Electromagnetic Coupling Between the Fluid Core and its Solid Neighbours By Mathieu Dumberry B.Sc. (Physics), Universite de Sherbrooke A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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 GRADUATE STUDIES DEPARTMENT OF EARTH AND OCEAN SCIENCES We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA March 1998 © Mathieu Dumberry, 1998 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying for scholarly purposes may be granted by the head of my 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. Ear th and Ocean Sciences The University of Br i t i sh Columbia 129-2219 M a i n MaU Vancouver, B C , Canada V 6 T 1Z4 Date: 11 Abstract A t low-frequency, the nearly geostrophic force balance in the fluid core constrains axisymmetric fluid motions to be purely azimuthal and independent of position along the rotation axis. F lu id motions can thus be described by a set of concentric rigid cylinders, which are free to rotate about their common axis. When these cylinders are coupled by a magnetic field, the associated restoring forces give rise to torsional oscillations. These waves are thought to cause the observed fluctuations in the length of day by transferring angular momentum to the mantle and the inner core. The theory of torsional oscillations assumes that the stresses at the fluid-solid boundaries, which transfer angular momentum, do not alter the rigid nature of the fluid cylinders. This assumption is probably valid at the base of the mantle where the magnetic field is not large enough to alter the geostrophic balance in the fluid near the boundary. However, it is not valid at the inner core boundary ( ICB) where higher field strengths are likely to perturb the geostrophic balance. In this case, the Lorentz force has to be retained in the momentum balance. A complete analytical solution is given for the influence of Lorentz forces in the core. The model problem involves a conducting and rotating fluid between two plane conducting boundaries in the presence of a background magnetic field. The solution gives us a clear view of how boundary layers form near the solid and how the coupling to the mantle and the inner core occurs.' More importantly, we can directly see how the inclusion of Lorentz forces alters the velocity from rigid rotation and how this velocity differs from that obtained with the torsional oscillation theory. For cylinders that terminate on the inner core, it is found that the rigid rotations are perturbed for a range of background magnetic field strength and frequencies. A t decade periods, there is insufficient inertia to disrupt rigid rotations. However, for annual fluctuations, departures in the fluid velocity from rigid rotations are significant, which implies that the Lorentz force can not be neglected in the dynamics of the fluid core when calculating the electromagnetic coupling at the I C B . Ill Contents Abstract i i Table of contents i i i List of figures v List of tables v i i Acknowledgments . . . v i i i Introduction 1 Formulation of the problem 13 2.1 Geometry and basic equations of the model 13 2.2 Form of the solution of the equations . 20 2.3 Boundary conditions 24 Single-plate problem 29 3.1 Introduction ' . • 29 3.2 Solution of the problem 30 3.3 Results and discussion . . 34 3.3.1 Physical description of the solution 35 3.3.2 Coupling mechanism . 40 3.3.3 Variations of parameters and regime of solutions ." 41 Two-plate problem 51 4.1 Introduction . . . .. 51 4.2 Solution of the problem 51 4.3 Results and discussion 57 4.3.1 Physical description of the solution . 58 4.3.2 Transfer of angular momentum 61 Table of contents iv 4.3.3 Relative magnetic stresses at each boundary 63 4.3.4 Rig id rotations 65 4.3.5 Conclusions 67 Transfer of angular momentum 74 5.1 Introduction 74 5.2 Angular momentum of rigid rotations 75 5.3 Comparison of theories as a function of Bz 76 5.4 Comparison of theories as a function of oscillation period 79 5.5 Influence of waves on angular momentum transfer 82 5.5.1 Standing waves on the fluid cylinder 82 5.5.2 Complications when the plate velocities are different 86 5.5.3 Relative contribution of each plate to the total torque on the fluid column 86 Rigid rotations from a concentrated torque 93 6.1 Introduction 93 6.2 Model for a concentrated torque inside the fluid 95 6.2.1 System of equations 95 6.2.2 Influence of a prescribed force on a fluid column 97 6.2.3 Boundary conditions 99 6.3 Results and discussion 102 6.3.1 Solution with no magnetic stress at the ends of the fluid column . . 102 6.3.2 Solution including the magnetic stress at the ends of the fluid column 105 Conclusions 113 References 121 List of Figures 1.1 Length of day variations for the past 15 years 2 1.2 Rig id rotations in the Earth's fluid core 3 2.1 Schematic diagram of the model for the fluid core 14 2.2 Identification of the bounding solids 15 3.1 Relation between the secondary magnetic field and the induced electric current 39 3.2 Solution for the scalar potentials in the fluid and the solid at frequency to = ft/104 and magnetic field Bz = 5G 46 3.3 Solution for the scalar potentials in the fluid and the solid at frequency u = ft/102 and magnetic field Bz = 5 G 47 3.4 Solution for the scalar potentials in the fluid and the solid at frequency uj = ft/106 and magnetic field BZ = 5G 48 3.5 Solution for the scalar potentials in the fluid and the solid at frequency u> = ft/104 and magnetic field B2 = 0.5G 49 3.6 Solution for the scalar potentials in the fluid and the solid at frequency u> = ft/104 and magnetic field Bz = 50 G 50 4.1 Toroidal acceleration of the current free region due to fluid circulation . . . 60 4.2 Transfer of angular momentum from the plates to the boundary layer regions by an exchange of charges 62 4.3 Solution for the scalar potentials of the velocity in the fluid at a magnetic field BZ = 5G 68 4.4 Solution for the scalar potentials of the magnetic field in the fluid at a magnetic field Bz = 5 G 69 4.5 Solution for the scalar potentials of the velocity in the fluid at a magnetic field BZ = 35G 70 4.6 Solution for the scalar potentials of the magnetic field in the fluid at a magnetic field Bz = 35 G 71 4.7 Solution for the scalar potentials of the velocity in the fluid at a magnetic field Bz = 100 G 72 4.8 Solution for the scalar potentials of the magnetic field in the fluid at a magnetic field Bz = 100 G 73 List of figures vi 5.1 Transfer of angular momentum to the fluid as a function of magnetic field strength at a period of 200 days 88 5.2 Transfer of angular momentum to the fluid as a function of magnetic field strength at a period of 400 days 88 5.3 Transfer of angular momentum to the fluid as a function of the period of oscillation for Bz = 50 G 89 5.4 Departures of our model from rigid rotations 89 5.5 Solution for the toroidal fluid velocity at large values of magnetic field. . . 90 5.6 Resonant excitation of standing waves on the fluid column (part I) 91 5.7 Resonant excitation of standing waves on the fluid column (part II) 91 5.8 Resonant excitation of standing waves on the fluid column for unequal plate velocities (part I) 92 5.9 Resonant excitation of standing waves on the fluid column for unequal plate velocities (part II) 92 6.1 Schematic diagram of the model, with the presence of a concentrated torque. 95 6.2 Symmetric profile of a concentrated force in the disc 100 6.3 Solution for the scalar potentials of the velocity at a magnetic field Bz = 5 C7.107 6.4 Solution for the scalar potentials of the magnetic field at a magnetic field BZ = 5G 108 6.5 Solution for the scalar potentials of the velocity at a magnetic field Bz = 50 G. 109 6.6 Solution for the scalar potentials of the magnetic field at a magnetic field B. = 50(3 110 z 6.7 Solution for the scalar potentials of the velocity, when the coupling at the fluid-solid boundaries is included I l l 6.8 Solution for the scalar potentials of the magnetic field when the coupling at the fluid-solid boundaries is included 112 Vll List of Tables 3.1 Parameters used in calculation of the electromagnetic coupling between the fluid and a single plate 35 4.1 Parameters used in calculation of the electromagnetic coupling between a cylinder of fluid and two bounding plates 58 5.1 Parameters used in comparison of the angular momentum transfer between the rigid rotation model and the more complete model 77 6.1 Parameters used in calculating the torque on the fluid column due to a concentrated force 103 6.2 Parameters used in calculating the torque on the fluid column due to com-bined effects of electromagnetic coupling with the plates and the presence of a concentrated force 106 VIU Acknowledgments There are so many people that have made the last two and a half years of my life so enjoyable that I have to use single spacing to fit it all on one page. First of all, a mill ion thanks to my advisor Bruce Buffett, for his incredible patience and extraordinary mentorship. I've learned so much during the countless hours I spent in his office, appreciating his approach to science and to life in general. Y o u were a dream advisor, as you gave me the perfect balance between guidance and freedom. You have been a model that I have little hope to live up to. Thanks to everyone in the geophysics building, starting with the faculty members, for providing a work atmosphere free of internal politics and therefore setting the tone of the department. Thanks to the postdocs and the graduate students for creating this remarkably stimulating environment. A l l along, you made me feel as i f I were part of a great family. Thanks to everyone else, simply for being who you were. For the individual honours, thanks Karen Wijns for being the child you are and the child you wi l l always be. Thanks Tiah Goldstein, you gave me a glimpse of life outside the department. Thanks Col in Farquharson for your calm, your quietness, and for the boots! Thanks Dave Hildes, for your constant enthusiasm and joviality. Thanks Shawn Marshal l for being an inspiration in many domains. Thanks Gwenn Flowers,. for your radiating happiness and great companionship. Many thanks Kev in Kingdon, for your reassuring presence. Y o u won the contest of the person who disturbed me the most often in my work, and believe me, it was always appreciated. I find it impossible to dissociate you from my stay in Vancouver. A n d big thanks for letting me finish before you! A n d thank you infinitely Camille L i , for the numerous never-ending conversations, for your clever analysis of people and amazing ability to dissect all aspects of mundane day-to-day life. Y o u might dream of being a lighthouse keeper, but you sure were a lighthouse in many ways. In the "everyday life" category, thanks to Calhoun's for letting me use their tables as my unofficial second office. Thanks to Yoka's, for the excellent coffee beans and the warming smiles (...and yes, it 's for a plunger!). Thanks to Tanglewood books for providing most of the food for my delirious brain. I don't think I've ever walked in front of that shop window during open hours, and not entered, sometimes just for a few seconds, to enjoy the atmosphere, the music and the soothing smell of old books. Other random thanks to room 253 (where I've had my most brilliant ideas!), to the Highway 99 (the most gorgeous holiday road), to the yellow couch in the reading room (always comfortable after 2 a.m.!), to Wrang and Inverse (my two most intimate compan-ions in the department!), to my commuting partner "Hard Rock" and my riding partner V i t a l i , to L a Biciclet ta, to Hollyburn, to the ski-car, to the $5 ladies-series, to Oatmeal Ci ty and Friday afternoon soccer, to Dr . McConkey, to Broadway Produce, to Benny's jalapeno bagels, to the bulk candy section at Safeway, to the Hollywood theater and the Ridge, to Prussin music, to Shelagh Rogers and C B C Radio 2. IX "Neither you or anybody else know anything about the real state of the Earth's interior." - Professor von Hardwigg ( J4 Journey to the center of the Earth, Jules Verne) 1 Chapter 1 Introduction Studies of the rotation of the Ear th over the past three hundred years show that the length of the day (or L O D ) is not constant [McCarthy and Babcock 1986]. The Ear th as a whole is exchanging angular momentum with the moon by means of t idal friction. The energy lost by the Ear th from tidal dissipation reduces its angular velocity. The time the Ear th takes to complete one full rotation around its axis, the length of the day, is then slowly increasing on the order of 0.7 milliseconds per century [Stephenson and Morr ison 1984]. The length of day data also reveal small oscillations about that slowly increasing trend. In figure 1.1 we present the set of data for the last 15 years, obtained by Very Long Baseline Interferometry(VLBI) [T. Herring (pers. comm.)]. Two periodicities are evident from these data; one period is about one year and the other period is on the order of 30 years. These oscillations suggest that the angular momentum of the solid Ear th (or the mantle) is not constant. Hence, the coupling between the mantle and the other constituents of the Ear th permits an exchange of angular momentum. The yearly period of oscillation can be explained by the coupling between the mantle and the atmosphere [Rosen 1993]. The angular momentum carried by the winds in the atmosphere account for almost all of the changes in mantle's angular momentum at annual periods. The oceans are also believed to play a role, although its significance has not been assessed quantitatively yet [Ponte 1990]. It is assumed that the core is uncoupled from the mantle at these frequencies and does not participate in the angular momentum exchange [Chao 1994]. Chapter 1: Introduction 2 I S S O 1 9 8 5 I S S O 1 S 9 5 D a t e ( y e a r ) Figure 1.1: Length of day variations from the V L B I data for the past 15 years. There is not enough power in the angular momentum fluctuations in the atmosphere to explain the long period oscillation of the L O D variations. Since no known exterior torque acts on the Ear th at such frequency, the transfer of angular momentum can only be explained by internal coupling between the core and the mantle. The flow in the core is not directly observable, so we must rely on theoretical models to assess the angular momentum carried by the core. A t time scales much longer than the period of rotation of the Ear th , and in the absence of magnetic forces, the flow in the outer l iquid core is geostrophic. That is, the Coriolis force is balanced by the pressure gradients. The flow is everywhere perpendicular to the pressure gradient and, by the nature of the Coriolis force, also perpendicular to the Earth 's rotation axis. Hence, the flow is necessarily two-dimensional and the gradients in velocity along the rotation axis are necessarily zero. This result is known as the Proudman-Taylor theorem [Proudman 1916; Taylor 1917]. For an axisymmetric flow and an axisymmetric container, the velocity remains constant on an imaginary line aligned in the direction of the axis of rotation. The Proudman-Taylor theorem constrains the flow to a set of rigid cylindrical shells aligned with the axis of rotation (figure 1.2). When the magnetic force (often referred as the Lorentz force) is reinstated, the basic flow in the outer core Chapter 1: Introduction 3 is magnetogeostrophic. If this Lorentz force is small, the Proudman-Taylor theorem still applies and the rigid rotations remain a valid description of the fluid motions. Figure 1.2: The Proudman-Taylor theorem applied to the Earth's fluid core restricts the motion to rigid rotations aligned with the rotation axis. Jault et al. [1988] and Jackson et al. [1993] have computed the angular momentum carried by the core at decade time scale using this model for the flow in the core. The zonal velocity arising from the motion of individual cylindrical shells was deducted from the variations over time (or secular variations) of the magnetic field at the surface of the Ear th . The decade time scale of these variations is much shorter than the diffusion time for the magnetic field in the core, which is on the order of 10 4 years. The core can then be approximated as a perfect conductor. In the absence of diffusion, the magnetic field lines are attached or "frozen" to conductors and wil l follow their motion. This is known as Alfven's frozen-flux hypothesis. The magnetic field in the core is then assumed to be frozen to the fluid parcels. If we assume that the mantle is an insulator and does not disturb the magnetic field, we can thus infer the fluid motion at the core-mantle boundary ( C M B ) by monitoring changes in the magnetic field at the surface of the Ear th . The use of the "frozen-flux" hypothesis to infer fluid velocities was first proposed by Roberts and Chapter 1: Introduction 4 Scott [1965]. In the studies by Jault et al. [1988] and Jackson et al. [1993], the flow inferred from the changes in the magnetic field was averaged over the longitude to yield the axisymmetric component of the fluid flow. Averaging the axisymmetric flow over an interval of latitudes gives the motion attributable to a specific cylindrical shell. Used in conjunction with' the Proudman-Taylor theorem, the frozen-flux hypothesis can thus yield angular velocities for a set of cylindrical shells within the core. The total angular momentum of the core is then obtained by adding the angular momentum carried by all the individual shells. The variations in time of this total angular momentum reproduces fairly well the changes of angular momentum in the core required to explain the long period oscillation in the LOD fluctuations. Hence, this confirms that the exchange of angular momentum between the core and the mantle is indeed responsible for these decade fluctuations. The second conclusion that can be drawn from these studies is that the system of rigidly rotating cylindrical shells adequately models the flow in the core at decade time scales. However, the nature of the coupling between the core and the mantle responsible for this angular momentum exchange is still unsolved. Several coupling mechanisms have been proposed and the most cited ones are viscous, gravitational, topographic and electromag-netic. The gravitational coupling between the fluid and the mantle arises from aspherical density variations due to convective fluctuations [Jault and Le Mouel 1989]. This coup-ling might play a role at periods of several hundred years but does not provide a viable explanation for variations with decade periods [BufFett 1996]. Similarly, the poorly known kinematic viscosity of the fluid core is believed to be too small to generate an efficient cou-pling mechanism [Kuang and Bloxham 1993]. The topographic coupling originates from the pressure that the fluid exerts on "bumps" or other topographical features at the base of the mantle [Hide 1969]. The topography at the base of the mantle is poorly resolved Chapter 1: Introduction 5 by seismology, which impairs the assessment of a reliable quantitative coupling strength. Electromagnetic coupling at the C M B depends on the electrical conductivity of the mantle, but it remains a viable mechanism to exchange angular momentum. Conductors tend to drag the magnetic field lines with their motion. A shear in veloc-ity between two conductors wi l l stretch the magnetic field lines and induce a secondary magnetic field in the direction of the shear. This induced field generates a magnetic force that opposes the relative motion between the two conductors. In the context of the L O D fluctuations, relative angular motion between the cylinders in the core and the mantle at the C M B wi l l stretch the magnetic field lines and set up a restoring magnetic force that op-poses the relative motion. This phenomenon is called the "w-effect" and is the mechanism that the electromagnetic coupling at the C M B depends upon. To correctly assess the angular momentum in the core, we need a dynamical description of the fluid core. Whi le the Proudman-Taylor theorem constrains fluid motions to occur as rigid cylinders, it is possible for angular momentum to be transferred between cylinders. Braginsky [1970] developed a theoretical model to quantitatively assess the coupling be-tween the cylindrical shells. This model is based on an electromagnetic coupling between cylinders and is presented as the Torsional Oscillation ( T O ) theory. The cylindrical shells in the fluid are not free to rotate independently of one another when their surfaces are permeated by a radial magnetic field. Relative displacements between adjacent shells wi l l deform the radial magnetic field. The resulting Lorentz force produces a torque that acts to return the cylinders to their undisturbed configuration (e.g. one in which the radial magnetic field is undisturbed). This electromagnetic coupling between cylinders is then based on the same principles as the w-effect described above. The magnetic field provides a restoring force and the cylindrical shells wil l undergo oscillations. The strength of the radial field is readily adjusted to oscillation periods of 30 years, comparable to the period Chapter 1: Introduction 6 of L O D changes. The coupling of these torsional oscillations with the mantle is accomplished through electromagnetic coupling associated with the u>-effect. Braginsky [1970] provided a quant-itative assessment of this coupling, described in terms of magnetic stresses created by the differential motion between the core and the mantle at the C M B . It is assumed that the stress at the ends of the fluid columns wi l l not be strong enough to break the rigid nature of the fluid motions. In other words, the deformation of the magnetic field in the fluid does not alter the geostrophic force balance near the ends of the fluid columns. The amplitude of the magnetic stress is proportional to the strength of the magnetic field that permeates the boundary. A t the C M B , the magnetic field is believed to be weak (~ 5 G [Langel and Estes 1985]). The Lorentz force wil l then be small and the Proudman-Taylor theorem is valid. The assumption that the magnetic stress would not deform the fluid at the C M B is then justified. The magnetic field at the inner core boundary ( ICB) can be substantially larger. Recent dynamo simulations [Glatzmaier and Roberts 1996] suggest fields that can be as large as 100 G. The electromagnetic coupling of the cylindrical shells that terminates on the surface of the inner core wi l l be strong. (We refer to the cylinder that is tangent to the equator of the inner core as the tangent cylinder (see figure 1.2). Those cylinders that terminate on the inner core are inside the tangent cylinder.) We also note that a strong field in a fluid is not easily deformed. In the T O theory, the whole tangent cylinder is assumed to rotate as a rigid column because the strong coupling at the I C B prevents any differential motion. The angular momentum exchange between the tangent cylinder and the mantle is then fully assessed through the electromagnetic coupling by the magnetic stresses at the C M B . Chapter 1: Introduction 7 A recent study on the gravitational coupling between the inner core and the mantle [Buffett 1996] alters this picture. There exists a minimum gravitational potential energy between an heterogeneous mantle and the inner core. Misalignments between the mantle and the inner core wi l l increase the gravitational potential energy and wi l l produce a restoring force that tends to realign them. This coupling proves to be very efficient and the realignment wi l l occur on a time scale one order of magnitude smaller than the decade periods of the L O D fluctuations. In the model of the Earth's interior at decade periods, the angular motion of the mantle and the inner core are then assumed to be gravitationally "locked" together. The inner core can no longer be assumed to be simply embedded in the tangent cylinder and follow its toroidal motion. Its presence and motion relative to the fluid core has to be accounted for. Hence, the cylindrical shells inside the tangent cylinder can no longer be as a single column with an embedded inner core. Instead, the columns must now be separated in two distinct regions above and below the inner core. These cylinders wi l l be bounded by the mantle at one end and the inner core at the other, unlike the single-column approximation which terminates at the mantle at both ends. We now face the problem of assessing the electromagnetic coupling of these cylindrical shells at the I C B . The conductivity of the inner core is similar to the conductivity of the fluid and thus much larger than the conductivity of the mantle. As we have already pointed out, the magnetic field at the I C B is potentially very strong. The coupling between the inner core and the fluid wi l l then be very efficient on account of this large field and the good conductivity of the inner core. It seems unlikely that rigid rotations can be assumed inside the tangent cylinder be-cause the large magnetic field at the I C B wil l generate a large Lorentz force that can no Chapter 1: Introduction 8 longer be neglected in the force balance in the fluid. This wi l l invalidate the Proudman-Taylor constraint and disrupt the rigid character of the fluid motions. The coupling model developed by Braginsky [1970] for the C M B , where the rigid rotations were a priori as-sumed, can not be applied at the I C B . Hence, to assess the contribution of the cylindrical shells located inside the tangent cylinder for the exchange of angular momentum, a new coupling model must be developed. This is the purpose of the present work: to develop a model that describes the behav-ior of the fluid as a result of the electromagnetic coupling with moving conducting solid boundaries. In this model, oscillations in the angular velocity of the mantle and the inner core (necessary to explain the L O D fluctuations) serve as a forcing at the ends of the fluid cylinders. We want to assess the behavior of the fluid in response to this forcing. The Lorentz force is kept in the force balance and the deformation of the magnetic field at the boundaries wi l l influence the dynamics of the fluid. A t the same time, the fluid flow tends to carry the field lines and thus, gives a feedback to the magnetic field. A solution for the complete fluid behavior enables us to make a direct comparison wi th Braginsky's model of electromagnetic coupling. The angular momentum transferred from the oscillating solid to the fluid can be easily computed with my model and compared wi th the angular momentum transferred to the fluid by assuming rigid rotations. A full description of the fluid velocity can also explicitly show how departures from rigid rotations in the fluid arise. The deformations in the fluid cylinders are strongest at the boundaries and this raises concern on another issue. The calculation of angular momentum in the core is based upon the assumption that the zonal velocities at the top of the core are equivalent to the velocities in the middle of the fluid columns. Large velocity gradients in Chapter 1: Introduction 9 the direction of the rotation axis, particularly in the vicinity of the solid boundaries, would invalidate this assumption. The more rigorous treatment of the fluid behavior enables one to assess the validity of this assumption under different conditions. Our main concern remains to assess the electromagnetic coupling of the cylindrical shells inside the tangent cylinder. This is where problems with the use of rigid rotations are expected to be most serious. However, this same model will also be applicable to cylindrical shells outside the tangent cylinder that extend from the mantle to the mantle with suitable changes in the material parameters. The generality of my model also has the advantage of being applicable to situations other than the assessment of the angular momentum transfer at decade periods. For in-stance, there have been no extensive studies on the nature of the electromagnetic coupling at yearly periods. Can the fluid motions at annual periods be described by rigid rotations? The contribution of the core in the angular momentum balance of the short period oscilla-tion of the LOD fluctuations has been proposed by Zatman and Bloxham [1997] although uncertainties in the behavior of the fluid at these frequencies limit the conclusions of this study. The electromagnetic coupling mechanism developed in my model does not rely on rigid rotations and the angular momentum transfer can be computed for any frequency. One can thus simultaneously evaluate the angular momentum transferred to the core at yearly period and verify if rigid rotation coupling remains a good approximation at these time scales. Improvements in the quantitative assessment of the electromagnetic coupling between the mantle and the core results in a better understanding of the parameters and physical processes in the Earth's deep interior. The parameters of my model can be scaled in order to match the amplitude of the angular momentum transfer required to explain the Chapter 1: Introduction 10 L O D fluctuations. Constraints on the magnitude of these parameters, for which we lack observations, can thus be inferred. For instance, the strength of the magnetic field at the I C B required to satisfy the data can thus be used as a constraint in the dynamo simulation models. The correlation between the data and the results of an electromagnetic coupling model can also yield conclusions on the importance of the electromagnetic coupling versus other coupling mechanisms. The second chapter of this work describes how the problem is formulated. It explains the geometry of the model, and the basic forces that influence the dynamics of the system. The equations that describe the evolution of the velocity in the fluid and the evolution of the magnetic field in the fluid as well as in the solid are developed. Simplifications of these equations are achieved by representing the fields in terms of poloidal and toroidal potentials. The final system of coupled equations in the fluid and the solid is presented and the general form of the solution of these equations is given. Finally, the boundary conditions that link the solutions in the fluid to the solutions in the solid are developed. In the third chapter, I consider the simplified problem of an electromagnetic coupling between a fluid which overlies a single oscillating boundary. This experiment is useful as it unravels the physical processes that occur with the electromagnetic coupling. Among other things one gains familiarity with the concepts of boundary layers that form in the fluid as a result of the adjustment of the fluid to the oscillating boundary. The principles of fluid transport, electrical current loop and induced magnetic field in the boundary layer are examined in detail. In the fourth chapter the second boundary is reinstated. We obtain solutions in the intended geometry; a fluid cylinder trapped by two conducting solid layers. The electro-magnetic coupling between the fluid and the solid occur at each boundary. The physical Chapter 1: Introduction 11 insight gained in the previous chapter prove very useful to explain the behavior of the fluid when the coupling occurs at both ends. Near each solid, there is a boundary layer adjustment region. The presence of these boundary layers is essential for the transfer of angular momentum from the solids to the whole column of fluid. Two distinct regimes of solution in the fluid are identified in the analysis; a diffusive regime and a wave regime. Rig id rotations persist in the diffusive regime, where the boundary layers are small and the amplitude of the fluid and magnetic disturbances in the layers are also small. The assumption of rigid rotation breaks down in the wave regime, where waves travel through the whole column. The main parameter that regulates the regime of the fluid solution is the strength of the magnetic field. A magnetic field strong enough to elevate the magnetic forces to the same order as the other forces in the momentum balance wi l l propel the solution into a wave regime. The fifth chapter assesses the transfer of angular momentum to the fluid by the os-cillating boundaries. It is possible to compare my results with the angular momentum transferred when rigid rotations are assumed in the fluid. It is shown how my results dis-agree with rigid rotations for a range of magnetic field strength and a range of frequencies of oscillation. The sixth and last chapter introduces a third forcing term on the fluid; an electro-magnetic torque that results from relative motion between adjacent cylindrical shells (e.g. the restoring force which is responsible for torsional oscillations). The theory of T O as-sumes that the net restoring torque on each cylinder is independent of how this torque is distributed over the surface of the cylinder. To test this idea, we want to confirm that a height varying torque wil l still produce a rigid rotation. If the coupling between adjacent cylindrical shells does not produce rigid rotations, it is futile to develop a coupling model with the solid boundaries that relies on rigid rotations. This assumption can be verified by Chapter 1: Introduction 12 incorporating in my model a torque concentrated at a particular height in the fluid column. The presence of this torque is modelled by modifying the original equations of the model and introducing additional boundary conditions. This study is motivated by potential inconsistencies in assuming rigid rotations in the fluid for the purpose of assessing the electromagnetic coupling at the I C B . In the final chapter, we wil l summarize the results of each chapter and devote special attention to the implications of this work for the Earth's core. It is found that at decade periods the electromagnetic coupling does not disrupt rigid rotations even inside the tangent cylinder where the magnetic field is large. A t these low frequencies, there is not enough inertia in the fluid to generate waves of sufficient amplitude. Hence, a rigid rotation model for the flow is sufficient in the fluid core at decade periods. A t annual periods, the waves have larger amplitudes and the fluid motion inside the tangent cylinder can no longer be described by rigid rotations. The Lorentz force must be included in the momentum balance to properly describe the fluid motion at annual periods. 13 Chapter 2 Formulation of the problem 2.1 Geometry and basic equations of the model A t very low frequencies, the Proudman-Taylor condition constrains fluid motions to two dimensions with no variations in the direction of rotation. Axisymmetr ic motions are characterized by relative rotations of fluid cylinders aligned with the rotation axis. We model the fluid outer core as a conducting layer of fluid bounded by plane boundaries (figure 2.1). For the region outside the tangent cylinder (the cylindrical surface tangent to the equator of the inner core), the two planar boundaries represent the mantle (figure 2.2 A ) whereas for the region inside the tangent cylinder, the fluid cylinders are bounded between the mantle and the inner core (figure 2.2 B ) . We wil l always assume as a convention that the inner core, when it is present, represents the bottom boundary. In other words, we only treat the fluid cylinders above the inner core and assume that the fluid cylinders under the inner core wil l present symmetric solutions. Curvature effects at the I C B and C M B are neglected as they do not alter the physical mechanism of the electromagnetic coupling. The axisymmetry of the system would be pre-served in the curved geometry. The axisymmetrical shear in the velocity at the boundaries of the fluid can be adequately modelled by plane boundaries. A different geometry which encompasses the effects of curvature would alter the amplitude of the fluid motions only by small amounts. However, our geometry of plane boundaries normal to the axis of rotation is inadequate in the equatorial regions of the I C B and the C M B , where the real surfaces are nearly aligned with the rotation axis. The vertical extent of the solid above and below the fluid is assumed to be infinite. Chapter 2: Formulation of the problem 14 Each solid "plate" can be considered either as a conductor with a uniformly distributed conductivity or as an insulator with a thin finite conducting layer adjacent to the fluid. The whole geometry is in rotation at a frequency aligned with the axis of the fluid cylinder (normal to the plates). The rotation rate of the conducting plates fluctuates about the average value f2 with a frequency ui and their motion wi l l perturb the fluid as a result of the magnetic coupling. We assume that a constant magnetic field B = Bzz, generated by the dynamo, per-meates the fluid and the plates. Only the vertical (or axial) component of the background magnetic field contributes to the electromagnetic coupling by the w-efFect. The other com-ponents of the field wi l l alter the momentum balance in the fluid but not the coupling mechanism. They can be neglected altogether, as results of recent dynamo simulations [Kuang & Bloxham 1997] suggests that the poloidal magnetic field at the I C B is mainly aligned with the rotation axis. z=0 z,Cl,B Solid «* Fluid .Solid Figure 2.1: The fluid outer core is modelled as a conducting layer of fluid bounded by plane conducting boundaries. We assume the fluid is homogeneous and incompressible ( V • v = 0). The fluid is in a hydrostatic ini t ial state. Perturbations from this basic state are governed by the Navier-Stokes equation, which gives the momentum balance and describes the evolution of the Chapter 2: Formulation of the problem 15 M A N 1 LI." M A N T L E F L U I D F L U I D M A N T L b INNER C O R E A B Figure 2.2: Identification of the bounding solids depending on the location of the cylinder. A : F lu id cylinders outside the tangent cylinder. B : F lu id cylinders inside the tangent cylinder. velocity field in the fluid [Gubbins and Roberts 1987], <9v / P\ 1 — + v V v + 2 f i x v = - V - + — ( V x B ) x B + i / V 2 v , (2.1) dt \p) pn where v is the velocity field, B is the magnetic field, 0 is the frequency of rotation of the Ear th , P is the effective pressure including the effects of the centrifugal force, p is the density of the fluid, fi is the permeability of free space and v is the kinematic viscosity. We neglect the effect of viscosity {y = 0) in the momentum balance as the E k m a n number (importance of the viscous forces) in the core is many orders of magnitude smaller than the Elsasser number (importance of the magnetic forces) [Gubbins and Roberts 1987]. We also assume that the fluid has uniform density. So, apart from the basic hydrostatic balance, there is no gradient in density and thus no buoyancy force. The induction equation describes the evolution of the magnetic field [Gubbins and Roberts 1987]: — B = V x (v x B ) - V x (T/V x B ) , (2.2) where r) = 1/o-fU- is the magnetic diffusivity and cr^ the electrical conductivity of the fluid. The velocity field v in equations (2.1) and (2.2) represents the perturbation from a rotating ini t ial state. The perturbation of the magnetic field from the background poloidal Chapter 2: Formulation of the problem 16 Bz field is denoted by b and the total magnetic field wi l l be B = Bzz-\-h. The magnetic field induced in the azimuthal direction by the a>-effect can be large, but the axial component of the field wi l l be significantly smaller than Bz. We assume that both v and b are small so that non-linear terms in the perturbations can be neglected in both the momentum and the induction equation. This assumption can be verified subsequently by comparing the relative importance of the neglected terms. The simplified linearized momentum and induction equations become: d 1 —v + 2S7 x v = - V T T + — ( V x b ) x 5 2 i (2.3) ot p/j, d —b = V x (v x Bzz) - V x ( i / V x b ) , (2.4) where 7r = P/p. Periodic motions of the solid boundaries produce an axisymmetric torque on the fluid, which causes axisymmetric perturbations v and b wi th an equivalent period. The geometry of the system suggests the use of cylindrical coordinates (£, (p, z) and a convenient way to express the perturbation fields is to separate the azimuthal part (toroidal part) and poloidal part using scalar potentials, v = [ 4 + V x ( ^ ) ] e - i r f b =[T4> + Vx (Pf)] e —iu>t where the scalar potentials u, ip, T and P are function of s and z only, u is the angular velocity and T the toroidal magnetic field, ip and P can be interpreted as stream functions of the poloidal velocity and poloidal magnetic field. These definitions satisfy the divergence-free condition on both v and b. Substituting these definitions into the momentum and induction equation and taking the curl of the momentum equation to eliminate the gradient of pressure yields a system of Chapter 2: Formulation of the problem 17 6 equations for the scalar potentials. However, the s and z component of the induction and momentum equation are redundant. For example, the s and z component of the induction equation are _d_ dz ld_ t s ds d n n d , (did n d2 > -P - B—ip - V — -£—sP + —P dt dz \ds s ds dz2 > 9 n „ d , ( did n d2 > = 0 = o (2.6) (2.7) We thus have equations of the form -§-z[M] = 0 and \f-ss [M] = 0. It follows that M has to be a constant with respect to z and we are left with J^s [M(s)] = 0, which implies that M(s) = C/s, where C is a constant. This solution is singular at s = 0 unless C = 0. Since there is no reason why the magnetic field or the velocity field should present a singularity at s = 0, we have to assume that C = 0 and hence M = 0. The s and z components combine to generate a single poloidal equation. For this reason, the number of equations reduces from 6 to 4; one poloidal and one toroidal equation for the evolution of the velocity and the magnetic field perturbation: -T - B—u - VAT = 0 dt dz S 2S19 B £ S T = 0 dt dz pfi dz -Ai, + 20—u - —AP = 0, dt dz pp, dz (2.8) (2-9) (2.10) (2.11) where: A(-) d i d . , d2 . , ds s ds Chapter 2: Formulation of the problem 18 If we assume that the geometry extends to infinity in the radial direction, we can simplify the problem by using a von Karman similarity solution. This solution supposes that all toroidal and poloidal scalars are linearly dependent on s (e.g. ifi(z,s) = stp(z)). Using this representation in (2.5) yields 1 d z * ' (2.12) b = [ - f j s i + fsj> + 2Pz] e-^ for the cylindrical components of the perturbed fields. The poloidal potentials (or stream functions) ib and P can be interpreted as the axial component of their respective fields, while the derivative with respect to z represents the radial component of the fields. We note that only the axial component is independent of s. The toroidal and radial compo-nents increase with the radius and hence, the toroidal and radial part of each field are unbounded as s —• oo. This behavior is expected because the rigid rotation velocity of the infinite plates wi l l also be unbounded and wil l induce infinite velocities in the fluid. The advantage of extending the solution to infinity is that we don't have to worry about the boundary conditions associated with the radial surface of a finite cylinder. The von Karman similarity solution also represents a good approximation of the physical behavior of the system because the departures from a linear gradient in 5 between the variables of adjacent fluid shells in the radial direction is expected to be small when the cylinders are threaded with a magnetic field. The mathematical advantage of the similarity solution is that it eliminates the radial variable from the system of equations. The only s-dependence in equations (2.8) to (2.11) occurs in the operator A . B y assuming that each scalar has the form ib(z,s) = stp(z), we eliminate the s derivatives in A , leaving only the second derivative wi th respect to z. A constant factor of s can be eliminated from every term of every equation, which yields the Chapter 2: Formulation of the problem 19 following four coupled equations for the scalars P(z),T(z),u(z) and V>(z): %-P - B.§j -V^P=0 (2.13) —T - B —u - ri-Ot zdz 'd ~ -u-v—f ^0 (2.14) £ i _ - ^ | - f = 0 (2.15) dt oz pfi oz + a n * . - 5 . ^ = 0 (2.i6) ai oz2 oz pp, oz3 The result is dependent on time t and height z only. This implies that the solutions for the scalars are valid at any radius. The fields at specific radii can be evaluated with (2.12). Knowing that the departures from a linear increase in radius between adjacent shells are small, this infinite layer of fluid model can therefore be used to evaluate the coupling of each individual cylindrical shell to the solid boundaries. We apply the same poloidal-toroidal representation for the fields in the conducting region of the solid plates (either the entire plate or a thin layer of finite conductivity). However, we don't need the momentum equation as the toroidal velocity of the rigid plates is prescribed. The absence of deformation in the solid reduces the induction equation to diffusion equations for the poloidal and toroidal components: Ttp--i'&p-=° <2'17> = ° - ( 2 - i 8 ) where the subscript s denotes a quantity in the solid. This system is valid for both the mantle and the inner core; the only difference between them is the value of the diffusivity Chapter 2: Formulation of the problem 20 2.2 Form of the solution of the equations Since we are imposing a periodic time dependence on the motion of the plates and, hence, the fluid perturbations, the set of equations for the fluid and the solid becomes a set of O D E ' s in z wi th fixed coefficient. The fluid variables accept solutions of the form Y = Y0eiXz-iut, (2.19) where Y is a generic variable and Y0 its associated amplitude. A solution of this k ind wi l l generate oscillatory solutions in both time and distance. The A's are the eigenvalues of the system which, as we wil l see, have both real and imaginary parts. Thus the eigenvalues describe the wavelength of the spatial oscillations and their attenuation or growth. The solution in the fluid is obtained by solving the four coupled equations in (2.13) to (2.16). Substituting the known time and spatial exponential dependency of (2.19) for every variable in these equations yields a set of algebraic conditions: -icvP0 - i\BjQ + A V „ = 0 (2.20) -iwf0 - i\Bzu0 + \2r,f0 = 0 (2.21) B - , -ium0 - U20V>o - i\—T0 = 0 (2.22) pp -iu\2i>0 - i\2Qu0 - a 3 ^ P Q = 0 , (2.23) Chapter 2: Formulation of the problem 21 where the common time and spatial dependence has been eliminated. The above system can be written in matrix form as: 7 7 A 2 — iuj 0 0 r/X2 — iu 0 -iX 3 ^ PI* 0 -i\Bz 0 -IX2Q, -iu>X2 0 Po -i\Bz T0 —iw -iX2Q U) 0 = 0. (2.24) The algebraic conditions in (2.24) admit a t r ivial solution for the amplitudes of the perturbation, but this is of no interest. Non-zero solutions can be obtained only i f the matr ix is singular, which requires its determinant to be zero. Hence, the system can be treated as an eigenvalue problem, where the A's are the eigenvalues and the vector of perturbation amplitudes are the eigenvectors. The final solution wi l l then be the sum of all the eigensolutions, which we describe as the normal modes of the system. We note that (2.24) is not a conventional eigenvalue problem, since the matrix is not of the standard form A — XL For our system, the eigenvalues are not l imited to the d iagona land bear higher powers in some occasions. Equating the determinant to zero gives a cubic equation in A 2 : A 7 / 2 (2Q) 2 + -ir)u> + B 2 \ 2N + A 4 (-2^-oj2 - i2ur)(2n)2 + i2o>3?7 + A 2 (-u> 2(2Q) 2+u> 4) = 0 , (2.25) from which the A's (the eigenvalues) can be found. It is interesting to note that this system is of order 6. Adding the highest power of every equation suggests that the order of the system should be 8, but the coupling between these equations reduces the order by 2. It is obvious that two of the eigenvalues wil l be zero. The remaining eigenvalues can be found by numerically solving a quadratic equation in A 2 . Chapter 2: Formulation of the problem 22 The eigenvalues include both real and imaginary parts, which indicates that the so-lutions are oscillatory and dissipative. The magnetic diffusivity is the sole source of dis-sipation in the system and when it vanishes, the fluid becomes a perfect conductor and the waves do not suffer any ohmic dissipation. Setting 77 = 0 in the dispersion relation in (2.25) removes all the imaginary terms and reduces the dispersion relation to A 6 (j^j + A 4 ( - 2 ^ 2 ) + A 2 ( -u , 2 (2f t ) 2 .+V) = 0 , (2.26) which gives purely real solutions for A. According to the spatial dependence assumed in (2.19) we obtain oscillatory solutions which are undamped. This last equation is consistent wi th the dispersion relation for ideal M C waves [Gubbins and Roberts 1987], so called because of the basic pressure-Magnetic-Coriolis force balance in effect. We note that these waves can be viewed as Alfven waves (Magnetic-pressure force balance) modified by the rotation of the Ear th . When dissipative effects are included the nature of the eigenvalues wi l l include 2 grow-ing and 2 decaying waves. The growing solutions present a problem because the dissipative effects should attenuate the disturbances, not intensify them. But i f we track the solution in z from one boundary towards the second boundary, we can recognize that what appears to be a growing solution starting from one boundary can also be interpreted as a decaying solution at the second boundary. The final solution may be expressed as a sum of all the eigensolutions or modes of the system: - , 6 Y = YjCnYneiXn{z-z°)-iu\ (2.27) n = l where zQ is the position of one of the plates. Yn represents the eigenvector associated wi th the eigenvalue A n and Cn is an integration constant. The structure of the eigenvector for Chapter 2: Formulation of the problem 23 the A = 0 mode for the single plate case and for the two-plate geometry wi l l be developed in chapter 3 and chapter 4, respectively. This mode wi l l generate spatial solutions that are either constant or linearly dependent on z. The remaining modes wi l l be characterized by attenuated oscillations in z. Solutions of the diffusion equations in the solid (2.17 and 2.18) are also of the form: Y. = Y„. ^ ' " ^ > (2-28) but unlike the fluid equations, the equations in the solid are uncoupled and we can directly solve for the common eigenvalues Xs for both the poloidal and toroidal perturbations: A, = ± ( i + 0 2 ^ (2.29) The inverse of \ s gives the wavelength of the oscillation in the solid. This wavelength is also known as the magnetic skin depth, which represents the typical propagation length scale for the magnetic perturbation. The solution in the solid is the sum of a growing oscillation and a decaying one: ' p / J,X$(z-z0)-iut _|_ f: J.\a (z-z0)-iwt (2.30) Since the fluid-solid interface is located at z = z0, the 4 constants ( P s + , P~, T+ and r~) are in essence the amplitude of the magnetic field scalars at the boundary. B o t h of these independent solutions must be kept when we allow for a thin conductive layer in the plate, whereas when we assume the conductive region to extend to infinity, the growing solution must be eliminated to have a finite solution at ± o o . The integration constants Cn in (2.27) are then the only remaining unknowns in the fluid, while the amplitude constants in (2.30) are the only unknowns in the solid. A l l these Chapter 2: Formulation of the problem 24 constants are constrained by the boundary conditions, which are examined in the next section. These boundary conditions link the solution in the solid plates to the solution in the fluid. 2.3 Boundary conditions Solutions for the fluid and solid systems can be found in terms of a number of arbitrary constants. To specify these constants, we have to impose boundary conditions at the fluid-solid interface. The component of the magnetic field perpendicular to the boundary, the axial magnetic field, must be continuous across the interface. The solenoidal condition on the electric current also implies continuity on the axial component of the current across the same interface. In the magnetohydrodynamics ( M H D ) approximations, the velocities are much smaller than the speed of light. It follows [Gubbins and Roberts 1987] that Ampere's law can be expressed as V x B = fiJ. Hence, the continuity of the axial current implies the continuity of the toroidal magnetic field. Therefore, P and T are both required to be continuous across the fluid-solid interface. In addition, we require continuity of the tangential electric field, but an equivalent condition on the magnetic field may be inferred from the induction equation. However, this derivation is deferred unti l after we have dealt with the conditions on the fluid velocity. Since there is no flow across the boundary, the vertical component of the fluid velocity vanishes at the fluid-solid interface. Viscous stresses at the boundary cause the fluid to follow the motion of the plate. This condition is called a no-slip condition. Hence, the toroidal velocity in the fluid at the boundary has to equal the rotation velocity of the plate, and the radial flow must vanish at the boundary. Chapter 2: Formulation of the problem 25 The solutions of the velocity in the fluid and the magnetic field perturbations in both the fluid and the solid must satisfy these boundary conditions. Viscous and magnetic diffusion are important near the boundary, but both of these effects are less important in the interior of the fluid. The adjustment of the fluid velocity to the no-slip condition forms a viscous boundary layer in the fluid, while the adjustment of the ambient magnetic field produces a magnetic boundary layer in both the fluid and the solid. For typical estimates of the viscosity and conductivity of the Earth's core, the thickness of the viscous layer wi l l be several orders of magnitude smaller than the magnetic skin depth, which is the boundary layer originating from the magnetic interaction. This can be seen through a simple analysis of the equations. In the induction equation, the basic balance in the vicinity of a boundary gives a diffusion equation for each component of the magnetic field. We can infer the electromagnetic length scale, or magnetic skin depth, from that balance: sjlnjui. The basic balance in the momentum equation near the boundary is between the Coriolis force and the viscous force. From this force balance, the typical thickness of the viscous layer is given by ^Ju/2Q. The kinematic viscosity is poorly known, but it is believed to be roughly 1 0 - 6 m 2 s _ 1 [Gubbins and Roberts 1987]. For phenomenon occurring during a time scale on the order of days ( f t - 1 ) , the viscous layer wi l l then be at least 3 orders of magnitude thinner than the magnetic diffusion layer. A t the time scale of the oscillation of the mantle and the inner core ( L O D fluctuations), 10 4 - ft-1, the magnetic diffusion becomes thicker and the difference from the viscous layer becomes even bigger (at least 5 orders of magnitude). The viscous layer is so thin compared to the magnetic diffusion layer that its influence on the dynamics of the fluid can be approximated with a velocity discontinuity. This is the equivalent of imposing a viscous stress-free boundary condition. The axial velocity must still vanish at the fluid-solid interface, but the horizontal velocity can be discontinuous Chapter 2: Formulation of the problem 26 across the boundary. This type of boundary condition is referred to as a free-slip condition. The presence of the viscous layer is encompassed by the velocity discontinuity at the boundary. Diffusion of the magnetic field rapidly smooths vertical variations over length scales which are comparable to the thickness of the thin viscous layer. Hence, the axial and toroidal magnetic field can be approximated as continuous through the viscous layer. The discontinuity in the tangential velocity does not affect the continuity conditions on P and The continuity of the magnetic field scalars has to be preserved in time. Hence, the time-evolution of these scalars should also be continuous. The evolution in time of both magnetic field scalars is given by the induction equations (2.13) and (2.14). The integration of these two equations across the fluid-solid interface, and thus across the velocity discon-tinuity, give two more boundary conditions, which relate the jump in the velocity to the vertical gradient of the magnetic field. B y Faraday's law, V x E = — these conditions can also be interpreted as the requirement of continuity of the tangential electric field. Altogether, we have a total of 5 boundary conditions which are listed below: the absence of flow through the boundary; the continuity of the poloidal and toroidal magnetic field; and the jump in the gradient of the magnetic field in relation with the jump i n velocity, which are obtained from the poloidal and toroidal part of the induction equation: T. 1> = 0 (2.31) P = Ps (2.32) T = TS (2.33) Chapter 2: Formulation < In the fifth boundary condition, the discontinuity in the toroidal velocity across the boundary introduces the velocity of the plate, u3. This plate velocity is a parameter of the system and may be viewed as the forcing term. The amplitude of the fluid perturbation wi l l be directly proportional to its value. The coupling between the plates and the fluid is due solely to magnetic forces. If the plates were insulators, there would be no electric currents in the solids. This means there can be no force on the solids and hence, no reaction force on the fluid. The plates and the fluid would then be totally decoupled. Another way to see this is based on the idea of frozen flux when the conductivity is high. A good conductor wi l l carry the magnetic field. Conversely, the magnetic field wi l l slip easily through a poor conductor. When the conductivity of the plate drops to zero, the Bz field is undisturbed by the motion of the plate. Since we have neglected the effects of viscosity, the fluid wi l l not be perturbed. The model of the conductive plates that extend to infinity requires both Ps and T3 to vanish at distances far from the fluid-solid boundary; the perturbation of the magnetic field is caused by the motion of the fluid relative to the solid and this magnetic perturbation diffuses away with distance. Some studies [Glatzmaier and Roberts 1995] assume the mantle to be an insulator but for a thin conductive layer at the C M B . To model this case, we have to split the solid into two distinct regions; a conductive layer adjacent to the fluid and an insulator that extends to infinity. The boundary conditions developed above for the fluid-solid interface are still valid at the fluid-conductive layer interface. In addition, the equations developed for the conductive solid (2.30) also apply in the conductive layer, but the continuity of the magnetic field requires the solution to match the scalar potentials of the field in the insulator at the conductive layer-insulator interface. A l l electrical currents r the problem d - d -•" V-Ts~rj-T oz oz = 0 27 (2.35) Chapter 2: Formulation of the problem 28 must vanish in the insulator, which in our notation translates into the conditions: Tj = 0 and -Q^- = 0. The scalar PT could be represented by a linear term in z plus a constant, but the requirement that the solution remains finite at infinity requires the perturbation of the axial magnetic field in the insulator to be simply Pj = C, where C is a constant. This spatially constant field wi l l oscillate in time with the imposed periodicity. The continuity of the magnetic field at the boundary between the conductive layer and the insulator imposes three conditions: Ps = P r , ^ = ^ t and T5 = TT. Knowing the solution in the insulator, these conditions can also be written: P . = C, ^ p 1 = 0 and T„ = 0. These two models for 3 f (jz S the solid are interchangeable in the calculations and the choice of one over the other can affect the solution in the fluid. Chapter 3 Single-plate problem 29 3.1 Introduction The first case we wil l study is the response of a conducting fluid above a single oscillating plate. We remove the top plate from the original configuration (figure 2.1) and allow the fluid to extend to infinity in z. This simple case is useful for developing insights into the electromagnetic coupling between a fluid and a moving boundary. As we have already hinted by physical arguments, and by the nature of the solution, the perturbation in the fluid is periodic in time and in space. The attenuation of the waves with distance from the boundary defines a region called a boundary layer, where the flow adjusts to a boundary perturbation. The presence of boundary layers in fluid reflects the enhancement of diffusion near the boundaries. It is useful for future reference to briefly summarize the different types of boundary layers that can develop. First , the Ekman layer is a result of a viscous friction between the fluid and the boundary when the whole system is in rotation (the Coriolis force is important in the force balance) [Pedlovsky 1987]. This boundary layer has been extensively studied and used in oceanography. If the fluid and the boundary are subject to electromagnetic coupling in addition to the viscous coupling, then the fluid velocity wi l l be influenced by an additional adjustment of the magnetic field. The imposed magnetic field is dragged through the fluid by perturbations in the motion of the plate. The magnetic forces alter the force balance in the boundary layer, which, in the absence of rotation, is often called the Hartmann layer [Gubbins and Roberts 1987]. When the system is in rotation, this layer can be regarded as an Ekman layer with the influence of an imposed Chapter 3: Single-plate problem 30 magnetic field or a Hartmann layer to which we have added the effect of rotation. This kind of layer is often called an Ekman-Hartmann boundary layer [Gubbins and Roberts 1987]. Loper [1970] studied the hydromagnetic boundary layer induced by a rotating conduct-ing plate. His calculation shows the presence of an Ekman-Hartmann layer in the fluid adjacent to the solid boundary as the fluid is spun up. He also indicates that when the conductivity of the plate is of the same order as that of the fluid, the magnetic diffusion region extends much further away from the solid boundary than the viscous diffusion re-gion, hence, beyond the Ekman-Hartmann layer. This suggests that the magnetic stress at the boundary is much larger than the viscous stress. We can thus impose a viscous stress-free condition at the boundary and neglect the presence of the Ekman-Hartmann layer by making suitable adjustments to the boundary conditions, as we argued in the pre-vious chapter. The coupling between the fluid and the solid is then solely accomplished by magnetic interaction and the ensuing boundary layer region in the fluid represents strictly a magnetic diffusion region. 3.2 Solution of the problem We set z = 0 as the position of the boundary between the plate and the fluid. Positive values of z correspond to the fluid and negative values are in the solid. The solution in the fluid is given by (2.27). Without the second plate bounding the fluid vertically, the solutions that grow endlessly towards infinity are excluded. Hence, the two eigensolutions with eigenvalues Im(\) < 0 are eliminated from the final solution. The expansion of (2.27) Chapter 3: Single-plate problem is then the sum of three normal modes: 31 • p Pi f e~iwt + C2 u u2 AXr.z—iuit e 2 V>3 i\~z—iwt e 3 (3.1) where C1,C2 and C 3 are the unknown integration constants. The modes wi th the non-zero eigenvalues wi l l give rise to oscillatory solutions that decay with z. The physical significance of the mode A = 0 can be inferred by looking at the structure of the eigenvector. Substituting A = 0 in (2.24) gives -iu> which yields: 0 0 0 0 0 0 A 0 0 +i 1 « i 0 0 Pi 0 — IU! 0 (3.2) (3.3) for the first eigenvector in (3.1). From the representation of the velocity in (2.12), we conclude that the A = 0 mode represents a constant axial velocity. This axial velocity is present through the whole geometry, even at infinity. It wi l l be shown that this axial velocity is connected to a radial transport in the vicinity of the plates. Because the fluid is incompressible, a sink of fluid calls for a source elsewhere in the cylinder (or vice-versa) Chapter 3: Single-plate problem 32 to conserve fluid mass. The eigensolution with A = 0 is necessary to balance the radial fluid transport in the boundary layer. This phenomenon is often referred to as pumping. The amplitude of this pumping wil l follow the oscillating behavior imposed by the periodic motion of the plate. It can be directed towards or away from the plate, depending on the direction of the rotation of the plate. We also need the solution in the solid to impose boundary conditions on the magnetic field and solve for the integration constants Cn in (3.1). If we assume a uniform conducting infinite plate, then we have to eliminate the eigenvalue \ 3 wi th i m ( A 3 ) > 0 in (2.29) so that solution in the solid decays towards z = —oo. The solution in the solid (2.30) is then: -P. where A, = - d + i ) ^ . (3.5) We have a total of 5 unknowns in the solutions (3.1) and (3.4): there are 3 unknown integration constants (C1, C2, C3) in the fluid; and the amplitude P~ and T~ of the perturbed magnetic field in the solid at the boundary. The boundary conditions listed in (2.31) to (2.35) provide 5 equations that link these 5 unknowns together and uniquely determine the whole system. The boundary conditions can be represented in matr ix form J.\,z—iut (3.4) Chapter 3: Single-plate problem 33 as V>2 0 0 n Pi P* -1 0 0 fl 0 -1 0 v\Pi 0 A" 0 -7i - 7 2 - 7 s 0 where 7n = z?7AnTn + 752'un. Presented in this form, it is clear that the prescribed oscillation of the plate (us) is driving the whole system. Once (3.6) is solved for the 5 unknown constants, we have the complete description for fluid and solid variables at all heights. To get the time dependency, we have to multiply the final solution with the periodic time dependence: e~lu>t. Several comments can be made on the boundary conditions. We have explicitly elim-inated the exponentially growing eigensolution in the fluid. We could have accomplished the same task by requiring the solution to be defined at infinity. These regularity condi-tions at infinity would require an undisturbed magnetic field. Hence, P = 0 and T — 0 would then be two of these conditions. Since the order of the full system is 6 and the solution of the single-plate case without the exponentially growing solution is of order 3, a th i rd condition is required. The additional condition is imposed on the velocity at infinity. If the magnetic field is undisturbed, then the main field Bzz cannot be disturbed by any velocity. Consequently, the radial velocity must vanish: |j = 0 at z = oo, although an axial velocity, parallel to Bzz, is stil l permitted. This last condition could also be written as u = 0, using the relation in equation (2.22) in the absence of a toroidal magnetic per-turbation T. These three conditions force the amplitude factor of the two exponentially growing solutions to equal zero. Chapter 3: Single-plate problem 34 A complication is introduced if one adopts a conductive layer in the solid instead of a uniform conductivity through the infinite plate. The number of boundary conditions would increase by 3. We have already developed these three conditions at z = — A , where A is the thickness of the conductive layer: Tt = 0, -gf = 0 and Pa equal to a constant value (say K) in the insulator. The three additional unknowns would have been P + , T s + and K. The dimension of the matrix in (3.6) would then be 8 x 8. 3.3 Results and discussion The amplitude of the solution in the fluid is determined by the amplitude of the plate motion, while the form of the solution depends on the parameter values that appear in the matr ix in (3.6). Since we are more concerned with the qualitative aspect of the solution than the quantitative description, the amplitude of the plate motion is somewhat arbitrary. For simplicity, we have imposed an angular oscillation with a 1 degree ( = 7r/180 radians) amplitude. The motion of the plate is: d<[> 7T _ . t u=---= OJ e (3.7) 8 dt 180 V ; and the maximum velocity of the plate occurs at t = 0. For an ini t ial example of the solution, it is convenient to use a set of parameters which are not likely to break the Proudman-Taylor constraint. The physical processes respon-sible for the coupling are more easily identified in this regime. We can then modify the parameters to alter the relative importance of these processes and see how the flow departs from rigid rotations. This ini t ial solution wil l be calculated with Bz = 5 C7, comparable to the radial field at the top of the core as inferred from a continuation of the magnetic field at the surface of the Ear th [Langel and Estes 1985] (The same value was used by Chapter 3: Single-plate problem 35 Braginsky [1970] in the approximations that lead to the TO). The period of oscillation has been chosen to correspond to the period of LOD fluctuations: 10 000 days or about 30 years. The conductivity in the solid is an important factor in determining the strength of the coupling and can be chosen to represent either the mantle or the inner core. Since our concerns originate with the validity of the rigid rotations inside the tangent cylinder because of the strong field at the ICB and the large conductivity of the inner core, we assume that the plate represents the inner core. The conductivity in the solid is taken to be equal to that of the fluid, as both are nearly pure iron and any small difference between them will not alter the solution very much. The parameters used in this calculation are listed in table 3.1. p 1.1 x 104 kgrn'3 4-7T x. 1 0 - 7 kg m A~2s~2 ZA x 10s Sm'1 3.4 x l 0 B 5 m " 1 7.27 x l O - 5 ^ - 1 BZ 5G u> O/10 4 = 7.27 x I O " 9 * - 1 WTT/180 = 1.26 x I O - 1 0 * " 1 Table 3.1: Parameters used in calculation of the electromagnetic coupling between the fluid and a single plate. 3.3.1 Physical description of the solution In figure 3.2 we present the solution for each variable. A l l these plots show the real part of the solution at t = 0 (or at any integer multiple of t = 2TT/U>). Throughout the analysis, we have to keep in mind that the solutions we are showing are always snap shots in time. If we evolve these solutions, then the disturbance will propagate away from the boundary. The most relevant plot for applications of LOD variations is u/us; the toroidal velocity Chapter 3: Single-plate problem 36 reflects the angular momentum distribution in the fluid, which is the quantity we want to investigate to deepen our understanding of momentum transfer in the different regions of the Earth's interior. The amplitude of u varies periodically with z, as the disturbance due to past plate movements propagates away from the boundary. The force responsible for the coupling between the fluid and the plate is a magnetic stress at the fluid-solid interface. The propagation of the disturbance away from its source at the boundary is l imited to a finite region by diffusive losses. This boundary layer region in the fluid can also be referred to as the magnetic diffusion region, to follow the terminology used by Loper [1970]. The same study labels the unaffected fluid above this magnetic diffusion region as the "current free region". We identify this regime of solution as the diffusive regime. The decaying-oscillating behavior in this boundary layer is attributable to the second and third mode of the solution in (3.1). The oscillation is characteristic to a wave propagating away from the boundary, an M C wave. This type of wave, also known as "slow" wave, represents an Alfven wave strongly modified by the rotation of the Earth . A l l 4 scalars in the fluid have similar curves, which is expected given the form of the solution in (3.1). A boundary layer is also formed in the solid but it involves only the magnetic field scalars. The amplitude of the magnetic field scalars in both the fluid and the solid are of the same magnitude, as required by the boundary conditions. They also both follow a solution governed by diffusion; in the solid, because the magnetic field is represented by a diffusion equation; and in the fluid, because we are in the diffusive regime. The first mode in (3.1), the constant vertical velocity, only affects the solution of ip. We can see that this scalar does not decay to zero with distance but approaches a constant velocity. Since the eigenvalue of this first mode is zero there is no oscillating or decaying Chapter 3: Single-plate problem 37 behavior. Instead, the velocity is maintained throughout the whole space in z. The behavior of ip can also be related to the solution of u wi th simple physical ar-guments. In the boundary layer, the oscillation of ip represents vertical fluid movement. As equation (2.12) indicates, the derivative of ip wi th respect to z represents the radial velocity, so we can deduce from the spatial variation in ip that the radial flow wi l l also vary with z in the boundary layer. The fluid is incompressible, so to conserve mass, the divergence of the velocity must equal zero. In other words, there can be no sources or sinks of fluid. This is why vertical pumping must accompany radial flow; i f we "isolate" a section of fluid, say a disc, where fluid is being transported radially inward, this fluid must go somewhere and it is evacuated from the disc in the vertical direction. Radial motion also affects the toroidal displacement of the fluid by conservation of angular momentum. For example, when the radial pumping brings in some fluid from a larger radius, then conservation of angular momentum causes these parcels of fluid to increase their angular velocity. In detail, the angular momentum conservation is slightly more complicated, be-cause of magnetic forces which can transfer angular momentum (see equation (2.22)) . The effects on the toroidal velocity by the radial pumping are also felt a quarter of a period out of phase. Nevertheless, we can still picture how ip has to be consistent wi th the oscillating behavior of u through angular momentum conservation and fluid circulation. We now have the physical insight needed to explain the first mode. The fluid adjust-ment to the electromagnetic coupling with the moving plate produces a net radial transport of fluid in (or out through) the boundary layer. This is compensated by a vertical pumping of equivalent mass out of (or into) the layer. But there can be no vertical transport at the boundary with the solid; the totality of the vertical pumping has to occur at the top of the boundary layer, where it smoothly blends with the bulk of the fluid (the current free region). Outside the layer the vertical pumping stays constant and extends to infinity. Chapter 3: Single-plate problem 38 This constant vertical pumping represents the first mode and is required to balance the radial fluid transport in the boundary layer. We recall that the solutions presented are snap shots in time. This pumping oscillates with time and half a period later, it wi l l be completely reversed as the radial pumping in the boundary layer wi l l also be reversed. The circulation of the conducting fluid in the boundary layer distorts the original magnetic field configuration and thus creates secondary fields, which can be seen on the plots of P and T. The toroidal velocity u wi l l stretch the Bz field in that same direction, setting up a magnetic field that wi l l produce a force that opposes the deformation. This secondary magnetic field is toroidal (T) and consistently with the oscillating u, it presents an oscillating-decaying behavior. It is also common to represent this secondary field in terms of electric current, which mathematically is expressed as the curl of the magnetic field. Associated with T is a vertical current aligned with Bz (figure 3.1 A ) . This current points upwards (downwards) for positive (negative) T, and follows a similar oscillating-decaying behavior. The variation of T along z demands the presence of a radial current (|jj, figure 3.1 B) which has a similar oscillating behavior in z. In the same way that the vertical and radial velocities are connected with one another to satisfy the fluid mass conservation, the vertical and radial currents are also related. The total charge has to be conserved for a given volume, thus the divergence of the current for that volume must equal zero. Inward radial currents in a "disc" volume wil l be compensated by outward vertical currents and vice versa. The magnetic force results from the cross product between the induced current and the magnetic field, which is approximated by the background field Bz in the linearized case. The radial current (figure 3.1 B) creates the expected toroidal force but the vertical current (figure 3.1 A ) is parallel to Bz, nullifying the cross product. Hence, only the radial Chapter 3: Single-plate problem 39 component of this poloidal current wi l l contribute to the induced magnetic force. The presence of P can be understood in a similar way. The radial motion in the boundary layer deforms the Bz field which sets up a poloidal secondary magnetic field that induces a counteracting radial force. The curve of P follows the behavior imposed by ip. Again , we can look at it also from the point of view of secondary electric currents. The induced poloidal magnetic field is associated with a toroidal current (-3-3-) figure 3.1 C ) . This toroidal current satisfies charge conservation. The cross product of this current with Bz wi l l indeed give a force in the radial direction which opposes the deformation of the main field by the radial flow. c Figure 3.1: The secondary magnetic field is grey shaded and the associated electric current is black. A : The toroidal magnetic field (T) is associated with an axial electric current. B : The vertical gradient of the toroidal field is associated with a radial current. C : The vertical gradient of the radial field ( § j r ) is associated with a toroidal current. In summary, fluid motions in the boundary layer induce currents which in turn alter fluid motions. The toroidal motion u sets up a toroidal magnetic field T and consequently a poloidal electrical current to counteract this toroidal motion. The poloidal motion ip generates a poloidal magnetic field P and thus a toroidal current that produces a radial force to counteract this poloidal motion. These motions are connected through the conservation Chapter 3: Single-plate problem 40 of angular momentum. The vertical current generates no force and no current is able to generate a vertical force. However a vertical flow develops in order to conserve fluid mass, and similarly a vertical current is needed to conserve charge. 3.3.2 Coupling mechanism Now that the connection between the different scalar potentials has been established by physical arguments, it is easier to describe the coupling mechanism between the solid and the fluid. We confine our attention to the transfer of angular momentum between the solid and the fluid by electromagnetic interactions. The shear in the toroidal velocity near the plate deforms the background axial magnetic field, creating a secondary toroidal magnetic field (T). This field wi l l induce an axial electric current, which does not vanish at the boundary but permeates the interface. This implies an exchange of charge between the fluid and the solid. Since the current vanishes away from the boundary, an axial current through the boundary implies a net radial current in the boundary layer in the fluid and an equal but opposite net radial current in the diffusion boundary layer of the solid in order to conserve charge. The net radial current in the fluid interacts with Bz to produce a toroidal force that accelerates the fluid. Angular momentum is then transferred from the solid to the fluid by the exchange of charges between them. Without viscous effects, electromagnetic in-teractions between the fluid and the solid are solely responsible for the coupling. If the axial current vanishes, there can be no net radial current in the fluid and the toroidal Lorentz force also vanishes. This result is in agreement with the study of Loper [1970] for the coupling of a rotating, electrically conducting plate, where the current permeating the boundary is identified as a Hartmann current. Chapter 3: Single-plate problem 41 The force due to the current across the fluid-solid interface is often described in terms of the magnetic stress imposed on the fluid by the moving boundary. The magnetic stress is proportional to the product of the axial magnetic field (Bz) and the toroidal deformation of that field (T). Hence, the magnetic stress is proportional to the exchange of charge between the fluid and the solid, or equivalently, proportional to the amplitude of the disturbance of T in the boundary layer. The radial transport of fluid in the boundary layer is a consequence of the combination of the Lorentz force and the Coriolis force. The toroidal acceleration induced by the Lorentz force in the boundary layer is subject to the Coriolis force, which tends to deflect the motion in the radial direction. 3.3.3 Variations of parameters and regime of solutions In the diffusion regime, the magnetic forces are small. The Elsasser number gives the ratio between the magnetic force and the Coriolis force in the fluid, A = %• ^ W i t h a weak magnetic field (Bz = 5G), this dimensionless number is small. The force balance in the fluid is primarily between the Coriolis force and pressure gradients. The source term for the magnetic field perturbation, according to the induction equation, is the strain of Bz by the velocity, but this source is negligible compared wi th the diffusion. Similarities between the purely diffusive solution in the solid and the solution in the fluid indicates that the fluid motions have little influence on the magnetic field. Al ter ing the frequency u> wil l alter the inertia of the fluid. Although, when the fre-quency is small compared with the rotation rate, the force balance remains unchanged. Chapter 3: Single-plate problem 42 The magnetic Reynolds number (ratio of advection over diffusion of the magnetic field) wi l l also be unaffected. Hence, the solution remains in a diffusive regime and the shape of the curves for every variable wi l l be identical. In figure 3.3 and 3.4 we present results obtained respectively with m = O/10 2 and ui = fi/106. The curves differ only in the scale of the disturbance in z. The magnetic skin depth (y2ri/tjj) depends on the frequency, and slower plate movement wil l permit a deeper propagation of the perturbation in both the fluid and the solid. This can also be understood through simple physical reasoning: if the plate oscillates at a longer period, diffusion has time to penetrate deeper. Conversely, for a shorter period the boundary layer thickness wi l l decrease. Changes in inertia can be seen by looking at the amplitude of the velocity scalars. The absolute amplitude of the toroidal velocity in the fluid wi l l be different, but its value relative to the plate motion remains unchanged. The strength of the magnetic coupling that allows the plate motion to drag the fluid in the toroidal direction depends on the strength of Bz and not on the frequency. The absolute amplitude of ip varies with the changes in the absolute amplitude of u. For a higher (lower) frequency, the pumping has to increase (decrease) to accommodate the increase (decrease) in the toroidal velocity. However, the change in amplitude of ip is not directly proportional to the variation in u>. Conservation of angular momentum, which relates ip to u, does involve u>. Consequently, the frequency wi l l alter the ratio between the toroidal velocity and the radial velocity. The changes in inertia wi l l also affect the secondary magnetic field. For a higher fre-quency, the induced magnetic force that opposes the toroidal motion has to be stronger. The absolute amplitude of the secondary toroidal field T required to produce this force has to increase. The increase in the radial velocity associated with a faster plate motion necessitates a stronger radial magnetic force. However, the absolute amplitude of the sec-ondary poloidal field P stays constant for variations in frequency. A n increase in frequency Chapter 3: Single-plate problem 43 narrows the boundary layer and the increase in the radial magnetic force is accomplished solely through this change in vertical length scale. A lower frequency has exactly the opposite effect on both the toroidal and poloidal secondary magnetic field. For phenomenon with periods similar to the rotation of the Ear th , inertial effects wi l l become much more important. When the force balance is between the Coriolis force and the inertia, inertial waves wi l l be generated in the system. Our general model can still yield the accurate solution for this case, but the physics at these fast frequencies is of less interest to us as we are mainly concerned with phenomenon that occurs wi th small temporal Rossby number. The relative importance of the Lorentz force versus the Coriolis force wil l be changed if Bz is altered. For Bz smaller than 5 G , the momentum equation continues to balance Coriolis force and pressure gradients, while the magnetic solution wi l l stil l be in the diffusion regime (figure 3.5 with Bz = 0.5 G). The coupling between the plate and the fluid wi l l be less efficient and the amplitude of each variable wi l l decrease. But the shape of the curves wi l l not be altered because the magnetic skin depth which defines the characteristic length of the disturbance remains unchanged. When Bz is stronger, the force balance is altered because the Lorentz force becomes more important. The characteristics of the solution change when the Lorentz forces are comparable to the Coriolis force. In other words, when the Elsasser number (3.8) is roughly equal to 1. This happens when the magnetic field is R3 20 G. The presence of a restoring force in the momentum balance alters the fluid motions, which induce a magnetic field perturbation and overcome diffusive losses. The time scale for the attenuation of the disturbance is now several periods of oscillation of the plate (figure 3.6 wi th Bz = 50 G). Thus, an M C wave can be maintained for a longer vertical length scale. The solution is Chapter 3: Single-plate problem 44 therefore in a wave regime. The velocity of these waves wi l l increase with a stronger Bz\ the perturbation propagates deeper in the fluid because the waves are moving faster and because the wavelength is longer (so diffusive losses are reduced). The magnetic field scalars in the solid continue to obey a diffusion equation. The amplitude of Ps and Ts, however, wi l l increase in step with the changes in the amplitude of P and T. Substituting the model of the uniformly conducting infinite plate with the model of an insulating infinite plate under a conductive layer wi l l not alter the solution i f the thickness of that layer is larger than the magnetic skin depth in the solid. However, i f the thickness of the conductive layer is smaller than the skin depth, the coupling is reduced. The effect is similar to a simple decrease in the conductivity of an infinite conducting plate. B y reducing the thickness of the conducting layer, the total conductance of the solid is reduced. It is relevant to emphasize that the transition between the diffusion regime and the wave regime is only a function of Bz, as long as the temporal Rossby number is small. When Cl, p and <7y are given, the Elsasser number (3.8) depends only on Bz. The value of Bz at which the Elsasser number roughly equals unity defines the point where the Lorentz forces wi l l be comparable to Coriolis force in the fluid. In the diffusive regime, the amplitude of u is always a few orders of magnitude smaller than the movement of the plate. This suggests that the coupling is rather weak, even at the I C B , when the field is weak. Increasing the conductivity of the plate strengthens the coupling only to a certain l imit , which is set by the conductivity of the fluid. This does not invalidate the idea that the inner core and the fluid column above and below it are perfectly coupled together in the T O theory. We have to remember that the perfect coupling is achieved by a large magnetic field. The weak field solution is more representative Chapter 3: Single-plate problem 45 of what is happening at the C M B . In the wave regime, the amplitude of the fluid's toroidal motion can be of the same order than the amplitude of the plate. A n d the oscillating nature of the flow introduces large vertical gradients in these toroidal motions. This is inconsistent with the idea of rigid toroidal motions. The potential problems with the T O theory can thus be expected when we are in a wave regime. Since the imposed magnetic field is the parameter that defines the regime of the solution, it supports the idea that the rigid rotations do not describe the flow accurately for strong magnetic fields, because the Proudman-Taylor constraint is broken. What is also clear from this study is that the transfer of angular momentum to the fluid from a single plate is l imited to a boundary layer. Hence, even i f the rigid rotations are in theory valid for weak magnetic field (in the diffusive regime), we have not yet reproduced such motion from the electromagnetic coupling between the fluid and the solid because the fluid column is assumed to be infinite. In this single-plate coupling, fluid can escape at infinity, which differs from the real geometry. The study must then be extended to include the coupling with a second plate that bounds the fluid. The two plate model is developed in the next chapter. Chapter 3: Single-plate problem 46 U / U< 0 2 (m/s) x 10"8 200 (G) x 10" 5 0 5 (G/m) x 1 o"7 Figure 3.2: Solution for the scalar potentials in the fluid and the solid, obtained wi th the set of parameters listed in table 3.1. The vertical axis of every graph represents the vertical distance z. The position of the fluid-solid interface is at z = 0. The solution presented is at t = 0. Chapter 3: Single-plate problem 47 Figure 3.3: Solution obtained with the set of parameters listed in table 3.1, but substituting u> by ft/102. The vertical axis of every graph represents the vertical distance z. The position of the fluid-solid interface is at z = 0. The solution presented is at t = 0. Chapter 3: Single-plate problem 48 u / u< ^-1000 -20 -15 -10 -5 0 5 (G/m) xio"8 -2000 -15 -10 5 0 5 (G/m) xio'8 Figure 3.4: Solution obtained with the set of parameters listed in table 3.1, but substituting oj by fi/106. The vertical axis of every graph represents the vertical distance z. The position of the fluid-solid interface is at z = 0. The solution presented is at t = 0. Chapter 3: Single-plate problem 49 Figure 3.5: Solution obtained with the set of parameters listed in table 3.1, but substituting Bz by 0.5 G. The vertical axis of every graph represents the vertical distance z. The position of the fluid-solid interface is at z = 0. The solution presented is at t = 0. Chapter 3: Single-plate problem 50 2000 u / u< 2000 -2000 -0.01 0.03 -2000 -0.5 0 0.5 1 (G/m) x 10"5 0 5 (G/m) xio" 6 Figure 3.6: Solution obtained with the set of parameters listed in table 3.1, but substituting B by 50 G. The vertical axis of every graph represents the vertical distance z. The position of the fluid-solid interface is at z = 0. The solution presented is at t = 0. Chapter 4 Two-plate problem 51 4.1 Introduction One of the dominant features that comes out of the single-plate analysis in the previous chapter is the constant axial pumping outside the boundary layer region. The pumping extends to infinity in z and presents a clear problem for the model of the Earth's interior. The next level of improvement in the model is to introduce a second oscillating plate that wi l l bound the fluid in z. This wi l l then confine the vertical pumping to a finite region and give a more sensible model for fluid motions in the outer core. This second plate has the same characteristics as the first; the same options for the conductivity of the solid can be applied independently to each plate. A n electromagnetic torque wi l l arise on the fluid near the second plate and we can therefore anticipate the formation of a second boundary layer. Each boundary layer wi l l now be influenced by the axial pumping generated from the other layer, so we can not simply superimpose the single plate analysis solution from each plate. In fact, superimposing the two solutions would necessarily imply the existence of a flow through the fluid-solid interfaces. The solution in one boundary layer has to be consistent with the solution in the other boundary layer. Hence, the solution in this finite column of fluid must encompass this idea of communication between the boundary layers in order to yield the correct fluid circulation. 4.2 Solution of the problem W i t h the top plate reinstated, the geometry returns to the original model of a fluid layer enclosed by two infinite conducting plates, as presented in figure 2.1. We define z = 0 at Chapter 4: Two-plate problem 52 the fluid-solid boundary of the bottom plate and suppose the plates are separated by a distance h, so the fluid-solid boundary of the top plate is located at z = h. The plates are oscillating with the same prescribed frequency u>, which imposes the same periodic time dependence on the fluid solution. The amplitude of oscillation of the second plate may be freely chosen and introduces a second forcing term on the fluid motion. The motion of the fluid is still described by the coupled system of equations in (2.13) to (2.16). As before, we are looking for exponential solutions in space with a known time-dependence of the form given by (2.19). The development of the coupled equations, after substituting this known form, still yields the same eigenvalue problem presented in (2.20) to (2.23). Hence, the eigenvalues and eigenvectors of the double plate system wi l l be identical to those found for the single plate analysis. W i t h the presence of a second boundary layer, the exponentially growing solutions must now be retained in order to satisfy boundary conditions. A fourth and a fifth independent solution must be added to (3.1). We must also deal with the degenerate eigenvalue at A = 0 in order to obtain two independent eigenvectors. This wi l l add a sixth independent solution to (3.1), reinstating the six independent solution required by (2.27). Two independent solutions are required for the eigenvalue A = 0 to ensure that there is no flow across the fluid-solid boundaries. The first independent solution is a constant axial pumping between the boundary layers. In the case of a single plate, this constant pumping was a consequence of the radial transport in the boundary layer. In the double-plate geometry, the. normal vectors from each plate are opposite but the direction of the axis of rotation is the same at each boundary. If the direction of the pumping from each boundary layer is exactly opposite (e.g. when the two plates are identical, have the same amplitude of motion and oscillate in phase), the net axial pumping is then zero, which Chapter 4: Two-plate problem 53 implies the absence of a net radial transport in the boundary layers. Consequently the net toroidal acceleration of the boundary layer must be zero, which implies that there is no axial current accross the boundary and therefore no electromagnetic coupling. A constant axial pumping is, therefore, not sufficient; there has to be a second independent solution associated with A = 0 to properly describe the flow outside the boundary layers. Mathematically, a second independent solution can be constructed using the first mode of the repeated eigenvalue A = 0. If we let Yx in (3.3) be this first solution, then the second independent solution is given by ( f 6 + zYx) e** , (4.1) where Y6 is an unknown constant vector. Inserting the second independent solution in the system of equations gives us the vector 0 ^6 0 u6 (4.2) We can verify that the second independent solution was indeed zero for the single plate case, as P, T and u had to vanish at infinity. The addition of the second solution wi th constant vector Y6 introduces the possibility of a non-zero P and u outside the boundary regions as well as a linear variation with z in the vertical pumping (ip) associated wi th zY1. The full solution in the fluid is expressed as a linear combination of its 6 independent Chapter 4: Two-plate problem 54 parts as: • • p A PG Pi P2 f Tx e~iwt + C6 fi e~iut + C2 f2 e ~ i w t + zC6 i>i i>i u U~6 u2 +c* T3 giA3z—iujt + c 4 e i A 4 ( z - / i ) - i w t I Q V>5 = * A 5 ( z — h)—iuit (4.3) where the C ' s are integration constants. We recall that the pair of eigenvalues A 2 and A 4 are equal but opposite, where i\2 has a negative real part and zA 4 has a positive real part. The same can be said about the pair A 3 and A 5 . The 6 integration constants are determined with the boundary conditions, which in-troduce additional constants for the solution in the solid plates. The magnetic field in each plate is modelled by the diffusion equations (2.17) and (2.18). Each plate has its own conductivity, which determines the strength of the magnetic coupling and thus the torque imposed on the fluid layer at each boundary. The solution in each plate is given by (2.30). If we assume that both plates are uniformly conductive and extend to infinity, then we ehminate the exponentially growing solution at z = oo and z = — oo. This means neglecting the positive root of A s (2.29) for the bottom plate and the negative root of \ s for the top plate. If we denote by A and B the bottom and the top plate variables, the solution in the solid is expressed as: pA OS ei\fz-i*t a n d pB OS rjhA s rjhA OS rpB s rpB OS J\f(z-h)-iwt (4.4) Chapter 4: Two-plate problem , 55 where: With the 6 coefficients in the fluid and the two constants for the magnetic variables in each plate, we have a total of 10 unknowns: 5 more than for the single plate case. The boundary conditions that must apply at z = 0 are exactly the same than those applied in the single plate analysis, which are listed in (2.31) to (2.35). An equivalent set of boundary conditions must also be applied at z = h, which brings the number of boundary conditions to 10. The solution for all the unknowns is obtained by solving the following system: A2 CA pA _A3 A4 CB F B where the A n ' s are matrices. The matrices Ax and A4 represent the single plate coupling matrix for the bottom and the top plate. Hence, the electromagnetic coupling at one boundary without taking account the presence of the second boundary. Matrix Ay is presented in (3.6), and is repeated here for convenience, 1>1 i>2 4>3 0 0 Pi p3 - 1 0 0 - 1 v\Pi 7/A3P3 0 -ll - 7 2 -7s 0 where jn = ir)\nfn + Bzun. Matrix A4 is equivalent to Alt but is evaluated at the top Chapter 4: Two-plate problem plate and involves only the scalars that decay away from this boundary: 56 V>4 V<5 Tp6 + hfa 0 0 h p* Pe + hPi - 1 0 f 6 + hft 0 - 1 T]X1P6 - ivP1 0 - 7 4 - 7 e - 0 Matrices A2 and A3 represent the influence of the second plate on the single plate problem. They are given by V>4e4 - P 4 e 4 T 4 e 4 4 e 5 ipe + h^ 0 0 P6 + hP1 0 0 fe + h^ 0 0 7 7 A 4 P 4 e 4 v^5pse5 V^Pe - " 7 - P i 0 0 - 7 4 e 4 ~%e5 ~le -V?! 0 0 P& 1>i 4>2ei ^ 3 e 3 0 0 Pi P2et -P3e3 0 0 As = T2e2 ^ 3 e 3 0 0 v\Pi rj\2P2e2 r]\3P3e3 0 0 - 7 i -%4 -%4 0 0 where e± = e±iX"h. Vectors CA,CB, FA and FB are, respectively, the vector of integration constants and the forcing vector for the single plate problem at each boundary: CA — [Cl->C2i ^3) Posi ^ M ] cB = [c4,c5,c6,PB,f0Bs}T FA= [ 0 , 0 , 0 , 0 , -Bzuf]T FB= [0 ,0,0,0,-B s u?] T Chapter 4: Two-plate problem 57 We have seen in the analysis for a single plate that adopting a conductive layer rather than assuming a uniform conductivity in the plate adds three unknowns to the system and three boundary conditions. We can implement the conductive layer model for either or both of the plates. The required matrix system is modified accordingly. 4.3 Results and discussion W i t h the bounded fluid, we can now explicitly investigate the parameters under which the rigid rotations wi l l be broken. As in the single plate case, it is more convenient to begin the analysis with a set of parameters that reproduce rigid rotations. These solutions are obtained in a diffusive regime, which implies that the imposed magnetic field is weak. We model a fluid column inside the tangent cylinder, so the bottom and top plate wi l l take the conductivity of the inner core and the mantle. B o t h of these conductivities are poorly known, but we expect the conductivity of the mantle to be several orders of magnitude smaller than that of the highly conducting iron core. We use af = 3.4 x 10 5 S m~l and o~f = 3.4 x 10 3 S ' m - 1 in our ini t ial example solution, and assume that both plates are uniformly conducting and extend to infinity (e.g. we solve the matr ix system in equation (4.5)). The moment of inertia of the mantle is roughly a thousand times that of the inner core. For convenience, we have selected the amplitude of oscillation of the mantle to be one order of magnitude smaller than the inner core. The other parameters are the same as those used to produce the ini t ial solution for the single plate case, and are listed in table 4.1. Chapter 4: Two-plate problem 58 p 1.1 x IO4 kgm-3 /* Air x I O - 7 kgm A'2 s~2 0~ 3.4 x 10s Sm'1 °* 3.4 x 10 5 Sm'1 *? 3.4 x 10? Sm-1 7.27 x I O " 5 * " 1 h 2000 km Bz 5G ft/104 = 7.27 x l O - 9 ^ - 1 O;TT/180 = 1.26 x I O - 1 0 * - 1 IO" 1 WTT/180 = 1.26 x I O " 1 1 s-1 Table 4.1: Parameters used in calculation of the electromagnetic coupling between a cylin-der of fluid and two bounding plates. 4.3.1 Physical description of the solution In figure 4.3 and 4.4 we present, respectively, the solution of the fluid velocity and secondary magnetic field, generated with the set of parameters shown in table 4.1. Only the real part of each variable is plotted at t = 0 (or any integer multiple of t = 27T/<*;). We can immediately appreciate the presence of the predicted boundary layers adjacent to each plate in the fluid. These boundary layers are magnetic diffusion regions, as i n the single plate analysis. The wavelength of the oscillation is identical in each boundary layer. Mathematically, this is a consequence of the pairs of equal but opposite eigenvalues that govern the spatial dependence of the solution. It can also be understood physically in terms of the magnetic skin depth which gives the scale of the propagation of the magnetic perturbation by diffusion. For a constant fluid conductivity, the magnetic skin depth has a common value throughout the fluid. The magnetic diffusion regions at each boundary thus have identical thicknesses, regardless of the conductivity of the adjacent solid. Sim-ilarly, the depth at which the magnetic perturbation propagates in the solid depends on the conductivity in this solid and not on the conductivity in the adjacent fluid. The mag-Chapter 4: Two-plate problem 59 netic diffusion regions in each solid have different thicknesses on account of their different conductivity. One striking difference from the single plate solution is that u and P are now both non-zero outside the boundary layers. In addition ip shows a linear variation in z. These features represent the characteristics of the second independent solution associated wi th A = 0 (see (4.1) and (4.2)). The region between the boundary layers was designated as the current free region by Loper [1971]. W i t h the physical insights gained in the single plate case, we are in position to explain physically the behavior of these variables in this current free region. We recall that the radial transport in the boundary layer near one plate is accompanied by a vertical transport or pumping. If we transpose this idea to the double plate geometry, the same phenomenon must happen in both boundary layers. Hence, at each boundary, there is an exchange of fluid mass between the boundary layer and the current free region. When there is a net amount of fluid mass transferred from the two boundary layers to the current free region (or vice-versa), a radial transport in the current free region is required to balance the fluid flow. The vertical pumping, ip, varies linearly with z in the current free region as dictated by (4.1). The radial transport, which according to (2.12) is defined as the gradient of ip in z, is then constant everywhere in the current free region. As shown in the cartoon of figure 4.1A, a poloidal circulation is then set up in the whole column of fluid. The second independent solution associated with A = 0 (4.1) is then responsible for the fluid circulation required to conserve fluid mass. The first independent solution for A = 0, which represents a constant axial velocity in z (3.3), fits the pumping variations in the current free region to the appropriate values of pumping at each end of this region (at the limits of the magnetic diffusion regions). Chapter 4: Two-plate problem 60 We have already discussed in the analysis of the single plate that the radial trans-port in the boundary layers is associated with a toroidal acceleration to conserve angular momentum. The same concept applies in the current free region, where we now have a constant radial transport. Hence, the whole region of fluid between the boundary layers wil l experience a toroidal acceleration to satisfy the principles of conservation of angular momentum (figure 4.IB). This explains the presence of a constant toroidal velocity u in the current free region. A B Figure 4.1: Toroidal acceleration of the current free region due to fluid circulation. A : The magnetic stress at the boundaries produces a radial transport in the boundary layers (black arrows). A poloidal circulation is required in the current free zone to conserve mass. B: The radial transport in the current free region produces a toroidal acceleration by conservation of angular momentum. In the current free region, there is no shear in the toroidal velocity and hence, no deformation of Bz. The constant radial transport wi l l either concentrate the Bz field lines together or "move them apart". This modifies the intensity of the axial magnetic field and causes a perturbation to Bz. This explains the presence of the constant P in the current free region. We can note that this constant axial field does not create any electric current and this region is still "current free". The currents are confined to the magnetic diffusion regions in the fluid and the plates. Chapter 4: Two-plate problem 61 4.3.2 Transfer of angular momentum A t each end of the column, electromagnetic coupling can transfer angular momentum from the solid to the fluid. The coupling mechanism is unchanged; an exchange of charge between the solid and the fluid is essential to accelerate the boundary layer region by a toroidal Lorentz force. When the net radial current in one boundary layer is not equal and opposite to the net radial current in the other boundary layer, there is an overall net radial current through the fluid column. Hence, there is an overall net exchange of charges between the solid plates and the fluid, which corresponds to an overall transfer of angular momentum. If the magnetic torques at each end of the fluid column are exactly equal but opposite, the net radial current through each boundary layer are also equal but opposite. Hence, there is no net overall exchange of charges between the solid and the fluid and no net torque is imposed on the fluid column. This result can be correlated with Taylor's theorem [Taylor 1963], which stipulates that in the absence of exterior torques, 11 the couple exerted by the magnetic forces on any cylinder of fluid co-axial with the axis of rotation must vanish": In our axisymmetrical geometry, B = Bz z so only the radial current wi l l interact wi th this magnetic field to produce a torque. When the magnetic torques at each end do not cancel each other, there is a net torque imposed on the fluid column generated by a net radial current. The angular momentum is then transferred from the plates to the boundary layers, where the radial currents are confined (figure 4.2). Another fundamental feature of the coupling described in the single plate analysis is a net radial transport of fluid in or out through the boundary layers in the fluid. As explained (4.6) Chapter 4: Two-plate problem 62 A B Figure 4.2: Transfer of angular momentum from the plates to the boundary layer regions by an exchange of charges. A : The exchange of charge at each boundary requires a net poloidal electric current (black arrows), which is l imited to the magnetic diffusion regions in both the fluid and the solid. B : The net radial current in the boundary layers in the fluid produces a torque, which accelerates the fluid toroidally. in the previous subsection, conservation of mass requires a constant radial transport in the current free region (figure 4.1 A ) . B y conservation of angular momentum, the current free region is toroidally accelerated (figure 4.1 B ) . Because the Bz field is undeformed in the current free region, this toroidal acceleration can only be rigid. As the fluid in the boundary layers adjusts to the electromagnetic coupling with the moving boundary, angular momentum is then transferred from the plates to the whole column of fluid. This result is fundamental for the transfer of angular momentum from the plates to the fluid or vice-versa. In fact, it removes any doubts on the way that the angular momentum can be transferred instantly to the whole column. In contrast, the single plate analysis would have lead us to think that the toroidal fluid motion induced by the motion of the plates was limited to the magnetic diffusion regions. Not only is the transfer of angular momentum not instantaneous, but it is only transferred as far as the magnetic skin depth in the fluid. This picture presents an inconsistency with the idea of exciting torsional oscillations from the magnetic coupling at the ends of the cylindrical shells. However, Chapter 4: Two-plate problem 63 fluid circulation in the current free zone ensures that the angular momentum is transferred instantly and removes the apparent inconsistency. This concept of angular momentum distributed over the whole length of the fluid column was used by Greenspan [1963] in his analysis of the viscous spin-up of a fluid layer enclosed by two rotating flat plates. The spin-up time of the whole fluid was derived from the Ekman pumping of the single plate problem. Loper [1971] proceeded similarly to find the spin-up time of a conducting fluid layer by two conducting plates. The rotation of the whole column is a consequence of the difference between the pump-ing at each boundary layer, the net exchange of fluid mass between the boundary layers and the current free region. If the pumping at both boundaries is equal and oriented in the same direction, then there is no need for a radial transport in the bulk of the fluid. A n d thus, no toroidal acceleration in the current free region. Again , this wi l l happen when the magnetic torques at each end are equal but opposite (e.g. when the plates have the same conductivities and are oscillating with the same amplitude but a phase difference of exactly 180 degrees). The angular momentum carried by each boundary layer is equal and opposite, so the net radial transport in the boundary layers is ehminated. F l u i d mass is conserved without requiring a radial transport in the current free region. 4.3.3 Relative magnetic stresses at each boundary The amplitudes of the exchange of charge at each boundary are not independently deter-mined. The fluid circulation enables the plates to communicate with each other. Thus the amount of charge exchanged at each boundary wil l adjust to the total angular momentum transferred to the fluid column. It is possible to express the intensity of the axial current at one boundary in terms of Chapter 4: Two-plate problem 64 parameters of our system. When the magnetic field is small, the magnetic restoring force is negligible in the momentum balance and the magnetic field perturbation is adequately described by a simple diffusion equation. B y using the solution of the toroidal part of that amplitude of the axial current at the boundary in terms of the parameters of our model. This relationship is: where oc denotes proportionality, and conductivities ers and o-j refer to the solid and the fluid, respectively. The quantity u — ua represents the differential rotation between the fluid and the solid at the interface, with u being the velocity of the rigid rotation in the fluid. Thus, the amplitude of the exchange of charge at each boundary is function of the toroidal velocity of the current free region. Suppose that strong coupling at one boundary forces the fluid to follow the motion of that plate. We refer to this as efficient coupling (e.g. a measure of how closely the fluid follows the motion of the plate). Efficient coupling at one boundary does not imply that the magnetic stress is significantly different at each boundary. For instance, in figures 4.3 and 4.4, the conductivity of the bottom plate is much larger. The coupling strength between the fluid and this bottom plate is larger than the one at the top plate boundary and the toroidal velocity of the fluid follows the amplitude of motion of the bottom plate. However, the top plate having a toroidal velocity 10 times smaller than that of the bottom plate, the shear in velocity is much larger at the top boundary. The decreased conductivity of the top plate is compensated by larger jump in velocity to produce a magnetic stress similar to the one at the bottom boundary. This is apparent i i i figures 4.3 and 4.4, where the diffusion equation (T) , the solution of the toroidal part of the diffusion equation in the solid (Ts), wi th the boundary conditions (2.35) and (2.33), we find a relationship for the (4.7) Chapter 4: Two-plate problem 65 amplitude of the disturbances at each boundary are similar. This is another example of the communication between the plates. In the single plate solution, a decreased conductivity in the solid would necessarily decrease the magnetic stress at the boundary. A smaller toroidal velocity of the plate would also have the same effect. But in the bounded geometry, the magnetic stress at one boundary is not inde-pendent of the magnetic stress at the other boundary. They both adjust to the angular momentum transferred to the fluid column as a result of the magnetic torques imposed at each boundary. 4.3.4 Rigid rotations In figure 4 .3 , we clearly see the only departures from a rigidly rotating column are l imited to the magnetic diffusion regions.. However, the thickness of these boundary layers is very small versus the height of the column. Consequently, they represent a small fraction of the flow. Even the amplitude of the oscillations in the toroidal velocity is a few orders of magnitude smaller than the rigid toroidal motion. The thickness of the magnetic diffusion regions can be extended by reducing the period of oscillation of the plates, as we have shown in the previous chapter. However, the amplitude of the oscillations wil l stil l be negligible compared with the rigid rotation velocity. Rig id rotations still represent an accurate description of the flow. W i t h a weak magnetic field, the solution is in a diffusive regime. Changes in the magnetic field give only a small feedback to the momentum balance. In other words, the Lorentz force is small. We might imagine that the magnetic restoring force in the fluid enables waves to propagate although the velocity of their propagation is slow and they are rapidly attenuated. The Proudman-Taylor constraint is stil l applicable and the basic Chapter 4: Two-plate problem 66 toroidal motion in the fluid is largely a rigid body rotation. In order to alter the flow from a rigid rotation, the force balance i n the fluid, which is between the Coriolis force and the pressure gradients in the diffusive regime, must be changed. The Lorentz force plays a more active role in the force balance when it becomes comparable to the Coriolis force or, equivalently, when the Elsasser number (3.8) is m 1. This condition is reached when the magnetic field Bz is increased to ~ 20 G. W i t h a sufficiently large Lorentz force, waves can propagate deeper in the fluid before being attenuated. The solution wil l then be in a wave regime, as we have seen in the single plate solution. In figures 4.5 and 4.6, we have used the set of parameters presented in table 4.1, but have replaced the magnetic field with 35 G. In figures 4.7 and 4.8, we have increased the magnetic field to 100 G. W i t h Bz = 35 G, the waves are sustained for many periods of oscillation and can propagate deeper in the fluid. The magnetic diffusion regions are extended and represent a good fraction of the flow in the column. The amplitude of the waves is still smaller than the rigid rotation in figure 4.5, but large enough to cause a significant deviation in u from a rigid rotation. This confirms the results of the single plate analysis, which stated that the rigid rota-tions might not be an accurate description of the flow when the magnetic field is strong. Again , this is consistent with the idea that the Proudman-Taylor constraint is broken by a strong magnetic field. This is even more evident in figure 4.7 where Bz = 100 G. The cur-rent free region has been swallowed by the extended propagation of the waves. The solution oscillates about an average rigid rotation velocity, but the amplitude of these oscillations are comparable to the amplitude of the average rotation. Chapter 4: Two-plate problem 4.3.5 Conclusions 67 A t each boundary, the motion of the plate imposes a torque on the fluid by electromagnetic coupling. A non-zero net torque (the sum of the magnetic torques from each boundary) is necessary to transfer angular momentum from the plates to the fluid. If the sum of the magnetic torques is zero, there is still a boundary layer adjustment at each end of the column, but the current free region is not accelerated toroidally. The column of fluid reacts to the net magnetic torque by undergoing rigid rotation, which is achieved through fluid circulation. However, if the magnetic stress on the fluid becomes too large, the Proudman-Taylor constraint is broken and rigid rotations no longer describe the fluid motion. The requirements for a large magnetic stress on the fluid are: a large conductivity in the solid and a strong magnetic field. If the magnetic field is weak, the Elsasser number (3.8) is small and the solution stays in a diffusive regime; rigid rotations wil l be generated even wi th a strong conductivity in the solid. If the conductivity of the solid is weak, a strong magnetic field wi l l enable the wave to propagate but the amplitude of these oscillations wi l l be negligible versus the rigid rotation velocity. Hence, for the cylindrical shells outside the tangent cylinder, the conductivity of the mantle is weak and so is the magnetic field at the C M B . Rigid rotations are then applicable. However, for a cylindrical shell inside the tangent cylinder, the conductivity of the inner core is large and the magnetic field at the I C B is probably large enough to yield solutions in the wave regime. Rigid rotations would fail to describe the fluid motions above and below the inner core. Chapter 4: Two-plate problem 68 (m/s) xio" Figure 4 .3 : Solution for the scalar potentials of the velocity in the fluid obtained wi th the set of parameters listed in table 4.1. The vertical axis of every graph represents the vertical distance z. The bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. The solution presented is at t = 0. Chapter 4: Two-plate problem 69 4000 B 4000 1 0 1 ^ « (G/m) x io"7 -1 o 1 (G/m) xio"7 Figure 4.4: Solution of the scalar potentials of the magnetic field in the fluid and the solid obtained with the set of parameters listed in table 4.1. The vertical axis of every graph represents the vertical distance z. The bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. The solution presented is at t = 0. Chapter 4: Two-plate problem 70 u / u s A (m/s) xto" Figure 4.5: Solution of the fluid velocity obtained with the set of parameters listed in table 4.1, but substituting Bz by 35 G. The vertical axis of every graph represents the vertical distance z. The bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. The solution presented is at t = 0. Chapter 4: Two-plate problem 71 4000 4000 2000 2000 0 5 (G/m) x 10"7 -5 0 5 (G/m) xio' 7 Figure 4.6: Solution of the secondary magnetic field obtained with the set of parameters listed in table 4.1, but substituting Bz by 35 G. The vertical axis of every graph represents the vertical distance z. The bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. The solution presented is at t = 0. Chapter 4: Two-plate problem 72 U / U S A Figure 4.7: Solution of the fluid velocity obtained with the set of parameters listed in table 4.1, but substituting Bz by 100 G. The vertical axis of every graph represents the vertical distance z. The bottom fluid-solid interface is at z — 0, while the top fluid-solid interface is at z = 2000. The solution presented is at t = 0. Chapter 4: Two-plate problem 73 -0.02 0.06 -200 0 2 (G/m) x10" s Figure 4 .8: Solution of the secondary magnetic field obtained with the set of parameters listed in table 4.1, but substituting B by 100 G. The vertical axis of every graph represents the vertical distance z. The bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. The solution presented is at t = 0. 74 Chapter 5 Transfer of angular momentum 5.1 Introduction The motivation for this work is to look at the electromagnetic coupling between the fluid core and the surrounding solids, the mantle and the inner core. In other words, to in-vestigate the exchange of angular momentum through electromagnetic interactions. This problem is modelled by considering the fluid core as a set of rotating cylindrical shells. In the theory of torsional oscillations (TO) [Braginsky 1970] the fluid cylindrical shells are coupled to the solid by electromagnetic forces. The relative motion between the fluid and the solid strains the ambient magnetic field, which creates a magnetic torque on the fluid, proportional to the relative motion at the boundary. In the T O theory, the toroidal velocity of the fluid at the C M B is equal to the average toroidal velocity of the whole fluid cylindrical shell. This approximation was made with the assumption that the magnetic stress at the ends of the cylindrical shell (at the C M B ) was not sufficient to alter the rigid nature of the toroidal motion. The results of the previous chapter confirm this approximation when the field is weak. Another assumption was that the strong magnetic field at the I C B would force the inner core to be perfectly coupled to the fluid cylinders inside the tangent cylinder. The whole region inside the tangent cylinder can thus be regarded as a single, rigid cylinder. However, the previous chapter suggests that the strong coupling at the bottom plane boundary, and hence the I C B , wi l l generate waves that wi l l break the rigid rotations of the cylindrical shells above and below the inner core. Hence, this raises questions about the use of rigid rotations inside the tangent cylinder Chapter 5: Transfer of angular momentum 75 for the purpose of assessing the exchange of angular momentum between the fluid and the solid. The purpose of this chapter is to investigate the transfer of angular momentum from the plates to the fluid, and to estimate how the results depart from the theory of Braginsky's where rigid rotations are assumed. 5.2 Angular momentum of rigid rotations If we assume rigid rotations in the fluid and neglect the coupling between adjacent cylindrical shells, the angular momentum balance relates the changes in the angular ve-locity u(s) of the shell at radius s to the torques due to magnetic stresses —fi(s) and —fm(s) at the I C B and C M B , respectively. When the angular velocity is proportional to exp(—iu;t), the angular momentum balance is expressed as [Braginsky, 1970] -u>2m(s)u(s) = iwfi(s) + iufm(s). (5.1) The moment density, m(s), of a cylindrical shell of height h(s) is defined as m(s) = ATvps3h(s). (5.2) Since the magnetic torques are proportional to the relative motion between the fluid and the solid, we can express fi(s) and fm(s) as / . ( , ) = T ^ ' ) [«( , ) - u<] (5.3) / « ( * ) = T m ( * ) [*(*) - « J . (5-4) where u{ and um are the angular velocities of the inner core and the mantle. Buffett [1997] developed the amplitudes and T m , which can be expressed in our geometry as: T i (* ) = « 8 ( l + 0 B , V / (5.5) Chapter 5: Transfer of angular momentum 76 (5.6) The difference between these amplitudes arises because of the different conductivity model used at each boundary. It has been assumed that the conductivity of the mantle crm is confined to a layer of thickness A at the C M B . The thickness of this layer is assumed to be much smaller than the typical magnetic skin depth in the mantle. The inner core is assumed uniformly conductive with a value identical to that of the fluid cr,. The skin depth The amplitude of the rigid velocity u reflects the angular momentum transferred to the fluid by the solid. Isolating u(s) from equation (5.1) gives which is readily evaluated in terms of the parameters of our system; ui and um are forcing 5.3 Comparison of theories as a function of Bz The angular momentum transferred from the plates to the fluid is given by (5.7), when the toroidal velocity of the fluid is described by a rigid rotation. This estimate can be compared wi th the angular momentum transferred using the model from the previous chapter, where rigid rotations are not assumed. Instead, a complete solution for the fluid motion is used to calculate the coupling. For weak magnetic field, we should expect little or no difference because our model reproduces rigid rotations. A t stronger field, the rigid rotations are broken inside the tangent cylinder and the angular momentum transfer from our model is likely to be different (5.7) B\ [(l + i)7r/180 = 6.34 x I O - 9 * - 1 1 0 - W / 1 8 0 = 6.34 x I O - 1 0 s'1 Table 5.1: Parameters used to compare the transfer of the angular momentum. In figure 5.1, we show the real part of the ratio between the toroidal velocity in the fluid and the velocity of the inner core as a function of Bz. The solid line represents the toroidal velocity of rigid rotations obtained with (5.7), while the dashed line is the velocity from the more complete model, averaged over the cylinder. This plot reflects the angular momentum transferred to the fluid from each model. The behavior exhibited by Braginsky's theory (solid line) can be easily explained. We can imagine that the fluid and both solids are three rigid cylinders on top of one another. The magnetic torque at the top plate boundary is very weak due to the low conductivity and the thin conductive layer of the mantle. The top cylinder is then totally uncoupled from the fluid cylinder for any value of Bz. We can imagine that the bottom .cylinder and the fluid cylinder are joined together with elastic threads. The elasticity of the threads is analogous to the strength of the magnetic field, and therefore the strength of the coupling between the fluid and the plates. For a very weak magnetic field, the threads have virtually no elastic strength and are more like loose strings. The movement of the bottom plate will Chapter 5: Transfer of angular momentum 78 not transmit any angular momentum to the fluid column. In this case, u = 0 and the ratio u/u{ is also zero. For a slightly stronger magnetic field, the elastic strength of the threads is detectable, but still weak, and the motion of the bottom plate produces a small torque on the fluid column. The toroidal velocity of the column is only a fraction of that of the plate and out of phase by almost 180 degrees. As the magnetic field increases, the threads become stiffer, permitting a larger angular momentum transfer to the fluid and a smaller phase difference. W i t h a very large magnetic field, the threads are now like metal rods and the fate of the fluid column is to follow unconditionally the movement of the bottom plate. For magnetic field not large enough to provide a perfect coupling (i.e. u/ui = 1) but sufficiently large to ensure that the motion of the fluid cylinder is almost in phase wi th the motion of the bottom plate, the velocity of the fluid cylinder overshoots the velocity of the bottom plate. The ratio u/ui is larger than 1 and this signifies that the inertia transmitted to the fluid cylinder is large. The transfer of angular momentum estimated using the more complete model is similar to the result of Braginsky's theory, but there are some important differences. For weak magnetic fields, the Proudman-Taylor constraint is valid and our model gives rigid rota-tions. The angular momentum transferred to the fluid is then identical to that obtained using rigid rotation. However, the magnetic force becomes important in the force balance at roughly 20 G, and the departures with Braginsky's theory coincide with the onset of waves which cause departures of the fluid from rigid rotations. For large magnetic field, though, the curve from my model converges back towards the rigid rotation curve. A large magnetic field stiffens the fluid, prohibiting large amplitude motion. The column of fluid once again behaves like a rigid body. A t such large magnetic field, the coupling between the fluid and the solid is very strong. Chapter 5: Transfer of angular momentum 79 Both curves in figure 5.1 have a similar shape, but the one from my model appears to be "stretched" towards higher magnetic field values. Thus, the general effect of the fluid deformation is to reduce the coupling strength between the fluid and the solid, as a larger Bz is required to obtain the same coupling. A n d as a result, there is a window of magnetic field values for which there is a significant difference in the angular momentum transferred to the fluid. 5.4 Comparison of theories as a function of oscillation period In the previous section, we have used an oscillation period of 200 days for the motion of the plates. The choice of this particular period was merely a convenience since it emphasized the correlation between the departures of angular momentum transfer and the departures of the fluid from rigid rotations. However, the frequency of oscillation of the plates has an important influence on the strength of the coupling with the fluid. In figure 5.2, we have used a longer period of oscillation (lower frequency): 400 days. Results from both curves, Braginsky's theory and the more complete model, have the same structure as in figure 5.1, but the curves appear to be "compressed" towards weaker magnetic field strength. Less inertia is transmitted to the fluid and a weaker magnetic field is required because the corresponding oscillations in the angular momentum are smaller (e.g. smaller torques are required). For a given magnetic field, the ratio u/ui increases at larger periods and decreases at shorter periods. The coupling is thus enhanced with a longer period and, conversely, a shorter period would reduce the coupling. Similarly, the height of the fluid column wi l l alter the effective strength of the coupling because it modifies the moment of inertia of the fluid. A n higher fluid column requires a larger magnetic field to produce the same ratio u/ui. Chapter 5: Transfer of angular momentum 80 To evaluate the departures between Braginsky's theory and the more complete model in terms of the frequency of oscillation of the plates, we compare the models over a range of frequencies for a constant magnetic field. To observe any differences as the frequency is varied, the magnetic field must be above 20G, where the Elsasser number is roughly equal to one. In figure 5.3, the main field is chosen to be Bz = 50G. The other parameters are the same as those presented in table 5.1. Once again, the solid line represents the angular momentum transferred to the fluid with the assumption of rigid rotations (equation 5.7), while the dashed line is the integrated velocity from the more complete model. The solid curve can again be explained with simple arguments. We refer back to the simple model of three rigid cylinders, where the bottom cylinder and the fluid cylinder are joined by elastic threads. In this case though, the elasticity of the threads is fixed. If the bottom plate is oscillated at high frequency (low period), the fluid column wi l l not have time to adjust to the movement of the plate. The elastics are stretched over a time scale which is too short to overcome the inertia of the fluid cylinder. The amplitude of the toroidal motion of the fluid cylinder wi l l only be a fraction of the amplitude of the plate and their motion wi l l be out of phase. When the period of oscillation is increased, the coupling becomes more efficient. The fluid cylinder has more time to adjust to the torque imposed by the stretched elastics. For very long periods, the fluid cylinder follows closely the motion of the bottom plate. Only small differential rotations are required to produce the small torques that are needed to cause the slow changes in angular momentum. The dashed curve, representing the more complete model, is similar to the solid curve, obtained with Braginsky's theory, but there is a. clear discrepancy between the curves. For a fixed J5Z, my model requires a lower period to reach the same ratio ujui than that required by the rigid rotation model. This reflects that the deformation in the fluid reduces the strength of the coupling. Chapter 5: Transfer of angular momentum 81 The difference between our model and the rigid rotation model at a fixed magnetic field is plotted in figure 5.4. W i t h Bz = 50 G, my model generates a solution where waves are present (e.g. wave regime), so departures from rigid rotations are expected. However the departures in angular momentum are confined to a range of frequencies. For high frequencies, the coupling is inefficient in both models and the differences between them are not important. For very low frequencies, there is not enough inertia in the fluid to generate waves with amplitudes sufficiently large to break the rigid rotation. The plot of differences between the two models in figure 5.4 is reminiscent of a resonance phenomenon. For each magnetic field strength, there is a peak frequency at which the difference between the angular momentum transferred to the fluid from each model wi l l be maximum. Weaker magnetic fields wil l shift the position of the resonance towards longer periods and broaden the peak, while larger magnetic fields wi l l shift the position of the resonance towards shorter periods and sharpen the peak. When the magnetic field is smaller than the required value to break rigid rotations (fti 20C7), the amplitude of the resonance is negligible. We can now go back to interpret the results of figure 5.1 in the previous section from the perspective of this frequency analysis. The window of magnetic field values for which there is a discrepancy between the models is dependent on the frequency of oscillation of the plates. The lower limit of magnetic field at which a discrepancy wi l l occur is always defined by the value required to break the rigid rotations ( « 20 G). The magnetic field value at which the fluid column stiffens to a rigid rotation at this given frequency defines the upper bound of the window. In figure 5.1, a higher frequency wi l l require a stronger magnetic field to generate rigid rotations, extending the window of discrepancy. A lower frequency on the other hand would decrease the magnetic field necessary to stiffen the fluid column and the width of the window is reduced. If the frequency is low enough that this Chapter 5: Transfer of angular momentum 82 magnetic field value is smaller than m 20(7, the curves will be identical and the window of discrepancy disappears. Our original interest was to look at the validity of rigid rotations for the exchange of angular momentum at decade periods. With 10 000 days 30 years), a magnetic field of 2 0 G is well above the field required to stiffen the fluid column and guarantee rigid rotations. From the point of view of the resonance curve (figure 5.4), the peak is at a period of ~ 200 days with a magnetic field of 50 G; At 10 000 days, the difference is negligible. A smaller Bz would displace the resonance towards longer periods, but eventually Bz becomes too small to break the Proudman-Taylor constraint. To achieve a resonance at decade periods, the field falls below 20 G and waves are suppressed. Hence, the amplitude of the resonance in the discrepancy at decade period would be basically zero. The transfer of angular momentum from the plates to the fluid at decade periods is then correctly assessed with the assumption of rigid rotations. However, for phenomena associated with shorter periods, there is a clear discrepancy between the electromagnetic coupling of the rigid rotation model and our model. For instance, if one is interested in including the contribution of the core in the angular mo-mentum balance for the short period oscillation in the L O D variations ( w 1 year), rigid rotations inside the tangent cylinder can not be assumed for the purpose of calculating the electromagnetic coupling with the mantle and the inner core. 5.5 Influence of waves on angular momentum transfer 5.5.1 Standing waves on the fluid cylinder When the solution is in a wave regime, it consists of a linear superposition of solid body rotation and a wave bounded by the plates. When the coupling is "efficient" . at each Chapter 5: Transfer of angular momentum 83 boundary, the waves in the fluid column are much like the standing waves on an oscillating string. The frequencies and structure of these waves are constrained so that a finite number of half-wavelengths fit perfectly between the boundaries. The fundamental mode consists of a single half-wavelength and the higher modes are the harmonics of this fundamental mode. When the forcing frequency equals one of the modes, then a resonance occurs. To avoid confusion, it is important to make a distinction between the resonance effect that we are describing here and the resonance effect presented in figure 5.4 of the previous section. That latter resonance is defined as the frequency at which the discrepancy between the two models is maximum. In this present section we are considering the excitation of standing waves on the fluid column. The effects of the standing waves are shown in figure 5.5, which is the same graph presented in figure 5.1, but extending the range of magnetic field values. For Bz larger than 120 G, the more complete model (dashed line) indicates rapid fluctuations in u/v-i which are absent from the rigid rotation model (solid line). These oscillating features result from resonant excitation of standing waves. In figure 5.5, the coupling at the C M B is very weak. It is as if the string was not attached at that boundary, and was free to oscillate. The sharpness of the resonances is increased when the string is attached at both ends. Hence, for the analysis of the resonant excitation of standing waves, we increased the coupling at the C M B . This can be accomplished by increasing the thickness of the conductive layer in the mantle to A = 2000/sm. The period of oscillation is chosen to be 200 days as before, and we use um = u{ (tt m = c*>7r/180) to keep the solution as simple as possible. The other parameters used in the calculation are the same as those presented in table 5.1. Chapter 5: Transfer of angular momentum 84 Figures 5.6 and 5.7 show the results obtained using the two models as Bz is increased. The solid curve continues to represent the results obtained with the assumption of rigid rotations. Since the conductivity in the mantle is no longer confined to a thin layer, it is not possible to use (5.6) to describe the torque on the mantle. The resulting expression for the torque on the mantle has the same general form as that on the inner core 4TTS 3 (1 + i)B\ HV2OJ (5.8) where 7/n is the magnetic diffusivity of the inner core or the mantle and r}f is the magnetic diffusivity in the fluid. We note that (5.8) reduces to (5.5) when nn = Vf- Using (5.8) in the angular momentum balance (5.1) gives u, the angular velocity of the rigid rotations K + K — iphoj u (s) = K + (5.9) where K = B2z(l + i) The dashed line in figures 5.6 and 5.7 is again the result of the more complete modeL For Bz larger than 100 G, there are fluctuations due to the excitation of standing waves on the fluid column. At fixed frequency, the strength of the magnetic field determines which mode is excited. A large magnetic field increases the velocity of the wave, which increases the wavelength. Consequently, lower modes are excited as the magnetic field is increased; The resonance at the largest magnetic field value (at w 2200 G) corresponds to the fun-damental mode, the one in which a single half-wavelength fits between the boundaries. Increasing the magnetic field will keep the wave solution in this fundamental mode, but the amplitude of this half-wave will decrease as we go away from the resonance. This is Chapter 5: Transfer of angular momentum 85 consistent with the stiffening of the fluid column due to the strong magnetic field. The fluid motion begins to approximate a rigid rotation and it is no accident that the solution converges back towards the rigid rotation solution. Decreasing the magnetic field, from the position of the fundamental mode, wi l l decrease the wavelength of the wave. We wi l l thus hit the successive higher harmonics. A s the wave-length decreases, diffusion becomes more important, broadening the peaks and lowering the amplitude of the resonances. This effect is most obvious on figure 5.6. When the magnetic field is low enough, it is no longer possible to distinguish the different harmonics. The amplitude of these resonances is also a function of the frequency of oscillation. For instance, the resonances that are observable in figure 5.5 at a period of 200 days have disappeared at a period of 400 days in figure 5.2. As we have seen in section 5.4, the amplitude of the waves in the fluid depends on the frequency of oscillation. If the frequency is such that waves of large amplitude are generated, the standing waves are more efficiently excited. O n the other hand, if the frequency does not induce waves of large amplitudes, the amplitude of the resonances wil l also be l imited. For decade periods, where there is not enough inertia in the fluid to generate waves of significant amplitude, the frequency is too low to excite the resonances. Hence, the natural modes in the fluid solution do not threaten the validity of the rigid rotations at decade periods. Furthermore, this phenomenon does not introduce additional departures from the rigid rotations at shorter periods. The largest departures brought by the standing waves occur at very large magnetic fields, which are greater than the plausible values of the magnetic field in the core. Chapter 5: Transfer of angular momentum 5.5.2 Complications when the plate velocities are different 86 Standing waves are excited more efficiently when the plates have different amplitude of motions. This is particularly evident for strong coupling because the fluid velocity at each boundary tends to follow the motion of the plates. If the velocity of each plate is different, this implies a shear in the toroidal velocity in the fluid from one plate to the other, which cannot be described with a rigid rotation. The amplitude of the wave part of the solution increases in order to satisfy the boundary conditions, and the amplitude of the standing waves increases. This effect is shown in figure 5.8 and 5.9, where the amplitude of motion of the mantle is 10% of the amplitude of motion of the inner core and the thickness of the conducting layer of the mantle is A = 2000 km. The period of oscillation is 3000 days. If the amplitude of motion of each plate were identical at this period, then the amplitude of the waves would be small. However, the different amplitude of motion requires waves of large amplitude in order to satisfy the boundary conditions. The dashed line, which represents the more complete model, exhibits the resonant excitation of the standing waves. Again , the discontinuity at the stronger magnetic field (~ 700 G) is the location of the fundamental mode, and the higher harmonics are found with decreasing Bz. 5.5.3 Relative contribution of each plate to the total torque on the fluid column Figures 5.8 and 5.9 identify further differences between Braginsky's model and the more complete model. These results were obtained with a strong torque at each boundary and different amplitude of motions for each plate, The average velocity in my model (dashed line) at large magnetic field does not converge to the velocity of rigid rotations obtained Chapter 5: Transfer of angular momentum 87 with (5.9) (solid line). In the rigid rotation model, u at large field is closer to ui than um because the torque at the I C B is stronger than that at the C M B . However, the relative strength of the torque at each boundary is unchanged as Bz is varied. A larger magnetic field wi l l increase the magnetic torque at each boundary by the same amount. Thus the total torque on the fluid cylinder increases, but the relative contribution to the torque from each boundary is unaltered. In the more complete model, at large magnetic field, the ratio u/uf converges towards 0.55, which is the average value of the motion of each plate (uf = 1.0, uf = 0.1). Each plate is contributing almost equally to the total torque on the fluid. A t large field, the coupling at each boundary is strong, which forces the fluid to be "locked" to the motion of the plates. This is the case even at the C M B where the conductivity of the mantle is smaller. W i t h different plate motions, a wave wi l l connect the velocity at one boundary .to the velocity at the other. The velocity of the wave averaged over the cylinder represents the average value of the fluid velocity at each boundary, hence, the average value of the plate velocities. We recall, however, that the departures presented here are a direct consequence of the strong coupling at each boundary. But as the thickness of the conductivity layer at-the base of the mantle is believed to be on the order of a few hundred meters [Buffett 1992], the coupling at the I C B wil l always be significantly stronger even for very large magnetic fields. O n the other hand, magnetic fields exceeding 100 G are at the l imit of plausible values for the magnetic field at the I C B , and such fields are definitely too large for the C M B . Hence, the solutions with strong coupling at both ends of the fluid cylinder presented here are physically plausible, but may not represent the physical conditions that apply.in the core. Consequently, the effects of standing waves, particularly with unequal solid velocities, pose no problem to the validity of rigid rotations at decade periods. Chapter 5: Transfer of angular momentum 88 1.4 _0 4 I 1 1 1 1 J : 1 0 20 40 60 80 100 120 Figure 5.1: Transfer of angular momentum to the fluid as a function of magnetic field strength at a period of 200 days. The solid curve represents the toroidal velocity of rigid rotations obtained with (5.7). The dashed curve is the average velocity from the results of my model. The parameters used in the calculation are given in table 5.1. 1.4 -0.4 1 ' ' ' ' ' ' 1 ' 1 1 0 10 20 30 40 50 60 70 80 90 100 Bz ( G> Figure 5.2: Transfer of angular momentum to the fluid as a function of magnetic field strength at a period of 400 days. The solid curve represents the toroidal velocity of rigid rotations obtained with (5.7). The dashed curve is the average velocity from the results of my model. Apart from the period, the parameters used in the calculation are given in table 5.1. Chapter 5: Transfer of angular momentum 89 1.4 (Days) Figure 5.3: Transfer of angular momentum to the fluid as a function of the period of oscillation for Bz = 50 G. The solid curve represents the toroidal velocity of rigid rotations obtained with (5.7). The dashed curve is the average velocity from the results of my model. The other parameters used in the calculation are given in table 5.1. 0.9 ,-0.8 -0.7 -CO 0.6 -\ 0.5 -1 3 0.4-0.3 -0.2 -0.1 -1.1 I ' ' ' ' ' ' ' ' • 1 0 100 200 300 400 500 600 700 800 900 1000 (Days) Figure 5.4: Departures of our model from rigid rotations. This graph represents the difference between the two curves presented in figure 5.3. The difference is reminiscent of a resonance phenomenon. We used Bz = 50 G, and the other parameters used in the calculation are given in table 5.1. T Chapter 5: Transfer of angular momentum 90 Figure 5.5: Solution for the toroidal fluid velocity at large values of magnetic field. Var i -ations in the amplitude of u with Bz reflect the resonant excitation of standing waves on the fluid column. The parameters used in the calculation are given in table 5.1 Chapter 5: Transfer of angular momentum 91 50 100 150 200 250 300 350 400 450 500 (G) Figure 5.6: Resonant excitation of standing waves on the fluid column (part I). The solid curve represents the toroidal velocity of rigid rotations obtained with (5.9). The dashed curve is the average velocity from the results of my model. Apar t from um = u- and A = 2000 km, the parameters used in the calculation are given in table 5.1. Refer to figure 5.7 for magnetic field values above 500 G. I 3 I 3 Bz (G) Figure 5.7: Resonant excitation of standing waves on the fluid column (part II). Refer to figure 5.6 for magnetic field values between 0 and 500 G. Chapter 5: Transfer of angular momentum 92 Figure 5.8: Resonant excitation of standing waves on the fluid column for unequal plate-velocities (part I). The solid curve represents the toroidal velocity of rigid rotations obtained with (5.9). The dashed curve is the average velocity from the results of my model. Apart from u> = ft/3000 (3000 days), the parameters used in the calculation are given in table 5.1. Refer to figure 5.9 for magnetic field values above 200 G. I =3 „ , 1100 1200 (G) Figure 5.9: Resonant excitation of standing waves on the fluid column for unequal plate velocities (part II). Refer to figure 5.8 for magnetic field values between 0 and 200 G. 93 Chapter 6 Rigid rotations from a concentrated torque 6.1 Introduction Long period oscillations in the length of day can be explained by the exchange of angular momentum between the fluid core, the inner core and the mantle. A commonly proposed mechanism for the coupling is through electromagnetic stresses between the fluid and the solid. The previous chapter confirmed that rigid rotations in the fluid can be used to model the transfer of angular momentum between the solids and cylindrical shells of fluid at decade periods. Angular momentum can also be transferred between fluid shells through magnetic coupling. To fully assess the angular momentum budget of a single cylindrical shell, we have to incorporate this additional torque. Magnetic coupling between adjacent cylindrical shells provides the restoring force for the torsional oscillations. When a cylindrical shell is rotated out of alignment wi th the others, this wi l l deform the radial magnetic field that threads the surface of the cylinders. B y Lenz ' law, the deformation of this magnetic field wi l l generate a magnetic force that opposes the deformation. The radial magnetic field provides a restoring force which permits oscillations between adjacent cylindrical shells. The dynamo of the Ear th should excite these waves at their natural frequencies. Braginsky [1970] calculated the restoring force by averaging the torque over the surface of the cylinder. This averaged torque is distributed uniformly over the whole length of the fluid column and, consequently, generates rigid rotations. These rigid rotations are also coupled to the solids by magnetic stresses. It is then assumed that the torque due to a radial magnetic field that varies with height may be calculated from the average value of Chapter 6: Rigid rotations from a concentrated torque 94 this radial field over the length of the cylinder. Since a uniformly distributed averaged field would tend to produce a rigid rotation of the fluid column, it follows that the equivalent (but variable) field must also produce a rigid rotation. This concept may seem questionable at first glance. Lets suppose for instance that there is an axisymmetric radial field confined to a volume that approximates a disc at a particular height in the column. When fluid cylinders are differentially rotated, a magnetic torque is exerted on the fluid in the disc by the distortion of the radial magnetic field. It is intuitive to think that only the region in the vicinity of the disc wi l l respond to the torque, and that the rest of the fluid should not be altered. Therefore, it seems that the motion of the fluid column should reflect the variations of the radial magnetic field with height and hence, that rigid rotations can hardly be generated from a height dependent Bs. However, this simple intuition does not take into account the effects of fluid circulation. The physical insights gained in the previous chapters suggest that a torque confined to a narrow disc within the fluid column causes an adjustment in the fluid in the vicinity of the disc. We might then expect the presence of boundary layers. The radial transport in those boundary layers would generate a radial transport in the current free region to conserve fluid mass. B y angular momentum conservation, a rigid body rotation could be induced in the whole column of fluid. The goal of this chapter is to demonstrate that we can indeed transfer angular mo-mentum to the whole column of fluid when the torque is applied to a th in disc at an arbitrary height. It is possible to construct more general solutions by adding together the torques on small disc sections along the length of the column. Corresponding velocities due to each individual disc may be added to obtain a final rigid rotation velocity. Thus, by showing that rigid rotations are obtained from a torque confined to a small section of Chapter 6: Rigid rotations from a concentrated torque 95 the column, we prove that a z-dependent torque will indeed yield a rigid rotation. Hence, the effect of the electromagnetic coupling between adjacent shells can be captured by using the average value of B3 to estimate the torque, as Braginsky [1970] did. 6.2 Model for a concentrated torque inside the fluid 6.2.1 System of equations We want to model the influence of a torque which is concentrated at a particular height in the fluid column. This disc will separate the fluid column in three regions: the cylinders of fluid above and below the disc and the volume of the disc itself. To clarify the terminology, a fluid cylinder refers now to the reduced region between the disc and a solid boundary. The fluid region between the solid boundaries, containing the disc and both fluid cylinders, will be referred as the column of fluid (figure 6.1). t o p p l a l e u p p e r f l u i d c y l i n d e r f l u i d d i s c > f l u i d c o l u m n l o w e r f l u i d c y l i n d e r I b o t t o m p l a l c Figure 6.1: The fluid outer core is modelled as a conducting layer of fluid bounded by plane conducting boundaries. The disc region separates the fluid column in two fluid cylinders. The system of equations presented in (2.13) to (2.16) still apply in each fluid cylinder. In the disc, however, the additional torque from the magnetic coupling between adjacent cylindrical shells will modify the system of equations. Braginsky [1970] expressed the torque Chapter 6: Rigid rotations from a concentrated torque 96 due to the radial magnetic field in terms of the differential rotation of fluid cylinders. We assume that this torque can be captured by an azimuthal body force, F = F^e~%u't+ - { » - £ * - ) + 2 0 ( i + - * - ) - 1 ( l > - j ? p - ) = ° • (6-io> Chapter 6: Rigid rotations from a concentrated torque 98 where the superscripts "+" and "-" refer to boundary values evaluated at the top and bottom of the disc. The average quantities { P } , {T}, and {u} are the only terms that require a detailed knowledge of the fields inside the disc. However, when the disc is vanishingly thin these terms depend on A z , which vanishes. We assumed that is constant when A 2 —> 0, so the body force increases as A z —* 0 like a delta function. Before evaluating the jump conditions, it is useful to interpret the terms in equation (6.9), which may be viewed as an angular momentum budget for the fluid in the disc. The first term on the left hand side represents the change in the angular momentum of the disc. The infinitesimal thickness implies that the disc carries a negligible amount of angular momentum. The second term involves the Coriolis force and relates to the transfer of fluid mass between the disc and the surrounding fluid. The jump (tp+ — tp~) in the axial velocity on either side of the disc represents the net axial flow pumped into or out of the disc. This axial pumping wi l l be accompanied by a radial transport, which redistributes angular momentum to the whole column of fluid. The third term involves the Lorentz force and expresses the angular momentum trans-ferred to the fluid by the exchange of charges. The jump in the axial electrical current on either side of the disc (T+ — T~) represents the net amount of charges injected into (or extracted from) the disc. The axial electrical current requires a radial current in the fluid column. The cross product between the background axial magnetic field and this radial current wi l l generate a Lorentz force that wi l l accelerate the fluid toroidally. Hence, the torque applied to the disc (e.g. /^) is balanced by an equal torque applied on the over and underlying fluid cylinders, which is due to the exchange of fluid mass and Chapter 6: Rigid rotations from a concentrated torque 99 electric charges between the disc and the fluid. Since the disc is infinitely thin and carries no angular momentum, the angular momentum transferred to the fluid column is directly proportional to the amplitude of the concentrated force f^. 6.2.3 Boundary conditions The effect of a concentrated force on the fluid cylinders is modelled by introducing jump conditions on the fluid solution at the position of the disc. We require separate solutions above and below the disc, each with 6 integration constants. Consequently, to solve the problem, we require 6 additional boundary conditions at the position of the disc, one for each component of the magnetic field and the velocity. If there were no forcing, the presence of the disc would not induce any perturbation in the fluid so all three components of the velocity and the induced magnetic field have to be continuous across the disc. Because the disc is thin, we can assume that any background field from another source (say the coupling between the fluid and the plates) remains constant over the thickness of the disc. Equations (6.7) to (6.10) represent a set of conditions that relate the amplitude of to the discontinuities in the fluid variables across the disc. We can exploit symmetries to simplify these discontinuity conditions. The concentrated force is assumed to be symmetric about the center plane of the disc (see figure 6.2 A ) . This assumption imposes symmetry conditions on the other scalar variables. The toroidal velocity perturbation produced by this force wi l l also be symmetric about the center plane of the disc as shown in figure 6.2 B . The radial transport generated in the disc also has to be symmetric (figure 6.2 C) simply because there is nothing to cause an asymmetry in this flow. Chapter 6: Rigid rotations from a concentrated torque 100 fluid fluid _ A B C Figure 6.2: Symmetric profile of a concentrated force in the disc. A : concentrated force profile. B: toroidal velocity profile. C: radial velocity profile. The symmetry on the toroidal and radial velocity implies the following jump conditions: u+ - u~ = 0 (6.11) | " ^ + " | ^ ~ = °- (6.12) dz dz Similar symmetry conditions can be obtained for the induced magnetic field. The radial transport caused by concentrates or spreads the Bz field fines apart. This perturbs the axial magnetic flux, which is described by P. Symmetry of the radial transport about the center plane of the disc ensure a similar symmetry in P. Consequently, P+ - p- = 0. (6.13) The deformation of the background magnetic field by u and (figure 6.2 B and C), will induce electric currents in the disc. However, the symmetry of these deformations will induce symmetric radial ( | j j ) and toroidal (§^r) electrical currents. This may be inferred from the discontinuity conditions (6.8) and (6.10) respectively, once (6.11) and (6.12) have been applied. We see that and are continuous, which means that the radial and Chapter 6: Rigid rotations from a concentrated torque 101 toroidal currents are continuous across the disc. Since a jump in current would imply a sheet current in the disc, we can infer that there is no net current through the disc. There is no need for any exchange of charge between the disc and the surrounding fluid in order to conserve charge. This gives us the following jump condition on the axial electrical current, ''• T • - ' / '- = 0 (6.14) Implementing condition (6.14) in equation (6.9) implies, then that, the concentrated force can not transfer angular momentum to the fluid column by an exchange of charge. Hence, the exchange of fluid mass between the disc and the fluid is solely responsible in generating the torque on the .fluid column. Equation (6.9) gives us then the jump condition on the axial velocity: •2Q(V" K i>~) - : (6.15) One final condition, on the third component of the induced magnetic field, the radial component, is-required. It is given by equation (6.7),. which relates the jump in the radial induced magnetic field to the jump in the axial velocity: -Bz(V-r) - V ( | p t - J | p - ) = 0 ' . (6.16) A jump in the axial velocity implies that there is a net radial transport within the disc, which will concentrate or spread apart the Bz field lines. A net.axial induced magnetic field is caused in the disc by this rearrangement of the magnetic flux. Since the induced magnetic field is divergence free and cannot extend to infinity because it is subject to diffusion, an axial magnetic field generated by must be accompanied by a radialmagnetic field above and below the disc. The radial field on one side of the disc is equal and opposite to the Chapter 6: Rigid rotations from a concentrated torque 102 radial field on the other side, and the amplitude of the jump (^fj- — ~f~~) is proportional to the jump in the axial velocity. Addi t ional conditions arising from equations (6.8) and (6.10) have not been used be-cause they are redundant. These additional conditions represent the continuity on the radial and toroidal electric currents, respectively, they are already included through the conditions on the velocity and the magnetic field. 6.3 Results and discussion In order to isolate the effects of a concentrated force on the fluid cylinders, we consider the case where there is no coupling to the solid boundaries. This is done by making the boundaries insulators. Once the influence of is understood, we can add the effects of the electromagnetic coupling at the plates. The method of solution is a straightforward extension of the approach described in the previous chapters. The solution in both fluid cylinders is given by equation (4.3). The induced magnetic field in each solid plate is described by equation (4.4) if we assume that the conductivity is uniformly distributed through the solid. The boundary conditions at each fluid-solid interface are given by equations (2.31) to (2.35), while the boundary conditions at the position of (z = zd) are listed in (6.11) to (6.16). We now have a system of 16 equations and 16 unknowns, with 3 forcing terms: the amplitude of motion of each plate and the forcing at z = zd. 6.3.1 Solution with no magnetic stress at the ends of the fluid column To ehminate the electromagnetic coupling at the fluid-solid boundaries and isolate the effect of f^, the conductivity of the solids must vanish. A weak magnetic field (Bz = 5 G) Chapter 6: Rigid rotations from a concentrated torque 103 is used in the first example. The period of oscillation imposed by is 10 000 days (pa 30 years). The disc is located in the middle of the fluid column, separating it in two fluid cylinders of identical height | . The strength of may be chosen arbitrarily since the solution scales linearly with To be explicit, we choose f(f> = h- 1 0 - 1 2 Nm~z. The list of parameters used in the calculation is given in table 6.1. p 1.1 x 10 4 kgm-z p 47T X 1 0 - 7 kg m A'2 s"2 0~ 3.4 x l O ^ m - 1 1 x n r 1 5 Sm'1 1 x I O " 1 5 Sm'1 ft 7.27 x I O " 5 * - 1 h 2000 km BZ 5G on ft/104 = 7.27 x lO^s'1 uos WTT/180 = 1.26 x l O - 1 0 ^ - 1 «f. OS a;7r/180 = 1.26 x I O - 1 0 * " 1 f h • 10~ 1 2 Nm~3 zd 1000 km Table 6.1: Parameters used in calculating the torque on the fluid column due to a concen-trated force. Figures 6.3 and 6.4 show the profiles of the scalars u, rp, P, and T for this calculation. The plot of the toroidal velocity u is now divided by (f^/utp), instead of the amplitude of the plate motion, as it was the case in the previous chapters. The adjustment in the fluid to is similar to what has been observed before with the electromagnetic couphng at the fluid-solid boundaries. The oscillating forcing creates a boundary layer region where the amplitude of the disturbance in the fluid decreases away from the position where the force is applied. The oscillating behavior in this boundary layer is governed by the same processes that apply in the fluid-solid boundary layers. The main difference resides in the couphng mechanism. In the case of the fluid-solid couphng, the adjustment in the fluid was necessary to Chapter 6: Rigid rotations from a concentrated torque 104 respond to the motion of the plate, which carried the Bz field lines with it. A transfer of charges between the fluid and the solid exerted a magnetic stress on the fluid and produced the fluid motion. In the present case, no net electric currents are created in the disc by the concentrated force and hence, no charges exchanged between the fluid and the disc. The force is communicated to the fluid cylinders through a transfer of fluid mass, which amplitude is proportional to f^. The adjustment in the fluid is not initiated by a toroidal acceleration in the boundary layer, as it was the case for the coupling at the fluid-solid boundaries. The transfer of fluid mass out of (or into) the disc implies a net radial transport in the disc. Consequently, there is an induced axial magnetic field. This magnetic perturbation diffuses out of the disc over a distance determined by the magnetic skin depth in the fluid. The poloidal magnetic field in the boundary layer is associated with a toroidal electric current (see figure 3.1 C ) . The Lorentz force results from the interaction of this current with Bz and accelerates the fluid in the radial direction. The adjustment of the fluid in the boundary layer region is thus initiated by a radial acceleration. Moving to a different height wi l l produce the same rigid rotation, as the angular momentum in the fluid column is proportional to and is independent of its position. The amplitude of this rigid rotation is out of phase by 90 degrees with and varies between ±(f^/pui) ( ± 1 on the plot of u). We are showing the solution at t = 7r /4. We expect the rigid rotations to be altered when the magnetic field is strong enough to cause waves. In figures 6.5 and 6.6 we present the solution obtained with the same set of parameters used in figures 6.3 and 6.4 but with a magnetic field of Bz = 50 G. Aga in , the results are similar to those observed earlier with electromagnetic coupling at the fluid-solid boundary. The disturbance propagates further away from the interface and the amplitude Chapter 6: Rigid rotations from a concentrated torque 105 of the perturbation is increased relative to the amplitude of rigid rotation. For the fluid-plate coupling, the amplitude of the disturbance was dependent on the frequency of oscillation of the plates. A t very low frequency, not enough inertia was transferred to the fluid and at very high frequency, diffusion effects were important and the coupling was inefficient. There was a range of frequencies in between, for which the amplitude of the waves were sufficiently large to destroy the rigid rotations. Similarly, in the present case of a concentrated force, at low frequency, the amplitude of the waves are small and at very high frequency, the diffusion restricts the disturbances to a very thin boundary layer. Hence, we can expect that the solution wi l l also depart from rigid rotations for a specific range of frequencies. 6.3.2 Solution including the magnetic stress at the ends of the fluid column The main goal of this chapter is to demonstrate that the angular momentum transferred to the fluid column in response to a concentrated torque is redistributed over the whole length of the fluid column. The goal of this section is to show that the inclusion of the coupling at the fluid-solid boundaries does not change this picture. To reinstate the magnetic torque at the ends of the fluid column, the conductivity of the plates has to be increased. A 30-year period of oscillation is chosen to represent a long period variation in the L O D . We have selected a magnetic field of 30 G in order to reduce the damping of the waves and increase the thickness of the boundary layers. This wi l l facilitate the comparison between the boundary layers at the fluid-solid interfaces wi th those at the position of f^. The list of parameters used in the calculation is presented in table 6.2. In figures 6.7 and 6.8, we present the solution in both fluid cylinders obtained wi th this Chapter 6: Rigid rotations from a concentrated torque 106 p 1.1 x 104 kgm-3 47r x IO - 7 kgm A'2 s~2 0~ 3.4 x 105 Sm'1 °i 3,4 x 105 5 m - 1 3.4 x - l O ' S m " 1 ft 7.27 x l O - 5 ^ - 1 h 2000 km 30 G U) . ft/104 = 7.27 x IO" 9*- 1 ui WTT/180 = 1.26 x . l f r 1 0 * - 1 IO"1 WTT/180 = 1.26 x IO"11 s'1 u l f(j> h-lQ-12Nm~3 Zd 1000 km Table 6.2: Parameters used in calculating the torque on the fluid column due to combined effects of electromagnetic coupling with the plates and the presence of a concentrated force. set of parameters at t = 7r/4. The toroidal velocity (it) is divided by uf. We can observe the presence of boundary layers on both sides of the disc, as well as boundary layers at the fluid-solid boundaries. The departures from rigid rotations are typical of what we have observed in the previous chapters for large magnetic fields. A weak magnetic field accelerates the fluid column as a rigid body. Hence, this proves that a z-dependent torque will yield a rigid rotation. It thus validates the use of an average value of Bs to estimate the electromagnetic couphng between adjacent shells. A final rigid rotation of the fluid column from a z-dependent torque can thus be computed by decomposing this torque into several small disc-like sections. This final rigid rotation velocity is obtained by summing the rigid rotation velocities resulting from each individual concentrated torque. However, we have not proven that this final rigid rotation velocity is equivalent to the rigid rotation velocity obtained with the average value of this z-dependent torque. This could be the subject of further investigations. The focus of this work is on the couphng between the fluid and the solid and not between the fluid cylindrical shells. Chapter 6: Rigid rotations from a concentrated torque 107 070726 0.7073 0.70734 -5 0 5 (m/s) x10" 7 Figure 6.3: Solution for the scalar potentials of the velocity for the set of parameters in table 6.1. The bottom pate is at z = 0, the top plate is at z = 2000, and is at z = 1000. The superscripts A and B refer to the solution in the fluid cylinder below and above the concentrated force. Chapter 6: Rigid rotations from a concentrated torque 108 2000 2000 1000 ~ A (G/m) x 10-T (G) X10' 1 2 (G/m) x 10"12 Figure 6.4: Solution for the scalar potentials of the magnetic field for the set of parameters in table 6.1. The bottom pate is at z = 0, the top plate is at z = 2000, and is at z = 1000. The superscripts A and B refer to the solution in the fluid cylinder below and above the concentrated force. Chapter 6: Rigid rotations from a concentrated torque 109 0.704 0.706 0.708 0.71 0.712 -2 -1 0 1 2 0.704 0.706 0.708 0.71 0.712 -2 -1 0 1 2 (I7l/S) x10"6 Figure 6.5: Solution for the scalar potentials of the velocity. Apar t from Bz = 50 G , the parameters used in the calculation are given in table 6.1. The bottom pate is at z = 0, the top plate is at z = 2000, and is at z = 1000. The superscripts A and B refer to the solution in the fluid cylinder below and above the concentrated force. Chapter 6: Rigid rotations from a concentrated torque Figure 6.6: Solution for the scalar potentials of the magnetic field. Apart from Bz = 50 the parameters used in the calculation are given in table 6.1. The bottom pate is at z = the top plate is at z = 2000, and is at z = 1000. The superscripts A and B refer to t solution in the fluid cylinder below and above the concentrated force. Chapter 6: Rigid rotations from a concentrated torque 111 ~ A- ~ A u / u< 0.5 0 0.5 1 A (m/s) x i o " 6 0.45 0.5 . 0.55 0.6 0.65 0.7' (m/s) x10 T Figure 6.7: Solution for the scalar potentials of the velocity when the coupling at the fluid-solid boundaries is included. The parameters used in the calculation are given in table 6.2. The bottom pate is at z — 0, the top plate is at z = 2000, and is at z = 1000. The superscripts A and B refer to the solution in the fluid cylinder below and above the concentrated force. Chapter 6: Rigid rotations from a concentrated torque 112 - 2 - 1 0 1 -4 -2 0 2 4 (G) x 10"5 . (G/m) x 10"1° Figure 6.8: Solution for the scalar potentials of the magnetic field, when the couphng at the fluid-solid boundaries is included. The parameters used in the calculation are given in table 6.2. The bottom pate is at z — 0, the top plate is at z = 2000, and is at z — 1000. The superscripts A and B refer to the solution in the fluid cylinder below and above the concentrated force. 113 Chapter 7 Conclusions The purpose of this work is to develop an improved model for electromagnetic coupling between the fluid core and the rest of the Ear th . A good description of the exchange of an-gular momentum is required in order to explain the observed variations in the length of the day ( L O D ) . This work is motivated by potential problems that arise wi th electromagnetic coupling inside the tangent cylinder. Exist ing models rely on the assumption that fluid motions in the core are described by rigid rotations of cylinders and that this motion is not altered by the electromagnetic coupling at the core-mantle boundary ( C M B ) . Hence, that the velocity determined at the boundary is the same as the velocity inside the core. Using this approach, it is possible to calculate the angular momentum of the core using these same velocities. Agreement between the inferred changes in core angular momentum and the observed changes in the length of day is encouraging, but it is important to assess the validity of the underlying approximations, especially inside the tangent cylinder. The calculation in this thesis confirm that the assumption of rigid rotations is valid at decade periods or when the magnetic field is weak (e.g. Bz