateral variations in the physical properties in this region [Lay et al., 1998]. Much narrower regions of heterogeneity have been detected within a few tens of kilometers of the  Chapter 1.  INTRODUCTION  6  C M B i n large areas [Garnero and Helmberger, 1995, 1996]. These regions are known as ultra-low velocity zones ( U L V Z ) , because they are characterized by decreases i n P and S velocities up to 30%. Roughly one-third of the investigated areas display wave velocity depressions [Garnero and Jeanloz, 2000]. A map of regions investigated for U L V Z s is displayed i n Figure 3.3 [Garnero et al, 1998]. Regions i n red denote U L V Z detections, whereas regions i n blue indicate no evidence for U L V Z structures. T h e thickness of these anomalous structures is variable (5 - 40 k m ) and has been determined using waveform modelling [Garnero and Helmberger, 1996]. T h e origin of U L V Z structures m a y be the partial melt of mantle material [Williams and Garnero, 1996], due to thermal anomalies such as enhanced heat flux from the core [Williams et al, 1998], or viscous heating [Steinbach and Yuen, 1999]. In addition to partial melt, changes i n bulk composition may also decrease the seismic velocities [Garnero, 2000]. Possible explanations for deviations i n bulk composition include chemical reactions between the silicate mantle and the liquid core alloy [Knittle and Jeanloz, 1989, 1991], iron enrichment of partial melt under lower mantle conditions [Knittle, 1998], or grain boundary wetting of the silicate minerals by the iron alloy originating i n the outer core [Poirer and Le Mouel, 1992; Poirer et al, 1998]. These mechanisms predict a complex chemical constitution of the lower mantle, and i m p l y that the lowermost part of the mantle is composed of conducting material. However, the predicted values of electrical conductivity i n D" and thickness of the conducting region vary from one work to another. Buffett et al. [2000] attempt to explain the complex structure at the C M B by suggesting the presence of silicate sediments accumulating at the top of the core as the result of secular cooling and solidification of the core. T h e chemical equilibrium of the liquid core is disturbed by the segregation of lighter elements as the (solid) inner core grows. This equilibrium could be re-established by the expulsion of silicates from the liquid core. T h e buoyancy of silicates i n liquid iron suggests that sediments accumulate i n the  Chapter 1.  INTRODUCTION  7  topographic highs, where the boundary is displaced radially towards the mantle. These regions act as basins from the standpoint of the core material and are predicted to exhibit high conductivities, owing to the presence of interstitial liquid iron in the sediment layer. The iron content of the layer can be as large as 50%, which yields conductivities of the order of 10 S/m. Alternatively, the model predicts that regions where the boundary is 5  pushed towards the core are not favourable for sediment accumulation. These regions would have electrical conductivities more like typical mantle materials [Shankland et al, 1993], which are orders of magnitude lower than the values in the sediment layer. Accordingly, the sediment model predicts the existence of a layer with variable conductivity and relates the position of the conducting regions to the CMB topography. The presence of a region of enhanced conductivity near the core-mantle boundary is compatible with observations of the Earth's nutation [Buffett, 1992] and could account for the velocity drop in the ULVZs.  Outer Core  Figure 1.2: Possible complex structures at the CMB [Garnero, 2000]. The ULVZ is associated with partial melt and chemical heterogeneity. The sharpness of the CMB is constrained to be less than 4 km.  In this work, I assume the presence of a laterally varying conducting layer at the  Chapter 1.  INTRODUCTION  8  base of the mantle. During a magnetic polarity reversal, the dipole component of the Earth's magnetic field decreases i n time and therefore electrical currents are generated i n the conductive parts of the layer.  Circulation of these currents i n the basal layer  induces a secondary magnetic field. Superposition of the secondary field on the magnetic field generated by the geodynamo changes the characteristics of the field and might yield preferred reversal paths due to the new configuration of the transition field. T h e goal of this thesis is to assess the effects of the secondary field on the V G P paths. T h e next chapter of the thesis presents the theroretical framework for this problem and provides the basic formulae, including the definition and calculation of the V G P . T h e t h i r d chapter formulates the problem from a conceptual point of view and develops quantitative solutions. It explains the geometry and parameters of the model and the physics of this problem. T h e interaction between the electric and magnetic fields are described by Maxwell's equations.  Simplifications of these equations are achieved by assuming a  low frequency approximation. The fields are represented i n terms of poloidal and toroidal potentials.  The scalar and vector functions i n this study are described employing the  spherical harmonic expansion w i t h Schmidt quasi-normalized coefficients. T h e solution to the magnetic induction problem is found using a thin sheet approximation. ^Computations show that the secondary field depends on the physical and geometrical parameters of the layer and on the variation of the dipole field. T h e field generated by the layer is characterized by a finite sum of spherical harmonic components.  T h e most important  terms are upward continued to define the magnetic field at the surface during a polarity reversal. T h e fourth chapter uses the calculated magnetic field to determine synthetic paleomagnetic data, i.e., the magnetic inclination and declination, at a set of random locations around the globe. The synthetic data yields the V G P at each site. T h e resultant V G P paths are binned i n histograms of 20° longitude. To gain a better understanding of how  Chapter 1.  INTRODUCTION  9  the perturbation affects the V G P paths, I consider the effect of each t e r m of the perturbation field separately. T h e results show that the distribution of the V G P reversal paths is different for each component of the perturbation field. W h e n the entire perturbation is considered, the histograms of computed V G P longitudes reveal preferred paths covering A m e r i c a n and Greenland longitudes. To improve the fit to the data, the initial model is refined based on modern observations. I consider the secular variation of the internal field and compare it w i t h the model predictions of the magnetic field induced i n the layer. I assume that the observed secular variation signal includes the signature of the perturbation generated i n the conductive layer at the base of the mantle. T h e model parameters are adjusted such that the predicted coefficients reproduce the same variability as the observations.  T h e constraints deduced from fitting the observations are adopted i n a  new conductivity model which is now constrained not only by the geodynamic topography parameters but also by the secular variation data. Subsequently, a new dataset of paleomagnetic observations is generated for the same random locations. T h e distribution of V G P reversal paths obtained with the new conductivity model reveals preferred paths covering N o r t h A m e r i c a n longitudes, i n partial agreement w i t h the observations, which usually find two preferred paths. However, non-uniform sampling, which mimics the distribution of sites i n the paleomagnetic database, leads to two reversal paths, i n agreement w i t h the observations. In the last chapter the transition field is modelled by considering other non-dipolar terms besides the perturbation field.  I use the secular variation statistical model of  Constable and Parker [1988] to describe the non-dipolar terms.  These terms define a  'background' field from the standpoint of the magnetic perturbation.  I am interested  i n determining whether the perturbation field rises above the natural variability of the Earth's magnetic field during a reversal. The transition field is devised such that each different realisation of the statistical model represents a different reversal of the magnetic  Chapter 1.  INTRODUCTION  10  field. Statistics show that during a reversal, when the dipole field weakens or disappears completely, the amplitude of the perturbation field is big enough to leave its signature on the V G P paths. The distribution of the V G P paths i n this case reveals again the preferred paths through N o r t h America, confirming the results i n the previous chapter. In addition, the results i n the fifth chapter show that the duration of the reversal and the number of reversals are other factors which influence the distribution of the V G P reversal paths.  Chapter 2  THEORETICAL BACKGROUND  2.1  Definition and Calculation of V G P  T h e V i r t u a l Geomagnetic Pole corresponds to the pole of a geocentric dipolar field which gives the direction of magnetization at an observation site [Jacobs, 1984].  Therefore,  the calculation of V G P assumes a dipolar field, which is not strictly true. O n the other hand, the V G P can be defined from the direction of magnetization at a single site. T h e geographic co-ordinates of the V G P (latitude, A , longitude, (j) ) are calculated using the 1  1  co-ordinates of the observation site (latitude, A, longitude, <j>) and the paleomagnetic direction measured at the site (declination, D , m  inclination, I ). m  F r o m the spherical  geometry, the following relationships are obtained [Merrill and McElhinny,  sin A  =  1  sin A cos 9 + cos A sin 9 cos D ,  1  = (p + 8,  (p  1  = (j> + 180 — 3,  sin/?  =  for ( - 9 0 ° < A < + 9 0 ° ) 1  m  (p  when cos 9 > sin A sin A  1  when cos 9 < sin A sin A  sin0 s i n D / c o s A , 1  m  1983]  1  for ( - 9 0 ° < 3 < + 9 0 ° ) .  (2.1)  T h e paleo-colatitude 9 of the site is defined relative to the position of the V G P and is determined by the paleomagnetic inclination I . m  T h e f and 0 components of a magnetic  dipolar field B are defined i n terms of the paleo-colatitude 9 by 2/^op cos 9  m  B  r  =  l ^ r -  11  • <-) 2  2  Chapter 2.  THEORETICAL  12  BACKGROUND  and B = e  po P  sin 0,m  (2.3)  AirR  3  where p is the dipole moment, po — 4 x 7rl0~ H/m is the permeability of free space, 7  R is the distance to the source and 9 is the magnetic co-latitude. Paleo-magnetists m  often assume that 9 is a reasonable estimate of the co-latitude 0 because statistics m  show that inclinations from the Earth's magnetic field (which is predominantly dipolar) cluster around the values expected for a geocentric axial dipolar field [Tauxe, 2002]. From Figure 2.1, it follows that t a n / tan I  m  =  m  = B /Be, r  and a straightforward substitution leads to  2 cot 9 . m  North (geographic) Magnetic meridian  East  Figure 2.1: The main elements of the magnetic field B . The horizontal component H of the magnetic field can be resolved into two components, X = —Be, northwards, Y = B^,, eastwards. The vertical component of B is Z = — B . The deviation of a compass needle from the true north is the declination D . The angle J between the magnetic field B and the horizontal component H is called inr  m  m  z  clination  Downwards  or dip.  Chapter 2.  2.2  THEORETICAL  BACKGROUND  13  Electromagnetic Background: Maxwell's Equations  Magnetic and electric fields are governed by Maxwell's equations. Historically, the four equations known as Maxwell's equations existed as experimental laws, in fragmentary forms, long before him. The laws were combined by Maxwell [1873], who extended the theory and postulated a set of differential equations which apply to all macroscopic electromagnetic phenomena. In the following, E and B are the electric and magnetic fields, J is the electric current density vector, D is the electric displacement, and cr, / i , e are material constants repre0  0  senting the electrical conductivity of the material, the permeability and permittivity of free space. The first equation, Faraday's law, relates the magnetic field and the electric field at a point in space, stating that changes in time of a magnetic field generate an electric field' <9B  —  = - V x E.  (2.4)  Ampere's law determines the magnetic field associated with an electric current V x H = J + ^ ,  (2.5)  where H is the magnetic field strength, which is related to the magnetic (induction) field through the constitutive relation B = noH.  (2.6)  D = e E.  (2.7)  The electric displacement is given by 0  For low frequencies, dD/dt is significantly smaller than J in conductors, even in poor conductors such as the mantle. In addition, if the magnetic permeability is constant throughout the media, (2.5) can be written as V X B  = /J0J.  (2.8)  Chapter 2. THEORETICAL  BACKGROUND  14  Since there are no magnetic monopoles, the magnetic field lines are continuous and close onto themselves. Therefore the continuity equation for the magnetic field is V B  (2.9)  = 0.  The set of equations is completed by the constitutive equation that relates the electric current to the electric field J = <rE.  (2.10)  Like other physical properties, the value of the electric field depends on the frame of reference in which it is measured. In this study, the stationary reference frame of the mantle is assumed. 2.3  Potential Fields, Spherical Harmonics and Legendre Functions  The current density J in (2.8) is negligible within the mantle and between the surface of the Earth and the ionosphere. From (2.8) it follows that B may be taken to be curl-free. This allows the field to be represented as the gradient of a scalar potential B = -VV.  (2.11)  In spherical co-ordinates ( r = radius, 9 — co-latitude, <$> = longitude) the gradient of a scalar field takes the form „ dV ldV* 1 dV VV = —r+-—0 + -7-^-wr<p. Or r o9 r sin 9 0<f> T  r  A  ^ . (2.12)  7  / n  n  Combining equations (2.9) and (2.II) yields Laplace's equation VV = 0  (2.13)  for the scalar potential V. Laplace's equation can be written in spherical co-ordinates as ld {rV)  l  2  r  Or  2  dr.  r sin 9 d9 2  v  „dV^, 06 '  l  dV  ,  2  0  r sin 9 dq> 2  2  Chapter 2.  THEORETICAL  15  BACKGROUND  This equation can be solved using the method of separation of variables, which assumes a solution of the form V = R(r) Y(9, <f>). Substituting R(r) Y(6, </>) in (2.14) yields Euler's differential equation for R(r), with solutions R(r) = r and R(r) = l  r~( \ l+1  where 1(1 + 1) is the separation constant. To represent the magnetic field of internal origin we keep only the solution that vanishes when r goes to infinity, i.e., R(r) =  r~( \ l+1  The method of separation of variables is applied to find the form of Y(9, cp), by assuming that Y(0, <p) = P(9)E((p).  This yields Legendre's differential equation for P(6) and  a second order ODE for E(<p), with the separation constant m . 2  The solution of the  Legendre differential equation has the general form P(9) = P / , . The P / , polynomials m  m  are called associated Legendre functions and their general expression is given by  where fi = cos 6. Hence, the general solution for the potential of the magnetic field is CO  I  F = E E  (2.16)  A r-^exp(im<p)P , l>m  l<m  1=1 m=-l  where Ai  iTn  are complex coefficients. A series representation such as the equation (2.16),  is called a spherical harmonic expansion and the quantities Ai  iTn  are called spherical  harmonic coefficients, whereas the functions Yi (0,4>) — exp(im(p) Pi iTn  <m  (cos 9) are called  spherical harmonics. In this notation / is the degree and m is the order of the spherical harmonic. The term / = 0 and m = 0 corresponds to a magnetic monopole, and is therefore excluded from the expansion to be consistent with (2.9).  Chapter 2. THEORETICAL  2.4  BACKGROUND  16  Orthogonality of Spherical Harmonics, Schmidt Normalization and Gauss Coefficients  Spherical harmonic functions are orthogonal basis functions for expanding scalar functions on a sphere. T h e orthogonality condition on the unit sphere is  / .  r  i  - ^  <  m  = ( 2 ? ' l ' x " l ) l * - )  f  ( 2  '  1 7 )  where * represents complex conjugation and dQ, represents the solid angle, dQ, = sin 9 d9 d(f>. T h e orthogonality property becomes important when we want to expand a function i n a series of spherical harmonics. A p p l y i n g the orthogonality of spherical harmonics we can determine the amplitude of the expansion coefficients. T h e spherical harmonics are often normalized, and several normalizations are used i n geomagnetism. Fully normalized Y,  satisfy  m  / Y Y *dSl m  l  t  s  = 6 ,6 . l  (2.18)  rnt  T h e Schmidt quasi-normalized form was introduced by Schmidt [1935] and is the most widely used i n geomagnetism. T h e orthogonality relationship i n this case is given by  l > Y  m Y  '-"  d a  =  WTi '- <*>s  6  T h e relationship between the un-normalized spherical harmonics Yi  ( 2 1 9 )  <m  and the Schmidt  normalized spherical harmonics Y™ is  2(Z-m)!  (l + m)\  1/2  Y , lm  form>0.  (2.20)  T h e Schmidt normalized coefficients are frequently used i n the expression of the magnetic potential V i n (2.16). Since the magnetic potential is a real scalar, the expansion of  Chapter 2. THEORETICAL  BACKGRO  UND  17  V i n (2.16) can be re-defined i n terms of real quantities. It is also customary to multiply the expression by the mean radius of the E a r t h surface, a, such that the spherical harmonic coefficients have the same dimension as the field B. Re-arranging constants and using the Schmidt normalization for the Legendre functions, the general solution for the potential of the magnetic field of internal origin used almost unanimously is N  1  /a\'  V=aJ2J2[-)  + 1  (gr cos ™ <f> + h? sin m^Pf  (cos 6).  1  (2.21)  1=1 m=0 T h e coefficients g™ and h™ i n this representation are called Gauss coefficients i n the honour of the well known mathematician and philosopher C a r l Friedrich Gauss, who was the first to employ the spherical harmonic expansion to describe the magnetic potential in 1839.  2.5  Vector Spherical Harmonics  A s mentioned before, scalar functions on a sphere can be expanded i n surface spherical harmonics by exploiting the orthogonality of the basis functions Y J  /(M) =  oo  I  E  E  4 ir(M), m  m  (2-22)  (=0 m=-l  where A™ is a complex constant.  T h e expansion theorem can be extended to vector-  valued functions on a spherical surface.  T h e approach relies on representing a vector  field i n terms of three scalar potentials U = UT + VIV + (-r xVi)W, where  is the tangential derivative.  (2.23)  Chapter 2. THEORETICAL  BACKGROUND  18  The unit radial vector r, the tangential derivative V i and (—r x V i ) are orthogonal and determine a vector basis for u. The scalar potentials U, V, W can be expanded in spherical harmonics. Using the definition of vector harmonics RT  = rYr(e,<j>),  - r x V j ^ M )  (2.25)  Ar(r)R?(0,<p) + Br(r)Sr(8,i) + Cr(r)Tr(d,4>).  (2.26)  =  the field u can be expressed as I  oo  u =E  E  1=0 m=-l  The S f and R ™ parts are often described as the spheroidal part of the field, whereas T™ 1  is the toroidal part. In terms of the spherical co-ordinates, the new vector basis can be expressed as  Rr  = *ir(M),  sr  =  TP  = 6— dtYr{0,<l>)-4>d6Yr{eA)-  odeYrie^y+t^d+Yrie^), sin (7 a  (2.27)  When the vector field is solenoidal (i.e., V • u = 0), an alternative representation for u relies on the fact that there are unique scalar fields V and r such that u = V x AV + A T ,  (2.28)  where A = V x f. This representation is very useful since it leads to a great deal of simplification during computations and therefore was often used in this work. The field V x A ? is called the poloidal part of u. ThefieldA r is called the toroidal part of u. Using the definition of the vector harmonics, the toroidal and poloidal parts of a field can be written as V x r r = -*,, T,, , r m  T O  (2.29)  Chapter 2. THEORETICAL  BACKGROUND  M±H r  l  V x V x P r  =  z  pi  19  jcT +  'dr  SJ".  r  (2.30)  The vector spherical harmonics are orthogonal and this property is used to expand vector-valued functions on a sphere. When using the Schmidt normalized Legendre functions, the orthogonality relations for the vector spherical harmonics are: pzir  pit  / / RT R * sm6d6d<p = Jo Jo l  s  4 7T  ——8i 8 ,  zt+ 1  / / S? S ,* sin6 d6d<p = l{l + Jo Jo 1  /•2ir  fir  / / T?T\* Jo Jo  sin6d9d<f> =  s  mt  l) r —Si S 7  2.1 -\- l  T  4 7T  1(1+1)——Si,8 . 2/ + 1  mt  a  mU  ,  *  (2-31)  Chapter 3 THE PROBLEM  3.1  A Conceptual Model  The sediment model of Buffett et al. [2000] suggests that solidification of the core causes chemical dis-equilibrium which is restored by precipitation of lighter elements from the liquid core. Deposition of these lighter elements occurs with some initial porosity, but viscous compaction of the layer gradually decreases the porosity and expels the interstitial liquid iron. A n important issue is whether the liquid iron remains interconnected in the silicate matrix or if isolated pockets of metallic liquid develop as the porosity decreases. In this study it is assumed that the liquid iron remains interconnected and a layer with variable conductivity develops at the C M B . According to the predictions of the sediment model, silicates tend to accumulate in the topographic highs of the boundary, which can be viewed as basins from the standpoint of the light material in the outer core. As a result of sediment accumulation, these regions exhibit high porosities. The presence of residual liquid iron raises the bulk conductivity of such regions close to core values. In contrast, regions where the boundary is depressed may have no sediments, or sediments with very low porosity because of low accumulation rates. In either case, the solid side of the C M B (including the sediment layer) lacks interstitial iron. In this context, the variable iron content of the silicate layer yields lateral variations in the electrical conductivity (Figure 3-1)A geodynamic estimate of topography at the C M B is shown in Figure 3.3 b. [Forte  20  Chapter 3. THE  PROBLEM  21  h = CMB  thickness  boundary  of the layer  Figure 3.1: Schematic illustration of the thin layer model, showing undulations of the boundary topography. The thickness of the layer, his assumed to be 1 k m . Conductivities w i t h i n the layer i n regions of topographic highs are assumed to be 2 x.10 S / m , whereas the conductivities i n depressed regions are 1 S / m . 5  et al,  1995].  This estimate of C M B topography shows positive, radial displacement  underneath the Pacific, whereas the boundary i n the surrounding regions is displaced towards the core. Since the accumulation of sediments is related to the topography, it is reasonable to assume that the distribution of conductivity emulates the C M B topography. T h e configuration of the boundary topography is used to devise a model for lateral variations i n electrical conductivity at the C M B . The dominant pattern of boundary topography is described i n terms of spherical harmonics by the second degree terms, of which the sectoral Y (6,<f)) term is predominant. 2  2  T h e coefficients of the spherical  harmonic representation of the conductivity model are assumed to be proportional to the coefficients of Forte et al. [1995] topography model. T h e region under the Pacific becomes anomalously conductive, alternating with low conductivities i n the surrounding regions. This choice is based on geodynamic predictions of positive radial displacement of the boundary, but it is also compatible with the detection of the U L V Z s . Estimates for the physical properties of the layer are inferred from the predicted profiles of porosity and electrical conductivity of the sediment model.  According to  Chapter 3. THE  Buffett et al.  PROBLEM  22  [2000], the largest porosity i n the silicate layer occurs at the interface  where the sediments are deposited. The porosity might be as large as 50%, but decreases rapidly w i t h the distance into the sediment layer. Beyond a distance of 1.5 k m from the core-sediment interface, the porosity is almost invariant and has a residual value of less than 5% . Hence, the largest volume of interstitial iron occurs over a 1.5 k m region adjacent to the sediment interface. Here, the conductivity is expected to be on the order of 10 S / m . 5  T h e conductivity model assumes a sediment layer w i t h a thickness h of 1 k m , and a depth-averaged conductivity o~ that varies between 1 and 2 x 10 S / m , depending on 5  the geographic position. The spatial distribution of a is described using the spherical harmonic representation oo  /  a(9,<f ) = a + J2Y, >  «m  0  co m<p + <r£ sinm<j>) P?(cos6), S  m  (3.1)  1 = 1 771=0  where cr is the mean value and o~f and c\ 0  m  m  give the amplitude and phase of the lat-  eral variation of o~(0, <p). (Schmidt normalization [Langel, 1987] is used for the associated Legendre function Pp .) 1  Since the conductivity configuration is proportional to the spher-  ical harmonic representation of the C M B topography of the Forte et al. [1995], I have restricted the calculations to the second degree terms of the erf  and <r£ series, as these m  have the largest contribution to the topography function. In addition, there is no need to consider the zonal term as it has no longitudinal dependence and therefore does not affect the V G P longitude. Consequently, the following particular configuration is considered for the conductivity model (Figure 3.2) 2  a(6,<l>) = <r + £ 0  771 =  «  m  c o s m ^ + ^  i m  sinm^)P  m 2  (cos^).  (3.2)  1  M y choice of parameters yields an average conductivity within the layer of order 10  5  S / m . This yields a total conductance of order 10 S, i n agreement w i t h the predictions 8  Chapter 3. THE  PROBLEM  0.2  0.4  23  0.6  0.8  1  1.2  1.4  1.6  1.8  2 X 1 0 S/m S  Figure 3.2: T h e conductivity model ( T C ) based on the topography of the core-mantle boundary [Forte et ai, 1995].  of the sediment model. Is T h e existence of a conductive region i n the lowermost mantle is generally accepted, however, the total conductance of such a region depends on the nature of material transport across the C M B . Several transport processes are thought to contribute to the enrichment i n core material of the core-mantle interface.  These  include the flow of liquid iron through capillary percolation or convective entrainment [Knittle, 1998], transport of core material along grain boundaries [Poirer et al., 1998], and underplating of the C M B by accumulation of lighter core constituents as described by the sediment model of Buffett et al. [2000]. Separately, or i n combination, these different processes could influence the composition and therefore the conductivity of the lower mantle over distances as large as 100 k m [Stevenson, 2003] above the C M B . T h e total conductance of the lower mantle differs from one model to another.  A s an  example, the model of Poirer et al. [1998] predicts a conductance which is several orders of  Chapter 3. THE  PROBLEM  24  magnitude lower than the values predicted by the sediment model, but the mechanism of iron infiltrating the mantle through grain wetting differs significantly from the mechanism of sediment accumulation. The difficulty i n devising a conductivity model lies i n the fact that currently there are few constraints to model the conductivity of the C M B region. A d d i t i o n a l constraints on the amplitude and phase of the conductivity model will be inferred later i n my study by analyzing the present-day secular variation of the Earth's magnetic  field.  T h e influence of the conductive layer on the magnetic field is studied i n the context of a polarity reversal. T h e transition field models that have been proposed i n various studies attempt to relate the mechanism of reversals to the dynamics of the outer core.  One  model assumes that the dipole disappears during reversals, as suggested by the reduction i n remanent magnetization before and after the change i n field polarity [e.g., Opdyke et al, 1973; Gubbins and Sarson, 1994; Olson, 2002]. Other studies suggest that the field retains a primarily dipolar character [e.g., Creer and Ispir, 1970; Hammond et ai, 1979] and that it reverses not by disappearing or by becoming non-dipolar, but through the rotation of the dipole axis [Runcorn, 1992]. Some studies have found evidence that the transitional fields were predominantly characterized by higher-order non-zonal harmonics [Clement, 1991; Hoffman, 1992], or even zonal, remaining axisymmetric but not dipolar [Hoffman and Fuller, 1978; Fuller et al,  1979].  In this work, the polarity transition  is modelled by assuming that the axial dipole field decreases i n intensity to zero and increases gradually i n the opposite direction. T h e Gauss coefficient  which defines the  axial dipole, decays to zero and builds up i n the opposite direction, at a prescribed rate. T h e non-dipole terms are stationary for the duration of one reversal and are determined from a statistical field model [Constable and Parker, 1988]. T h e duration of the reversal is typically a few thousand years. T h e model I adopted for the transition field relies on observations of secular variation of the internal field, which reveal a precipitous decline  Chapter 3. THE  PROBLEM  25  i n the dipole over the past 150 years. This decrease could eventually lead to a reversal, having the same characteristics with the last known reversal [Hulot et al., 2002].  3.2  Formulation of the Problem  Temporal variations i n the axial dipole induce an electric field i n the layer, which generates electric currents i n the conductive parts of the layer. Circulation of currents w i t h i n the layer generates a secondary magnetic field which perturbs the total field.  Calcula-  tions of the secondary field are used to evaluate the V G P paths and to assess whether the amplitude of the perturbation is sufficient to rise above the natural variability i n the non-dipole field, thereby leaving its signature on the V G P paths. Magnetic and electric fields inside and above the conductive layer are governed by Maxwell's equations i n a low frequency approximation [Gubbins and Roberts, 1987] described i n the previous chapter. T h e time derivative of the axial dipole i n (2.4) is estimated by assuming that the axial dipole decays and grows i n the opposite direction over the time interval of a reversal. It is also assumed that the initial and final amplitude of the dipole equals the present-day value. Using an imposed value for dT5/dt at the C M B , (2.4) is solved for the electric field E and the current J i n the layer is later determined from (2.10). Finally, the current J in (2.8) is used to solve for a secondary magnetic field B, whose effect is added to the magnetic field of core origin.  T h e method of solving for the secondary field B uses a  t h i n sheet approximation. Temporal fluctuations i n the axial dipole during a reversal are probably very complicated, but for the purpose of predicting V G P paths the model assumes that the rate of change of the dipole is constant.  Hence, b o t h E and B i n  the layer are independent of time. T h e absence of temporal variations i n B means that the variation on the left-hand side of (2.4) is due solely to changes i n the dipole. A s a  Chapter 3. THE  PROBLEM  26  consequence, the electric field E is not altered by the generation of secondary field B.  3.3  Boundary Conditions  T h e boundary conditions on B and J are derived by considering the behaviour of the magnetic field at an interface between two media of different, but finite electrical conductivities. Let n be the normal to the interface. Integrating the solenoidal condition (2.9) throughout a small volume, shows that B • n is continuous. Likewise, applying the divergence operator on (2.8) leads to the solenoidal condition on J (i.e., V - J = 0), which implies that J • n must be continuous. A p p l y i n g Stokes' theorem i n (2.4) indicates that the tangential component of the electric field E • t is also continuous across the boundary. Gauss' law V •  E=  —  S,  (3.3)  where e is the electric permittivity of free space and S is the electric charge density, shows 0  that E • n is discontinuous i f a charge develops. This discontinuity is also indicated by Ohm's law (2.10): i f J - n is continuous, but a is not, then E • n must be discontinuous. To determine the boundary condition on the tangential component of the magnetic field, consider a small volume that encloses a portion of the interface. A p p l y i n g Stokes' theorem to (2.8) for a rectangular surface perpendicular to the interface layer and making the height of the rectangle infinitesimally small, shows that B • t is discontinuous across the boundary. A l l these conditions can be summarized as follows:  [B]+ • n = 0 [J]+ • n = 0 [E]+ • t = 0  Chapter 3. THE  PROBLEM  27  [E]+ • n / 0 [B]+ • t ^ 0, where [  denotes the jump across the interface. The continuity of the normal (radial)  component of the magnetic field is used to match the field outside the layer when one medium is an insulator (i.e., mantle). 3.4  Calculation of the Electric Current in the Layer  The solenoidal condition in (2.9) means that B can be expressed in terms of a vector potential. Hence, the dipole part of the field is represented as Bj = V x A.  (3.4)  Substituting B° in (2.4) and interchanging the order of differentiation we get „  _  OA  _  Vx^- = -VxE, and further Vx  ~0A  + E  = o.  The quantity between brackets is irrotational, thus conservative, so it can be written as the gradient of a scalar potential 8A ^_ +  E  =  (3.5)  where (j) is an arbitrary scalar. The solution for the electric field is then E = - ^ - V 0 .  (3.6)  In the previous chapter it has been shown that the magnetic field can be decomposed into poloidal and toroidal parts (2.28). Because the axial dipole B° is purely poloidal,  Chapter 3. THE  PROBLEM  28  let  B° = V x V x (Vr), where  V is a poloidal scalar given by V = p°Y°.  (3.7)  Equating (3.7) and (2.11) gives the  relation between the coefficient of V and the corresponding Gauss coefficient  P? = -v* ? •  (3-8)  A t the Earth's surface, V = a g° cos 6. T h e vector potential A is found by equating B° 2  from (3.7) and (3.4)  X  A = V x (Vi).  (3.9)  Substituting A i n (3.6) we get E  =  - V x (Vr)  - V<p,  (3.10)  where the dot represents differentiation w i t h respect to time. Ohm's Law (2.10) yields the density current i n the layer  J = - <rV x (Vr) - <rV<f>,  (3.11)  responsible for the secondary field B. The scalar V depends on the rate of change of the dipole field and is treated as a known parameter. Therefore the term — crV x (VT) is readily computed while the current due to the potential <p remains to be determined. T h e term o~V(p is evaluated by imposing the condition that the electric charge is conserved  V-J  = 0.  (3.12)  Substituting (3.11) into (3.12) and applying vector identities yields (Va)  • V x (Vr)  = - V • <TV0.  (3.13)  Chapter 3. THE  PROBLEM  29  The term aV(f> can be directly determined from (3.13). Since our interest lies mainly i n the current crV</>, this is a faster approach than to compute the potential G6 and then determine the current due to this potential. To do this, we first compute the left hand side of (3.13). Since the V x (Vr) has only a <p component, we only consider the <p component of the conductivity gradient i n evaluating the dot product on the left hand side of (3.13). T h e resulting scalar has a spherical harmonic expansion which is identical to that of the conductivity distribution. For instance, a lateral variation i n cr with a Y (0,d)) term gives 2  2  2a • ( W ) - V x (Vr) = —g°{o-2 3  t2  cos m < £ - c r f  sin rn <p) P (cos 6).  (3.14)  2  ) 2  2  A similar result is obtained with a distribution of conductivity that varies as Y (6,(f>). 2  It is important to note here that the spherical harmonic correspondence between cr and the left-hand side of (3.13) is only possible because the poloidal scalar V represents the axial dipole, with 1 = 1 and m = 0. The spherical harmonic representation of ( V c ) • V x (Vi) establishes the form of the potential (f>. Because (V<r)- V x (Vi) has the same angular order as <r, and this order must equal the sum of the angular orders of cr and c6, by virtue of orthogonality of the spherical harmonics c6 must be zonal (i.e., with angular order m = 0). Since <j> has no longitudinal dependence, the gradient Vc6, evaluated as i n (2.12), has only a 0 component. Let now o-V<f> = f 0.  (3.15)  e  We can evaluate f by substituting (3.15) into (3.13), which gives e  -±-l( in9f ) r sin v o$ s  e  = -(Va) - V x (Vi).  (3.16)  M u l t i p l y i n g (3.16) by (r sin#) and integrating with respect to 8 determines f (and hence e  <rV<f>).  Chapter 3. THE  PROBLEM  30  A s an example, when (Ver) • V x (VT) has a Y  2 2  dependence, the integral over 6 i n  the right hand side of (3.16) has the form F(6)  = j P (cosrJ) sin 6d0 = ^ . ^ f l  _^  2  2  0  COS  +  C  ,  (3.17)  where the value of C is chosen to ensure that <TV<^> vanishes at the poles. Expressing f  8  i n terms of F(6) gives  fe = —  9i « \  2  •  cos 2 0 - <rf sin 2 <p) - A - f . sine' |2  (3.18)  7  The total electric current J , computed as above from (3.11) has the form 2a  3  .0  ^fT  2 2  COS 2 (j) — (T 2 2)  s  m  2 <f)  ——— 9  sin 6^  —  (o"2,2  C  O  S  2 <j> +  0~2 2  S  m  2 4>)  s  m  ^ ^  (3.19) A similar treatment is used to compute the potential term and later the total electric current when the conductivity distribution varies as Y . X  2  T h e magnetic field generated by the current J is computed from (2.8), i.e., V x B = /io J - T h e poloidal part of the magnetic field is associated with toroidal electric currents, whereas the toroidal part is related to poloidal currents i n the layer. A t the E a r t h' s surface we are able to measure only the poloidal part of the magnetic field, and although a toroidal field may be generated i n the layer due to poloidal currents, it cannot contribute to the V G P paths. We confine our attention to the part of the current J that contributes to the magnetic potential field at the surface, which is associated w i t h the toroidal vector harmonic, defined by (2.29). The toroidal part of J is decomposed into spherical harmonic components CO  Jr =  I  EEirTr,  (3.20)  7=1 m=0  where the coefficients jf  1  are evaluated using the orthogonality property of the vector  harmonics (2.31). The first non-zero coefficient of the expansion of J is found to be jl  Chapter 3. THE  when u has a Y  2  PROBLEM  31  dependence, and j\ when cr has a Y  2  dependence. Terms w i t h higher  order I cause magnetic perturbations which are more rapidly attenuated by the upward continuation to the surface and contribute much less to the V G P paths. So far we have determined the horizontal current which arises i n the conductive layer from the time variation of the axial dipole. Secular variation models of the geomagnetic field show that the equatorial dipole component may also vary with time during a reversal. This component is represented by the terms g\ and h\ for the spherical expansion of V i n (2.21). To determine the electric current which arises from the variation of the equatorial dipole, the development of the equations is the same as before and the current is still given by (3.11), but now since the equatorial dipole has a latitude and longitude dependence, the poloidal scalar V is given by V = p\ Y^-\- complex conjugate. Similar calculation as i n the vertical dipole case gives the expression for the real coefficients of the poloidal scalar i n terms of Gauss coefficients: Pii  =  a  3  ,  — 9i r 3  Pn = - h \ .  (3.21)  T  W i t h these notations, V at the Earth's surface becomes V =  (g\ cos (J)-\-h\ sin  sin 6.  In this case, V has a more complicated form than that for the axial dipole and therefore the determination of the current is slightly different as we cannot determine directly the term crV</> from the current conservation equation (3.13). T h e calculations get a little bit tedious when it comes to evaluating the potential (f>. A s previously, the toroidal part of the current is decomposed into spherical harmonic components. T h e lowest harmonics for the electric current generated by the variation of the equatorial dipole are Y  2  and Y£.  These terms have smaller amplitudes than those i n the previous case because the rate of change for the equatorial dipole is smaller than the rate of change for the axial dipole (g\ = 1 5 n T / y and h\ = 3 5 n T / y ) . A s a consequence, the contribution of these terms to  Chapter 3. THE  PROBLEM  32  the current J can be ignored and was not incorporated i n further calculations.  3.5  Solution for the Magnetic Perturbation Using a Thin Sheet Approximation  T h e presence of the electric current i n the conductive parts of the layer induces a secondary magnetic field which superimposes on top of the existing field. T h e method of solving for the secondary field B uses a thin sheet approximation because the time scale for magnetic difussion throughout the layer is short compared with the duration of a reversal. T h e time delay introduced by the electrically conductive layer is given by the diffusion time of the magnetic field i n the layer  d Ho o-,  (3.22)  2  where d represents a scale of the conductive layer and rj  m  is the magnetic diffusivity.  For a layer w i t h a thickness of 10 m and a conductivity of 10 S / m , the characteristic 3  5  decay time is roughly three days. Since the reversal time is a few thousand years, a time delay of three days introduced by the conductive layer is not important i n calculating the field B . Therefore, it can be safely assumed that the rate of change of the dipole i n the layer is given by the downward continuation of the rate of change of the dipole at the surface. Another reason to use the thin layer approximation derives from the fact that the thickness of the layer is three orders of magnitude smaller than the thickness of the overlying mantle. Consequently, the solution to (2.8) for the field at the surface should not depend on the radial distribution of J inside the layer. To determine the secondary magnetic field produced by J i n the layer, (2.8) is i n tegrated over the volume of a small pill box constructed to contain the t h i n layer. T h e upper and lower surfaces of the box coincide with the surfaces of the mantle and core  Chapter 3. THE  PROBLEM  33  respectively. T h e integral  J (V xB)dV = fi J 3dV  (3.23)  0  is transformed using Gauss' theorem to give  ••J^nxBdS  = poj JdV,  (3.24)  where n represents the (outward) normal to the surface of the pill box and dS indicates the surface element. If the height h of the pill box becomes vanishingly small, the sides of the pill box contribute nothing to the surface integral, so the only contributions are due to the area of the top <S and bottom S~ of the pill box. Combining the two contributions +  into a single integral, where r is the outward normal on the top surface and — f is the outward normal on the bottom surface, we obtain /  (r x B  where the notations B , B +  _  +  - r x B~) dS = fioh f  JdS,  (3.25)  indicate the magnetic fields on the top and bottom of the  conductive layer. T h e R H S of (3.25) is a consequence of assuming that the current J does not vary radially, hence the approximation dV ~ hdS may be used. Because (3.25) is valid for an arbitrary surface area, the integrands are equated to obtain f x (B+ - B ) -  The  = pohZ.  (3.26)  region above the layer (i.e., the mantle) can be regarded as poor conductor  [Gubbins and Roberts, 1987], sufficiently poor so that V x B is effectively zero. Hence, the magnetic field within the mantle B w i t h radial solution r~( \ l+1  +  is a potential field, described by a potential  For the magnetic field B  _  below the conductive layer (i.e.,  w i t h i n the core), there are two bounding cases. In one case it is assumed the core is a perfect conductor. This means that the magnetic field at the top of the core is a few orders of magnitude stronger than the field generated i n the layer, therefore the magnetic  Chapter 3. THE  PROBLEM  34  perturbation does not diffuse into the core and B ~ = 0 i n (3.26). T h e other bounding case treats the core as a perfect insulator.  In this case the magnetic field i n the core  is described by a magnetic potential (with radial dependence r ). l  T h e amplitude of the  perturbation i n the case of a perfectly insulating core is approximately 50% less than that for the perfectly conducting core. However, calculations show that for this value the perturbation does not alter the V G P paths. Hereafter the case of the perfectly conducting core is discussed. T h e field above and below the layer must satisfy the boundary condition i n (3.26). In the case of a perfectly conducting core, (3.26) reduces to r x B where B  +  +  = ftohJ,  (3.27)  = - V $ . The series representation (2.21) is employed to describe the potential  $ by introducing the Gauss coefficients g™ and h™. B y the virtue of the orthogonality of the spherical harmonics, the coefficients j™ of the toroidal current J determine the Gauss coefficients  and h™ w i t h the same degree and order. T h e coefficients g™ and  h™ are evaluated using (3.27) and upward continued to the surface to obtain ST  =  K  =  Q  N?  \ -j  bphg"*^,  bphg r , 0  h  l(  2 m  (3.28)  where b is the core radius and N™ is a constant depending on the degree and order of the Gauss coefficient. To give an order of magnitude estimate of the coefficients I assume a present-day dipole that reverses i n one thousand years and a 1 k m thick layer w i t h a conductivity variation of order 10 S / m . The largest coefficients i n the potential field are 5  at I = 3, m = 2 and I = 1, m = 1 and have amplitudes of roughly several hundred n T . T h e coefficients of $ i n (3.28) are used to generate synthetic data by evaluating the magnetic field components i n different locations on the Earth's surface. T h e components  Chapter 3. THE  PROBLEM  35  of the magnetic perturbation are added to the field of core origin to predict the declination and inclination at each site.  Chapter 3. THE  PROBLEM  36  Ultra Low Velocity Zones No ULVZ Detected  ULVZ Detected  (Garnero et al. 1998)  C M B Topography  (Forte et al. 1995)  -6  mm  m  m  +6  km  Figure 3.3: Top: Location of ultra-low velocity zones (ULVZ) at the core-mantle boundary. Bottom: Core-mantle topography inferred from geodynamical calculations. Negative values indicate positive radial dispacement of core material into the mantle, positive values indicate regions where the boundary is pushed towards the core.  Chapter 4  CONDUCTIVITY MODELS A N D RESULTS  4.1  V G P Paths Based on C M B Topography  To determine the influence of the perturbation field on V G P paths, I start w i t h a very simple reversal field i n which the axial dipole decays close to zero from its present value and grows i n the opposite direction i n one thousand years. T h e variation i n time of the axial dipole induces a magnetic perturbation due to the lateral conductivity at the C M B . T h e conductivity model i n Figure 3.2 is devised using the coefficients of order I — 2 of the Forte et al. [1995] topography model, since these coefficients are the highest.  The  term w i t h equatorial symmetry I = 2, m = 0 is not considered as this term does not alter the V G P paths. T h e Y  and Y  2  2 2  terms i n the conductivity model a are responsible  for generating the magnetic perturbation, which can be represented at the surface by the potential $ . A s shown earlier, the leading terms of $ are described by the Y* and Y$ Gauss coefficients. T h e model is used to calculate the synthetic V G P paths for 500 sites randomly distributed over the Earth's surface (Figure 4.1). T h e magnetic inclination and declination at each site are calculated from the superposition of the axial dipole and the magnetic perturbation due to  I compute the histogram of the V G P paths by binning the results  into intervals that span 20° of longitude. To gain a better understanding of the effect of different components of the perturbation on V G P positions, it is instructive to consider the influence of the Y  2  and Y  2 2  terms of the conductivity function separately.  37  Chapter 4. CONDUCTIVITY  MODELS AND  RESULTS  38  Figure 4.1: T h e V G P paths have been computed for a random distribution of 500 sites.  T h e perturbation given by the Y  2  part of the conductivity cr is defined by the g\  and h\ coefficients. If taken alone, this part of the perturbation yields V G P paths which concentrate i n one single path around 320° ( 4 0 ° W ) . This result is expected, because the perturbation describes an equatorial dipole. Consequently the superposition of the axial dipole and the perturbation is a purely dipolar field, so the orientation of the V G P at any location is identical to the dipole orientation. W h e n the Y  2 2  part of cr is considered,  the field is composed of the axial dipole and a non-dipole part described by the g\ and k|  coefficients. T h e histogram of V G P longitudes computed for this field reveals four  preferred paths, two of them lying on American and Asian longitudes and two covering mid-Pacific and Western African longitudes (Figure 4.2 top). T h e result is encouraging because these paths coincide with the observations. Besides the two well-known A s i a n and A m e r i c a n paths, the other two paths have also been encountered i n the paleomagnetic database, though i n a much smaller number of records [Herrero-Bervera et al, 1987; Liddicoat, 1982]. W h e n the two effects are added together, the non-axial-dipole field is composed of the  Chapter 4. COND UCTIVITY MODELS AND RES ULTS  coefficients g\,h\,g\  and h\.  39  Superposition of these components causes most of the V G P  paths to fall on American and Greenland longitudes (Figure 4.2 bottom). More than half of the total number of paths fall i n the Eastern region of N o r t h A m e r i c a . T h i s result is reminiscent of studies that show more frequent paths over the Americas compared w i t h the A s i a n paths [e.g., Trie et al., 1991; Aurnou et al., 1996]. The prevalence of the paths through Eastern parts of N o r t h A m e r i c a i n my results is due to the size of the equatorial dipole terms g\ and h\ relative to the non-dipole terms g\ and h\. Y  2  E v e n though the  part of the conductivity configuration a contributes less to the total conductivity  than the Y  2 2  part (the Y^ part is about 25% of the total), the coefficients g\ and h\ are  larger because they represent a longer wavelength field and are attenuated less by upward continuation.  801  1  1  1  1  50  100  150  200 Longitude  250  300  1  1  1  1  1  1  r  50  100  150  200 Longitude  250  300  350  r  60 -  0  1001  0  350  Figure 4.2: Top:VGP paths obtained for a field which contains the axial dipole and the Y£ part of the perturbation. Bottom:VGP paths obtained when adding the Yj part of the perturbation to the above field. 1  Chapter 4. CONDUCTIVITY  MODELS AND  RESULTS  40  T h e results obtained so far show that the magnetic perturbation induced i n the layer can affect the V G P paths during the reversal of the axial dipole. T h e model yields the preferred paths through America, partially resolving the observations. However, these paths are slightly shifted from the preferred American paths given by the observations. One possible interpretation is that the geographical confinement of the V G P during a reversal is caused by a perturbation field with a slightly different geometry. A s seen from the analytic solution, the degree and order of the Gauss coefficients of the perturbed field are functions of the degree and order of both the time-varying magnetic components and conductivity geometry (see Equation 3.28). A different geometry of the perturbation field can be related to the distribution of the conductivity at the C M B . A s shown at the beginning of this section, higher order terms have a very small contribution to the topography function, therefore they were not incorporated i n the conductivity model. T h e feature that could make a difference i n the geometry of the V G P paths is the conductivity pattern given by order 1 = 2 terms. In the following, I attempt to derive new constraints for the phase and amplitude of the conductivity model using the secular variation ( S V ) data. A s seen later, the new constraints yield a model which produces a better agreement between the observed and predicted V G P paths.  4.2  Observational Constraints on the Conductivity Model  Decreases i n the dipole field induce a secondary magnetic field i n the conductive layer at the C M B . The magnetic potential of the perturbation is described at the E a r t h ' s surface primarily by the Y% and Y^ terms, and these terms are well constrained i n the present-day field. Historical records over the last hundred years show that the amplitude of the axial dipole is decreasing at a rate which is comparable to the value assumed i n  Chapter 4. COND UCTIVITY  MODELS AND  RESULTS  41  modelling the magnetic polarity reversal.  x 10  1900  1920  1940  1960  1980  2000  1900 6000 r  -1600  1920  1940  1960  1980  2000  . _  -1800 -2000  5500  N N  -2200 -2400 1900  1920  1940  1960  1980  2000  1300  400  a  1920  1940  1960  1980  2000  1920  1940 1960 Year  1980  2000  r  /  3  /  1250  1200 1900  5000 1900  1920  1940 1960 Year  1980  2000  1900  Figure 4.3: Secular variation of the geomagnetic field - I G R F [Langel, 1992]. These observations have been used to derive new constraints on the conductivity model.  Whether the present rate of decay of the dipole is representing an early stage of a reversal or excursion is uncertain at this moment [Hulot et al., 2002; Olson, 2002]. Nevertheless, at this rate of decrease we might expect to see evidence of the perturbation $ , and hence of the layer, i n the secular variation observations. Therefore, I attempt to derive constraints for the conductivity model using modern observations. To constrain the conductivity model, I use the International Geomagnetic Reference F i e l d ( I G R F ) [Langel, 1992] which describes the secular variation of the magnetic field over the last century (Figure 4.3).  T h e variations that provide the most information  on the presence of the conductive layer at the C M B occur i n the Y% and Y J Gauss 1  Chapter 4. COND UCTIVITY  MODELS  AND  RESULTS  42  coefficients. I compute the rate of change of the axial dipole from the I G R F model and use this value as input i n (3.28) to predict the coefficients of the perturbed potential It is important to note that a constant rate of decrease i n the axial dipole causes constant perturbations i n the Y% and Y* parts of the field, which would be difficult to distinguish from the direct contributions due to the geodynamo. However, fluctuations i n the rate of change of the dipole decrease (see Figure 4.3) cause fluctuations i n the Y% and Y* terms of the magnetic perturbation, which might be identified i n the secular variation of the field. T h e model previoulsy developed to predict the Gauss coefficients of the perturbation field is not entirely valid for the subsequent analysis, since the input dB/dt  i n (2.4) is  no longer constant i n time. This fluctuation will contribute to dTZ/dt i n (2.4), so we can no longer assume that the electric field is due solely to changes i n the axial dipole. However, the time variations i n the magnetic perturbation do not directly contribute to the K) and Y 2  x  parts of the potential, field at the surface, so my simplified model produces  a good approximation for the perturbed coefficients. The goal is to assess whether the predicted fluctuations are evident i n the I G R F coefficients and if this provides additional constraints on the structure of the conducting layer at the C M B . I assume that the magnetic field at the surface includes a large contribution from the interior of the core due to the geodynamo and a smaller contribution from the magnetic induction i n the conducting layer at the base of the mantle.  Since we are looking for evidence of the  conducting layer i n the fluctuations of the geomagnetic field, it is reasonable to remove the mean from the I G R F coefficients. I first refer to the non-zonal octupole field coefficients.  Figures 4.4 and 4.5 show  a comparison of the observed variations i n the Gauss coefficients w i t h the predicted variations caused by the observed non-constant decrease i n the dipole field. To facilitate comparison of these variations on the same scale, the mean is removed from both the observations and the predictions.  Chapter 4. COND UCTIVITY MODELS  -20  -  -30  -  -40  -  -50 1900 L  ' 1910  .o  1920  o  AND RESULTS  43  observed predicted best fit  o 1930  1940  1950 Year  1960  1970  1980  1990  2000  Figure 4.4: Comparison between the observed variations in gl Gauss coefficients [ I G R F , Langel, 1992] and predicted variations caused by the fluctuations i n the axial dipole field.  T h e dashed line represents the predictions obtained using the conductivity model based on the C M B topography of Forte et al. [1995]. I subsequently refer to this model as the T C model (standing for the topography-based conductivity). W e see that the amplitude of the predicted fluctuation i n gl is somewhat smaller than the amplitude of the observed  fluctuation.  O n the other hand, the observed changes i n the hi coefficient  show almost no evidence of decadal fluctuation, which is clearly seen i n the predicted variation. It is possible that the nearly linear increase i n the observed hi is due to changes i n the geodynamo. Consequently, I remove the linear trend and compare the resulting observations (circles) against the model predictions i n Figure 4.5. T h e amplitude of the detrended fluctuation i n the hi is now much smaller than the predicted variation (unlike the case w i t h gl). Therefore the phase of the conductivity pattern must be adjusted to diminish the response i n the hi coefficient and increase the response i n gl while keeping the amplitude of the conductivity variation identical to that predicted using the topography  Chapter 4. CONDUCTIVITY  MODELS AND  o o o  RESULTS  44  /, .—-o0  o observed observed(detrended[) predicted best fit  1900  1910  1920  1930  1940  1950 Year  1960  1970  I960  1990  2000  Figure 4.5: Comparison between the observed variations i n h\ Gauss coefficients [ I G R F , Langel, 1992] and predicted variations caused by the fluctuations i n the axial dipole field.  model of Forte et al.  [1995].  Better agreement between the predicted and observed  fluctuations i n the Gauss coefficients is obtained when the conductivity pattern of a\ is rotated clockwise by an angle of 20°. W h e n the same procedure is applied to the equatorial dipole coefficients g\ and h\, the results are inconclusive because the observed variations are substantially larger than the predicted variations (Figure 4.6).  This suggests that changes i n the  geodynamo  at g\ and h\ overwhelm any signature that arises from lateral variations i n electrical conductivity.  T h e most that can be said about the g\ and h\ coefficients is that the  model predictions are not incompatible with the observed variations. Similar arguments could be made about the g\ and h\ coefficients because the observed variations may be unrelated to the effects of the lateral variations i n electrical conductivity. O n the other hand, the conductivity of the sediment layer could not be much larger than the values adopted here since the predicted variations would be too large to be compatible w i t h the  Chapter 4. CONDUCTIVITY  MODELS  AND RESULTS  45  observations. However, lower values of conductance cannot be ruled out i f the observed variations i n gl and hi have another source.  2000  2000  Figure 4.6: Comparison between the variations of the equatorial dipole coefficients [ I G R F , Langel, 1992] and predicted variations caused by the fluctuations i n the axial dipole field.  T h e validity of these assumptions has been tested on a data set which covers a longer period of time. T h e secular variation model gufm 1 [Jackson et al., 2000] provides data for the last 400 years, however, only the last half of it can be used i n m y analysis. This happens because absolute intensity of the magnetic field was not measured prior to 1832 and therefore the time variation of the axial dipole g° cannot be computed for a period prior to 1832 (Figure 4.7). T h e analysis of the data has been done i n the same way as i n the previous case, but  Chapter 4. COND UCTIVITY  MODELS  AND  RESULTS  46  x 10  -20 1600  1700  1800  1900  2000  1700  1800  1900  2000  1700  1800 Year  1900  2000  6000  -4000 1600  1700  1800  1500  1900 •  2000  1600 400  —  1000  500 1600  1700  1800 Year  1900  2000  Figure 4.7: Secular variation model gufm 1 [Jackson et al., 2000] used to test the constraints on conductivity pattern.  this time I have used the time-derivative of the axial dipole provided by the gufm 1 model [Jackson et al, 2000], rather than compute it. The same steps were followed, removing the mean and linear trends when necessary. The predictions of Gauss coefficient variations were computed using the conductivity distribution of the topography-based model and subsequently of the distribution constrained by the S V data. Inspection of the variation of the gl coefficients i n Figure 4.8 (top) shows that the variability observed i n the I G R F model is present at earlier times. I n addition, it appears that the constraints derived earlier are compatible w i t h the longer dataset. The variation of the hi coefficients (Figure 4.8 bottom) shows that the linear trend observed i n the last century is part of a larger variation. Small-scale variations are buried i n the much larger  Chapter 4. CONDUCTIVITY  MODELS AND RESULTS  47  100r  50 •  v  11  \  S' ' ,  '  •\'s --.>*-  /  - \ /  y i-  ^ \  ~  /  -50 •  -100' 1800  ' 1820  1  1840  1  1860  1  1880  1  1900  1  1920  1940  1960  Year  1980  2000  Observed Predicted Best fit  80 60 40 , 20 "  Jl  0 -20 -40 -60 1800  1820  1840  1860  1880  1900  1920  1940  1960  1980  2000  Year  Figure 4.8: Comparison between the variations i n the g\ and h\ Gauss coefficients of the gufm 1 model [Jackson et al, 2000], and predicted variations caused by the fluctuations i n the axial dipole field. (Note: Linear trends have been removed from the observations.)  linear trend.  A s i n the case of the gl coefficients, the same remark can be made for  the hi coefficients, that the variability observed i n the I G R F model is present at earlier periods. T h e gufm 1 model shows that both Y£ coefficients, which represent an octupole effect, have a large range of variation over the last four centuries. However, following the same analysis as for the I G R F model, once the linear trends are removed and the derived constraints are applied, a reasonable fit to the data is obtained (Figure 4.8). It is interesting to notice the apparent time delay between the observed and bestfitting models for the octupole field. T h e ten-year lag, present i n both analyses is too  Chapter 4. CONDUCTIVITY  MODELS AND  RESULTS  48  large to be attributed to differential diffusion times between the 1 =  1 and I =  3  components of the magnetic field [Smylie, 1965]. It is more likely that other processes contribute to the <?| and hi coefficients. For example, variations i n the toroidal field at the C M B , possibly linked to variations i n the axial dipole, can contribute to the gl and hi coefficients through magnetic scattering [Koyama, 2002]. Referring now to the changes i n the equatorial dipole, the high amplitude variation observed i n the previous analysis is evident over the past four centuries.  Predicted  Secular variation and model prediction for g]  -300 1800  1820  1840  1860  1880  1900 Year  1920  1940  Secular variation and model prediction for hj  1960  0  1980  2000  observed predicted  200  -200  -400  -600 1800  1820  1840  1860  1880  1900 Year  1920  1940  1960  1980  2000  Figure 4.9: Comparison between the variations of the equatorial dipole coefficients of the gufm 1 model [Jackson et al., 2000], and predicted variations caused by the fluctuations i n the dipole field.  coefficients have a much smaller variation than the historical secular variation model, which also displays rapid variations due to geomagnetic jerks [Bloxham et al., 2002].  Chapter 4. COND UCTIVITY MODELS  AND RESULTS  49  Consequently it is not possible to decipher the variations induced by the conductive layer (Figure 4.9).  Increasing the amplitude of the predicted variation i n the g\ and  h\ coefficients would require an increase i n the conductance of the layer beyond the predictions of the sediment model of Buffett et al. [2000]. Fine tuning was possible i n the case of the octupole coefficients gl and hi because the applied treatment affected only the phase of the conductance, and not its amplitude, and therefore the constraints imposed by the topography and sediment models were preserved. T h e attempt to fit the model predictions to the observed variability i n the historical field  slightly changes the phase of the conductivity model T C and consequently the  configuration of the perturbation field $ .  The new constraints are sought i n order to  improve the fit between the observed and predicted V G P paths during a reversal. To obtain a different configuration for the perturbation field, we could also assume that nonaxial-dipole terms of the geodynamo field are inducing secondary field components i n the layer. Observations indicate complex temporal variation i n all components, however, this variation is merely a random process that cannot necessarily be linked to the decay of the axial dipole (which presumably causes the polarity reversal i n the magnetic field). O n the other hand, higher order components of the magnetic field would yield perturbations which are very rapidly attenuated by the presence of the mantle.  For these reasons,  the effect produced i n the layer by temporal variation of non-dipole components of the magnetic field has not been assessed. This doesn't rule out possible strong effects of non-dipolar transition fields of core origin on reversal paths, as pointed out by a number of papers [e.g. Clement, 1991; Bogue, 1991; Gubbins and Kelly, 1993]. In the subsequent discussion I assume that the fluctuations i n gl and hi are caused by lateral variations i n electrical conductivity and adopt the best-fitting conductivity model for the Y  2 2  Y  2  part of a. For the sake of consistency the same phase shift is adopted for the  part of a. This conductivity model is called the C C model, standing for constrained  Chapter 4. CONDUCTIVITY  MODELS AND  RESULTS  50  conductivity. The resultant pattern of conductance is shown in Figure 4.10. The total conductance of the layer varies from 1 to 1.976 x 10 S. This constrained parameter 8  is close to the 1.7 x 10 S value, predicted by the sediment model and is sufficient to 8  explain the nutation observations [Buffett et ai, 1992]. The conductivity parameters used for both models are presented in Table 4.1, whereas Table 4.2 contains the values for the computed Gauss coefficients during a reversal. The new configuration is adopted to predict V G P paths during a reversal. Since the phase shift of the model is small (about 20° in longitude), the layer preserves an anomalous conductive region below the Pacific region, where the sediment model based on dynamic topography of the boundary predicts the presence of iron-enriched material.  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2 X10  5  S/m  Figure 4.10: The constrained conductivity model (CC) obtained using observations of secular variation from historical magnetic field.  Chapter 4. COND UCTIVITY  4.3  MODELS  AND  RESULTS  51  V G P Paths Obtained with a Constrained Model  Figure 4.11: Top:VGP paths obtained for a field which contains the axial dipole and the Y% part of the perturbation. Bottom:VGP paths obtained when adding the Y] part of the perturbation to the above field. Perturbation given by the constrained conductivity model C C . 1  T h e effect of the constrained model C C is tested on the same 500 sites randomly distributed on the Earth's suface.  The effects of the Y* and Y% components of $ are  tracked separately as i n the case of the T C model. A s expected, the paths are shifted slightly i n longitude by the angle of rotation of the conductivity pattern.  W h e n the  effects of the YJ and X , parts of $ are combined, the histograms of the V G P longitudes 1  2  show a concentration of paths along American longitudes (Figure 4.11), due to the Y^  1  part i n the perturbation field. T h e effect of the equatorial dipole (i.e., Y*) term of $ is overwhelming. In one realisation of the random sites, 200 of the 500 V G P paths fall between the 280° and 340° longitudes. A p a r t from the preferred paths through N o r t h  Chapter 4. CONDUCTIVITY  MODELS AND  RESULTS  52  America, the rest of the paths are scattered and no other geographical confinement of paths emerges. Clearly, when the conductivity pattern is constrained by the secular variation, the predicted American paths are in better agreement with the observations.  Chapter 4.  COND UCTIVITY  MODELS  AND RESULTS  Table 4.1. Conductivity coefficients Model  -9  °2,2  Total R M S  ^2,1  [S/m]  TC  55.86 x 1 0  CC  128.53 x 1 0  137.92 x 1 0  3  75.0 x 1 0  3  3  3  -40.65 x 10  3  26.91 x 1 0  3  19.76 x 1 0  4  -29.79 x 10  3  38.59 x 1 0  3  19.76 x 1 0  4  Table 4.2. Gauss coefficients of the perturbation $ Model  9l  h\  9\  h\  Total R M S [nT]  TC  92.91  -229.39  -156.04  103.25  434.61  CC  -213.76  -124.73  -114.33  148.11  434.61  Chapter 5 ADDITIONAL RESULTS  5.1  Tests on the Paleomagnetic Database  T h e analysis of synthetic data from 500 randomly distributed sites represents an ideal situation for the study of the paleomagnetic field, because the pseudo-record contains a large number of sites evenly spread over the Earth's surface. In reality, observations come from a smaller number of rocks unevenly sampled around the globe. Consequently, it is of interest to ask how the conclusions of the previous chapter change i f the analysis is restricted to sites i n existing databases. A s a first test, I take a distribution of sites from the volcanic database [Love, 1998], which contains 63 magnetic measurements of lavas sampled i n Iceland, Greece, B r i t i s h Columbia, Hawaii and Africa. I r u n the reversal simulation and use the perturbation field given by the C C model. This non-uniform sampling yields a different distribution of paths from the one obtained from the uniform spreading of sites (Figure 5.1). T h e distribution shows preferred paths along the A s i a n and American longitudes as well as mid-Pacific. T w o of the paths are i n agreement w i t h the paths found by Love i n analyzing the data.  I also examine  the paths for prominent sites i n the database. T h e predicted paths for several sites i n Iceland are confined to longitudes through A s i a , which is i n agreement w i t h the observations. In fact, the large number of Icelandic observations i n the volcanic database shifts the preferred paths i n Figure 5.1 to longitudes through Asia. In contrast, the preferred paths inferred from a more uniform distribution of sites were confined only to longitudes  54  Chapter 5. ADDITIONAL  RESULTS  55  20 Icelandic Paths  15 cu E 10 3  5 0 O  6 0  0  E  1 2 0  0  E  1 8 0  0  W  1 2 0  0  W  6 0  0  W  0 °  Longitude Figure 5.1: V G P paths for a distribution of sites i n volcanic database (Love, 1998). T h e dashed lines on the upper panel indicate the preferred paths obtained from the volcanic database. T h e lower panel gives the predictions of the model when the same sites are used.  through N o r t h America. T h e effect of the perturbation field was also tested on a data set containing magnetic measurements of lake and oceanic sediments. For this, I use a compiled database consisting of 21 records of the same reversal, mostly sampled from A t l a n t i c and Pacific O D P locations [Clement, 1991]. (Clement's study has also implied the presence of preferred VGP  paths through the Americas and Asia.) The histogram showing the distribution  of the V G P paths obtained with my model reveals geographical confinement of paths  Chapter 5. ADDITIONAL  RESULTS  56  O D P database (Clement, 199  60°S  180°W  120°W  60°W  120 W  60 W Longitude  60 E  120 E  180°W  60  120 E  180 W  0  0  6 5 4 01 JO  E  z  3  2 1 180 W 0  0  0  n  c  0  0  Figure 5.2: V G P paths for a distribution of sites i n oceanic sediments database (Clement, 1991). T h e dashed lines on the upper panel indicate the paths obtained from the sediment database. T h e lower panel gives the predictions of the model when the same sites are used.  through N o r t h A m e r i c a and Asia. (Figure 5.2). T h e American paths cover a wide sector of longitude, peaking at approximately 100° W , i n good agreement w i t h the observations. Finally, I consider a data set compiled by Gubbins and Coe [1993], which include both volcanic and sedimentary data. To this dataset I add several sites (Iceland, Greece) to define a representative collection of sites.  The locations have provided paleomagnetic  measurements on lavas (e.g., Hawaii, Iceland, B r i t i s h Columbia) and deep-sea sediments (e.g., Atlantic, Greece, Pacific).  M y model yields reversal paths which are w i t h one  Chapter 5. ADDITIONAL  RESULTS  57  exception confined to longitudes which cover A s i a and N o r t h A m e r i c a (Figure 5.3). T h e plot shows a geographical confinement around the Pacific r i m , as was very often suggested by the observations. T h e bi-modal distribution of paths has prompted various suggestions for the mechanism that generates the preferred V G P paths, most of them based on the apparent symmetry of the preferred paths. The A s i a n paths i n m y model occur due to sampling of European sites. A s it is possible that records coming from this part of the world have been extensively used for paleomagnetic measurements, I speculate that the preferred A s i a n paths might be an artifact of uneven sampling.  Figure 5.3: V G P paths for twelve sites compiled from the palaeomagnetic database.  It is clear from these examples that sampling can significantly influence the interpretation of transition fields. M y results are i n agreement w i t h the observations because N o r t h A m e r i c a n and A s i a n paths emerge when site locations are chosen to mimic the paleomagnetic databases. T h e V G P paths i n m y model are dependent on the sampling location because the perturbation $ contains non-axial-dipole terms, which are superimposed on the time-varying axial dipole. A s seen earlier i n this study, the spectrum  Chapter 5. ADDITIONAL  RESULTS  58  of the perturbation field $ depends on the structure of reversing magnetic field and the geometry of the conducting layer. It is equally interesting to see if the perturbation is large enough to be detectable when we allow for variability of the main magnetic field. To test this, the perturbation $ is superimposed on a transition field which includes both dipole and non-dipole components. The non-axial-dipole part of the transition field is represented as a realisation of a Gaussian process, based on the geomagnetic secular variation model of Constable and Parker [1988]. This part of the field, which was not present in my previous analyses, is referred to as the 'background' field. The presence of the background field accounts for the natural variability of the geomagnetic field. In the next sections I describe the implementation of the statistical model and the results obtained when the background field is incorporated in the transition field. 5.2  Brief Description of the Transition Field Model  The secular variation of the magnetic field encompasses changes on time scales from 10 to 10 years, reflecting variations in the mechanism producing the magnetic field. 4  Reversals of the field polarity represent a particular feature of the secular variation and, as mentioned earlier, have a duration of several thousands of years. To determine the influence of the perturbation field over the geodynamo field during a reversal, I seek to model a polarity reversal in which the main dipole field remains axial but decreases in strength during a transition so that the non-dipole part of the main field becomes significant in determining the orientation of the VGP. The question that is asked is whether the perturbation $ is large enough to leave its signature on the field in the intermediate stages of a reversal. The temporal variability of the axial dipole term is modelled as in the previous section,  Chapter 5. ADDITIONAL  RESULTS  59  by allowing a decrease i n intensity of the g° Gauss coefficient followed by an increase i n intensity i n the opposite field direction i n a time interval equal to the duration of the reversal. field,  For the rest of the coefficients I make use of a statistical description of the  which is an effective way to describe the characteristics of the variability i n the  geomagnetic field, and simpler than attempting to physically model the field at each instant. T h e statistical model proposed by Constable and Parker [1988] separates the behaviour of the dipole and non-dipole parts i n the Earth's magnetic field. It is assumed that the dipole part of the field is statistically independent from the non-dipole part.  The  spherical harmonic coefficients of the non-dipole field are regarded as realisations of a Gaussian distribution. A t a fixed radial distance r, the quantity / n  \ 2(1+2)  C+  Ri=(-) V  r  /  I i ) £  [(sH + (fcr) ], 2  2  (5-i)  m=0  known as the Mauersberger-Lowes spectrum [Constable and Parker, 1988], defines the power i n the magnetic field associated w i t h each degree /. T h e geomagnetic field spectrum decays rapidly w i t h increasing I. T h e usual interpretation of the spectrum is that the field described by terms w i t h degree 1 < / < 12 originates i n the core, whereas the field described by terms w i t h I > 16 has a crustal origin. A t r ~ c the spectrum Ri is not very dependent on I, therefore indicating a 'white noise' source at the C M B . Observations have shown that the power i n Ri is evenly distributed across the angular orders m and variable i n time. Therefore, the squared amplitude (g™) + (h™) is also 2  treated as a random variable.  2  T h e model proposed by Constable and Parker ( C P 8 8 )  treats every spherical harmonic coefficient as a normally distributed, independent random variable and uses the Lowes spectrum as a guide i n constructing the statistical model. For the non-dipole terms, the model spectrum is devised to be exactly flat at the surface  Chapter 5. ADDITIONAL  RESULTS  60  of the core. The expectation of the spectrum for the non-dipole terms of the field is (5.2) where a is a fitted paramenter. T h e coefficients w i t h 1^1  are independent samples of a  single zero-mean Gaussian process w i t h variance (07) , where 2  (5.3) In the C P 8 8 model, the axial dipole is treated differently since a Gaussian distribution for this term is incompatible w i t h the observations.  O n the other hand, the equatorial  dipole can be well fit by the Gaussian process. The terms g\ and h\ i n m y transition field model have been incorporated using the same statistical description as the non-dipole terms. I have used the statistical model C P 8 8 to generate a number of realisations of the Gaussian process, which were employed to describe different realisations of the non-axialdipole background field. The spectrum of the coefficients is plotted i n Figure 5.4. T h e value for the fitted parameter a, which gives the variance of the distribution, has been taken from the study by Constable and Johnson [1999], correcting the value given i n the C P 8 8 model. T h e field thus constructed simulates the geomagnetic field at the Earth's surface (up to degree and order 8). T h e transition field is described by the fluctuations of the dipole which decreases i n intensity to zero and increases i n the opposite field direction. T h e intensity of the axial dipole term (g°) before the onset of the reversal is given by the present value of the Gauss coefficient of the geomagnetic field. T h e intensities of the equatorial dipole terms (g\, h\) right before the reversal are given by the statistical model. T h e non-dipole part of the field is given by the C P 8 8 model and is considered stationary over the period of the reversal.  This is consistent w i t h field orientation  observations  which suggest that significant changes i n the non-dipole field have a time constant of  Chapter 5. ADDITIONAL  RESULTS  61  order 100 - 200 k y r [Elmaleh et al, 2003], much longer than the duration of one reversal. Therefore my model for a transition field makes a clear distinction between the dipole terms and the non-dipole terms of the geomagnetic field. T h e dipole terms are allowed to vary i n time, while the non-dipole terms are constant during a reversal. T h e perturbation $ is superimposed on the transition field.  2  4 6 Harmonic degree I  Figure 5.4: The geomagnetic field spectrum (normal and semi-log scale) at the Earth's surface, for I = 1 to 8, excluding the g\ term. T h e spectrum refers to the coefficients computed using the C P 8 8 field model.  5.3  V G P Paths Obtained When a Background Non-Dipole Field Is Included  T h e V G P paths that arise from the above field are computed for the 500 randomly distributed sites previously considered.  A t the beginning of the reversal process, the  dipole components are close to their initial values and the longitude of the V G P paths is not affected by the presence of the perturbation field. This is because the amplitude of  Chapter 5. ADDITIONAL  RESULTS  62  the perturbation is less than 10% of the field intensity before the reversal starts. W h e n the dipole terms have almost vanished, the orientation of the field is given by the nondipole field and the effect of the perturbation 4? becomes more significant because $ has the same order of magnitude as the non-dipole terms.  M y model computes the V G P  paths at the moment when the dipole vanishes. T h e results of a reversal with one non-dipole realisation of the magnetic field are shown i n Figure 5.5. T h e histograms of the V G P longitudes are plotted for the transition field w i t h and without the perturbation due to $ . The plots clearly show that the distribution  0  50  100  150  200 Longitude  250  300  350  0  50  100  150  200 Longitude  250  300  350  Figure 5.5: T o p : V G P paths obtained without the magnetic perturbation. For the non-dipole terms, a realisation of the statistical model CP88 is used. B o t t o m : V G P paths obtained when the magnetic perturbation is included.  of paths is affected by the presence of  Inasmuch as the statistical field does not yield  paths on A m e r i c a n longitudes, the perturbation is affecting the distribution of paths,  Chapter 5. ADDITIONAL  RESULTS  63  and moves the paths to the Americas. It is noticeable from the plots that there is also a preferred path through Asia, given entirely by this one realisation of the statistical model. 150 -  100 -  Longitude  Figure 5.6: Top: VGP paths obtained from a statistical field. Bottom: VGP paths obtained when the perturbation $ is added to the above field.  Figure 5.6 shows the outcome of the experiment on another statistical realisation. This unusual example was chosen because this particular realisation of the field already gives the appearance of the two preferred paths. Superimposing the effect of the perturbation $ displaces paths from the Asian longitudes and enhances the American path. The effect of the perturbation has been tested for more realisations of the statistical model, which can be viewed as individual reversals. For all cases the results have shown that the paths are confined to American longitudes. This experiment was concluded by averaging the results of 20 realisations of the  Chapter 5. ADDITIONAL  RESULTS  64  statistical field on 40 sites randomly distributed over the Earth's surface. A s expected, when the field is unperturbed, the distribution of the V G P longitudes is nearly flat as a result of averaging a number of realisations of the statistical field model. W h e n the perturbation is added to the field, the only outstanding feature i n the distribution of the V G P paths is the concentration of paths through the A m e r i c a n longitudes.  These  experiments show that the secondary magnetic field is strong enough to influence the transition field during a reversal when the dipole is decreasing i n intensity. T h e V G P longitudes are mostly controlled by the Y* geometry of the secondary field. Consequently, my results indicate that when the perturbation caused by the conductive basal layer is superimposed on a transition field, it is almost certain that preferred reversal paths through the Americas will be recorded. T h e final question I consider is how the duration of the reversal affects the distribution of the V G P paths. In my model, the period of time i n which the field completes a polarity reversal determines the rate of change i n the dipole field. It is therefore possible that for a lower rate of change i n the dipole, the secondary field will not stand out from the background field.. A s an example, I take a representative site i n the paleomagnetic database (Iceland) and run several reversal simulations for different durations of the reversal. I consider first that the reversal is happening i n one thousand years and then the duration of reversal is increased to four thousand years.  For each case I average the V G P paths over 20  and later 40 reversals. The results of this experiment show that the distribution of the V G P longitudes during a reversal is sensitive to the time interval i n which a reversal is happening (Figure 5.7). For a slower reversal, the amplitude of $ is not high enough to yield preferred V G P paths. T h e preferred path through A s i a is obvious for a faster reversal, whereas for a slower reversal the distribution of V G P paths is more uniform. I conclude that preferred V G P paths are more likely to be recorded when the reversal is  Chapter 5. ADDITIONAL  RESULTS  65  Site: Iceland R e v e r s a l Time 1000 years 20 realizations  100  200 Longitude 20 realizations  100  40 realizations  300  0  100  R e v e r s a l T i m e 4 0 0 0 years  200 Longitude  100  200 Longitude 40 realizations  200 Longitude  Figure 5.7: VGP paths obtained for different durations of polarity reversals.  rapid (i.e., 1000 - 2000 years). In order for slower reversal to produce preferred paths, my model requires an initial dipole intensity higher than the present-day value, so that the rate of change in g° is sufficiently large. The same experiment shows that the probability of recording preferred VGP paths increases with the number of reversals. Figure (5.7) displays the histograms of VGP paths recorded in the same location for 20 and 40 reversals. For a faster reversal, the preferred paths through Asia are more evident when the number of reversals increases. For the slower reversal case, the distribution of paths is quite flat when only 20 reversals  Chapter 5. ADDITIONAL  RESULTS  66  are taken into account. However, increasing the number of reversals tends to emphasize the paths through Asia.  Chapter 6  DISCUSSION A N D C O N C L U S I O N S  The goal of this thesis was to study the effect of a highly conducting layer at the base of the mantle on the magnetic field during a polarity reversal. The existence of preferred V G P paths during a reversal has received much interest among the paleomagnetic community and the model proposed in the present study provides a plausible explanation for this phenomenon. M y study has been equally interesting, challenging and rewarding and presents opportunities for further research. The main findings of the thesis are summarized below.  Preferred Reversal Paths The model developed in this work has shown that induction of a secondary magnetic field in a highly conductive layer at the base of the mantle can alter the non-axial dipole part of the Earth's magnetic field and hence produce preferred reversal paths. The secondary field is generated by electric currents in the conducting layer, which are induced by temporal variations in the dipole part of the magnetic main field. • The perturbation is characterized at the Earth's surface by equatorial dipole (Y^ ) and non-dipolar (Y ) 1  2  3  terms. When the polarity reversal occurs through a decrease in the dipole field, in the absence of the perturbation, the positions of the V G P are determined by the non-dipole components of the field. The non-dipole part of the Earth's magnetic field is believed to be mainly isotropic [Constable and Parker, 1988; Constable and Johnson, 1999; Quidelleur and Courtillot, 1996] and my study treats the g™ and h™ Gauss coefficients with I > 2 as 67  Chapter 6. DISCUSSION  AND  CONCLUSIONS  68  zero-mean, normally distributed, independent variables. The V G P paths obtained for an unperturbed transition field were randomly distributed in longitude, with no persistent geographical confinement from one reversal to another. The addition of the secondary field to the non-dipole field systematically affects the transition field at the surface. perturbation.  The largest effect is due to the Y  x  component of the  Superposition of the perturbation field over the pre-existent non-dipole  field makes the geomagnetic field anisotropic during a reversal which yields preferred locations of the magnetic pole. A perturbation generated by temporal variations of the dipole in a variably conducting layer with a configuration close to the C M B topography yields preferred reversal paths through the American longitudes.  One or more preferred paths?  The research presented in this thesis provides a physical explanation for producing preferred V G P paths during a reversal. The results show that the reversal paths given by a uniform distribution of sites have an asymmetric pattern, revealing a strong concentration of paths on the American longitudes. When using a non-uniform sampling of sites (reproducing the uneven distribution of real data), the model yields Asian and American paths. These distributions of paths are in agreement with the observations, which generally find two preferred paths, one of them following American longitudes. The bi-modal distribution of paths through the Americas and Asia was documented in the MatuaymaBrunhes transition [e.g., Laj et al., 1991], whereas grouping of reversal paths through the Americas only was observed in the Upper Olduvai reversal [e.g., Trie et al, 1991]. M y results showing longitudinal confinement of paths are consistent with the observations, but also show that non-uniform sampling can influence the interpretation of the reversal paths distribution.  Chapter 6. DISCUSSION AND  CONCLUSIONS  69  Persistence of equatorial dipole terms during a reversal Previous works have shown that the equatorial dipole part of the present field (or of a paleo-field with a similar configuration) has an important influence on the reversal paths during a reversal [Constable, 1992; Gubbins and Sarson, 1994]. My tests have revealed that for a reversal in which the equatorial dipole persists through a reversal, the additional field caused by the conducting layer is too small to affect the distribution of the reversal paths. However, a transition field characterized by decreases in both the axial and equatorial dipoles, is significantly affected by the magnetic perturbation, causing a geographical confinement of VGP paths to American longitudes. The clustering of paths over the Americas in my model is due to the Y± component of the secondary field produced in the conductive layer by decreases in the axial dipole. This term of the perturbation has the same geometry as the equatorial dipole. Additional computations have shown that temporal variations in the equatorial dipole (say during a reversal) also induce perturbations with Y] terms, although of smaller amplitude than those caused by decreases in the axial 1  dipole. A persistent perturbation with Y, dependence suggests that an equatorial dipole 1  could be present in the geomagnetic field at the surface,, even if the axial and equatorial dipoles of the geodynamo normally vanish during the reversal. The persistence of equatorial dipole terms in the field is important because these long wavelength terms strongly influence the field direction, thereby producing preferred reversal paths.  Duration of the reversal and the strength of the dipole Computations show that the magnetic field generated by variations in the axial dipole g° has the largest influence on the VGP paths. The amplitude of the perturbation field in my model depends on the initial value of the axial dipole intensity and the duration of  Chapter 6. DISCUSSION  AND  CONCLUSIONS  70  the reversal, which define the rate of change i n the magnetic field. Temporal variations i n the non-axial-dipole components produce much smaller perturbations due to the smaller rate of change. T h e results i n the previous chapter show that the V G P preferred paths are more likely to occur when the magnetic field reverses itself i n about 2000 years or so. For this situation, the rate of change i n the axial dipole is sufficient to cause a perturbation that rises above the natural variability i n the non-dipole, background field. A s an example, preferred A s i a n paths were detectable for sampling sites located i n Iceland for a reversal that occured i n 1000 years. O n the other hand, the tests have shown that for slower reversals, the rate of change i n the axial dipole decreases to the point where the perturbation has a weak influence on the V G P paths. For Icelandic sites, some preferred V G P paths were detectable only if a larger number of reversals were averaged. Real data sets do not usually provide a large number of reversals for the same site. Based on m y model, we can conclude that for slower reversals, which generate weaker perturbations, the chance of recording preferred paths is smaller. The model can however yield preferred V G P paths for slower reversals if the dipole field at the onset of reversal is stronger than the present-day value.  Conductivity at the base of the mantle T h e occurrence of preferred V G P paths i n my model is due to the magnetic perturbation generated within the t h i n , conducting layer i n the lowermost mantle. Lateral variations i n conductivity at the core-mantle interface are given by the compositional heterogeneity as a result of secular cooling of the core [Buffett et al, 2000]. The conductivity model devised for the study of V G P paths is constrained by the predictions of the sediment accumulation model and the dynamic model of C M B topography [Forte et al, 1995].  Chapter 6. DISCUSSION AND CONCLUSIONS  71  A d d i t i o n a l constraints on the conductivity structure are sought i n order to improve the fit between predictions and observations of V G P reversal paths.  For this, I use the  variability i n the secular variation ( S V ) data to detect the signature of the conductive layer i n modern observations. The analysis of S V data supports the interpretation that fluctuations i n the non-dipole field can be attributed to the effect of fluctuations i n the axial dipole i n a conductive layer at C M B . However, the model constrains the conductance of the layer to be no more than 1.9 x 10 S i n order to fit the amplitude of the secular 8  variation observations. This upper bound for the conductance is i n agreement w i t h the predictions of the sediment model [Buffett et al, 2000] and is sufficient to explain the nutation observations [Buffett, 1992].  Configuration of the C M B and time constraints on lower mantle dynamics T h e influence of the conductive layer on the V G P paths can persist for several hundred million years before the conductivity pattern changes at the base of the mantle. T h e time scale for altering the conducting layer is probably related to the time scale for changing topography at the C M B . Thereby changes i n the distribution of the conducting material would take place on a time scale of 10 years. In this case, the geometry of the preferred 8  paths can be the same for several reversals but we might find gradual changes i n the preferred paths over time scales longer than 10 years. 8  T h e length of the period for  which the same pattern for reversal paths is maintained can provide time constraints on the lower mantle convection and hence contribute to the understanding of the core-mantle interactions. 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